control handbook
TRANSCRIPT
Handbook of
Pi and P I D Controller Tuning Rules
2nd Edition
Aidan O*Dwyer
Imperial College Press
Handbook of
PI and PID Controller Tuning Rules
2nd Edition
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Handbook of
P I and P I D Controller Tuning Rules
2nd Edition
Aidan O'Dwyer Dublin Institute of Technology, Ireland
ICP Imperial College Press
Published by
Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE
Distributed by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
HANDBOOK OF PI AND PID CONTROLLER TUNING RULES (2nd Edition)
Copyright © 2006 by Imperial College Press
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 1-86094-622-4
Printed in Singapore by B & JO Enterprise
Dedication
Once again, this book is dedicated with love to Angela, Catherine and Fiona and to my parents, Sean and Lillian.
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Preface
Proportional Integral (PI) and Proportional Integral Derivative (PID) controllers have been at the heart of control engineering practice for seven decades. However, in spite of this, the PID controller has not received much attention from the academic research community until the past fifteen years, when work by K.J. Astrom, T. Hagglund and F.G. Shinskey, among others, has sparked a revival of interest in the use of this "workhorse" of controller implementation.
There is strong evidence that PI and PID controllers remain poorly understood and, in particular, poorly tuned in many applications. It is clear that the many controller tuning rules proposed in the literature are not having an impact on industrial practice. One reason is that the tuning rules are not very accessible, being scattered throughout the control literature; in addition, the notation used is not unified. The purpose of this book is to bring together and summarise, using a unified notation, tuning rules for PI and PID controllers. The author restricts the work to tuning rules that may be applied to the control of processes with time delays (dead times); in practice, this is not a significant restriction, as most process models have a time delay term.
It is the author's belief that this book will be useful to control and instrument engineering practitioners and will be a useful reference for students and educators in universities and technical colleges.
I would like to thank the School of Control Systems and Electrical Engineering, Dublin Institute of Technology, for providing the facilities needed to complete the book. Finally, I am deeply grateful to my mother, Lillian and my father, Sean, for their inspiration and support over many
VII
viii Handbook of PI and PID Controller Tuning Rules
years and to my family Angela, Catherine and Fiona, for their love and understanding.
Aidan O'Dwyer
Contents
Preface vii
1. Introduction 1 1.1 Preliminary Remarks 1 1.2 Structure of the Book 2
2. Controller Architecture 5 2.1 Introduction 5 2.2 PI Controller Structures 6 2.3 PID Controller Structures 7
2.3.1 Ideal PID controller structure and its variations 7 2.3.2 Classical PID controller structure and its variations 11 2.3.3 Non-interacting PID controller structure and its
variations 12 2.3.4 Other PID controller structures 17 2.3.5 Comments on the PID controller structures 19
2.4 Process Modelling 20 2.5 Organisation of the Tuning Rules 24
3. Tuning Rules for PI Controllers 26 3.1 FOLPD Model 26
3.1.1 Ideal controller - Table 3 26 3.1.2 Ideal controller in series with a first order lag
-Table 4 61 3.1.3 Ideal controller in series with a second order filter
- Table 5 62 3.1.4 Controller with set-point weighting - Table 6 63 3.1.5 Controller with proportional term acting on the output 1
- Table 7 65 3.1.6 Controller with proportional term acting on the output 2
- Table 8 66
ix
x Handbook of PI and P1D Controller Tuning Rules
3.2 FOLPD Model with a Positive Zero 67 3.2.1 Ideal controller - Table 9 67 3.2.2 Ideal controller in series with a first order lag
- Table 10 68 3.3 FOLPD Model with a Negative Zero 69
3.3.1 Ideal controller in series with a first order lag -Table 11 69
3.4 Non-Model Specific 70 3.4.1 Ideal controller - Table 12 70 3.4.2 Controller with set-point weighting - Table 13 75
3.5 IPD Model 76 3.5.1 Ideal controller - Table 14 76 3.5.2 Ideal controller in series with a first order lag
-Table 15 83 3.5.3 Controller with set-point weighting - Table 16 84 3.5.4 Controller with proportional term acting on the output 1
- Table 17 85 3.5.5 Controller with proportional term acting on the output 2
-Table 18 87 3.5.6 Controller with a double integral term - Table 19 88
3.6 FOLIPD Model 89 3.6.1 Ideal controller - Table 20 89 3.6.2 Controller with set-point weighting - Table 21 92 3.6.3 Controller with proportional term acting on the output 1
- Table 22 94 3.6.4 Controller with proportional term acting on the output 2
- Table 23 101 3.7 SOSPD Model 102
3.7.1 Ideal controller - Table 24 102 3.7.2 Controller with set-point weighting - Table 25 120
3.8 SOSIPD Model - Repeated Pole 122 3.8.1 Controller with set-point weighting - Table 26 122
3.9 SOSPD Model with a Positive Zero 123 3.9.1 Ideal controller - Table 27 123
3.10 SOSPD Model (repeated pole) with a Negative Zero 125 3.10.1 Ideal controller in series with a first order lag
- Table 28 125 3.11 Third Order System plus Time Delay Model 126
3.11.1 Ideal controller - Table 29 126 3.11.2 Controller with set-point weighting - Table 30 128
3.12 Unstable FOLPD Model 130 3.12.1 Ideal controller-Table 31 130
Contents XI
3.12.2 Controller with set-point weighting - Table 32 135 3.12.3 Controller with proportional term acting on the output 1
- Table 33 137 3.12.4 Controller with proportional term acting on the output 2
- Table 34 140 3.13 Unstable FOLPD Model with a Positive Zero 141
3.13.1 Ideal controller - Table 35 141 3.13.2 Ideal controller in series with a first order lag
-Table 36 142 3.13.3 Controller with set-point weighting - Table 37 143
3.14 Unstable SOSPD Model (one unstable pole) 145 3.14.1 Ideal controller-Table 38 145
3.15 Unstable SOSPD Model with a Positive Zero 147 3.15.1 Ideal controller - Table 39 147 3.15.2 Controller with set-point weighting - Table 40 148
3.16 Delay Model 149 3.16.1 Ideal controller - Table 41 149
3.17 General Model with a Repeated Pole 152 3.17.1 Ideal controller - Table 42 152
3.18 General Model with Integrator 153 3.18.1 Ideal controller - Table 43 153
4. Tuning Rules for PID Controllers 154 4.1 FOLPD Model 154
4.1.1 Ideal controller - Table 44 154 4.1.2 Ideal controller in series with a first order lag
-Table 45 181 4.1.3 Ideal controller in series with a second order filter
-Table 46 183 4.1.4 Ideal controller with weighted proportional term
-Table 47 185 4.1.5 Ideal controller with first order filter and set-point
weighting 1 - Table 48 186 4.1.6 Controller with filtered derivative - Table 49 187 4.1.7 Blending controller - Table 50 193 4.1.8 Classical controller 1 - Table 51 194 4.1.9 Classical controller 2 - Table 52 208 4.1.10 Series controller (classical controller 3) - Table 53 209 4.1.11 Classical controller 4 - Table 54 211 4.1.12 Non-interacting controller 1 - Table 55 212 4.1.13 Non-interacting controller 2a - Table 56 213 4.1.14 Non-interacting controller 2b - Table 57 215
Handbook of PI and PID Controller Tuning Rules
4.1.15 Non-interacting controller based on the two degree of freedom structure 1 - Table 58 217
4.1.16 Non-interacting controller based on the two degree of freedom structure 2 - Table 59 222
4.1.17 Non-interacting controller based on the two degree of freedom structure 3 - Table 60 223
4.1.18 Non-interacting controller 4 - Table 61 224 4.1.19 Non-interacting controller 5 - Table 62 226 4.1.20 Non-interacting controller 6 (I-PD controller)
- Table 63 227 4.1.21 Non-interacting controller 7 -Table 64 230 4.1.22 Non-interacting controller 11 - Table 65 231 4.1.23 Non-interacting controller 12 - Table 66 232 4.1.24 Industrial controller - Table 67 233 Non-Model Specific 235 4.2.1 Ideal controller - Table 68 235 4.2.2 Ideal controller in series with a first order lag
- Table 69 241 4.2.3 Ideal controller in series with a second order filter
-Table 70 243 4.2.4 Ideal controller with weighted proportional term
- Table 71 245 4.2.5 Controller with filtered derivative - Table 72 246 4.2.6 Ideal controller with set-point weighting 1 - Table 73 . . . . 252 4.2.7 Classical controller 1 - Table 74 253 4.2.8 Series controller (classical controller 3) - Table 75 254 4.2.9 Classical controller 4 - Table 76 255 4.2.10 Non-interacting controller based on the two
degree of freedom structure 1 - Table 77 256 4.2.11 Non-interacting controller 4 - Table 78 257 4.2.12 Non-interacting controller 9 - Table 79 258 BPD Model 259 4.3.1 Ideal controller - Table 80 259 4.3.2 Ideal controller in series with a first order lag
-Table 81 262 4.3.3 Ideal controller with first order filter and set-point
weighting 2 - Table 82 263 4.3.4 Controller with filtered derivative - Table 83 264 4.3.5 Controller with filtered derivative and dynamics on
the controlled variable - Table 84 265 4.3.6 Classical controller 1 - Table 85 266 4.3.7 Classical controller 2 - Table 86 268
Contents xiii
4.3.8 Classical controller 4 - Table 87 269 4.3.9 Non-interacting controller based on the two
degree of freedom structure 1 - Table 88 270 4.3.10 Non-interacting controller based on the two
degree of freedom structure 3 - Table 89 272 4.3.11 Non-interacting controller 4 - Table 90 273 4.3.12 Non-interacting controller 6 (I-PD controller)
- Table 91 274 4.3.13 Non-interacting controller 8 -Table 92 276 4.3.14 Non-interacting controller 10 -Table 93 277 4.3.15 Non-interacting controller 12 -Table 94 278
4.4 FOLIPD Model 279 4.4.1 Ideal controller - Table 95 279 4.4.2 Ideal controller in series with a first order lag
- Table 96 282 4.4.3 Ideal controller with weighted proportional term
-Table 97 284 4.4.4 Controller with filtered derivative - Table 98 285 4.4.5 Controller with filtered derivative with set-point
weighting 1 - Table 99 286 4.4.6 Controller with filtered derivative with set-point
weighting 3 - Table 100 287 4.4.7 Ideal controller with set-point weighting 1 - Table 101 . . . 289 4.4.8 Classical controller 1 - Table 102 290 4.4.9 Classical controller 2 - Table 103 292 4.4.10 Series controller (classical controller 3) - Table 104 293 4.4.11 Classical controller 4 - Table 105 294 4.4.12 Non-interacting controller based on the two
degree of freedom structure 1 - Table 106 295 4.4.13 Non-interacting controller 4 - Table 107 297 4.4.14 Non-interacting controller 6 (I-PD controller)
- Table 108 298 4.4.15 Non-interacting controller 8 - Table 109 303 4.4.16 Non-interacting controller 11 - Table 110 304 4.4.17 Non-interacting controller 12 - Table 111 305 4.4.18 Industrial controller - Table 112 306 4.4.19 Alternative controller 1 - Table 113 307 4.4.20 Alternative controller 2 - Table 114 308
4.5 SOSPD Model 309 4.5.1 Ideal controller - Table 115 309 4.5.2 Ideal controller in series with a first order lag
-Table 116 332
Handbook of PI and PID Controller Tuning Rules
4.5.3 Ideal controller in series with a first order filter -Table 117 336
4.5.4 Ideal controller in series with a second order filter -Table 118 337
4.5.5 Controller with filtered derivative - Table 119 338 4.5.6 Controller with filtered derivative in series with
a second order filter - Table 120 339 4.5.7 Ideal controller with set-point weighting 1 - Table 121 . . . 340 4.5.8 Ideal controller with set-point weighting 2 - Table 122 . . . 341 4.5.9 Classical controller 1 - Table 123 342 4.5.10 Classical controller 2 - Table 124 351 4.5.11 Series controller (classical controller 3) - Table 125 352 4.5.12 Non-interacting controller 1 - Table 126 354 4.5.13 Non-interacting controller based on the two
degree of freedom structure 1 - Table 127 363 4.5.14 Non-interacting controller 4 - Table 128 368 4.5.15 Non-interacting controller 5 - Table 129 370 4.5.16 Non-interacting controller 6 - Table 130 371 4.5.17 Alternative controller 4 - Table 131 372 I2PD Model 374 4.6.1 Controller with filtered derivative with set-point
weighting 2 - Table 132 374 4.6.2 Controller with filtered derivative with set-point
weighting 4 - Table 133 376 4.6.3 Series controller (classical controller 3) - Table 134 378 4.6.4 Non-interacting controller based on the two
degree of freedom structure 1 - Table 135 379 4.6.5 Industrial controller - Table 136 380 SOSIPD Model (repeated pole) 381 4.7.1 Non-interacting controller based on the two
degree of freedom structure 1 - Table 137 381 SOSPD Model with a Positive Zero 383 4.8.1 Ideal controller - Table 138 383 4.8.2 Ideal controller in series with a first order lag
-Table 139 385 4.8.3 Controller with filtered derivative - Table 140 386 4.8.4 Classical controller 1 - Table 141 387 4.8.5 Series controller (classical controller 3) - Table 142 388 4.8.6 Classical controller 4 - Table 143 389 4.8.7 Non-interacting controller 1 -Table 144 390 4.8.8 Non-interacting controller based on the two
degree of freedom structure 1 - Table 145 391
Contents xv
4.9 SOSPD Model with a Negative Zero 392 4.9.1 Ideal controller - Table 146 392 4.9.2 Controller with filtered derivative - Table 147 393 4.9.3 Classical controller 1 - Table 148 394 4.9.4 Classical controller 4 - Table 149 395 4.9.5 Non-interacting controller 1 - Table 150 396 4.9.6 Non-interacting controller based on the two
degree of freedom structure 1 - Table 151 397 4.10 Third Order System plus Time Delay Model 398
4.10.1 Ideal controller - Table 152 398 4.10.2 Ideal controller in series with a first order lag
- Table 153 399 4.10.3 Controller with filtered derivative - Table 154 400 4.10.4 Non-interacting controller based on the two
degree of freedom structure 1 - Table 155 401 4.11 Unstable FOLPD Model 403
4.11.1 Ideal controller - Table 156 403 4.11.2 Ideal controller in series with a first order lag
- Table 157 408 4.11.3 Ideal controller with set-point weighting 1 - Table 158 . . . 410 4.11.4 Classical controller 1 -Table 159 414 4.11.5 Series controller (classical controller 3) - Table 160 415 4.11.6 Non-interacting controller based on the two
degree of freedom structure 1 - Table 161 416 4.11.7 Non-interacting controller 3 - Table 162 420 4.11.8 Non-interacting controller 8 - Table 163 421 4.11.9 Non-interacting controller 10 - Table 164 423 4.11.10 Non-interacting controller 12 - Table 165 425
4.12 Unstable SOSPD Model (one unstable pole) 427 4.12.1 Ideal controller - Table 166 427 4.12.2 Ideal controller in series with a first order lag
- Table 167 430 4.12.3 Ideal controller with set-point weighting 1 - Table 168 . . . 431 4.12.4 Classical controller 1 - Table 169 432 4.12.5 Series controller (classical controller 3) -Table 170 433 4.12.6 Non-interacting controller 3 - Table 171 434 4.12.7 Non-interacting controller 8 - Table 172 439
4.13 Unstable SOSPD Model (two unstable poles) 442 4.13.1 Ideal controller - Table 173 442 4.13.2 Ideal controller with set-point weighting 1 - Table 174 . . . 444
4.14 Unstable SOSPD Model with a Positive Zero 445
xvi Handbook of PI and PID Controller Tuning Rules
4.14.1 Ideal controller in series with a first order lag -Table 175 445
4.14.2 Non-interacting controller based on the two degree of freedom structure 1 - Table 176 447
4.15 Delay Model 449 4.15.1 Ideal controller - Table 177 449 4.15.2 Ideal controller in series with a first order lag
-Table 178 450 4.15.3 Classical controller 1 -Table 179 451
4.16 General Model with a Repeated Pole 452 4.16.1 Ideal controller - Table 180 452 4.16.2 Ideal controller in series with a first order lag
-Table 181 453 4.17 General Stable Non-Oscillating Model with a Time Delay 454
4.17.1 Ideal controller - Table 182 454 4.18 Fifth Order System plus Delay Model 455
4.18.1 Ideal controller - Table 183 455 4.18.2 Controller with filtered derivative - Table 184 457 4.18.3 Non-interacting controller 7 - Table 185 460
5. Performance and Robustness Issues in the Compensation of FOLPD Processes with PI and PID Controllers 462 5.1 Introduction 462 5.2 The Analytical Determination of Gain and Phase Margin 463
5.2.1 PI tuning formulae 463 5.2.2 PID tuning formulae 466
5.3 The Analytical Determination of Maximum Sensitivity 469 5.4 Simulation Results 470 5.5 Design of Tuning Rules to Achieve Constant Gain and
Phase Margins, for all Values of Delay 475 5.5.1 PI controller design 475
5.5.1.1 Processes modelled in FOLPD form 475 5.5.1.2 Processes modelled in IPD form 477
5.5.2 PID controller design 480 5.5.2.1 Processes modelled in FOLPD form
- classical controller 1 480 5.5.2.2 Processes modelled in SOSPD form
- series controller 482 5.5.2.3 Processes modelled in SOSPD form with a
negative zero - classical controller 1 482 5.5.3 PD controller design 483
5.6 Conclusions 484
Contents xvii
Appendix 1 Glossary of Symbols Used in the Book 485
Appendix 2 Some Further Details on Process Modelling 493
Bibliography 507
Index 535
Chapter 1
Introduction
1.1 Preliminary Remarks
The ability of proportional integral (PI) and proportional integral derivative (PID) controllers to compensate most practical industrial processes has led to their wide acceptance in industrial applications. Koivo and Tanttu (1991), for example, suggest that there are perhaps 5-10% of control loops that cannot be controlled by single input, single output (SISO) PI or PID controllers; in particular, these controllers perform well for processes with benign dynamics and modest performance requirements (Hwang, 1993; Astrom and Hagglund, 1995). It has been stated that 98% of control loops in the pulp and paper industries are controlled by SISO PI controllers (Bialkowski, 1996) and that, in process control applications, more than 95% of the controllers are of PID type (Astrom and Hagglund, 1995). The PI or PID controller implementation has been recommended for the control of processes of low to medium order, with small time delays, when parameter setting must be done using tuning rules and when controller synthesis is performed either once or more often (Isermann, 1989).
However, despite decades of development work, surveys indicating the state of the art of control industrial practice report sobering results. For example, Ender (1993) states that, in his testing of thousands of control loops in hundreds of plants, it has been found that more than 30% of installed controllers are operating in manual mode and 65% of loops operating in automatic mode produce less variance in manual than in automatic (i.e. the automatic controllers are poorly tuned). The situation
l
2 Handbook of PI and PID Controller Tuning Rules
does not appear to have improved in recent years as Van Overschee and De Moor (2000) report that 80% of PID controllers are badly tuned; 30% of PID controllers operate in manual with another 30% of the controlled loops increasing the short term variability of the process to be controlled (typically due to too strong integral action). The authors state that 25% of all PID controller loops use default factory settings, implying that they have not been tuned at all.
These and other surveys (well summarised by Yu, 1999, pages 1-2) show that the determination of PI and PID controller tuning parameters is a vexing problem in many applications. The most direct way to set up controller parameters is the use of tuning rules; obviously, the wealth of information on this topic available in the literature has been poorly communicated to the industrial community. One reason is that this information is scattered in a variety of media, including journal papers, conference papers, websites and books over a period of seventy years. The author has recorded 408 separate sources of tuning rules since the first such rule was published by Callender et al. (1935/6). In a striking statistic, 293 sources of tuning rules have been recorded since 1992, reflecting the upsurge of interest in the use of the PID controller recently.
The purpose of this book is to bring together, in summary form, the tuning rules for PI and PID controllers that have been developed to compensate SISO processes with time delay. Tuning rules for the variations that have been proposed in the 'ideal' PI and PID controller structure are included. Considerable variations in the ideal PID controller structure, in particular, are encountered; these variations are explored in more detail in Chapter 2.
1.2 Structure of the Book
Tuning rules are set out in the book in tabular form. This form allows the rules to be represented compactly. The tables have four or five columns, according to whether the controller considered is of PI or PID form, respectively. The first column in all cases details the author of the rule and other pertinent information. The final column in all cases is labelled "Comment"; this facilitates the inclusion of information about the tuning
Chapter 1: Introduction 3
rule that may be useful in its application. The remaining columns detail the formulae for the controller parameters.
Chapter 2 explores the range of PI and PID controller structures proposed in the literature. It is often forgotton that different manufacturers implement different versions of the PID controller algorithm (in particular); therefore, controller tuning rules that work well in one PID architecture may work poorly on another. This chapter also details the process models used to define the controller tuning rules.
Chapters 3 and 4 of the book detail, in tabular form, tuning rules for setting up, respectively, PI controllers and PID controllers (and their variations), for a wide variety of process models. P controller and I controller tuning rules are also defined (as subsets of PI controller tuning rules), and PD controller tuning rules are defined (as a subset of PID controller tuning rules), for processes whose model includes an integrator. One hundred and eighty-three such tables are provided altogether. To allow the reader to access data readily, the author has arranged that each table start on its own page of the book; each table is preceded by the controller used, together with a block diagram showing the unity feedback closed loop arrangement of the controller and process model.
In Chapter 5 of the book, analytical calculations of the gain and phase margins of a large sample of PI and PID controller tuning rules are determined, when the process is modelled in first order lag plus time delay (FOLPD) form, at a range of ratios of time delay to time constant of the process model. Results are given in graphical form.
An important feature of the book is the unified notation that is used for the tuning rules; a glossary of the symbols used is provided in Appendix 1. Appendix 2 outlines the range of methods that are used to determine process model parameters; this information is presented in summary form, as this topic could provide data for a book in itself. However, sufficient information, together with references, is provided for the interested reader.
Finally, a comprehensive reference list is provided. In particular, the author would like to recommend the contributions by McMillan (1994), Astrom and Hagglund (1995), Shinskey (1994), (1996), Tan et al. (1999a), Yu (1999), Lelic and Gajic (2000) and Ang et al. (2005) to the
4 Handbook of PI and PID Controller Tuning Rules
interested reader, which treat comprehensively the wider perspective of PID controller design and application.
Chapter 2
Controller Architecture
2.1 Introduction
The ideal continuous time domain PID controller for a SISO process is expressed in the Laplace domain as follows:
U(s) = Gc(s)E(s) (2.1)
with
Gc(s) = Kc(l + -^- + Tds) (2.2)
and with Kc = proportional gain, Tj = integral time constant and Td = derivative time constant. If T; = oo and Td - 0 (i.e. P control), then it is clear that the closed loop measured value, y, will always be less than the desired value, r (for processes without an integrator term, as a positive error is necessary to keep the measured value constant, and less than the desired value). The introduction of integral action facilitates the achievement of equality between the measured value and the desired value, as a constant error produces an increasing controller output. The introduction of derivative action means that changes in the desired value may be anticipated, and thus an appropriate correction may be added prior to the actual change. Thus, in simplified terms, the PID controller allows contributions from present controller inputs, past controller inputs and future controller inputs.
Many variations of the PI and, in particular, the PID controller structure have been proposed. As Tan et al. (1999a) suggest, one
5
6 Handbook of PI and PID Controller Tuning Rules
important reason for the non-standard structures is due to the transition of the controllers from pneumatic implementation through electronic implementation to the present microprocessor implementation. The variations in the controller structures are detailed below.
2.2 PI Controller Structures
Tuning rules have been detailed for seven PI controller structures:
1. Ideal controller Gc(s) = K( i+-L V T i s 7
(2.3)
2. Ideal controller in series with a first order lag:
Gc(s) = Kc 1 + T:S
1
1 + Tfs (2.4)
3. Ideal controller in series with a second order filter (also labelled the 'generalised PID' controller (Lee and Shi, 2002)):
Gc(s) = Kc 1 + 1
v T>sy
2 ^ 1 + bfls + bf2s
l + afls + af2s j (2.5)
4. Controller with set-point weighting:
U(s) = Kc 'i-L^ V T i s /
E(s)-aK cR(s) (2.6)
5. Controller with proportional term acting on the output 1:
U(s) = ^ E ( s ) - K c Y ( s ) Ts
(2.7)
6. Controller with proportional term acting on the output 2: r
U(s) = K 1 \
V 1+
Tfsy
E(s)-K,Y(s) (2.8)
7. Controller with a double integral term:
Chapter 2: Controller Architecture 7
Gc(s) = Kc ' 1 1 ^ 1+ +
V ^ilS Ti2S j
(2.9)
2.3 PID Controller Structures
Forty-six such structures have been considered. The labelling of these structures has not been consistent in the literature; for example, the first PID controller structure considered (labelled the ideal controller below) has also been labelled the 'non-interacting' controller (McMillan, 1994), the 'ISA' algorithm (Gerry and Hansen, 1987) or the 'parallel non-interacting' controller (Visioli, 2001). The controller structures have been divided into four types, as described below.
2.3.1 Ideal PID controller structure and its variations
1. Ideal controller: Gc(s) = Kc
f 1 A
1 + — + Tds V T . s j
(2.10)
A variation of the controller is labelled the 'parallel' controller structure (McMillan, 1994). This variation has also been labelled the 'ideal parallel', 'noninteracting', 'independent' or 'gain independent' algorithm:
Gc(s) = K c + - i - + Tds (2.11) TiS
The controller structure is used in the following products: (a) Allen Bradley PLC5 product (McMillan, 1994) (b) Bailey FC19 PID algorithm (EZYtune, 2003) (c) Fanuc Series 90-30 and 90-70 independent form PID algorithm
(EZYtune, 2003) (d) Intellution FIX products (McMillan, 1994) (e) Honeywell TDC3000 Process Manager Type A, non-interactive
mode product (ISMC, 1999) (f) Leeds and Northrup Electromax 5 product (Astrom and
Hagglund, 1988)
Handbook of PI and PID Controller Tuning Rules
(g) Yokogawa Field Control Station (FCS) PID algorithm (EZYtune, 2003).
2. Ideal controller in series with a first order lag: f \ 1
Gc(s) = Kc l + + Tds v T i s ; Tfs + l
(2.12)
3. Ideal controller in series with a first order filter: /
Gc(s) = Kt 1
1 + — + Tds V T i s J
bfls + l
afls + l (2.13)
4. Ideal controller in series with a second order filter: /
Gc(s) = Kt 1 \
1 + — + Tds v T i s J
l + bfls
l + a n s + af2s (2.14)
5. Ideal controller with weighted proportional term: ( 1 A
Gc(s) = Kt b + — + THs V T i S
J (2.15)
6. Ideal controller with first order filter and setpoint weighting 1: r
U(s) = Kc 1 >
i + — + T d s V T i s J
1 Tfs + 1
R ( S ) 1±^_ Y ( S ) 1 + sT
(2.16)
7. Ideal controller with first order filter and setpoint weighting 2: /
U(s) = Kc 1 \
V T i s
1
Tfs + 1 R<.)l±2^i-Y(.,
l + sTr
K0Y(s)
(2.17) 8. Ideal controller with first order filter and dynamics on the controlled
variable: /
U(s) = Kt 1 A
1 + — + Tds _J^TV t ( l ± K) _JC_ Tfs + 1 • Ts + 1 Ts + 1
9. Controller with filtered derivative:
Chapter 2: Controller Architecture 9
Gc(s) = Kc 1 + — + - d
T*s l + s ^ N
(2.19)
This structure is used in the following products: (a) Bailey Net 90 PID error input product with N = 10 (McMillan,
1994) and FC156 Independent Form PID algorithm (EZYtune, 2003)
(b) Concept PIDP1 and PID1 PID algorithms (EZYtune, 2003) (c) Fischer and Porter DCU 3200 CON PID algorithm with N = 8
(EZYtune, 2003) (d) Foxboro EXACT I/A series PIDA product (in which it is an
option labelled ideal PID) (Foxboro, 1994). (e) Hartmann and Braun Freelance 2000 PID algorithm (EZYtune,
2003) (f) Modicon 984 product with 2 < N < 30 (McMillan, 1994;
EZYtune, 2003) (g) Siemens Teleperm/PSC7 ContC/PCS7 CTRL PID products with
N = 10 (ISMC, 1999) and the S7 FB41 CONTC PID product (EZYtune, 2003).
10. Controller with filtered derivative in series with a second order filter: ( \
Gc(s) = Kt T>s 1 +
T< V N
l + bfls + bf2s
l + afls + af2s2 (2.20)
11. Controller with filtered derivative with setpoint weighting 1: f \
Gc(s) = K( , 1 TdS 1 + — + %r
T:S - L l + ' d
V s
N j
l + bfls
l + afls R(s)-Y(s) (2.21)
12. Controller with filtered derivative with setpoint weighting 2:
10 Handbook of PI and PID Controller Tuning Rules
f
Gc(s) = Kc T . s l + ^ s
N
l + b f ls + b f2s"
l + a f ls + a f 2s2 -R( s ) -Y(s ) (2.22)
13. Controller with filtered derivative with setpoint weighting 3: r \
Gc(s) = K( 1 + — + — h -Xs . X
v 1 + ^ s
N ;
1 + b » S + b " S2
2 + b » S3
3 R ( s ) - Y ( s ) A
(2.23)
14. Controller with filtered derivative with setpoint weighting 4:
Gc(s) = Kc 1 + — + • d
Xs , . Td 1 + ^ s N j
' 1 + b " S + b ^ + b " S 3 + b " S4
4 R ( s ) - Y(s) ' 1 + Sf-jS H-S-f-^S ~H S ^ T S + 3 . ^ 4 8
(2.24)
15. Controller with filtered derivative and dynamics on the controlled variable:
Gc(s) = Kt 1 1 T d s
1+- --^ d Xs , . . Xd 1 + s-
N
E ( s ) - K 0 Y ( s ) (2.25)
16. Ideal controller with setpoint weighting 1:
U(s) = Kc (FpR(s) - Y ( s ) ) + ^ ( F i R ( s ) - Y(s))+ KcXds(FdR(s) - Y(s))
(2.26) 17. Ideal controller with setpoint weighting 2:
( U(s) = K,
1 \ 1 + — + Tds
V T i s j
1
T i V + T j S + 1 R ( s ) - Y ( s ) (2.27)
Chapter 2: Controller Architecture 11
18. Blending controller:
Gc(s) = Kc
f 1 >
Ts - (2.28) s
This structure is used in the Yokogawa EFCS/EFCD Field Control Station product (Chen et al., 2001).
2.3.2 Classical PID controller structure and its variations
19. Classical controller 1: This controller is also labelled the 'cascade' controller (Witt and Waggoner, 1990), the 'interacting' or 'series' controller (Poulin and Pomerleau, 1996), the 'interactive' controller (Tsang and Rad, 1995), the 'rate-before-reset' controller (Smith and Corripio, 1997), the 'analog' controller (St. Clair, 2000) or the 'commercial' controller (Luyben, 2001).
Gc(s) = Kt ' 1 ' l + STd (2.29) l + J-Ts T
N The structure is used in the following products: (a) Honeywell TDC Basic/Extended/Multifunction Types A and B
products with N = 8 (McMillan, 1994) (b) Toshiba TOSDIC 200 product with 3.33<N<10 (McMillan,
1994) (c) Foxboro EXACT Model 761 product with N = 10 (McMillan,
1994) (d) Honeywell UDC6000 product with N = 8 (Astrom and
Hagglund, 1995) (e) Honeywell TDC3000 Process Manager product - Type A,
interactive mode with N = 10 (ISMC, 1999) (f) Honeywell TDC3000 Universal, Multifunction and Advanced
Multifunction products with N = 8 (ISMC, 1999) (g) Foxboro EXACT I/A Series PIDA product (in which it is an
option labelled series PID) (Foxboro, 1994).
12 Handbook of PI and PID Controller Tuning Rules
20. Classical controller 2: Gc(s) = Kc 1 + -T,Sy
l + NTds
1 + Tds j (2.30)
21. Series controller (Classical controller 3): This controller is also labelled the 'interacting' controller, the 'analog algorithm' (McMillan, 1994) or the 'dependent' controller (EZYtune, 2003).
Gc(s) = Kt 1 + T.Sy
(l + sTd) (2.31)
The structure is used in the following products: (a) Turnbull TCS6000 series product (McMillan, 1994) (b) Alfa-Laval Automation ECA400 product (Astrom and
Hagglund, 1995) (c) Foxboro EXACT 760/761 product (Astrom and Hagglund,
1995).
22. Classical controller 4: This controller is also labelled the 'interacting' controller (Fertik, 1975).
Ge(s) = Kt 1
l + - TdS
1 + s ^ N J
(2.32)
The structure is used in the following products: (a) Bailey FC156 Classical Form PID product (EZYtune, 2003) (b) Fischer and Porter DCI 4000 PID algorithm (EZYtune, 2003)
2.3.3 Non-interacting PID controller structure and its variations
23. Non-interacting controller 1: This controller is also labelled the 'reset-feedback' controller (Huang et al., 1996).
U(s) = Kt 1 + ]_
T i S ,
E(s) % - Y ( s )
1 + -N
This structure is used in the following products:
(2.33)
Chapter 2: Controller Architecture 13
(a) Bailey Fisher and Porter 53SL6000 and 53MC5000 products with N = 0 (ISMC, 1999).
(b) Moore Model 352 Single-Loop Controller product (Wade, 1994).
24. Non-interacting controller based on the parallel controller structure: 24a. Non-interacting controller 2a:
f 1 A U(s) = Kt 1 + E(s)-
Tds
N
Y(s)
24b. Non-interacting controller 2b: /
U(s) = 1 A
v
K c + TjSy
E(s)-Tds
N
Y(s)
(2.34)
(2.35)
25. Non-interacting controller based on the two degree of freedom structure 1: This controller is also labelled the 'm-PID' controller (Huang et al, 2000) the 'ISA-PID' controller (Leva and Colombo, 2001) and the 'P-I-PD (only P is DOF) incomplete 2DOF algorithm' by Mizutani and Hiroi (1991).
U(s) = Kt , 1 TdS
v N
E(s)-K c ct + PTds
1 + ^ N
R(s) (2.36)
This structure is used in the following products: (a) Bailey Net 90 PID PV and SP product (McMillan, 1994) (b) Yokogawa SLPC products with <x = - l , (B = - 1 , N=10
(McMillan, 1994) (c) Omron E5CK digital controller with p = 1 and N=3 (ISMC,
1999).
26. Non-interacting controller based on the two degree of freedom structure 2:
14 Handbook of PI and PID Controller Tuning Rules
U(s) = K( 1 + - U - ^ T « 8 l + ^ s
I N ,
E(s)-K (
f \ PTds X
a +
V l + ^ s 1 + T i s
N
R(s)
(2.37)
27. Non-interacting controller based on the two degree of freedom structure 3:
U(s) = K j ( b - l ) + (c-l)Tds]R(s)
( 1 + K, 1 + — + Tds
v T . s R(s)
(l + sTf): -Y(s) (2.38)
28. PD-I-PD (only PD is DOF) incomplete 2DOF algorithm (Mizutani andHiroi, 1991):
f V
U(s) = Kc a-_ 1 _ %Tds 1
1+-5 -S ^ N
Td_ 1 + Xs R(s)
• K , , 1 TdS 1 + — + — -
T ' s 1 + ^ s N
Y(s) (2.39)
However, no tuning rules are available for this controller structure.
29. PI-PID (only PI is DOF) incomplete 2DOF algorithm (Mizutani and Hiroi, 1991):
U(s) = K( a + 1 P
T;s TiS(l + TiS). R(s)-K£ 1+ + =7-
T;s 1 + ^ - s N
Y(s)
(2.40) However, no tuning rules are available for this controller structure.
Chapter 2; Controller Architecture 15
30. Super (or complete) 2DOF PID algorithm (Mizutani and Hiroi, 1991):
U(s) = Kc a- Ts 1 — J _ + _£?>_
1 + Ts I+3L N
R(s)
- K ,
N v
Y(s) (2.41)
Mizutani and Hiroi (1991) state that 0 < a < 1, p > 1 and % ^ 1 , and suggest starting values of a = 0.4, P = 1.35 and % = 1.25. However, no tuning rules are available for this controller structure.
31. Non-interacting controller 3:
U(s) = Kt 1 + — . TiSy
E(s)-KcTds
1 + ^ L N
Y(s) (2.42)
This structure is used in the following products: (a) Allen Bradley SLC5/02, SLC5/03, SLC5/04, PLC5 and
Logix5550 products (EZYtune, 2003). (b) Modcomp product with N = 10 (McMillan, 1994).
32. Non-interacting controller 4: This controller is also labelled the 'PI+D' controller (Chen, 1996).
U(s) = Kt 1 + E(s)-KcTdsY(s) (2.43)
This structure is used in the following products: (a) ABB 53SL6000 Product (ABB, 2001). (b) Genesis product (McMillan, 1994) (c) Honeywell TDC3000 Process Manager Type B, non-interactive
mode product (ISMC, 1999) (d) Square D PIDR PID product (EZYtune, 2003)
16 Handbook of PI and PID Controller Tuning Rules
33. Non-interacting controller 5:
U(s) = Kc b + -T,s
E(s)-(c + Tds)Y(s) (2.44)
34. Non-interacting controller 6 (I-PD controller):
U(s) = - ^ E ( s ) - K c ( l + Tds)Y(s) T:S
(2.45)
This structure is used in the Toshiba AdTune TOSDIC 211D8 product (Shigemasa et al, 1987) and the Honeywell TDC3000 Process Manager Type C non-interactive mode product (ISMC, 1999).
35. Non-interacting controller 7:
K „ U(s) = - ^ E ( s ) - K ,
T:S n T's
1 - T " N
Y(s) (2.46)
36. Non-interacting controller 8:
U(s) = K c | l + - j - + Tds |E(s)-K i(l + Tdis)Y(s) (2.47) V T,s
37. Non-interacting controller 9: /
U(s) = Kt 1 \
1 + — + Tds v T.s ;
E(s)--^sY(s) N
(2.48)
38. Non-interacting controller 10:
U(s) = Kc
\ T i sy E(s)-K f -^-s + 1
VTm J Y(s) (2.49)
39. Non-interacting controller 11:
Chapter 2: Controller Architecture 17
U(s) = Kt T=s
(l+Tds)E(s)-K i(l + Tdis)Y(s) (2.50)
40. Non-interacting controller 12: f 1 A
U(s) = Kc V T i s j 1 + Tds
E(s)-K,Y(s) (2.51)
41. Non-interacting controller 13:
U(s) = Kc
v T;sy E(s)- Tds
1 + I l s + I^s2 Y(s) (2.52)
N 2N This structure is used in the Foxboro IA PID product, with 10 < N <50(EZYtune, 2003); however, no tuning rules are available for this controller structure.
2.3.4 Other PID controller structures
42. Industrial controller (Kaya and Scheib, 1988): ( ^
U(s) = Kc 1+ — . T,Sy
1 +T s R(s)-i±MY(s)
V N
(2.53)
This structure is used in the following products: (a) Fisher-Rosemount Provox product with N = 8 (ISMC, 1999;
McMillan, 1994) (b) Foxboro Model 761 product with N=10 (McMillan, 1994) (c) Fischer-Porter Micro DCI product with N = 0 (McMillan, 1994) (d) Moore Products Type 352 controller with 1 < N < 30 (McMillan,
1994) (e) SATT Instruments EAC400 product with N=8.33 (McMillan,
1994) (f) Taylor Mod 30 ESPO product with N=16.7 (McMillan, 1994)
18 Handbook of PI and PID Controller Tuning Rules
(g) Honeywell TDC3000 Process Manager Type B, interactive mode product with N = 10 (ISMC, 1999)
43. Alternative controller 1: Gc(s) = Kc
f V
1 + Ts
1 + ^ v N
1 + Tds
y\ N j
(2.54)
44. Alternative controller 2:
Gc(s) = Kc 1 + TjS
T l + - ^ s
N j \
2 „ 2 l + 0.5xms + 0.0833xm s
N
(2.55)
45. Alternative controller 3:
K „
/
U(s) = - ^ R ( s ) - K ( T:S
1+ . T,sy
l + sTd
N ;
Y(s) (2.56)
This structure is used in the following products: (a) Honeywell TDC3000 Process Manager Type C, interactive mode
product with N = 10 (ISMC, 1999) (b) Honeywell TDC3000 Universal, Multifunction and Advanced
Multifunction products with N = 8 (ISMC, 1999) However, no tuning rules are available for this controller structure.
46. Alternative controller 4:
U(s) = Kt
N
E(s) + R(s) (2.57)
Chapter 2: Controller Architecture 19
2.3.5 Comments on the PID controller structures
In some cases, one controller structure may be transformed into another; clearly, the ideal and parallel controller structures (equations (2.10) and (2.11)) are very closely related. It is shown by McMillan (1994), among others, that the parameters of the ideal PID controller may be worked out from the parameters of the series PID controller, and vice versa. The ideal PID controller is given in Equation (2.58) and the series PID controller is given in Equation (2.59).
f 1 ^ 1 + -Gcp(s) = Kcp
Gcs(s) = Kc 1 + T~I
+ Tdps
(l + sTds)
(2.58)
(2.59)
Then, it may be shown that (
K c p = l + _d!L
X. ^•cs ' *ip ~~ V i s + * d s / ' *dp _ X.
X.S+X*
Similarly, it may be shown that, provided X- > 4T( dp •
K„5 = 0.5K„ 1 +Jl-4-l d P
X. X. = 0.5X, 1+11-4 -
X dp
Xds=0.5Xdp 1 - 1 - 4 -LdP
Astrom and Hagglund (1996) point out that the ideal controller admits complex zeroes and is thus a more flexible controller structure than the series controller, which has real zeroes; however, in the frequency domain, the series controller has the interesting interpretation that the zeroes of the closed loop transfer function are the inverse values of Tis and Tds. O'Dwyer (2001b) developed a comprehensive set of tuning rules for the series PID controller, based on the ideal PID controller; as these are not strictly original tuning rules, only representative examples are included in the relevant tables.
In a similar manner, the parameters of an ideal controller in series with a first order filter may be determined from the parameters of the classical controller 1 (Witt and Waggoner, 1990) and the parameters of
20 Handbook of PI and PID Controller Tuning Rules
the non-interacting controller 10 structure may be determined from the parameters of the ideal PID controller (Kaya et al, 2003).
2.4 Process Modelling
Processes with time delay may be modelled in a variety of ways. The modelling strategy used will influence the value of the model parameters, which will in turn affect the controller values determined from the tuning rules. The modelling strategy used in association with each tuning rule, as described in the original papers, is indicated in the tables (see Chapters 3 and 4). These modelling strategies are outlined in Appendix 2. The following models are defined:
K e~STm
1. First order lag plus time delay (FOLPD) model ( G m (s) = — )
2. FOLPD model with a positive zero ( G m ( s ) = - mV m3J
l + sTml
K m ( l + sTm 3)e-3. FOLPD model with a negative zero ( G m (s) = — ^ ^ )
1 + sTml
4. Non-model specific
K e"SXm
5. Integral plus time delay (IPD) model ( G m (s) = —2 ) s
6. First order lag plus integral plus time delay (FOLIPD) model
(Gm(S)= v r v s(l + s T j
7. Second order system plus time delay (SOSPD) model K e~SXm K e~SXm
(0- (' )=T.1v;^T . , .+ ,°r0-w-(1+TjxuT. I .)> 8. Integral squared plus time delay ( I 2 P D ) model ( G m ( s ) = —m- )
s 9. Second order system (repeated pole) plus integral plus time delay
(SOSIPD) model ( G m (s) = fm& "" ) s(l + sT m ) 2
Chapter 2: Controller Architecture 21
10. SOSPD model with a positive zero
( G (s)- Kj l -sT^y"- Km(l-sTm3)e-( m ( ) (l + sTml)(l + s T m 2 ) ° r G m ( S ) T m l V + 2 ^ T i n l s + l )
11. SOSPD model with a negative zero
( G m ( s ) =K - ( l + S > > - " ; o r G m ( s ) , Km(l + STm3)e
- S T „
(l + sTml)(l + sTm2) m Tml
2s2+2^mTmls + l
12. Third order system plus time delay model
,^ , -. " m \ - • J,S + b , S 2 + b . S , .
(Gm(s)= mA ' 2 , 3 A or Km(l + b1s + b2s2+b3s3)e"ST"
(l + a,s + a2s2 + a3s3)
K e""m
G m ( S ) = ( l + sTml)(l+msTm2)(l + S T m 3 ) )
13. Fifth order system plus delay model
Km(l + bIs + b 2 s 2 + b 3 s 3 + b 4 s 4+ b s s 5 ) e - ^
(Gm(s) = r j 5 4 n ' \1 + ats + a2s + a3s + a4s + a5s J
K e~STm
14. Unstable FOLPD model (Gm(s) = —* ) Tms-1
15. Unstable FOLPD model with a positive zero Km(l-sTm 3)e-
(Gm(s)= m \ mi ) Tms-1
16. Unstable SOSPD model (one unstable pole)
K e"STm
( G m ( s ) = 7 - ™ ) (TmlS-lX1 + sTm2)
17. Unstable SOSPD model with a positive zero
' ~ T m l V - 2 S m T m l s + l (Gm(s)= J Y ^ T m 3 i e . )
18. Unstable SOSPD model (two unstable poles)
K e~STm
(G (s)= -^- ) ( m ( ) (T m l s- lXT m 2 s- l ) )
19. Delay model (Gm(s) = Kme_s'm )
K e"s'tm
20. General model with a repeated pole (Gm(s) = — ) (l + s T J n
22 Handbook of PI and PID Controller Tuning Rules
21. General model with integrator
K n(Tm , i s + i)n(T
m / s 2 + 2^„iTm 2 i
s + i)
j i
22. General stable non-oscillating model with a time delay
Tables 1 and 2 show the number of tuning rules defined for each PI or PID controller structure and each process model. The following data is key to the process model type in Tables 1 and 2: Model 1: Stable FOLPD model Model 2: Non-model specific Model 3: IPD model Model 4: FOLIPD model Model 5: SOSPD model Model 6: Other models
Table 1: PI controller structure and tuning rules - a summary
Process model Controller Equation
(2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) Total
1
172 3 2 7 1 2 0
187
2
42 0 0 2 0 0 0
44
3
46 2 0 4 9 1 1
63
4
10 0 0 3 14 1 0 28
5
32 0 0 3 0 0 0
35
6
62 2 0 13 7 2 0 86
Total
364 7 2
32 31 6 1
443
Chapter 2: Controller Architecture 23
Table 2: PID controller structure and tuning rules - a summary
Process model Controller Equation
(2.10,2.11) (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) (2.21) (2.22) (2.23) (2.24) (2.25) (2.26) (2.27) (2.28)
Subtotal
(2.29) (2.30) (2.31) (2.32)
Subtotal
(2.33) (2.34) (2.35) (2.36) (2.37) (2.38) (2.42) (2.43) (2.44) (2.45) (2.46) (2.47) (2.48)
1 2 3 4 5 6 Total
Ideal structure + variations 100 7 0 2 2 1 0 1 16 0 0 0 0 0 0 0 0 1
130
51 2 0 2 2 0 0 0 11 0 0 0 0 0 0 1 0 0 69
22 1 0 0 0 0 1 0 2 0 0 0 0 0 1 0 0 0
27
12 5 0 0 1 0 0 0 2 0 1 0 2 0 0 1 0 0
24
55 13 1 1 0 0 0 0 3 1 0 0 0 0 0 1 2 0 77
36 9 0 0 0 0 0 0 5 0 0 1 0 2 0 9 0 0 62
276 37 1 5 5 1 1 1
39 1 1 1 2 2 1 12 2 1
389 Classical structure + variations
54 1 6 2 63
5 0 5 1
11
8 2 0 1
11
5 1 1 1 8
20 2 7 0 29
9 0 6 2 17
101 6
25 7
139 Non-interacting structure + variations
1 5 6 12 4 2 0 10 2 10 1 1 0
0 0 0 1 0 0 0 3 0 0 0 0 1
0 0 0 5 0 1 0 4 0 8 0 1 0
0 0 0 6 0 0 0 2 0 11 0 2 0
5 0 0 7 0 0 0 3 1 1 0 0 0
2 0 0 13 0 0 4 0 0 0 1 4 0
8 5 6
44 4 3 4 22 3
30 2 8 1
24 Handbook of PI and PID Controller Tuning Rules
Process model Controller Equation
(2.49) (2.50) (2.51)
Subtotal
(2.53) (2.54) (2.55) (2.57)
Subtotal
1 2 3 4 5 6 Total
Non-interacting structure + variations (continued) 0 0 1
55
0 0 0 J
1 0 1
21
0 1 1
23
0 0 0
17
2 0 1
27
3 1 4
148 Other controller structures
8 0 0 0 8
0 0 0 0 0
0 0 0 0 0
1 2 1 0 4
0 0 0 2 2
1 0 0 0 1
10 2 1 2 75
Total | 256 | 85 | 59 | 59 | 125 | 107 | 691
Table 1 shows that 82% of tuning rules have been defined for the ideal PI controller structure, with 42% of tuning rules based on a FOLPD process model.
The range of PID controller variations has lead to a less homogenous situation than for the PI controller; Table 2 shows that 40% of tuning rules have been defined for the ideal PID controller structure, with 37% of tuning rules based on a FOLPD process model. Many of the controller structures are used in a variety of manufacturers' products (as outlined in Section 2.3); clearly a range of tuning rules are available for use with these products.
2.5 Organisation of the Tuning Rules
The tuning rules are organised in tabular form in Chapters 3 and 4. Within each table, the tuning rules are classified further; the main subdivisions made are as follows: (i) Tuning rules based on a measured step response (also called process
reaction curve methods), (ii) Tuning rules based on minimising an appropriate performance
criterion, either for optimum regulator or optimum servo action, (iii) Tuning rules that give a specified closed loop response (direct
synthesis tuning rules). Such rules may be defined by specifying the
Chapter 2: Controller Architecture 25
desired poles of the closed loop response, for instance, though more generally, the desired closed loop transfer function may be specified. The definition may be expanded to cover techniques that allow the achievement of a specified gain margin and/or phase margin,
(iv) Robust tuning rules, with an explicit robust stability and robust performance criterion built in to the design process,
(v) Tuning rules based on recording appropriate parameters at the ultimate frequency (also called ultimate cycling methods),
(vi) Other tuning rules, such as tuning rules that depend on the proportional gain required to achieve a quarter decay ratio or to achieve magnitude and frequency information at a particular phase lag.
Some tuning rules could be considered to belong to more than one subdivision, so the subdivisions cannot be considered to be mutually exclusive; nevertheless, they provide a convenient way to classify the rules. In the tables, all symbols used are defined in Appendix 1.
Chapter 3
Tuning Rules for PI Controllers
3.1 FOLPD Model Gm(s) = K„e"
l + sT„
3.1.1 Ideal controller Gc(s) = Kt
v T> sv
Table 3; PI controller tuning rules - FOLPD model Gra(s) = l + sT„
Rule
Callender et al. (1935/6).
Model: Method 1
Kc Ti Comment
Process reaction '0.568/Kmxm
20.690/Kmxm
3.64xm
2-45xm
^=- = 0.3 T
m
Decay ratio = 0.015; Period of decaying oscillation = 10.5xm
' Decay ratio = 0.043; Period of decaying oscillation = 6.28xn
26
Chapter 3: Tuning Rules for PI Controllers 27
Rule
Ziegler and Nichols (1942).
Model: Method 2
Hazebroek and Van derWaerden(1950).
Model: Method 2
Oppelt(1951). Model: Method 2
Moros (1999). Model: Method 1
Chienefa/. (1952)-regulator.
Model: Method 2;
0.1 <^2-< 1.0 T
Chiene/a/. (1952)-servo.
Model: Method 2;
0.1 <^2-< 1.0 T
Kc
0.9Tm
K m t m
x l T m
Km tm
T;
3.33xm
x 2 x m
Comment
Quarter decay ratio.
T
Coefficient values
T
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x l
0.68 0.70 0.72 0.74 0.76 0.79 0.81 0.84 0.87
x2
7.14 4.76 3.70 3.03 2.50 2.17 1.92 1.75 1.61
x, = 0 . 5 ^ + 0.1
3KC(1)
0.8Tra/Kmxm
0.91Tm/Kraxm
0-6Tm
K m T m
0.7Tm
Kmxm
0.35Tm
0-6Tm
K r a x m
x m
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
x 2 = -
x i
0.90 0.93 0.96 0.99 1.02 1.06 1.09 1.13 1.17
x2
1.49 1.41 1.32 1.25 1.19 1.14 1.10 1.06 1.03
*m
•6tm-1.2Tm
1.66xm
3.32Tra
3tm
3.3xm
4xm
2-33xm
1.17Tm
Tm
T
2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
x i
1.20 1.28 1.36 1.45 1.53 1.62 1.71 1.81
x2
1.00 0.95 0.91 0.88 0.85 0.83 0.81 0.80
^2->3.5
xm » Tra
^ m « T m
attributed to Oppelt
attributed to Rosenberg
0% overshoot
20% overshoot
0% overshoot
20% overshoot
3 K,(1> K „
. 5 7 0 8 ^ + 1 I recommended K (0 K „
0.77-
28 Handbook of PI and PID Controller Tuning Rules
Rule
Reswick(1956). Model: Method 2;
K c , Tj deduced
from graphs.
i s - = 6.25 T
m
Cohen and Coon (1953).
Model: Method 2
Two constraints method - Wolfe
(1951). Model: Method 4
Two constraints criterion - Murrill (1967) -page 356. Model: Method 5
Heck (1969). Model: Method 2
Kc
0.20/Km
0.30/Km
0.15/Km
0.20/Km
J _ 0 . 9 ^ + 0.083
" XlTm
KmTm
Ti
0.20xm
0.22Tm
0.16xm
0.14xm
4 T ( 2 ) i
X 2 T m
Coefficient values
T m
0.2
0.5
0.925
* 1
4.4 1.8
x2
3.23 2.27
/ N 0.946
l X m J
0.8Tm
K m T m
0
T
1.0 5.0
*1
0.78 0.30
x 2
1.28 0.53
x ( A0 '583
1.078 [Tm J
3 i m
x 2 Km T m
Comment
0% overshoot -regulator
20% overshoot -regulator
0% overshoot - servo
20% overshoot -servo
Quarter decay ratio;
0 < - ^ _ < 1.0 T
Decay ratio is as small as possible;
minimum error integral (regulator
mode).
Quarter decay ratio; minimum error
integral (servo mode).
0.1<-^2L<1.0 T
m
Integral controller
Representative coefficient values - from graph - "Stoning"
1B. T
0.11
0.13 0.14
0.17
X 2
2.94 2.86
2.78 2.63
T m
0.20
0.25 0.29 0.33
x 2
2.50
2.38 2.35 2.30
IHL T
1 m
0.40 0.50 0.56
0.63
x 2
2.17
2.13 2.08 2.00
T
0.71
x 2
1.96
• ( 2 )
3.33- -0.31
1 + 2.22-
2\
Chapter 3: Tuning Rules for PI Controllers 29
Rule
Heck (1969)-continued.
Fertik and Sharpe (1979).
Sakai et al. (1989). Model: Method 2
Borresen and Grindal (1990).
Model: Method 2
Klein et al. (1992). Model: Method 1
McMillan ( 1 9 9 4 ) -page 25.
Model: Method 5 St. Clair ( 1 9 9 7 ) -
page 22. Model: Method 5
Hay (1998) -pages 197-198.
Model: Method 1;
Coefficients of KC ,T;
deduced from graphs
Shinskey (2000), (2001).
Model: Method 2
Kc Tf Comment
Representative coefficient values - from graph - "Fuhrung" T m
T
0.11
0.13 0.14
0.17
x2
3.92
3.85 3.64
3.51
T m
0.20
0.25 0.29
0.33
0.56
K m
1.2408Km
T T
1.0Tm
K m t m
0.28Tm
K m ( t m + 0 . 1 T r a )
3
0.333Tm
K m T m
x , / K m
x2
3.33
3.08 2.94
2.82
T
0.40
0.50 0.56
0.63
x2
2.63 2.47 2.41 2.35
0.65Tm
0.5xm
3^ r a
0.53Tra
m
T
X 2 t m
T
0.71
x2
2.27
Model: Method 3
Also described by ABB (2001)
Time delay dominant processes
l a . > 0.33 T
Coefficient values - servo tuning
Tm/Tm
0.1 0.125
0.167 0.25 0.5
x i
6.0 4.5 3.0 2.0 0.8
x2
9.5 7.5 5.4 3.4 1.7
"Cm Am
0.1 0.125
0.167 0.25
0.5
x l
4.0 3.2 2.2 1.3 0.5
x2
10.0 8.0 6.0 4.0 2.0
Coefficient values - regulator tuning
^ m / T m
0.1 0.125 0.167 0.25
0.5
X I
9.5 7.0 5.0 3.4 1.8
0.667Tm
KmTm
x2
3.2 3.2 3.0 2.8 2.0
tm/Xi 0.1
0.125 0.167 0.25 0.5
3.78xra
x i
6.8 5.6 4.3 2.8 1.2
x2
2.8 2.8 2.6 2.3 1.6
T m
30 Handbook of PI and PID Controller Tuning Rules
Rule
Liptak(2001). Model: Method 2
Minimum IAE -Murrill(1967)-pages 358-363.
Minimum IAE -Frank and Lenz
(1969). Model: Method 1
Minimum IAE -Pemberton (1972a), Smith and Corripio (1997) -page 345.
Minimum IAE - Yu (1988).
Model: Method 1. Load disturbance
K e~STu
model = —- . l + sTL
Kc
0.95Tm
Kmxra
T i
4^m
Comment
Minimum performance index: regulator tuning f N 0.986
0-984 (lm\
— | x , + x 2 - ^
T
0.608
( \
T
0.707
x 2 I T m J
Model: Method 5;
0 . 1 < ^ - < 1 . 0 T
Representative coefficient values - deduced from a graph
* m / T m
0.2 0.4 0.6 0.8
x l
0.37 0.42 0.44 0.45
T
KmTm
5KC(3)
6 K c ( 4 )
x 2
0.77 0.85 0.91 0.98
Tm/Tm
1.0 1.2 1.4
T
T ( 3 )
T ( 4 )
x i
0.46 0.46 0.46
x 2
1.05 1.11 1.17
Model: Method 1;
0.1 <-^2-< 0.5 T
m
^ - < 0.35 Tm
5 K <3> = 0.685
K „ T V m
T „
X ( 3 ) = -0.214 T,
\ 1.977-!=—0.55 . T I
K W _ 0.874
K „
N -0.099+0.159—
T V 1 m J
m J
N - 1 . 0 4 1
T •^-1 , ^ < 2.641^2- + 0.16.
T T
T V m 7
-4.515—+0.067,
- ( 4 )
0.415 T T , 2.641-SL + 0 . 1 6 < - ^ - < l .
T T
Chapter 3: Tuning Rules for PI Controllers 31
Rule
Minimum IAE - Yu (1988)-continued.
Minimum IAE -Shinskey(1988)-
page 123. Model: Method 1
Minimum IAE -Shinskey(1996)-
page 38. Model: Method 1 Minimum IAE -Mar l in (1995) -pages 301-307.
Model: Method 1.
K c , Tj deduced from
graphs: robust to
±25% process model
parameter changes.
Minimum IAE -Edgar e/a/ . ( 1 9 9 7 ) -
pages 8-14,8-15. Model: Method 1 Minimum IAE -
Edgar e/a/ . ( 1 9 9 7 ) -page 8-15.
Kc
7 K C( 5 >
8 j / (6)
1.00Tm /Kmxm
1.04Tm /K r axm
l - 0 8 T m / K m t m
1.39Tm /Kmxm
0.95T m /K m T m
0.95T m /K m T m
l-4/Km
l-8/Km
l-4/Km
l-0/Km
0.8/Km
0.55/Km
0.45/Km
0.35/Km
0-4/Km
0-94Tm /KmTm
0 .67T m /K m x m
T,
T ( 5 ) i
T ( 6 ) i
3.0tm
2.25im
l-45t ra
"Cm
3.4xm
2-9xm
0.24xm
0.52tm
0.75tm
0.68xm
0-71im
0.60t ra
0.54tm
0.49xm
0.5xm
4xB
3-5im
Comment
^ r a / T m > 0 . 3 5
T m / T m = 0 . 2
T m / T r a = 0 . 5
^ m / T m = l
^ m / T m = 2
T r a / T m = 0 . 1
T m / T m = 0 . 2
* m / T m = 0 . 1 I
t m / T r a = 0 . 2 5
t m / T m = 0 . 4 3
t m / T m = 0 . 6 7
x m / T m = 1 . 0
x m / T r a = 1 . 5 0
x m / T m = 2 . 3 3
T m / T m = 4 . 0
Time delay dominant
Time constant dominant
Model: Method 2
K (5) 0.871 f ^ -0 .015+0 .384^ f _ ^-1.055
Ti
K „ T T V m J
<(5)
0.444 T,
\ -0.217-!!— 0.213/ % 0.867
I T" I X ) T ,
T T
K (6) _ 0.513 K „ T V T m
» i 0.670 T
V m J
-0.003—-0.084,/
T V m y
32 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum IAE -Huang et al. (1996).
Model: Method 1
Minimum IAE -Shinskey(1988)-
page 148. Model: Method 1
Minimum IAE -Shinskey(1994)-
page 167.
Minimum IAE -Shinskey(1996)-
page 121. Minimum ISE -
Hazebroek and Van derWaerden(1950).
Model: Method 2
Kc
9KC(7>
0.58KU
0.54KU
0.48KU
0.46KU
Ku
T 3 .05-0 .35-^
0.55KU
. . K c ( 9 ,
Ti
T ( 7 ) i
0.81TU
0.66TU
0.47TU
0.37TU
10 j (8)
0.78TU
1.43Tm
Comment
0 . 1 < ^ - < 1 Tm
i m / T m = 0 . 2
xm /Tm=0.5
v / L =i fm/T m =2
Model: Method 1
Model: Method 1; x m /T m =0.2
^ m /T m <0.2
' K ™ K „
6.4884 + 4 .6198^- + 0.8196 ^ T
V m
K „
-5.2132
-7.2712 -0.7241e' 'm J
•y (') T-1
0.0064 + 3 . 9 5 7 4 ^ - 6 . 4 7 8 9 ^ 2 - 1 +9.4348
10 y (8) _ ~ 0 .87-0 .855-^ + 0.172
12\
T -10.7619
f \4
T V ra J
+ T „ 7 . 5 1 4 6 ^ •l.lttb f \ 6
T
» K ' 9 ) = ^ M o . 7 4 + 0.3-T"
Chapter 3: Tuning Rules for PI Controllers 33
Rule
Hazebroek and Van derWaerden(1950)-
continued.
Minimum ISE -Haalman(1965). Model: Method I Minimum ISE -Murrill(1967)-pages 358-363.
Model: Method 5
Minimum ISE -Frank and Lenz
(1969). Model: Method 1
Minimum ISE -Zhuang and Atherton
(1993). Model: Method I
Minimum ISE - Yu (1988).
Model: Method 1.
Kc
x l T m
K m T m
T;
^ m
Comment
Coefficient values
Li. T
0.2 0.3 0.5
X l
0.80 0.83 0.89
x2
7.14 5.00 3.23
0.67Tm
Kmxm
f N 0.959
1.305 f T O
Km IT J
i f T ^ — x 1 + x 2 - s -
Xm
T
0.7 1.0 1.5
x i
0.96 1.07 1.26
x2
2.44 1.85 1.41
Tm
T (
0.492 [
T m
X 2 v
T m "
T
.0.739
x , + x 2 ^ l
T
2.0 3.0 5.0
x l
1.46 1.89 2.75
x2
1.18 0.95 0.81
Mmax =1.9; Am =
2.36; *„, = 50°
0.1 < ^ s - < 1.0 T
Representative coefficient values - deduced from a graph
TmAm
0.2 0.4 0.6 0.8
1.279
1.346
x l
0.53 0.58 0.61 0.62
/ \ 0.945
—l / s 0.675
—] 12 K (10)
x 2
0.82 0.88 0.97 1.05
Tm/Tm
1.0 1.2 1.4
0-535 \jmj
T ( m
0.552 [
0.586
\0.438
T ( 1 0 )
x l
0.63 0.63 0.62
x 2
1.12 1.17 1.20
0 .1<^_<1 .0 T
\.\<^<2.0 T
^S-<0.35 T
K e~ 12 Load disturbance model = —-— 1 + sT,
K (10) 0.921 ( TL
0.181-0.205—,- N -1.214
T
0.430
TL
N 0.954—-0.49
V m J
, - ^ < 2.3 1 0 ^ L +0.077. T T
34 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ISE - Yu (1988)-continued.
Minimum ITAE -Murrill(1967)-pages 358-363.
Model: Method 5
Minimum ITAE -Frank and Lenz
(1969). Model: Method 1
Kc
13 K (11)
14 K (12)
15 K (13)
/ N 0.977
0-859 (lm\
— (x, + x 2 i -
Ti
T ( i i )
T ( 1 2 ) 1 i
T ( 1 3 )
, .0.680
Tm O 0.674 | j m J
^ f x 1 + x 2 ^ x2 V ' » ;
Comment
-^2- < 0.35 T
-^2- > 0.35 T
0 .1<^L<1.0 T
Representative coefficient values - deduced from a graph T m / T m
0.2 0.4 0.6 0.8
x i
0.32 0.35 0.37 0.38
x 2
0.77 0.80 0.86 0.93
Tm/Tra
1.0 1.2 1.4
x l
0.38 0.39 0.39
x 2
1.00 1.05 1.10
1.157 K „ T
T ( U ) = .
0.359 T,
-2.532-=!-0.292 .
T V 1 m J
, 2 . 310^ + 0.077 < i < l T T
H K (12) 1.07 , N -0.065+0.234-5!- , N-1.047
T
T„,
T V 1 m J
,(12)
0.347 T
-1.112—!!!—0.094 ,
v T l<-^-<3.
15 K (13) 1.289
K „ T V m
0.04+0.067-Tm T
T(13) X > i
(T \
0.596
0.372-55-- 0.44 ^ .0.46
T V m J , T V m y
Chapter 3: Tuning Rules for PI Controllers 35
Rule
Minimum ITAE - Yu (1988).
Model: Method 1. Load disturbance
K e"STL
model = — . l + sTL
Minimum ITSE -Frank and Lenz
(1969). Model: Method 1
Kc
16 ^ (14)
17 K (15)
18 K (16)
19 K (17)
\ ( T ^ — X , + X 2 ^
Tj
j ( 1 4 )
T ( 1 5 )
T ( 1 6 )
T ( 1 7 )
x2
( T ^ X, +X2^2-
T
Comment
^ - < 0 . 3 5 T
m
•^E->0.35 T
m
Representative coefficient values - deduced from a graph
Tm/Tm
0.2 0.4 0.6 0.8
x i
0.46 0.50 0.53 0.55
x 2
0.84 0.89 0.97 1.04
^m/Tm
1.0 1.2 1.4
x l
0.55 0.54 0.54
x 2
1.09 1.15 1.20
0.598 K „
/ N 0.272 -0.254-
T T V m J
.(14) T „ /• s 0.304-5^-0.112^ x 0.196
0.805 > T ^ < 2 .385-^ + 0.112.
17 j r (15) 0.735 , x-0.011-1.945-
K, T V m J
T
-1.055
'(15) T m T,
18 K (16) _ 0.787 T T L
Km {Tm
0.425
0.084+0.154
T V m J
-5.809^+0.241 , N 0.901
2 .385^- + 0 . 1 1 2 < ^ < l . T T
T
, N - 0 . 1 4 8 - ^ - 0 . 3 6 5 / N 0.901
-(16)
0.431 , T T V l m J
, l < i . < 3 . T
19 K (17) 0.878 (T L
Km [Tm T V 1 m
. (17) T m
/ \C228-15—0.257
0.794 T T
36 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ISTSE -Zhuang and Atherton
(1993). Model: Method 1
Minimum ISTSE -Zhuang and Atherton
(1993).
Minimum ISTSE -Zhuang and Atherton
(1993).
Minimum ISTES -Zhuang and Atherton
(1993). Model: Method I
Nearly minimum IAE, ISE, ITAE -
Hwang (1995).
Kc
1.015
K m
1.065
/ \ 0.957
V T nJ r T \0-673
VTnJ
20 K (18)
21 K (19)
1.021
1.076
K m
, N 0.953
Tjn.1 V T m J
(T ^ 0.648
22 K (20)
T i
T
0.667
f x 0.552
V m j
T /" ^ - 4 2 7
0-687 {Tm j
T ( 1 8 ) i
T ( 1 9 ) 1 i
T x m
0.629
T
0.650
( \ T m
f \
T
0.546
0.442
K c( 2 0 ) / u 2 K u co u
Comment
0 .1<^2L<1 .0 T 1 m
1.1 < ^ s - < 2.0 T Am
Model: Method 1;
0.1 < ^ s - < 2.0 T
m Model: Method 31;
0 . 1 < ^ S - < 2 . 0 T Am
O . l ^ ^ - S l . O T
m
L I S — < 2 . 0 T
Model: Method 35;
0 . 1 < ^ - < 2 . 0 T Am
20 K ( ,8 )_f l -892K m K u +0.244 , y T ( 1 8 )
3.249KmKu +2.097 Ku, V
K, (19)
A
0.706KmKu-0.227 \
0.7229KmKu+1.2736 J
4.126KmKu-2.610
5.848KmKu-1.06 K u , ^
(19) _ 5.352KmKu-2.926
x5.539KmKu + 5.036
22 ^ = ( 1 - ^ , , * = 1 .14( l -0 .482 m .T m ^0 .068a ,^ ) K u K m / l + KuKm
0.0694(-l + 2.1muTm-0.367co2uT^)
T u .
H2 K u K m / 1 + K u K n
Decay ratio = 0.15.
Chapter 3: Tuning Rules for PI Controllers 37
Rule
Nearly minimum IAE and ITAE - Hwang
and Fang (1995). Model: Method 25
Gerry (2003) -minimum error: step
load change. Model: Method I
Minimum IAE or ISE - ECOSSE Team
(1996a). Model: Method 1.
Coefficients of Kc, Tj deduced from
graphs
Minimum IE -Devanathan (1991). Model: Method 1.
Coefficients of Kc, Tj deduced from
graphs
Kc
23 K (21)
0.3
Kra
x,/Km
Ti
T ( 2 1 ) i
0.42xm
X 2 ( t m + T m )
Comment
0.1 < -^2- < 2.0; T
m
decay ratio = 0.12
x ±ZL>5 T
Coefficient values
fm/Tm
0.11 0.25 0.43 0.67 1.0 1.50 2.33 4.0
X l
1.1 1.8 1.1 1.0 0.8 0.59 0.42 0.32
24 K (22)
x2
0.23 0.23 0.72 0.72 0.70 0.67 0.60 0.53
f m / T m
0.11 0.25 0.43 0.67 1.0
x 2 T u
x i
5.8 3.1 2.1 1.7
0.91
x2
0.5 0.6 0.7 0.8 0.9
Coefficient values
t m / T m
0.2 0.4 0.6 0.8 1.0
x2
0.35 0.29 0.26 0.23 0.21
^m/ T m
1.2 1.4 1.6 1.8
x2
0.19 0.18 0.18 0.17
0 .515-0 .0521^- + 0.0254 T
2 ^
K u ,
, ( 2 1 ) (K c( 2 1 ) /K u « u )
0.0877 + 0.0918-2L-0.0141 T T
m _
2 ^
2; cos 24 K (22)
tan 1.57DD
Kmcos tan
0.16
6.28T
38 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum IAE -Gallier and Otto
(1968). Model: Method 47
Minimum IAE -Rovira etal. (1969).
Model: Method 5
Minimum IAE - Bain and Martin (1983). Model: Method 1 Minimum IAE -Marlin(1995)-pages 301-307.
Model: Method 1
Minimum IAE -Huang et al. (1996).
Model: Method I Minimum IAE -
Smith and Corripio {1991)-page 345. Model: Method 1 Minimum IAE -Huang and Jeng
(2002). Model: Method 1
Kc
Minimum x, /Km
Ti Comment
performance index: servo tuning x 2 ( t m + T m )
Representative coefficient values - deduced from graphs
fmAm
0.053 0.11 0.25
x l
12 6.4 2.9
, % 0.861
0.758 fT m ^ K m l T m J
1
Km
0.3 + 0.3^2-
1.4
Kra
x2
0.95 0.91 0.84
•Cm/Tm
0.67 1.50 4.0
T m
1.020-0.323^-Tra
Tm+0.4xm
0.72tm
Xl
1.15 0.65 0.45
x2
0.75 0.66 0.55
0 .1<Is-<1.0 T
m
-^ ->0 .2 T
^ - = 0.11 T
m Kc , Tj deduced from graphs; robust to ±25% process model
parameter changes
25 K (23)
0-6Tm
KmTm
0.59Tm
K r atm
T ( 2 3 ) i
T m
T m
0.1<^-<1 T
0.1<^-<1.5 T
m
Minimum IAE = 2.1038xm;
Tm /Tm<0.2
25 K (23) _ _ j _ •13.0454 - 9.0916— + 0.3053 -T T
1 +
K „
N - 1 . 0 1 6 9
•1.1075 f N3.5959
-2.2927
T ,
f •, 3.6843
Tm 4.8259eTm
T V m
0.9771 - 0.2492 - ^ + 3.4651 T T
-7.4538 ^ M + 8.2567 ^
-4.7536 U H L ] + 1 .1496^-
Chapter 3: Tuning Rules for PI Controllers 39
Rule
Minimum IAE -Tavakoli and Fleming
(2003). Model: Method 1
Small IAE - Hwang (1995).
Model: Method 35
Minimum ISE -Zhuang and Atherton
(1993). Model: Method 1
Minimum ISE - Khan and Lehman (1996).
Model: Method 1
Kc
26 K (24)
Ti
T ( 2 4 )
Minimum Am = 6dB ; minimum
27 K (25)
f x 0.892
0-980 (lm\
1.072
Km
(T ^ m
l T m J
0.560
28 ^ (26)
Kc(25)/n2Ku<»u
Tm
0.690-0 .155^-T
T m
0.648-0.114-^-T
m
T ( 2 6 ) i
Comment
0.1<^2_<10 Tm
* m =60°
Decay ratio = 0.1,
0 . 1 < ^ L < 2 . 0 T
m
0.1< — < 1 . 0 T
1.1<^2.<2.0 T
0.01 < ^2-< 0.2 T
m
26 j , (24)
K „ 0.4849-2- + 0.3047 T (24) _ T
' i n 0.4262^2. + 0.9581
T
r = 0.5;
27 Kc(25) = (i _ ^ ) K u , n, = 1-07(l-0.466muTm+0.0667co2ux^) ^
K u K m / 1 + K u K m
_ 0.0328(- 1 + 2.21couTm - 0.33&a>lz2m)
^ ~ K u K m / l + K„Km
l.ll(l-0.467<ouTm + 0.0657co2,T2n)
K u K m / 1 + K u K m
^ = 0.0477(-1 + 2.07c)uTm - O . ^ ^ x j ) ^_ K u K m / 1 + K u K m
1.14(l-0.466ffluTm+0.0647ffl2,T2J K u K m / 1 + K u K r a
= 0.0609(-l + 1.97c i)uTm-0.323(n;T2n)>r_1|
K u K m/ 1 + KuKm
r = parameter related to the position of the dominant real pole.
0.7388Tm+0.3185xm
= 0.75;
28 K (26) f 0.7388 ^ 0.3185 "I T m T ( 2 6 ) = T
tm T IK ' ' -0.0003082Tm +0.5291x„
40 Handbook of PI and PID Controller Tuning Rules
Rule
Khan and Lehman (1996) - continued.
Minimum ITAE -Rovirae/a/. (1969).
Model: Method 5
Approximately minimum ITAE -
Smith (2002).
Minimum ISTSE -Zhuang and Atherton
(1993). Model: Method 1
Minimum ISTSE -Zhuang and Atherton
(1993).
Minimum ISTES -Zhuang and Atherton
(1993). Model: Method 1
Kc
29 K (27)
0.586 (Tm^ K m V T m ;
0.916
0.586 Tm
0.712 fTm "> K m V T m ;
0.921
, N 0.559
0-786 (lm\
0.361KU
31 K (29)
, s 0.951
0-569 (lm\
Km I T J
0.628
Km
, N 0.583
KXm J
Ti
T ( 2 7 ) i
T m
1.030-0.165 — T •* m
T
T
0 .968-0 .247^-T
m
T
0.883-0.158^2-T
m
30-r, (28) M
T ( 2 9 )
T
1.023-0.179^-T
T 1 m
1.007-0.1 6 7 ^ L T
Comment
0.2 < i s - < 20 T
0 .1<i2 .<1.0 T
m
Model: Method 1
0.1<is -<1 .0 T 1 m
1.1< — < 2 . 0 T
Model: Method 1;
Q.\< — <2.0 T
m Model: Method 31;
0.1 < i = - < 2.0 T
0.1 < i 2 - < 1.0 T
m
1.1< — < 2 . 0 T
m
^0.808 0.511 0.255 29 j , (27)
VT~T7
- ( 2 7 ) . 0.808Tm+0.51lTm-0.255VT~^
0.095Tm+0.846xm-0.38lVf^
30 T (28) T/28 ' =0.083(l.935KmKu+l)Tu
31 K (29)
f A A
1.506KmKu-0.177
3.341Km K„ + 0.606
K u , T(29) = 0.0551 3.616Km Ku+1 |TU
Chapter 3: Tuning Rules for PI Controllers 41
Rule
Nearly minimum IAE and ITAE - Hwang
and Fang (1995). Model: Method 25
Simultaneous servo/regulator tuning
- Hwang and Fang (1995).
Model: Method 25
34 Minimum performance index -
Wilton (1999). Model: Method I.
Coefficients of Kc
estimated from graphs
Kc
32 K (30)
T(
T ( 3 0 )
Comment
0.1 < -^2- < 2.0; T
decay ratio = 0.03
Minimum performance index: other tuning
33 K (31)
x,/Km
T ( 3 1 ) * i
T
0 . 1 < - i < 2 . 0 ; T
m
decay ratio = 0.05
Coefficient values
^m/Tm
0.1 0.1 0.1 0.1
0.4 0.4 0.4 0.4 0.4
x i
750.0 17.487 4.8990 2.1213
1.6837 1.5297 1.2247 0.9192 0.7668
b
0.0001 0.04 0.2 1
0.008 0.04 0.2 1 5
fm/Tm
0.2 0.2 0.2 0.2 0.2
0.5 0.5 0.5 0.5
x i
3.3675 2.2946 1.7146 1.1313 0.8764
150.01 7.1386 2.6944 1.1314
b
0.008 0.04 0.2 1 5
0.0001 0.04 0.2 1
32 K (30) 0.438-0.110-2-+ 0.0376 T
•}2\
Ku ,
- ( 30 ) (Kc(30)/Kucou)
0.46-0.0835-21-+ 0.0305 T
2 ^
K „
p (31) = _
0.0388+ 0.108-2!--0.0154 T
(KC(31,/KU(DU)
2 ^
0.0644 + 0.0759— - 0.0111 — T T
2V
' Performance index = r[e2( t ) + b2Km2(y(t)-yocl)
2]dt
42 Handbook of PI and PID Controller Tuning Rules
Rule
Wilton (1999)-continued
Minimum ITAE -ABB (2001).
Model: Method 7
Bryant et al. (1973). Model: Method 1
CKmetal, (1973). Model: Method 1
Mollenkamp et al. (1973).
Model: Method I
Kc T i Comment
Coefficient values - continued
t m / T m
0.6 0.6 0.6 0.6 0.6
1.0 1.0
0.8591 r
*1
1.1225 1.1218 0.9798 0.7778 0.6572
72.004 1.6657
N 0.977
b
0.008 0.04 0.2 1 5
0.0001 0.2
1.4837Tm
Tm/Tm
0.8 0.8 0.8 0.8 0.8
1.0 1.0
T
0.68
x.
0.8980 0.9178 0.7348 0.7071 0.6025
3.6713 0.8485
b
0.008 0.04 0.2 1 5
0.04 1
^ - < 0 . 5 T
Direct synthesis: time domain criteria 0.37Tm
K m t m
0.40Tra
K m T m
0
T
T
x2KmT r a
Coefficient values - deduced fr
^m/X 0.33 0.40 0.50 0.67 0.33 0.40 0.50 0.67
x2
12.5 11.1 11.1 9.1 7.1 6.3 5.6 4.3
^Tm
Kjl + XxJ
^m/Tm
1.0 2.0 10.0
1.0 2.0 10.0
x 2
6.7 4.2 3.0
3.2 2.1 1.6
T
X = 1.0 , OS<10%, Model: Method 10 (Po 2004).
T m
K m(T C L+0 T m
Critically damped dominant pole
Damping ratio (dominant pole) = 0.6
om graph
Critically damped dominant pole
Damping ratio (dominant pole) = 0.6
X variable; suggested values: 0.2,
0.4,0.6, 1.0. ulin and Pomerleau,
Appropriate for "small xm"
Chapter 3: Tuning Rules for PI Controllers 43
Rule
Keviczky and Csaki (1973);
OS values in first row; Tm/Tm values
in first column; K m = l .
5% overshoot - servo - Smith e< a/. (1975) - deduced from graph
Model: Method 1 1% overshoot - servo - S m i t h s a / . (1975) - deduced from graph
Bain and Martin (1983).
Model: Method 1
Suyama (1992). Model: Method 1
5% overshoot - servo - Smith and Corripio (1997) page 346. Model: Method 1
Viteckova (1999), Viteckova et al.
(2000a) (Model: Method 9 -
Viteckova et al. (2000b)).
Kc
0 Ti
* 2 * m
Comment
Model: Method 1
x2 coefficient values (estimated from graph)
^m/Tm OS
0.1 0.2 0.5 1.0 2.0 5.0 10.0
0%
30 18 8 5 4 3
2.5 0.52Tm
KmTm
5%
22 11 6 4 3
2.4 2.1
10%
16 9 5
3.5 2.5 2
1.8
T
15%
12 7.5 4 3
2.2 1.8 1.7
20%
10 6
3.5 2.5 2
1.6 1.5
0.04 < i s - < 1.4 T
Also given by Bain and Martin (1983).
0.44Tm
KmTm
0.43Tra
K m T m
1
0.5Tm
K m t m
0-5Tm
T m
T
T
T
T
Model: Method 1;
0.04 < i s - < 1.4 T
1 % overshoot - servo
- i s - > 0.2 T
"Very conservative tuning for small
delay"
OS = 10%
<j>m = 60° ; 0.25 < i s - < 1 (Voda and Landau (1995)) m
* i T m
Kmxm T
Closed loop response overshoot (OS)
shown Coefficient values
x i
0.368 0.514 0.581 0.641
OS (%)
0 5 10 15
Xl
0.696 0.748 0.801 0.853
OS (%)
20 25 30 35
x i
0.906 0.957 1.008
OS (%)
40 45 50
44 Handbook of PI and PID Controller Tuning Rules
Rule
Mann etal. (2001). Model: Method 31 or
Method 33; Closed loop response overshoot (step input)
< 5 %
Viteckova and Vitecek (2002)
Model: Method 1
Pinnella e? a/. (1986). Model: Method 20
Kc
0.5 lTm
K r
35 K (32)
T,
Tm
T ( 3 2 )
x, = 0.4, x2 =3.68
x,=0.8, x2 = 3.28
x,=1.2, x2 = 2.88
0
37 K (34)
36 j <33> •M
T ( 3 4 )
Comment
0<-^2-<2
K t m / T m < 2
2 < T m / T m < 4
4 < ^ m / T m
For 0.05 < — < 0.8 , T; = Tm ; overshoot is under 2% (Viteckova
and Vitecek (2003))
0 38 K (35) Critically damped
servo response
35 K (32)
K „ 0.56 + 1.02-
, ( 3 2 )
1.0404 ^ - - x , x„.
+ x ,
0.56 + 1.02- I m _ h . 0 4 0 4 ^ 2 - - x , + x 2
0 . 0 4 - 1 . 0 2 ^ + 1 ^ 2 - 1 . 0 4 0 4 ^ - x , | + x, M
• ( 3 3 ) K „
1 0.25 1 0.5 - + X " T ' T
m m m m
T T -J2- +0.25 - - = - + 0.5
37 K (34) ^ - [ T m T m x 12 + ( 2 T m + x m ) x 1 + l ] e ^ >
2 0.5 2 0.25 „, X, = + T + r , T :
(34)
T T V T 2
T 2
\ 2
^ m T m x 12 +(2T m +T m )x l +l
X ^ m 1 ™ * ! +Tm + 1 J
38 K (35)
4Kmxm ' T T m J
T J 2 -T T
2
+ 1 0.5
e
Chapter 3: Tuning Rules for PI Controllers 45
Rule
Schneider (1988), Bekker etal. (1991), Khan and Lehman
(1996). Model: Method 1
Regulator - Gorecki et al. (1989).
Model: Method 1
Hang ef a/. (1991). Model: Method 1
McAnany (1993). Model: Method 24
Sklaroff(1992), Astrom and Hagglund (1995) -pages 250-
251.
Kc
0.368 T m
Km tm
0.4 T
13 m
Kraxm
T 1.571 m
39 K (36)
40 K (37)
T,
Tm
T
T
T (36) 1 i
T ( 3 7 )
Comment
4 = 1
4 = 0.6
4=0.0
Pole is real and has maximum attainable
multiplicity
0.16 < ^2-< 0.96 T
Servo response: 10% overshoot, 3% undershoot 41 K (38)
3
Km fl + ^ 1
T ( 3 8 ) i
T m
TCL = 1.67Tm
Model: Method 28; Honeywell UDC 6000 controller
39 K (36> = 2 T
2 + 2T
V m
- 1 2Tm 2Tm
. ( 3 6 ) 2T
3 + v 2T , 2T
2 + 2T
V m
2 + 2T
40 K (37) = :
12 + 2
1 1 - ^ + 13 T
3 7 ^ - - 4 T
15 + 14
1 1 ^ - + 13 T
3 7 ^ - - 4 T
K U ,T , (37) _ 0.2
11^2- + 13 T
3 7 ^ - 4 m
+ i
y
41 K (38) _ (l-44Tm +0.72TmTro -0.43xm -2.14) (3g)
K ^
Km(5
72xm+0.36Tmz+2 4T +1 .28T-2 .4
46 Handbook of PI and PID Controller Tuning Rules
Rule
Fruehauf et al. (1993).
Model: Method 2
Kamimura et al. (1994).
Model: Method 1
Zhang (1994). Model: Method 1
Davydov et al. (1995).
Model: Method 6
Kuhn(1995). Model: Method 14
Kc
0.5556Tm
TmKm
0.5Tm
T m K m
42 K (39)
43 K (40)
44 K (41)
45 K (42)
0.5
1
T i
5tm
Tm
T ( 3 9 )
T ( 4 0 ) 1 i
T ( 4 1 )
T
T ( 4 2 )
0.5(Tra+Tra)
0.7(Tm+xm)
Comment
i a -<0 .33 T
-^2-> 0.33 T
m Servo response: 10%
overshoot Servo response: 0%
overshoot "Quick" servo
response: "negligible" overshoot
"Good" servo response; xm is
"small" 4 = 0.9;
0 . 2 < x m / T m < l .
"Normale Einstellung"
"Schnelle Einstellung"
42 K (39) 0 - 5 ( T m + Q
Km [0 .5xm (2Tm+xJ + 0.10]
0.375(Tm+Tm)
T (39) _ T
' li 1 i r 0.5xm(2Tm+xm)+0.10
Tm + t m
43 jr (40) _
44 K (41)
Km [o .5xm (2Tm+xJ + 0.0781]
0-425(Tm+Tm)
Km[0.5xm(2Tm+xm)+0.083] :
T ( 4 0 ) _ T + T 0.5xm(2Tm+xm)+0.0781
T-(41)=T +x 0.5x r a(2Tm +xm) +0.083
45 K (42) ,(42)
K „ 1.905--- + 0.826 T
0.153-^ + 0.362 T
T .
Chapter 3: Tuning Rules for PI Controllers 47
Rule
Khan and Lehman (1996).
Model: Method 1
Abbas (1997). Model: Method 1
Valentine and Chidambaram
(1997a). Biefa/.(1999).
Model: Method 13
Bi et al. (2000). Model: Method 13
Ettaleb and Roche (2000).
Model: Method 29
Kc
46 K (43)
47 K (44)
48 j r (45)
1 0.43-0.97^2-
T
0.5064Tm
K m T m
0.5064Tm
Kraxra
1
Ti
T (43) i
T (44 ) i
Tm+0.5xm
0.413 + 0.8^2-T
m
Tm
TmKm
0.5064
T m + T m
Comment
0.01 < ^ L < 0.2 T
0.2 < -^2. < 20 T
0 < V < 0.2 ;
0 . 1 < ^ - < 5 . 0 T
Model: Method 1. \ = 0.707 49
TC L = Tm + ^m
46 T^ (43) _ _J_ni
K „ MM1,0,723 ^
-0.404tm2+0.723Tmxm-0.3852Tm
2
-0.525Tm2+0.4104Tmtm-0.00024Tm
47 K (44)
K „
0.404 0.256 0.1275 T_ VTmT.r
0.404Tm+0.256tm-0.1275VTV^ 1 0.0808Tm +0.719xm - 0 . 3 2 4 7 ^
48 K (45)
0.148 + 0.186 - ^
Km (0.497- 0.464V0590)
s -1.045
49 ±1% Ts = 2.5Tra , 0.1 < -^s. < 0.4 ; ±1% Ts = 5Tm , 0.5 < - ^ - < 0.9 .
48 Handbook of PI and PID Controller Tuning Rules
Rule
Chen and Seborg (2002).
Skogestad (2003). Model: Method 45
Gorez (2003). Model: Method 1
Kc
50 K (46)
0.5Tm
51 K (47)
52 p, (48)
Ti
T ( 4 6 ) 1 i
min(Tm,8Tm)
T
( l - V > m + T m
Comment
Model: Method 1
o < ^ = — < i m ^ m + T m
I < T m < 1 ^m + Tm
50 K (46) 1 TmTm + 2 T mT C L ~ T c L T (46) _ J m ^ m + . 2 T m T C L - T C L
Km (TC L +xJ2 T m + T m
0<T r < T m + ^ T + T x T/^se -v
d desired K c ( T C L S + l ) 2
XK m T„ with
1 1 . 5 M / - 2 x = — - + M s ( M s - l ) 0 . 3 2 M S
2 ( M S - 1 x + T
V 5x„
t m +
0.5
MS(MS
TmX/M7j
2.5x„
v (xm+TmyMsy
52 K (48)
XKmTn withv = 1-0.5 T m + T m
-0.4JM
1-0.4, M
0.1M„
M - 1 7.5-
-0.4, M
1-0.4, M
Chapter 3: Tuning Rules for PI Controllers 49
Rule
Sree et al. (2004), Chidambaram et al.
(2005). Model: Method I
Haen (2005). Model: Method 1
Hougen(1979)-page 335.
Model: Method 1; Coefficients of Kc
estimated from graphs.
Regulator - Gorecki etal. (1989).
Model: Method 1
Kc
, x-0.8915
0 . 9 1 7 9 ( 0
54 K (50)
T i
5 3 T ( 4 9 )
0.96Tm+0.46xm
Comment
0 . 1 < ^ - < 1 T
m
0 < I H L < 4 T
Direct synthesis: frequency domain criteria x,/Km T
Coefficient values
OTm
0.1 0.2 0.3 0.4 0.5
X l
7.0 3.5 2.2 1.8 1.4
55 K (51)
0 T m 0.6 0.7 0.8 0.9 1.0
x l
1.1 1.0 0.8 0.75 0.7
T ( 5 1 ) i
Maximise crossover frequency
Low frequency part of magnitude Bode
diagram is flat
53 j (49) _ j 10.59 -J2- - 2 . 3 5 8 8 - ^ + 0.8985 T T
V m J m
0.1 < - 2 - < 0.4; T
0.7719 -3.6608 + 6.5791 O -5.1652-^1- + 2.8059 T J T
0.4<-J2-<1.5 T
54 v (50) _ 1
K . K „ 0.407+ -
6.282
1 + 14.956 T
V in
55 K (51) 1 l + 3(TmAm)+6(Tm /xm)2 + 6 ( T m / Q :
K „ l + 3(Tm/Tm)+3(Tm/Tm)2
T ( 5 , ) =x 1 + 3 T, „A m )+6(T m /T m ) 2 +6(T m /x m ) 3
1 + 2 (Tm/xm)+2(Tm/xm)2]
50 Handbook of PI and PID Controller Tuning Rules
Rule
Somaraeffl/. (1992). Model: Method 1;
Suggested A m :
1.75 < A m <3 .0
Hang et al. (1993a), (1993b) -pages 61-
65.
Kc
* , / K m A m5 6
T,
x 2 * m
Comment
Coefficient values X m / T m
0.0375 0.1912
0.3693 0.5764 0.8183 1.105 1.449
x i
36.599 7.754
4.361 3.055 2.369 1.950 1.672
57 K (52)
o p T m
K m A m
x2
3.57 3.33 3.13 2.94 2.78 2.63 2.50
T m / T m
1.869 2.392
3.067 3.968 5.263 7.246 10.870
T (52)
58j(53)
x i
1.476 1.333 1.226 1.146 1.086 1.042 1.011
x2
2.38 2.27 2.17 2.08 2.00 1.92 1.85
o.\<^-<\ T
m Model: Method 31
or Method 33
• F o r i a L < 1 , K e . o ^ i Tm K m A m \(
• + 1 or
1.886 + 4—si- -1 .373 T
K, 0.9806
1 + 1 .886^1 ; f o r ^ > l , K * - ^ l + 8. VTr. K r a A m
669
57 K (52)
K m A m
0.80 + 1.33- T (52) __ 5T^
1.04 0.80 + 1.33-
1
2co p- + — P n T
; - ^ < 0.3 , Yang and Clarke (1996).
Chapter 3: Tuning Rules for PI Controllers 51
Rule
O'Dwyer (2001a). Model: Method 1;
i s -> 0.3, T
m
Yang and Clarke (1996).
O'Dwyer (2001a). Model: Method 1
Chen et al. (1999b). Model: Method 1
Gain margin and maximum closed loop magnitude - Chen et
al. (1999a). Symmetrical
optimum principle -Voda and Landau
(1995). Model: Method 1
Kc
K m T m
T,
Tm
Comment
Am = 7t/2x, ;
4m = V 2 - x , 5 9
Representative results x i
1.048
0.7854
Am
1.5
2
x l T m K u T u Tm VV+^X 2
x j m
^mKm
<L 30°
45°
Xi
0.524
0.393
Tra
Tm
Am
3
4
<L 60°
67.5°
Am = V 2 x i ;
<t>m = V 2 - X l
Coefficient values
Xl
0.50
0.61
0.67
0.70
Am
3.14
2.58
2.34
2.24
•I'm
61.4°
55.0°
51.6°
50.0° r l » r T m
Km
1
3.5C }P ( j ° w )
1
2.828 G p ( J ° W )
Ms
1.00
1.10
1.20
1.26
x i
0.72
0.76
0.80
Am
2.18
2.07
1.96
1
4 2 1
4.6
4
ffli 35°
<L 48.7°
46.5°
44.1°
Ms
1.30
1.40
1.50
Model: Method 1 60
^ - < 0.1 T
0 - « / n u T
'Given Am , ISE is minimised when <|>ra = 68.8884-34.3534Am + 9.1606-
for servo tuning (Ho et al. (1999)); 2 < Am < 5 ; 0.1 < Tm/Tm < 1 :
Given Am , ISE is minimised when <|>m = 45.9848Ama2677(Tm/Tm)a2755 for
regulator tuning (Ho et al. (1999)); 2 < Am < 5 ; 0.1 < Tm/Tm < 1
7t 2r ,An
4 x „ , A m ) 2 - l
52 Handbook of PI and PID Controller Tuning Rules
Rule
Voda and Landau (1995)-continued.
Friman and Waller (1997).
Model: Method I
Schaedel (1997), Henry and Schaedel
(2005) -Optimal servo
response.
Henry and Schaedel (2005).
Model: Method 22; Optimal regulator
response; —— < 0.2
Kc
61 K (54)
0.1751
Gp(j»135<.)
62 0.5T,<55»
K m ( T , - T < 5 5 > )
63 0.375T/56 '
K ^ T . - T , ^ )
0.375Ti(57)
K m ( T l - T < " > j
64 K (58)
T,
T ( 5 4 ) i
1
03135°
T ( 5 5 ) i
T ( 5 6 ) i
63 T (57) i
T ( 5 8 ) * i
Comment
0 . 1 5 < ^ - < 1
^ > 2 . A m > 3 m
"Normal" design -Butterworth filter. Model: Method 21
"Sharp" design -Tschebyscheff filter
(0.5dB). Model: Method 21
Minimum ITAE (Henry and Schaedel,
2005). Model: Method 22
"Normal" design -Butterworth filter
61 ^ (54) ___
4.6
1 , ( 5 4 )
Gp(ja>11,„)-0.6Kn
1.15JGp(jm135o)| + 0.75Kn,
(Bl35„[2.3|GI)(j(D135„)|-0.3Kra]'
6 2 T i( 5 5 ) = A / T , 2 - 2 T 2
2 ,
1 + 3.45
— + Tm,T22=1.25Tmia
m + I n L _ T m + 0 . 5 T m2
; 0 < ^ < 0.104;
-3.45-T
0.7Tm
Tm-0.7xam
2 0.7T, - T m > T 2 = T m T m +
Tm-0.7x^ -0.5tm
2 , i s . > 0.104.
= T i - ^ f - > Tj = -0 .64T 1 + 1 .64T , J l -1 .2 | ^ -
64 j , (58) 1 0.5 ,(58)
2 -\
- 1 T Z / ' T 2 ^
T, I T,2 j
Chapter 3: Tuning Rules for PI Controllers 53
Rule
Henry and Schaedel (2005) - continued.
Leva (2001). Model: Method 1
Minimum §m = 50°
Potocnik et al. (2001).
Model: Method 39
Kc
65 £ (59)
66 £ (60)
67 £ (61)
68 j ^ (62)
69 K (63)
70 £ (64)
T ;
T< 5 9 )
rp(60)
T
T ( 6 3 ) i
T ( 6 4 )
Comment
"Sharp" design -Tschebyscheff filter
(0.1 dB) Minimum ITAE
20 M c T m + 5 T m
20 c x +5T
i s -< 0.1 T
0.1 < i s . < 0.15 T
65 ^ (59)
66 £ (60)
0.7 •(59) 2 "\
, T: (59) T = 3 . 1 1 - ^
T, 1-1.43 V
K „
68 „ (62)
0.69
V
"0.69£
K m
"rp (59) "
. T 2
2 "J
- 1
J
20
j (60) _
T m
Tm K m ( T m + 5 T m ) _
' 0.698 lTm
K m T m I<
50T„, ]
' (r *• in \ m
+ 5Tm)J
3 . 8 6 ^ 1 - 1 . 4 6 ^
J
• i
69 T ^ (63) 0.5
|Gp(jW135»] 1.14 + 13.24
• (63)
G p ( jw , 3 5 o)
J135°
1 + 2.48 *M#). in AnpMA ^ + 17.40
70 K (64) K -2|Gp(ico135„]
2 K m [ K m - 2 | G p ( J a W ) | ] '
-(64) . [ K m + V2|G„ (jco135 .)[ ] [2|GP( j iQ135 . )| ~ K m ]
Q)135»|Gp(j£0135»)|[|GP ( jCD135»>|-Kn1 l
54 Handbook of PI and PID Controller Tuning Rules
Rule
Potocnik e< a/. (2001) - continued.
Cluett and Wang (1997), Wang and
Cluett (2000) -page 177.
Model: Method 1;
0.01<-^2-<10 T
m
Kc
71 K (65)
72 jr (66)
73 v (67)
74 jr (68)
75 jr (69)
76 K (70)
T,
T(65) i
T(66)
T(67) i
T(68) i
T(69)
T(70)
Comment
0 . 1 < ^ - < 1 Tm
TCL = 4xm .
Am e [7.50,8.28],
(t)me[78.50,79.70]
TCL = 2xm.
A m e [4.35,5.02],
<S)m e\l0.l\74.0°]
TCL = 1.33xm.
Am e [3.29,3.89],
(|)me[64.40,70.60]
TCL = Tm •
A m e [2.74,3.30],
(t)mg[57.80,68.50]
TCL = 0.8tm .
A m e [2.39,2.93],
* m e [49.6°,67.1°]
71 K (65) 2K„ 0.75 + 1.15
|GpOroi35°) K „
T: (65) 0.75 + 1.15^ = G°w)
K „
72 K (66, _0.019952Tm+0.20042Tm ^ m
73 K (67) _0.055548Tm+0.33639Tm T_(67)
74 K (68) _0.092654Tm+0.43620Tm j m
Kmxm ' '
75 K (69) _0.12786Tm+0.51235Tm T m =
Kmxm
76 Kc(70) _0.1605lTm+0.57109Tm > T
K m T m
0.099508Tm+0.99956Tm
0.99747xm -8.7425.10"5Tm
0.16440Tm +0.99558Tm
0.98607Tm-1.5032.10"4Tn,
0.20926Tm+0.98518Tm
0.96515Tm+4.2550.10"3Tm
0.24145Tm+0.96751Tm
0.93566xm +2.2988.10"2Tm
m T ( T O ) _ 0.26502Tm+0.9429 lTm
0.89868xm+6.9355.10"'Tm
Chapter 3: Tuning Rules for PI Controllers 55
Rule
Cluett and Wang (1997), Wang and
Cluett (2000) -continued.
Modulus optimum principle - Cox et al.
(1997). Model: Method 16
Smith (1998). Model: Method 1
Wang et al. (2000a). Model: Method 31 or
32
Kc
77 K (71)
78 K (72)
0.5Tm
0.35
Km
79 K (73)
T|
T ( 7 1 ) 1 i
T ( 7 2 ) i
Tm
0.42xm
Tm
Comment
TC L=0.67Tm .
AmG [2.11,2.68],
4>me[38.1°,66.3°]
t —2L<1 T
m I » L > 1
T
6 dB gain margin -dominant delay
process
1.3 <MS <2
77 K (71) 0.19067tm +0.61593Tm . ( 71 ) _ 0.28242Tm + 0.9123 lTm
0.85491-r +0.15937T„
0.5 T mJ + V T m + 0 . 5 T m T ^ + 0 . 1 6 7 T ,
Tm2Tm+TmTm
2+0.667Tm3
3 ^
• (72) _ / T m
3+ T m
2 T m + 0 .5T m x m2
+ 0 .167T m3 ^
Tm2+TmTm+0.5xm
2
79 K (73) 1.451-1.508
M.
56 Handbook of PI and PID Controller Tuning Rules
Rule
Wang and Shao (2000a).
Model: Method I Chen and Yang
(2000). Model: Method 8
Hagglund and Astrom (2002).
Model: Method 5; M.=1.4
Hagglund and Astrom (2002).
Model: Method I
Kc
80 K (74)
0 7T
v . / i m
KmTm
0.25Tm
^ m T m
0.1Tm 0.15 — +
Kmt r a Km 0.15
Km
81 K (75)
Ti
T . ( 74 )
Tm
0.8Tm
0.3xm+0.5Tm
0-3Tm
T ( 7 5 )
Comment
A.e [1.5,2.5]
Ms =1.26;
A m = 2 . 2 4 ;
* m = 5 0 °
0 . 2 < ^ L < 1 T
Tm
T m
Ms =1.4;-is->0.5
80 K (74) 1
*-fl (WQn« ) f 2 ( W Q n o ) - -
fl (ffl90» ) :
K „
fr + W } ' -[(Tm + { 1 + (O90l>2Tm2}'tm)Sm(-CO90»'Cm ~ t a n ' " V 7 ™ ) ]
(l + co90„2T, 2 ,"1 ,V.5 ko°Tm2 cosC-^so"1", " tan"1 co90„Tm)J,
f2(« 9 0 «) :
1 + ^ X 2 [(Tra +{l + co9o0
2Tm2}Tm)cot(-co90„xm -tan"1 co90„Tm)J
«9o»lTn1+(l + ©Qn»2Tm
2)xm]cos0 + (l + 2co9Oo2Tm2)sin(
m 9 0 o T n i i , 2 T 2 1 + CV T
m
, (74) _ w90° L m 90u m / m J
-co „3Tm2cos9 + co9no
2[Tm +(l + co9no2Tm
2)Tm]sin(
81 j r (75) ,
K „
90 m y m J
0.14 + 0 . 2 8 ^ - 1 , T;(75) = 0.33xm + 6 - 8 T ™ T "
- C 0 9 0 o T n , - t a n " l c 0 9 0 » T m -
10xm+Tm
Chapter 3: Tuning Rules for PI Controllers 57
Rule
Leva et al. (2003). Model: Method 1
Matausek and Kvascev (2003). Model: Method 1
Kristiansson (2003). Model: Method 11
Leva et al. (1994). Model: Method 43
Kc
82 K (76)
83 K (77)
x l T m
Kraxra
85 ^ (78)
86 K (79)
Ti
Tm
T
T 1 m
1.25 T
m T ( 7 9 )
Comment
minimum <j>m ;
maximum coc
minimum ()>m = 50
x ^ O . S r t - ^ 8 4
1.6 <MS <1.9
82 K ( 7 6 ) = minimum ~Tm(0.5n-$m) 5T„
83 K (77) _ mm lmum
0.698T„,
KmTCL
SaT,,,
Kmxra ' K m ( T m + 5 T j ,T C L = T m + 5 T m
; a e [ 4 , 1 0 ] .
84 'Acceptable' values of x, : 0 . 5 < x , < 3 . 'Recommended' values of x, :
0.32 <x , < 0.54 (59° < <|>m <72°). Choose x, = 0.32 when large variations in
process parameters are expected; choose x, =0.54 to give 6%<OS<13% and
1.4 <M < 2 .
85 K (78) _ ^150°
K „ 0.2 + -
0.075 K , , „ o
K „ + 0.05
86 j ^ (79) _ M c n x i l + «c„2Tm2
Km ii+ajnr]]2
tan , (79)
- ^ + T m < B c n + t a n " ' ( M c n T m )
2.82- • tan
with a>„
58 Handbook of PI and PID Controller Tuning Rules
Rule
Tan et al. (1996). Model: Method 30 Xu et al. (2004).
Model: Method 31
Huang et al. (2005). Model: Method 42
Rivera etal. (1986). Model: Method 1
Chien (1988). Model: Method 1
Brambilla et al. (1990).
Model: Method 1
Kc
87 jr (80)
T
^Kn^m
0.50Tm+0.22xra
KmTm
Ti
T ( 8 0 ) 1 i
Tm
0.9Tm+0.4xra
Comment
X e [1.5,2.5]
Robust
^Km T
m
A>1.7xm,
A>0.1Tm .
A, > 1.7xm , X > 0.2Tm (Luyben (2001)); X = 2xm (minimum ISE -
deOliveirae/a/., (1995)) 2Tm+xm
2XKm
Tm
Km(xm+A)
Tm+0.5xra
Km(A + xra)
Tm+0.5xm
Tm
Tm+0.5xm
A>1.7xm,
A>0.1Tm .
A = Tm (Chien
(1988))88
No model uncertainty: A, = xm , 0.1 < xm/Tm < 1 ;
A = [ l -0 .51og 1 0 (x m /T m ) ]x r a ) l<x m /T m <10.
K (80) .
Am )/l + (pTi(80)coJ ' ' K tanl-tan-1 pTmco<, -pxmco<, -<t>j'
T
co,,, < cou. (3 = 0.8,is- < 0.5 ; p = 0.5,^2- > 0.5 . ra ^m
X = 2xm (Bialkowski (1996)). X > Tm + xm (Thomasson (1997)). X e [0.45xm,0.8xm]
(Huang et al. (1998)). A = 2xm (aggressive, less robust tuning) (Gerry, (1999));
X = 2(Tm + xm) (more robust tuning) (Gerry, (1999)); A = a.max(Tm,xm): a > 3 ...
slow design, 2 < a < 3 ... normal design, 1 < a < 2 .. .fast design, a < 1 ... faster design (Andersson (2000) -page 11). Model: Method 10.
A 6 [0.1Tm,0.5T ra],^ < 0.25 ; A = 1.5(xm+Tm),0.25 < ^n_ < 0.75 ;
* = 3(xm + T m ) , ^ - > 0.75 (Leva (2001)). A e [Tm ,xJ (Smith (2002)).
Chapter 3: Tuning Rules for PI Controllers 59
Rule
Pulkkinen et al. (1993).
Ogawa(1995). Model: Method 1;
Coefficients of Kc ,Tj
deduced from graphs
Thomasson (1997). Model: Method 1
Chen etal. (1997). Model: Method 26
Lee etal. (1998). Model: Method 1
Isaksson and Graebe (1999).
Model: Method 1
Chun et al. (1999). Model: Method 34
Zhong and Li (2002). Model: Method 1
Smith (2002). Model: Method 1
T
0.5
1.0
0.5 1.0 0.5 1.0
0.5 1.0
2K
K ra
Tm
K
T
Tm(x
K r
K
Kc
0.7Tra
K m T m
x,/Km
M
Tf
^ m + T m )
X 2 Tm
Coefficient values
x i
0.9
0.6
0.7 0.47 0.47 0.36
0.4 0.33
x 2
1.3 1.6
1.3 1.7
1.3 1.8 1.3 1.8
*m
m ( t m + ^ )
0-5Tm
max(im ,0.5)
2
1 T m
2(A. + 0
+ 0.25xm
m + 2 X ) - ^ 2
„ (x m +X) 2
Tm
ra(a + 0
1 T
T
2.0
10.0
2.0 10.0 2.0 10.0
2.0 10.0
minim
T m
T
Tm(x
*1
0.45 0.4
0.4 0.35 0.32 0.3 0.3
0.29
x 2
2.0
7.0 2.2
7.5 2.4
8.5 2.4
9.0
0-5xm
um(T m ,6T m )
3
2
1 T m
2(* + 0 + 0.25im
m+2X)-X2
x m + T m
Tm
Tm
Comment
Model: Method 1
20% uncertainty in process parameters
33% uncertainty in process parameters
50% uncertainty in process parameters
60% uncertainty in process parameters
*m » Tm i
^ = TCL
x r a > 0 . 5
T m < 0 . 5
Desired closed loop
specified
Representative A.:
X = 0.4Tm
a e [At m ,1 .4At m ] ;
Axm = time delay
uncertainty.
60 Handbook of PI and PID Controller Tuning Rules
Rule
McMillan (1984). Model: Method 2 or
Method 44
Boe and Chang (1988).
Model: Method I
Geng and Geary (1993).
Model: Method 1
Feric etal. (1997). Model: Method 40
Matsubaefa/. (1998). Model: Method 1
Huang et al. (2005). Model: Method 42
Kc T, Comment
Ultimate cycle
89 K (81)
0.54KU
0.7 lTm
K m ^ m
Ku tan2 4>m
Vl + tan2c()m
0-99 JX^ 0.9
90 K (82)
T ( 8 1 ) 1 i
1.935Tm [ Xm
T
0.654
3.33xm
0.1592 Tu
tan<t>m
2 . 6 3 T J ^ \ lm J
0.097
T (82)
Tuning rules developed from
K u ' T u
Quarter decay ratio
i a - < 0.167 T
m
0.1 < ^ 2 - < 1.0 T
m
89 ^ (81) 1-881 Tm
K m Tm
L T ;( 8 1 , = 1 . 6 7 T J I + | T "
90 K (82) _ 0-55
Km
0.9, K„ - 1
TC - tan K „ - 1
+ 0.4
X ( 8 2 ) =0.159T„ 0.9, K„ - 1 + 0.4-U - tarT K„ - 1
Chapter 3: Tuning Rules for PI Controllers 61
3.1.2 Ideal controller in series with a first order lag
Gc(s) = K, 1 + -T j S ,
1 Tfs + 1
R(s) > E(s)
y i
K, 1
1 + Tfs
U(s)
• Kme-"»
l + sTm
V
^
Table 4: PI controller tuning rules - FOLPD model Gm (s) = K„,e~
l + sT„
Rule
Umetal. (1985). Model: Method I
Rivera and Jun (2000).
Model: Method 1
Kc T; Comment
Direct synthesis: time domain criteria
Tm
Km(24TCL2+xm) T T 2
j XCL2 1 2 ^ T C L 2 + - C
m
Robust
Tm
Km(2Tm+*.)
T T f =
TmX ; 2Tm+X
X not specified
Desired closed loop transfer function = TC L 2V+2^TC L 2s + l
62 Handbook of PI and PID Controller Tuning Rules
3.1.3 Ideal controller in series with a second order filter
Gc(s) = Kc
E(s)
R(s) ®+K + X
1 +
U(s)
J_ T i S y
2 A
l + b f l s + b f 2 s
1 + -T i S y
l + b f l s + b f 2 s 2 ^
v 1 + 3. j-jS H~ 3. j -2^ /
K e"
l + sTm
Y(s)
Table 5: PI controller tuning rules - FOLPD model Gm (s) = Kme~ l + sT„
Rule
Tsange/a/. (1993). Model: Method 15
Lee and Shi (2002). Model: Method I
Kc Ti Comment
Direct synthesis: time domain criteria x l T m
K r a x m T 2
Coefficient values x i
1.851 1.552 1.329 1.160
\ 0.0 0.1 0.2 0.3
x l
1.028 0.925 0.841 0.768
\ 0.4 0.5 0.6 0.7
x l
0.695 0.622 0.553
\ 0.8 0.9 1.0
Robust 3 K (83) T
Y
2T l < y < ^ 2 _
2 b f l = 0.5xm , b f 2 = 0.0833xm2, afI = 0.2xm, af2 = 0.01xm
2
3 K (83) «>bw T m
Km(cobwyTm+lJl + ^ ^ t a n
tm+2cobwTmTm+2Tm
;b f l =T m +0.5T m ;b f 2 =0.5T m T r a ;
2 KwY^ m +l ) ;a f2
Tmxro
2(cobwYTm +1) , suggested co =
0.6
Chapter 3: Tuning Rules for PI Controllers 63
3.1.4 Controller with set-point weighting
U(s) = K
<xK„
+ R(s) t ~ E(s) K,
v T i s ;
v T>sy E(s)-aKcR(s)
+® + U(s)
Kme~ 1 + sT
Y(s)
Table 6: PI controller tuning rules - FOLPD model G ra (s) = Kme-
l + sT„
Rule
Servo/regulator optimisation
- Taguchi and Araki (2000).
Model: Method 1
Chidambaram (2000a).
Model: Method 1
Kc T i Comment
IVlinimum performance index: other tuning
4 K (84) T(84)
Tm /Tm<1.0.
Overshoot (servo step) <20%
a = 0.6830 - 0.4242-21- + 0.06568 -^-
2
Direct synthesis: time domain criteria T-i 'T' (85)
T 2 +T ( 8 5 ) x 1 C L + M xm
5 j (85) Choose TCL,^
K (84)
K „ 0.1098 + -
0.7382
2™ T
• 0.002434
. ( 8 4 ) = T „ 0.06216 + 3.171—s-- 3.058
T T _ m _
2
+ 1.205 T
_ m _
3 \
J
5 j (85) = TmTm + 2 T C L ^ - TC L Tj(85) - £Jt
, ( 8 5 ) ^ o r a = "C"J!1 K . K J T ^ ' + T J - T / 8 5 '
-(85)
Handbook of PI and PID Controller Tuning Rules
Rule
Srinivas and Chidambaram (2001).
Astrom and Hagglund (1995) - dominant pole design-page
204-208. Model: Method 5 or
14
Astrom and Hagglund (1995)-modified Ziegler-
Nichols - page 208.
Hagglund and Astrom (2002).
Model: Method 5; Ms =1.4;
0 < a < 0 . 5
Hagglund and Astrom (2002).
Model: Method 1
Kc
6 0.9Tra
K m T m
Ti
3.33xm
Comment
Model: Method 1
Direct synthesis: frequency domain criteria
0.29e-2'7t+3jT!Tm
K m x m
g _ 9 T i n e -6 .6 , + 3.0T» Q r
0.79Tme-L4l+2-4t2
0.14<^=-<5.5; Tra
M.=1.4
a = l-0.81e°'73,+L9T2
0.78e-4lT+5'7t2Tm
K m t m
8.9xme^+3-0T2 or
0.79Tme-L4x+2-4T2
0 . 1 4 < ^ L < 5 . 5 ; Tm
M s = 2 . 0
a = l-0.44ea78t-°-45t2
0-4Tm
K m T m 0.7Tm
0.1 < — <2 T
m a =0.5; Model: Method 5 or 14
0.35Tm 0.6
0.35Tm 0.6 K m T m K m
0.25Tm
7 R (86)
7xm
0.8Tm
0.8Tm
T ( 8 6 )
-^=-<0.11 T
m
0.11<^=-< 0.17 T
0 . 1 7 < ^ - < 0 . 2 T
^2-<0.5 T
m Ms =1 .4 ;0<a<0 .5
Sample K c , Tj taken by the authors (corresponding to the process reaction curve
tuning rules of Zeigler and Nichols (1942)); oc = - — - — = 0.30-1.11-^2 K c K m T m
for
these tuning rules.
7 K (86) — 0.14 + 0 .28^ -1 , T-Km 1 T
(86) : 0 .33 t „ 10xm+Tm
Chapter 3: Tuning Rules for PI Controllers 65
3.1.5 Controller with proportional term acting on the output 1
K „ U(s) = - ^ E ( s ) - K c Y ( s )
T:S
R(s)
+ ?) • F- E(s)
Kc
T;S
U(s) +
P V s J - *
Kc
t
Kme-™
l + sTm
Y(s)
^-
Table 7: PI controller tuning rules - FOLPD model G ra (s) = K e-
l + sT„
Rule
Minimum ITAE -Setiawan et al.
(2000). Model: Method 1
Chien etal. (1999). Model: Method 1
l a T
0.5 0.75 1.0 1.5 2.0
Kc Ti Comment
Minimum performance index: other tuning x l T m
Km tm
x l
0.554 0.570 0.594 0.645 0.720
x2
1.805 1.373 1.117 0.822 0.679
Direct si
8 K (87)
^ m
Coefficient values
Ira. T
3 4 5 6
X l
0.900 1.090 1.290 1.520
x2
0.547 0.480 0.445 0.428
l a T
m 7 8 9 10
x i
1.740 1.930 2.120 2.410
x2
0.413 0.398 0.385 0.389
,'nthesis: time domain criteria
T (87) Underdamped system response - ^ = 0.707 .
*•» > 0.2Tm
K (87) -TCL/+1.414TCL2Tm + tmTm r(87)
K m T f , , /+1 .414T r , ,T m +T„
-1.414TCL2Tm+Tmxn
T „ + T „
66 Handbook of PI and PID Controller Tuning Rules
3.1.6 Controller with proportional term acting on the output 2
U(s) = Kc
V T , s 7 E(s)-K,Y(s)
R(s) E(s)
>6* » + 1 i
( i "1 K c 1 +
U(s) +
T
K m e-""
l + sTm
Y(s)
' P"
Table 8: PI controller tuning rules - FOLPD model Gm (s) = Kme"
1 + sT,.
Rule
Lee and Edgar (2002).
Model: Method I
Kc T, Comment
Robust
9 K (88) T (88) Kj =0.25KU
K, K m (^ + Tm)
Tm-0.25KuKmTm ^ T
2(X + t m )
. ( 8 8 ) m -0 .25K u K m x m
l + 0.25KuKm 2{X + zJ'
Chapter 3: Tuning Rules for PI Controllers 67
3.2 FOLPD Model with a Positive Zero Gm(s) K m ( l - s T m 3 ) e -
1 + sT, ml
3.2.1 Ideal controller Gc(s) = Kc
v T i s y
R(s) E(s)
< 2 > — Kc v T i sy
U(s) . Km ( l -sTm 3>-"-
l + sT„
Y(s)
Table 9: PI controller tuning rules - FOLPD model with a positive zero
K m ( l - s T m 3 > - ^ Gm(s) =
l + sT„
Rule
Sree and Chidambaram
(2003a). Model: Method 1
Kc Ti Comment
Direct synthesis: time domain criteria i YTml
K m T m 3 T (89)
K „ T m 1 y = value of the inverse jump of the closed loop system; y<— m m3
T m i
• (89) _ T m 3
^ [ T m 3 ( l - x l ) - 0 . 5 x m ( l + x,)]
yTm
Lm3
- ( l - X , ) - X ,
x, e 0 . 9 5 — — ^.98—ITJHI— Y T
m l + T m 3 Y T m l + T m
YTml 0 3_ YTm, 0 . 1 -
yTmi+Tm3 yTmi+Tm
0.1 < - = ^ < 0.3;
- > 1
68 Handbook of PI and PID Controller Tuning Rules
3.2.2 Ideal controller in series with a first order lag
Gc(s) = Kc 1 + — 1
R(s) E(s)
•®->K
U(s)
v Ti sv 1 + Tfs ^ K ro(l-sTm3)e-""
TjsJl + TfS
Y(s)
l + sT„
Table 10: PI controller tuning rules - FOLPD model with a positive zero
K m ( l - s T m 3 ) e - ^ Gm(s) =
1+sT^
Rule
Sree and Chidambaram
(2003a). Model: Method I
Kc T; Comment
Direct synthesis: time domain criteria
Tmi Km(2Tm3+X + Tm) Tmi
Tf = T m 3 ^ - 0
2Tm 3+^ + tm
Chapter 3: Tuning Rules for PI Controllers 69
3.3 FOLPD Model with a Negative Zero Gm(s) _ K m ( l + sTm3)e-
1 + sT, ml
3.3.1 Ideal controller in series with a first order lag
Gc(s) = Kc
f 1 A
1 + — v T iSJl + Tfs
1
^ 6 ? S K v T i s ;
l
1 + T fs
U(s) K m ( l + sTra3)e-
l + sT„
Y(s)
Table 11: PI controller tuning rules - FOLPD model with a negative zero
Km(l + s T m 3 > - ^ Gm(s) =
l + sTm
Rule
Change/al. (1997). Model: Method 2
Kc
Tm, Km(TCL + Tra)
Ti
Robust Tml
Comment
Tf = Tm3
70 Handbook of PI and PID Controller Tuning Rules
3.4 Non-Model Specific
3.4.1 Ideal controller Gc (s) = Kc
v T i sy
R(s) l> m> y *
L
Kc f i+-Ll I TiSJ
U(s) Non-model
specific
Y(s)
Table 12: PI controller tuning rules - non-model specific
Rule
Ziegler and Nichols (1942).
Hwang and Chang (1987).
Parr (1989) -page 191.
Parr (1989)-page 192.
Hanged a/. (1993b)-page 59.
Pessen(1994).
McMillan (1994)-page 90.
Astrom and Hagglund (1995) -pages 141-
142.
Kc Ti Comment
Ultimate cycle 0.45KU
0.45KU
0.83TU
1 ("5.22 5.22'
P. 1 Tu T i , T
Quarter decay ratio
p, ,T, = decay rate, period measured under proportional control
when Kc = 0.5KU
0.5KU
0.33KU
0.25KU
0.25KU
0.3571KU
0.4698K„
0.1988KU
0.2015KU
0.43TU
2T
0.25TU
0.167TU
T
0.4373TU
0.0882TU
0.1537TU
dominant time delay process
dominant time delay process
A m = 2 , ( j ) m = 2 0 0
Am = 2.44,
Am = 3.45,
4>m = 46°
Chapter 3: Tuning Rules for PI Controllers 71
Rule
Calcev and Gorez (1995).
Edgar e/a/. (1997)-page 8-15.
ABB (2001).
Robbins (2002).
Wang et al. (1995b).
VranCicefa/. (1996), Vrancic(1996)-
page 121.
Vrancic(1996)-page 140.
Friman and Waller (1997).
Kc
0.3536KU
0.59KU
0.5KU
0.3KU
0.3KU
Ti
0.1592TU
0.81TU
0.8TU
1 T (90)
1 T(91)
Direct synthesis 2 ^ ( 9 2 )
0.5A3
(A ,A 2 -K r a A 3 )
0.45KU
0.4830
Gp(j f f l150»)
T(92)
A3
A2
A3
A2
3.7321
Comment
4>m= 45°, smallxra
(j)ra= 15°, largexm
Minimum IAE -regulator
Minimum IAE -servo
Good response -regulator
Am>2,<t>m>60°
(Vrancic et al. (1996))
Modified Ziegler-Nichols method
A m > 2 ,
K > 15°
1 T/90 ' = (0.24 + 0.11 lKuKm )TU ; T;(91) = (0.27 + 0.054KuKm )ru . 2 For a stable process, Km is assumed known; for an integrating process, Km and i
are assumed known.
Gp(jfflCL){l-GCL(jcoCL)}J'
GP(JCOCL){1-GCL(JCOCL)}_
J W C L G C L G«>CL) I Gp(jffi>cL){l-GcL(jfflCL)}_
K c ( 9 2 ) = ^ _ I m
. (92)
Im
Rl
72 Handbook of PI and PID Controller Tuning Rules
Rule
Kristiansson and Lennartson (2000).
Kristiansson and Lennartson (2002).
Kristiansson (2003).
Aikman(1950).
Rutherford (1950).
MacLellan (1950).
Young (1955).
Kc
1.18KuKm-1.72
K u K m
0.50KuKm-0.36
K U Km
2 0 K 1 3 5 „ K m - 1 6 0
K,35oKm
5.4K 1 3 5 „K m -13 .6
K135 o Km
0.33KuKm -0.15
Km(0.18KuKm+l)
5KC<97»
6 K (98)
Ti
3 T (93) i
3 T (94)
4T(95) i
4 T (96)
KuKm
cou(0.18KuKm+l)
T(97)
T(98)
Comment
0.1<K uK r a<0.5
K u K m >0.5
K u K m < 0 . 1 ;
K135oKm<0.1
K u K m < 0 . 1 ;
K135oKm>0.1
K u K m <10
K u K m >10
1.6 <MS <1.9
Other tuning rules
0-81K37%
[0.63K33„/o,0.97K33„/o]
0.90K37%
1.19K37%
1.02K37%
^ 3 7 %
1.29T37% J
[1.1T33%>1.3T33%
T37%
0.92T37%
0.92T37%
<1.1TU
37% decay ratio
33% decay ratio
37% decay ratio
37% decay ratio
37% decay ratio
37% decay ratio
3 j (93) _ 1.18KuKm -1.72 (94)
• (0.33KuKm-0.17)cou ' ' 0.50KuKm-0.36
(0.33KuKm-0.17)o)u
4 T ( 9 5 )
5 v (97)
2 0 K 1 3 5 o K m - 1 6 0 - ( 9 6 ) 5 .4K 1 3 5 „K m -13 .6
K
(0.315K1 3 5„K r a-0.175) f f l u (1 .32K 1 3 5 .K m -3 .2 )m u
0 . 0 9 K , „ „ K m + 0 . 2 5 135° 0 .09K,„ 0 K m +1.2
, ( 9 7 ) K135oKm
,5ol0.09K1 3 5„Km +1.2)
6 K (98)
K „ 0.2+ -
0.075
150- +0.05 K „
>T i( 9 8 )=<D 150°
0.06 + 1.6-K,
Kn
-0.06 ^ 1 5 0 °
V K m J
Chapter 3: Tuning Rules for PI Controllers 73
Rule
Chesmond(1982) -page 400.
Parr (1989) -page 191.
McMillan ( 1 9 9 4 ) -pages 42-43.
Wade (1994) -page 114.
ECOSSE team (1996b).
Hay (1998) -page 189.
Bateson (2002) -page 616.
Astrom (1982).
Leva (1993).
Astrom and Hagglund (1995) -page 248.
Hagglund and Astrom (1991).
Cox et al. (1994).
Jones ef a/. (1997). Model: Method 2
Kc
K x %
0.5 + 0.3611nx
0.999K25%
0.667K25%
0.42K25%
0.33K25%
0.7515K25%
0.65K50%
K 2 5 %
K2 5%
sin(|)m
G p ( j a v )
7 K C< 9 9 )
0.5
Gp(jra135o)
0.25
Gp(jco135„)
0 . 2 5 K U
0.2hT u s i n ( t> m
A P U
0.5787 A P
Ti
Tx%
1.2Vl+0.0251n2x
0.814T25„ /o
T25%
^25%
^25%
0.7470T25„ /o
0 .7917T 5 0 %
^25%
T
tan(|)m
co90»
T (99) 1 i
4
1.6
co 1 3 5 „
0.2546 Tu
0.16Tu tan*m
aTu
Comment
x = 100(decay ratio)
Example: x = 25 i.e. quarter decay ratio
Quarter decay ratio
'Fast ' tuning
'Slow' tuning
'Modified' ultimate cycle method
tan((j)m - (j)ra - 0.5TI)
> 0 Alfa-Laval
Automation ECA400 controller
Alfa-Laval Automation ECA400
controller - large delay
Model: Method 2
a e [0.7958,1.5915]
tan(<L-<!>. • , -0-5rc) T ( •») = tan((|)m - <|)m - 0.5TC)
| G p ( j a ) ) ^ t a n 2 ( ^ - < ! . , „ - 0 . 5 T I ) ' ' m
74 Handbook ofPIandPID Controller Tuning Rules
Rule
NI Labview (2001). Model: Method 2
Tang et al. (2002). Model: Method 4
Kc
0.4KU
0.18KU
0.13KU
2.5465hsin<f>m
Apx, co90o Am
^
0.8TU
0.8TU
0.8TU
tan<|)m
C 0 9 0 °
Comment
Quarter decay ratio
'Some' overshoot
'Little' overshoot
0.81 <x, < 1 ;
x, = 1, 'small' xm;
x, =0.91, 'large'Tm
Chapter 3: Tuning Rules for PI Controllers
3.4.2 Controller with set-point weighting
75
U(s) = Kc
aK„
v Tisv E(s)-aK cR(s)
+ <?>
R(s) - E(s)
*K, v T i s / + U(s)
Non-model specific
Y(s)
Table 13: PI controller tuning rules - non-model specific
Rule
Astrom and Hagglund (1995) -page 215.
Vrancic (1996)-page 136-137.
Kc Ti Comment
Direct synthesis
0.053Kue2-9K-2-6K2 0.90T„e-4'4K+2'7K2 Ms =1.4
a = l - l . l e - 0 0 0 6 l K + L 8 K 2 ; 0<K m K u <oo .
0.13KueL9K-L3Kl O^OT^-4 '4"2-7^ Ms = 2.0
a = 1 - o.48e0'40^017K2; 0 < KmKu < oo .
8 K (100) rp (100)
a = [0.2,0.5] - good servo & regulator action (Vrancic
(1996), page 139)
8 K (100)
c " ( l - b 2 K 2 A 3 +
A , A 2 - K m A 3 - x
A, - 2 K m A , A 2 J
. K m A 3 - A , A 2 < 0 ;
K (100) A , A 2 - K r a A 3 + x
( l - b 2 J ( K m2 A 1 + A , 3 - 2 K m A 1 A , ^m " 3 T ^ l y
K m A 3 - A , A 2 > 0 ; b = l - a ;
X = V ( K m A 3 - A 1 A 2 ) 2 - ( l - b 2 ) A 3 ( K m2 A 3 + A 1
3 - 2 K m A 1 A 2 ) ;
A, . (100)
K m + -K (I00)K 2
2K (100) (,-„>)
76
3.5 IPD Model Gm(s)
Handbook of PI and PID Controller Tuning Rules
3.5.1 Ideal controller Gc(s) = Kc T,Sy
R(s)
+
E(s) K 1 + -
Xs
U(s) Kme" Y(s)
Table 14: PI controller tuning rules - IPD model Gm (s) = Kme"
Rule
Ziegler and Nichols (1942).
Model: Method 2
Two constraints method - Wolfe
(1951). Model: Method 2
Coon (1956), (1964). Model: Method I
Astrom and Hagglund (1995) -page 13. Model: Method 1
Hay (1998)-page 188.
Hay (1998)-page 199.
Model: Method 1 Kc and T>
deduced from graphs
Kc T,- Comment
Process reaction 0.9
Kmxm
0.6
K m t m
0.87
K m t m
3.33xm
2.78xm
4.35xm
Quarter decay ratio
Decay ratio = 0.4
Decay ratio is as small as possible
Minimum error integral (regulator mode). 1.0
0.63 K m T m
0.42
K m X m
7.5
3.5
2.5
2.2
0
3-2xm
5.8xm
4.5Kmxm
Quarter decay ratio; Kc,Tj deduced from
graphs Ultimate cycle Ziegler-Nichols
equivalent
Model: Method 3
K m t m = 0 . 1
K m x m =0.2
KmTm=0.3
K m x m =0.4
Chapter 3: Tuning Rules for PI Controllers 77
Rule
Hay ( 1 9 9 8 ) -continued.
Bunzemeier (1998). Model: Method 4
c Kp
" s(l + sT p ) n
Minimum IAE -Shinskey ( 1 9 8 8 ) -
page 123.
Minimum IAE -Shinskey ( 1 9 9 4 ) -
page 74.
Minimum IAE -Shinskey ( 1 9 8 8 ) -
page 148. Model: Method 1 Minimum ISE -
Hazebroek and Van derWaerden(1950).
Minimum ISE -Haalman(1965). Model: Method 1
Minimum ITAE -Poulin and Pomerleau
(1996). Model: Method 1
Skogestad (2001). Model: Method 1
1
1
* i
0.18
0.24 0.22
0.21
0.20 0.20
(
1
(
1
C
(
(
(
Kc
2.0
1.7
* i
^ m
x 3
( J n ,
x 2
1.350
3.050 1.910
1.690 1.584 1.557
Minimu
).9524
).9259
.61KU
1.5
).6667
X5264
).5327
K m T r a
Minir
0.28
K m T m
Ti
X 2 T m
Comment
K m t m = 0 . 5
K m x m = 0 . 6
0% overshoot (servo)
10% overshoot (servo)
Coefficient values
x 3
0.23 0.40 0.30
0.28 0.26
0.26
n
0 1 2
3 4
5
x i
0.20
0.19 0.19
0.18 0.18
x 2
1.477
1.463 1.453
1.385 1.378
X 3
0.25 0.24 0.24
0.24 0.24
n
6
7 8
9 10
m performance index: regulator tuning
num
4T
4T
Tu
5.56xm
0
4.5804xm
3.8853xm
Model: Method 1
Model: Method 1
Also specified by Shinskey (1996),
page 121.
Model: Method 2
M , = 1 . 9 ,
A m = 2 . 3 6 ,
* m = 5 0 °
Process output step load disturbance
Process input step load disturbance
performance index: other tuning
7 t m Mm a x =1.4
78 Handbook of PI and PID Controller Tuning Rules
Rule
Skogestad (2003). Model: Method 1
Skogestad (2001). Model: Method 1
Tyreus and Luyben (1992). Model:
Method 1 or 9
Fruehauf et al. (1993).
Rotach(1995). Model: Method 5
Wang and Cluett (1997).
Model: Method 1
Cluett and Wang (1997).
Model: Method 1
Poulin and Pomerleau (1999).
Model: Method 1
Kc
0.404
K m T m
0.49
T,
7xm
3.77xm
Comment
Mm a x=1.7
Mmax = 2.0
Direct synthesis: time domain criteria 0.487
Kraxm
0.31KU
0.5
0.75
Kmxm
i K (ioi)
2 K (102)
0.9588/Kmxm
0.6232/Kmxm
0.4668/Kmxm
0.3752/Kmxm
0.3144/Kmxm
0.2709/Kmtm
2.13 0.34KU or
KmTu
8.75xm
2.2TU
5xm
2-4lTra
T ( i o i ) i
•j- (102)
3.0425xm
5.2586xm
7.2291xm
9.1925xm
11.1637xm
13.1416xm
1.04TU
Maximum closed loop log modulus
= 2dB; TCL =
Model: Method 2
Damping factor for oscillations to a
disturbance input = 0.75.
4 = 0.707;
T c L ^ m ^ x J
4 = 1; TCL 6k.>16xJ
TcL = Tm
TCL = 2xm
TCL = 3xm
TCL = 4tm
TCL = 5xra
TCL = 6xm
Mmax = 5 dB
i K (ioi)
Kmim(0.7138TCL+0.3904)
1
Kmxm(0.5080TCL+0.6208)
, T,m) = (l.4020TCL +1.2076)xm .
Tj002 ' = (l.9885TCL +1.2235)xm .
Chapter 3: Tuning Rules for PI Controllers 79
Rule
Viteckova(1999), Viteckova et al.
(2000a). Model: Method 1
Chidambaram and Sree (2003).
Model: Method I
Huba and Zakova (2003).
Model: Method I
Skogestad (2003), (2004b).
Model: Method 1
Hougen(1979)-page 333.
Model: Method 1 Chidambaram (1994),
Srividya and Chidambaram (1997).
Gain and phase margin - Kookos et
al. (1999). Model: Method 1
Kc
X l / K m T m
x l
0.368 0.514 0.581 0.641
OS
0% 5% 10% 15%
1.1111
Km tm
0.23 K m T m
0.281
K m T m
1
K m ( T C L + 0
0.5
m m
T; 0
Coefficient values x i
0.696 0.748 0.801 0.853
OS
20% 25% 30% 35%
4-5xm
2.914xm
3.555xm
4?2(TCL+0
8tm
Comment
x i
0.906 0.957 1.008
OS
40% 45% 50%
Suggested % = 0.7 or 1
'good' robustness -
T C L = x m , 5 = l
Direct synthesis: frequency domain criteria 3 K (103)
0.67075 K r a T m
m p
AmKm
0-942/Km xm
0.698/Kra xm
0.491/Km rm
0.384/Kmxm
0
3.6547tm
1
C0p (0.571-COpTm)
Representative results 4.510xm
4.098tm
6.942xm
18.710xm
Maximise crossover frequency
Model: Method 6; A m = 2
Am =1.5;
<L = 22.5°
A m = 2 ; 4 , m = 3 0 °
Am = 3; *„,= 45°
Am = 4 ; ^ m = 6 0 0
3 Values recorded deduced from a
^ m ^ m
K (103)
0.1
7.0
0.2
3.5
0.3
2.2
graph: 0.4
1.8
0.5
1.4
0.6
1.1
0.7
1.0
0.8
0.8
0.9
0.75
1.0
0.7
80 Handbook of PI and PID Controller Tuning Rules
Rule
Chen et al. (1999a). Model: Method 10
Cheng and Yu (2000).
Model: Method 1
O'Dwyer (2001a). Model: Method 1
Kc
•L+fta-i) K m T r a (A m
2 - l )
5 K (105)
0.5236/Km xm
1 * 1
< m T m
T;
4rp(104)
j ( 1 0 5 )
8^m
* 2 * m
Comment
f 1.0607~] Am
A m =2 .83 ;
* m = 46.1°
[6]
Representative coefficient values
* 1
0.558
0.484
0.458
x2
1.4
1.55
3.35
Am
1.5
2.0
3.0
4>m
46.2°
45.5°
59.9°
x i
0.357
0.305
x2
4.3
12.15
Am
4.0
5.0
4>m
60.0°
75.0°
4 T (104) __n_ A mz - 1
• 0 . 5 i r ( A m - l ) 0.57t-
+ 0 . 5 7 t ( A m - l )
5 r , (105) T l 7 1 ! 2 ^ , , , - l ) - ( 1 0 5 ) _ _
4 K m x m ( r 12 A m
2 - l ) ' ; C r 12 A m ( r 1 A m - 2 X 2 r 1 A m - l )
A m - 1
^ ( r . ' A ^ - l ) 2
A „
x 2
4 x ,
Tt i" . ,
IT--\
7t
x2
2
1 + -71 7t 7t
"2 +^T_^7
<t>m = t a n ' 1 ^0 .5x ! 2 x 22 + 0 . 5 x 1 x 2 ^ / x , 2 x 2
2 + 4 0.5x,2 + 0 . 5 ^ J - A / x 12 x 2
2 + 4 M
Chapter 3: Tuning Rules for PI Controllers 81
Rule
O'Dwyer (2001a). Model: Method 1
Chien (1988). Model: Method I
Ogawa(1995).
Model: Method 1;
Coefficients of Kc
and Tj deduced from
graphs
Alvarez-Ramirez et al. (1998).
Model: Method 1 Zhong and Li (2002).
Model: Method 2
K c
aK„
T,
bTu
Comment
[']
Robust
1 ( 2X + zm ^
X l / K m Tra
x l
0.45
0.39
0.34
0.30
0.27
x 2
11
12
13
14
15
K m lT C L TmJ
2 a + i m
K m ( a + t m ) 2
2X + rm
X 2 t m
8
20% uncertainty in process parameters
30% uncertainty in process parameters
40%) uncertainty in process parameters
50% uncertainty in process parameters
60% uncertainty in process parameters
T m + ^ m
2 a + xm
TCL > t m
a not specified
7 Am =
2b Tea
71
4b
1 + -71
—+ 2
7t
4b
<h tan 27:2a2b2+27tabV47t2a2b2+4 - . . 0 .125TCV +—V47i 2 a 2 b 2 +4 11 16b
X = [l/Km ,xm] (Chien and Fruehauf( 1990)); X = 3xm (Bialkowski (1996));
I > tm +Tm (Thomasson (1997)); I = [l.5xra,4.5Tm] (Zhang era/. (1999)). From a
graph, X = 1.5tm ....Overshoot = 58%, Settling time = 6xm
,....Overshoot = 35%, Settling time = 1 lxm
, ....Overshoot = 26%, Settling time = 16xm
.Overshoot = 22%, Settling time = 20xm .
X = emax /100Km ; emax = maximum output error after a step load disturbance
(Andersson (2000) -page 12);
X = 2.5Tm
X = 3.5Trn
X = 4.5xm
„„V/100K„
Xse
Kc(^s + l) • (Chen and Seborg (2002)); X = xra (Smith (2002)).
82 Handbook of PI and PID Controller Tuning Rules
Rule
Smith (2002). Model: Method 1
Skogestad (2004a). Model; Method 1
Penner(1988). Model: Method 1
Kc
1
Km^m
9 K (106)
Ti
%
4
Kc(1°6)Kra
Comment
Other methods 0.58
KmTm
0.8
KmTm
10tm
5.9xra
Maximum closed loop gain = 1.26
Maximum closed loop gain = 2.0
9 K (106) > l A d 0 l
I A Y max I
|Ad0| = maximum magnitude of a sinusoidal disturbance over all frequencies,
I Ay maX | = maximum controlled variable change, corresponding to the sinusoidal
disturbance.
Chapter 3: Tuning Rules for PI Controllers
3.5.2 Ideal controller in series with a first order lag
Gc(s) = K(
U(s)
83
1+-T i S y
1
R(s) E(s) K, 1 + —
v T i s /
1
1 + Tfs
Kme"
1 + Tfs
Y(s)
Table 15: PI controller tuning rules - IPD model Gm (s) = Kme"
Rule
H„ optimal -
Tan et al. (1998b). Model: Method 1
Rivera and Jun (2000).
Model: Method 1
Kc
0.463X. +0.277
Kmxm
1 K (107)
Ti
Robust
t m
0.238X +0.123
2(x1B+X)
Comment
A. = 0.5;
' 5.750A. + 0.590
X not specified
1 £ (107) 2{Tm+X) , T f = -
x ^
Km(2xm2+4zmX + X2)' f 2xm
2+4TmX + X2
84 Handbook of PI and PID Controller Tuning Rules
3.5.3 Controller with set-point weighting
U(s) = Kc
KaK V T > s /
E(s)-aKcR(s)
R(s)
K8> - E(s)
K, v T . sy
xg> U(s)
Kme" Y(s)
Table 16: PI controller tuning rules - IPD model Gm (s) = -Kme"
Rule
Taguchi and Araki (2000).
Model: Method I
Minimum ITAE -Pecharroman (2000). Model: Method 10
Chidambaram (2000b).
Model: Method 1
Hagglund and Astrom (2002).
Model: Method 2
Kc T i Comment
Minimum performance index: servo/regulator tuning
0.7662
KmTm
x,Ku
4.091xm
X 2 TU
a = 0.6810, Tm /Tm<1.0.
Overshoot (servo step) < 20%
K m = l
Coefficient values
* i
0.049
0.066
0.099
0.129
0.159
0.189
x2
2.826
2.402
1.962
1.716
1.506
1.392
a
0.506
0.512
0.522
0.532
0.544
0.555
0-64/Kra xm
0.487/Km xm
In general, a ~ 0.
0.35 K T
4>c
-164°
-160°
-155°
-150°
-145°
-140°
*1
0.218
0.250
0.286
0.330
0.351
x2
1.279
1.216
1.127
1.114
1.093
a
0.564
0.573
0.578
0.579
0.577
4>c
-135°
-130°
-125°
-120°
-118°
Direct synthesis 3.333Tta
8.75xm
a = 0.65
a = 0.557
6 (on average) or a = 0.'
7?m
T i
Ms = 1.4;
0 .3<a<0.5
Chapter 3: Tuning Rules for PI Controllers 85
3.5.4 Controller with proportional term acting on the output 1 K „
U(s) = -£-E(s)-K cY(s)
R(s) • 6 ^ *\ F- E(s)
Kc
T,s ~ L 6 ? V — * -+ U(s)
Kc
4
Kme-"» s
Y(s)
•
Table 17: PI controller tuning rules - IPD model Gm (s): Kme"
Rule
Minimum ISE -Arvanitis et al.
(2003a).
Arvanitis et al. (2003a).
Model: Method 1
Minimum ISE -Arvanitis et al.
(2003a).
Arvanitis et al. (2003a).
Model: Method 1
Kc T; Comment
Minimum performance index: regulator tuning 0.6330
Kra^m
2 0.6114
KmTm
3 0.6146
Kmxm
Minimum | 0.6270 K m x m
2 0.6064
K m t m
3 0.6130 K m T r a
2.3800im
2.7442xm
2.6816xra
Model: Method I
jerformance index: servo tuning
2.4688xm
2.8489tm
2.7126tm
Model: Method I
1 Minimum f[e2(t) + K m V(t ) ]d t .
Minimum 1 rdu^2
e 2 ( t ) + K - 2 U t dt.
86 Handbook of PI and PID Controller Tuning Rules
Rule
Chiene/a/. (1999). Model: Method 2
Arvanitis et al. (2003a).
Model: Method 1
Arvanitis et al. (2003a).
Model: Method 1
Kc
4 j r (108)
4
(8-P)Kmxm
5 K (109)
Ti
Direct synthesis 1.414TCL2+tm
4t m
P
Robust
0.(712 - a j
Comment
Underdamped system response - £, = 0.707 .
P%Al
4 v (108)
^r'=—r 1.414TCU+Tm
J . , , 2 +1 .414T r , I ,T m +x„ 2 ) ' CL2
5 K (109) 4K1
[(8-a>i2 +a2]Kr
CL2 '•m T l m
with 0 < a < %2 :
recommended a = — 2
1 - 1 ' T I 2 ( 4 5 2 + I )
, £> 0.3941
Chapter 3: Tuning Rules for PI Controllers 87
3.5.5 Controller with proportional term acting on the output 2
R(s) E(s)
+ * K, 1 + -
T;s
U(s) = Kc 1 +
U(s)
Ti^y E(s)-K,Y(s)
"KgH-K„e- S T ™
Y(s)
K,
I Table 18: PI controller tuning rules - IPD model Gm (s) =
K „ , e "
Rule
Lee and Edgar (2002).
Model: Method 1
Kc Ti Comment
Robust
6 K (HO) T(110) i
K, =0.25KU
6 K (HO) 0.25K,,
fc + O K u K m 2(X + T m ) T(110) _ _ i _ _ T , Tm
KuKm m 2(\ + T J "
88 Handbook of PI and PID Controller Tuning Rules
3.5.6 Controller with a double integral term
Gc(s) = K
E(s)
' l l ^ 1 + — + — -
T;,s X9s2
R(s) < g ^ K, \ 1 1 ^ 1 + +
V Tiis T i 2s2y
U(s) K„e" Y(s)
Table 19: PI controller tuning rules - IPD model Gm (s) = K „ e -
Rule
Belanger and Luyben (1997).
Model: Method 2
Kc Ti Comment
Ultimate cycle
0.33KU
T„ = 2.26TU ;
Ti2 = 20.5TU2
Maximum closed loop log modulus =
+2dB.
Chapter 3: Tuning Rules for PI Controllers 89
3.6 FOLIPD Model Gm(s) - K e "
>(l + sTm)
3.6.1 Ideal controller Gc (s) = K(
v T i sy
R(s) ?) E ( s ) , y *
k
Kc
I TisJ
U(s) K m e " ^
s(l + sTm)
Y(s)
Table 20: PI controller tuning rules - FOLIPD model Gm (s) = K„e"
s(l + sTm)
Rule
Coon (1956), (1964). Model: Method 1; Coefficients of Kc
deduced from graphs
Minimum IAE -Shinskey(1994)-
page 75. Minimum IAE -
Shinskey(1994)-page 158
Kc
x i
K m (^ m +T m )
^m/Tm
0.020 0.053 0.11
x i
5.0 4.0 3.0
Minimum pe 0.556
K m (x m +T m )
0.952
K m ( T m + T j
^ Comment
Process reaction
0 Quarter decay ratio
criterion
Coefficient values
t m / T m
0.25 0.43 1.0
*1
2.2 1.7 1.3
tm/Tm
4.0
x i
1.1
rformance index: regulator tuning
3.7(xra+Tm)
4(Tm + 0
Model: Method I
Model: I Method 1
90 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ITAE -Poulin and Pomerleau
(1996). Model: Method 1
Process output step load disturbance
Process input step load disturbance
Hougen(1979)-page 338.
Model: Method 1 Poulin and Pomerleau
(1999). Model: Method 1
Kc
i K (I")
Ti
x , (x m +T m )
Comment
Kc and Ti
coefficients deduced from graph
Coefficient values Tm/Tm
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
*1
5.0728 4.9688 4.8983 4.8218 4.7839 3.9465 3.9981 4.0397 4.0397 4.0397
x2
0.5231 0.5237 0.5241 0.5245 0.5249 0.5320 0.5315 0.5311 0.5311 0.5311
tm Am 1.2 1.4 1.6 1.8 2.0 1.2 1.4 1.6 1.8 2.0
* 1
4.7565 4.7293 4.7107 4.6837 4.6669 4.0337 4.0278 4.0278 4.0218 4.0099
x2
0.5250 0.5252 0.5254 0.5256 0.5257 0.5312 0.5312 0.5312 0.5313 0.5314
Direct synthesis
2 1.26X! K m T m
2 13 0.34K,, or
KmTu
0
1.04TU
Maximise crossover frequency
Mmax = 5dB
1 K (111) X 2 T m
° _ K m ( x m + T m ) ^ X l ( T m + T m ) 2
2 x, values recorded deduced from a graph:
* m / T m
x,
1
0.31
2
0.19
3
0.14
4
0.11
5
0.09
6
0.078
7
0.068
8
0.06
9
0.053
10
0.05
Chapter 3: Tuning Rules for PI Controllers 91
Rule
Huba and Zakova (2003).
Model: Method 1
McMillan (1984). Model: Method 1
Peric etal. (1997). Model: Method 8
Kc
3 K (112)
^ 0
Comment
Representative tuning rules; tuning rules for other xm/Tm values
may be obtained from a plot.
0.193/Kraxm
0.207/Kmtm
0.219/Kmxm
3.717xm
3.386xm
3.127xm
t m / T m = 0 . 5
Tm Am =1
^ m / T m = 2
Ultimate cycle 4 K (113)
Ku tan2 <L
yj\ + tan2 <L
-p (113)
0.1592 Tu
tan<L
Tuning rules developed from
KU,TU
3 v (112) - 2 T m + ^ / x m2 + 4 T m
2
JS.. — 1— 2Tm+xm->/Tni
2+4Tm
4 K . ( H 3 ) _ l - 4 7 7 T m
K m x m2
1
1 + | —
• , Ti1"-" = 3.33xm 1 + ri Y'65
m I
92 Handbook of PI and PID Controller Tuning Rules
3.6.2 Controller with set-point weighting
U(s) = Kc 1 + -TiS,
E(s)-aKcR(s)
* aK c
R(s) + T E(s) K, 1 + -
Xs + U(s)
K„e"
s(l + sTm)
Y(s)
Table 21: PI controller tuning rules - FOLIPD model Gm(s) = Kme"
s(l + sTm)
Rule
Taguchi and Araki (2000).
Model: Method I
Minimum ITAE -Pecharroman (2000).
Model: Method 4
Km = l ; Tm = l
0.05 <Tm <0.8
Kc
Minimum
5
Xl
0.049
0.066
0.099
0.129
0.159
0.189
K c 0 M ,
X 1 K U
x2
2.826
2.402
1.962
1.716
1.506
1.392
Tj Comment
performance index: servo/regulator tuning
T(114)
x2Tu
^n/T r a <1.0 . Overshoot (servo
step) < 20%
Coefficient values a
0.506
0.512
0.522
0.532
0.544
0.555
4>c
-164°
-160°
-155°
-150°
-145°
-140°
x i
0.218
0.250
0.286
0.330
0.351
x2
1.279
1.216
1.127
1.114
1.093
a
0.564
0.573
0.578
0.579
0.577
i -135°
-130°
-125°
-120°
-118°
K (114)
K „ 0.1787 + -
0.2839
-0.001723
. ( 1 1 4 ) V " " =4.296 + 3.794-^- + 0.2591 - ^ T T
a = 0.6551+ 0.01877-
Chapter 3: Tuning Rules for PI Controllers 93
Rule
Astrom and Hagglund (1995) -pages 210-
212. Model: Method 3
0.14 <^2-< 5.5 T
Kc
Q 4 1 e -0 .23T+0.019t 2
K m ( T m + i J
a g l e - l . l t + 0 . 7 6 T '
K m (T m + T m )
Ti
Direct synthesis
5.7xrae,J-°-69T2
a = l -0 .33 e2 5 ' - '^ 2
3 4 T e ° - 2 8 i - 0 0 0 8 9 ^
a=l-0.78e-L 9 T + 1 '2 x !
Comment
Ms =1.4
M s = 2 . 0
94 Handbook of PI and PID Controller Tuning Rules
3.6.3 Controller with proportional term acting on the output 1 K„
U ( s ) = - * - E ( s ) - K c Y ( s ) T:S
R(s)
+ T d • y ' F - E(s)
K«
T,s ^ 6 ? i k-. f U(s)
Kc
T
K m e ^
s(l + sTm)
Y(s)
•
Table 22: PI controller tuning rules - FOLIPD model Gm (s): Kme"
s(l + sTra)
Rule
Arvanitis et al. (2003b).
Model: Method 1
Kc Ti Comment
Minimum performance index: regulator tuning 4
(8-p)Kmxm
4xm
P
Minimum ISE 6
3P = 2.6302-2.1248 i
, T + 2.5776
T •1.8511
T V m J
+ 0.82557 T
-0.23641 • I ' m /
-1.7164.10" , T V m J
-0.044128 -si-VTm J
+ 3.1685.10"7
-0.0053335^2- + 0.00040201^1-V m J V m 7
10
±2L , 0 < i s - < 4 . 1 ; T T
P = 1.7110073-0.003554 v T
-4.1 |, -21-> 4.1.
Chapter 3: Tuning Rules for PI Controllers 95
Rule
Arvanitis et al. (2003b).
Model: Method 1 (continued)
Kc Ti Comment
Minimum performance index 7 ,8
Minimum performance index 9 , l 0
7 Performance index = He2 (t) + Km V (t)Jdt .
!B = 2.1054-0.76462^=- +0.79445 f^=- -0.51826
K IT, ' T + 0.21771
-0.06 T
-4.132.10"
+ 0.010927 ' T *
T -0.0013006
T -9.717.10"
T
T
('_ A + 7.6235.10"
T , 0 < - S - < 4 . 6 ;
T * m
B = 1.69667833406 - 0.00148768 T
V m
-4.6 •>4.6.
Performance index: e2(t) + K m2 f dt
6 = 1.6505623 - 0.0034983 - ^ - - 8 > 2 ;
8 = 2.1806-0.6027: V m J
+ 0.3875 X
T -0.14955
T + 0.034535
T V m
-0.0042243 T
V, m J
+ 8.2395.10" > T V m
+ 5.0912.10"5 |^- -7.283.10" T
V m J
+ 4.2603.10"7 (, \9
T V m J
-9.5824.10 I — , -21- range not defined.
96 Handbook of PI and PID Controller Tuning Rules
Rule
Arvanitis et al. (2003b).
Model: Method 1
Kc
II K (H5)
T i
T ( H 5 )
Comment
Minimum ISE 12
Minimum performance index '3, 14
n K ( i i 5 ) _ . 12 .5664(T m +Tj-4
Km[2(xm +Tm)(l2.5664(xm +Tm j - 4 ) - (l.5Tra2 + 3xmTm + 2Tra
2^J
with a = [0.5708 +
12 % = 0.99315 + 1.9536
3.4674Tm+3.1416Tm- l k ; T | ( 1 1 5 ) _ 1 2 . 5 6 6 4 ( x m + T m ) - 4
, T -2.5157
T 1.9163
( \3
T 0.8893
, T , V m /
+ 0.26167 - E -
+ 2.0083.10"
-0.049809-^2-V m J
+ 0.0061098-2-T
- 0 .00046593^-V m J
T - 3.7368.10-7
, T 0<^_<6 .0 ;
T
T Xm 6 1, -^2- > 6.0 . X = 1.96705757 + 0.01308189
' Performance index = l[e2 (t) + KmV(t)]di
X = 0.73114 + 2.346ip2- -2 .8509PS Un, J ( j r a
+ 2.0965 A-1
(~ \ -0.95252
m / T
V m J
+ 0.27639 ^ L _ 0.052089 — l T m J {Tm j
+ 0.0063413 f ^7
- ^ L | -0.00048064 T, >J
V m J
+ 2.0611.10~5 f. V
T , -3.8178.10~7
fx^
T V m J
X = 1.916423116 + 0.018175439^- -4 .5 V ' i n
, 0<-!2-<4.5 ; T
• > 4.5 .
Chapter 3: Tuning Rules for PI Controllers 97
Rule
Arvanitis et al. (2003b).
Model: Method 1 (continued)
Arvanitis et al. (2003b).
Model: Method 1
Kc T, Comment
Minimum performance index 15, 16
Minimum i 4
(8-p)KmTm
performance index: servo tuning 4T
P
Minimum ISE "
' Performance index = f e2(t) + Km2 — I dt
J raUJ| o L J
X = 0.60896+ 3.0593
+ 0.38696
, T -3.9339 ' T *
, T , V m /
+ 2.933 V m
-1.3353 T
V m
T V m J
-0.07277: r \6
x„
T + 0.0088395
T -0.0006686 it
-2.8623.10" T
V m
-5.2941.10' > T
0<-m_<4.5; T
X = 1.9283751 + 0.016846129 T
V m
-4.5 - ^ - > 4 . 5 . T
17 p = 1.997- 0.82781| - ^ - |+1.0002 ^ *
T V m J
0.72106 M M +0.32269 [ ^ T , T
-0 .092452^- +0.017215 v T T
-0.0020701 \TmJ
-0.00015505 T
-6.5691.10" T
V m J
+ 1.2025.10" T
V m J
, 0 < ^ - < 5 . 5 ; T
(3 = 1.6281745-0.001171 T
- - 5 . 5 ->5.5.
98 Handbook of PI and PID Controller Tuning Rules
Rule
Arvanitis et al. (2003b).
Model: Method 1 (continued)
Kc Ti Comment
Minimum performance index l8,19
Minimum performance index 20, 21
00
1 Performance index = f[e2 (t) + K m V (t)]dt
'(3 = 1.5693 + 0.1066^2-1-0.10101 f \ 2
T V 1 m J
+ 0.05217 -0.016147 e \ 4
+ 0.0030412 — I -0.00032187 V m
-^2-1 +1.2181.10"5
VTm , T
T V m J
+ 9.9698.10"
•1.2274.10"7
T
r \ + 3.7365.10"
T
10
, 0 < - ^ - < 4 . 6 . T
T
Performance index = *"**.'i£ dt
P = 0.780651417 + 4.7464131 -^--0.1 T
0<^_<0.3; T
P = 1.729934 -0.051535 -^--0.3 T
V 1 m J
, 0.3<-SL<2. T
Chapter 3: Tuning Rules for PI Controllers 99
Rule
Arvanitis et al. (2003b).
Model: Method 1
Kc
22 K (116)
Ti
i
Comment
Minimum ISE 23
Minimum performance index 24,25
22 v (116) 12.5664(xm+Tm)-4
Km[2(xm +TmXl2.5664(Tm + T m ) -4)- ( l .5x m2 +3xmTm + 2Tm
2>]
wkha = [0.5708 + i p Z l ^ H l l K E i ] x , T(..« = 12.5664(xm+Tm)-4
23 x = 0.79001 + 2.114|^2-Tmy
-2 .5601P 2 - ! +1.8971
IT,
r A3
T V 1m J
0.86871 f \ 4
x „
T V m y
+ 0.25371 -0.048068
.9231.10" T
V m
T
-3.5718.10"
+ 0.0058771 • ^ M -0.00044707 ^ M Y m 7 V m 7
T V 1m J
0< JL<4 .5 ; T
X = 1.903177576 + 0.0181584p2--4.5 , ^ - > 4 . 5 . V m / m
00
1 Performance index = f[e2(t) + K r aV(t)jdt .
25 x = 0.48124 + 2 .5549^-1-2 .896
\5
T + 2.0495
vTm
+ 0 . 2 6 0 9 4 ^ -Y m )
- 0.048775 ' T ^
i T , V m J
+ 0.0059064 T
-0.911151 —E-
-0.00044618
+ 1.9095.10" T V m j
-3.5335.10" T
V 1m J
0<-^ -<4 .0 ; T
X = 1.87054524 + 0.021802| — - 4 ->4 .0 .
100 Handbook of PI and PID Controller Tuning Rules
Rule
Arvanitis et al. (2003b).
Model: Method 1 (continued)
Arvanitis et al. (2003b).
Model: Method 1
Kc T i Comment
Minimum performance index 26,27
Direct synthesis 4
(8-P)Kmxm
28K ( U 7 )
4t m
p
(l + 4 ? 2 ) k + T j
p = ^
00 f
' Performance index = I e2(t) + K m2 —
X = 0.31373 + 3.7796 ' x ^
T V m J
- 5.0294 T
V m
dt.
+ 3.8589 f ^ I™. T
1.7956 ( \4
T
+ 0.52906 r A5
T 0.10079 -S2-
\
+ 4.0678.10-5|-E- -7.5694.10
+ 0.012369 ^ 2 - | -0.00094357 f \
T
T V 1 m J
T 0 < - E - < 3 . 5 ;
T
X = 1.86144723+ 0 . 0 2 1 9 8 9 ^ - 3 . 5 , — > 3.5 . V * m j *m
(l + 4 4 2 k + T m )
K r a k 2 ( 0 . 5 + 842)+xraTm(l + 16^ )+8^ 2 T m2 ] '
28 j r (117)
Chapter 3: Tuning Rules for PI Controllers 101
3.6.4 Controller with proportional term acting on the output 2
( 1 1 U(s) = Kc 1 + — E(s)-K,Y(s) V T.sy
R(s) b(s)
k I TisJ
+ vy
f 1
t
»;
•"W
K m e - "
s(l + sTm)
Y(s)
w
Table 23: PI controller tuning rules - FOLIPD model Gm(s) = K e~
s(l + sTm)
Rule
Lee and Edgar (2002).
Model: Method 1
Kc Ti Comment
Robust
29 K (118) T ( 1 1 8 ) K, =0.25KU
29 ^ (118) 0.25K,,
(A- + 0 KuKm 2(X + T„ >T;
(118)
2( + 0 '
102 Handbook of PI and PID Controller Tuning Rules
3.7 SOSPD model Kme" K e"
or-T m l V + 2 ^ T m l s + l (l + TmlsXl + Tm2s)
3.7.1 Ideal controller G (s) = K 1 + — V T i s 7
R W ^ E ( S ) . + 1 i
Kc f.+-L) U(s) SOSPD model
Y(s)
Table 24: PI controller tuning rules - SOSPD model
K - e - " - o f K m e - ' -
T m l V+2i ; m T m l s + l (l + TmlsXl + Tm2s)
Rule
Minimum IAE -Shinskey(1994)-
page 158.
Minimum IAE -Shinskey (1996)
page 48. Model: Method 1
-^2-= 0.2 Tml
Kc Ti Comment
Minimum performance index: regulator tuning
1 K (119)
0.77Tm,
0.70Tml
K m X m
0.80Tml
K m T m
0.80Tral
Kmxm
T ( 1 1 9 ) i
2-83(Tm+Tm2)
2.65(Tm+Tm2)
2.29(tm+Tm2)
l-67(Tm+Tm2)
Model: Method 1
T - H i = 0.1 Tml
^Jsl = 0.2 Tml
^ 2 1 = 0.5 Tml
T i==- = 1.0 Tml
1 K (119) 100T„.
K m (x m +T m 2 ) 50 + 55 ! _ e ^ + T -
-019) _ . 0.5 + 3.5 1-e
3Tm,
Chapter 3: Tuning Rules for PI Controllers 103
Rule
Minimum IAE -Huang et al. (1996).
Model: Method 1
Kc
2 K (HO)
T,
T (120)
Comment
0 < I n i 2 . < i ; T 1 m l
0.1<-±2-<l Tra,
K
Note: equations continued into the footnote on page 104
(120) _ 1
Kn
1
Km
J_ Km
1
Km
J_
1
K „
6.4884 + 4 .6198 -^2 - - 3.49i_i!i2__ 25.3143 TmT™2
T m l T m l T m l
( _ A -0.9077
0.8196 V T m i y
•5.2132
-0.063
-7.2712
f \ 0.5961 X,
Lml J V T m i y
-18.0448
•sO.7204 , x
^L\ + 5 . 3 2 6 3 ^ -vTmi
0 . 4 9 3 7 ^ - + 1 9 . 1 7 8 3 ^
l T r a l
^ n , |
1.0049 , \ 1.005
+ 13.9108—212-
+ 12.2494-VTml V
8.4355-1 m 2
' m l
-17.6781-T V ^ m l
-0.7241eT"» -2.2525eT m l +5.4959eT""2
, (120) = T t m , „ im2 <c /1-709 J E " 0.0064 + 3.9574—2- + 4.4087-2^- - 6.4789 T T
+ T„
+ Tml
+ T„
+ Tml
-12.8702 T m T m 2 -1.5083 T 1 ml
' ^ 2 i l | +9.4348 V ^ m l V^ml
17.0736-VTml 7
+ 15.9816-VTml J
- 3 . 9 0 9 T "l 1 m 2
VTml y
•10.7619
-6.6602
V •'ml
x„ X
•10.864-Tm, Tm,
-22 .3194 V ^ m l V ^ m l
^ml V t"1 J
+ 6.8122|- i2l T,
^ + 7 .5146-21
ml J ml /
104 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum IAE -Huang etal. (1996).
Model: Method 1
Kc
3 K (121)
T,
T ( 1 2 1 )
Comment
0.4 < 5m < 1;
0.05 < — <1
+ T„
+ Tm
+ T„
+ T„.
2.8724-
.2174,
T V ml
11.4666 V *-m\
2 , N 3
V ^ m l
+ 11.1207 \2 ( V
^m t m T
V ^ m l
l ml J
-4.3675 ml J
5
(T x m 2
V * m l
- 0 . 1 1 2 ^ - ^ 2 - + 1 . 0 3 0 8 - ^
-2.2236
•1.9136 V ^ m l V ' ' m l
-3.4994U^3- I 1-=^-
lT, , , T ml y \ xml y
-1.5777 T
T-m2 I + 1.1408^4 i 2 -^m] V ml .
' Note: equations continued into the footnote on page 105.
K (121)
K „ 10.4183 - 20.9497^2- - 5.5175^m - 26.5149^m - ^
l m l ITT
K „ 42.7745
s N 1.4439 T
V lml
+ 10.5069 f \0.1456
Tm ! +15.4103[ T" V * m l
+ — [34.32364m3'7057 -17 .8860C 5 3 5 9 -54.0584^m
L9593] K m
-0.0541 f -\ 4.7426
+ — I 22.4263^mf ^=_) +2.7497^, ' T' K „
K „
T
50.2197^m'-8i88 - 2 - -17.1968—=s-4m2-7227 + 1.0293£,m -JBL
Kn
5 m ^
, ( 1 2 1 )
•16.7667e'ml +14.5737e?m -7.3025e
11.447 + 4.5128-S- - 75.2486^m - 110.807 V ^ m l
12.282^n
+ T„. 345.3228^m2 +191.9539 ^ | +359.3345^n
Chapter 3: Tuning Rules for PI Controllers 105
Rule
Minimum IAE -Shinskey(1996)-
page 121. Model: Method 1
Minimum ISE -McAvoy and Johnson
(1967). Model: Method 1;
Coefficients of Kc and Tj deduced
from graphs
^
1 1 1
Kc
0.48KU
* i / K m
Ti
0.83TU
X2^m
Coefficient values T 1 m l
0.5 4.0 10.0
x i
0.8 5.7 13.6
X 2
1.82 12.5 25.0
^
4 4
Tm,
^m
0.5 4.0
x i
4.3 27.1
x2
3.45 6.67
^m
7 7
Comment
^ - = 0.2, Tm, T - ^ - = 0.2 Tm i
Iml
•Cm
0.5 4.0
x i
7.8 51.2
x2
3.85 5.88
+ T„.
+ T„
+ T„
+ T„
-158.7611—4m2 -770.2897^m
3 -153.633 Tm i
-412.5409£,n l m l J
-414.7786^m2 U * - + 4 8 5 . 0 9 7 6 ^ m
3
864.51954m +55.4366 - s - | + 222.2685i;n
275.116^ 21 Tm ' ' - 2 0 5 . 2 4 9 3 ^ - \J -479.5627 Um, ' ' m l V^ml
-473.1346^m -6.547 5 c e n _Tn
T - 43 .28224 m | - ^
' m l
A + 99.8717£,n
+ Tm
+ Tm
- 7 3 . 5 6 6 6 1 ^ 1 4m2 "56.4418 ^ - j ^J -37.4971 ^ | ^
1 6 0 . 7 7 1 4 ^ m5
' m l
106 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ITAE -Lopezes / . (1969). Model: Method 7;
Coefficients of Kc and T; deduced
from graphs
Minimum ITAE -Chao etal. (1989). Model: Method 1
Minimum ITAE -Hassan (1993).
Model: Method 7
Kc
x,/Km
^
x2Tm
Comment
Coefficient values
lm
0.5 0.5 0.5
Tml
0.1 1.0 10.0
*1
3.0 0.2 0.3
x2
2.86 0.83 4.0
4 K (122)
Sm
1 1 1
Tml
0.1 1.0 10.0
x l
7.0 0.95 0.35
x2
2.00 2.22 5.00
T(122) i
4-
4 4 4
*m
Tm, 0.1 1.0
10.0
Xl
40.0 6.0
0.75
x2
0.83 3.33 10.0
0.7 < £ < 3 ; 0.05 < Tm < 0.5 2^mTml
5 K (123) T (123)
0.5<£. m <2,
0 . 1 < ^ S - < 4 Tra,
4 R ( m ) _ 0.09578+ 0.6206£,m-0.1069i;n
K „
-1.108+0.2558l;m-0.0579l;m
-0.6441+0.76265m-0.11935„'
V^Sm ml J
T,<122) = 2i;mTml(l.31-0.0914£.m + 0.07464J;m2 J - I s _
V^Sra 'ml ,
5 Formulae correspond to graphs of Lopez et al. (1969). Kc(123) is obtained as follows:
log[KmKc<123)]= -0.0099458 + 1.9743700E,m - 3.8984310^2- - 1.1225270E,m
2
,(123) ,
+ 2.3493280 — VTml
+ 1.9126490§m -is_ + 0.0754568£,m2
V ml .
-0.5594610^, f \2
VTml J - 0.7930846£,iT1
2 - i * - + 0.2412770£,m3
Tmi
-0.3567954 V 'ml J
-0.0024730£,n
-0.1354322E.m3-^--1.1756470£.m
3
' m l vTmi
+ 0.04785934n
- 0.0096974^,
/
V ' m l .
3 X„
ml /
T; is obtained as follows (equation continued into the footnote on page 107):
• (123) '
log = 0.9321658-0.7200651E,ra -3.4227010^2- + 0.0432084£,m2
Tmi
Chapter 3: Tuning Rules for PI Controllers 107
Rule
Minimum ITSE -Chao etal. (1989). Model: Method 1
Nearly minimum IAE, ISE, ITAE -
Hwang (1995). Model: Method 10
Kc
6 K (124)
Tf
T(124)
Comment
0.7 < £m < 3 ; 0.05 < — ^ — < 0.5
7 K (125) T (125) i
Decay ratio = 0.2
T V m
-1.6011510£,n
+ 1.4965440-^2- +5.3636520^m-^--0.0848431£m2
f ^
V^ml - 2.1777800^m
2 -2- + 0.0404586^m3
-0.09120084m -21--0.1769621-^2-l T n , l ,
+ 0.3033341^ra3 -^2- + 0.1753407^m
3
Tml
+ 0.15014744n 2 t „
K (124) 0.3366 + 0.6355E,m -0.1066^n
K „
r^-) -0.0622614^m3f^2
V Iml 7 \lmlJ
-1.0167+0.1743^1T,-0.03946^m
2^mTm
T/124' =2^mTml(l.9255-0.3607^m +0.1299^n 2 ^ T r a
-0.5848+0.72945m -0.1136l;m
7 0.2 < —— < 2.0 , 0.6 < £ra < 4.2 . Equations continued into the footnote on page 108.
K (125) _ 0.622[l-0.435coHTm +0.052(coHTm)2 [
KHKm / ( l + K H K m ) K t
. (125) Kc(125)(l + K H K m )
K (125)
0.0697fflHKra(l
0.724[l-0.469coHTm +0.0609(coHTm)2|
KHKm / ( l + K H K m )
l + 0.752coHTm-0.145coH2xm
2 •, s < 2.4 ;
K,
,(125) Kc(125)(l + K H K m )
0.0405(DHKm(l + 1.93(oHTm-0.363coH2Tm
2j 2 . 4 < e < 3 ;
K (125) 1.26(0.506)mHl'"[l-1.07/s + 0.616/s2]'
KHKm / ( l + K H K m ) K^
108 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum IAE -Galher and Otto
(1968). Model: Method 19.
Minimum IAE -Huang et al. (1996).
Model: Method 1
Kc ^ Comment
Minimum performance index: servo tuning x, /Km
x 2( Tm i + T m 2 + x m ) T - T
' m l x m2 Representative coefficient values - deduced from graphs
T m
2Tml
0.053 0.11 0.25
x l
3.6 2.3 1.35
8 j r (126)
x2
1.35 1.17 0.93
^m
2Tml
0.67 1.50 4.0
T (126)
Xl
0.76 0.53 0.43
x2
0.72 0.60 0.51
0 < I E 2 . < I ;
Tm,
0 . 1 < ^ - < 1 T
.(125) KC(I25)(1 + K H K J
K (125)
-. = =rf- r—-5, 3 < s < 20 : 0.066 koHKm(l + 0.824 ln[coHtm])(l + 1.71/s-1.17/E2 j
x 1.09[l-0.497a)Hxm+0.0724((oHTm)2lN
K „ K j ( l + K H K j K H ,
X (125) Kc<125)(l + KHKm)
0.054fflHKm(-1 + 2.54a,HTm -0.457coH2xm
2)
' Note: equations continued into the footnote on page 109.
, 8 > 20 .
K (126)
K,
1
1
K m
1 Km
1
1
,T, -13.0454 - 9.0916^2- + 2 . 6 6 4 7 - ^ + 9.162 m2 T T
0.3053 + 1.1075
' m l
3.5959
•2.2927
-31.0306 U ^ -l T m l
13.0155|^sl X ml /
l m l J
+ 9.6899 i ^ -l T m l
- 0.6418^2-+ 18.9643^2 I xm T T ' m l V ' m l
-39.7340-' m l V ml .
28.155^-1 -V^- -2.0067-T T
V ' m l J
T„, T „ , ; 1 m T m .
4.8259eT"" + 2.1137eT"" +8.45lie T""2
Chapter 3: Tuning Rules for PI Controllers 109
Rule
Minimum IAE -Huang et al. (1996).
Model: Method 1
Kc
9 R (127)
Ti
T ( I 2 7 ) 1 i
Comment
0 - 4 < ^ m < l ;
0.05 < - ^ 2 - <1 Tml
.(126) 0.9771 - 0 . 2 4 9 2 - ^ - + 0 . 8 7 5 3 - ^ + 3.4651 - ^ ' m l ' m l V rr
3
+ 1 1 . 6 7 6 8 - ^ Tmi / Tr
- 3.8516 T m T T 2
7.5106 T
-7.4538 V ^ m l
+ T„
-10.9909-
-18.1011-
^ - 1 -16.1461 - ^ + 8 . 2 5 6 7 ^ V ^ m l ) V^n-v"Tmi
V ^ m l
+ 6.2208 V ^ m l
-21.9893-V T m l .
15.8538 V ^ m l
-4.7536 + 14.5405-
+ T„
+ Tml
+ Tm
-2.2691 V T m l J
m
l ml J
-8.387
16.651 T T
V ml ) \ 1 m l
ml J
4
2 f ^ 3
t m 1 T„
' m l / i ' m
5
Is!_ If I s L I -7.1990 I n n ] + u496p2_
-4.728-T
V ml
+ 1.1395 V ' m l
-0.6385 - ^ s -l T m l V ^ m l
.08851 ^ _ I Y^A + 3.16151 ^ Tm, J I Tral J { T „
2 / r 1 m 2
V ' m l ml / V ml
' Note: equations continued into the footnote on page 110
+ 4.5398-
K (127) — - 10.95 - 1 . 8 8 4 5 ^ 2 - - 3.41234m + 4.59544m - ^ L . - 1.7002
T V ml
\ Xml J
K „ - 2 . 1 3 2 4 1 J J - * -
^ml y
-14.4149^m2 | ^ 1-0.7683^, 3
110 Handbook of PI and PID Controller Tuning Rules
K„
/• N0.421
7 . 5 1 4 2 ^ 2 -VTml j
+ 3.7291 V^ml
lis-1 +5.3444| J V ml j
+ J _ [ _ 0 . 0 8 1 9 ^ m1 9 ' 5 4 1 9 - 3 . 6 0 3 ^ m
1 0 7 4 9 + 7 .1163^ m
1 1 0 0 6 ]
K„ 3.206^n -7.8480£n - ^ 5 - | + 11.3222!; L9948
V *ml V ml .
K„
T (127) _ T 2i _ 1ml
2.4239eT»' + 3.4137e^ +1.0251e T°" - 0 . 5 5 9 3 ^ ^ 2 1
2.4866 - 2 3 . 3 2 3 4 - ^ - + 5.3662£m + 65.6053 Tmi
r \2
Tml
+ T„
+ Tm
+ T„
+ T„
+ T„
+ T„
+ T m
+ Tm
29 .0062^ m ^- -24 .1648) ; m2 - 8 3 . 6 7 9 6 ^
Tml I Tm
-135.9699£n + 43.1477-^a-^m2 + 5 1 . 9 7 4 9 ^
Lml J
86.0228 Tral
+ 70.4553£„ r A3
VTml^ + 153 .4877^ 2 * n
Trol
- 1 2 5 . 0 1 1 2 ^ ^ m3 - 6 8 . 5 8 9 3 ^ m
4 -62 .7517 Tmi T
V ml
2 7 . 6 1 7 8 ^ m - J 2 -f \ 3
-152.7422^ n l m l J VTml J
- 2 0 . 8 7 0 5 ^ = - ] £„
54.0012 U ^ i - C + 5 8 . 7 3 7 6 C + 1 3 . 1 1 9 3 | ^ 'ml .
2 0 . 2 6 4 5 ^ | ^ | - 2 3 . 2 0 6 4 4 m6 - 6 1 . 6 7 4 2
V Xml
f \ 3
136.2439 V^mlV
f - \ Z.J -95 .4092
Tm, C+20 .4168^C
lml
Chapter 3: Tuning Rules for PI Controllers 111
Rule
Minimum ISE -Keviczky and Csaki
(1973). Model: Method 1
Representative K,.,^ coefficients
estimated from graph; Kra = 1
Minimum ITAE -Chao etal. (1989). Model: Method 1
Minimum ITSE -Chao etal. (1989). Model: Method 1
^m/X 0.1 0.2 0.5 1.0 2.0 5.0 10.0
Tm, 0.2 0.5 0.5 1.0 1.0
10
l i
Kc
x i
Ti
x 2 * m
Coefficient values
%
I
0.1
x i
0.26 0.21 0.15 0.12 0.1 0.1 0.15
x2
1.5 1.3 1.2 1.1 1.0 1.2 1.6
0.2
X l
0.55 0.4
0.25 0.2 0.16 0.16 0.2
x2
2 1.5 1.3 1.2 1.2 1.4 2.0
0.5
x i
2.2 1.5 0.8 0.65 0.4 0.4 0.45
x 2
3.5 2.8 2
1.8 1.9 3.0 5.5
Comment
1.0
x i
6 3 2 1.2 0.9 0.65 0.6
Other coefficient values
x i
30 13 30 7 14
x2
15 14 30 14 30
K (128)
\
5.0 5.0 10.0 5.0 10.0
Tml
2.0 2.0 5.0 5.0 10.0 10.0
x i
4 8
1.9 3.4 1.1 2.0
T (128)
0.7<^m < 3 ; 0 . 0 5 < — ^ — <0. 2i;mTml
K c ( . 2 9 , x (129)
0.7<£m < 3 ; 0 . 0 5 < — ^ — <0. 2^mTml
x2
6 4.5 3.5 3.3 3.5 5.0 7.5
2.0
x i
14 9 5 3
1.8 1.0 0.7
x2
9 8 7
6.5 7 8 10
x2
15 30 17 32 19 34
4 5.0 10.0 5.0 10.0 5.0 10.0
5
5
10 K (128) 0.2763 + 0.3532^m - 0.069554n
It T
0.3772-0.24141;m +0.033421;m
ml J
Tj"28 ' = 2^mTml (o.9953 + 0.07857^m + 0.005317£,m2| — -
-0.3276+0.3226^,-0.05929^„
11 v (.29) 0.3970+ 0.43764m-0.08168^ K „ ^ m T m
- ( 1 2 9 ) ^ r a T r a l ( l 1.3414-0.1487^m +0.07685E,n
2^mTral
0.5629-0.11874m+0.01328^„,2
-0.42B7+O.I981^m-0.01978E.m
^mr„
112 Handbook of PI and PID Controller Tuning Rules
Rule
Nearly minimum IAE, ISE, ITAE -
Hwang (1995). Model: Method 10
Kc
12 ^ (130)
T, T ( 1 3 0 )
i
Comment
For^m <0.776 + 0.0568-^- + 0.18 ' m l V
Coefficient values x i
0.822 X 8
-0.441 X15
-0.986 x 2 2
-0.466
x2
-0.549 x9
0.0569 X16
0.558 x 2 3
0.0647
X 3
0.122 X 10
0.0172 X 17
0.0476 x 2 4
0.0609
x4
0.0142 x n
4.62 x I 8
0.996 x 2 5
1.97
X 5
6.96
X 1 2
-0.823 X | 9
2.13 X 26
-0.323
X 6
-1.77 X 13
1.28 X 2 0
-1.13
X 7
0.786 X14
0.542 X 21
1.14
12 Decay ratio = 0.1; 0.2 < -^2- < 2.0 ; 0.6 < £,m < 4.2 . Tmi
( K (130) Xl[l + x2a)HTm +x3(coHTm)2J
K H K m / ( l + K H K j K t
• ( 1 3 0 ) Kc(l30)(l + K H K m )
X4aHKm(l + x5coH
r, E < 2 . 4 ;
t m +X 6 f f l H T„
K (130) X7[l + X8(0HTm+X9(cOHTm)2J
KHKm / ( l + K H K m ) K H .
. (130) Kc(l30)(l + K H K m )
K (130)
f ^ ^ 7—r\, 2.4 < E < 3 ; X l 0 « H K m l l + X„C0HT r a + X | 2 C 0 H Tm j
x13x14m"T"'[l + x l 5 /s + x 1 6 / e 2 f
KHKm /( l + K H K m ) K H ,
•(130) Kc(130)(l + K H K m )
X 1 7 ( 0 H K m ( 1 + XI8 l n[C OH-Cm]X1 + X 1 9 / S + X 2 o / s 2 ) 3 < s < 20 ;
K (130) X21[l + X22(0HTm +X23(cOHTm)2J
KHKm / ( l + K H K m )
,(130)
Kv
KC(130)(1 + K H K J
^ H K r a ( - 1 + X25COHTm + X26COH2Tm
2 j e>20.
Chapter 3: Tuning Rules for PI Controllers 113
Rule
Nearly minimum IAE, ISE, ITAE -Hwang (1995)-
continued
Kc T i
For lm > 0.889+ 0 . 4 9 6 ^ - + 0.26 Tmi
Comment
2
Coefficient values x i
0.794 X 8
-0.415 X 15
-0.959 x 2 2
-0.466
x2
-0.541 X 9
0.0575 X 16
0.773 X 2 3
0.0667
X 3
0.126 X 10
0.0124 X 17
0.0335 x 2 4
0.0328
x4
0.0078 x u
4.05 X 18
0.947 X 25
2.21
x 5
8.38 x 1 2
-0.63 x19
1.9 x 2 6
-0.338
X 6
-1.97 X 13
1.15 x 2 0
-1.07
x 7
0.738 X 1 4
0.564
x2)
1.07
For £m >0.776 + 0.0568^2- + 0.18 — M and - ' i l l i-p ryi
ml V 1ml J
Em < 0.889 + 0 . 4 9 6 ^ - + 0 . 2 6 ^ - :
Coefficient values x i
0.789 x 8
-0.426 X15
-0.978 X 2 2
-0.467
X 2
-0.527 x9
0.0551 X16
0.659 X 23
0.0657
x 3
0.11 x 1 0
0.0153 X17
0.0421 x 2 4
0.0477
x4
0.009 x u 4.37
X 18
0.969 X 25
2.07
X 5
9.7 X 1 2
-0.743 x19
2.02 X 2 6
-0.333
X 6
-2.4 X 13
1.22 X 2 0
-1.11
X 7
0.76 X 1 4
0.55 X21
1.11
114 Handbook of PI and PID Controller Tuning Rules
Rule
Bryant et al. (1973). Model: Method 1
K c T| Comment
Direct synthesis 0
x i
K m X m
x 2 K m T m l
Tm ,
T m i > T m 2
Tmi > T m 2
Coefficient values - deduced from graph
tm/Tml
0.33
0.40 0.50 0.67 0.33
0.40 0.50
0.67
x i
0.08
0.09 0.09 0.11 0.14 0.16
0.18 0.23
tm/Tml
1.0
2.0 10.0
1.0 2.0 10.0
x i
0.15
0.24 0.33
0.31 0.47 0.64
Critically damped dominant pole
Damping ratio (dominant pole) = 0.6
13 Representative x2 values, summarized as follows, are deduced from graphs:
x 2 value
^
0.4 0.6 0.8 0.9 1.0 1.1 1.2
1.5 1.9
•tmAml
0.5
----
7.94 8.93
9.80 12.2 15.4
1.0
-
5.13 5.99 5.81 9.01 9.90
10.8 13.3 16.7
2.0
5.81 6.94 7.30 7.30 11.5 12.3 13.2
15.4 18.9
3.0
7.46 8.40 9.17 10.0 13.9 14.5
15.4 18.2 21.3
4.0
9.17 10.0 11.9 14.1 16.4 16.9
17.9 20.0 23.8
5.0
10.9 11.8 15.4 19.6 18.9 19.6
20.4 22.7 25.6
6.0
12.7 14.1
20.0 28.6
---
-
-
Chapter 3: Tuning Rules for PI Controllers 115
Rule
Bryant e/a/. (1973) -continued
Hougen(1979)-pages 345-346.
Model: Method 1
Kc
1 4 X l / K m
15 K (131)
T| x2KmTml
T ( 1 3 1 )
Comment
Tmi > Tm2
Maximise crossover frequency
1 Representative x, values, summarised as follows, are deduced from graphs:
x, value
km 0.4 0.6 0.8 0.9
T m / * m l
1.0
--
0.055 0.139
2.0
0.006 0.039 0.110 0.147
3.0
0.076 0.093 0.127 0.122
4.0
0.103 0.114 0.114 0.093
5.0
0.115 0.122 0.093 0.070
6.0
0.122 0.120 0.075 0.052
Representative x 2 values, summarised as follows, are deduced from graphs:
x2 value
5m 0.4 0.6 0.8 0.9
* m / T m l
1.0
-5.13 5.99 5.81
2.0
5.81 6.94 7.30 7.30
3.0
7.46 8.40 9.17 10.0
4.0
9.17 10.0 11.9 14.1
5.0
10.9 11.8 15.4 19.6
6.0
12.7 14.1 20.0 28.6
15 K (131)
K „ 1+-
1ml *m2
0.7Tm2(Tml+T ra2)+Tm, 2~\
Tm, +0.7T„
Ti ( 1 3 1 )=(Tm l+0.7Tm 2 | 1 + 0.1
x, values are obtained as follows:
x, value
Tm2/Tml 0.1 0.3 0.5 1.0
' m2 /Tml 0.1 0.3 0.5 1.0
0.1
0.31 0.6 0.65 0.7
0.6
0.085 0.2 0.29 0.34
0.2
0.19 0.42 0.5 0.6
0.7
0.075 0.18 0.26 0.32
c r a / ^ m l
0.3
0.15 0.35 0.45 0.5
C m / ' m l
0.8
0.07 0.16 0.24 0.31
0.4
0.11 0.28 0.46 0.45
0.9
0.065 0.155 0.23 0.29
0.5
0.09 0.23 0.32 0.38
1.0
0.06 0.15 0.22 0.28
116 Handbook of PI and PID Controller Tuning Rules
Rule
Hougen(1988). Model: Method I
Somanie?a/. (1992). 17
Kc
16 K (132)
X | / K m A m
T i
T (132) i
x 2 T m
Comment
Five criteria are fulfilled
Coefficient values - see page 117
Equations for Kc deduced from graph;
K (>32,=_1_ ] 0! 731og,0 -=H +0.65
:0.1
K (132) 0.611ogl0 -1H1 +0.29
K „ = 0.2;
K c < 1 3 2 » = ^ 1 0 0.48 l o g j - ^ +0.05
= 0.5;
K (132)
For
K „
K , 25TmlT.
Ti
0.9806
K m A j
-10 0.41 l o g j - m i -0.06
^ . = l.T i(132)==Tml+0.7Tm2+0.12Tn
' m l
« 1 .
<X ^ 2.944 - tan"
1 m l Am
1 + 2.944 - tan" J_m_L + J_m2_
K * ^ - U i + ' T . "2
2.944 -T T
m ''m
1 + T T
2,944 - ml - m2
For 25T ,T
X • » 1, K
0.9806 c K A „
1 + (T
0.5 V ml T m 2 j
+ 0.945+ ,0.25 V^ml Tm2
- 0 . 9 4 5 ^ + — l T m l T m 2 J
0.5 + m , wm
T T V ml 1 m 2 J
+ 0.945-,|0.25| ^is_ +JEJS_ Tmi Tm2 j
-0.9451 -^=- + -^2-Trai Tm2
Chapter 3: Tuning Rules for PI Controllers 117
Rule
Sorrnmi etal. (1992) - continued.
Model: Method 1;
Suggested A m :
1.75 < A m <3 .0
Tan et al. (1996). Model: Method 2
?ou\in etal. (1996). Model: Method 1
Kc Ti Comment
Coefficient values
^ m
Tm2
0.1 0.1 0.1 0.1 0.1 0.1
0.125 0.125
0.125
0.125 0.125 0.125 0.25
0.25 0.25
0.25 0.25
0.25
0.5 0.5 0.5 0.5 0.5
0.147 0.440 0.954
1.953 4.587
13.89 0.121 0.404
0.897 1.842
4.219 11.628 0.064
0.256 0.667
1.406 3.021
6.452
0.062 0.376
0.909 1.880
3.401
x i
11.707 10.091 10.315
11.070 12.207
13.337 11.318 8.613
8.510 8.997 9.840
10.707 14.494
6.504 5.136 4.961 5.174
5.502
14.918 4.353
3.274 3.025
3.038 18 jr (133)
Tm, K m ( T m l + T m )
x2
12.5 8.33 6.25
5.00
4.17 3.70 12.5 8.33
6.25 5.00 4.17
3.70 11.11
8.33 6.25
5.00 4.17
3.70
8.33 6.25
5.00 4.17
3.70
Tra2
1.00 1.00 1.00
1.00
1.67
1.67 1.67 1.67
1.67 2.5 2.5 2.5 2.5 5.0 5.0 5.0 5.0 5.0 10.0 10.0
10.0 10.0
10.0
T m
0.082
0.437 1.016
2.075
0.169 0.581 1.242 2.445
4.651 0.156
0.338 0.559 1.182
0.075 0.238
0.433 0.667
0.951 0.073 0.235 0.424 0.649
0.917 -p (133)
T ml
x i
12.389 3.462
2.370
2.087
6.871 2.773 1.930 1.624
1.957 7.607
4.015 2.797
1.848 16.187 5.623 3.448 2.52
2.014
17.650 5.996 4.524 2.641
2.093
x2
6.25 5.00 4.17
3.57
5.00
4.17 3.57 3.13
3.13 4.55
4.17 3.85 3.33 4.17 3.85
3.57 3.33
3.13 3.85 3.57
3.33 3.13 2.94
T - T iml 'm2
Tmi > 5Tm2 ;
Min lmum §„ = 58°
18 K (133) P ^ S V 1 ^ 7 - " * ) ' t p _ a s > t m
Am)/l + (3T<'33>coJ ' " T - >
118 Handbook of PI and PID Controller Tuning Rules
Rule
Schaedel (1997). Model: Method 1
Pomerleau and Poulin (2004).
Model: Method 5
Leva et ah (2003). Model: Method 1
Brambilla et ah (1990).
Model: Method 1
X values obtained from graph
Kc
19 K (134)
20 K (135)
Tml
K m (T m l +t n J Tml
2K m (T m 2 +T,J
Tm l+Tm 2+0.5Tm
Kmxm(2A + l)
Tm
'ml + T m 2
0.1
0.2
0.5
1.0
Tj
T (134) i
T (135) i
l-STml
4(Tm 2+xm)
Robust
Tm l+Tm 2+0.5Tm
Coefficient values
X
3.0
1.8
1.0
0.6
*m
Tm ,+T,
2.0
5.0
10.0
Comment
"Normal" design
"Sharp" design
Tm l=Tm 2 ;OS<10%
' m l >> Tm2 + Tm '
Symmetrical optimum principle
0.1 < Tm <10 Tmi + Tm2
n2 X
0.4
0.2
0.2
19 K (134) (134) 0.5T;
Ti°34) = V & m T m l + x , J -2(T ra l2 +2^mTm l tm +0.5T,,,2).
20 K (.35) _ 0.375 4Z,m'Tml* +2^mTmlTm +0.5T,/ - T m l2
Km Tm l2+24mTm lTm+0.5Tm
2
.(135) . 4 s m ' m l + ^ S m ' m l T m + ^ - 5 Tm ' m l
2smTmi + Tm
Chapter 3: Tuning Rules for PI Controllers 119
Rule
Krist iansson (2003) . Model: Method 1
Panda et al. (2004) . Model: Method 1
Kc
21 K (136)
22 K (137)
23 K (138)
25 K (139)
T i
2 ^ m T m l
j(138)
T(139)
C o m m e n t
Suggested M m a x = 1 . 7 , p = 10
24
21 K (136)
22 v (137)
2^mT„ a - la2 - 1 + -
M „ , a = l+ - 1 - 1
M „
K 2^raTral
Kmxra
1 + 0 . 1 9 1 3 ^ - - J o . 3 4 6 + 0.38261s!2-+ 0.0366-is2-
23 v (138) _ 8 ^ m T r o l + 2 ^ m T m + T r o l T ( n 8 ) _ 4 ^ r o T m l + 4 m T m + 0 - 5 T m l K „
X, > max imum
2MC
0 . 8 2 8 + 0 .812-2! 2 - + 0 .172T m l e
2 ^ m
-(69T„,2/Tml) , 7
0.828 + 0 . 8 1 2 ^ ^ + 0 .172T m l e -(6.9Tm;/Tml) , 0 -558T m 2 T m
25 v (139) _ K
A.K„ T m l + 1 . 2 0 8 T m 2 | 0.Sxn
T i(139)=0.828 + 0 . 8 1 2 ^ + 0.172Tm,e-(6-9T^ /T-'+ ° - 5 5 8 T ^ T " " + 0 . 5 t n
Tml Tm,+1.208Tm2
120 Handbook of PI and PID Controller Tuning Rules
3.7.2 Controller with set-point weighting
U(s) = Kc
v T i s y E ( s ) - a K c R ( s )
* a K „
+ T R(s)
E(s) * K ,
v T.sy + U(s)
K e"
T ^ V + l ^ T ^ s + l
Y(s)
Table 25: PI controller tuning rules - SOSPD model Kme"
Tml2s2+2i;mTmls + l
Rule
Taguchi and Araki (2000).
Model: Method I
Kc T i Comment
Minimum performance index: servo/regulator tuning
1 K (140) T (140) 5m = i ;
T m / X < 1 . 0 ;
Overshoot (servo step) < 20%
K (140)
K „ 0.3717 + -
0.5316
+ 0.0003414
•(140) 2.069-0.3692-S1- + 1.081 T
-0.5524
f ~ \ a = 0.6438 - 0.5056^2- + 0.3087
T T -0.1201
U ^ T
V X m J
Chapter 3: Tuning Rules for PI Controllers 121
Rule
Taguchi and Araki (2000) - continued
Minimum ITAE -Pecharroman (2000),
Pecharroman and Pagola (2000).
Model: Method 11
K m = 1 i T m i = 1 •>
Kc
2 ^ (141)
x,Ku
x i
0.147
0.170
0.1713
0.195
0.210
0.234
0.249
0.262
0.274
x2
1.150
1.013
1.0059
0.880
0.720
0.672
0.610
0.568
0.545
1
T ( 1 4 1 ) 1 i
x2 T
Coefficient values a
0.411
0.401
0.4002
0.386
0.342
0.345
0.323
0.308
0.291
i -146°
-140°
-139.7°
-133°
-125°
-115°
-105°
-94°
-84°
* i
0.280
0.291
0.297
0.303
0.307
0.317
0.324
0.320
x2
0.512
0.503
0.483
0.462
0.431
0.386
0.302
0.223
Comment
^ m =0.5
a
0.281
0.270
0.260
0.246
0.229
0.171
0.004
-0.204
4>c
-73°
-63°
-52°
-41°
-30°
-19°
-10°
- 6 °
2K <141> - -K „
0.1000 + -0.05627
- +0.06041]2
4.340-16.39^2-+ 30.04 T
-25.85
a = 0.6178 - 0.4439-^ - 7.575 T
f \ l
3 r - i4^
+ 8.567 - ^ T
3
T + 9.317
vTm
•3.182 T
V i m .
122 Handbook of PI and PID Controller Tuning Rules
3.8 SOSIPD Model - Repeated Pole s(l + Tmls)2
3.8.1 Controller with set-point weighting
U(s) = Kc ' .+-L' v T i s y
E(s)-aK cR(s)
aK
R(s) &
E(s) *K,
V T i S V x8>
U(s)
K „ e "
s(l + Tmls)2
Y(s)
•
Table 26: PI controller tuning rules - SOSIPD model Gm (s) = K e"
Rule
Taguchi and Araki (2000).
Model: Method 1
Minimum ITAE -Pecharroman (2000).
Model: Method 2
Km = 1; Tml = 1
0.1 <xm <10
Kc
Minimum
4
Xl
0.049
0.066
0.099
0.129
0.159
0.189
K c ( . 4 2 ,
x,Ku
x2
2.826
2.402
1.962
1.716
1.506
1.392
T;
s(l + TraIs)
Comment
performance index: servo/regulator tuning
8.549 + 4 . 0 2 9 ^ -Tm,
X 2 T u
^ / T m l < 1 . 0 ;
Overshoot (servo step) <20%
Coefficient values a
0.506
0.512
0.522
0.532
0.544
0.555
* c
-164°
-160°
-155°
-150°
-145°
-140°
x i
0.218
0.250
0.286
0.330
0.351
x2
1.279
1.216
1.127
1.114
1.093
a
0.564
0.573
0.578
0.579
0.577
4>c
-135°
-130°
-125°
-120°
-118°
K (142)
K „ 0.07368 + -
0.3840
- + 0.7640 , a = 0.6691+ 0.006606-
Chapter 3: Tuning Rules for PI Controllers 123
3.9 SOSPD Model with a Positive Zero Gm(s) = K m ( l -T m 3 s)e-^
(l + sTmlXl + sTm2)
3.9.1 Ideal controller Gc (s) = Kc ' i+-L^ v T i s y
R(s)
+ < ^
E(s) c i N\
K, 1+-T j S y
U(s) Km(l-Tm3s)e-
(l + TralsXl + sTm2)
Y(s)
— •
Table 27: PI controller tuning rules - SOSPD model with a positive zero
K m ( l - T m 3 s ) r ^ Gm(s) =
(l + sTmlXl + sTm2)
Rule
Pomerleau and Poulin (2004).
Model: Method 3
Kc Tj Comment
Direct synthesis: time domain criteria 1 Tml
KmTm l+T r a 3+Tm l-5Tml OS<10%;Tm l=Tm 2
124 Handbook of PI and PID Controller Tuning Rules
Rule
Khodabakhshian and Golbon (2004).
Model: Method 1
Luyben (2000). Model: Method 1
T m 3 / T m i v a l u e s
x, values
Kc ^ Comment
Direct synthesis: frequency domain criteria 5 K (143) T (143) Recommended
M r a a x =0dB;
Tmi > Tm2 Ultimate cycle
Ku
x i Km
0.2
3.9
3.8
4.1
4.3
4.8
5.2
5.5
6.0
0.4
3.3
3.4
3.6
3.8
4.3
4.6
5.0
5.4
6 Tu
T1
0.8
2.8
3.0
3.2
3.4
3.7
3.9
4.4
4.7
1.6
3.0
2.9
3.1
3.2
3.3
3.4
3.7
3.8
Tmi - Tm2
T ra/T ra3=0.2
^m /T r a 3=0.4
xm /T r a 3=0.6
xm /T r a 3=0.8
xm/Tm3=1.0
Tm/Tm3=1.2
x r a/Tm3=1.4 X J T m3 =1-6
5 K (143) 4 ' f f l 2 T,"43) J(T r o lTm 2)V+(Tm l
2+Tm 22)to
Km \ ( T m J m 3 ) 2
r a4 + ( T m , 2 + T m 3
2 > 2 + l with o> given by solving
•10 -0.1M„„ A
= 0.5TI + tan"1 (Ti<143)co)- tan"1 (Tm3co)- tan"1 (Tm2co)
-tan-'(Tmlco)-Tmco:
.(143) __J_ i T
1 + 0.175—S2- + 0.3| A m l
+ 0.2- -<2:
,(143) (T \
0.65 + 0.35—— + 0.3 VTml J
+ 0.2- ->2.
0.5 + 1 . 5 6 ^ - + 3.44-1.56-Tm3 ' Tm
Chapter 3: Tuning Rules for PI Controllers 125
3.10 SOSPD Model (repeated pole) with a Negative Zero Km(l + Tm3s)e-
Gm(s) : (l + sTml)
2
3.10.1 Ideal controller in series with a first order lag
Gc(s) = Kc 1 + -T|S y 1 + Tfs
R(s) ^ E ( s ) - •
Table 28: PI controller tuning rules - SOSPD model with a negative zero
Km(l + T r a 3 s>-^ Gm00 =
(l + sTml)2
Rule
Pomerleau and Poulin (2004).
Model: Method 3
Kc T i Comment
Direct synthesis: time domain criteria 1 Tml
K m T m l + T m l-5Tml Tf =Tm3;OS<10%
126 Handbook of PI and PID Controller Tuning Rules
3.11 Third Order System plus Time Delay Model
Gm(s) = Kr l + b,s + b2s2 +b3s3
l + a,s + a2s3 +a3s
Gm(s) =
'or
K „ e "
(l + sTmlXl + sTm2Xl + sTm3)
3.11.1 Ideal controller Gc(s) = Kc 1 + —
E(s)
R(s)
+ %-* ?_
Kc \ 0 1 + — I TisJ
U(s) Third order system
plus time delay model
Y(s)
Table 29: PI controller tuning rules - Third order system plus time delay model
Kme-"" „ , , „ l + b,s + b,s 2 +b-,s3 _,, _ , N
Gm(s) = Km ! 2~ ^ e st" or Gm(s) = l + a,s + a,s +a,s
(l + sTmlXl + sTm2Xl + sT m 3 ) -
Rule Kc T, Comment
Direct synthesis
Hougen (1979) -pages 350-351.
Model: Method 1
0.7
Km \Xm )
0.333
I K (H4)
T ( H 4 )
^ - > 0.04 ;
T m l - '•ml - T m 3
•^2- < 0.04 ; Tml
T m l > T m 2 > T m 3
1 K (144)
2Kn
0.7|i=i- + 0.8 Tmi + Tm2 + Tm3
vTmiTm2Tm3 r
Chapter 3: Tuning Rules for PI Controllers 127
Rule
Vrancic e/a/. (1996), Vrancic (1996)-
page 121. Model: Method 1
Vrancic et al. (2004a).
Model: Method 1
Vrancic et al. (2004b).
Kc
0.5A3 2
(A ,A2-K m A 3 )
0.5A3
( A , A 2 - K m A 3 )
0
4 R (146)
T;
A3
A2
3 x (145) i
A,A3
A[A2 - KmA3
T (146)
Comment
A m > 2 , 4 L > 6 0 0
(Vrancic et al. (1996))
Model: Method 1
2 A, = K r a ( a 1 - b 1 + x m ) , A 2 = K m ( b 2 - a 2 + A , a , - b , t m + 0 . 5 x m2 ) ,
A3 = Km(a3 - b 3 + A2a, - A,a2 +b2xm -0.5b,x ra2 +0.167xm
3).
A,A3(A,A2 - KmA3)) 3 T (145)
4 K (146)
(A,A2 - K m A 3 ) 2 +KmA 3 (A IA 2 - K m A 3 ) + 0.25Km2A3
2
A,A2 - A 3 K m -[sign(A,A2 - A3Km)]A,VA22 - A,A3
A , K m2 - 2 A , A , K m + A , 3
-(146) _ 2A, A, A2 - A3Km - [sign(A1 A2 - A3Km )]A, JA2
2 - A, A3
(A3Km2 - 2A,A2Km + A,3)(l + KC
(146)KJ2
12 8 Handbook of PI and PID Controller Tuning Rules
3.11.2 Controller with set-point weighting
( 1 "1 U(s) = Kc 1 + — V T i s J
aK„
R(s) E(s)
K, v T i sy
U(s)
E(s)-aKcR(s)
Y(s)
Third order system plus time delay model
Table 30: PI controller tuning rules - Third order system plus time delay model
O . W K • * b - + - - : + b - : . - - . , o . w . l + a,s + a2s3 +a3s3 l + sTmlXl + sTm2Xl + sTm3) '
Rule
Taguchi and Araki (2000).
Model: Method 1
Kc Ti Comment
Minimum performance index: servo/regulator tuning 5 K (147) T (147)
i ^2-<1.0 T
Overshoot (servo step) < 20% ; Tra] = Tm2 = Tm3
5 K (147)
K „ 0.2713 + -
0.7399
- + 0.5009 , a = 0.4908-0.2648-^ + 0.05159
T
f \2
T V m J
T (147) = T 2.759-0.003899-^2-+ 0.1354 T
Chapter 3: Tuning Rules for PI Controllers 129
Rule
Vrancic(1996)-page 136-137.
Model: Method 1
Kc T; Comment
Direct synthesis
6 K (148) T (148) i
b = [0.5,0.8]-good servo and regulator
response (page 139); b = l-oc
6 j r (148)
V^¥ A , A 2 - K r o A 3 - X K m A , - A , A , < 0 ;
n2 A 3 + A 1
3 - 2 K m A 1 A 2 j •~mni n \ n l
K (148)
¥^¥
X = V(KmA3 - A,A2)2 - ( l -b2]A3(K r a2A3 + A,3 - 2 K m A , A 2 ) ;
A , A 2 - K m A 3 + x K m A 3 - A , A , > 0 . 2 A , + A , 3 - 2 K „ A , A , ) ' ~ ™ - 3 — ' " 2
.(148)
K m + -
m rt3-r^i.| z . x v m ^ , r v 2
A,
1 K <148>K J with
2K (148) &-„')
• K . ( b ' ) . A, = K m ( a i - b , + T m j , A2 =K m ^b 2 - a 2 +A 1 a 1 -b 1 T r a +0 .5T m
A3 =Km(a3 - b 3 + A2a, - A,a2 +b2xm - 0 . 5 b , x j +0.167xm3).
130 Handbook of PI and PID Controller Tuning Rules
Kme-sx™ 3.12 Unstable FOLPD Model Gm(s)
T „ s - 1
3.12.1 Ideal controller Gc(s) = Kc
R(s)
1 + -TiSy
?) E ( S ) r i
Kc M I Ti sJ
U(s) K m e - -
T r a s -1
Y(s)
Table 31: PI controller tuning rules - unstable FOLPD model Gm (s) = -Kme-
Tms-1
Rule
Minimum ITSE -Majhi and Atherton
(2000). Model: Method 1 Minimum ITSE -
Majhi and Atherton (2000).
Model: Method 3
Kc
Minimum i
1 K (149)
2 K (150)
T, Comment
Derformaiice index: servo tuning
T (149) i
x ( I S O ) i
0<I™-< 0.693 T
0 < I s . < 0.693 ; Tm
K = - A p / K m h
1 v (149) _ _J_ K „
- ' A 0.889 +
- 0.064
v eT-/T™ - 0.990
2 K (150) J_ 0.889 + 1
K(I + K) T (150)
T(i49) _2.6316Tm(eT-/T- - 0 96e)
' (e- ' -^"-0.377)
TmK(l + K)
0.5(l-0.6382K)tanh_1 K '
Chapter 3: Tuning Rules for PI Controllers 131
Rule
De Paor and O'Malley (1989). Model: Method 1
Venkatashankar and Chidambaram (1994).
Model: Method 1 Chidambaram
(1995b). Model: Method 1
Chidambaram (1997).
Valentine and Chidambaram (1998).
Model: Method 1
Kc T; Comment
Direct synthesis
3 K (151)
4 K (152)
i f T ^ — 1 + 0.26-^-K „ A ^J
1.678, In
Km
2 .695 -1.277Tra/Tm
Km
T ( 1 5 1 ) i
25(Tm-xJ
25Tm-27xm
0.4015Tme5'8t'"/T"'
0.4015Tme58T"'/T"'
Ara = 2 ; ^ < l m
^S-<0.67 T
m
- ^ - < 0 . 6 T
m
Model: Method 1
5 ^ = 0.333;
^2-<0.7 T
K (151)
K „ cos
/ T m jT m \ t „
1--=- | sin T, T T
-(151) , <|> = t a n T V Tmy
4 ^ (152) = _
K m |
/ 0.98J1 +
0.04T^
( T m - x m ) 2
25 |p(Tm-xm)
1 + P2T2
l + P 2 ^ | ( T m - x J 2
B = 1.373, — < 0.25 ; B = 0.953, 0.25 < — < 0.67 .
5 +1% Ts = 7.5Tm , xm/Tm = 0.1; ±1% Ts = 8.75Tm, xm/Tm = 0.2 ;
±1% Ts = H.25Tm, xm/Tm =0 .3 ; ±1% Ts = 15.0Tm, xm/Tm =0.4;
±1% Ts= 17.5Tm , xm/Tm =0.5 ; ±1% Ts = 20.0Tm , xm/Tm =0.6.
132 Handbook of PI and PID Controller Tuning Rules
Rule
Ho and Xu (1998). Model: Method 1
Luyben(1998). Model: Method I
Gaikward and Chidambaram (2000).
Model: Method I Chidambaram
(2000c). Model: Method 1
Kc
» p T m
AmK.m
Ti 6 T ( 1 5 3 )
i
Comment
^ < 0.62 T
m
Representative results 0.5775Tm
K m T m
0.6918Tm
K r a T m
x ] T m / K m xm
T T
0.3212Tm-xm
T T
0 3359T - T
-T
Good servo response. Am = 2.3 ,
* m =25°
Good regulator response. Am = 1.9 ,
<L=22.5°
Other representative results: coefficient values x i
1.0472
0.7854
Am
1.5
2
0.31KU
K 30°
45°
X l
0.5236
0.3927
2.2TU
Am
3
4
*m
60°
67.5°
TCL = 3.16xm
Maximum closed loop log modulus = 2 dB
7 K (154)
8 ^ (155)
x (154)
T (155)
0.1<^2-<0.6 T
'small' ^ -Tm
6 j (153)
1.57cop-cop xm - —
7 v (154) _ _J_ Km
K „ 0.2619 + 0.7266-^- ,(154) = T 0.193 + 4.19—2-4-8.214 T
r \1
T
8 K (155) -(155)
0.04TS242+Ti(155)tr
- T-(155) _ 0 .04T S
2 ^ 2+ T m t m + 0 .4T S ^
Chapter 3: Tuning Rules for PI Controllers 133
Rule
'Chandrashekar et al. (2002).
Model: Method 1
Coefficients of
Kc and Tj recorded
in tables; TCL2 as in
footnote
Sree et al. (2004). Model: Method 1
Rotstein and Lewin (1991).
Model: Method 1
Kc
49 .535 e-21.644xm/Tm
Km
/• \ -0.8288
0.8668 f x j
Km tTmJ
Ti
0.1523Tme7'9425T">^
Comment
i s - < 0.1 T
tn
0.1 < i s - < 0.7 T
Alternative tuning rules 6.0000/Km
3.2222/Km
2.2963/Km
1.8333/Km
1.5556/Km
1.3265/Kra
1.1875/Kn
0-8624 f x j
Km | j m J
-0.9744
0.6000Tm
1.4500Tm
2.6571Tm
4.4000Tm
7.0000Tm
14.6250Tm
31.0330Tm
10 T (156)
xm /Tm=0.1
Tm /Tm=0.2
xm /Tm=0.3
x m /T m =0.4
* m /T m =0.5
t m / T m = 0 . 6
Xm /Tm=0.7
0.01 < i a - < 0.6 T
m
Robust
Tra^
^2Km
Km uncertainty =
50%
X
^m/T r a =0.2
A. obtained graphically - sample
values below
Ae[0.6Tm,1.9Tm]
1 Desired closed loop transfer function = fas+!>-"- , (TCL2S + 0 2 '
0.025 + 1.75- T m , i 2 . < 0 . 1 ; T C L 2 = 2 x m , 0 . 1 < i 2 . < 0 . 5 ;
T C L 2=2x m +5T m is-_o.5 , 0.5<is-<0.7. T T
10 J (156) _ rj, 143.34 -73.912 + 19.039 - 2 - -0.2276 T
134 Handbook of PI and PID Controller Tuning Rules
Rule
Rotstein and Lewin (1991) - continued
Tan et al. (1999c). Model: Method 1
Kc
Km uncertainty =
30%
11 K (157)
Ti
*m /Tm =0.2
*m /Tm =0.4
^m /Tm =0.6
T ( 1 5 7 )
Comment
^G[0.5Tm,4.5Tm]
^e[1.5T r a ,4.5TJ
^6[3.9Tm,4.1Tra]
^*-<0.5 T
K (157) , ( 1 5 7 )
K „ 0.1486-^ + 0.6564 T
12.7708—2- + 3.8019 T
0.1486-2!- +0.6564 T
Chapter 3: Tuning Rules for PI Controllers
3.12.2 Controller with set-point weighting
135
E(s)-aKcR(s)
Table 32: PI controller tuning rules - unstable FOLPD model Gm (s) = Kme-
T m s -1
Rule
Chidambaram (2000c).
Model: Method 1
Jhunjhunwala and Chidambaram (2001).
Model: Method 1
Kc T, Comment
Direct synthesis 12 K (158)
13 2 . 6 9 5 -1.277tm/Tm
Kra
14 K (159)
f \ -0.898
0 . 8 7 6 ^
T (158) i
0.4015Tme5-8x™/T'"
T (159) _ i
6.461tm
0.324Tme T"
'small' xm/Tm
'Dominant pole placement' method;
E, = 0.333
^2_<0.2 T
m
0.2 < -is- < 0.6 T
m
12 j r (158) -(158)
O.O^s^HTi"58 '!,, • . T i
(.58) 0 .04T s ^ 2 +T m T m + 0.4T s 4 2
<x = l-0.2T s4
2(Tm-Tm)
0.04TS2^2+Tmxm+0.4TS^
a = 0.733+ 0.6^2--0.36 T
r \2
T V ra J
0.1 < - 2 - < 0.6. T
14 v (159) _ 1 K„ 10.7572-35.
K c " " ' K m ( 2 - T j + ( K c ' l " ' K m - l ) r i " ^
2K<159)K
136 Handbook of PI and PID Controller Tuning Rules
Rule
15 Chandrashekar et al. (2002).
Model: Method I
Coefficients of Kc and Tf recorded
in tables; TCL2 as in
footnote
Hagglund and Astrom (2002).
Model: Method 1; M, =1.4;
0 .3<a<0.5
Kc
4 9 . 5 3 5 -21.644Tm/Tm
e ra' m
Kra
0.8668 f O
Km [ T J
Kc
6.0000/Km
3.2222/Km
2.2963/Km
1.8333/Kra
1.5556/Km
1.3265/Km
1.1875/Km
0.28/Kml
0.2/KmT
0.28/KJ
0.13/KJ
0.28/KJ
0.28/KJ
-0.8288
Ti
0.1523Tme7'9425t"/T™
Comment
i s - < 0 . 1 T
m
0.1 < -^s- < 0.7 T
Alternative tuning rules
Ti
0.6000Tm
1.4500Tra
2.6571Tm
4.4000Tm
7.0000Tra
14.6250Tra
31.0330Tm
m
m
m
m
m
m
a
0.6667
0.7246
0.7742
0.8182
0.8571
0.8974
0.9323
8T m
7.8xm
l l ^ m
10tm
7tm 19tm
135tm
^ m /T m =0.1
^ m /T m =0.2
xm /Tm=0.3
i m / T m = 0 . 4
t m /T m =0 .5
^ m /T m =0.6
t m / T m = 0 . 7
•^2- - 0.02 T
m
i ^ - 0 . 0 5 T
i s - - o . l O Tm
^m /Tm=0.15
15 Desired closed loop transfer function = — — (TCLZS + I ) 2
TCL2 = 0.025 + 1.75- T m , ^ < 0 . 1 ; T C L 2 = 2 x m , 0 . 1 < ^ < 0 . 5 ;
T C L 2 = 2 t m + 5 x m | ^ - 0 . 5 , 0 . 5 < - ^ < 0 . 7 . T
a = 0.505 + 3 . 4 2 6 ^ -10.57 ^ T T
• < 0 . 2 :
a = 0.713+ 0.232-^-, 0 .2<i3-<0.7 T T
Chapter 3: Tuning Rules for PI Controllers 137
3.12.3 Controller with proportional term acting on the output 1
U(s) = ^ E ( s ) - K c Y ( s ) T:S
+ d r - E(S)
Kc
T i s
^6?V——fc.
- X u(s)
t
Kme-St™
T m s - 1
Y(s)
•
Table 33: PI controller tuning rules - unstable FOLPD model Gm (s) = Kme"
Tms-1
Rule
Minimum ISE -Paraskevopoulos et
al. (2004). Model: Method 1
Kc T; Comment
Minimum performance index
1 K (160) i . 9 i 6 , m +4 - ; ^ 2
•'rn
0.88--^-T
2.725xra + L ? ^ 2
0 . 8 8 - ^ T
m
0.05 < ^ - < 0.85; T
regulator tuning
0.05 < ^ s - < 0.85 ; T
servo tuning
£ (160) _ / ] £ (min)j^
with K (min)
K „
m m i n T i ) l + m m i n 2 T m 2 R (max) _ fi>maxTi ^ " m a x ^ m 2 ^
l + m m i „ T i K „ 1 + w m a x 2 T i 2
T T +T . ™n
K~
2xm
T rt
T i ^m
T i + ^ m T m j
T i ^
138 Handbook of PI and PID Controller Tuning Rules
Rule
2 Paraskevopoulos et al. (2004).
Model: Method 1
f Minimise
iraskevopoulos al. (2004).
dt
et
Paraskevopoulos et al. (2004).
Model: Method I
Kc
Minimum |
1 K (160)
(page 137)
1 j , (160)
(page 137)
1 K (160)
(page 137)
X l
0.7269 1.7016 3.0696
x2
2.5362 3.0585 3.7509
Ts Comment
performance index: other tuning 3 T ( 1 6 0 )
i
3 r-p (161) i
2.58,m + 3 - 6 V
T
0 . 8 8 - ^ -m
Direct synthesis - j . (min) 4
i
. l-35xm
V m J
X l ^ m + X 2
2
VTmTm
Coefficient values
4 0.5 0.75
1
x i
6.975 12.452
0.01 < — < 0.3 T
m
0.3 < i s - < 0.85 T
0.03 < ^ - < 0.85; T
m
Model: Method 1
0.02 <^S-< 0.7 T
m
0 . 1 < ^ - < 0 . 9 T
i s - < 0.3 T
x2
5.7053 8.4358
S 1.5 2
: Minimise performance index = l[e2(t) + (u ( t ) -u„ ) 2 jit
3 j (160) _ ~ 0.747 + 3 . 6 6 - ^ + 18.64
T
f \ 3
T V m J
. T ( 1 6 1 ) _ _
1 (% \ 4.52-!2- +0.476 - ^
T T
0.88-
Ti(min) =Tm [1.3146-^- - 2 . 3 8 0 5 ^
( - \ + 57.622
T V m J
( . -158.99
T
+ 173.41 ' ^
T
Chapter 3: Tuning Rules for PI Controllers 139
Rule
Paraskevopoulos et al. (2004) -continued.
Paraskevopoulos et al. (2004).
Model: Method 1
Zhang and Xu (2002). Model: Method 1
Kc 1 K (160)
(page 137)
Ti
Xi t m +x 2 T m + 6 T -5
X 3 T m 1 m
Comment
0.2 < ^2- < 0.7 T
Coefficient values X l
4.59 6.16 8.21
x2
-0.19 0.028 0.37
X 3
8.28 9.90 16.88
1 K (160)
(page 137)
I 0.5
0.75 1
x i
12.41 19.91
x2
1.89 3.44
5 T (162)
X 3
51.95 95
4 1.5 2
^2L<0.2 T
Coefficient values x i
0.409 0.717 1.142
x2
0.2817 0.6542 1.177
1 K (160)
(page 137)
TcL2 0.5 0.75
1
T
x i
2.353 4.07 8.88
x, +x 3
r \
T
3~
x2
2.67 4.76 10.73
TcL2 1.5 2 3
0.2 < ^ 2 . < 0.7 T
m
Coefficient values x i
1.518 3.34 5.81
x2
10.54 16.89 29.17
TcL2 0.5 0.75
1
x i
13 23.08 51.94
x 2
49.66 82.87 176.8
*CL2
1.5 2 3
Robust
6 K (163) T (163) ^ < Tmi i for
^2 -<0 .5 , T
X. = 2 -5x m
• (162) = T „
T
6 v (163) ^ + 2XTm + T m T m (163) i V ; T-, > ^i
^+2XTm+TmTm
K ^ + T J
140 Handbook of PI and PID Controller Tuning Rules
3.12.4 Controller with proportional term acting on the output 2
E ( s ) - K , Y ( s )
Table 34: PI controller tuning rules - unstable FOLPD model Gm(s) = Kme-Tms-1
Rule
Chidambaram (1997). Model: Method 1
Lee and Edgar (2002).
Model: Method 1
Kc T, Comment
Robust T m - t m
8 K (165)
7 j (164) i
T (165)
X > 1 . 7 ; i s - < 1 T
m
7 j (164) _
8 K (165)
-p (165)
(T ' T - - 5 5 -
1 111
0.5
K / l 6 5 , K m - l
Km(X + Tm)
T m - K , ( I 6 5 ) K
n ^T v 1 .
1 v . - , . m , x M
L Kr>Km-i 2
T T
K , ( 1 6 5 ) K m - l
K (165) 1
K m
2(X
C ° S \
+ ' J '
T T V m J * m
/ „ \0 .5
x m2 1
2(^ + TJ_
1
'
T m , H i
Chapter 3: Tuning Rules for PI Controllers
3.13 Unstable FOLPD Model with a Positive Zero
Gm(s) =
141
Km(l-sTm3)e-
T m l S - 1
3.13.1 Ideal controller Gc (s) = K
E(s)
1 + -Ti^y
R(s) <g)— + T
f i >\
K, 1
V T i S V
1 + -U(s)
K r a ( l - sT m 3 ) e -
T m l s - 1
Y(s)
Table 35: PI controller tuning rules - unstable FOLPD model with a positive zero •
K m ( l - s T m 3 > - " -Gm (s) = ~ :
Rule
Sree and Chidambaram
(2003c). Model: Method 1
Kc Ti Comment
Direct synthesis: time domain criteria
1 Y T ml
K r a T r a 3 T (166)
y = the negative of the value of the inverse jump of the closed loop system; y > —I! -Tmi
YTm -(166) _ i m 3
-iTm3(l-x,)-0.5Tm(l + x,)]
^ - ( l - x j + x , , X , >
YTrol
YTml-Tm3
142 Handbook of PI and PID Controller Tuning Rules
3.13.2 Ideal controller in series with a first order lag
G c ( s ) = Kc 1 + — V T>SV 1 + T fs
R(s) E(s)
+ T
U(s)
K, 1 + v T i s y
l
1 + T fs K m ( l - s T m 3 ) e -
T m , s - 1
Y(s)
Table 36: PI controller tuning rules - unstable FOLPD model with a positive zero •
Rule
Sree and Chidambaram
(2003b). Model: Method 1
Kc T; Comment
Direct synthesis: time domain criteria
2 K (167) T (167) ISL. - JEHL. < o.62 Tmi Tm3
( l - s T )(l + T.(167)s)e~ 2 Desired closed loop transfer function = -—*—^-^——-1 EL, (I + T&SJ(I + T&S) '
K (167) .(167)
K r a lT( 'L>2 +T<2L)2 +Tm
, (167) )•
. (167) _ Tml jTCL2TCL2 ~ Tm3Tm + [TCL2 + TCL2 + Tm3 + Xm Fml j Tml2+T r a3Xm-(-C r a+Tm 3)Tm l
T _ T(167) 1m3Tmxi
T (T(l) + T ( 2 ) + T + T _ T ( 1 6 7 ) V iml^CLJ + XCL2+ 1 m 3 + T m li )
Tf =
Chapter 3: Tuning Rules for PI Controllers
3.13.3 Controller with set-point weighting
U(s) = Kc
v T i s y
143
E(s)-aK cR(s)
aK„
R(s) + T - E(s) Kr
v T i sy
U(s)
v8» K m ( l - s T m 3 ) e -
T m l s - 1
Y(s)
Table 37: PI controller tuning rules - unstable FOLPD model with a positive zero •
K m ( l - sT m 3 ) r "» Gm(s) = -
Tm ls-1
Rule
Sree and Chidambaram
(2003b). Model: Method 1
Kc Ti Comment
Direct synthesis: time domain criteria
3 K (168) ry (168) -^s—-^2-< 0.62 Tmi Tm3
(1_ST )(l + T.(168)sVst 3 Desired closed loop transfer function = -—*—m3?\ y '—,„; \
(l + T&sJ l + T&s)
K (168) .(168)
K ^ l (i) , T < 2 ) 'TCL2 + T C L 2 + T m 3 + T m ~ *i
. (168)
. (168)
T, =
T ( T ( D T ( 2 ) _ T T , | T ( l ) + T ( 2 ) + T + t ( r ) 1 m U 1 C L 2 1 C L 2 1 m 3 1 : m + I ^CI^ + XCL2 + i m 3 + xm\lm\]
f = :a
Tmi + T m 3 t m - ( x m + T m 3 J T m |
T T ( 1 6 8 )
a = 1-0.5 IT") + T ( 2 ) + T + T _ T ( 1 6 8 )
i 1 C L 2 + ^CL2 + im7,+Xm l\ )
T 4- T^1) 4- T ( 2 ) i m 3 + XCL2 + XCL2 ,(168)
V
144 Handbook of PI and PID Controller Tuning Rules
Rule
Sree and Chidambaram
(2003c). Model: Method 1
Kc
4 yTm ,
KraTra3
T|
T (169)
Comment
4 y = the negative of the value of the inverse jump of the closed loop system;
^ - [ T m 3 ( l - x 1 ) - 0 . 5 T m ( l + x1)] > T m 3 ^, (169) _ T n l 3
_ T m 1 ' ' " ^ - ( l - x j + x , 1 m 3
yTm
yT m l -T m
ct = l -0 .5 1 m 3 "*" XCL2 "*" 1 CL2
T <169)
Desired closed loop transfer function = (l-sTn3)(l4.T,»»),)b-".
(Sree and Chidambaram (2003b)).
Chapter 3: Tuning Rules for PI Controllers
3.14 Unstable SOSPD Model (one unstable pole)
Gm(s) =
145
K e"
(Tmls-l)(l + sTm2)
3.14.1 Ideal controller Gc(s) = K(
R(s) . o . E(s)
1 + -^J
K e ~ (Tmls-lXl + Tm2s)
Y(s)
Table 38: PI controller tuning rules - unstable SOSPD model (one unstable pole)
Kme"s t" Gm(s) =
(Tmls-lXl + sTm2)
Rule
Minimum ITAE -Poulin and Pomerleau
(1996). Model: Method 1
Process output step load disturbance
Process input step load disturbance
Kc Ti Comment
Minimum performance index: regulator tuning 1 K (170)
Tm/Tm
0.05 0.10 0.15 0.20 0.25
x i
0.9479 1.0799 1.2013 1.3485 1.4905
4Tm l(Tm+Tm 2) X l T m l - 4 ( T m + T m 2 )
x2
2.3546 2.4111 2.4646 2.5318 2.5992
tm/Tm 0.30 0.35 0.40 0.45 0.50
Coefficients of KC,T;
deduced from graph
x i
1.6163 1.7650 1.9139 2.0658 2.2080
x2
2.6612 2.7368 2.8161 2.9004 2.9826
0.05 0.10 0.15 0.20 0.25
1.1075 1.2013 1.3132 1.4384 1.5698
2.4230 2.4646 2.5154 2.5742 2.6381
0.25 0.35 0.40 0.45 0.50
1.6943 1.8161 1.9658 2.1022 2.2379
2.7007 2.7637 2.8445 2.9210 3.0003
X 1 ! „ .
K m ( x l T r r
4(Tm+Tm2
-4 [xm+T^
146 Handbook of PI and PID Controller Tuning Rules
Rule
Pouline/a/. (1996). Model: Method 1
McMillan (1984). Model: Method 1
Kc
<°Tml
3.53
K r a
3 K (172)
T;
Direct synthesis 2 T ( 1 7 1 )
i
Tml
Ult imate cycle T(172)
Comment
minimum i?m = 25
T m 2 + t m <0.16TmI;
<t>me[25°,90°]
Tuning rules developed from
KU>TU
2 x( '71) 4Tm l (Tm+T r o 2 )
1 .3T m l -4 (T m + T m 2 )
where phase of Gm(jco)Gc(j(n) is maximum
. 0.16Tm, < Tm2 + xm < 0.3Tml ; co = frequency
3 £ (172) 72) 1.477 T^T K m Xm
2
1 + 0.65
l ^ ro l + ^m2/^ml ^m2
(Tml _ Tm2 XTral _ Tm i1!! _
Umi + T m 2 j T m l T m 2 T:(172) = 3.33xj l + lTml T m 2 j ( T m l - T m JT r
Chapter 3: Tuning Rules for PI Controllers 147
3.15 Unstable SOSPD Model with a Positive Zero
Gm(s) _ Km( l-Tm 3s>-
2 „ 2 T m l V - 2 2 ; m T m l s + l
3.15.1 Ideal controller G c ( s ) = K
R(s)
1+ —
+ 3 E ( s ) > f- Kc
U(s) Km( l -Tm 3s)e-^
Tml2s2-2^mTmls + l
Y(s)
— •
Table 39: PI controller tuning rules - unstable SOSPD model with a positive zero
K m ( l -T m 3 s ) e -^ G n ( s ) =
Tm l2s 2 -2^mTm ls + l
Rule
Sree and Chidambaram (2004).
Model: Method 1
Kc Ti Comment
Direct synthesis: time domain criteria
4 R (173) T (173)
( l - sT yi + T('73)s)e"STm
4 Desired closed loop transfer function = -f —^ 7-7-17 r . (l + s T ^ l + s T ' ^ l + sx,)
(173)
J (173) _ Tml \TCL + TCL + X l + Tm3 + Tm j ~ TCLTCL X l ^ ^
I i r ^ m ~~ ^ml
x T m , (T^+Ta ) +T m 3 +T m ) ( -2^ m T m 3T m +T m l (T 3 + T m ) ) - x 2
X ' (- 2^raTml + Tra3 + xm Jr^Tg, ' - Tml2 + (Tra3xm - Tm l
2)(Tg + T<2> + 2^mTm ]) '
and with x2 = (Tm3xm - T ^ ) ( T # T & > -TmTm 3 ) ; 2^m
V Tmi < 0.62 .
1m3 /
148 Handbook of PI and PID Controller Tuning Rules
3.15.2 Controller with set-point weighting
U(s) = Kt
a K v T ; s y
E(s)-aK cR(s)
+ R(s)
®— E(s)
K, v T i s y
• < g ^ U(s)
K m ( l - s T m 3 ) e - " -
T m l2 s 2 - 2 ^ m T m l s + l
Y(s)
Table 40: PI controller tuning rules — unstable SOSPD model with a positive zero
K m ( l - T m 3 s ) r ^ Gm(s) =
T m ,V-25 r a T m I s + l
Rule
Sree and Chidambaram (2004).
Model: Method 1
Kc T; Comment
Direct synthesis: time domain criteria
5 K (174) T (174) i
1 Desired closed loop transfer function = (l + s T ^ l + s T ^ l + sx , ) '
K (174) _
-(174)
K ^ | T S + T g . ) + x 1 + T i n 3 + T i n - T i (174)
'ml T m 3 y
< 0.62 ;
T 2 ( T ( 1 ) +T < 2 ) + X 4-T + T 1 - T ( I ) T ( 2 ) I •p (174) _ ' m l VXCL + JCL + X 1 + ^ 3 +Xm) XCL XCL X l ^
*m3 X m — ' m l
., T m , ( T ^ + T g + T m 3 + T m ) ( - 2 ^ T m 3 T m + T m l ( T + T J ) - X 2
(- 2^mTml + Tm3 + xm )r^>Tg.> - Tml2 + (Tm3xm - Tml
2 )(T<'L> + T<2L> + 2^mTm l) '
a n d w i t h x 2 4 m 3 T m - T m , 2 f e ^ ^ ^
Chapter 3: Tuning Rules for PI Controllers 149
3.16 Delay Model Gm(s) = Kme
3.16.1 Ideal controller Gc(s) = K
R(s)
'i+-L^ v T i s y
Table 41: PI controller tuning rules - G m (s) = K me
Rule
Callender et al. (1935/6).
Model: Method 1
Two constraints method - Wolfe
(1951). Model: Method 1
Haalman(1965)-minimum ISE.
Model: Method 1 Gerry (1998)-
minimum error: step load change.
Gerry and Hansen (1987) - minimum
IAE.
Kc Ti Comment
Process reaction '0.568/KmTm
20.65/Kraxm
30.79/KmTm
40.95/Kmtm
0.2 K m T m
3.64xm
2.6xm
3.95im
3.3xm
0.3im
Representative results deduced from graphs
Decay ratio is as small as possible
Minimum error integral (regulator mode)
Minimum performance index: regulator tuning 0 l-5xm
M , = 1 . 9 ; A m = 2 . 3 6 ; i)m = 50°
0.3
Km
0.345
Km
0.42tm
0.455im
Model: Method I
Model: Method 1
Decay ratio = 0.015; Period of decaying oscillation = 10.5xn
2 Decay ratio = 0.1; Period of decaying oscillation = 8xm
3 Decay ratio = 0.12; Period of decaying oscillation = 6.28xm
4 Decay ratio = 0.33; Period of decaying oscillation = 5.56im
150 Handbook of PI and PID Controller Tuning Rules
Rule
Shinskey (1988)-minimum IAE - page
123. Shinskey (1994)-
minimum IAE. Model: Method 1 Shinskey (1988)-
minimum IAE -page 148.
Shinskey (1996)-minimum IAE -page
167.
Minimum IAE -Huang and Jeng
(2002). Model: Method 1 Minimum ISE -
Keviczky and Csaki (1973)
Model: Method 1
Fertik(1975). Model: Method I
Astrom and Hagglund (2000).
Model: Method 1
Skogestad (2001). Model: Method 1 Skogestad (2003). Model: Method 1 Skogestad (2001). Model: Method 1
Astrom and Hagglund (2004).
Model: Method 1
Kc
0.43
Km
0-4/Km
0
0.4255
Km
0.4KU
^
0.5xra
0.5xm
1.6Kmxm
0.25TU
0.25TU
Comment
Model: Method 1
page 67
page 63
Model: Method 1
Model: Method 1
Minimum performance index: servo tuning
0.36
Km
0
0.5
Minimum i 0.35/Km
0.25
Km
0.16/Km
0.200/Km
0.26/Km
x,/Km
0.47xm
1.33xm
0.635xm
Minimum IAE = 1.377xm. A r a = 2 ;
<L=60° .
Minimum ISE = 1.53xm; K r a = l
K m = l
jeri'ormance index: other tuning 0-4xm
0.35xm
0.340xra
0.318xm
0.306xm
x2 tm
Kc,Tj values
deduced from graph Maximise
Kc/T i subject to
M m a x =2
Mm a x=1.4
Mm a x=1.6
Mm a x=2.0
Coefficient values
* i
0.057 0.103
0.139
0.168
0.191
x2
0.400 0.389
0.376
0.363
0.352
Mmax
1.1 1.2
1.3
1.4
1.5
*1
0.211 0.227
0.241
0.254
0.264
x2
0.342 0.334
0.326
0.320 0.314
M max
1.6 1.7
1.8
1.9 2.0
Chapter 3: Tuning Rules for PI Controllers 151
Rule
Van der Grinten (1963).
Model: Method 1
Buckley (1964). Model: Method 1
Bryant et al. (1973). Model: Method 1
Keviczky and Csaki (1973).
Model: Method 1; Representative Kc,Ti values
estimated from graphs; Km = 1
Kamimura et al. (1994).
Model: Method 1
Hansen (2000). Model: Method 1
Bequette (2003) -page 300.
Skogestad (2003). Model: Method 1
Skogestad (2003).
Kc Ti Comment
Direct synthesis 0-5/Km
5 K (175)
0
0.075/Km
0.53/Km
0.55/Km
0
0
0
0.5xm
T (175) i
l-45Kmxm
0-Hm
T m
1.59im
2-70Kmxm
2-50Kmim
X2^m
Step disturbance
Stochastic disturbance Mm a x=2dB
Representative results;
Mm a x=2dB
Critically damped dominant pole
Damping factor (dominant pole) = 0.6
Coefficient values x2
0.64
OS
105%
x2
1.00 x i
OS
50%
x2
1.5
OS
17%
2xm
x2
2
OS
4%
Coefficient values x>
0.68
OS
5%
x i
0.72 0
0
0
0-2/Km
OS
10%
x! 0.75
OS
15% 2Km tm
2.6667Kmtm
2.3529Kmxm
0.3tm
x i
0.78
OS
20% OS = 10%
OS = 0%
"Quick" servo response; "negligible"
overshoot
Robust
V Km(2^ + xm)
0.294/Km
0.5xm
0.5xm
Model: Method 1
IMC based rule
Other tuning 0.25/Km 0.333xm Model: Method 1
(•75) = e - m ^ A p s e - ^ J T (175) = _ L s _ (t _ 0.5e-»^»).
152 Handbook of PI and PID Controller Tuning Rules
3.17 General Model with a Repeated Pole
/ 3.17.1 Ideal controller Gc (s) = Kc
1
R(s) <S>
E(s)
+ T K,
f 1 ^ 1 + -
T;s
1 +
U(s) Kme"
d + sTm)"
Y(s)
Table 42: PI controller tuning rules - general model with a repeated pole
K m e - n -Gm(s) =
d + sTm)n
Rule
Schaedel (1997). Model: Method 2
Kc T i Comment
Direct synthesis: frequency domain criteria 0.5
Km[r
0.375
Km
. 0 5 - l J "n + l"
_ n - l .
x +T m ar
(n + l)(Tm+Tar)
2n
'Normal' design
'Sharp' design
Chapter 3: Tuning Rules for PI Controllers 153
3.18 General Model with Integrator
n ( Tm , i s + i ) n ( T
m , 2 s 2 + 2 ^ i Tm 2 i S + i )
Gm(s) = K,
s n(Tm3i s+ i)n(T
m/ s2+2^diTm4is+i)e
3.18.1 Ideal controller Gc (s) = K
E(s)
1 + -T i s y
Table 43: PI controller tuning rules - general model with integrator
n(T^ s + in(T- / s 2 + 2^ i T^ s + i) Gm(s) =
K „
s n (T->'S+^n t v + 2 ^ « . . i T m , i s + ! ) Rule
Loron(1997). Model: Method 1
Kc Ti Comment
Direct synthesis 6 K (176)
K c ( 1 7 7 ,
a J Q
Symmetrical optimum method Non-symmetrical optimum method
6 K (176)
KmVo7TQ
l + sin<t
cos<|>n
T Q = ^ T m j i + 2 ^ ^ m d i T m 4 i - ^ T m | i + 2 ^ ^ m n i T , + T „
K (177) X =
KmV a2TQ
M „
M / - 1 , Mc = desired closed loop resonant peak,
1 + SlnA^l .A^cos^A).
Chapter 4
Tuning Rules for PID Controllers
4.1 FOLPD Model Gm(s) = K e~
l + sT„
/ 4.1.1 Ideal controller Gc(s) = Ki
1 \
V T i s J
R(s) ,o E ( s )
Table 44: PID controller tuning rules - FOLPD model G m (s) = Kme~
l + sT„.
Rule
Callender et al. (1935/6).
Model: Method 1 Ziegler and
Nichols (1942). Model: Method 2
Kc Ti Td Comment
Process reaction
i 1.066 K m T m
aTm
KmXm
a s [1.2,2]
1.418xm
2 T m
0.353xra or
0.47xra
0.5xm
^ - = 0.3 T
Quarter decay ratio
Decay ratio = 0.043; Period of decaying oscillation = 6.28xn
154
Chapter 4: Tuning Rules for PID Controllers 155
Rule
Chien et al. (1952)-regulator.
Model: Method 2
Chien et al. (1952)-servo.
Model: Method 2
Cohen and Coon (1953).
Model: Method 2
Three constraints method - Murrill
(1967) -page 356.
Model: Method 5
Astrom and Hagglund(1988) -pages 120-126.
Parr (1989)-page 194.
Borresen and Grindal (1990).
Kc
0.95Tm
K m t m
l-2Tm
Kmxm
0.6Tm
K m x m
0.95Tm
1 K (178)
1.370
Km —
l T nJ
Ti
2.38xm
2xra
Tm
1.36Tm
T ( 1 7 8 ) i
T
1.351
r x 0.738
x m J
Td
0.42xm
0.42xm
0.5xra
0.47xm
0.37tm
1 + 0 . 1 9 ^ -T
m
2 T (179)
Comment
0% overshoot;
0.1<^2-<1 T
m
20% overshoot;
0 . 1 < ^ - < 1 T
m
0% overshoot;
0 . 1 < ^ - < 1 T
20% overshoot;
0 .1<-^L<1 T
Quarter decay ratio.
0 < ^ - < l Tm
KcKraTd = Q
T
0 .1<^-<1 T
m Quarter decay ratio; minimum integral error (servo mode). 3
Km
^90% 0.5Tm Model: Method 23
T90% = 90% closed loop step response time (Leeds and Northrup
Electromax V controller). 1.25Tm
K m t m
Tm
2.5xn
3^m
0.4tm
0.5tm
Model: Method 2
Model: Method 2
K (178)
K „ 1.35—!2- + 0.25
,(178)
2 . 5 - ^ + 0.46 T
rx ^
T \ m /
1 + 0.61-
2 ^ ( 1 7 9 ) = 0 3 6 5 T I I
156 Handbook of PI and PID Controller Tuning Rules
Rule
Sain and Ozgen (1992).
Conne l l (1996) -page 214.
Model: Method 2
Hay ( 1 9 9 8 ) -pages 197.198.
Model: Method 1
Coefficients of
K c ,T j ,T d
deduced from graphs
Moros (1999). Model: Method 1
Liptak(2001).
ControlSoft Inc.
(2005).
Kc
3 K (180)
l-6Tm
K m T n i
x , / K m
T
0.125
0.167
0.25
0.5
x i
6.5
5.0
3.5
1.8
IlIL T
0.125
0.167
0.25
0.5
x i
9.8
7.2
4.5
2.2
l-2Tm
K m x m
1.2Tm
Kmx r a
0.85Tra
K m x m
2
K m
Ti
T (180) i
1.6667xm
x 2 T m
Td
T (180) *d
0-4Tm
X 3 t m
Coefficient values - servo tuning
x 2
9.0
6.5
4.2
2.2
x 3
0.47
0.45
0.40
0.35
Coefficient values
x2
2.0
1.8
1.6
1.4
x 3
0.52
0.50
0.45
0.35
2 t m
2^m
l-6xm
^ m + T m
T
0.125
0.167
0.25
0.5
x l
5.0
3.7
2.5
1.1
- regulator tuning
T
0.1
0.125
0.167
0.25
0.5
x i
10.0
8.0
6.0
4.0
2.0
0.42Tm
0.44xm
0.6xm
4 T (181) : d
Comment
Model: Method 24
Approximate quarter decay
ratio
x 2
9.8
7.0
4.8
2.5
x 3
0.34
0.32
0.30
0.27
x 2
2.2
2.2
2
1.8
1.4
x 3
0.35
0.35
0.34
0.32
0.28 attributed to
Oppelt
attributed to Rosenberg
Model: Method 2
Model: Method 10
K „ 0.6939-=!-+ 0.1814 | , T,
(180) 0.8647Tm +0 .226T„
^ + 0.8647
. (180) 0.0565T^
4 -T, (181)
Ty ' =max
Tm Tm
0.8647^-+ 0.226
(slow loop); Td =min tm Tm (fast loop).
Chapter 4: Tuning Rules for PID Controllers 157
Rule
Minimum IAE -Murrill(1967)-pages 358-363.
Minimum IAE -Marlin(1995)-pages 301-307.
Model: Method 8
Minimum IAE -Peng and Wu
(2000). Model:
Method 17
Minimum IAE -Pessen (1994).
Model: Method 1 Modified
minimum IAE -Cheng and Hung
(1985).
Kc Ti Td Comment
Minimum performance index: regulator tuning
1.435
Km —]
l-3/Km
l-0/Km
0.85/Km
0.65/Km
0.45/Km
0-4/Km
T i ^ 0 • 7 4 9
0.878 [xm J
0.7xm
0.68xm
0.65xra
0.65xm
0.55xm
0.50tm
5 T (182) x d
0.01xm
0.05xm
0.08xm
0.10xm
0.14xm
0.21xm
Model: Method 5;
0 . 1 < ^ - < 1 Tm
xm /Xm=0.43
xm/Xm=0.67
^ / T m = 1 . 0
T m / T m = 1 . 5
xm/X r a=2.33
*m/Tm=4.0
Coefficients of K c , T ; , Xd estimated from graphs; robust to ±25%
process model parameter changes 6.558Kra
2-31 lKm
1.479Km
1.117Km
0.947Km
0.825Km
0.731Km
0.695Kra
0.630Km
0.651Km
0.7KU
3 K m
'T ^ m
X
0.921
0.925Xm
0.677Xm
0.633Xm
0.647Xm
0.705Xm
0.752Xm
0.792Xm
0.871Xm
0.895Xm
1.030Xm
0.4XU
T ( A0'749
0-878 \Tm )
0.061Xm
0.189Xm
0.233Xm
0.250Xm
0.268Xm
0.289Xm
0.312Xra
0.316Xm
0.326Xm
0.306Xm
0.149XU
6 T (183)
xm/X r a=0.1
i m /X m =0.2
xm /Xm=0.3
xm /Xm=0.4
t m /X m =0.5
xm /Tm=0.6
xm /Xm=0.7
x r a/Xm=0.8
V / T r a = 0 . 9
*m/Tm=1.0
0.1<^2-<1.0 X
m
Model: Method 12
5 T (182) f N1137
T
6 T d d S 3 ) = 0 4 g 2 T n
X V m
158 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ISE -Murr i l l (1967)-pages 358-363.
Minimum ISE -Zhuang and
Atherton(1993). Model: Method 1
Minimum ISE -Ho et al. (1998). Model: Method 1
Minimum ITAE -Murr i l l (1967) -pages 358-363.
Minimum ISTSE - Zhuang and
Atherton(1993). Model: Method I
Kc
1.495 f T j
Kra l x m J
1.473
K m
1.524
K m
0.945
, N 0.970
VTnJ
(T ^ m
0.735
8 K (186)
1.357 ( T J Km {zmj
1.468 f T j K m \xm)
1 .515 fT j
Km W)
0.947
0.970
0.730
T ,
T x m
1.101
T m
1.115
T Am
1.130
r \
T
0.771
( V' 7 5 3
XJ , N 0.641
T ( l 8 6 ) i j
T Am
0.842
/• \ 0.738
V 1 m V
Tm O 0-942 ( j m J
T ( A0598
Tm O 0.957 Tm J
Td
0.56Tm
, x 1.006
r 7 T (184)
i d
7 T (185) i d
T (186) i d
9 T (187) Xd
10 T (188) i d
10 rp (189) i d
Comment
0 . 1 < ^ s - < l ; T
Model: Method 5
0 . 1 < ^ - < 1 . 0 T
1.1 < ^ 2 - < 2.0 Tm
A ra 6 [2,5],
<L40 0 , 60° ] ,
0.1 < ^ 2 - < 1.0 T Am
Model: Method 5
0 . 1 < ^ s - < l T
m
0.1 < - ^ < 1.0 T
m
l . l < ^ 2 - < 2 . 0 T
7 ^ ( 1 8 4 ) = 0 _ 5 5 0 T n
/ \ 0.948
T V 1m J
, T d( 1 8 5 ) =0.552T n
T
8 K (186) _ 1-0722 <|>n
Km A . T T-
.86) l-2497Tm^n
0.2099 T
(186) _ 0.4763Tmf -0.328
0.0961 T A m V *m J 0.2035 / / T , \0.3693
Given A r a , ISE is minimised when <t>m = 46.5489Am°'2035(Tm/Tn ,)°
(Hoe / al. (1999)); 2 < A m < 5 ; 0.1 < t m / T m < 1.0 .
9 T (187) = 0.381Tm | ^ s _
10 T (188) = Q . 4 4 3 T
T V i m
, Td(189) = 0.444T;
T
Chapter 4: Tuning Rules for PID Controllers 159
Rule
Minimum ISTSE - Zhuang and
Atherton(1993).
Minimum ISTES - Zhuang and
Atherton(1993). Model: Method 1
Minimum error -step load change -Gerry (1998).
Nearly minimum IAE, ISE, ITAE -Hwang (1995).
Kc
11 K (190)
12 K (191)
1.531
1.592
Km
VXm J
(T ^
0.960
0.705
0.3
16 ^ (194)
Ti
•p (190)
T ( 1 9 1 ) i
Tm
0.971
T
0.957
, x 0.746
, N 0.597
0.5xm
T (194)
Td
0.144TU
0.15TU
15 T (192)
15 T (193) x d
T
0.471KU
K m © u
Comment
0.1 <^2-< 2.0; T
m
Model: Method 1
0.1 < -^s- < 2.0; T
m
Model: Method 31
0.1<i3-<1.0 T
m
1.1 < ^ s - < 2.0 T
m
T i s - > 5 ; T
Model: Method 1 6, < 2.4
Model: Method 35; Decay ratio = 0.15; 0.1 < i s - < 2.0 T
„ K (190) 4 .434KmKu- 0.966 (I90)
5.12KmKu+1.734
12 (191) 6 .068KmK u -4 .273* „ ( , 9 „ 1.1622Km K „ - 0.748 « J v c — K u , l j — ~ l u .
5.758KmKu-1.058
15T<192)=0.413T, r A 0.933
2.516KmKu-0.505
N 0.850
T , Td
(193)=0.414Tn T
V 1 m J
16 K (194) ' 0.674JL - 0.447coHltm + 0.0607fflHI2Tm
2 f Km/(l + KH 1Km)
,(194) . K.r>(l + K H I K m ) coH1Km0.0607(l + 1.05(DHItm-0.233coH,2Tm
2)'
160 Handbook of PI and PID Controller Tuning Rules
Rule
Nearly minimum IAE, ISE, ITAE -Hwang (1995)
- continued.
Nearly minimum IAE, ITAE -
Hwang and Fang (1995). Model:
Method 25
Kc
17 K (195)
18 K (196)
19 K (197)
20 j ^ (198)
T i
T (195) i
T ( 1 9 6 ) i
T (197)
T ( 1 9 8 ) 1 i
Td
T (198) ' d
Comment
2.4 <e, <3
3<e , <20
e, >20
0 . 1 < - ^ - < 2 . 0 ; T
decay ratio = 0.12
, 7 K < 1 9 5 ) = k , 0.778[l-0.467(oH,Tm + 0.0609coH/x 2T 2 h HI ^m
Km/(l + KH1Km)
18 ^ (196)
j(195) Kc (l + K H 1 K m )
»HIKm0.0309(l + 2.84WH1xm - 0.532o>H12Tm
2)"
1.3l(0.519)m"|T" [1-1.Q3/S) +0.514/e l2f
Km /(l + KH ,Km) J'
KC"96)(1 + KH1K )
)H1Km0.0603(l + 0.9291n[coH1xm])(l + 2.01/s1-1.2/8 l2) '
(196)
19 K (197) K t
izi 1.14|l-0.482coH,xm +0.068(DH,2Tm2]N
Km/(l + KH1Km)
• (197) Kc(""(l + K H 1 K j
)H,Km0.0694(-l + 2.1coH,Tm-0.367WHI2Tm
2)'
20 K (198) 0.802 -0.154 — + 0.0460 T
' x ^
i T , V m J
K „
,(198) K (198)
K „ 0 ) u 0.190 + 0.0532-52- - 0.00509 T
f ^2
T V 1m
• (198) . 0.421 + 0.00915-21--0.00152P!! T I T
Chapter 4: Tuning Rules for PID Controllers 161
Rule
O'Connor and Denn(1972).
Model: Method 1; K m = T m ;
^ 2 - < 0 . 2 T
m
Syrcos and Kookos (2005).
Model: Method 1
Minimum IE -Devanathan
(1991). Model:
Method 1; Coefficients of
K c , T; provided
are deduced from graphs
Kc
2i axm2 +
(2 + x , )
Ti QO
Minimum 0.5
0
2x,
22 K (199)
Minimum
Td Comment
; ' " K ^ axm
2 + 2 X l
2Tma
T (199) 1 i
x l x m
xm2a + 2x,
T (199) Xd
performance index - constraints are input and output signals
23 K (200)
Tm/Tm
0.2 0.4
0.6
0.8
1.0
(
x 2
0.31 0.29
0.29
0.29
0.28
132TU
* 2 T u
0.08TU
x3T„
Coefficient values
x 3
0.08 0.07 0.07
0.07
0.07
T m / T m
1.2 1.4
1.6
1.8
Recommended
a e 0.6 1
2 ' 2
2 < I i 2 L < 5 T
placed on the process
x 2
0.26 0.26 0.24
0.24
x 3
0.07
0.07 0.06
0.06
2 + -
22 K (199) = _
K „ 0.31 + 0.6- x (199) _ T
^COS
23 K (200)
tan 1.57D„ 0.16
D R J
K cos tan 6.28Tm
0.777 + 0.45-2!-T
(199) (~ 0.44 - 0.56
162 Handbook ofPIandPID Controller Tuning Rules
Rule
Minimum IAE -Gallier and Otto
(1968). Model:
Method 1
Minimum IAE -Rovira et al.
(1969).
Minimum IAE -Mar l in (1995) -pages 301-307.
Model: Method 1
Minimum IAE -Sadeghi and Tych (2003).
Wang et al. (1995a).
Model: Method 1
0.05 < ^ - < 6 "T
m
Kc Ti Td Comment
Minimum performance index: servo tuning
x , / K m x2(Tm + O x3(Tm + 0
Representative coefficient values - deduced from graphs
tm/Tm
0.053
0.11 0.25
* 1
12.3
6.8 3.4
/ \ 0.869
1 0 8 6 | T 1
Km m
l-3/Km
l-0/Km
0.8/Km
0.7/Km
0.55/Km
0.48/Km
0-4/Km
Coefficients of
25KC<202)
26 K (203)
26 K (204)
x2
0.95
0.93 0.88
x3
0.0215 0.041 0.083
T
0 .740-0.13—
0.93xm
0.80xm
0.7hm
0.73xra
0.64xm
0.56xm
0.52xm
Cc ,Tj ,Td estima
process model pa
T (202) i
Tm + 0.5xm
•tm/Tn,
0.67 1.50 4.0
* i
1.45 0.85 0.57
24 T (201)
0.02xm
0.05xm
0.06xm
0.09xm
0.12xm
0.15xm
0.21xm
x2
0.75 0.63 0.53
X 3
0.155 0.21
0.19
Model: Method 5
0.1< — < 1
xm /Tm=0.25
xm /Tm=0.43
xm /Tm=0.67
xm /Tm=1.0
t m /T m =1 .5
xm /Tm=2.33
t m / T m = 4 . 0
ted from graphs; robust to ±25%
rameter changes.
1.45933xm
^ - + 3.27873 Tm
0.5Tmxm
Tm + 3.5xm
Model: Method 1
0 . 1 < ^ - < 1 . 0 T
Minimum IAE
Minimum ISE
24 T (201)
25 v (202)
= 0.348T„
f -,0.914
T
K
26 K (203)
K „ 0.26266 + 0.82714- T (202) T
0.7645 + 0.6032
* m / T m
0.28743-S L +1.35955 .
(T m +0.5x m )
Km(Tm + xm) >KC
(204)
0.9155 +
T
0.7524 (Tm+0.5xra)
K ra(Tm + xm)
Chapter 4: Tuning Rules for PID Controllers 163
Rule
Minimum ISE -Zhuang and
Atherton (1993). Model: Method 1
Minimum ISE -Hoe/al. (1998). Model: Method 1
Minimum ISE -Sadeghi and Tych (2003).
Minimum ITAE - Rovira et al.
(1969).
Kc
1.048
1.154
fry } m
X
0.897
f \ 0.567
-
29 j r (207)
3oK c( 2 0 8»
0.965 i- \ 0.85
, X m J
T,
27 T (205)
28 j (206)
T (207) i
T (208)
31 T (209)
Td
T (205)
T (206) *d
T (207)
1.93576xm
^ + 3.83528 T
T (209) x d
Comment
0 .1<^_<1 .0 T
1.1 < ^s_ < 2.0
Ame[2J5],
0 . 1 < ^ - < 1 . 0 . T
0 . 1 < ^ - < 1 . 0 ; T
Model: Method 1
0 . 1 < i s - < l ; T
m
Model: Method 5
27 T (205 ) _
1.195-0.368-., Td
(205) = 0.489Tn T
V m J
28 ry (206)
1.047-0.220-
(206) _ 0.490T„ ' O
T V 1 m J
29 K (207) 1.8578 <|>n
Km Am° ,T, (207) 0.4899Tm<t>m
0.1457 ( \ 1.0264
A „ T
T(207) 0.021 lTm[l + 0.3289Am +6.4572^)m +25.1914(xm/Tm)],
' l + 0.0625Am-0.8079(),m+0.347(xm/Tm)
Given Am , with 2 < Am < 5 and 0.1 < xm/Tm < 1.0 , ISE is minimised when
^ ^ 2 9 . 7 9 8 5 + ^ l ^ +4 0 - 3 1 8 2 T - - ^ g ^ - ( H O e / a / . ( 1 9 9 9 ) ) .
K „ 0.40455 + 0.96441- ry (208) _ ry 0.43770-2-+ 1.39588
T
31 j (209)
0.796-0.1465-
Td<209) = 0.308Tn
r N 0.929
X„
T V my
164 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ITAE - Wang et al.
(1995a). Model: Method 1 Minimum ITAE - Sadeghi and Tych (2003).
Modified minimum ITAE
- Cheng and Hung (1985).
Modified minimum ITAE - Smith (2003).
Minimum ISTSE - Zhuang and
Atherton (1993). Model: Method 1
Kc
32 K (210)
3 3 K C < 2 > "
11
0.965
1.042
T 1
(T N
m T
3.855
0.855
/ x 0.897
—1 i.i42 rTm
N 0.579
T,
Tm+0.5xm
T ( 2 1 1 ) i
3 4 T ( 2 1 2 ) i
1.26Tm
35 j (213)
36 q-. (214)
Td
0-5TmTm
Tm+0.5xm
1.55237Tm
^ - + 4.00000 Tm
T (212)
0.308xm
T (213) 1 d
T (214) 1 d
Comment
0.05 < ^ - < 6 T
m
0 . 1 < - ^ - < l ; T
m
Model: Method 1 £, = 0.707;
Model: Method 12
Model: Method 1
0.1 < ^2-< 1.0 T
m
1.1 < i=-< 2.0 T
32 K (210)
0.7303-0.5307
t m /T m
(Tm+0.5xm)
33 K (2") = _ 1 K „
34 ~ (212)
K m (T m +x m )
0.20360 + 0.80451- T,(211) =T I 0 . 2 9 3 4 9 ^ + 1.29110
0.796-0.147-T <212) = 0.308Tn
, \ 0.929
T
35 j (213)
0.987-0.238-TH
(213) = 0.385T„ / \ 0.906
, T V. m
36 j ( 2 1 4 ) , (214)
0.919-0.172-, Td
l i" ' = 0.384Tn
/ -,0.839
T ,
Chapter 4: Tuning Rules for PID Controllers 165
Rule
Minimum ISTSE - Zhuang and
Atherton(1993).
Minimum ISTES - Zhuang and
Atherton(1993). Model: Method I
Nearly minimum IAE, ISE, ITAE -Hwang (1995).
Model: Method 35
Kc
0.509KU
0.604 Ku
/• \ 0.904
0.968 Tm)
1.061
Km
/ „ \ 0.583
—l 40 ^ (219)
41 K (220)
T,
37 T (215)
37 T ( 2 1 6 )
38 T (217)
39 T (218)
T (219) i
T (220) i
Td
0.125TU
0.130TU
T (217)
T (218) 1 d
0.471KU
K m » u
Comment
0 .1<^S-<2 .0 ; T
Model: Method 1
0 . 1 < ^ - < 2 . 0 ; T
Model: Method 31
0.1 <-^s_< 1.0 T
l . l < i s - < 2 . 0 T
m
8, <2.4
2 .4<6 ,<3
37 Tj(2i5) = 0 .05l(3.302KmKu+l)Tu , T^2'6' =0.04| 4.972Kra Ku + 1 |T„.
T f \ 38 T ( 2 1 7 ) _
0.977-0.253-, T d
u l " =0.316Tn
0.892
39 T (218) _ • T
(218) :0.315T„
T
, N 0.832
0.892-0.165-T
V m J
40 K (219) 0.822[l-0.549coH1Tm+0.112a)H12Tm
2]N
Km/(l + KH 1Km)
,(219) K,<"»>(1 + KH 1KB)
2JT+
41 K (220) 2 l\\
coH1Km0.0142^ + 6.96o)HITra-1.77coH1 T„
( 0.786[l-0.441fflH1Tm +Q.Q569coH1
KH1
Kc'22°>(l + KH1Km)
K J ( I + K „ , K J
T (220) _
J
coH1Km0.0172(l + 4.62coH1xm-0.823a)H,2T„21 u H l l n T
166 Handbook of PI and PID Controller Tuning Rules
Rule
Nearly minimum IAE, ISE, ITAE -Hwang (1995)
- continued
Nearly minimum IAE, ITAE -
Hwang and Fang (1995).
Kc
42 K (221)
43 j , (222)
Ti
T (221)
T (222) i
Td Comment
3 < e, < 20
E, >20
Decay ratio = 0.1; 0.1 < — < 2.0 m
44 K (223) T (223) T (223) Model: Method 25
0.1 < ^ - < 2.0 ; decay ratio = 0.03
42 K (221) 1 .28(0 .542) ' ° H ' X ' " [ I -0 .986 /E 1 +0.558 /S 12 ]
H' Km/(l + KH 1Km)
43 K (222)
44 K (223)
T (221) K c ( 1 + KH|Km)
raH1Km0.0476(l + 0.9961n[coHITm])(l + 2.13/s1 -1 .13/e , 2 ) '
' K 1.14[l-0.466mH1Tm+0.0647coH12Tm
2lN
Km/(l + KH 1Km)
.(222) Kc (l + KH 1Km)
' coH1Kra0.0609(-l + 1.97coH1Tm-0.323coH12xm
2)'
K „ , 0.537 - 0.0165^2- + 0.00173 -^ T T
T, (223) K (223)
K u » u 0.0503 + 0.163—m-- 0.0389 T
f \ 2
T V 1 m J
• (221) _ 0.350 - 0.0344-^- + 0.00644 - ^ T T
Chapter 4: Tuning Rules for PID Controllers 167
Rule
Simultaneous Servo/regulator tuning - nearly minimum IAE, ITAE - Hwang
and Fang (1995).
Servo or regulator tuning - minimum IAE
- Huang and Jeng (2003).
Astrom and Hagglund (2004).
Model: Method 1
Kc Tj
Minimum performance
45 K (224)
0.36 + 0 . 7 6 ^
46 £ (225)
x l ^ r o + X 2 T m
X l
0.057
0.103
0.139
0.168
0.191
x2
0.139
0.261
0.367
0.460
0.543
_
T (224)
0.47xm+Tm
T (225)
x 3 T m + x4*m
Coefficient values x3
0.400
0.389
0.376
0.363
0.352
x4
0.923
0.930
0.900
0.871
0.844
X 5
0.012
0.040
0.074
0.111
0.146
Td Comment
index: other tuning
T (224)
0.47Tmtm
0-47Tm+Tm
T (225)
XeTm^m
t m + X 7T m
X 6
1.59
1.62
1.66
1.70
1.74
X 7
4.59
4.44
4.39
4.37
4.35
0 . 1 < ^ - < 2 . 0 ; T
m
decay ratio = 0.05;
Model: Method 25 Model: Method 37;
^2- < 0.33 T
1SL>O.33
M m a x = l . l
Mm a x=1.2
Mm a x=1.3
Mm a x=1.4
Mm a x=1.5
45 K (224) 0.713-0.176-2!-+ 0.0513 T
- (224)
f ^ Is. T
K u ,
K (224)
Kumu 0.149 + 0.0556-21- - 0.00566 T
( \2 x.
T V m J
0.371 - 0.0274 -2- + 0.00557 T I T co„K
(224)
46 K (225) 0.8194Tn+0.2773Tn. v 0.9738T 0.0262
, T^"> =1.0297Tm + 0 . 3 4 8 4 T „
T (225) _ 0.4575Tm+0.0302Tn.
1.0297Tm+0.3484t my
168 Handbook of PI and PID Controller Tuning Rules
Rule
Astrom and Hagglund (2004)
- continued.
Van der Grinten (1963).
Model: Method 1
Regulator -Gorecki et al.
(1989). Model: Method 1
Kc Ti Coefficient values
X l
0.211
0.227
0.241
0.254
0.264
x2
0.616
0.681
0.740
0.793
0.841
x 3
0.342
0.334
0.326
0.320
0.314
x4
0.820
0.799
0.781
0.764
0.751
X 5
0.179
0.209
0.238
0.264
0.288
Td
X 6
1.78
1.81
1.84
1.87
1.89
X 7
4.34
4.33
4.32
4.31
4.30
Comment
Mm a x=1.6
M r aax=1.7
M r aax=1.8
Mm a x=1.9
Mm a x=2.0
Direct synthesis: time domain criteria
_ L ( o . 5 + ^ l
47 K (226)
48 K (227)
Tm+0.5tm
T (226)
T (227)
Tra^m ^ m +2T m
T (226)
T (227)
Step disturbance
Stochastic disturbance
Pole is real and has maximum
attainable multiplicity; "tm/Tn, <2
47 K (226) _ e_
K „
- (226)
In
1 - 0.5e"°,',x"' +-la-e-<»d^ x „
(226) (l - 0.5e""jT- )ln
T
'K (227) _ 2 \
2T„ - x , - 9 -
v 2 Tm y 2T
2Tm
T ( 2 2 7 ) = T 1 i L n
21 + 3 ^ - + T
1
2T
— 1 N 2 T J _
X
x, - 9 - - ^ - -2T
, - 36 -4 .5 - ^ -T
'O l 2 T m ;
(2T
2
^
r A3
—
l2 TJ T < 2 2 7 ) = 0 . 5 x „
A A 2
3 +
6 + -2T
x , - 9 -2T 2T
2T
Chapter 4: Tuning Rules for PID Controllers 169
Rule
Saito et al. (1990). Model:
Method 18
Suyama(1992). Model: Method 1
Fruehauf et al. (1993).
Model: Method 2
Juang and Wang (1995).
Model: Method I
Abbas (1997) Model: Method 1
Camacho et al. (1997).
Kc
0 . 2 1 5 t m + T m
1.37Kmxm
50 K (229)
0.56Tm
TmKm
0-5Tm
TmKm
51 K (230)
52 £ (231)
1 T m + x m
T|
0 .315T m +T m
T m + 0 . 3 0 9 t m
5 i m
Tm
T (230) • M
T m + 0 . 5 t r a
4Tmxm
T r a + x m
Td
0.315Tmxm
0.315x r a+Tm
49 x (228)
T (229)
<0.5Tm
<0.5xm
T (230)
T m * m
2 Tm + tm
T m+*m
Comment
2 H L < 1
T m
1 < ^ - < 1 0 T
m OS=10%
^ 2 - < 0.33 T
m
— >0.33 T
TCL = « T m ,
0 < a < l
0 < V < 0 . 2 ;
0.1 < -^2- < 5.0 T
m
Model: Method 1
( 2 2 8 ) _ 0 . 3 1 5 x m T m + 0 . 0 0 3 T „
0 .315T r a +T m
50 v (229) _ 1
K . K „ 0 . 7 2 3 6 - 2 - + 0.2236 . T,
(229) 2-236TraTm
51 K (230) T T
K m a
7 .236T m +2.236x n
,2
. (230)
a + IsL + o.5 U s T I T
a + -
(230) _ V i m J V 0.5 a + ^ - O . S a ^
T T
a + i s_ a + Ii!L + o.5 Tm Tm
2 ^
52 K (231)
0.177 + 0.348 i "
Km(0.531-0.359Va7 '3)
170 Handbook of PI and PID Controller Tuning Rules
Rule
Cluett and Wang (1997), Wang
and Cluett (2000). Model:
Method 1;
0.01 < -^2- < 10 T
Kc
53 K (232)
Ti
T (232)
Td
T (232) ld
Comment
TCL = 4xm , Am E [7.97 ,8.56], *m e [79.67° , 79.70°]
54 ^ (233) T (233) T (233) ld
T C L = 2 x m , A r a e [4.84,5.31], *„ e
55 K (234) T (234) i
73.9°,74.14°
T (234) ld
TCL=1.33Tm,AmE[3.81,4.16],<t.ra6
56 j , (235) T (235) i
69.84°,70.70°
T (235)
TCL= Tm, Am G [3.30,3.56], <))m e [65.86°,68.50°
57 K (236) T (236) T (236)
T C L =0 .8 t m , Am 6 [3.00,3.19], * m e 60.60°,67.21°
53 „ (232) _0.019952xm+0.20042Tm (232)
54 v (233)_0.055548Tm+0.33639Tm (233)
0.099508xm + 0.99956Tm
0.99747x-8.7425.10_5T„
(232) -0.0069905xm+0.029480T„
0.029773xm +0.29907Tm
0.16440xm +Q.99558Tm
0.98607x-1.5032.10~4T„, m
(233) -0.01665 IT m + 0.09364 lTm
0.093905xm +0.56867T„
55 v <234) _0.092654xm+0.43620Tm T (234)
K m T m 'K ,
0.20926xm + 0.98518Tm
0.96515xm+4.2550.10-3T„
(234) . -0.024442xm +0.17669Tm
0.17150xm +0.80740T„
56 K (235) _0.12786xm+0.51235Tm ^ T<235)
K m T m
57 „ (236) _0.1605lTm+Q,57109Tm T(236) K c . l i
0.24145xm+Q.9675 lTm
0.93566xm + 2.2988.10~2T„,
(235) -0.030407xm +0.27480Tm
0.25285xra+1.0132Tm
0.26502xm +0.94291Tm
0.89868xm+6.9355.10"2Tm m '
T (236) = -0.035204Tm+0.38823Tm d 0.33303xm+1.1849Tm
Chapter 4: Tuning Rules for PID Controllers 171
Rule
Cluett and Wang (1997), Wang
and Cluett (2000) - continued.
Valentine and Chidambaram
(1997a). Morilla et al.
(2000). Model:
Method 28
Kc
58 K (237)
T|
T (237) *i
Td
T (237) *d
Comment
TCL= 0.67xm, Am e [2.80,2.95], ^m e [52.35°,66.45° j
59 K (238)
60 K (239)
T (238) i
T (239)
T (238)
al™
\ = 0.707 ; Model: Method I
a = 0.1; 60 = [0.2,0.5]
58 v (237) 0.19067tm+0.61593Tm T<237) _ 0.28242Tm +0.91231Tm ^ Kmx„ ' ' 0.85491T+0.15937T n K „
(237) -0.039589Tm +0.5194 IT,,
0.40950T„ + 1.3228Tm
59 K (238) _ 1 0.746-1.242 In T
T
rr\ {£Jd) rr\ 0.3106 +1.372-^1- _ 0.545 T
r ^2
T V m J
-T, (238) T
16.015Tm -11.702x„
±1% Ts = 2.5Tm, 0 . 1 < - ^ - < 0 . 5 ; ±1% Ts = 3.0Tm, - ^ - = 0.6; +1% Ts
3.75Tm , ^=- = 0.7; ±1% Ts = 4.3Tm ,^- = 0.8; ±1% Ts = 5.0Tm , ^ = 0.9 .
60 K (239) T^'V , (239)
K r a [28 0 + r o n 0 (x r a -T j( 2 3 9 ) ) l ' ' 1 + V T ^ 4 ^ '
• 5„Tm + V» 02 T m » + T m (T B , -T i '
r o >)-ar r ,<° ! "] '
( ^ - T ^ j - a P ^ ] 2 Tm (?m " T i
with 80
1 + 2TT
logc[b/a]
2 a = desired closed loop response decay ratio.
172 Handbook of PI and PID Controller Tuning Rules
Rule
Mann et al. (2001). Model:
Method 31 or Method 33
Chen and Seborg (2002).
Kc
61 j r (240)
62 ^ (241)
«K C < 2 4 2 >
Ti
T (240) i
T (241) i
T (242) i
Td
T (240)
x (241)
T (242)
Comment
OS < 5%;
0 < * m / T m < l
OS < 5%; l ^ m / T m < 2
Model: Method 1
61 K (240)
0.770+ 0.245^2-l T n W
0.854"
Tm
T ( 2 4 0 ) ' i 1.262 + 0 . 1 4 7 ^
V ir
0.262 + 0.147 (240) _
, N 0.854
V m J
0.770 + 0 . 2 4 5 ^
62 j , (241)
0.603 + 0.275 T
V m ,(241)
K m ' C n i
1.1618 + 0.165 r N-2.4
T V m
T m '
(241) _
0.1618 + 0.165|-^
0.603 + 0.275
63 K (242) 1 (2TmTm + 0.5xm2)(3TCL + 0 . 5 x J - 2 T C L
3 -3TCL2xm
Km 2TC L3+3TC L
2xm+0.5xm2(3TC L+0.5xJ
T(242, _ (2Tmtm +0.5Tm2)(3TCL + 0 .5xJ -2T C L
3 -3TCL2xn
t m ( 2 T m + T m j
Td(242) . 3TcL2xmTm + 0-5TmTm
2 (3TCL + 0.5xm)- 2(Tm + xm )TC
(2Tmxm +0.5xm2)(3TCL + 0 .5xJ -2T C L
3 -3TCL2xm
T<242>s(l + 0 .5x m s ) r -
K ^ ' ^ s + l)3
Chapter 4: Tuning Rules for PID Controllers 173
Rule
Viteckova and Vitecek (2002).
Model: Method 1
Gorez (2003). Model: Method 1
Sree et al. (2004).
Model: Method 1
Kc
64 K (243)
T,
T (243) i
Td
T (243) l6
Comment
For 0.05 < S- < 1.6 , T; = 1.2Tm ; overshoot is under 2% (Viteckova
and Vitecek (2003)).
( l -v )x m +T r a
XKmTm
1 [
1.377
T 1 -^2-+ 0.5 T m J
(T ^
T V m J
0.8422
( l - v ) r m + T m
Tm+0.5xm
67 T (245)
( l -v)xmTm
0-v)xm+Tm
66Td< 2 4 4 )
T (245)
65
0 . 1 < ^ _ < 1 T
m
K w = ^ l [ T m 2 T m X | 3 + ( 3 T m T m + T m2)X i2 + T m X ] _ 1 ] ; 64 v (243) _ e
K „
3 0.5 | 3 0.25 x, = +
"m ^m x m T m A | T
2 T 2
m m
• (243)
(243)
^ m ' ^ X , 3 + ( 3 T m T m + T m2 ) x 1
2 +T X, - 1
("cm2Tmx, + 2 t m T m +xj)
T/"J '=-0.5 ^ m
2 T m x , 2 +(4x m T m + x m2 ) x 1 + 2 x m + 2 T m
T m 2 T m X l 3 + ( 3 t m T m + T m2 ) x , 2 +TmX, - 1 '•m 1 m T Lm r-\ T v m ^ l
' F o r O ^ T " <r i V = > T m + T m2 + 0 . 5 x m
2
X =
F o r T m < ^ ^ < 1 , v = ^ Tm
+ T ™ 2 + 0 - 5 ^ 2
( ^ m + T r a ) 2
1 | ( 1 .3M S2 -1 ) 1.56Tm
2(0.5xm+3T,
2 M , ( M , - l ) 2 M . ( M . - l ) ( i m + T m ) 3
0.65M,
( ^ m + T m ) 2 M . - l
66 x (244) 0 . 5 i m ( T m + 0 . 1 6 6 7 t m )
T m + 0 . 5 x m
67 T < 2 4 5 > = i . 0 8 5 T „ T
V m J
,T, (245) _ . 0 . 3 8 9 9 - ^ + 0.0195
T
174 Handbook of PI and PID Controller Tuning Rules
Rule
Regulator -Gorecki et al.
(1989).
Somani et al. (1992). Model:
Method 1; Suggested A m :
1.75 <Am <3.0
Kc ^ Td Comment
Direct synthesis: frequency domain criteria 68 j r (246) T (246) T (246
Low frequency part of magnitude Bode 69 K (247)
^m/Tm
0.0375 0.1912 0.3693 0.5764 0.8183 1.105 1.449
X 2 T m * 3 * m
Coefficient values x2
3.57 3.33 3.13 2.94 2.78 2.63 2.50
X 3
0.71 0.67 0.63 0.59 0.56 0.53 0.50
* m / T m
1.869 2.392 3.067 3.968 5.263 7.246 10.870
Model: Method 1
diagram is
x2
2.38 2.27 2.17 2.08 2.00 1.92 1.85
flat.
x 3
0.48 0.45 0.43 0.42 0.40 0.38 0.37
68 K (246)
K m 2 _ x„
- (246) - + 1
V "m /
7 + 42-^ - + 135 T ( 2 4 6 ) - T
T m
rT ^ + 240
(T + 180
(y \4
15
T (216) _
T 2 - ^ - + l l + 3 ^ _ + 6
/ T 'N2
1 + 7 - ^ + 27 xm
+ 60 ' T „ °
+ 60
7+ 42-2!-+ 135 xm
(T \ + 240 + 180
fy A4
For -EL < 1 , K T
(247) 0.7809
KmAm
+ 1 or
|0.803 + 4^s-+0.896
K (247) 0.7809
KmAm1 1 + 0.803 , T v K (247) 0.7809
KmAm1
1 + 0.455 • > 1 .
Chapter 4: Tuning Rules for PID Controllers 175
Rule
Chidambaram (1995a). Model:
Method 1
Chidambaram (1995a).
Model: Method 1
Kc
70 ^ (248)
*i/KmAm
Iss. T
0.062 0.083 0.150 0.196 0.290 0.339
x l
29.770 22.100 12.470 9.670 6.686 5.784
1.20Tm
K m t m
1.03Tm
K m T m
Ts
T (248)
X 2 t m
Td
T (248)
x 3 ^ m
Coefficient values
x2
2.34 2.33 2.29 2.27 2.23 2.21
X 3
0.37 0.37 0.37 0.36 0.36 0.35
2-4xm
2-4xm
ISL m
0.389 0.543 1.03 2.92 3.69
x i
5.104 3.781 2.262 1.209 1.105
0.38xm
0.38tm
Comment
Alternative
x2
2.19 2.14 2.00 1.72 1.67
Ara
Am =
X 3
0.35 0.34 0.32 0.28 0.27
= 1.5
= 1.75
70 j r (248)
•(248)
KmAm
3.42 • + 0.85
4.45 + 4 -S - -2 .11
5Tm
0 - 8 5 , A AC
or ,1+4.45
5T„
/ T \2
1.17 3.42
+ 0.85
T <248> ~ yd
4.45 + 4 -5 i - -2 . i l Tm
0.8Tm
1.17 3.42
+ 0.85
4.45 + 4-21--2.11
(T \ 4.45 -0.15
0.8T
4.45 / T \2
0.15
176 Handbook of PI and PID Controller Tuning Rules
Rule
Chidambaram (1995a)-continued.
Branica et al. (2002). Model:
Method 1; M.=1.8
Minimum ISTSE - Zhuang and
Atherton(1993). Model: Method 1
Kc
0.90Tm
KmTm
71 K (249)
72 ^ (250)
73 K (251)
7 4 mK u cosfaJ
Ti
2.4xm
T (249) i
T ( 2 5 0 )
T ( 2 5 1 ) i
T ( 2 5 2 ) 1 i
Td
0.38xm
T (249) Xd
T (250)
T (251) x d
T (252) i
a
Comment
A m =2.0
0.1< — <0.5 T
0.5 <-^S- < 1.1 T
1.1 <^2-< 2.0 T
* m = 60°
71 v (249) = 1 K, 0.021 + 0.788- , T i
( 2 4 9 ) =1 .534T n, T "
K „ 72 v (250) _ _J_[ 0.292 + 0.671-^2- | , Ti(250) =1 .239T„
(249)
f \ -0.554
/- \ -0.055
v T V m J
(250) : 0 .230T„
T V 1 m J
r \ -0.069
T
73 v (251) _ 1 K „ 0.338 + 0.623^2- T<25» = 1.228xm - ^
x m T „
\-0.480
Td(251)=0.23kJ^H
T
i = 0.614(1 - 0.233e-°'347K"'K»), <|>m = 33.8°(l - 0.97e"°'45K"'K"), 0.1 < ^ - < 2.0 ,
a = 0.413(3.302KmKu +1), T; tan(c|>J+ - + tan 2(<|>J
+1), T/252' = a 1 « 2co„
Chapter 4: Tuning Rules for PID Controllers 111
Rule
Minimum ISTSE - Zhuang and
Atherton(1993) - continued.
Uet al. (1994).
Model: Method 1
Tan et al. (1996). Model:
Method 30
Kc
7SmKucos(4)m)
76 j r (253)
<L 15°
20°
25°
Ara
2
1.67
1.67
n 2.8
3.2
3.5
4h
7tApAm
-7—cos<t>m
Ara
Ti T (252)
(page 176)
4 T d ( 2 5 3 ,
4>m
30°
35°
40°
Am
1.67
1.43
1.43
il
3.8
4.0
4.2
0.3183T
aTd(254>
78 ry (255)
Td
T (252) i
a
T (253) *d
4>m
45°
50°
55°
Am
1.25
1.25
1.25
n 4.6
4.9
5.2
0.0796T
77 x (254) ld
T (255) *d
Comment
A m = 2 ,
* m = 6 0 °
4>m
60°
65°
Am
1.11
1.11
TI
5.4
5.5
simplified algorithm
a chosen arbitrarily
Arbitrary A m ,
<L at <»,,,
'Model: Method31; m = 0 .613( l -0 .262e" u ^^" ) , <(>m = 33.2U l-1.38e -0.68KmK„
a = 1.687KmKu, 0.1 < - ^ < 2.0 .
76 K (253) 4hcos0 (253)
7 l A „ fiT^ + T*™ 471 T1 + J T 1 2 + -
tan(|)m -_
}*[*,
(b/VA.'-b1
^ . • - " • • J .
JJ with ±b = deadband of relay, T = limit cycle period.
77 T (254) _ T
i n — I n
K 2 tancj>m + J — + tan i
47C I recommended Am = 2 , d)m = 45
178 Handbook of PI and PID Controller Tuning Rules
Rule
Friman and Waller (1997).
Model: Method 1;
A m > 2
Rivera et al. (1986).
Model: Method 1
Brambilla et al. (1990).
Model: Method 1
Lee et al. (1998) Model: Method 1
Gerry (1999). Model-
Method 10
Kc
0.25KU
0.4830
GpO r a .5o°]
T;
1.1879TU
2.6064
Td
0.2970TU
0.6516
Comment
t m / T m < 0 . 2 5 ,
$m > 60°
0.25 < -i=- < 2.0 Tm
* m > 4 5 °
Robust
T m + 0 . 5 t m
K m ( ^ + 0.5xm) T m + 0 . 5 T m
Tn^m 2 T r a + x m
X > 0 . 1 T m ,
^ > 0 . 8 x m .
0.5xm < X < 3xm (Zou et al. (1997))
79 Tm + 0-5xm
Km(A. + i J
T m +0 .5x r a
2 T m + x m
0.1 < -^s -< 10 T
m No model uncertainty - X « 0.35xm
T (256)
K ra(X + x m )
81 K (257)
Tm
K m ( ^ + 0.5xm)
80 y (256)
T (257)
T
T (256)
T (257) 1 d
0-5xm
T C L - ^
X = 2xm (aggressive, less robust tuning);
X = 2(Tm + x m ) (more robust tuning)
X£[0.1Tm ,0 .5Tm] ,^<0.25;
^ = 1 .5(x m +T m ) ,0 .25<^< 0.75 ; X = 3 ( x r a + T m ) , ^ > 0.75 (Leva (2001)).
80 y (256)
81 K (257)
= Tm + 2(^ + t m )
, (257)
(256)
2{X + Tm) 1- -
. (256) 3T-V J 1 i J
K r a ( 2 T C L + x r a - a ) ' l i
0 1fi7r 3 -T m a -
T / ^ T + a ^ + a ^ - 0 . 5 x n
2TCL + xm - a
0.167xm3-0.5axm
2
(257) 2Trl + xm - a T r l 2 + axm - 0.5x 2
, (257) 2Tr, +xm - a
a = T, m - T j l - ^ M e
2 _ISL
Chapter 4: Tuning Rules for PID Controllers 179
Rule
McMillan (1984).
Model: Method 2 or Method 44
Geng and Geary (1993).
Model: Method 1
Wojsznis and Blevins(1995).
Peric et al. (1997). Model:
Method 40
Matsuba et al. (1998).
Model: Method 1
Wojsznis et al.
(1999). Model: Method 1
Kc T, Td Comment
Ultimate cycle
82 K (258)
0.94Tm
Kmx r a
T m + 0 . 5 x m
a K r a ^ m
K u « > S ( | > m
84 K (260)
Kucos(|)m
A m
T (258) 1 i
2 t m
T m + 0 . 5 t r a
aT d( 2 5 9 )
T (260) i
85 j (261)
T (258)
0.5xm
T m t m
2 T m + x m
83 T (259)
x (260) Xd
a T . ( 2 6 1 )
Tuning rules developed from
Ku>Tu
•^2- < 0.167 T
1.3<a < 1.6;
Model:
Method 36
Recommended
ot = 4
0.1 < ^ s - < 1 . 0 T
82 K (258) _ 1.415 Tm
1 +
. T.(258) = x 1 + T + x
(258) = 0 . 2 5 T J 1 +
T +x V m m y
/ 83 x (259)
84 K (260)
tan<L + J— + tan2 <|>n a
1.31 f T, ,0.9
Km Vt m y
2.19 IT,
K r a ^ m
T
47t
0.9
T i( 2 6 0 )=1.59x„
T <260) = 0.40x„
= (tan<j)m + y 4 a + tan2<|> I—-J 4na
180 Handbook of PI and PID Controller Tuning Rules
Rule
Wojsznis et al. ( 1 9 9 9 ) -
continued.
Wojsznis et al. (1999).
Model: Method 36
Kc
0.5Kucos<t>m
0.38KU
0.27KU
87 K (263)
T| 86ry (262)
i
1.2T U
0.87TU
ry (263) i
Td
0.15Ti(262)
0.18TU
0.13TU
0.125Tj<263)
Comment
Default values
^m /Tm<0.2
A m = 3 ,
4>m=33°;
xm /Tm*0.25
a, e [0.3,0.4];
a2 E [0.25,0.4];
0.2<^-<0.7 T
m
T i( 2 M ) = 0 . 5 3 T „ ( t m ( | > m + 7 a 6 + • tan
87 ^ (263) _ K 1.0-a, 0.6
1 + e
r-p (263) T-. 0.6
- - ^ - 7 . 0
l + e ^ T m -j
Chapter 4: Tuning Rules for PID Controllers 181
4.1.2 Ideal controller in series with a first order lag f
G c ( s ) = Kc 1 \
V T i s j
1
l
1 + Tfs
U(s)
• K m e ' " "
l + sT„
T fs + 1
Y(s)
Table 45: PID controller tuning rules - FOLPD model G m (s) = K m e
l + sT„
Rule
Schaedel (1997). Model:
Method 21
Horn et al. (1996).
Model: Method 1
Kc Ti Td Comment
Direct synthesis: frequency domain criteria 1 K (264) T,2 - T 2
2
T T (264)
T (264) Tf =Td(264)/N
5 < N < 20
Robust Tm+0.5xm
Tm+0.5xm
T r a ^ m
2Tm+Tm
T ^ m
' 2^ + x J ' ^e[xm,Tra]
X = max(0.25Tm,0.2Tm)(Luyben(2001)); X>0.25tm (Bequette,
(2003)).
1 ^ (264) 0.375T (264)
K „ T, +-(264)
N , (264)
_ j (264) _ T j T 3
^ ' d - T, T22
1 + 3.45-
- + T m , T 2 = 1 . 2 5 T m T > - T „ , + 0 . 5 T ' T , = 0 ,
1 + 3.45-
0.7T -+^m -T2 = T mC +
0.7T - + 0.5t„
T„-0.7xam "" ' T m - 0 . 7 <
0.952Tm2(C-0. l) , 0-35Tm
2tm2
0<-^-< 0.104: T
0.167xJ, — > 0.104. T „ , - 0 . 7 T * T m - 0 . 7 <
182 Handbook of PI and PID Controller Tuning Rules
Rule
H^ optimal -
Tan et al. (1998b).
Model: Method I Lee and Edgar
(2002). Model: Method 1
Huang et al. (2002). Model:
Method 1
Zhang et al. (2002).
Kc
2 K (265)
Tj
Tm+0.5xra
Td
x T
^m+2Tm
Comment
X = 2 : 'fast' response; X = 1: 'robust' tuning; X = 1.5 : recommended
3 K (266)
Tm+0.5xm
Kj^ + xJ
T (266) M
Tm+0.5xm
T (266) Xd
Tm
2Tm+xm
i-p r-p (266) l f - l d
' 2(X + xJ X = 0.65xm - minimum ISE; X. = 0.5xm - overshoot to a step input
< 5% , Am = 3 . In general, Ara = 2{X/xm +1).
4 K (267) Tm+0.5xm 2Tm+x ra Model: Method 1
2K(265)_0-265X + 0.307rTm
K„ lx m
1
. T f 5.314X. +0.951
3 K (266)
Km(^ + xm) Tm+- T 3 T m - x m xn
"X„ I—7- r + -2( \ + T J m\6(*. + x J 4( + x r a ) 2
. (266) _ry X „ 3 T m - ' c m + . t „
2(X + xm) m ^ + x J 4(^ + x r a)2 '
T (266) _
[6(X + x J 4(X + xJ 2
K (267) _ _ 1 Tm+0.5x„
Km 2X. + 0.5X. 2X. + 0.5x„ orTf_AE^m
x m + 2 T
Chapter 4: Tuning Rules for PID Controllers 183
4.1.3 Ideal controller in series with a second order filter r 1 A
Gc(s) = K£
V T i s J
l + bfls
l + afls + af2s
R(s) E(s)
K, l + — + Tds V T i S
u(S; l + b f ls
l + af,s + af2s
K „ e "
l + sT„
Y(s)
Table 46: PID controller tuning rules - FOLPD model Gm (s) = K„e"
l + sT„
Rule
Kristiansson (2003). Model:
Method 11
Kc Ti Td Comment
Direct synthesis: frequency domain criteria
5 K (268) rp (268) T (268) 0.05<^S-<2; T
m
1.6 <MS <1.9
' Equations continued into the footnote on page 184;
K (268) _
l . l ^ - O . l 0.06 + 0 . 6 8 — - 0.12 T
f \2
T,
T
K „ f ~ \
0.68+ 0.84-=!--0.25 T T
2T„
. (268)
0.06 + 0.68 -5=- -0.12 T T
0.68 + 0.84 -^- -0.25 T T
184 Handbook of PI and PID Controller Tuning Rules
Rule
Kristiansson (2003) -
continued.
Horn et al. (1996).
Model: Method 1
Kc
6 K (269)
7 K (270)
Ti
T (269) i
Td
T (269)
Robust
Tro+0.5Tm
T m T m
^ m +2T m
Comment
0.7 < — < 2 ; T
m
1.6<MS <1.9
*• > ^ - *• < T m •
(268) _
0.68 + 0 . 8 4 ^ - 0 . 2 5 ^ T T
T r = -
0.06 + 0 . 6 8 ^ - -0.12 T
f \ 2 x
ra )
-|2
b l f = 0 , a l f =0.8T f , a 2 f =T f with
T 1 . 1 ^ - 0 . 1
6 K (269)
. (269)
min-l 5+ 2 - ^ , 2 5 x „
0.6667(Tm+Tm 1.1 - - 0 . 1 X m
0.68+ 0.84-2!--0.25 T
' O 2
0.6667(Tm+xm
0.68+ 0 .84^ - -0 .25 T
m
) ( ^2
\LmJ
T V m /
n . T , (269) 0.1667(Tm+xm)
0.68 + 0.84 ^~ T
(. \ 2 ]
-°ii) \ (Tm +xJ2
b l f = 0 , a,,- = 0.8Tf, a2f = Tf with Tf = -
1 . 1 ^ - 0 . 1 xm
9T„ 5 + 2-V m y
7 K (270) 2T m +x m _A.2xm+2Tmxm(xm-A.) , 2*,(2Tm->.) - , D f l - -, ; +
2(2A. + x m - b „ ) K n Tm(xm+2X) (xm+2X) '
2XTm+2X2+bf,Ta, « f l — TT~. : \ — i df
^ x „ fl 2(2X + x r a - b f l ) ' f2 2(2X + T m - b f i :
Chapter 4: Tuning Rules for PID Controllers
4.1.4 Ideal controller with weighted proportional term
185
Table 47: PID controller tuning rules - FOLPD model Gm (s): Kme"
l + sT„
Rule
Astrom and Hagglund(1995) -pages 208-210.
Model: Method 5 or Method 14
Gorez (2003). Model: Method 1
Kc
3 . 8 e - " , + 7 J t I T m
K m T m
8.4e-* 6 t + M ' 2Tm
K m T m
0 - v > m + T m
xKraTm
Ti Td
Direct synthesis
5 . 2 x m e - — '
or
0.46Tme2 '8T-2 'u2
0.89Tme-037t-4w2
or
0.077Tme50t-4 '8l !
b = 0.40e0'18t+2'8x2
3.2Tme-'-5T-a93T2
or
0.28Tme18T"''6t2
0.86Tme-'9t-04412
or
0.076Trae3 '4t-MT!
b = 0.22ea 6 5 , + 0 0 5 l T 2
0-v) t m +T m ( l - v > m T m
( l - v ) x m + T m
b _ ( X - I ) t m - T m : 1
( l - v ) t m + T m
Comment
M , = 1 . 4 ;
0.14 < ^ 2 - < 5.5 Tm
M s = 2 . 0 ;
0.14 < ^ 2 - < 5.5 T
8
' For 0 < -^m+Tm
• < T V =
Fori < V < 1 ; V _^T m + T m2
+ 05T m2 _
TmTm+Tm2+0.5Tm
2
(^m+Tm)2
1 { (1.3MS2-1) 1.56Tm
2(0.5Tm+3Tm),
2 M , ( M . - l ) 2 M . ( M , - l ) (Tm+Tm)3
0.65M„
^m+Tm frm+Tj M - 1
186 Handbook of PI and PID Controller Tuning Rules
4.1.5 Ideal controller with first order filter and set-point weighting 1
f
K(s)
w
l + 0.4Trs
1 + Trs
U(s) = K
E(s)
1 \ 1+ — + Tds
V T > S J
1
Tfs + 1
D , , l + 0.4Trs R(s) — ^ J - - Y(s)
^ - ^
( K„
1 \ 1 + — + Tds
V T i s J
1
1 + Tfs
1 + sT,
U(s)
K e~
l + sT„
Y(s)
Table 48: PID controller tuning rules - FOLPD model G m (s) = K e -
l + sTm
Rule
Normey-Rico et al. (2000).
Model: Method I
Kc Ti Td Comment
Direct synthesis
0.375(xm +2Tj K m T r a
Tm+0.5xm
T m ^ r a
2Tm+xm
Tf=0.13xra
T r=0.5xm
Chapter 4: Tuning Rules for PID Controllers 187
4.1.6 Controller with filtered derivative Gc(s) = K£ , l T d s
T;s . X,
Table 49: PID controller tuning rules - FOLPD model G m (s) l + sT„
Rule
Tavakoli and Tavakoli (2003).
Model: Method 1;
0 . 1 < i s - < 2 ; T
N = 1 0
Kc T( Td Comment
Minimum performance index: servo tuning 1 K (271)
2 K (272)
1
Km
( \
0.8
i s - + 0.1 T
T (271)
T ( 2 7 2 )
T m
( T ^ 0.3 +-si-
l Xm J
0.01 HTm
x m
( \
0.06
i s - + 0.04 T
Minimum IAE
Minimum ISE
Minimum ITAE
1 K (271)
2 K (272)
1
" K r o
1
~ K m
1
i s - + 0.2 T
V m )
0.3i°- + 0. T
i=- + 0.0; T
T < 2 7 ' ) ->
75
)
T ( 2 7 2 )
0 . 3 - ^ + 1.2 T
I Tm
> x i ^ - -"-m
+ 0.08 J
f
1
T
"\
4
m
0.06
i s - + 0.04 T
Handbook of PI and PID Controller Tuning Rules
Rule
Davydov et al. (1995).
Model: Method 6
Kuhn(1995). Model:
Method 14; N = 5
Kc
3 K (273)
4 K (274)
VKm
2/Kra
T, Td Comment
Direct synthesis T(273)
T (274) i
0.66(Tm+Tm)
0 .8 (T m +Tj
T (273)
T (274) 1 d
0.167(xm+Tm)
0.194(xm+Tm)
^=0.9;
0 . 2 < t m / T m < l
"Normale Einstellung"
"Schnelle Einstellung"
3 K (273)
K J 1.552-^ + 0.078
, (273) 0.186-E- + 0.532 T
Td(273) = 0.25 0 .186-^ + 0.532 |T„
•(274)
K „ 1.209^2-+ 0.103 T
0.382-^s- + 0.338 Tm T
(274) 0.4 0.382^2-+ 0.338 T
Chapter 4: Tuning Rules for PID Controllers 189
Rule
Schaedel (1997)
Goodwin et al. (2001) -page
426.
Kc
5 K (275)
6 j , (276)
Ti
T (275)
T (276) i
Td
T (275)
T (276)
Comment
Model: Method 21
Model: Method 1
0.375T (275)
T
T, + -(275)
N
,(275)
•(275)
1 + 3.45-
- + T m , T / = 1 . 2 5 T n X +
T Z ryi Z FT* £, r-p J
1 ~ X2 T (275) l2 h T T (275) ' *d T T 2
- T m + 0 . 5 T m M 3 = 0 ,
1 + 3.45-
0 < ^ - < 0.104: T
0.7T, 2
? r a , V = T m x am
0.7T„
Tm-0.7x...
3_0.952Tm2(xa
m-0.l)
T -0.7xa •+o.V,
TV Tm-0.7x a
+ TmTmTan + °-35Tro Tm +0.167Tm
3, ^ - > 0 . 1 0 4 m m m _ A _ a m ' r p
T m - 0 - 7 T m lm r (275)
N: .(275) A
iii^l+o. . (275)— (275)
3 7 5 - ^ Kv
0.05 < a < 0.2 , Kv = prescribed overshoot in the manipulated variable for a step
response in the command variable.
6 v (276) . _ (4f ;TC L 2+Tm ) (xm+2TJ-4TC L 22
Km(l6^2TCL22+8^TCL2x ra+Tm
2) '
T (276) _ (4^TCL2 +Tm)(xm +2T m ) -4T C L 2 , +i_ ^
2(l6i;2TCL22+8i;TCL2Tm+Tm
2) ^
Td( 2 7 6 )=(l6^2TC L 2
2+84TC L 2xm+T r a2)
l(4^TCL2 + xm )2imTm - 2TCL22(4^TCL2 + xm Xxm + 2Tm)+ 8TC
(4^TCL2 + xm )3 [(4^TCL2 + xm Xxm + 2T m ) - 4TCL22 ]
N = 4 4 T C L 2 ^ m T ( 2 7 6 ) w , t h G c L ( s ) = . '
2Tr TCL2V+2!;TCL2s + l
190 Handbook of PI and PID Controller Tuning Rules
Rule
Chien (1988). Model: Method 1
Morari and Zafiriou(1989).
Model: Method 1 Gong et al.
(1996). Model: Method 1 Maffezzoni and Rocco (1997).
Kasahara et al. (1999).
Kc Ti Td Comment
Robust Tra+0.5xm
Km(^ + 0.5xm)
7 j r (277)
8 K (278)
9 K (279)
10 K (280)
Tra+0.5xm
Tm+0.5xm
Tm+0.3866xm
T (279)
T (280)
Tmtm
2Tm+xm
Tm^m
2Tm+xm
0.3866Tmxm
Tm+0.3866xm
T (279)
T (280)
(Chien and Fruehauf
(1990)); N=10 ^.>0.25xm,
\>0.2Tm
N = [3,10]
Model: Method 12
Robust to 20% change in plant
parameters
Model: Method 1; N = 10; 0 < ^ - < 1 T
K (277) _ 1 T m +0.5x„
8 j , (278)
K (279)
... . ^ + Tm
Tm+0.3866xm
Km(x + 1.0009xm)
= 2 T m ( T m + X ! ) + T m
2 K m ( t m + x , ) 2
N = 2Tm(^ + xm)
^ (2T m +x m ) '
. (0.1388+ 0.1247N)Tm+0.0482NTn , *• = — r - r . . . _ . - T „
- T (279)
0.3866(N - l)Tm +0.1495Nxn
2
= T + 2 ( T n + x , ) '
T(279)_ xm2[2T ( x m + x , ) - x m x 1 ] M _ 2T m (x m +x 1 ) -x m x 1
2( t m +x , ) | 2T m (x r a +x 1 )+x m2 J ' x,[2Tm(xm+x1)+xm
2]
x Xj = 0.25tm for uncertainty on xm only; xx >0.1Tm>—— <0.4 for uncertainty on the
minimum phase dynamics within the control band.
10 K (280> = 0.6K, 0.718-0.057- , T^8U ' = 0.5TU 0.296 + 0.838-
Td(280) = 0.125Tu 0.635-1.091-^
T
Chapter 4: Tuning Rules for PID Controllers 191
Rule
Kasahara et al. (1999)-
continued.
Leva and Colombo (2000).
Kc
11 K (281)
12 K (282)
13 K (283)
14 K (284)
Ti
x (281)
T (282)
T (283)
2
T I T m
m 2(X + T J
Td
T (281)
T (282)
T (283)
T (284) x d
Comment
Robust to 30% change in plant
parameters Robust to 40% change in plant
parameters Robust to 50% change in plant
parameters A. not specified;
Model: Method 1
11 K<281)=0.6K, 0.690-0.076- , Ti(281) = 0.5TU 0.274 + 0.788-
Td(281) = 0.125TU 0.526-0.057-!2-
T
12 v (282) K/"" ; =0 .6K, 0.331 + 0.149- • (282) _ 0.5T„ 0.041 + 0.857-E-T
Td(282) =0.125TU 0.495-0.064-
13 K <283) = 0.6K, 0.195 + 0 .266-^ T
X(283) = 0.5T„ -0.013 + 0.731-
(283) :0.125T„ 0.489-0.065-52-
T
14 v (284) _ 1 2 ( X + T m ) T m + T m2
(284) _ Tm{l + tm)
Km 2(X + TmY [2(^m)Tm+Tra2]'
N = 2Tm(X + Tm)2
x[2(A. + Tm)Tm+Tm2J'
192 Handbook of PI and PID Controller Tuning Rules
Rule
Leva (2001). Model: Method I
Leva et al. (2003).
Zhang et al. (2002).
Kc
15 j r (285)
16 j , (286)
1 7 j , (287)
Ti 2
T T i m m 2(* + TJ
2 T
T i m
m 2(x + xJ
T (287)
Td
T (285)
T (286) 1 d
T (287) x d
Comment
Model: Method 1
Model: Method 1
5 v (285) _ 1 2(X + Xm)Tm +T n K
K m 2(X + Xm)2 N:
2Tm(X + t J 2
«.[2(X + T , ) T m + x „
V2S5)=rJ ^ Y } 2 ^ ' ^ 2 |A e [0.1Tm ,0.5Tm] ,^0.25; 2(X + Tm)[2Tm(^ + Tm)+Tm
2J Tm
>. = 1 . 5 ( t m + T m ) , 0 . 2 5 < ^ - < 0 . 7 5 ; A. = 3 ( T m + T m ) , i s - > 0 . 7 5 .
16 v (286) _ 1 2(^ + Tm)Tm + T m T (286) __ TmTm(X + Tm) K „
Km 2(X + T m ) 2 [2(^ + x l )T m + T r a
2 ] ;
N = r 2T™(X + X™> ^ . Xe[0.1Tm,0.5Tm], i 2 - < 0 . 3 3 ; 42(^ + x m )T m + t m
2 Tra
X = 1 . 5 ( x m + T m ) ) 0 . 3 3 < ^ < 3 U = 3 ( x m + T j , ^ > 3 .
0.5xm+T -~A-17 K (287) m m
(287) d
N Km(2* + 0.5xJ
, (287) :0.5xm+Tm
(287)
N
(287) _ l m i m x m T „ T, (287) T (287)
2T(287) N ' N 2?,+0.5x„ orN=10.
Chapter 4: Tuning Rules for PID Controllers 193
f 4.1.7 Blending controller Gc (s) = Kc
E(s)
1 \
V T i S J
R(s) ®— + T
f K,
1 \ 1 + — + Tds
U(s) K e"
1 + sT
Y(s)
Table 50: PID controller tuning rules - FOLPD model Gm (s) = l + sTm
Rule
Chen et al. (2001).
Model: Method 1
Kc Ti Td Comment
Direct synthesis: frequency domain criteria 18 K (288) T (288) T (288)
Representative X values
^ = 9.56tm) Mmax =1.20; ^ = 4.68xra, Mmax =1.26;
^ = 2.19xm, Mmax =1.50;^ = 1.40tm, Mmax =2.00;
^ = l . l lT r a ,M r a a x =2.50;X = 0.95Tm, M r a a x=3.00.
K (288) 1 T + T + 2 A . ,(288) _
K „ (*• + *.? T m +T r a +2X ,T (288)
T„+Tm+2X.
194 Handbook of PI and PID Controller Tuning Rules
4.1.8 Classical controller 1 Gc(s) = K
Table 51: PID controller tuning rules - FOLPD model G m (s) = K m e "
l + sTm
Rule
Witt and Waggoner
(1990). Model: Method 2
Witt and Waggoner
(1990). Model: Method 2
Kc Ti Td Comment
Process reaction
K m T m
X! 6 [0.6,1.0]
1 K (289)
*m
T ( 2 8 9 )
X m
T (289)
Equivalent to Ziegler and
Nichols (1942); N e [10,20]
Preferred tuning; N e [10,20]
1 Tuning equivalent to Cohen and Coon (1953).
K (289)
1 . 3 5 0 ^ _ + 0.25+ ^ . 0 . 7 4 2 5 + 0 .0150^3-+ 0.0625 V2
T
2K„
*(289)
1.350^2- + 0 . 2 5 - i s - 0.7425 + 0.0150-^=- + 0.0625 ^ t m Tm V Tm T
(289)
1 . 3 5 0 - ^ + 0.25 + - E - 0 . 7 4 2 5 + 0.0150^ ! ! -+ 0.0625 t m xm V Tm
r \2 xn
T V 1 m J
Chapter 4: Tuning Rules for PID Controllers 195
Rule
Witt and Waggoner (1990)
(continued).
St. Clair (1997) -page 21.
Model: Method 2
Harrold(1999). Model:
Method 10; N not specified
Shinskey (2000). Model: Method 2
Minimum IAE -Huang and Chao
(1982).
Kc
2 K (290)
Tm
K m x m
0-5Tm
K m x m
1/Km
0-5/Km
0.25/Km
0.889Tm
K m x m
T,
T ( 2 9 0 )
5xra
5xm
Tm
l-75xra
Minimum performance
0.817
Km \Xm J
0.982
3 T ( 2 9 1 )
Td
x (290)
0.5xm
0.5xm
<0.25Tm
0.70xm
Comment
Alternate tuning; N e [10,20]
'aggressive' tuning;
N > 1
'conservative' tuning;
N > 1
^m/T r a^0.25
x m /T m «0.5
x m /T r a >l
^2-= 0.167; T
N not specified
index: regulator tuning
x (291) Xd
N = 8; Model: Method 1
2 K (290)
1 . 3 5 0 ^ - + 0.25 - ^ i 0.7425 + 0 . 0 1 5 0 ^ - + 0.0625 x m x m V T m
f \2 x„
VTmV
2K„
X (290)
1 . 3 5 0 ^ + 0.25 + — . 0 . 7 4 2 5 + 0 .0150-^ -+ 0.0625 xm xm V Tm
f \ 2 x„
T
(290)
1.350^2- + 0.25 - — .0 .7425 + 0 . 0 1 5 0 ^ + 0.0625 x„ xm V Tm T
s 0.954
3 T(291) = Q.903T T
, Td( z y" = 0.602T,,
, T ,
196 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum IAE -Kaya and Scheib
(1988). Model: Method 5 Minimum IAE -
Witt and Waggoner
(1990). Model: Method 2
Minimum IAE -Shinskey(1988)
-page 143. Model: Method 1
Kc
4 ^ (292)
5 K (293)
6 j r (294)
T i
T (292) 1 i
T (293)
T (294)
Td
T (292) 1 d
T (293)
x (294)
Comment
0 < i l L < l ;
T
N=10 Preferred tuning
Alternate tuning
0.1<-^2- < 0.258; N e [ l 0 , 2 0 ] m
0 .95T m /K m i r a
0.95T m /K m x m
1.14Tm /KmTm
1.39T ra/K raTm
1.43Tm
l-17Tm
1.03xm
0.77xm
0.52xm
0.48tm
0.40xm
0.35xm
^ / T r a = 0 . 2
xm /Tm=0.5
tm/Tm =1
T m /T m =2
4 K (292) _ 0.98089 (^ ^ 0.76167
K „
, (292) T , N 1.05211
0.91032 T V m J
Trf<292) = 0.59974T I s
5 £ (293) 0.718 | T„ K m VTmy
f ,. A 1 + , 1 - 1.693
0.964T„
T V 1 m J
r A 1 1 3 7
, (293) VTm V
1 - , 1-1.693 IlIL , T
6 ^ (294) 0.718
K „ T
-0.921
1 - , 1-1.693
0.964T„ r(294)
T V m J
f \1.137
Im. T
0.964T„ (293)
l + J l + 1.693
0.964Tn
T \ m J
( \ 1 1 3 7
• (294) T
l - J l + 1.693
/• -,1.886
T V m J
l + J l - 1 . 6 9 3 T
Chapter 4: Tuning Rules for PID Controllers 197
Rule
Minimum IAE -Shinskey (1996)
-page 119. Minimum IAE -
Edgar et al. (1997) - pages 8-
14. 8-15. Minimum IAE -
Edgar et al. (1997) -page 8-
15. Minimum IAE -
Smith and Corripio(1997)-
page 345.
Minimum IAE -Shinskey (1988)
-page 148. Model-
Method 1
Minimum IAE -Shinskey (1994)
— page 167. Minimum ISE -Huang and Chao
(1982).
Minimum ISE -Kaya and Scheib
(1988). Model: Method 5
Kc
0.96Tm/Kmim
0.94Tm/Kmxm
0.88Tm/Kmxm
Tm
K m T m
0.56KU
0.49KU
0.51KU
0.46KU
Kuxm
3xm-0.32Tu
/ - m \ ° - 8 8 1
I.IOI r -o
9 K (297)
T i
l-43tm
1.5Tm
l-8xm
Tm
0.39TU
0.34TU
0.33TU
0.28TU
7 T (295) i
8 T (296) i
j (297)
Td
0.52xm
0.55xra
0.70xm
0-5xm
0.14TU
0.14TU
0.13TU
0.13TU
0.14TU
T (296)
T (297) i
Comment
Model: Method 1 x m /T m =0.2
Model: Method 1 Time constant
dominant; N=10 Model:
Method 2; N=10
Model: Method 1;
0.1 < ^2-< 1.5
x m /T m =0.2
xm /T r a=0.5
T m /T r a =l
^ m / T m = 2
Model: Method 1;
N not specified Model:
Method 1; N = 8
0 < - ^ - < l ; T
m
N=10
0.15—^-0.05
T ( 2 9 6 ) = U 3 4 T
> T
0.883
(296)
, K (297) 1.11907 TTm
K m {xm
• (297)
= 0.563T„
T
v T
0.7987 T
• (297) = 0.54766T, T
V ' • J
198 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ITAE - Huang and Chao (1982).
Minimum ITAE - Kaya and
Scheib (1988).
Minimum ITAE - Witt and Waggoner
(1990).
Kc
0-745 Tm^
11 K (299)
T i
10 T (298)
T (299)
Td
T (298) ' d
T (299)
Comment
Model: Method I;
N = 8
0 < i s - < l T
^=10; Model: Method 5 12 K (300)
13 K (301)
T (300)
T (301)
T (300) ld
T (301)
Preferred tuning
Alternate tuning
Model: Method 2; N e [l0,20]
> 0 T ( 2 9 8 ) = a 7 7 I t „ , Td(298)=0.597Trt
„ K (299) 0.77902 ( X
TT\? >T; (299) T
f \ 1.006
I™. , T
• A 0.70949 t .
1.14311 T
TH(299) = 0 . 5 7 1 3 7 T j i
12 K (300) _ 0-679
K „ T 1 + , 1-1.283
0.762T,
, 0.1 < -**- < 0.379; T
f N 0 . 9 9 5
.(300) T 0.762T„
/• -v 0.995
(300) T V 1m J
1-J l -1 .283 T
V m y
l + J l + 1.283 r V-7 3 3
13 K (301) _ 0.679
K „ T V 1 m J
1-J l -1 .283
0.762T„
-1.733
,(301) T 0.762T„
T V m J
, N 0 . 9 9 5
• (301) > T
1 -J l+ 1.283 1 + , 1-1.283 ( A1'733
T V ' m y
Chapter 4: Tuning Rules for PID Controllers 199
Rule
Minimum ITAE -Ou and Chen
(1995).
Minimum ITSE -Huang and Chao
(1982).
Minimum IAE -Huang and Chao
(1982).
Minimum IAE -Kaya and Scheib
(1988). Model: Method 5
Kc
Kc
1.357
Km
0.994
Km
(302) _
f N 0.947
Is]
(T v 0.907
Ti T (302) _
Tm O 0-842 [Tm J
15 j (303)
Td
14 T (302)
T (303) ' d
Comment
Model: Method 46
Model: Method 1;
N = 8
Minimum performance index: servo tuning f N-1.00
0.673 U \ Km I j J
K m ^ x m j
.04432
16 T (304)
17 T (305)
T (304)
X (3"5) lA
Model: Method 1;
N = 8
0 < ^ - < l ; T
N=10
14 x <302>
N = ± (302)x (302) 0.75K r aKcl^'Td
T i « 3 0 2 > + K m K c ^ ( T d ( 3 ° 2 ) + T i ^ - T m - T j N > 0 .
15 T ( 3 0 3 ) = 1 - 0 3 2 T Is. | , Td<303> = 0.570T„
V m T
16 T <304)
17 T . ( 3 0 5 >
0.148 'm
• > T , (304) = 0.502T,
+ 0.979 T
V m J
0.9895 + 0.09539-- .T,
(305) = 0.50814T v T
200 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum IAE -Witt and
Waggoner (1990).
Model: Method 2
Minimum IAE -Smith and
Corripio(1997)-page 345.
Model: Method 1
Kc
18 K (306)
19 K (307)
0.83Tm
K m x m
Ti
j (306)
T (307) i
T m
Td
T (306)
T (307)
0.5xm
Comment
Preferred tuning;
0 . 1 < ^ _ < 1 ; T
Ne[l0,20] Alternate tuning
0.1 < ^ - S 1.5; T
N not specified
18 ^ (306) 1.086
K „ T V ' m /
,(306)
1 + J l -1 .392 0.74-0.13-
0.696T , N 0 . 9 1 4
T V m J
1-J l -1 .392 / \u.»oy r ' T„ 1 I
T ^ 0 . 7 4 - 0 . 1 3 -
0.696T„ (306)
1 + J l -1 .392 , x 0.869
T
T V m J
0.74-0.13-
19 j , (307) 1 - J l - 1.392 , N 0.914
T
T V 1 m J
0.74-0.13-
0.696T . (307)
1 + J l -1 .392 0.74-0.13-
0.696T„ / \ 0.914
(307) T
l - J l - 1 . 3 9 2 f \ 0.869
T 0.74-0.13-
Chapter 4: Tuning Rules for PID Controllers 201
Rule
Minimum ISE -Huang and Chao
(1982).
Minimum ISE -Kaya and Scheib
(1988). Model: Method 5 Minimum ITAE
- Huang and Chao (1982).
Minimum ITAE -Kaya and Scheib
(1988). Model: Method 5
Kc
0.787 fTm^
21 K (309)
0.684
Km
/ \ 0.995
VXnJ
23 K (311)
Ti
20 ry (308)
T (309)
22-p (310)
T (311)
Td
T (308) x d
T (309)
T (310) 1 d
T (311)
Comment
Model: Method 1;
N = 8
0 < I H . < 1 ; T
N=10
Model: Method 1;
N = 8
0 < ^ 2 - < l ; T
m
N=10
20 T (308) 1.678T, TH(308) =0.594T„
f \ 0.971
T V ' m J
2, K (309) _ 0.71959 K „
T (309) _ _
1.12666-0.18145-
,0.86411
T <309) =0.54568Tm U *
22 T (310)
23 v (311) K
0.114| -!=-| + 0.'
1.12762 fTm
(310) = 0.491T„ v T \ m
0.80368
,(311)
0.99783 +0.02860-^ T
.(311)
202 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ITAE -Witt and
Waggoner (1990).
Model: Method 2
Kc
24 K (312)
25 K (313)
T;
T ( 3 1 2 )
T (313)
Td
T (312)
T (313)
Comment
Preferred tuning;
0 . 1 < - ^ - < l ; T
N g [10,20]
Alternate tuning
24 K (312) = 0-965 fxm
K m I T„ 1 + , 1 -1 .232
T i 0.796 -0.1465-JH-
0.616T, •(312)
1 - , 1-1.232
r ^0.85 X
T V 1m J
0 .796-0 .1465-
0.616T„ (312)
1 + J l - 1 . 2 3 2
, N 0 . 8 5
T V 1m J
0 . 7 9 6 - 0 . 1 4 6 5 - ^ -T
2 5 j c ,3,3) _ 0.965
K „ T 1 - J l -1 .232
f N 0.929 r
^ 2 - \ 0.796 - 0 . 1 4 6 5 ^
v T m J 1 Tra
0.616T„ .(313) T
V m
1 + , (1-1 .232
f N 0 . 8 5
T 0 .796-0 .1465-
0.616T„ (313)
( . \ T V m 7
1 - , 1-1.232 0 .796-0 .1465-
Chapter 4: Tuning Rules for PID Controllers 203
Rule
Minimum ITAE - Ou and Chen
(1995).
Minimum USE -Huang and Chao
(1982).
Tsang et al. (1993). Model:
Method 15
Tsang and Rad (1995). Model:
Method 15 Smith and
Corripio(1997)-page 346.
Model: Method 1
Kc
K c ( 3 1 4 , =
0.965 ( l ^
0.718 f
K m „
T 1 X m J
0.85
1.00
K r a tm
* i
1.6818 1.3829 1.1610
\ 0.0 0.1 0.2
0.809Tm
Km tm
0-5Tra
Kro^m
T i
26 T (314)
27 T (315)
Td
T (314) x d
T (315)
Direct synthesis
T 0.25xra
Coefficient values
* i
0.9916 0.8594 0.7542
\ 0.3 0.4 0.5
T
T
x i
0.6693 0.6000 0.5429
\ 0.6 0.7 0.8
0-5tm
0-5xra
Comment
Model: Method 46
Model: Method 1;
N = 8
N = 2.5
x i
0.4957 0.4569
£ 0.9 1.0
Overshoot = 16%;N=5
Servo - 5% overshoot; N not
specified
26 T ( 3 1 4 )
0.796-0.1465--. T,
(314)
T
N = ±-
•• 0 . 3 0 8 T „
0.75KmK ( 3 1 4 ) T (314)
(314) + KmKclJ""lV"+T i
27 T (315) T
(314) | T (314) + T ( 3 1 4 )
028
N > 0 , '^m-TrnJ
, ( 3 1 5 ) : 0.596T„.
0.063 ^ + 1.061 , T
204 Handbook of PI and PID Controller Tuning Rules
Rule
Schaedel(1997) Model:
Method 21
O'Dwyer (2001a).
Model: Method 1
Wang et al. (2002).
Model: Method 1
Kc 28 K (316)
X l T m
K m X . r
Ti
T 2
T , - - i 2 -T,
N
Td
-y/T,2-2T22
T
Representative result
0.079Tm
K m T m
1
K4 i+2r] 0-65 f - Q
1.04432
30 K (318)
0.1Tm
Tm
4 + ^ 29 j (317)
x (318)
Tm
0.5Tm
T (317)
T (318)
Comment
N not defined
A m = w / 2x ,N ;
K = 0.57c-x,N
A m = 2 . 0 ;
K = 45°
Labeled IMC; N = 1 0
Labeled IAE setpoint; N = 1 0
Labeled IAE load; N = 10
0.375 ( T 2A
28 K (316)
T,
T,
Kn
T
N T, v ' 2T,
J
1 + 3.45-
- + Tm,T,2=1.25Tmxam+ I i L _ T m + o . 5 T r a \ 0 < ^ < 0.104;
1 + 3 . 4 5 ^ -
L m > l2
T.= ° ' 7 T m + T m , T 22 = T m T a
m + _ ° - 7 T m _ + 0 . 5 x m2 , j ^ > 0.104.
Tm-0.7C T m - 0 . 7 C
2 9 ^ ( 3 1 7 ) (317) , Td
,J " =0.50814
, s 1.08433
0.9895 + 0.09539^2-T
T
30 K (3.8) _ 0,98089
K „ — I , Ti(318) =1.09851Trt
/• >, 1.05211
, T
, (318) : 0.59974Tm
Chapter 4: Tuning Rules for PID Controllers 205
Rule
Wang et al. (2002) -
continued.
Huang et al. (2005). Model:
Method 42
Kc
31 K (319)
32 K (320)
33 K (321)
34 K (322)
0.65Tm
Kmxm
T, -p (319)
ry (320) i
T (321)
T (322) M
T m
Td
T (319)
T (320) 1 d
T (321)
T (322) 1 d
0.4xm
Comment
Labeled ISE setpoint; N = 1 0
Labeled ISE load; N = 10
Labeled ITAE setpoint; N = 1 0
Labeled ITAE load; N = 10
N=20 A m = 2 . 7 ,
* m =65°
The above is a representative, default, tuning rule; other such rules may be deduced from a graph provided by the authors for other Am ,4)m
values.
31 K (319) : 0.71959 ( T,
^ •m v ' m ;
,(319) _
1.12666-0.18145-
32 „ (320) _ 1.H907 K r - -
K „
(320) = ] 252QT ' X n
T (319) ' d
N 0 . 9 5 4 8 0
: 0.54568 f \ 0.86411
Td<320) = 0.54766T,,
T V m J
f \ 0.87798
T V m J
33 K (321) m 1-12762
K „ , T= (321)
0.9978 + 0.02860-
34 ^ (322) ;
, x 1.06401 f
0.77902 T^ (322) = 0 8 7 4 8 1 T N
T (321) x d
0.70949
= 0.42844
Td(322) =0.57137Tm
206 Handbook of PI and PID Controller Tuning Rules
Rule
Harris and Tyreus(1987).
Model: Method 1
Chien (1988). Model: Method 1
Zhang et al. (1996). Model:
Method 38 Zhang et al.
(2002). Shi and Lee
(2002). Model: Method 1
Lee and Shi (2002).
Model: Method 1
Kc Ti Td Comment
Robust
Tm
K m ( x m + ^ )
Tm
K n (X + 0.5Tn)
0.5xra
K m ( ^ + 0.5xm)
35 K (323)
36 K (324)
37 K (325)
38 K (326)
39 g (327)
T
Tm
0.5xm
0.5xra
T m
T m
Tm m
T
40 j (328)
0-5xm
0.5xm
Tm
Tm
0.5xm
0.5xra
1 tan
» g I 2 J
0. " m
N = ( x m + ^ ) A ; X > max imum [0.45xm)0.1Tm]
(Chien and Fruehauf
(1990)); N=10
^e[0.2xm ,
Model: Method 1
Suggested 0.6
Suggested 0.6
35 K (323) _ _ 0 - 5 ^ N _ T m ( 2 X + 0.5xm) Km(o.5xm+2\)' tf
36 K (324) =
K m ( 0 . 5 x m + 2 ^ )
Tm
K m ( 2 ^ + 0.5xra)
, N = 0.5xm(2^ + 0.5xm)
N = 1 0 o r N = K
38 ^ (326)
39 v (327)
"bvyTm
K„ l + ^ t a n f e ^ zr , N = l +
0.5xm(2X + 0.5xm)
2co
< » „
-tan g O v 2 j
K J°bwTm
K „ 1 + ^ t a n V 2 ,
, N = 1 + ^ t a n [ ^
4 0 T ( 3 2 8 ) = J J ! L ^ 1 + T n L + _ l _
Tm m b w t r a
Chapter 4: Tuning Rules for PID Controllers 207
Rule
Huang et al. (2005). Model:
Method 42
Kc Ti Td Comment
Ultimate cycle
41 K (329) T (329) T (329)
N=20 A m = 2 . 7 ,
The above is a representative, default, tuning rule; other such rules may be deduced from a graph provided by the authors for other Am ,<|>m
values.
41 ^ (329) _ 0-65 K „ - 1
TI - tan" K,
,T il " ! "=0 .159T u J Ku - 1
Td(329) = 0.064 T„ 7t-tan ' J K„
208 Handbook of PI and PID Controller Tuning Rules
4.1.9 Classical controller 2 Gc (s) = Kc
E(s) R(s)
<g>-K 1 + -1 Yl + NTHs
XsA 1 + Tds
1
U(s)
T i s y
1 + NTds
V 1 + Tds j
K „ e '
l + sT„
Y(s)
Table 52: PID controller tuning rules - FOLPD model Gra (s) = -K„,e~
l + sT„
Rule
Hougen(1979)~ page 335.
Model: Method 1
Kc T; Td Comment
Direct synthesis: frequency domain criteria
1 j r (330) T 0.45^E-N
Maximise crossover frequency; N e [10,30]
1 Values deduced from graph. v (330) n A/T^ _ I T Kc
(330) =9.0/K ra,xm/Tm =0.1;KC(330) =4.2/Km,xm/Tm =
Kc(330)=2.8/Km,xm/Tm=0.3;Kc
(330»
K/330)=1.7/Km)x r a/Tm =
0.2;
K (330)
£ (330)
= 2.1/Km,xm/Tm=(
0.5;KC(330)
: m /T m =0 .8 ;
0.9;Kc(330 )=0.85/Km ,Tm/Tm=1.0.
.,l^m,.ml,m-^,^c =1.45/Km,Tm/T ra=0.6
= 1.2/K r a ,xm/Tm=0.7 ;Kc( 3 3 0 )=l.l/Km ,x
= 0.95/Km,Tm/Tm=0.9;Kc(330) =
Chapter 4: Tuning Rules for PID Controllers
4.1.10 Series controller (classical controller 3)
209
(l + sTd)
Table 53: PID controller tuning rules - FOLPD model G m (s) = Kme"
l + sT„.
Rule
Kraus(1986). Model:
Method 19 Tan et al.
(\999a) - page 25.
O'Dwyer (2001b).
Chien et al. (1952) equivalent
- regulator
O'Dwyer (2001b).
Model: Method 2
Kc
0.833Tm
Kraxm
0.6Tra
KmXm
x j m
Kmxm
* 1
0.7236
0.84
x2
1.8353
1.4
Representatii 2 K (331)
T; Td
Process reaction
l-5xm
*m
x 2 T m
0.25xm
^m
X 3 t m
Representative results X 3
0.5447
0.6
0% overshoot; 0.1 <
Comment
Foxboro EXACT controller pre-
tuning
Model: Method 2
Model: Method 2
^m/T m <l
20% overshoot; 0.1 < i m /T m < 1
/e results - Chien et al. (1952) equh T(331) T (331)
alent - servo 0% overshoot; Tm /Tm<0.5
K (331) 0.3T„ 1-J1
2x„ .(331) 1 + J l -
2x„
210 Handbook of PI and PID Controller Tuning Rules
Rule
O'Dwyer (2001b)-continued.
Tsang et al. (1993). Model:
Method 15
Pessen (1994). Model: Method 1
Kc
3 K (332)
x j r a
x l
1.8194 1.5039 1.2690
S 0.0 0.1 0.2
0.35KU
Ti
T (332)
Td
T (332) 1d
Direct synthesis
T m
x i
1.0894 0.9492 0.8378
\ 0.3 0.4 0.5
0.25xm
x i
0.7482 0.6756 0.6170
% 0.6 0.7 0.8
Ultimate cycle
0.25TU 0.25TU
Comment
20% overshoot; * m / T m < 0.7234
x i
0.5709 0.5413
I 0.9 1.0
0.1<^s-<l T
m
3 j , (332) _ 0 .475T,
Km tm
1+ 1-1.3824- T;(332) =0.68T„ 1 + . 1-1.3824-
Td(332) = 0.68Tn 1 - 1 - 1 . 3 8 2 4 -
Chapter 4: Tuning Rules for PID Controllers 211
4.1.11 Classical controller 4 Gc(s) = Kt 1 + - ^ L
1 + ^ N ;
Table 54: PID controller tuning rules - FOLPD model K „ e ~
l + sTm
Rule
Hang et al. (1993b) -page
76. Model: Method 2
Chien (1988). Model: Method 1
Kc
0.83Tm
K m T m
Tm
Km(X + 0.5Tm)
0.5xm
Km(X + 0.5xm)
Tf Td
Process reaction
l - 5 T m 0.25xm
Robust
Tm
0.5xm
0.5tm
Tra
Comment
Foxboro EXACT controller
'pretune'; N not specified
^ 6 [ T m. T m] (Chien and Fruehauf
(1990)); N=10
212 Handbook of PI and PID Controller Tuning Rules
4.1.12 Non-interacting controller 1
U(s) = K, v T i s y
E ( s ) — ^ - Y ( s )
N
R(s) E(s) K, '.+-0
V T i s /
U(s) — •
Ke~
1 + sT
Tds T
1 + ^ - s N
Y(s) — •
Table 55: PID controller tuning rules - FOLPD model Gm (s) = Kme^
l + sT„
Rule
Huang et al. (2000). Model:
Method I
Kc T i Td Comment
Direct synthesis
0.8(Tm+0.4xm)
Kmxm
0.6(Tm+0.4Tm)
""m m
Tm+0.4xm 0-4Tmxm
Tm+0.4Tm
N = min[20,20Td];
regulator tuning
N = min[20,20Td];
servo tuning
Chapter 4: Tuning Rules for PID Controllers
4.1.13 Non-interacting controller 2a
U(s) = K
213
v T i s y E(s) % - Y ( s )
1 + -N
R(s) / C > E(s) K,
U(s)
^ + < g > - 1 + sT
Tds T
N
Y(s)
Table 56: PID controller tuning rules - FOLPD model Gm (s) = K „ e "
l + sT„.
Rule
Minimum ISE -Zhuang and
Atherton(1993). Model: Method 1
Minimum ISTSE - Zhuang and
Atherton(1993).
Kc T( Td Comment
Minimum performance index: servo tuning
1.260
Km
/ \ 0.887
—] 1.295 f T j
0.619
/ x 0.930
1.053 fT r a j
Km [ x j
4 T ( 3 3 3 ) i
5 T (334)
6 ^ ( 3 3 5 )
T (333)
T (334)
T (335) Xd
0.1<^a-<1.0 T
1.1 < ^2 -< 2.0 T
m
0.1 <^S-< 1.0 ; T
m
Model: Method 1
4 T (333) _ _
0.701-0.147-, Td
(333) = 0.375T„ (. ^
T V 1 m j
,N=10.
5 T (334) _ _
0.661-0.110-
-,Td<334)=0.378Tm ' T" , N=10.
6 j ( 3 3 5 )
0.736-0.126-, Td
<335) = 0.349T,, / N0 .907
X,
T ,N=10.
214 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ISTSE - Zhuang and
Atherton (1993) (continued)
Minimum ISTES - Zhuang and
Atherton (1993). Model: Method I
Minimum ISTSE - Zhuang and
Atherton (1993).
Kc
1.120
0.942
1.001
(T ^
\ T m J
0.625
0.933
r N 0.624
10 £ (339)
11 K (340)
Ti
7 T (336)
8 j ( 3 3 7 ) i
9 T ( 3 3 8 ) i
•p(339)
0.271KuTuKm
Td
T (336)
T (337)
T (338) ld
0.112TU
0.116TU
Comment
1.1 < ^2-< 2.0; T
N=10
0.1 <-^=-< 1.0 T
1.1 < ^2-< 2.0 T
0 .1<^2-<2.0 ; T
m
Model: Method 1
0.1 <-^2-< 2.0; T
m
Model: Method 31
7 j (336)
• (337)
9 j (338)
0.720
0.770
-0.114^2-T
T
-0.130^2-T
T m
, Td<336) = 0.350T„ ' x . ^
(337) : 0.308T„
T V m J
, N 0.897 X„
T V ni J
N=10.
0.813
, Tdl"°' = 0.308Tm \—\ , N=10. (338)
0.754 -0.116^2-T
m
l l K c ( 3 4 o ) = 2 . 3 5 4 K m K u - 0 . 6 9 6 £ u ) N = 1 0
3.363KmKu + 0.517
Chapter 4: Tuning Rules for PID Controllers
4.1.14 Non-interacting controller 2b
215
R(s) E(s)
Xs
U(s) = Kc + E ( s ) - ^ ! _ Y ( s )
N
+ o U ( s )
*® > K„e"
1 + sT
Y(s)
T„s
T
N
Table 57: PID controller tuning rules - FOLPD model l + sTm
Rule
Minimum IAE -Kaya and Scheib
(1988). Minimum ISE -
Kaya and Scheib (1988).
Kc T( Td Comment
Minimum performance index: regulator tuning
12 K (341)
13 K (342)
T (341)
T (342) i
T (341) i d
X <342>
Model: Method 5
12 K (341) _ 1-31509 K „
13 K (342) _ 1.3466 [ Tra
x(3"l) _ 'm
'' 1.2587
(341)
T
T / " ' =0.5655Tm| —2-
^ 0.4576
0 < -SL < 1 ; N=10. T
Km VTmJ
,0.9308
. T i (342)
1.6585 T
. (342) T d ^ z ' = 0.79715T„
f x0.41941
T V 1m J
0<-!S-<l;N=10. T
216 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ITAE -Kaya and Scheib
(1988).
Minimum IAE -Kaya and Scheib
(1988). Minimum ISE -Kaya and Scheib
(1988). Minimum ITAE -Kaya and Scheib
(1988).
Kc
14 K (343)
T,
T (343) 1 i
Td
T (343)
Comment
Model: Method 5
Minimum performance index: servo tuning
15 K (344)
16 K (345)
17 K (346)
T ( 3 4 4 )
T (345)
T (346)
T (344)
T (345) ld
T (346) x d
Model: Method 5
14 K (343) 1.3176
K „
- (343)
1.12499 [Tm
15 K (344) 1.13031 I Tm
Km l T m /
Td(343) = 0.49547T„
• (344) _ _
, T 0 < - ^ - < l ; N = 1 0 .
T
5.7527-5.7241-
1.26239 (l„ . T j
(344)
(345)
/• \ 0.17707
T V m /
, 0 < - S - < l ; N = 1 0 . T
6.0356-6.0191-
Td( 3 4 5 )=0.47617Tm ;^ 0 < - ^ - < l ; N = 1 0 .
T
,7 „ (346) 0.98384 (Tm~)
K „ l T m , T-
(346) _ _
2.71348-2.29778-
• (346) 0.21443T,
T
N 0.16768
0 < - a - < l ; N = 1 0 . T
Chapter 4: Tuning Rules for PID Controllers 217
4.1.15 Non-interacting controller based on the two degree of freedom structure 1
f \ f \
U(s) = Kc 1 T„s
1+ — + -
v N
E(s)-Kc
J
a + PTds
1 + ^ s N
R(s)
( \ PTds
a + T 1 + ^ s
N ;
K
-V& +
R(s) E(s)
K, 1 T„s
1 + — + T's l + i s
N ;
U(s)
* g > — •
Y(s)
K^e" 1 + sT
Table 58: PID controller tuning rules - FOLPD model G m (s) = K m e "
l + sTm
Rule
Hiroi and Terauchi (1986)
- regulator. Model:
Method 2;
a = 0.6; (3 = 1
Kc T, Td Comment
Process reaction
0.95Tm
Km T m
1.2Tm
K m T m
2.38xm
2 t m
0.42tm
0.42tm
0% overshoot;
0 . 1 < ^ - < 1 T
m 20% overshoot;
0.1<-^=-<l T
218 Handbook of PI and PID Controller Tuning Rules
Rule
Taguchi and Araki (2000).
Shen (2002). Model:
Method 5; N=0, (3 = 0,
M s < 2
Kc Ti Td Comment
Minimum performance index: Servo/regulator tuning 1 £ (347) T (347)
i T (347) 1 d
Overshoot (servo step) < 20%
Minimum performance index: other tuning oo oo
Minimum f|r(t) - y(t)|dt + f|d(t) - y(t)|dt
0 0
2 K (348) T (348) T (348) 0 < I J 2 L < O O
T
1 v (347) _ _J_ K, 0.1415 + -
1.224
-0.001582
• (347) 0.01353+ 2.200-2--1.452 T
+ 0.4824
(347) 0.0002783 + 0.4119—E-- 0.04943 T
2~\
a = 0.6656 - 0.2786-2- + 0.03966 T
, N fixed (but not specified);
P = 0.6816 -0.2054-J2- + 0.03936 T
2 K (348)
.(348)
K m T m exp 2.94-11.63
T + T
•< 1.0. Model: Method 1.
+ 11.15 VTm+T
ra
(348)
: xm exp
: xm exp
1.88-3.63 V ^ m "•" ^m J
f + 0.86
VTm+T r a y
•0.25-0.06 VTm+Tm y
-1.99 ( , 1 m + ' n i ;
a = 1 - exp -0.22-0.90 ^ + T m ;
+ 1.45 * m + T m /
Chapter 4: Tuning Rules for PID Controllers 219
Rule
Kasahara et al. (1997).
Model: Method 1
Huang et al. (2000).
Model: Method 1
Srinivas and Chidambaram
(2001).
Kim et al. (2004). Model:
Method 1
Kc
3 K (349)
^Tm+0.4T t t
Kmxm
a
4 K (350)
x,/Km
T m / T m
0.1 0.2 0.3
T T
Ti Td
Direct synthesis •p. (349)
+ 0.4xm
T (349)
Comment
0-4Tmtra
Tm+0.4xm
Closed loop transfer function is 4th order; N = 0
For X = 0.80 , A r a=2.13
= ^ 1, P = 0 , N = min[20,20Tdl.
X]
12.5 6.1 4.1
T (350)
x2Tm
T (350)
x 3 T m
Coefficient values x2
0.22 0.41 0.57
x3
0.04 0.08 0.11
Model: Method 1;
N = oo
a
0.68 0.63 0.62
P 0.75 0.70 0.70
3 K (349)
K.
0.2xm3 + 1.2x 2T + 5 . 4 T T 2 + 9.6T 3
Tm(xm+3Tm)2
T i( 3 4 9 ) =1 .667x m i ! i + 3 T
-1.389T„
(349)
frm+3Tm)3
x m + 2 T m - " • - (x m + 2T m ) 4
T m2 T m (T m +3Tj
1 .2 (x m +2T r a ) 3 -x m (x m + 3T m ) 2 '
ct = l-0.723x„
P = l-0.261x r
^m+3Tm
V ^ + 2 T m
T „ +2T„
1.667x„ *m+3Tm) 1 „„ n T m
2 (x m +3T m ) 3
1%,+2T„ -1.389-
(xm+2Tm)4
1
X ( 3 4 9 ) T (349) •
'Sample Kc<350), j . * 3 5 0 ) ^ * 3 5 0 ' taken by the authors correspond to the process reaction
tuning rules of Zeigler and Nichols (1942); a =
T „ T „
— - 4 = 0.5-0.83^
and P = • (350) K„<350,Km 2T: (350)
T ( 3 5 0 )
= -0.16 for this tuning rule
220 Handbook of PI and PID Controller Tuning Rules
Rule
Kim et al. (2004) - continued
Leva and Colombo (2004).
Hang and Astrom (1988a).
Kc
T m / T m
0.4 0.5 0.6 0.7 0.8 0.9 1.0
OS < 20% (
5 K <35
6o.6K,
70.6K
* 1
3.1 2.5 2.11 1.82 1.61 1.44 1.33
(servo step ^hien et al.
)
i
T, Td
Coefficient values x2
0.71 0.83 0.94 1.05 1.13 1.22 1.26
x 3
0.15 0.18 0.21 0.24 0.28 0.31 0.34
Comment
a
0.59 0.58 0.56 0.54 0.51 0.48 0.49
P 0.69 0.69 0.69 0.69 0.68 0.69 0.66
i; settling time < that of corresponding rule of 1952); a cost function is minimized.
Robust T (351) T (351)
Ultimate cycle
0.5TU
ry (352)
0.1251
0.125T
Model: Method 41
u
u
Model: Method 2
5 K (351) Tm + 2(Tm+X) Km(xm+X)
T-(351) = Tm + 2 ( x m + ^ ) '
(351) = V . l ' , ^ ; " » N _ 2Tm(tm + f }
2Tm(T ra+X)+Tm2 2 (T m + X) ' x[2Tm{xm+X)+xm
2}
3AT a not specified; (3 = 1. A, > — , Aim = maximum variation in the time delay, with
only variation in the time delay considered (Leva and Colombo (2004)); X chosen so
that nominal cut-off frequency = <x>u (Leva, 2005).
a = l - 2 ( x - 0 . l ) - 1 , 6 6 T m , i s - < 0 . 3 ; a = l - 2 x - ^ - , 0 . 3 < ^ - < 0 . 6 ;
x = % overshoot. N=10, P = 1.
7 T (352 ) 0.5 1.5-0.83- T„. a = - ^ - - 0 . 6 , 0.6 < ^2-< 0.8; T T
a = 0.2, 0 .8<-S-<1 .0 . T
Chapter 4: Tuning Rules for PID Controllers 221
Rule
Hang and Astrom (1988a)-continued
Hang et al. (1991) - servo.
Hang and Cao (1996). Model:
Method 31
Kc
0.6KU
8 0.6K„ 9 0.6K U
0.6KU
T i
0.335TU
0.5TH
0.222KTU
10 y (353)
Td
0.125TU
0.125TU
0.125TU
T (353) 1d
Comment
-^2->1.0; T
a = 0.2 Model:
Method 2; (5 = 1
0 . 1 < ^ L < 0 . 5 ; T
N=10; (5 = 1
0 . 1 6 < ^ - < 0 . 5 7 ; N = 10. a = l r , 10% overshoot; Tm 15 + K
a = 1 5 i _ , 20% overshoot with K = 2\ 1 1 ^ - / ^ - " J + 1 3
27 + 5K l37 [T m /T m ] -4 /
9 0.57 < - ^ s - < 0.96 ; N =10. ce = l - — f - K ' + l 1, 20% overshoot, 10% undershoot. 1 7 1 9
10 T.(353) =1 0.53-0.22 T T U > X (
(353) l T
0 . 5 3 - 0 . 2 2 — — . a takes on values of L 4
1.0, 0.2 and 0.40(xm/Tm)2 -0.05(xm/Tm)+0.58 at different points on the open loop
process step response.
222 Handbook of PI and PID Controller Tuning Rules
4.1.16 Non-interacting controller based on the two degree of freedom structure 2
f \ f \
U(s) = Kc 1+ — + - a
X s , . Td
V 1 + ^ s
N ;
E ( s ) - K £
( \
pTds x__
l + ^ s 1 + T*S
v N
a + - K „
+
R(s) E(s)
K „ , l T d s
1+ — + - d T.8 l + i8
N .
a + PTds X
Td„ l + TiS 1 + ^ s N
U(s)
+
K e"
l + sTm
R(s)
Y(s)
Table 59: PID tuning rules - FOLPD model Gm (s) = Kme-l + sT„
Rule
Hiroi and Terauchi (1986) -
regulator. Model: Method 2
Kc
0.95Tm
K m T m
l-2Tm
Kmxm
a
Ti Td
Process reaction
2.38tm
2tm
0.42xm
0.42xm
= 0.6; p = l or (3 = -1.48 ; x =
Comment
0% overshoot;
0.1<^H-<1 T
m 20% overshoot;
0 . 1 < ^ - < 1 T
= 0.15
Chapter 4: Tuning Rules for PID Controllers 223
4.1.17 Non-interacting controller based on the two degree of freedom structure 3
U(s) = K c [ ( b - l ) + (c- l )T ds]R(s) + K ( Xs
R(s)-1
(l + sT f ): -Y(s)
K c[(b-l) + (c-l)Tds]
+ ir U(s) Y(s)
K e"
1 + sT
(l + sT f)2
Table 60: PID controller tuning rules - FOLPD model Gm (s): Kme"
l + sT„
Rule
Astrom and Hagglund (2004).
Astrom and Hagglund (2004).
Kc Ti Td Comment
Minimum performance index: other tuning 11 K (354) T (354)
i 0-5xmTm
°-3*m +Tm
Model: Method 1
No precisely defined performance specification 12 j r (355) T (355) T (355) Model:
Method 45 No precisely defined performance specification
11 K (354) = _
K „ 0.2 + 0.45-
Xm+0.1Tm m T
b = l , - 2 L > l ; c = 0; Tf = 0.1xn
12 K (355) = _
K „ 0.2 + 0.45 - T - + 0 - 5 r '
Tm+0.5Tf
, b = l , c = 0. Tf =-, (355) d
N ,Ne[4,10];
355) = 0 .4xm +0.8Tm +0.6T f
xm+0.1Tm+0.55Tf
T<355) _ 0.5(Tm+0.5Tf)(Tm+0.5Tf) . xm ? }
0.3xm+Tm+0.65Tf ' Tra
224 Handbook of PI and PID Controller Tuning Rules
4.1.18 Non-interacting controller 4
U(s) = Kc 1
v T>sv
R(s) E(s)
+ X K,
v T . sy
1 +
U(s)
E(s)-KcTdsY(s)
—+®~2—-K „ e ~
1 + sT
Y(s)
K J d s
Table 61: PID controller tuning rules - FOLPD model Gm (s) = Kme"
l + sT„
Rule
VanDoren (1998).
ABB (2001). Model: Method 2
Minimum IAE -Shinskey (1996)
- page 117.
Minimum IAE -Shinskey (1988)
-page 143. Model: Method 1
Minimum IAE -Edgar et al.
(1997) -pages 8-14, 8-15.
Minimum IAE -Shinskey (1988)
-page 148. Model: Method 1
Kc T, Td Comment
Process reaction l-5Tm
Kmxm
1.25Tm
Kraxm
2.5xra
2^m
0-4xm
0-5Tm
Model: Method 2
Minimum performance index: regulator tuning
1.32Tm/Kraxra
l-32Tra/Kratra
1.35Tm/KmTm
1.49Tm/Kmin
l-82Tm/K ratm
1.30Tm/KmTm
l-33Tm/Kmtm
1.80xm
l-77tm
l-43xm
1.17xm
0.92xm
1.8Tm
2.1xm
0.44xm
0.41xm
0.41xm
0.37xm
0.32xm
0.45xra
0.63xm
T m /T m =0 .1 ; Model: Method 1
x m /T m =0.2
i m /T m =0 .5
^ m / T m = l
^ r a / T m = 2
Model: Method 1
Model: Method 2
Minimum performance index: other tuning 0.77KU
0.70KU
0.66KU
0.60KU
0.48T„
0.42TU
0.38TU
0.34TU
0.11T„
0.12TU
0.12TU
0.12TU
^ m /T m =0.2
xm /Tm=0.5
x m / T m = l
x m / T m = 2
Chapter 4: Tuning Rules for PID Controllers 225
Rule
Shinskey (1994) -page 167.
ABB (2001).
Haeri (2002)
Kc
1 K (356)
2 j . (357)
3 K (358)
T;
T 2
0.125-^-
T (357)
Direct s T (358)
i
Td
0.12TU
T (357)
vnthesis T (357)
Comment
Minimum IAE; Model: Method I
Model: Method 7
Model: Method 1
1 K (356) K „
3 . 7 3 - 0 . 6 9 - i T
_ > i < 2 . 7 ; Kc<356)
T „ T „
K „
2 v (357) = 1.3570 (Tm K r T/3 5 7 ' =1.1760T„
2 . 6 2 - 0 . 3 5 - ^ *m
^ T
0.738
Xm
±->2.1.
(357)
N 0.995
: 0 . 3 8 1 0 T J - ^ - , ^ - > 0 . 5 .
3 K (358) = _J_ 6.84 - + 0.64
1 + 2.33 T
•7.82j
T (358) _ 0.95+ 2 .58 -S- +3.57 T
( \2
T V m J
0 < - ^ - < 0 . 2 9 , T
,(358) _ ,
(358)
T (358) _ T 1 d _ T n
2.15 - 0 . 7 6 ^ 2 - + 0 . 3 3 - ^ Tm T„.
0.29 ^m """ T m
+ 3.94 V T m + ^
0.29 < — < 4 ; T
- 4.65 T
m + t m
, 0 < - 2 L < 1 . 7 9 ; T
0.87| -^2- - 0 . 4 9 ^ - 1 +0.09 > T
1 . 7 9 < - 2 - < 4 . T
226 Handbook of PI and PID Controller Tuning Rules
4.1.19 Non-interacting controller 5
U(s) = K
R(s)
*-(b- l )K c
E(s)
+ T K,
v T i s v
b + — v T i s y
E(s)-(c + Tds)Y(s)
+
U(s)
>® K „ e "
l + sT„
Y(s)
(c + Tds)
I Table 62: PID controller tuning rules - FOLPD model Gm (s) = -
K„,e~
l + sT„
Rule
Chidambaram (2000a).
Model: Method I
Kc
4 K (359)
5 l-2xm
KmT ra
T| Td
Direct synthesis T(359)
0-5Tm
T (359)
0.15xm2
KraTm
Comment
c=Kc(359>
1.2T c - ra
K m T m
4 K (359) =
K „ 4.114-6.575-J2- + 3.342
T 'O*
T
,(359) 0.311 + 1.372^2--0.545 T
r \ 2
T V m J
(359) K (359)T(359)
16.016-11.7
b = 0.793-3.149^L + 8 . 4 0 5 ^ - -8.431 T T A m V m J
b = 0.0584 + 0.44^2. o.5 < - < o.9 . T T
, T •<0.5,
5 b = 0.3238 + 0.4332^- Im. < 0.8 ; b = 0.8507 , ^ - = 0.9 , T T T
m m Jm
Chapter 4: Tuning Rules for PID Controllers 227
4.1.20 Non-interacting controller 6 (I-PD controller)
R(s)
-J® E(s) K .
Xs
U(s) = E ( s ) - K c ( l + Tds)Y(s) T:S
U(s) Kme"
1 + sT
K c ( l + Tds)
Z3Z
Y(s)
Table 63: PID controller tuning rules - FOLPD model Gm (s) = Kme"
l + sT„
Rule
Suyama(1993).
Shigemasa et al. (1987),
Suyama (1993).
Kc Ti Td Comment
Direct synthesis: time domain criteria 6 R (360) rj, (360)
i T (360) *d
Closed loop transfer function is 4lh order 7 j r (361) T (361) T (361)
*d
Closed loop transfer function is 4th order Butterworth
Model: Method 1
Model: Method 1
6 K (360)
K „
0.2xm3 + 1.2xm
2Tra + 5.4xmTm2 +9.6Tm
3
^m(^m+3T ra)2
, (360) : 1 . 6 6 7 t mT m + 3 T m - 1 . 3 8 9 x m
2 j T m + 3 T m j ' m x m + 2 T m ( x m + 2T m ) 4
(360) T m2 t m ( t m +3T m )
1.2(xm+2Tm)3-xm(xm+3T r a)2
7 K (361) 1.3319(xm+2Tm)3 1
Kmxm(xm+3Tm) 2 Km '
T ( 3 6 l ) = 1 2 5 3 2 x m + 3T m _ L 6 9 ^ 2 ( x m + 3 T m )
(361)
T. .+2T, ™ ( T m + 2 T m r
1.5095(xm +2Tm)2 -(x r a +TmXtm +3Tm) : t m ( x r a + 3 T m ) 1.3319(x r a+2Tm)3-xm(xm +3T„
228 Handbook of PI and PID Controller Tuning Rules
Rule
Shigemasa et al. (1987),
Suyama (1993) (continued).
Model: Method 1
Kc
8 K (362)
Ti
T (362)
Td
T (362) *d
Closed loop transfer function is 4th order ITAE 9 K (363) T (363)
i T (363)
Closed loop transfer function is 4th order Bessel 10 K (364) T ( 3 6 4 ) T (364)
Closed loop transfer function is 4th order Binomial
Comment
Toshiba AdTune TOSDIC211D8
product
K (362) 2.6242(xm+2Tm)3 1
K m t r a (T m + 3T r a ) 2 Kr
T i( 3 6 2 ) = 1 . 8 8 9 8 x m i ^ ^
' m T J-9T
T (362) Xd
K (363)
.8898xm T m + 3 T m -0.720x 2 ym+yTm]t ,
^ r a + 2 T m (x m + 2T m ) 4
_ T (z + T r ^.3130(xm + 2 T m ) 2 - ( x m + T m X x m + 3 T m )
2 .6242(x m + 2T r a ) 3 -x m (T m + 3T m ) 2
0.9450(xm+2Tm)3 1
K m x r o ( x m + 3 T j 2 Km
T(363) = 3.3333xm T m + 3 T m
1 T + 2 T
333xm 1X12M._3.527xm2 .(Tm+fm|] ,
T(363) / , _ U.3444(x m +2T m ) 2 - (x m +1
1.9450(xm+2Tm)3-xmO
2 T m ) 2 - ( x m + T m X x m + 3 T m )
1.9450(xm+2Tm)3-xm(xm+3Tm)2
K (364)
m
_ 0 . 5 6 2 2 ( x m + 2 T j 3 1
K m x r a ( x m + 3 T j 2 Kr
T/364' = 5.3337xm T m + 3 T m - 9.488xm
2 ( T " + 3 T " )3
1 111 ."IT-. Ill / _
^ m +2T m (xm+2T,
T (364)
m x m + 2 T m ( x m + 2 T m ) 4 '
T (T + 3T ^ •1244(x m + 2T m ) 2 - (x m + T m XT m +3T r
0.5622(x r a+2Tm)3-xm(xm +3Tm)2
Chapter 4: Tuning Rules for PID Controllers 229
Rule
Chien et al. (1999).
Kasahara et al. (2001).
Model: Method 1
Ozawa et al. (2003).
Minimum ISE -Argelaguet et al.
(2000).
Kc
11 K (365)
12 j r (366)
13 K (367)
14 K (368)
T,
T ( 3 6 5 )
ry (366) i
T (367) i
T ( 3 6 8 ) i
Td
T (365)
T (366)
T (367) 1 d
T (368)
Comment
Model: Method 1
OS = 0%
OS = 10%
Model: Method I
OS e[0%,10%]
Minimum performance index: servo tuning 2 T m + *m
2K ratm Tm+0.5xm
T X
2T m +t m Model: Method 1
Underdamped system response - Z, = 0.707 . xm > 0.2T,,,
K (365) 1.414TC L 2Tm+TmTm +0.25Tm2-Tc
2
" ^ X.,,2+0.707T r , ,Tm+0.25Tm2
CL2 "
(365)
'CL2 l m
^ C L 2 T m "T l - m 1 m 1.414Tri,T +xmTm+0.25xmz
(365)
Tm+0.5xm
_ 0.707TroTCL2Tro +0.25Tmxm2 -0.5xmTCL2
2
Tmxm +0.25xm2 +1.414TCL2Tm -T C L 2
2
1 2 K ( 3 6 6 ) _ 0.28l(xm+2Tm)3 1
K m x m ( x m + 3 T m ) 2 Kn
, (366) :5.33x„ t m + 3 T „
-18.98xn (xm+3Tm)3
(366)
t m + 2 T m - ' ( x m + 2 T m ) 4 '
,0.562(xm + 2 T m ) 2 - ( x m + T m X x m + 3 T m ) : tm(^m+3Tm)-0.28l(xm+2T r a)3-xm(x r a+3Tn l
13 K (367) 1.066Tm -0.467xn •(367) _ 4.695xm(1.066Tm-0.467xm)
xm+2Tm
T (367) _ i d ~~ ln
0.331Tm-0.334xn
1.066T„ -0.467x„
K „ 14 v (368) .
1.6193(xm+2Tm)3 1
Kmxm(xm+3Tm) 2 Km ' xm + 3Tm
, (368) = 2.7051x„ •1.6706x„ (xm+3Tm
x m +2T m - •" ( x m + 2 T m ) 4 '
T <368> _ T IT J - T T )
1.7793(xm +2T m ) 2 - (x m + TmXx ro+3Tm) i d " T m l T m m j 1.6193(xm+2Tj3-xm(xm+3Tj2 '
230 Handbook of PI and PID Controller Tuning Rules
4.1.21 Non-interacting controller 7
U(s) = - ^ E ( s ) - K t
T,s 1 +
T -
R(s)
T
N
Y(s)
E(s) K
T iS
+ ^ U ( S ) Kme"
l + sTm
Y(s)
Table 64: PID controller tuning rules - FOLPD model l + sT„
Rule
Ogawa and Katayama
(2001).
Kc Ti Td Comment
Direct synthesis: time domain criteria 15 K (369) T (369)
i T (369) N=10
Minimum ISE - servo. Model: Method 1
15 e
Desired closed loop response = (l + sTCL)2 K (369) 1
~ K „
T „ T
XCL . T ^ C L
T T V ra x m J
j (369) _ T
<v T v XCL f 2 C L
T T V *m
l c k _ 2 ^ k + 4 m A
l CL + 4
V
( f
(369) _ . T T
m V m J
• T T IcL__2lck + 4 T T
^ e [0.01,1.00];
- ^ = -0.1902 T T
- 0.6974 - SL + 0.007393 , - ^ e fo.05,1.00l. 1 Tm T ' J
Chapter 4: Tuning Rules for PID Controllers
4.1.22 Non-interacting controller 11
231
U(s) = Kc
>
v T i s E(s)-K1(l + Tdls)Y(s)
j
R(s) E(s) 1 + x-> U ( s )
Kc(l+ — + ! » _±*(g) „ K e "
l + sTm
KiG + T^s)
Y(s)
Table 65: PID controller tuning rules - FOLPD model G m (s) = Kme"
l + sT„
Rule
Lee and Edgar (2002).
Model: Method I
Kc T, Td Comment
Robust
16 jr (370) T (370) T (370) Ki =0.25KU,
T d i =0
16K (370) l + 0.25KuKm
Kra(^ + t m )
Tm-0.25KuKmxm | xn
l + 0.25KuKra 2{X + xn
T(370) T m -0 .25K u K m x m | x 1 ; = — + l + 0.25KuKm 2(X + xm)'
T^10) _ l + 0.25KuKm 0.25KuKmxm2 3 4
Km(X + Tm) l2(l + 0.25KuKm) 6 (^ + T J 4{X + xm)2
, xm2 (Tm -0 .25K uKm im )
2(l + 0.25KuKmX^ + O '
232 Handbook of PI and PID Controller Tuning Rules
4.1.23 Non-interacting controller 12 f
U(s) = K{ T,s 1
Tds + 1 E(s)-K,Y(s)
R(s) E(s)
+ i i
Table 66: PID tuning rules - FOLPD model Gm (s) = Kme"
l + sT„
Rule
Lee and Edgar (2002).
Model: Method 1
Kc T, Td Comment
Robust
17 K (371) T (371) T (371) x d
K, =0.25KU
n £ (371) _ l + O^KyKn, fTm -0.25K l lKraTm
Km(^ + Tm) l + 0.25KuKm 2(^ + xm)
0.25KuKm im | xm2 | (Tm-0.25KuKmTm)Tm
2
2(l + 0.25KuKra 6(^ + Tm) 4(^ + Tm)2 2(l + 0.25KuKm)(X + xm)
2 T (371) _ Tm ~ 0 - 2 5 K u K m T m
l + 0.25KuKm 2(X + Tm)
0.25KuKm T m T „ (Tm-0.25KuKmT r a)in
2(l + 0.25KuKm 6(X + xJ 4{X + xm)2 2(1+ 0.25KuKm)(^ + xm)
(371
0.25KuKm (T ra-0.25K l lKmTm)xn
_2(l + 0.25K„Km 6(X + xm) 4(X + xm)2 2(1+ 0.25KuKra)(^ + Tm)
Chapter 4: Tuning Rules for PID Controllers 233
4.1.24 Industrial controller U(s) = K 1+ T,s
1 + T s R ( s ) - - ^ ^ Y ( s )
1 + ^ N
R(s) K, 1 + -
XS
U(s) K e~
l + sT„
Y(s)
1 + Tds
1 + ^ s N
Table 67: PID controller tuning rules - FOLPD model Gm (s) = Kme^
l + sT„
Rule
Kaya and Scheib (1988).
Model: Method 5
Minimum IAE Minimum ISE
Minimum ITAE
Kaya and Scheib (1988).
Model: Method 5
Minimum IAE Minimum ISE
Minimum ITAE
Kc Ti Td Comment
Minimum performance index: regulator tuning
x i M1
,0 T
X 3
f \
T
x 4
xsTm T
x6
0 < ^ - < l ; T
m
N=10 Coefficient values
x i
0.91 1.1147 0.7058
x2
0.7938 0.8992 0.8872
X 3
1.01495 0.9324 1.03326
x4
1.00403 0.8753
0.99138
X 5
0.5414 0.56508 0.60006
X 6
0.7848 0.91107
0.971 Minimum performance index: servo tuning
x, TT
K m l T m
V2
J
Tm
* 3 - X 4 ^
(O x5Tm - ^
" 6
0 < - ^ - < l ; T
N=10
Coefficient values x i
0.81699 1.1427 0.8326
x2
1.004 0.9365 0.7607
X 3
1.09112 0.99223 1.00268
x4
0.22387 0.35269 0.00854
x 5
0.44278 0.35308 0.44243
x 6
0.97186 0.78088 1.11499
234 Handbook of PI and PID Controller Tuning Rules
Rule
Harris and Tyreus (1987).
Model: Method 1
Hoetal. (2001). Model: Method 1
Kc T, Td Comment
Robust Tm
K m (0 .5 t m + ^)
18 K (372)
1.551Tm
K -m A m T m
T
T
T
0.5tm
0.5xm
0.5xra
X > max imum [0.8im,Tm];N not specified
N not specified
cog = frequency at which the Nyquist curve has a magnitude of unity.
Chapter 4: Tuning Rules for PID Controllers
4.2 Non-Model Specific
235
4.2.1 Ideal controller Gc (s) = K£
R(s)
+ 0 E(s) K.
A
1 + — + THs
V T,s
)
U(s) Non-model specific
Y(s)
Table 68: PID controller tuning rules - non-model specific
Rule
Ziegler and Nichols (1942).
Farrington (1950).
McAvoy and Johnson (1967).
Atkinson and Davey(1968).
Carr (1986); Pettit and Carr
(1987).
Parr (1989)-pages
190,191,193.
Tinham (1989).
Blickley(1990).
Kc T, Ta Comment
Ultimate cycle [ 0 . 6 K U , K J
[0.33KU,0.5KJ
0.54KU
0.25KU
0.5TU
Tu
T
0.75T„
0.125TU
[0.1TU,0.25TJ
0.2TU
0.25TU
Quarter decay ratio
20% overshoot - servo response
Ku
0.6667 Ku
0.5KU
0.5K„
0.5KU
0.5KU
0.4444KU
0.5KU
0.5T,
Tu
1.5TU
T
T 4 U
0.34TU
0.6TU
Tu
0.125TU
0.167TU
0.167TU
0.2TU
0.25TU
0.08TU
0.19TU
[0.125TU,
0.167TJ
Underdamped
Critically damped
Overdamped
£ =0.45
Less than quarter decay ratio response
Quarter decay ratio
236 Handbook of PI and PID Controller Tuning Rules
Rule
Corripio(1990)-page 27.
Astrom (1982).
Astrom and Hagglund (1984).
Hang and Astrom (1988b).
Astrom and Hagglund (1988)
- page 60.
Astrom and Hagglund (1995) -page 141-142.
DePaor(1993).
McMillan (1994) - page 90. Calcev and
Gorez(1995).
Kc
0.75KU
Ku cos<|>m
0.87KU
0.71KU
0.50KU
Ara
Kucos<j>m
Kusin<|)m
0.35KU
0.4698KU
0.1988KU
0.2015K„
0.906KU
0.866KU
0.5KU
0.3536KU
4>m
T:
0.63TU
1 T (373)
Td
0.1T„
aT,(373)
Representative results 0.55TU
0.77TU
1.19TU
arbitrary
aTd<374)
Tu(l-cos<|)m)
0.77TU
0.4546TU
1.2308TU
0.7878TU
0.5TU
0.5TU
0.5TU
0.1592TU
= 45° , small xm
0.14TU
0.19TU
0.30TU
T
47l2T;
2 T (374)
Tu(l-cos<|)ra)
47tsin(j>m
0.19TU
0.1136TU
0.3077TU
0.1970TU
0.125TU
0.125TU
0.125TU
0.0398T;
<L =15°, large
Comment
Quarter decay ratio
* m = 3 0 °
l m = 4 5 °
K = 60°
Specify Am
Specify <j>m
A m > 2 ,
* m ^ 4 5 0
Am = 2,
<L=20°
Am = 2.44,
* m = 6 1 °
Am = 3.45,
4>m = 25°
4>m = 30°
tm
, (373) V" = | tan<|)m + .J4a + t a n 2 c t > m ] — • a = 0.25 (Astrom, 1982); N J 4cac
2 T (374) . tan <L+V^ -tan
a = 0.5 (Seki et al., 2000); 0.1 < a < 0.3 (Seki et al., 2000). \T
Chapter 4: Tuning Rules for PID Controllers 237
Rule
ABB Commander
300/310(1996).
Luo etal. (1996).
Karaboga and Kalinli (1996).
Luyben and Luyben ( 1 9 9 7 ) -
page 97.
Tan et al. (1999a ) -page
26.
Tan et al. (1999a) -page
27.
Wojsznis et al. (1999).
Yu (1999) -page 11.
Tan et al. (2001).
Chau (2002) -page 115.
Robbins (2002).
Smith (2003).
Rutherford (1950).
Kc
0.625KU
0.48KU
|p .32Ku ,0 .6Ku]
0.46KU
0.4KU
0.5KU
T,
0.5TU
0.5TU
[0.213TU,
1.406TJ
2.20TU
0.5TU
Tu
Td
0.083TU
0.125TU
[0.133TU,
0.469TJ
0.16TU
0.125TU
0.125TU
Comment
'P+D' tuning rule
Based on work by Tyreus and Luyben (1992)
Tighter damping than quarter decay ratio tuning
0.4KU
0.33KU
0.2K„
K c (375 ,
0.33KU
0.2KU
0.45KU
0.45KU
0.75K„
0.333TU
0.5TU
0.5TU
3 T (375)
0.5TU
0.5TU
T ( 3 7 6 ) 4 l j
4 T (377) i
0.625TU
0.083TU
0.125TU
0.125TU
Q 2 5 T _(375)
0.333TU
0.333TU
0.25TU
0.25TU
0.1T„
Some overshoot
No overshoot
"Just a bit o f overshoot
No overshoot
Minimum IAE -step change in
setpoint "Good" response - step change in
load
Other rules
1.25K37%
! - 1 2 K 3 7 %
T37%
T37%
° - 1 2 5 T 3 7 %
0.25T37% 37% decay ratio
c „ „ „ „ , (375), 2 2 , ' i (375)
0.25[Ti ( 3 7 5 ) l2Q)„2+l = 0.3183T„tan
0.571 + Z G ; Q ( D U )
- (376)
Gc(jff lu) = desired frequency response of controller Gc(jffl) at cou
= (0.24 + 0.11 l K m K u )TU T/ 3 7 7 ' = (0.27 + 0.054KmK„ )TU
238 Handbook of PI and PID Controller Tuning Rules
Rule
MacLellan (1950).
Harriott (1964) -pages 179-180. Lip tak (1970) -pages 846-847.
Chesmond ( 1 9 8 2 ) - p a g e
400.
Parr ( 1 9 8 9 ) -page 191.
McMillan (1994) -page 43
Wade ( 1 9 9 4 ) -page 114.
Hay ( 1 9 9 8 ) -page 189.
ECOSSE team (1996b).
Bateson (2002) -page 616.
Rotach(1994).
Kc
1.77K37%
1.67K37%
1.49K37%
K 2 5 %
K 2 5 %
5 K (378)
0.999K25o/o
K 2 5 %
0-83K25%
0.67K25„/o
[1.002K25%>
1.671^] K 2 5 %
0.8497K50%
K 2 5 %
6 K (379)
Ti
1-29T37%
1.29T37„/o
° - 9 8 T 3 7 %
0.167T25./o
0.667T25%
j ( 3 7 8 )
0.488T25„/o
0.67T25%
0.5T25%
0.5T25%
0.45T25%
0.667T25%
0.475T50%
0.6667TU
T ( 3 7 9 )
Td
0.16T37./o
0.16T37„/o
0.24T37%
0.667T25%
0-167T25%
0.25Ti(378)
0.122T25%
0-17T25%
0.1T25o/o
0-lT25%
0.1125T25%
0.167T25„/o
0.1188T50%
0.1667TU
_0.5 ^ = -
Comment
37% decay ratio
Quarter decay ratio
Quarter decay ratio
x = 100(decay ratio)
Example: x = 25 i.e. quarter decay
ratio
Quarter decay ratio
'Fast' tuning
'Slow' tuning
'modified' ultimate cycle
5 K (378)
6 K (379)
0.5 + 0.3611nx
, (378) Tx%
2Vl +0.025 In2
M „
A/M^
• G . ( j < D M _ ) , Tj' (379)
dco.
Chapter 4: Tuning Rules for PID Controllers 239
Rule
Rotach(1994)-continued. Garcia and
Castelo (2000).
Lloyd (1994). Model: Method 3
Jones et al. (1997).
Kc
7 j , (380)
8 K (381)
0.4KU
0.4 Ku
0.7490
Ti
ry (380)
T (381)
0.3333TU
Tu
aT u ,
Td
T (380)
0.25Ti(381)
0.083TU
0.159TU
0.125TU
Comment
CO, < C 0 U
Self-regulating processes Non self-regulating processes
Model: Method 2
a e [0.2526,0.5053]
7 K (380) M „
Mraax - 1 •|Gp(jCDr)| .
, (380)
(380)
d ^ , . i f • -1 1 cor -•n + tan jt-sin + i
dcor Mmav
n 7 C -tan 7C-sin M „
0.5,
cor r dcor
1 + tan' 7i - s in + M,
tan 7t - sin Mn
K (381) cos(l80°+<t.m-ZGp(jco1)) ^ (381) 2[sin(l80° + +„ -ZGp( jm1))+ 1]
Gp(jco,) j(381) .
co1cos(l80°+(|)m-ZGp(jco1))
240 Handbook of PI and PID Controller Tuning Rules
Rule
Shin et al. (1997).
NI Labview (2001).
Model: Method 2
Tang et al. (2002).
Model: Method 4
Dutton et al. (1997).
Kc
9 j £ (382)
0.6 Ku
0.25 K„
0.15KU
2.5465hsin<j>m
PX1 a,9(f ^ m
T,
T (382) i
0.5TU
0.5TU
0.5TU
5^(383) ^
5 e [l.5,4]
Td
aT/382*
0.12TU
0.12TU
0.12T„
10 T (383) 1 d
Comment
Model: Method 2
Quarter decay ratio
'Some' overshoot
'Little' overshoot
0.81 <x, <1
x, = 1 , 'small' xm ; x, = 0.91, 'large' Tm
u0.5Keu
T e 1 u 0.25Tu
e pages 576-8.
y2 = — s i n | ZGp(jcou) | . Typical a : 0.1; typical i,: [0.3,0.7]. Ku
10 T (383) _ -cot4>m+Vcot2c|)m +4/5
2co9 0 ,
Te = 6.283
1 -55
1110 with r = ratio of the height of the
first peak to the height of the first trough of the closed loop response, when controller
gain Kc is applied; cod = damped natural frequency.
Chapter 4: Tuning Rules for PID Controllers 241
4.2.2 Ideal controller in series with a first order lag r
R(s) E(s)
- < g > K 1 + — + T « S
V T i S
Gc(s) = K£
U(s),
1 \ 1 + — + Tds
v T i s j
1
Tfs + 1 Non-model
specific
Tfs + 1
Y(s)
Table 69: PID controller tuning rules - non-model specific
Rule
Kristiansson (2003).
Kc Ti Td Comment
Other tuning: minimum performance index 12 K (384) T (384) T (384)
"Almost" optimal tuning rule; K)J0„ > 0.03 ; 1.6 < M s < 1.9
12 K (384)
,(384)
0.60
K„ K
150" K„
-0.07 0.32 + 1.6-K-,5no
K„
K 1 5 0 °
1.5
0.32 + 1.6-
(384)
(384) _ _
K 1 5 0 °
0.6667
-0.8 ^ 1 5 0 "
0.32 + 1.6-. K-nnO
K „ -0.8
K,
V K m J
0.4
K l 5 » ° - - 0 . 0 7 K „
K-,<„0 - J ^ l ; n > 0.32 + 1 . 6 — ^ _ - 0 . 8 |
K„ K „
-or
min-^ 6 -K„
K I 5 0 °
,25
(384)
20- 150° + 0.5 0.32 + 1 . 6 — ^ - - 0 . 8 | K,
K.
242 Handbook ofPIandPID Controller Tuning Rules
Rule
Wang et al (1995b).
Kc T: Td Comment
Direct synthesis 13 jr (385) T(385) T (385)
' d
13 For a stable process, Km is assumed known; for an integrating process, Km and T„
are assumed known. K (385) -Im • CL
JMCLGCLQCOCL)
G P O ^ C L ^ - G C L Q C O C L ) }
. (385)
Im 1
Rl
JCOCLGCL(jfflCL)
GpGttcLXi-GcLGaa.)}
J«CLGCL(J®CL)
_Gp(jfflCLXl-GCL(jQ)CL)}
1 + —
3
JMCLGCLCPCL) Rl •Rl Gp(j(ocJl-GCL(j«CL)jj 1 Gp 02(00. Xl-Go. 020)0.)]
j2coCLGCL(j2co
(385)
Rl f JMcLGcLptfcL) -Rl J2(»CLGCL(J2C0CL) 1
Gp(jcoCLIl-GCLOraCL)]J [Gp02coCL)[l-GCL02coCL).
Im JOJCLGCLO^CL)
GpOfflCLl1-GCLO(BCL)]j
Chapter 4: Tuning Rules for PID Controllers 243
4.2.3 Ideal controller in series with a second order filter f 1 A
G c ( s ) = Kc
V T i s J
l + bfIs
l + afls + af2s
R(s) E(s)
-+®-+ + x
U(s)
K, l + — + Tds V T i s
l + b f l s
l + a f ,s + a f 2s
Non-model specific
Y(s)
Table 70: PID controller tuning rules - non-model specific
Rule
Kristiansson et al. (2000).
Kc Ti Td Comment
Minimum performance index: other tuning l K (386) T (386) rj, (386) M . * l . 7
"Almost" optimal tuning rule
1.6
l ^ (386)
. (386)
Gp(j(Du) Gp(jcou) 1.6-L^ ^-L-2.3J—- L + l.l
K „ K „
Km
1.6
0.37 + Gp(jcou)|
K „
0.37 + G p ( jcoJ
=r,T, (386)
, b l f - 0 , alf - T f , a2f - Tf'
0.625
K „ 0.37 +
G p O u
K „
, ,M1.2 .3W+ 1 , K „ K „
0.37 + Gp(jcou)
K „ 13-20
Gp(ja>„)
Kn
|Gp(jcoj|
Km
< 0.5 or
! ,Ml_,3ra + 1 , K „
3co„ 0.37-Gp(j(Bu)
K~
K „
13-20 G P ( K )
Km
|Gp(jfflj|
Km
> 0 . 5 .
244 Handbook of PI and PID Controller Tuning Rules
Rule
Kristiansson (2003).
Kc
2 K (387)
Ti
T (387) i
Td
T (387) x d
Comment
"Almost" optimal tuning
rule; K,5 0„>0.03;
1.6 <MS <1.9
2 K (387) 0.60
K „ K 1 5 0 ° -0.07 0.32 + 1.6-
K,
K „ -0.8
V K m J
,(387)
(387)
1.5
0.32 + 1.6—^--0.8 Kn.
0.6667
K,
V K m J
0.32 + 1.6-K,,„o 150"
K „ -0.8 ^ 1 5 0 °
V Km J
b l f = 0 ,
0.8
2 0 ^ + 0.5 0.32 + 1 . 6 ^ ^ - 0 . ^ K l 5 0 ' ^ K „ K „
2 0 ^ 1 ^ + 0.5 0.32 + 1 . 6 ^ L - O.gf K ' s ° u
K „ K „
Chapter 4: Tuning Rules for PID Controllers
4.2.4 Ideal controller with weighted proportional term
Gc(s) = Kc
245
R(s)
K c ( b - 1 )
+ T
Y(s) E(s)
> K r 1 A
1 + — + Tds v T i s J
+ T U(s)
+ Non-model
specific
v T i s
Y(s)
Table 71: PID controller tuning rules - non-model specific
Rule
Astrom and Hagglund (1995)
-page 217.
Streeter et al. (2003).
Kc Ti Td Comment
Direct synthesis 3 K (388)
4 j r (389)
5 K (390)
ry (388)
T ( 3 8 9 ) i
T (390) 1 i
T (388)
T (389)
T (390)
0 < K m K u < c o
Mmax = 1.4
0 < K m K u < c o
Mmax =2.0
0 < K m K u < c o
Mmax =2.0
Kc(388) =(p.33e-°J l M 2)cu , T,(388) = (o.76e-L6K-036K2 )ru, (o^e"031*-*2 )KU
T ^ ^ O . n e ^ - ^ J i ^ b ^ . S S e - '
Kc<389) = (o.72e-L6K+1 2K2 )KU , Ti'389' = (o.59e-'-
3K+0-38K2 )ru ,
Td(389) =(o.l5e
1.3K+0.38
1.4K+0.56I
6
" % » = 0.25eu
Kc(390) =(o.72e-''6K+L2l<2)K:u -0.0012340TU -6.1173.10"
(390) _ (o.59e-'-3K+a38K2 ](o.72e-'6K+'-2lc2 )K„ - 0.0012340T - 6.1173.10"6 \ru
0.72e~
(390) 0.108e" KUTU - 0.00266401
0.040430e"1JK
log(l.6342'
•38K ] K U
w ° 0.72K,1e"L6K+,'2l<2 -0.0012340T,, -6.1173.10~6
K „
246 Handbook of PI and PID Controller Tuning Rules
4.2.5 Controller with filtered derivative Gc(s) = K t 1 + — + — —
R(s) E(s)
<8M + T K „
, 1 T d s
1 + — + d T*s l + ^ s
N .
U(s)
Non-model specific
N
Y(s)
Table 72: PID controller tuning rules - non-model specific
Rule
Leva (1993). N not specified
Yi and De Moor (1994).
Kc Ti Td Comment
Direct synthesis 6 R (391)
7 K (392)
8 K (393)
T {391)
aTd(392)
T (393) *i
6.283/Pco
T (392)
T (393) Xd
(3 = 10
6 < a < 1 0
N not defined
6 K (391) 0)T: (391)
Gp(j<4 1-coT, (391) 2TC
pj + W
2[T i<391)]2
X (391)
Y + ytan(<l>m-<l>o>-0-57c)
. <h> > < L " * •
7 K (392) COT: (392)
|Gp(ja,)|V(l-a,2T i^Td^)2+fl)
2[T i«M2>]2
(392) • aco + y a2©2 + 4aco2 tan2 (<(>m - <|>a - 0.5TC)
2aa>2 tan((j)m - <j>m - 0.57t)
K l J „ , 0.5cos[225°-ZGp(ja)^J T < 3 9 3 ) = ^ ( 393 ) (393)
Gp(jW^)
(393) tan[2250-ZGp(jffl^)]+A/4 + tan2[225°-ZGp(jffl^)
2®A
Chapter 4: Tuning Rules for PID Controllers 247
Rule
Vranfiic(1996)-page 130.
Vrancic (1996)-pages 120-121.
Vrancic (1996) -page 134.
Vrancic et al. (1999).
Kc T (394)
2(A,-T, (394 ))
10 J , (395)
1 1 K (396)
12 K (397)
T|
9 T (394)
T (395)
T (396) i
T ( 3 9 7 )
Td
T (394)
A 3 A 4 - A 2 A 5
A3 _ A , A 5
^ ( 3 % )
T (397)
Comment
N<10
N>10
N = 10;
1 = [0.2,0.25]
8 < N < 20
9 T (394)
A 2 - T d ^ > A , -(394)-.2
N
(394) -(A32 - A5A,) +J (A 3
2 - A 5 A J - 1 ( A 3 A 2 - A5XA5A2 - A4A3)
N ( A 3 A 2 - A 5 )
10 v (395) _ *r \."*> 0.5A3\A3 A,A5 j C ~ (A 3
2 -A 1 A 5 ) (A 1 A 2 -A 3 K r a ) - (A 3 A 4 -A 2 A 5 )A 12 '
T(395) = A 3 ( A 32 - A | A 5 )
' ( \ 32 - A , A 5 ) A 2 - ( A 3 A 4 - A 2 A 5 ) A 1
A2 - V A 22 - 4 X A , A 3 11 K (396)
12 K (397)
(396) 0.5T;
A . - K , , , ^ (396) T
(396)
2XA,
A 1 A 2 - A 0 A 3 - T d < 3 9 7 , A 12 - I I i -
- (397) .
N
(397)
-A0A, J
A - T ( 3 9 7 ) A [T, (397)-, 2
I J__
N
, T6 is obtained by solving:
K „
V M T (397) M , A ,A 3 L (397)}
W^ J + 1 ^ [ d J ApA5 A2A3
N [Td(397)f
+ (A 32 - A, A5lTd
<397)]+ (A2A5 - A3A4) = 0 .
248 Handbook of PI and PID Controller Tuning Rules
Rule
Lennartson and Kristiansson
(1997).
Kristiansson and Lennartson
(1998a).
Kristiansson and Lennartson
(1998b).
Kc
13 K (398)
14 j , (399)
15 K (401)
16 K (402)
Ti
T (398) i
T (399)
T (400) •M
T (401)
T (402)
Td
0 4 T . ( 3 9 8 )
0 4 T , ( 3 9 9 ,
0 4 T ( 4 0 0 )
04T . (40„
0.333T;(402)
Comment
KuKm>1.67
N = 2.5
13 v (398) v v 12KU Km -35K u K m +30 k c - N ^ n T ; ; 1 .
K„Km +2.5|l2K„2Km2 -35K u K m +30J
• (398) _
u m ' '— ' |_* "" * - u " m
K (398)
-0.053co„3 +0.47(0„2 -0.14co„ +0.11
N: 2.5 12 55_+ 30
KmKu K 2 K 2K 2
m K u
14 v (399) _ v v
, (399)
12K 2 K m2 - 3 5 K u K m + 3 0
K u K m + x , [ l 2 K u2 K m
2 - 3 5 K u K m + 3 o ] '
K (399)
-0.525CO, +0.473cou2 -0.143cou +0.113 KUK
±— < 0.4;
, (400) K (400)
-0.185couJ+1.052cou -0.854cou +0.309 KUK
, - ^ > 0 . 4 ;
N = - i l - 35 30 12 — + -
15 K (401)
Km K u Km
2Ku2
7.71 9.14
x , = 3 , K u K m > 1 0 ; x , = 2 . 5 , K u K m < 1 0 .
, , - + 3.14, KuKra > 1.67 , ii—>0.45; K 2K„2 KmK„ m K„K„ ••m " - U m " i
,(401) K, (401)
• 0.63co„3 + 0.39co„2 + 0.15co„ + 0.0082
16 v (402) 12K„3K 2 - 3 5 K „ 2 K m + 3 0 K „
" Km3Ku
3 +2.5[l2Ku2Km
2 -35K u K m +3o] '
,(402)
j Kc
(4°2)Km3Ku
2
cou[0.95Km2Ku
2-2Kr
2.5 — 1 , N =
Ku+1.4j KmKu
1 2 — 1 * - + - 3 0 K
m K u Km2Ku
2
Chapter 4: Tuning Rules for PID Controllers 249
Rule
Rristiansson and Lennartson
(2000).
Kc
17 K (403)
18 K (404)
T;
j (403)
T (404)
Td
T (403)
T (404) ld
Comment
0.1<KuKm
<0.5
K„K r a>0.5
'K (403) ( l . lK u2 K m
2 -2 .3K u K m + 1 .6 ) 2
K r a2Ku(-20 + 13KmKJ(l + 0.37KmKu)2
"l.6(-20 + 13KmKu)(l + Q.37KmKu)
l . lK m2 K u
2 -2 .3K m K u + 1 .6
-(403) _ K m K u ( l . lK u2 K r o
2 -2 .3K u K m +1.6)
cou(-20 + 13KmKJ(l + 0.37KmKu)2
1.6(-20 + 13KmKJ(l + 0.37KmKu) ^
U K m2 K u
2 - 2 . 3 K m K u + 1 . 6
(403) _ KmKu(l . lKu Km -2.3KuKm+1.6)
o)u(-20 + 13KmKu)(l + 0.37KmKu)2
(-20 + 13KmKu)2(l + 0.37KmKu)2
( l . lK m2 K u
2 -2 .3K m K u +1.6) 2
1.6(-20 + 13KmKu)(l + 0.37KmKJ
UK m2 K u
2 -2 .3K r a K u +1 .6
- -1 N: (403)
(403)
r (403) _ KmKu(l . lKu Km -2.3KuKm+1.6) f cou(-20 + 13KmKJ(l + 0.37KmKu)2
K (404) ( l . lK u
2 K m2 -2 .3K u K m +1.6) 2
3G)uKm2Ku
2(l + 0.37KmKu)2
4.8KmKu(l + 0.37KmKJ
U K 2K 2 •2.3K raKu+1.6
, (404) _ (UK u2 K m
2 -2 .3K u K m +1 .6 ) x2
(404) _
3cou(l + 0.37KmKur
( l . lK u2 K m
2 -2 .3K u K m +1.6)
3au(l + 0.37KmKu)^
4.8KmKu(l + 0.37KmKJ
l . lK m2 K u
2 -2 .3K r a K u +1.6
3Km2Ku
2(l + 0.37KmKJ2
( l . lK m2 K u
2 -2 .3K m K u +1.6) 2
4.8KmKu(l + 0.37KmKu)
l . lK m2 K u
2 -2 .3K m K u +1.6
(404)
N: Tf
(404) . T f (404) l . lK u
2 K m2 -2 .3K u K m +1.6
3cou(l + 0 .37K m KJ 2
250 Handbook of PI and PID Controller Tuning Rules
Rule
Kristiansson and Lennartson
(2000) -continued.
Kristiansson and Lennartson
(2002).
Kc
19 K (405)
20 K (406)
T;
j(405)
T (406)
Td
T (405)
T (406) Xd
Comment
K u K m <0.1
K u K m <10
19 K (405)
• (405)
(-6 + 3.7Kn,„Km)2 20.8(1.8+ 0.3KmK..,„)
13Kra(1.8 + 0.3KmK135o)2
(-6 + 3.7K135„Km)K135oKm
13co135o(1.8 + 0.3KmK1350)2
(405) _ ( _ 6 + 3-7 KmK135°)KmK- i35»
" 13o3135.(1.8 + 0.3KmK135„)2
(-6 + 3.7KmK135o)
20.8(1.8+ 0.3KmK135o)
(-6 + 3.7KraK1350)
' 169(1.8+ 0.3KmK135,)2
(-6 + 3.7KmKm„)2
20.8(1.8 + 0 . 3 K m K „ )
(405)
N = -(405) .Tf
(405)
l f
(-6 + 3.7KmK135„)
(-6 + 3.7KmK135o)KmK135„
13co135„(1.8 + 0.3KmK|35„)
- 1
2 '
(406) = Ku [l.5(KmKu +4)(0.4KmKu +0 .75)-K m K u X l ]x ,
Km (0.4KmKu+0.75)2(4 + KmK u )
(406) _ KmKu [l.5(KmKu +4)(0.4KmKu + 0.75)-KmK ux,]
CO
(406) K m K u
(0.4KmKu+0.75)2(4 + KmK u )
(4 + K m K J
u [l.5(4 + KraKu )(0.4KmKu + 0.75) - KmKux, \l + N" 1 ) '
N = -(406)
(406)
j (406) _ K m K u
(0.4KmKu+0.75)2(4 + K m K J
x, = 0.13 + 0.16KmKu -0.007Km2Ku
2 .
Chapter 4: Tuning Rules for PID Controllers 251
Rule
Kristiansson and Lennartson
(2002) -continued.
Kc
21 K (407)
Ti
T (407)
Td
T (407)
Comment
K uK r a>10
21 K (407) Kn5o [l.S(0.44KmK,35o +1.4)x2 -KmK,35oX3Jx3
K „ (0.44KmK135„+l.4)'x2
,(407) KmK135o [l.5(0.44KmK,35o +1.4)x2 - K m K u x 3 j
(407)
c o 1 3 5 „
K m K i 3 5 °
<B,
(0.44KmK,35o+1.4)2x2
J135" [l.5x2(0.44KmK135„+1.4)-KroK135„x3ll + N - 1 ) '
N: (407)
d (407) . T f
K 2K 2
(407) _ ^ m ^ 1 3 5 °
(0, (0.44KmK,1,„+1.4) ix2
x2 =min(14 + 0.5KmK135„,25),x3 =1.35 + 0.35KmK,35„ - 0 . 0 0 6 K m2 K | 3 /
252 Handbook of PI and PID Controller Tuning Rules
4.2.6 Ideal controller with set-point weighting 1
U(s) = K c (F pR(s)-Y(s))+^(F iR(s)-Y(s)) + KcTds(FdR(s)-Y(s)) 1 :b
Table 73: PID controller tuning rules - non-model specific
Rule
Mantz and Tacconi(1989).
Kc T i Td Comment
Ultimate cycle 0.6KU ;
Fp=0.17 0.5TU; 0.125T,;
Fd = 0.654
Quarter decay ratio
Chapter 4: Tuning Rules for PID Controllers 253
4.2.7 Classical controller 1 Gc(s) = K 1 T;S
1 + sT,
N
R(s) E(s)
K, 1 + — Xs T
N ;
U(s)
Non-model specific
Y(s)
Table 74: PID controller tuning rules - non-model specific
Rule
Minimum IAE -Edgar et al.
(1997) -page 8-15.
Kinney (1983).
Corripio(1990)-page 27.
St. Clair (1997)-page 17.
N > 1
Harrold(1999).
Kc T; Td Comment
Minimum performance index: regulator tuning
0.56KU 0.39TU 0.14TU N = 1 0
Ultimate cycle 0.25KU
0.6KU
0.5KU
0.25KU
0.67KU
0.5TU
0.5TU
1.2TU
1.2TU
0.5TU
0.12T„
0.125TU
0.125TU
0.125TU
0.125TU
N not defined
10 < N < 20
'aggressive' tuning
'conservative' tuning
N not defined
254 Handbook of PI and PID Controller Tuning Rules
4.2.8 Series controller (classical controller 3)
R(s) E(s) K,
v T i s y (l + sTd)
U(s) = K
U(s)
— •
' i+JL v T ;sy
(l + s T d )
Non-model specific
Y(s)
Ta
Rule
Pessen(1953).
Pessen (1954).
Tan et al. (1999a) -page
26. O'Dwyer (2001b)-
representative results - Astrom and Hagglund
(1995) equivalent -page 142.
Tan et al. (2001).
ble 75: PID controller tuning rules -
Kc
0.25KU
0.2KU
'optimum' re 0.33KU
0.3KU
0.2349KU
0.0944KU
0.1008KU
1 K (408)
Tj
- non-model specific
Td
Ultimate cycle 0.33TU
0.25TU
gulator response -0.5TU
0.15TU
0.2273TU
0.6154TU
0.3939TU
y (408) i
0.5TU
0.5TU
step changes 0.33TU
0.25TU
0.2273TU
0.6154TU
0.3939TU
0 2 5 T _(408 )
Comment
'optimum' servo response
'Some'overshoot
Ziegler-Nichols ultimate cycle
equivalent A m = 2 ,
*m = 20°
A r a=2.44,
<t»m=61°
Ara = 3.45,
+ m =46°
1 K (408) = _
, 4 0 8 ) ^ >u|Gc(jttu)|
.(408)
V[T i<408)]2cou
2+lVo.0625[T i(408)]2Q)u
2
6.25cou2 +4cou
2 tan2 2ZG;(jcou)+7t"
2
+ 1
-2.5cou
co„ tan 2ZG;(JCO U )+7I
Gc(jcou) = desired controller frequency response at oo = cou
Chapter 4: Tuning Rules for PID Controllers 255
/
4.2.9 Classical controller 4 Gc(s) = Kc
v T i s y 1 + - s T d
l + ^ L V N
R(s) E(s)
4- T K,
V T i s 7 1 + -
sT„
1 + s-N
U(s)
Non-model specific
Y(s)
Table 76: PID controller tuning rules - non-model specific
Rule
Hang et al. (1993b)
- page 58.
Kc Ti Td Comment
Ultimate cycle
0.35KU 1.13TU 0.20TU N not specified
256 Handbook of PI and PID Controller Tuning Rules
4.2.10 Non-interacting controller based on the two degree of freedom structure 1
f \ f >
U(s) = Kc 1 + — + - a
V
T . s 1 + ^ - s N j
E ( s ) - K c a + PTds
1 + ^ s
N ;
R(s)
( \
PTds
1 + ^ s N
a + K_
+
R(s) E(s)
K, 1+ — + - d
T . s 1 + ^ - s N .
U(s)
><%>-+
Y(s)
Non-model specific
Table 77: PID controller tuning rules - non-model specific
Rule
Shen (2002).
Kc Ti Td Comment
Minimum performance index: other tuning 2 K (409) T(409) T (409)
Ku > 1/Km
! Minimise Jr(t) - y(t)|dt + f|d(t) - y(t)|dt; N=0, (3 = 0, Ms < 2 :
K c( 4 0 9 '=K u exp 0.17-2.62/ KmK„ +1.79,
T, ( 4 0 9 )=Tuexp
Td( 4 0 9 )=Tuexp
-0 .02-2 .62/ K m K u +1.34,
1.70-0.59/ K K „ -0.25,
a = 1 - exp 0.30-0.48/ K m K u +0.93,
( . 2 ^
Km Ku
v j
f /> 2\
Km Ku
V J
f * 2 \ Km Ku
V J
If -2 Km Ku
Chapter 4: Tuning Rules for PID Controllers
4.2.11 Non-interacting controller 4
257
R(s) E(s)
+ X Kr
XS +
U(s) = K 1 + -V T i S /
E(s)-KcTdsY(s)
Non-model specific
Y(s)
K J d s
I Table 78: PID controller tuning rules - non-model specific
Rule
Minimum IAE -Edgar et al.
(1997)-page 8-15.
VanDoren (1998).
Kc Ti Td Comment
Minimum performance index: regulator tuning
0.77KU 0.48TU 0.11TU
Ultimate cycle 0.75KU 0.625TU 0.1TU
258 Handbook of PI and PID Controller Tuning Rules
4.2.12 Non-interacting controller 9
U(s) = Kc T:S
E(s)-^-sY(s) N
R(s) E(s>
+ X K „ 1+ — + Tds 3
U(s)
•
Non-model specific
Y(s)
N
3: Table 79: PID controller tuning rules - non-model specific
Rule
Bateson (2002) -pages 629-637.
Model: Method 1
Kc Ti Td Comment
Other tuning 3 K (410) T ( 4 1 0 ) 2
0 5 180° N = 1 0
3 K (410) = 1 0 » {{-K^Do^iGp^^H
dB;T i(410)=min[mg20,0.203,^. ]
' ppO^HO0] a n d | G P U ° W ) | are given in
4.3 IPD Model Gm(s)
Chapter 4: Tuning Rules for PID Controllers
K e"ST™
259
f 4.3.1 Ideal controller Gc(s) = Kc
1 A
v T i s j
Table 80: PID controller tuning rules - IPD model Gm(s) = Kme"
Rule
Ford (1953). Model: Method 3
Astrom and Hagglund(1995)
-page 139.
Hay (1998)-page 188.
Hay (1998)-page 199.
Model: Method i; KC!Td
deduced from graphs
Visioli (2001). Model: Method 1
Kc Ti Td Comment
Process reaction 1.48
K m x m
0.94
Km tm
2xm
2xm
0.37xm
0.5xra
Decay ratio 2.7:1
Model: Method 1
Ultimate cycle Ziegler-Nichols equivalent 0.4
Km 1™
10.0
4.0
2.5
2.0
1.8
3.2xra
3-2KmTm2
0-8xm
0.55xm
0.30tm
0.25tm
0.25Tra
0.25xm
Model: Method 3
K m x m =0.1
K r axm=0.2
K r axm=0.3
K r axm=0.4
Kmxm =0.5,0.6
Minimum performance index: regulator tuning X i /K r a x m x2Xm
x3xm
Coefficient values 1.37 1.36 1.34
1.49 1.66 1.83
0.59 0.53 0.49
Minimum ISE Minimum ITSE Minimum ISTSE
260 Handbook of PI and PID Controller Tuning Rules
Rule
Visioli (2001). Model: Method 1
Astrom and Hagglund (2004). Model: Method 1
Leonard (1994). Model: Method 1
Wang and Cluett (1997).
Cluett and Wang (1997).
Model: Method 1
Kc ^ Td Comment
Minimum performance index: servo tuning x l / K m T m
1.03 0.96 0.90
0 x 3 Tm
Coefficient values 0.49 0.45 0.45
Minimum ISE Minimum ITSE
Minimum ISTSE Minimum performance index: other tuning
X l / K m T m
X l
0.139 0.261 0.367 0.460 0.543
x2
76.9 23.3 12.2 7.85 5.78
0.74 K m T m
0.47KU
1 K (411)
2 K (412)
X i / K r a x m
X l
0.9588 0.6232 0.4668
x2
3.0425 5.2586 7.2291
X 2 t m x 3 T m
Coefficient values x3
0.346 0.365 0.378 0.389 0.400
Mmax
1.1 1.2 1.3 1.4 1.5
x i
0.616 0.681 0.740 0.793 0.841
x2
4.58 3.82 3.28 2.89 2.61
Direct synthesis
12.2tm
3.05TU
T ( 4 1 1 )
T (412)
0.4lTra
0.10TU
T (411)
T (412)
K u, Tu deduced from graph
X 2 t m X 3 t m
Coefficient values X 3
0.3912 0.2632 0.2058
x4
1 2 3
X l
0.3752 0.3144 0.2709
x2
9.1925 11.1637 13.1416
X 3
0.410 0.418 0.426 0.434 0.440
Mmax
1.6 1.7 1.8 1.9 2.0
OS (step input) < 10%; Minimum
IAE (disturbance ramp).
Model: Method 1
To. = X 4 T m
X 3
0.1702 0.1453 0.1269
x4
4 5 6
1 K (411)
KmTm(0.7138TCL+0.3904)
T (411) _ X
T/41" = (l.4020TCL +1.2076)rm ,
TCLe[T ra)16Tm], ^ = 0.707.
2 K (412) = 1 Kmim(0.5080TCL+0.6208)
• (412) .
1.4167TCL+1.6999
Ti(412) =(l.9885TCL+1.2235)rm,
.0043TCL +1.8194 T C L e k , 1 6 T m ] , %=\.
Chapter 4: Tuning Rules for PID Controllers 261
Rule
Rotach(1995). Model: Method 5
Chen et al. (1999a).
Chidambaram and Sree (2003).
Sree and Chidambaram
(2005b). Chen and Seborg
(2002).
Zou etal. (1997), Zou and Brigham
(1998). Model: Method 7
or Method 8
Kc
1.21 K m T m
Ti
1.60xm
Td
0.48xm
Damping factor for oscillations to a disturbance 1.661
A m K r r J m
1.2346 K m T m
0.896
Kmxm
5 K (415)
2
Km(X + 0.5Tm)
0
4-5xm
2.5xm
T (415) i
2 ^ + Tm
3 T (413)
4 T (414) ' d
0.45xm
0.55xm
T (415)
Robust
X + 0.25xm
— x 2X + xm
Comment
input = 0.75.
Model: Method 1
Model: Method 1
Model: Method I
Model: Method 1
0.5xm<^
^ 3 x m
Td<413) = 1.273-cJl- 0.6002Am (l.571 •
4 T / 4 , 4 ) = T J l . 2 7 3 2 riAm
1.6661
i(A _. T,(415)S(l + 0.5Tms)e-^ (4i5) _ T m (3TCL+Q.5tm)
' •' " """ " ' ' ~Km(TcL+0.5xmr v ^ / desired K c (TCL S + 1)'
. ( 4 1 5 ) = 3TCL+0.5Tm, Td
(415) 1.5T 2xm + 0 .75T r , x2 +0.125xm
3 - T 3
xm(3TCL+0.5xm)
262 Handbook of PI and PID Controller Tuning Rules
4.3.2 Ideal controller in series with a first order lag r
Gc(s) = Kc 1 \
1 + — + Tds V T i s j Tfs + 1
R ( V «•>. +1 k
Kc f 1 ^ I Ti s )
1
1 + Tfs
U(s) K m e " ^
s
Y(s)
Table 81: PID controller tuning rules - IPD model Gm (s) = Kme"
Rule
Rice and Cooper (2002).
Model: Method 1
Kc T i Td Comment
Robust
6 K (416) 2X + 1.5xm
0.5xm2+Xxm
2X + 1.5Tm
Suggested X = 3.162tm
6 j , (416) = _ 2X + 1.5tn
K, 1 ^ + 2 X x m + 0 . 5 t m2 )
. Tf = 0.5X2x„
^ + 2 ^ x m + 0 . 5 x m2
Chapter 4: Tuning Rules for PID Controllers 263
4.3.3 Ideal controller with first order filter and set-point weighting 2
U(s) = Kc
v T i s j
1
Tfs + 1
, l + 0.4Trs R(s) — - Y(s)
1 + sT
R(s)
— • 1 + 0.4Ts
1 + T s
+ E(s) f i A
1+ — + Tds V T i S J
1
Tfs + 1
+ U(s)
•K 0 Y(s)
Y(s)
+ & » Kme"
Kn
Table 82: PID controller tuning rules - IPD model Gm (s) = Kme-
Rule
Normey-Rico et al. (2000).
Model: Method 1
Kc Ti Td Comment
Direct synthesis
0.563
Kmt r a l-5xm 0.667tm
T f=0.13xm
Y ~ l
° 2Kraxm
T r=0.75xm
264 Handbook of PI and PID Controller Tuning Rules
4.3.4 Controller with filtered derivative Gc(s) = K(
R(s)
1 + — + d
N
?> E(s) » L
Kc 1, l , T « s
T>s l + ^ s V iN J
U(s)
s
Y(s)
—P-
Table 83: PID controller tuning rules - IPD model Gm (s) = K m e"
Rule
Kristiansson and Lennartson
(2002).
Chien (1988). Model: Method 1
Kc T, Td Comment
Direct synthesis 7 K (417) T (417) T (417)
Model: Method 1
Robust 8 K (418) 2X + xm Tm(X + 0.25xra)
2X + xm
N=10
7 K (417) 2.5(0,,
K „ .0.08 + ^ + ^
K, "i J
• (417) 2.5 2.3 - 2 K
_ (417) _ 2.5
N = -j (417)
(O .Q59 /K I2 )+ (0.055/K,)- 0.08
2.3-2K, 0.4(10+ 1/K,) —+ 1 N
T, (417) >T f
(417) (0.059/K!2)+ (0.055/K))- 0.08
0.16cou(lO + l/K!) , K, e [0.5,1].
8 K (418) 2>. + T„
Km(0.5^ + Tra)2 •; X e [l/Km,Tm] (Chien and Fruehauf (1990)).
2 (o.osVK^+fo.oss/Kj-o.os' 2.3 -2K, 0.4(10 + 1/K,)
2 (0.059/K)2)+ (0.055/K!)- 0.08"
0.4(10 + 1/K,)
Chapter 4: Tuning Rules for PID Controllers 265
4.3.5 Controller with filtered derivative and dynamics on the controlled variable
Gc(s) = Kc 1+ — + 5-
R(s) E(s)
+ T K,
N
N
+ U(s)
E(s)-K0Y(s)
* ® ^ K „
Y(s)
Table 84: PID controller tuning rules - IPD model Gm (s) = K m e ^
Rule
Normey-Rico et al. (2001).
Model: Method I
Kc Ti Td Comment
Direct synthesis - time domain criteria
9 K (419)
0.1716 K m T m
(1.5-yk,
*m
T (419)
0.5xra
Y - X
" ° 2 K m * r a
N = l;
Y - l
" ° 2 K m x m
Recommended y = 0.5 ; non-oscillatory response
•5-Y 4y + 1 - 4^/y2 +0.5y .T, (419)
1.5-y " Y ^ m . N •
l - r ( l •5-Y) (1.5 -y)y
266 Handbook of PI and PID Controller Tuning Rules
(
4.3.6 Classical controller 1 Gc (s) = Kc
V T i s V
1 + Tds
N J
R(s) &
E(s)
+ T
Y(s)
K „
v T i s y
l + sTd
l + s ^ N
U(s) Kme" Y(s)
Table 85: PID controller tuning rules - IPD model G m (s) = Kme"
Rule
Bunzemeier (1998). Model:
Method 4; G P =
K P
s(l + sT p )"
Minimum IAE -Shinskey (1988)
-page 143.
Minimum IAE -Shinskey (1994)
-page 74. Model:
Method 1
Kc
x i
K m T m m m
x4
K m x m
x, 0.21
0.68 0.47
0.35
0.30 0.27 0.25 0.23 0.22
0.22
^ 0.93
K r a x m
0.93
K m T m
0.93
K m x m
x2
0.730
1.570 1.200
0.981
0.866 0.828 0.801
0.782 0.767
0.756 linimum p<
1
1
Tf Td
Process reaction
X2 t ra
Coefficie
x3
0.280
0.796
0.669
0.587 0.530
0.487 0.452
0.424 0.401
0.381
;rformance
•57xm
•60xra
• 4 8 t m
X 3 t r a
nt values
x4
0.24
0.94 0.62
0.46
0.38 0.34
0.32
0.30 0.29
0.27
index: regi
0.58xn
0.58Tn
0.63xn
Comment
0% overshoot (servo)
10% overshoot (servo)
N
5.60
5.99 6.03
5.99
6.02 6.01 6.03
5.97 5.99
5.95
n
0 2 3 4 5 6 7 8 9 10
ilator tuning
i
i
i
Model: Method 2;
N not specified
N=10
N=20
Chapter 4: Tuning Rules for PID Controllers 267
Rule
Minimum IAE -Shinskey(1996)
-page 117. Minimum IAE -Shinskey (1996)
-page 121.
Chien (1988). Model: Method 1
Rice and Cooper (2002).
Model: Method 1
Luyben (1996). Model: Method 9
Belanger and Luyben (1997).
Model: Method 1
Kc
0.93 K m X m
0.56KU
10 K (420)
10 K (421)
11 K (422)
0.46KU
3.11KU
Ti
1.57xm
0.39TU
Td
0.56xm
0.15TU
Robust
0.5xm
2X. + 0.5xm
2X + im
2>. + 0.5xm
0.5xm
0.5xm
Ultimate cycle
2.2TU
2.2TU
0.16TU
3.64TU
Comment
Model: Method 2;
N not specified Model:
Method 1; N not specified
^ [ l / K m , x m ] (Chien and Fruehauf
(1990)); N=10 Suggested
k = 3.162xm
Maximum closed loop log modulus of +2dB ; N=10
N=0.1
10 K (420) _ 1
Km
0.5x„ K (421)
^ + 0.5xJ2J
K m ^ 2 +2^x m +0 .5x m2 )
K „
2X + 0.5xm
[ + 0.5xm]2
N = X.2+2A.xm+0.5x,
268 Handbook of PI and PID Controller Tuning Rules
4.3.7 Classical controller 2 Gc (s) = K
E(s) R(s)
<R)->K f 1 "\ l V l + NTds
v T i s A l + T a s J i + -
v T i s y
U(s)
l + NTds
K„e" Y(s)
Table 86: PID controller tuning rules - IPD model Gra (s) = Kme"
Rule
Hougen(l979)-page 333-334.
Model: Method 1
Hougen(1988). Model: Method 1
Kc ^
Direct synthesis - freq
12 K (423)
13 K (424)
00
oo
Td Comment
uency domain criteria
0.45^2-N
]QD°Sro im+0-65]
Maximise crossover frequency Ne[l0,30]
Five criteria are fulfilled; N = 10.
' Values deduced from graph:
Kmxm
K (423)
K r a T m
K (423)
0.1
9
0.6
1.45
0.2
4.2
0.7
1.2
0.3
2.8
0.8
1.1
0.4
2.1
0.9
0.95
0.5
1.7
1.0
0.85
Equations deduced from graph; Kc ' «—— 10 K „ T'
Chapter 4: Tuning Rules for PID Controllers 269
4.3.8 Classical controller 4 Gc (s) = Kc
f P + — v T i s y
l + ^ s
1-3* N
Table 87: PID controller tuning rules - IPD model Gm (s) = Kme"
Rule
Chien (1988). Model: Method 1
Kc T( Td Comment
Robust 1 K (125)
2 ^ (426)
2>. + 0.5Tm
0.5xm
0.5xm
2X + 0.5xm
^e[l/Km,tmJ (Chien and Fruehauf
(1990)); N=10
K (425)
K „
2? I + 0 . 5 T „
[X + 0.5T m j J
2 K (426) 1 I 0 .5 l „
Km{[x + 0.5xJ
270 Handbook of PI and PID Controller Tuning Rules
4.3.9 Non-interacting controller based on the two degree of freedom structure 1
f \ f ^
U(s) = K, R(s)
Table 88: PID controller tuning rules - IPD model G m (s) =
Rule
Minimum ITAE - Pecharroman
and Pagola (2000). Model:
Method 10
Kc Tf Td Comment
Minimum performance index: servo/regulator tuning x,Ku x2Tu * 3 T u
Coefficient values x i
1.672
1.236
0.994
0.842
0.752
0.679
0.635
0.590
0.551
0.520
0.509
x2
0.366
0.427
0.486
0.538
0.567
0.610
0.637
0.669
0.690
0.776
0.810
X 3
0.136
0.149
0.155
0.154
0.157
0.149
0.142
0.133
0.114
0.087
0.068
a
0.601
0.607
0.610
0.616
0.605
0.610
0.612
0.610
0.616
0.609
0.611
4»c
-164°
-160°
-155°
-150°
-145°
-140°
-135°
-130°
-125°
-120°
-118°
Chapter 4: Tuning Rules for PID Controllers 271
Rule
Taguchi and Araki (2000).
Model; Method 1
Hansen (2000). Model: Method 1
Chidambaram (2000b).
Model: Method 1
Chidambaram and Sree (2003). Model: Method 1
Kc
1
Km
f 1.253^
T,
2.388xm
Td
0.4137xm
Comment
^2-<1.0 T
m a = 0.6642 , p = 0.6797 , Overshoot (servo step) < 20% ;
N fixed but not specified Direct synthesis
0.938Kmtm
0.9425 K m T m
1.2346
Kmxro
2.7xm
2xm
4-5Tm
0.313tm
0.5xm
0.45Tm
N = 0; P = l ;
a = 0.833 N = oo; p = l ;
a = 0.707 or a = 0.65
N = 0; p = 0 ;
a = 0.6
272 Handbook of PI and PID Controller Tuning Rules
4.3.10 Non-interacting controller based on the two degree of freedom structure 3
U(s) = Kc [(b -1 ) + (c - l)Tds]R(s) + Kc
/ 1 \ 1 + + Tds
T s R ( s ) -
1
(l + sT f ): •Y(s)
K c[(b-l) + (c-l)Tds]
+ R(s)
-+<&->K ( 1 \
1 + — + Tds V T i s J
+ > +
U(s) Y(s)
K e"
(l + sT f)2
Table 89: PID controller tuning rules - IPD model Gm(s): Kme"
Rule
Astrom and Hagglund (2004). Model: Method 1
Kc T; Td Comment
Minimum performance index: other tuning
0.45
Km
8xm 0.5xra
No precisely defined
performance specification
b = 0, - ^ - < l ; b = l , ^ - > l . c = 0. Tf = 0 . k m m m
Chapter 4: Tuning Rules for PID Controllers
4.3.11 Non-interacting controller 4
273
U(s) = K v T i s y
R(s)
+
E(s) K,
f P 1 + -
v i i ° y Xs
+ >® U(s)
E(s)-KcTdsY(s)
Y(s) K „ e "
KcTds
T Table 90: PID controller tuning rules - IPD model
Kme"
Rule
Minimum IAE -Shinskey (1988)
-page 143. Model: Method 2 Minimum IAE -Shinskey (1994)
-page 74. Minimum IAE -Shinskey (1988)
- page 148. Minimum IAE -Shinskey (1996)
-page 121.
Kc T i Td Comment
Minimum performance index: regulator tuning
1.28 K m T m
1.28 K m T m
0.82KU
0.77KU
1.90xm
1.90xm
0.48TU
0.48TU
0.48xm
0.46tm
0.12TU
0.12TU
Also defined by Shinskey (1996),
page 117.
Model: Method 1
Model: Method 1
Model: Method 1
274 Handbook of PI and PID Controller Tuning Rules
4.3.12 Non-interacting controller 6 (I-PD controller)
U(s) = - ^ E ( s ) - K c ( l + Tds)Y(s)
R(s) >M + 1 k
E(s) K c
T iS
U(s)
r y y r
4 K c ( l + Tds)
t
K . e - ' -
s
Y(s)
— •
Table 91: PID controller tuning rules - IPD model Gm (s) = Kme"
Rule
Minimum ISE-Arvanitis et al.
(2003a).
Arvanitis et al. (2003a).
Arvanitis et al. (2003a).
Minimum ISE-Arvanitis et al.
(2003a).
Arvanitis et al. (2003a).
Kc Ti Td Comment
Minimum performance index: regulator tuning 1.4394 K m T m
1.2986
Kmxm
2.4569xm
3.2616Tm
0.3982xm
0.4234xra
Model: Method I
Model: Method 1
oo
Minimise performance index je2 (t) + Km2u2 (t)Jdt
0
1.1259
Kmxm 6.7092xm 0.4627xm
Minimise performance index
0
Model: Method 1
;'«^'(T]" dt
Minimum performance index: servo tuning 1.5521
Kmxm
1.4057
Km tm
2.1084tm
2.5986xra
0.3814xra
0.4038xm
Model: Method 1
Model: Method 1
oo
Minimise performance index [e2 (t) + Km2u2 (t)Jdt
0
Chapter 4: Tuning Rules for PID Controllers 275
Rule
Arvanitis et al. (2003a).
Chien et al. (1999).
Model: Method 2
Arvanitis et al. (2003a).
Kc
1.3332
Kmxm
Ti
3.0010xm
Td
0.4167xm
Minimise performance index 1
0
e2(t) + Km2 |
Direct synthesis
3 K (427)
16(2^2+l)
(32^2+4JKmtm
1.414TCL2+tm
N2+lk,
T (427)
1 6 ^ + 4 _
16 (2^ 2 +l j " m
Comment
Model: Method 1
'duf~ dt
Underdamped system response -
\ = 0.707
Model: Method 1
3 K (427) 1.414TCL2+xm
^ ( W + O J O T T c ^ + O ^ x , , 2 ' ' ±d r . (427) 0.25Tm^+0.707TCL2Tn
1.414TCL2+xm
276 Handbook of PI and PID Controller Tuning Rules
4.3.13 Non-interacting controller 8 f
U(s) = Kt 1 \
1 + — + Tds V T i s J
E(s)-K1(l + Tdis)Y(s)
R(s) E(s)
Table 92: PID controller tuning rules - IPD model Gm (s)
Rule
Lee and Edgar (2002).
Model: Method I
Kc T| Td Comment
Robust
4 K (428) T (428) T ( « 8 )
K; =0.25KU,
T d i =0
4 j , (428) _ 0 .25K U
(X- K ; „ K „
T ^ = 0 2 S ^ { Q 2_ T
2{X + xn
3
,(428)
K„Km ' 2(X + i J
xm2( l -0 .25K uKmTm) ,
fr + O 6(X + t r a ) 4{X + xmf 0.5KuKm(X + x J
Chapter 4: Tuning Rules for PID Controllers
4.3.14 Non-interacting controller 10
U(s) = Kc 1 + f- E(s)-K f
277
1+ I Tisy
-^-s + 1 vTm y
Y(s)
R(s) ^^y. E(s) K
V T , s 7
+ /cx U ( s ) K „ e "
Y(s)
K, i i s + l T
V m J I Table 93: PID tuning rules - IPD model Gm (s) =
K „ e "
Rule
Kaya and Atherton(1999),
Kaya (2003).
Kc Ti Td Comment
Direct synthesis
0.313Ku 0.733TU
5 T (429) Xd Model: Method 9
0.753
5 x (429)
( A A Km Ku
V T m T " )
0.333
^nJm K „
0.567
K „
( A A
Km Ku
V Tm T" J
• 0 . 3 1 3 K u
278 Handbook of PI and PID Controller Tuning Rules
4.3.15 Non-interacting controller 12 f
U(s) = Kc 1 A
V T i s j
1
Tds + 1 E(s)-K,Y(s)
R(s) E(s)
+ r — * ® - * Kc(l + -J- + Td
Table 94: PID controller tuning rules - IPD model Gm (s) = Kme-St™
Rule
Lee and Edgar (2002).
Model: Method 1
Kc T; Td Comment
Robust
6 K (430) T (430) T (430) K, =0.25KU
6 K (430) 0.25K,, • i . + -
, (430)
(X + xJ'KuKm "' 2{X + zm)
xm 0.5
4 x,
*m , V , (l-0.25K„Kn,Tni)
6ft+ 0 4(X + t m ) 2 0.5KuKm(X + Tm)
0.5
2{x+imy
0.5 ^ + - T" - +
T (430) _
( l-0.25KuKmxm)
6(^ + xra) 4(X + zm)2 0.5KuKm(^ + xm)
-i0.5
0.5 ^ + _ T" • + ( l-0.25KuKmxm)
6(X + xm) 4(X + xm)2 0.5KuKm(X + t m )
Chapter 4: Tuning Rules for PID Controllers 279
4.4 FOLIPD Model Gm(s) = K e"
<l + s T j
4.4.1 Ideal controller Gc (s) = Kc
f 1 A
1+ — + Tds T i S J
l ( !L^o E ( s )
+ T KJ1+ — + Tds
T:S
U(s) K e-sx"
s(l + sTm)
Y(s)
Table 95: PID controller tuning rules - FOLIPD model Gm (s) = Kme-
<l + sTm)
Rule
Minimum ISE -Haalman(1965).
Van der Grinten (1963).
Model: Method 1
Tachibana (1984).
Kc
Minim
2
3Kmxra
1 K m T m
e - 2 « ) d ^
K m t m
X
Km(l + Xxm)
Tj Td
um performance index: regulator
0 T
A m = 2 . 3 6 ; * m = 5 0 0 ; M 1 = 1 . 9 . Direct synthesis
0
0
0
Tm+0.5xm
1 x (431)
T
Comment
tuning
Model: Method 1
Step disturbance
Stochastic disturbance
Model: Method 5
(431) l -O-Se -^^+^ -e - " ' - t m e
280 Handbook of PI and PID Controller Tuning Rules
Rule
Chen et al. (1999a).
Viteckova (1999),
Viteckova et al. (2000a).
Model: Method 1
O'Dwyer (2001a).
Model: Method I
Chen and Seborg (2002).
Rivera and Jun (2000).
Model: Method I
Kc
rapTm
K A T <432)
riwrTm
v T (433)
x l / K m X r a
X,
0.368 0.514 0.581
OS
0% 5% 10%
X l T m
K m T m
x l
0.785
A
3 K (434)
Tm+T ra+2?t
Kra(Tm+X)2
Ti
0
0
0
Td
2 T (432)
2 T (433) ld
T
Coefficient values x l
0.641 0.696 0.748
OS
15% 20% 25%
0
x. 0.801 0.853 0.906
OS
30% 35% 40%
T
Representative coefficient values
m
) *m
45°
3TCL+Tra
Rot
T r a +T m +2^
x l
0.524
A
T (434)
)USt
Tm(xm+2X)
x m + T m + 2 ^
Comment
Model: Method 1
Model: Method 1
x. 0.957 1.008
OS
45% 50%
Ara = 0.5/x, ;
<t>m = 0.571-X,
m
s *m
60° Model: Method 1
X not specified
(432) 1
3(1
. ^ p ^ m 1 2co v- + —
P n T
(433)
4mr2xm [ 1
_ 0 25n(2r,A -1)
_(3T C L +x m >e-"" (434).. (3T C L + T m XT m +T m )
~Kc<->(TCLs + i r ' ~ (TCL+xJ3 •
X (434) 3T C L2 T m + 3 T C L T m T m -T C L
3 +T m T m2
(3TCL+xmXTm+Tm)
Chapter 4: Tuning Rules for PID Controllers 281
Rule
McMillan (1984).
Model: Method 1
Peric et al. (1997).
Model: Method 8
Kc T| Td Comment
Ultimate cycle
4 K (435)
KuCOS(|)m
T («5)
aTd<436)
0.25Ti(435)
5 T (436)
Tuning rules developed from
Ku>Tu
Recommended a = 4
K (435) 1-111 Tm • (435) 2 l m 1 +
5 T (436) tand)m +,|— + tan a
T 471
282 Handbook of PI and PID Controller Tuning Rules
4.4.2 Ideal controller in series with a first order lag r
Gc(s) = Kc 1 \
1 + — + Tds V T i s j
1
R(s) E(s)
+
r K,
1 \ 1 + — + Tds
v T i s J
1
Tfs + 1 K „ e "
s(l + sTJ
1 + Tfs
Y(s)
U(s)
Table 96: PID controller tuning rules - FOLIPD model G ra (s) = Kme"
<l + sTm)
Rule
H„ optimal-
Tan et al. (1998b).
Zhang et al. (1999).
Model: Method 1
Tan et al. (1998a). Model:
Method 1; X not specified
Kc Ti Td Comment
Robust 6 K (437) T ( 4 3 7 ) T (437) Model: Method I
A. = 0.5 (performance), A, = 0.1 (robustness), A. = 0.25 (acceptable)
7 j r (438) 3 - + Tm+xm
3^ + xm+Tm
1.5xm<A.
^4.5xm
X = 1.5xm , OS = 58%, Ts = 6xm ; X = 2.5xm, OS = 35%, Ts = 1 l i m ;
/. = 3.5xm,OS = 26%, T s= 16xm;A, = 4.5xm,OS = 22%, Ts = 20xm.
8 K (439)
8 ^ (440)
Tra+8.1633xra
Tm + 5 .3677xm
T m * m
0.1225Tm+xm
T T
0-1863Tm+Tm
Tf=0.5549xm
Tf=0.4482xra
6 v (437) (0.463X + 0.277)([0.238X + 0.123]Tm + T )
Kmx„/ 5.750A. +0.590
. (437)
7 K (438)
- T (437) Tmtm
0.238A. +0.123' d (0.238A,+ 0.123)Tm+xn
K ^ l 3^ + T r a+xm
3k'+3\xm+xm< i . T f = -
3k'+3\xm+xm'
K (439) 0.0337T„
K m x „ 1 + -
0.1225T„, , K
(440) 0.0754T„
Kmx m m V 0.1863T„
Chapter 4: Tuning Rules for PID Controllers 283
Rule
Tan et al. (1998a)-continued.
Rivera and Jun (2000).
Model: Method 1
Kc
9 K (441)
10 K (442)
Ti
Tm+3.9635xm
2(Tm+V) + Tm
Td
Tmtm
0.2523Tra+xm
2 T m ( t m + ^ )
2(xm+X) + Tm
Comment
Tf = 0.2863xm
X not specified
, K (44i) 0.1344Tm
K m T„ 1+-
0.2523Tm
10 K (442) _ 2(Tm+^)+Tm j _ Tm^
K r a K 2 + 4 T m X + X2)' f 2Tm2+4Tm^ + ^
284 Handbook of PI and PID Controller Tuning Rules
4.4.3 Ideal controller with weighted proportional term
Table 97: PID controller tuning rules - FOLIPD model Gm (s) = Kme"
s(l + sTm)
Rule
Astrom and Hagglund(1995) -pages 212-213. Model: Method 1
or Method 3
Kc
5.6e-8'8T+68,!
K r a(Tm+Tm)
8 6 e-7.1x+ 5 .4x ;
K m (T m +t m )
Ti Td
Direct synthesis
l . h r a e — ' ! 1.7tme-6-4T+20T2
b = 0.12e6'9t-6W
1.0xme"T-2^ 11 T (443)
Comment
Mmax =1.4
Mmax = 2.0
11 j (443) _ Q 3 C T 0.056T-0.60t2
Chapter 4: Tuning Rules for PID Controllers 285
4.4.4 Controller with filtered derivative G c ( s ) = Kc 1 + -Tds
R(s)
7 ^ E(s)
/
K, 1 + ^ + - ^
N
U(s)
T . S 1 + s ^ N
Y(s) Kme"CT
s(l + sTB)
Table 98: PID controller tuning rules - FOLIPD model G m (s) = K m e - "
s(l + sTm)
Rule
Kristiansson and Lennartson
(2002).
Chien (1988). Model: Method I
Kc
12 ^ (444)
2*. + ^ra+Tm
T, Td
Direct synthesis
T(444) T (444)
Robust
2^ + Tm+xm
Tm(2^ + xm)
2^ + T m +x m
Comment
Model: Method 1; K, e [0.5,1]
^e[Tm,Tm]
(Chien and Fruehauf (1990)); N = 1 0
12 ^ (444) 2.5co„
K „ •0.08 + ^ + ° ^
K, K i J
2 ( O . 0 5 9 / K I 2 ) + ( 0 . 0 5 5 / K I ) - 0 . 0 8
2.3-2K 0.4(10+ 1/K,)
,(444) 2.5 2.3-2K,
(0.059/K,2)+ (0.055/K,) - 0.08'
0.4(10 + 1/K,)
(444) 2.5 ( .2 ( O . 0 5 9 / K | 2 ) + ( 0 . 0 5 5 / K I ) - Q . 0 8 A
N = -(444)
d (444)
l f
2.3-2K, 0.4(10 +1/K,)
(o.osg/K^j+fo.oss/Kj-o.os 0.16cou(lO + l/K,)
N +
(444)
286 Handbook of PI and PID Controller Tuning Rules
4.4.5 Controller with filtered derivative with set-point weighting 1
Gc(s) = K£ , 1 T d s
TiS 1 + Td
v N
1 + b^R(s)-Y(s) . l + a f l s
R(s) l + b n s
1 + a,, s
+ E(s) /
K, 1 + -U-^ N ;
U(s) Kme"
sd + sTJ
Y(s)
Table 99: PID controller tuning rules - FOLIPD model G m (s) = -Kme-
'(1 + s T j
Rule
Liu et al. (2003). Model:
Method 1; recommended
A, = x
Kc
13 j r (445)
X
0-3Tm
0.5xm
t r a
l-5tm
^ Td Comment
Robust T(445) T (445)
Representative A values deduced from a graph
OS
= 57%
= 33%
= 7%
= 0%
Rise time
= 3Tm
= 3.5xm
= 5xm
- 8.5rm
A
2Tm
2-5Tm
3Tm
3-5Tra
OS
- 0 %
= 0%
= 0%
= 0%
Rise time
= 8Tm
= 10.5xm
= 12.5xm
= 14tm
n K (445) _{3X + xm+Tmi3X2+3Xim+zm2)-X3 ^
Km(3X2+3Xxm+xm2}
T (445) = (3X + xm+TmJ3X2 +3Xxm+ij)-X3
(3X 2 +3^ r a +T m2 ) '
T (445, Tm{3X + xmJ3X2+3Xxm+xm2) A3
(3A + Tm+Tm)(3A2+3Axm+Tm2)-A3 3A2+3ATm+Tm
2 '
Tm{3X + xm) _
^t(3A + T m + Tj(3A 2 +3AT m + T m2 ) -A 3 ]
N = -
Chapter 4: Tuning Rules for PID Controllers 287
4.4.6 Controller with filtered derivative with set-point weighting 3
Gc(s) = Kc 1 1 T
d s 1 + — + — = -
Ts - X
V 1 + ^ - s
N ;
ri+b f ls±b i ls2
+b f 3s 3_R ( s )_Y ( s )^ 1 + af ,s + af 2 s 2 + af 3s3
R(s)
l + b f l s + b f 2 s 2 + b f 3 s 3
l + a n s + a f 2 s 2 + a f 3 s 3
Table 100: PID controller tuning rules - FOLIPD model Gm (s) = Kme"
*(l + sTm
Rule
Liu et al. (2003). Model: Method 1
Representative
X values
(deduced from a graph)
K c
1 ^ (446)
a f l =
X 0-3Tm
0-5xm
t r a
l-5Tm
7X + xm; a
OS
= 43%
= 15%
= 0%
= 0%
T j Td
Robust r(446) rj- (446)
X d
f 2 = X(\6X + 4xm), af
Rise time
= 3.5xm
= 4.5Tm
= 8Tra
= 12-5Tra
X 2Tm
2.5Tm
3Tm
3.5Tra
Comment
= 4X2(3X + xm)
OS
= 0%
= 0%
= 0%
= 0%
Rise time
= 16.5tm
= 20xm
= 24t m
= 28Tm
• K (446, _ (3*. + t m + Tm J3X2 + 3Xxm +xm2)-X3
K ^ + S ^ + T , , , 2 ) 2
Ti(446) = {3X + xm+TmJ3X2 +3Xxm+xm2)-Xi
l 3X 2 +3^ m +T m2 )
(446) Tm{3X + xmJ3X2+3Xxm+xm2)
b f[ = 3X , b f2 = 3X , b f3 = A.
*3
(3X + T m +T m ) (3X 2 +3^ m +T m2 ) -X 3 3X2+3Xxm+Tm
2'
N = Tm(3X + T m ) t
X3 [(3X + xm + Tm )(3X2 + 3Xxm + xm2)- X3 ]
288 Handbook of PI and PID Controller Tuning Rules
Rule
Liu et al. (2003) - continued.
Kc Ti Td Comment
a f , = 4X + xm ; a f2 = X(3.25X + xm ) , a f 3 = 0.25A2 (3A. + xm )
Representative A. values (deduced from a graph)
A
0.3xm
0.5tm
V
l-5xm
OS
* 6 1 %
» 40%
- 1 5 %
* 7 %
Rise time
- 3 t m
- 3 i m
- 3 . 5 x m
- 5 x m
X
2xm
2.5xm
3xm
3-5xm
OS
* 4 %
* 4 %
* 1 %
» 0 %
Rise time
- 6 . 5 x m
* 8 x m
"10T m
* 1 2 x „
Chapter 4: Tuning Rules for PID Controllers 289
4.4.7 Ideal controller with set-point weighting 1
U(s) = Kc(FpR(s)-Y(s))+|^(F iR(s)-Y(s))+KcTds(FdR(s)-Y(s))
Table 101: PID controller tuning rules - FOLIPD model Gm (s) = Kme-
s(l + sTm)
Rule
Oubrahim and Leonard (1998).
Model: Method 1
Kc T, Td Comment
Ultimate cycle 0.6KU,
Fp=0.1
0.5TU,
F i = l
0.125TU,
Fd=0.01 0.05 < — < 0.8
T
20% overshoot
290 Handbook of PI and PID Controller Tuning Rules
4.4.8 Classical controller 1 Gc (s) = Kc
v T i s y
1 + Tds
N ;
R(s)
+
E(s) K,
V T i s 7
l + sTd
T 1 + s-i i
U(s) Kme"
s(l + sTm)
Y(s)
Table 102: PID controller tuning rules - FOLIPD model G m (s) = Kme"
S(l+sTm)
Rule
Minimum IAE -Shinskey (1994)
-page 75.
Minimum IAE -Shinskey (1994) pages 158-159.
Model: Method 1;
N not specified
Kc Ti Td Comment
Minimum performance index: regulator tuning 0.78
K m ( t m + T J
2 K (447)
3 K (448)
1.38(xm+Tm)
~ (447)
0-66(xm+TJ
0.56tra+0.75Tm
N=10. Model: Method 1
^-<2 T
IHL>2 T
2 j r (447)
3 v (448)
100
108KraTm 1.22-0.03-
- T ( 4 4 7 )=157x 1 + 1.2 1 - e T™
K 100
108Kmxm 1 + 0.4-
Chapter 4: Tuning Rules for PID Controllers 291
Rule
Minimum ITAE - Poulin and Pomerleau
(1996), (1997). Model:
Method 1
Process output step load
disturbance
Process input step load
disturbance
Poulin et al. (1996).
Model: Method 1
Chien (1988). Model: Method 1
Kc
4 R (449)
Ti
X 2 ^ m + T m )
Td
Tm
Comment
0< , T") , < 2 (Tm/N)
3.33<N<10 Coefficient values (deduced from graphs)
Xm
Tm/N
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
x i
5.0728 4.9688 4.8983 4.8218 4.7839 3.9465 3.9981 4.0397 4.0397 4.0397
x2
0.5231 0.5237 0.5241 0.5245 0.5249 0.5320 0.5315 0.5311 0.5311 0.5311
t m
Tm/N
1.2 1.4 1.6 1.8 2.0 1.2 1.4 1.6 1.8 2.0
Xl
4.7565 4.7293 4.7107 4.6837 4.6669 4.0337 4.0278 4.0278 4.0218 4.0099
x2
0.5250 0.5252 0.5254 0.5256 0.5257 0.5312 0.5312 0.5312 0.5313 0.5314
Direct synthesis
S K (450) 2l(0.2Tm+Tm) T N = 5.
Mmax = 1 dB
Robust
Tm
Kra( + 0 2
2X + xm
Km(X + t J 2
T
' *- + *m
2X + xm
Tm
^e[Tm ,Tm]
(Chien and Fruehauf
(1990Y>-N=10
292 Handbook of PI and PID Controller Tuning Rules
1 4.4.9 Classical controller 2 Gc(s) = K
E(s) R(s)
+ ®-K f , ~\
1+-V T i S y
1 + NTds
V 1 + T d « j
1+-v T i s y
U(s)
l + NTds
1 + Tds j
K „ e - " »
s(l + sTm)
Y(s)
Table 103: PID controller tuning rules - FOLIPD model Gm (s) = Kme-
<l+sTm)
Rule
Hougen(1979)-page 338.
Model: Method 1
Kc Ti Td Comment
Direct synthesis: frequency domain criteria
60.106x! K m T m
00 0.135Tma7tm
0-3
Maximise crossover frequency; N e [10,30]
x, values deduced from graph:
^m/Tm
x l
^m/Tra
X l
1
6
7
1.1
2
3
8
1.0
3
2.1
9
0.9
4
1.8
10
0.8
5
1.6
11
0.7
6
1.3
12
0.7
Chapter 4: Tuning Rules for PID Controllers
4.4.10 Series controller (classical controller 3)
Gc(s) = Kc
U(s)
293
1+ v Tisy
R(s)
~ &
E(s) K
v T i sy (l + sTd)
Kme" s(l + sTm)
(l + sTd)
Y(s)
Table 104: PID tuning rules - FOLIPD model Gm(s) =
Rule
Skogestad (2003).
Model: Method I
Kc T; Td Comment
Direct synthesis 0.5
K m X m 8tm T
294 Handbook of PI and PID Controller Tuning Rules
4.4.11 Classical controller 4 Gc(s) = Kc '.+-L' V T i s /
1 + •?*
1 + -N
R(s)
• &
E(s) Kr
V T i s / 1 + -
sTri
1 + s-N
U(s) Kme-
s(l + s T J
Y(s)
Table 105: PID controller tuning rules - FOLIPD model G m (s) = -Kme-
<l + sT j Rule
Chien (1988). Model: Method 1
Kc
2X + xm
Km(^ + xm)2
Tra
Kmfr + 0 2
Ti Td
Robust
2X + Tm
Tm
Tm
2X + xm
Comment
(Chien and Fruehauf
(1990)); N=10
Chapter 4: Tuning Rules for PID Controllers 295
4.4.12 Non-interacting controller based on the two degree of freedom structure 1
U(s) = Kc 1 + - U THS
TiS l + (Td/N> E(s)-Kc a +
PTds
l + (Td/N> R(s)
a + PTds
T
N ;
K „
R(s) + E(s)
K, T ' S l + ^ s
N
U(s)
+
K e"ST"
s(l + sT n )
Y(s)
Table 106: PID controller tuning rules - FOLIPD model Gm (s) = Kme-
<l + sTra)
Rule
Minimum ITAE - Pecharroman
and Pagola (2000). Model:
Method 4; Km = l ;
T m = i ;
P = 1,N=10;
0.05 <xm <0.8
Kc Ti Td Comment
Minimum performance index: servo/regulator tuning x,Ku x2Tu x3Tu
Coefficient values
* i
1.672
1.236
0.994
0.842
0.752
0.679
0.635
0.590
0.551
0.520
0.509
x2
0.366
0.427
0.486
0.538
0.567
0.610
0.637
0.669
0.690
0.776
0.810
X 3
0.136
0.149
0.155
0.154
0.157
0.149
0.142
0.133
0.114
0.087
0.068
a
0.601
0.607
0.610
0.616
0.605
0.610
0.612
0.610
0.616
0.609
0.611
*.
-164°
-160°
-155°
-150°
-145°
-140°
-135°
-130°
-125°
-120°
-118°
296 Handbook of PI and PID Controller Tuning Rules
Rule
Taguchi and Araki (2000).
Model: Method 1
Wang and Cai (2001).
Model: Method 1
Wang and Cai (2002).
Xu and Shao (2003a).
Xu and Shao (2003c).
Kc
1 K (451)
2 K (452)
0.6068
Kraxm
0.6189
5 K (455)
1.5xm+0.5Tra
2Kmxm2
Tj
T (451)
Direct s T (452)
5.9784xm
5.9112xm
2(Tm+1.5xm)
T m + t m
Td
T (451) x d
vnthesis T (452) *d
3 T (453)
4 T (454)
(Tm+0.5xm)2
(Tm+1.5Tm)
l-5Tmxm
1.5xra+0.5Tm
Comment
^s-<1.0 T
N=0; p = 0.
Model: Method 1
Model: Method 1
6
Model: Method I
K „ 0.7608+ -
0.5184
[-2- + 0.01308]z
,(451)
(451) = T
0.03330 + 3.997-
0.03432 + 2.058-
-0.5517
-1.774 -0.6878 3A
a = 0.6647 , P = 0.8653 - 0.1277(xm /Tm)+ 0.03330(xm /Tm f,
N fixed but not specified; OS < 20% .
K (452) 1.3608-1.2064
M „
T ,452) = xm(l.3608Ms-1.2064) 1 0.29M,-0.3016
(452) (Tm+0.1xmXl.450Ms-1.508) , a = -
0.2M, , 1.3<M < 2 .
1.3608MS-1.2064 ' 1.3608MS-1.2064
Td(453) = 0.8363(Tra + 0.1xm), a = 0.3296 , Ms = 1.6 (A m > 2.66 , <|)m > 36.4°)
Td(454) = 0.8460(Tm + 0.1xm), N=0, P = 0, a = 0.3232 , Am = 3 , <|>m = 60°.
K (455) 0.5(Tm+1.5x„ ;> a = : K m x m ( T m + x m ) ' Tm+1.5x
, P = 0.6667 , N= oo , Am > 2 , $m > 60°
N=co,p = 0 , A m > 2 , 4 m > 6 0 0
.5T+0.5X
Chapter 4: Tuning Rules for PID Controllers 297
4.4.13 Non-interacting controller 4
U(s) = Kc 1 + -T , sy
R(s) E(s)
+ X K, 1 + —
V T i S 7
U(s)
E(s)-KcTdsY(s)
Y(s) K m e-
s(l + sTJ
K cT ds
I Table 107: PID controller tuning rules - FOLIPD model Gm (s) =
Kme"
s(l + sTm)
Rule
Minimum IAE -Shinskey(1994)
-page 75. Minimum IAE -Shinskey(1994)
-page 159.
Kc T i Td Comment
Minimum performance index: regulator tuning 1.16
Km(x r a+Tm)
7 ^ (456)
l-38(tm+Tm)
-p (456) i
0.55(Tm+Tm)
0.48xm+0.7Tm
Model: Method 1
Model: Method 1
7 K (456) 1.28
Kmxm(l + 0 . 2 4 - ^ - 0 . 1 4
T ( 4 5 6 , = 1 . 9 x „
Tm "\
l + 0.75[l-e Tm]
298 Handbook of PI and PID Controller Tuning Rules
4.4.14 Non-interacting controller 6 (I-PD controller)
U(s) = - ^ E ( s ) - K c ( l + Tds)Y(s) T,s
R(s) >6* + 1 L
E(s) Kc
T iS
U(s)
' W "
Kc(l + Tds)
*
Kme-™
s(l + sTm)
Y(s)
— •
Table 108: PID tuning rules -FOLIPD model Gra (s) = Kme-
s(l + sTJ
Rule
Arvanitis et al. (2003b).
Minimum ISE 8
Kc T( Td Comment
Minimum performance index: regulator tuning 16
(16-3P)KmTm
4
(3Tra ( 8 - P ) T
16 m
Minimum j[e2(t) + KmV(t)]dt 9
0
Model: Method 1
P = 1.5841 + 0.3101| — +0.38841 - 0.4622 'O* T
+ 0.219181 —^
-0.060691 T
+ 0.010771 'O6
T V m j
•0.0012466 T
V m
-^H +9.1201.10"5 X,
T V 1m J
•3.8302.10" , T i*- +7.0316.10"8
f \10
T ,
P = 0.066485+ 2.0757 '^^ T
+ 1.0972 M -V m J
( - \ -2.7957
-0.68418 T
+ 0.14763 f ^
1M. , T
•0.019733
T
f ^
+ 1.9019 ( \4
T V m J
T + 0.0016016
t T
-7.2368.10" ^-| +1.397.10"6
T 0<-S-<4.0;
T„
P = 2.128638-0.005608121-^-4.0 •>4.0.
Chapter 4: Tuning Rules for PID Controllers 299
Rule
Arvanitis et al. (2003b).
Model: Method 1
Kc
10 j , (457)
T; T(457)
Td
T (457)
Comment
Minimum ISE ' '
Minimum performance index 12,13
10 K (457) 12.5664(xm +Tm)-4
Kmxm[25.1328(Tm+Tm)-8-(xm+2Tm)a]
r ^ ™ „ 3.4674xm+3.1416T - 1 , with a = [0.5708 + J ^ ^ ]% ,
T(457) . _ 12.5664(xm+Tm)-4 T <457> _ T T 2
' i - ' x d _ 1 m 1 r a 12.5664(xm+Tm)-4
11 x = 1.178 + 4.5663 't A
T V 'my
-7.0371 T
+ 5.5784 -^-\ -2.61031 V m V m
-0.7679 f \ 5
I™. T
V m
(~ \ -0.1458
T + 0.017833 'o7
> T , V m y
-0.0013561 H 2 -> T
+ 5.83.10" T
-1.0823.10" T
0<-2=-<3.5, T
X = 2.2036 - 0.0033077| ^ - - 6.5 , ^ - > 3.5 .
00
: Performance index = |je2(t) + KmV(t)]dt
13 x =-0.1185 + 4.781^2-]-3.9977 f \ x
+ 0.057708 / \5
X „
T V m y
0.0038421
T
V. 1 r a y
X
-1.733 X
T v m y
-0.42245 / \ 4
> T V m y
T V m y
+ 2.6386.10" T-i2-| +1.078.10"5' ^ s -vTm
-4.2486.10" T
V. m
300 Handbook of PI and PID Controller Tuning Rules
Rule
Arvanitis et al. (2003b).
Kc T, Td Comment
Minimum performance index: servo tuning 16
(16-3B)Kmxm
4
( 3 T m
(8-P) 16 m
Minimum ISE M
Minimum performance index 15, 16
Model: Method 1
14 p = 1.2171 + 0.7339Us- +0.083449 \—\ -0 .14905^2- + 0.030257^
lTJ ITJ \Jm) lTmJ + 0.0045567
/ ^5
T
T -0 .0029943p -0.00056821
T V m J
-5.4957.10"
+ 2.7505.lO"6!^1- -5.6642.10"8M2-
Performance index = |e2(t) + Km2u2(t)Jit
'P = 0.083709+ 2.28751 .18739 T
V m J
•1.0059PB IT.
-0.79718 vTJ
-0.3036^2- +0.067859-^2-
l T m J l T m y
(- \ -0.0093104
T + 0.00077161 ' T ,
A
T
-3.5473.10" 5 Lm + 6.9487.10" I™. , o<^2L<4.5
V T J Tm
6 = 2.444547-0.0146274714 •Is.-4.5 1 ^2_>4.5 . T I T
Chapter 4: Tuning Rules for PID Controllers 301
Rule
Arvanitis et al. (2003b).
Model: Method 1
Kc 1 7 K (458)
T;
T ( 4 5 8 )
Td
T (458) 1 d
Comment
Minimum ISE 18
Minimum performance index 19,20
17 K (458) 12.5664(xm+Tm)-4
KmTm[25.1328(xm + T m ) - 8 - ( x m +2Tm)a]
with a = [0.5708-3.4674xm +3.1416Tm - 1
- (458) 12.5664(Tm+Tm)-4 (458) = T - T m *-m 12.5664(x r a+Tm)-4
X = 1.1571 + 5 . 0 7 7 5 ^ . - 8 . 2 0 2 1 ^ - + 6 . 6 3 5 3 ^ - -3.1355
ITJ ITJ UJ + 0.92683-2!-
V m y -0.17635
vTm
f ^ X „
T
T V m J
+ 0.02158 -0.0016402 Xr
T V m y
]X
+ 7.0439.10" 5 l m -1.3056.10" 0<-in-<4,5
T
X = 2.124307 - 0.004844182| i= - - 4.5 , — > 4.5 .
' Performance index = |je2(t) + Km2u2(t)Jdt
20 x =-0.70861+ 7.4212| ^2-1-8.1214 'O' T
V m y
+ 4.8168 r A 3
x.
T V m y
-1.7217
+ 0.38998 ^ V ' m y
•0.056975 — +0 .0053141^- -0 .00030233^ V.T m J V m y V m y
+ 9.3969.10"6 - a - -1.1861.10" v 111 /
X = 2.1069751341 -0 .0017098-^--4 .5
0 < _ E - < 4 . 5 T
•>4.5 .
302 Handbook of PI and PID Controller Tuning Rules
Rule
Arvanitis et al. (2003b)
(continued).
Arvanitis et al. (2003b).
Model: Method 1
Kc
23 K (459)
24 K (460)
T| Td
Minimum performance index 21,22
Direct synthesis
k2+lk y (460)
16^2 +4
16(2i;2+l)Tm
T (460)
Comment
Performance index = e2(t) + K 2 — J m l dt
dt
22 % = 0.05486 + 0.816621 X
V m J -1.4989pa
V m J
-1.1955 p V m j
+ 0.11011 -0.013386-^-l T m J
^ -0.00059665 'x . N?
T
-0.48464p-
+ 3.8633.10"5 - ^ L
-5.307.10' T
V m J
+ 1.6437.10"7 0 < - 2 L < 6 . 0 ; T
X = 2.1111 -0.001025 -^S--6 T,
- > 6 .
23 JJ- (459)
24 v (460) K
16(2^2+l)
(3242 +4)KmTm '
^ + T m + 4 T m ^ 2
Kmtm2(l+8527
Tr ) ^ m +T m +4T r a ^ ( Tr 0 , = TmTm (l + 4^)
^ + T m + 4 t m ^
Chapter 4: Tuning Rules for PID Controllers
4.4.15 Non-interacting controller 8
303
/ U(s) = Kt
1 \
V T i s J
EC^-KiCl + T^^YCs)
Table 109: PID controller tuning rules - FOLIPD model G m (s) = K m e "
5(l + sT r a)
Rule
Wang et al. (2001b).
Model: Method 7
Lee and Edgar (2002).
Model: Method 1
Kc T( Td Comment
Direct synthesis 0.4189 K m T m
4-t 1.25(Tm+0.lTm)
Km^m
Td i=0; Am=3;<t .m=60 0
Robust
1 K (461) T (461) i
T (461)
Ki = 0.25KU ,
T d i =0
i „ (461) 0.25K,,
ft + O • x m +
2(A. + T „ )
, (461) t m +
2(^ + t r a ) :
(461) _ 0-25Ku 4Tm 2 Tm | Tm
T m2 ( l - 0 . 2 5 K u K m T m ) ,
0.5KuKm(^ + x J J
304 Handbook of PI and PID Controller Tuning Rules
4.4.16 Non-interacting controller 11
U(s) = K(
v T i sy (l + Tds)E(s)-K i( l + Tdls)Y(s)
R(s) E(s)
+
f 1 ^ <8^KC1 + -L(l + T d s ) ^ 0 > v Tisy
U(s) K „ e "
s(l + sTm)
Y(s)
Ki(l + Tdis)
I Kme"
Rule
Majhi and Mahanta(2001). Model: Method 6
Kc
0.5236
Kmtm
" ' & " " V U 1 " " "
T|
m v /
Td
Direct synthesis
2 x m e ^ T
s(l + sTm)
Comment
T d i =T m ;
K m T m
A m = 3 ;
4»m=60°
Chapter 4: Tuning Rules for PID Controllers
4.4.17 Non-interacting controller 12
305
f U(s) = Kc
1 \ 1 + — + Tds
V T i s j
1
Tds + 1 E(s)-K,Y(s)
R(s) E(s) 1
Tds + 1
+
w
U(s)
• § -K e"
(l + sTj
Y(s)
K,
I Table 111: PID controller tuning rules - FOLIPD model Gm(s) =
K . e " s(l + sTm)
Rule
Lee and Edgar (2002).
Model: Method 1
Kc Ti Td Comment
Robust
2 K (462) T(462) T (462) K, = 0.25KU
l K ^ , = 0 J 5 K J L _ J _ _ X m + _ L
^ + t m KuKm
4 T m - + 0.5xJ.
2(^ + xm)
K u K m
C , Tm4 f ( l-0.25KuKmxm)xm
6(^ + xm) 4(X + xm)2 0.5KuKm(^ + x J
-(462) 4 x - - T m + KuKm
2(X + T m )
4 T - - + 0.5xJ-KuK.m
V , C , ( l - 0 . 2 5 K u K m Q T m
6(^ + xra) 4(^ + x ra)2 0.5KuKm(X + xm)
(462) ^ - + 0.5x 2 - X" (l-0.25KuKmxm)x„
6(^ + xm) 4(^ + xm)2 0.5KuKm(A. + Tm)
306 Handbook of PI and PID Controller Tuning Rules
4.4.18 Industrial controller U(s) = K 1 + — R ( s ) - ^ Y ( s )
N j
R(s)
+ K, 1+-
T i S y
U(s) K e"
s(l + sTm)
Y(s)
1 + Tds
l + ^ s N
Table 112: PID controller tuning rules - FOLIPD model Gm (s) = K„e"
s(l + sTm)
Rule
Skogestad (2004b). Model:
Method 1
Kc Ti Td Comment
Direct synthesis: time domain criteria 1
Km (TC L +x r a ) ^ 2 ( T C L + T J T 'good'
robustness -
TcL = Tm ' 4 =0.7 or 1
N e [5,10]; N = 2 if measurement noise is a serious problem
Chapter 4: Tuning Rules for PID Controllers 307
4.4.19 Alternative controller 1 Gc (s) = Kc 1 + TjS
N j
1 + Tds
N ;
Table 113: PID controller tuning rules - FOLIPD model Gm (s) = Kme"
<l + sTm)
Rule
Tsang et al. (1993).
Model: Method 2
Tsang and Rad (1995).
Model: Method 2
Kc T, Td Comment
Direct synthesis x i
Km tm T 0.25xm N = 2.5
Coefficient values
* i
1.6818 1.3829 1.1610
% 0.0 0.1 0.2
0.809
x l
0.9916 0.8594 0.7542
% 0.3 0.4 0.5
T
Xl
0.6693 0.6000 0.5429
% 0.6 0.7 0.8
0.5xm
Xl
0.4957 0.4569
\ 0.9 1.0
Maximum overshoot =
16%; N = 8.33
308 Handbook of PI and PID Controller Tuning Rules
4.4.20 Alternative controller 2
Gc(s) = K£
\
1 + TjS
V N .
C 2 „ 2
l + 0.5Tms + 0.0833T„ s
N
K,
f \
1 + TjS
1 + ^ N
/ 2 „ 2
l + 0.5xms + Q.0833xm s
l + ^ s N
U(s) Kme"
s(l + sTm)
Y(s)
Table 114: PID controller tuning rules - FOLIPD model Gm (s) = Kme"
>(l + sTm)
Rule
Tsang et al. (1993).
Model: Method 2
Kc T i Td Comment
Direct synthesis
* i K
m T m T 0.25xm N = 2.5
Coefficient values
*1
1.8512 1.5520 1.3293
4 0.0 0.1 0.2
x l
1.1595 1.0280 0.9246
% 0.3 0.4 0.5
x i
0.8411 0.7680 0.6953
\
0.6 0.7 0.8
x i
0.6219 0.5527
\ 0.9 1.0
Chapter 4: Tuning Rules for PID Controllers 309
4.5 SOSPD Model
Gm(s) = K fi
l l + s T m l X l + s T m 2 ) orGm(s) =
Kme~
T m l2 s 2 +2^T m l s + l
4.5.1 Ideal controller Gc(s) = Kj 1 + — + Tds T:S
R ( s ) ^ E ( s )
+ S i Kc
{ 1 1 1 + — + Tds
I T i s J
U(s) SOSPD model
Vis)
w
Table 115: PID tuning rules - SOSPD model G m (s) = K m e "
(l + sTm,Xl + sTm2)
G m (s ) = K m e "
T m l2 s 2 + 2 ^ m T m l S + l
Rule
Auslander et al. (1975).
Kc Ti Td Comment
Process reaction 1 K (463) T (463)
i 0.25Tj(463) Model: Method 3
Note: equations continued into the footnote on page 310.
K (463) . 1.2 K m ( T n i l ~ T m 2 )
, Tm lTm 2 , Tjni T + In
111 rr^ ry^ r-r-> lm\ ~ Xra2 Xin2
( T m / Tm2
Tm l -Tm ; 1m2
T
Tml 1 Tml-Tm2
( T m l - T m
T , 1T„„-Tm! m2 ' 5 n 2 . 1 T - - T -
| ! Tm2 j Tm2
Tmi ~ Tm2 ^ Tm,
Trai I Tm2
Tmi Tm2 ^ T m l j
310 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum IAE -Wills (1962b) - deduced from
graph. Model: Method 1
Minimum I A E -Shinskey (1988)
-page 151. Model: Method 2
Minimum ITAE - Lopez et al.
(1969). Model:
Method 17
Kc T i Td
Minimum performance index: regulator
5.8
Km
5
Km
0.62KU
0.68KU
0.79KU
x , / K m
0.36TU
0.29TU
0.38TU
0.33TU
0.26TU
x 2Tm l
0.21TU
0.29TU
0.15TU
0.19TU
0.21TU
x3Tm l
Representative coefficient values (from graphs)
x i
25 0.7
0.35
25 1.8 9.0
x2
0.5 1.3 5
0.5 1.7 2
* 3
0.25 1.2 1.0 0.2 0.7
0.45
$« 0.5 0.5 0.5 1.0 1.0 4.0
^ m / T m i
0.1 1 10 0.1 1 1
Comment
tuning Representative tuning values;
= 0.1Tnl
T
Tm2 + T m
T m2 - 0 5 T m 2 + *m
T m 2 - 0 7 S
T m 2 + 1m
• (463)
2Tm lTm 2 Tml
Tm2 , T-ral T m 2
+ 2x „
1+-
2
( T m , - T m 2 )
x m2
V T m l )
T„,2
T m , - T m 2 (T ~\
1 m 2
VTml J
Tm, 1 T m , - T m 2
m2
Tml _ T m 2
J m 2
T V 'ml )
Tmi Tm2
Xm2
V T m l )
Chapter 4: Tuning Rules for PID Controllers 311
Rule
Minimum ITAE -Bohl and McAvoy (1976b).
Kc
2 K (464)
T,
T (464)
Td
T (464)
Comment
Model: Method I
T 2 _E!l = 0.12,0.3,0.5,0.7,0.9; - ^ - = 0.1,0.2,0.3,0.4,0.5
K, (464) 10-9507
K „
> -1.2096+0.1760 In 1
V Tml J
o^y
10-
0.1044+0.1806 In 10^=2- -0.2071 In 10-^=-l Tm,J { TmlJ
T i< 4 6 4 )=0.2979Tn l
N 0.7750-0.1026 In 10-
10-2 V Tml )
> 0.1701+0.0092 In
10- 2
T V ml /
J 1 (&L +0.0081 In [ 10 -^ -
Td< 4 6 4 )=0.1075Tn ,
v 0.6025-0.0624 In 10-
10-2
V Tml J
T „ ,
10-
0.4531-0.0479lnl 1 0 ^ - |+0.0128In| 10-T„ , T m i ;
' m l )
312 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ITAE -Hassan (1993).
Kc
3 R (465)
T,
T (465) i
Td
T (465)
Comment
Model: Method 7
J 0.5 < ^m < 2 ; 0.1 < — ^ < 4 . Kc,40:" is obtained as follows:
Tmi
log[KmKc(465)]= 1.9763370-0.6436825^m - 5.1887660-^2- + 0.43755584„
+ 2.9005550^2-] +3.14680104m^!_-0.16972214n V^ml
-0.81618084n T
1.2048220^m2 - 5 ! - - 0.08103734m
3
-0.4444091-^-1 +0.03194314n
f \ 3
VTml J + 0.1054399^
VTmiy
-0.16528074m3^i_ + 0.11759914m
3[^2-| -0.03752454n 3 ' T"
ml J
T:(4 5) is obtained as follows:
lot . (465)
-0.7865873 + 0.6796885^m + 2.1891540-^2-- 0.3471095^m2
Tmi
-1.9003610 - ^ M -0.70078014m-^- + 0.3077857^n ^Tmi J Tml VTm1 )
-0.8566974^ ' m l y
- 0.2535062^m2 ^ - + 0.0412943^m
3
Tmi
+ 0.3484161 ( \ x
VTml J -0.1626562^ ^ = - -0.06618994m
2
v 1mi y
+ 0.2247806£m3 - $ - - 0.2470783^n ^ + 0.049301 l^m
3U=
(465) Td is obtained as follows (continued into footnote on page 313):
IOE T, (465)
= -0.6726798 - 0.20724774m + 2.6826330^2- + 0.0807474^m2
-1.7707830 -^s- -1.6685140^m-^- + 0.0845958^m2
^Tml J Tml Tmi
Chapter 4: Tuning Rules for PID Controllers 313
Rule
Minimum ITAE - Sung et al.
(1996). Model: Method 9
Kc
4 j , (466)
T i
rj, (466)
Td
T (466)
Comment
0.05 < i s - < 2
(, \ + 0.7159307^n
VTmiy + 0.56314474m
2 i * - - 0.0225269£m3
Tmi
+ 0.2821874 -0.0616288^ m] /
i s - | -0.0626626^ 2
V^rol V^ml
- 0.0372784^m3 i s - - 0.0948097^m
3 i s - | +0.0272541^ 3
V ml .
Formulae continued into footnote on page 314.
K (466)
K „ -0.67 + 0.297
T V ml J
+ 2.189 r \ -0.766
1 „
VTml , Tm, •<0.9 or
K (466) _ __1_ -0.365 + 0.260 T
V iml
-1.4 +2.189 T
V 1 m l
->0.9;
. (466) 2.212 , N 0.520
VTml / -0.3 •<0.4 or
Ti(466) =Tm l {-0.975 + 0.910 •is--1.845 + a} , - i s - > 0.4 with VTml J Tml
a = 1-e 5.25-0.1
T (466) _
1-e 1.45 + 0.969 -1.9 + 1.576 V ^ m l
314 Handbook of PI and PID Controller Tuning Rules
Rule
Nearly minimum IAE, ISE, ITAE -Hwang (1995).
Model: Method 10
Kc
5 R (467)
6 K (468)
7 ^ (469)
Ti
T (467)
T (468)
T ( 4 6 9 )
Td
T (467)
Comment
s2 < 2.4
2.4 < e2 < 3
3 < e2 < 20
5 K (467) f K „ , - ^ ;l-0.435<DH,Tm +0.052a)H,2Tm2P
, (467)
Km/(l + K H 2 K m )
Kc'467)(l + K H 2 K j
Km0.0697(l + 0.752coH2tm-0.14503H22Tn,2j'
T d - = ^ ^ f l - ^ l ( l - 0 , 1 2 c o u x m + 0 , 0 3 c o u S m2 ) .
K m (»u I £ n
6 K (468) ' K 0.724[l-0.469(0H2Tm+0.0609coH22Tm
2lN
Km/(l + K H 2 K m )
.(468) K c « 6 8 > ( l + K H 2 K m )
o)H2Km0.0405^1 + 1.93oiH2Tm-0.363cDH2iTm
/
7 K (469) _ 1.26(0.506)'°"2T'"[l-1.07/e2+0,616/e22]N
Km/(l + K H 2 K m )
.(469) T l « " l K c (1 + K H 2 K m )
)H2Km0.066l(l+ 0.824 ln[coH2xm])(l + l . 7 l / E 2 _ l . l 7 / e 22 ) '
Chapter 4: Tuning Rules for PID Controllers 315
Rule
Hwang (1995)-continued
Minimum IAE -Wills (1962b) - deduced from
graph. Model: Method 1
Minimum IAE -Gallier and Otto
(1968). Model:
Method 1
Minimum ITAE -Wills (1962a)-deducedfrom
graph.
Kc
8 K (470)
Decay ratio
Ti
T (470) i
Td
~ (467)
(page 314) = 0.2; 0.2<xm /Tm l <2.0 and 0.6
Comment
e2 >20
<?im<4.2
Minimum performance index: servo tuning
6-5/Km
7/Km
x, /K m
Represent fm
2Tml
0.053 0.11 0.25
x i
6.7 4.15 2.35
18
Kra
1.45TU
0.95TU
9 T ( 4 7 1 )
0.14TU
0.22TU
T (471) x d
Representative tuning values;
Tm2 = Tm
= 0.1Tml
Tmi = Tm2
itive coefficient values - deduced from graphs x2
0.89 0.84 0.77
X 3
0.24 0.24 0.24
« T
V 2Tml
0.67 1.50 4.0
* 1
1.05 0.70 0.52
*0.2TU
x2
0.64 0.55 0.50
x3
0.24 0.26 0.20
Tmi = Tm 2 ;
^m=0.1Tm l ;
Model: Method J
K (470) _ ' 1.09[l-0.497(oH2Tm+0.0724a)H22Tm
2lN
H2 Km/(l + K H 2 K j
T(470) K c (l + K H 2 K m )
' fflH2Km0.054(-1 + 2.54aH2xm - 0.457a>H22xm
2)'
"if7" =x2(Tm l + T m 2 + x j , Td<471) =x3(Tml + T m 2 + xj.
316 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ITAE - Sung el al.
(1996). Model: Method 9 Nearly minimum IAE, ISE, ITAE -Hwang (1995).
Model: Method 10
Kc
10 £ (472)
11 j r (473)
Tj
j (472)
j (473)
Td
x (472) 1 d
0.471KU
Kmcou
Comment
0.05<-^2-<2
Decay ratio = 0.1;
0.2 < -^2- < 2.0; Tml
0 .6<^ m <4 .2 1 2
10 K (472) _ 1 - 0.04 + 0.333 + 0.949 / N-0.983
5 m <0.9 or
K (472) _ 1 -0.544 + 0.308—i2-+ 1.408 * . T i l i f ' 5" 5 m >0-9 .
Ti(472) =Tml [2.055 + 0.072(xm/Tml)Jm, Tm/Tml <1 or
T/472' =Tml[l.768 + 0.329(Tm/Tml)£m, xm/Tm, > 1 ,
(472)
T , „ i
J _ e 0.870 0.55 + 1.683
11 K (473) K 0.822t-0.549(oH2Tm+0.112coH22Tm
2]N
Km/(l + K H 2 K m )
- (473) K ( ^ ( 1 + KH2K,J (oH2Km0.0142(l + 6.96(DH2Tm-1.77o)H2
2xm2)'
12 ^< 0.613+ 0.613^21-+ 0.117 T ' m l
r ^2
vT m , y
Chapter 4: Tuning Rules for PID Controllers 317
Rule
Hwang (1995)-continued.
Hwang (1995)-continued.
Kc
13 j , (474)
15 K (475)
16 K (476)
17 K (477)
Ti
T (474)
T (475)
T ( 4 7 6 )
T (477)
Td
0.471KU
K m m u
0.471KU
K m f f l u
Comment
Decay ratio = 0.1;
0.2 < ^-2-< 2.0 ; Tm,
0 .6<^ m <4 .2 1 4
Decay ratio = 0.1;
0 . 2 < ^ - < 2 . 0 ; Tral
0 .6<4 m <4.2 1 8
13 v (474) _. „ 0-786]! - 0.441coH2Tm + 0.0569coH22Tm
2 h
' Km/(l + K H 2 K j
T(474) _ Kc (l + KH 2Km)
2Km0.0172(l + 4.62WH2xm -0.823coH22tm
2) '
E<0.613 + 0.613-^- + 0.117 U ^ Tml ^Tm
15 K (475) K t 1.28(0.542)c°H;tm[l-0.986/s2+0.558/e2
2}
, (475)
16 j . (476) K t
.H[I
Kc'475>(l + KH 2Km)
2Km0.0476(l + 0.9961n[coH2Tm])(l + 2.13/s2 -1 .13/e 22 ) '
1.1411-0.466coH2xm + 0.0647coH22Tm
2^
T;
Km/(l + KH 2Km)
(476) Kc (l + KH2Km)
17 K (477) *[l-0.: r-coH,Km0.0609 - 1 +1.97co„,xm - 0.323co
— 2 — 5 1 -H2 T m J
0.794|1 - 0.541coH2Tm + 0.126(oH22Tm
2 "
Km/(l + K H 2 K m )
.(477) K<477'(l + K H 2 K j
a>„2Km0.007811 + 8.38coH2Tm - 1.97coH2zxn
£> 0.649+ 0 . 5 8 ^ - - 0 . 0 0 5 Tm, VTml )
318 Handbook of PI and PID Controller Tuning Rules
Rule
Hwang (1995)-continued.
Hwang (1995)-continued.
Kc
19 j r (478)
21 j r (479)
22 K (480)
23 K (481)
T,
T (478)
T ( 4 7 9 )
T (480)
T (481)
Td
0.471KU
Kmwu
0.471KU
Comment
Decay ratio = 0.1;
0.2 < -^5- < 2.0 ; Tn,
0.6 < ^m < 4.220
Decay ratio = 0.1;
0.2 <-^2-< 2.0;
0 .6<^ m <4 .2 2 4
19 K (478) _ 0.738[l - 0.415ooH2xm + 0.0575coH22xm
2 \> j ^ H 2 . ,
Km/(l + KH 2Km)
T (478) K (l + K H 2 K m ) 1 aH2Km0.0124(l + 4.05coH2xm-0.63coH2
2xm2)'
X ( T ^ 20 £,> 0.649 + 0 .58-^- -0.005 —s
'ml V^miy
21 K (479) K t
1.15(0.564)m"iT"[l-0.959/e2+0.773/e22]"
Km/(l + K H 2 K m ) J'
T . (479) K c ( 1 + KH2Km)
coH2Km0.0335(l + 0.9471n[coH2xm])(l + 1.9/82-1.07/e22) '
22 K (480) _ i 1.07|l-0.466coH2xro+0.0667mH22xm
2]N
K H 2 - , J
v Km /(l + K H 2 K j
r(480) Kc'^(l + K H 2 K j
23 K (481) ?[i - o.:
mH2Km0.0328(-1 + 2.21C0H2X,,, - 0.338coH22Tm
2)'
0.789|l-0.527(DH2xm+0.11CoH22Tm
2l^ K H 2 —-• =
Km /(l + K H 2 K m )
.(481) K™(1 + K H 2 K B )
)H2Km0.009(l + 9.7co xm-2.4coH , 2Tm2) ' H2 lm
I < 0.649 + 0.58—— - 0.005 T VTml 7
,^>0.613 + 0.613^a- + 0.117 X „
VTml J
Chapter 4: Tuning Rules for PID Controllers 319
Rule
Hwang (1995)-continued.
Wilton (1999). Model: Method 1
Kc
25 K (482)
27 K (483)
28 K (484)
Ti
T (482)
rp (483)
T (484)
Td
0.471KU
Kmwu
Comment
Decay ratio = 0.1;
Q.2<^-<2.0; Tmi
0.6 <lm <4.22 6
Minimum performance index: other tuning
Minimum
x i
"[e2(t) + x 22Km
2 (y(t) -yJ 2 ]dt; Tm, >Tm2 •0
Tm l+Tm 2
Tm2
1 + Tm2/Tml
Coefficient values (obtained from graphs) T r a2/Tml=0.25
x i
1250.1 22.946 4.8990 2.1213
x2
0.0001 0.04 0.2 1
Tm2/Tml=0.5
x i
20.396 4.8990 2.1213
x2
0.04 0.2 1
Tm2/Tm, =0.75
x i
22.946 4.8990 2.1213
x2
0.04 0.2 1
xm /Tm l=0.1
25 K (482) _ ' 0.76[l-0.426o)H2Tm+0.0551mH22Tm
2lA
Km /(l + KH 2Km)
26 £,< 0.649 + 0.58-^—0.005 Tmi
• (482)
( \2
T
Kc«82>(l + K H 2 K m )
raH2Km0.01531 + 4.37coH2Tra-0.743»H2iTn
VTml V ,4>0.613 + 0 . 6 1 3 ^ - + 0.117
' x > '
VTmiy
27 K (483) 1.22(o.55)"'H'T" [l-0.978/s2 +0.659/e22 j
Km/(l + KH 2Km)
, (483)
28 K (484)
j t io j ; K c (1 + KH2Km )
coH2Km0.042l(l + 0.9691n[coH2xm])(l + 2.02/e2 - l . l l / s 22 ) '
' l.ll[l-0.467coH2Tm +0.0657coH22Tm
2lN
Km/(l + K H 2 K m )
T; (484) Kc'^(l + K H 2 K j
>H2Kra0.0477(-l + 2.07coH2Tnl-0.333fflH22Tm
2)'
320 Handbook of PI and PID Controller Tuning Rules
Rule
Wilton (1999) -continued.
Keane et al. (2000).
Model: Method 1
Huang and Jeng (2003). Model:
Method 13
Kc
x i
Ti
Tmi + Tm2
Td
Tm2
1 + Tm2/Tml
Coefficient values (obtained from graphs) Tm 2 /Tm l=0.25
x i
180.01 8.1584 2.9394 1.2728 90.004 4.5891 2.0331 0.8768
x2
0.0001 0.04 0.2 1
0.0001 0.04 0.2 1
29 K (485)
Tm2/Tml=0.5
x i
230.01 9.1782 3.1843 1.3435 100.00 4.6911 2.0821 1.0324
x2
0.0001 0.04 0.2
1 0.0001
0.04 0.2
1 T (485)
1 I
Tra2/TMl=0.75
x i
300.01 10.708 3.4293 1.4142 112.01 5.354
2.1311 1.0748
x2
0.0001 0.04 0.2
1 0.0001
0.04 0.2
1 T (485) ' d
Comment
Tm /Tm l=0.5
x m / T m l = l
Minimum ITAE servo plus ITAE regulator, with minimum Mmax and
with good noise attenuation; Tml = Tm2
30 j - (486) 2^mTml T (486) Sn.Sl-1
( l n K j T u + T m , ( l + lnKu)] 29 K (485)
-(485) _
T (485) _ T T ' u ' m l
T u+Tm l ( l + l n K u ) '
(lnKu)[Tu+Tm l(l + lnKu)]
(lnKu)(l + lnKu) + [ lnKu+ln(l + eT»-K-)][ln(Ku+1.33419Tm)-l]
30 K (486) _ 1.2858Tm^m [ T„
KmTm
(486) _ [(-0.0349^m +1.QQ64X +(0.4196^m -0.1100>mP
2T,^m
Chapter 4: Tuning Rules for PID Controllers 321
Rule
Huang and Jeng (2003) -
continued.
Hang et al. (1993a). Model:
Method 8;
^ 2 - > 0 . 3 Tml
(Yang and Clarke (1996)).
H o e ; al. (1994). Model: Method 1
Hoet al. (1995a). Model: Method 1
K c
31 K (487)
Ti
T (487) i
Td
T (487)
Comment
4m > 1-1
Servo or regulator tuning - Minimum IAE
Direct synthesis: Frequency domain criteria
^ m l
A m K r o T m 2Tml 0-5Tml Tml ~~ Tm2
Sample results
1.5708Tml
KmTm
1.0472Tml
K m x m
32 K (488)
M p T m l
An ,K i n
2Tml
2Tral
T ( 4 8 8 ) i
33 j (489)
0-5Tml
0-5Tml
T (488)
Tm2
A r a = 2 . 0 ,
A m = 3 . 0 ,
* m = 6 0 °
T m / T m < 2 ^ m
Tm, > T m 2 .
31 K (487) 0.5890
K m t m y ^ra V
T 0 . 0 0 5 2 - 2 ^ - + 0.8980Tm2 +0 .4877t m + T m l
T/ 4 8 7 ' = 0 . 0 0 5 2 - 2 ^ - + 0.8980Tm2 +0.4877xm + T m l ,
T (487) _ T
T 0.0052 - = ^ - + 0.8980Tm2 +0 .4877i n
0.0052 - = = - + 0.8980Tm2 +0.4877xm + T m l
32 T , (488) _ ' " " p 1 m l 2conTm,2 f „5
7 t A „ , 7 t - 2 0 ) T m
V«pTm,
T (488) _ 2 2 > M - ' m l
(488) _
J
f , ^ i s ! - + t - 2 « p x m
VTml«p TtT.
• + t T m l -2T m ] co p T m
2co„
1
4©D2Tm 1
322 Handbook of PI and PID Controller Tuning Rules
Rule
Ho et al. (1995a) - continued.
Ho etal. (1997). Model: Method 1
Leva et al. (1994). Model:
Method 15
Kc Ti Td Comment
Sample results 0.7854Tm,
K m T m
0.5236Tml
K m x m
34 K (490)
36 K (491)
Tml
Tm,
T (490) 1 i
T (491)
Tm2
Tm2
T (490)
O^TA491'
A m = 2 . 0 ,
<L=45°
A m = 3 . 0 ,
<K„=60o
35
* m = 7 0 °
(at least).
34 K (490)
tA m K m **>m+K-2-
l m l J
,Tr 0 , =-T m l fe m + , -2 , m ) , 7t
TtT, ml T (490) _ _
2 ( ^ m T m l + 7 t T m l - 2 T m ) ' T, Tm < 1 , 2£ > ^ 2 - .
8™ m ^ - ( A m
2 - l ) - 2 7 t A m ( A m - l )
4A„
36 v (491) _ W c n j i (491)
K l + »c„ 2Tm , 2
K m ^
) 2 +4^ m2 co c n
2 T m l
, (491)™ (4911 2
1-T^'V X" r+[Ti ] w r , ( 4 9 1 ) 0 ^ 2
4.07 - <|>ra + tan" i [2^x m T m l (0 .57 I - c t m )
(0.57t-<^ ra)2Tml2-xm
2
, (491) -tan 0-5 co c n T m +<|> m -0 .57t- tan"
2^mcocnTml
V ^ c n T m l - ! y
l m ^ o r ^ m > l (with 0 . 5 7 1 - ^ > 3 T „
^m+V^2-1
Chapter 4: Tuning Rules for PID Controllers 323
Rule
Leva et al. (1994)-
continued. Wang et al.
(1999).
Wang and Shao (1999). Model:
Method 12
Kc
37 K (492)
SmT r al
K m T m
1 K m X m
Ti
T (492) 1 i
2^raTm,
2^mTml
Td
T (492)
0-5Tml
4 m
T
Comment
Model: Method 12. 38
Some coefficient values
* i
1.571
1.047
Comment
A m = 2 , < L = 4 5 °
A m =3,< t , m =60°
* i
0.785
0.628
Comment
Am = 4 , ^ = 6 7 . 5 °
A r a=5,<j>m=72°
T 2 | C 0 c n 2 + - V ( 2 ^ V ^ 37 ^ (492) _ _ ^ c n j _ m l _ 1 m l V
( 4 9 2 > l l coj+x2
• 1 - 1
(0.5TI -(|>m) Tml - x m
co„
tan fflcnTml
^m+V^n
. (492) Tralz + U m + ^ m
2 - l (492) ,
Sm + ^ m2 - 0 Tmlz + Um + kJ-l
. 5m>l
with 3x„
£m "V^ir < 0.571 -< |> m < Tml
S m + A ^
^ m > 0.7071
or 0.05 < ° - 7 ° 7 h - <0.15, Q J Q 7 l T "- > U „ * 1 Tm l A /24m
2- l T m l ^ - 1
Tml
or 0.05 < % ^ s - < 0.15 , - ^ 2 - > l,^m < 1.
324 Handbook of PI and PID Controller Tuning Rules
Rule
Wang and Shao (2000b). Model:
Method 12
Chen et ah (1999b).
Model: Method 1
Leva et ah (2003).
Model: Method 1
Kc
39 ^ (493)
X lSmT m i
* i
1.00
1.22
1.34
1.40
1.44
1.52
1.60
^
2^mTml
2^m
Td
Tml
2^m
2^m
Comment
Suggested Mmax = 1.6
Comment
A m = 3 . 1 4 , <L = 61.4°, Ms =1.00
A m = 2 . 5 8 , ^ = 5 5 . 0 ° , M.=1.10
Am =2.34, <L = 51.6°, M. =1.20
Am =2.24, 4>m = 50.0° ,M, =1.26
A m = 2 . 1 8 , *m = 48.7° , M,=1.30
Am =2.07, <L=46 .5° ,M S =1.40
Am =1.96, 4)m =44.1°, M,=1.50
40 K (494)
41 K (495)
16(Tm 2+tm)
5Tm2+4Tm
4(Tm 2+xm)
T (495) x d
3 9 j£ (493) _ " » » m l 2^mTm .451-1.508 M „
40 K (494) :
8Km(T, 4(Tm , ) •
41 rr (495) _ Tm l ( 5 T m 2 + 4 T m ) T (495)
8Km(Tm 2 +Tm)2
4Tm2(Tm2+xm) 5Tm2+4Tm
T , > ' A m l —
8(Tm + x„
Chapter 4: Tuning Rules for PID Controllers 325
Rule
Wills (1962a). Model: Method 1 Servo response;
4 = 1-
•Cm = ° - 1 Tm l
Van der Grinten (1963).
Model: Method 1
Pemberton (1972b).
Pemberton (1972a), (1972b).
Model: Method 1
K T, Td Comment
Direct synthesis: time domain criteria X i / K r a x2Tu
X 3 T u
Representative coefficient values - deduced from graphs
* i
2.5 2.75 3.2 20 12 1.4 1.5
x2
2.9 2.9 2.9 2.9 2.9 1.4 1.4
* 3
0.035 0.073 0.143 0.285 0.57
0.035 0.073
42 £ (496)
43 j r (497)
2(Tml+Tm2)
3Kmxra
(Tm l+Tm 2)
Kraxm
* 1
1.6 16 10
0.42 0.4 10 2
x2
1.4 1.4 1.4
0.70 0.70 0.70 0.70
Tm l+Tm 2
+0.5Tm
T (497)
Tm l+Tm 2
T r a l+Tm 2
* 3
0.143 0.28 0.72 0.073 0.143 0.36 0.70
*1
0.2 8
0.09 0.1
0.11
T (496) 1 d
T (497)
TmiTm2
Tml +Tm2
TmiTm2
x2
0.48 0.48 0.28 0.28 0.28
* 3
0.143 0.36 0.073 0.143 0.28
Step disturbance
Stochastic disturbance
Model: Method I
0 . 1 < ^ - < 1 . 0 Tm2
0 . 2 < ^ - < 1 . 0 Tm2
42 K (496) = 1 f Q 5 i T m l + T m 2 I
K. (496) (Tm ,+Tm 2)xm+2Tm lT l 1 1 m 2
^ + 2 ( T m l + T m 2 )
43 K (497) _ e
K „ 1 - Q.5e-a'x- +-
T „ , + T „
• (497)
• (497)
Tml + T „
X
l-0.5e" ( T m l + T m 2 ) T
(Tml+Tm2) m /
l-0.5e~ *ml + ' m 2 e~MJTm
326 Handbook of PI and PID Controller Tuning Rules
Rule
Pemberton (1972b).
Model: Method 1
Chiu et al. (1973).
Model: Method 1
Mollenkamp et al. (1973).
Model: Method 1
Smith et al. (1975).
Model: Method 1
Suyama(1992). Model: Method 1
Wang and Clements (1995).
Gorez and Klan (2000).
Model: Method 1
Viteckova (1999),
Viteckova et al. (2000a). Model:
Method 1
Kc
2 ( T r a l + T m 2 )
3K m x m
^T m l + Tm2
K n ( l + XTm)
2smTm i
K r a ( T C L + x r a )
W m l
K m ( l + ^ t m )
X = pol
2 K m x m
2 ^ m T m l
K r a( l + ^ m )
44 K (498)
x , (T m l + T m 2 )
K m t m
x i
0.368 0.514
0.581 0.641
OS (%)
0 5
10 15
Ti
Tmi +Tm2
Tmi + T m 2
2smTm i
Tml
e of specified FO]
Tmi +Tm2
2smT r a i
2smTm i
Tmi + Tra2
x,
0.696 0.748 0.801
0.853
OS (%)
20 25
30
35
Td
T r a , + T m 2
4
TmlT,n2
T m l + T m 2
Tml
2^ m
Tm2
Comment
0 . 1 < ^ k . < 1 . 0 Tm2
0.2 < ^ 2 - < 1.0 Tm2
A. variable; suggested values are 0.2, 0.4, 0.6
and 1.0.
Appropriate for "small xm "
Tm, > Tm2
J D closed loop response.
TmiTm2
T m l + T m 2
24m
Tml
2^ m
Tm iTm 2
T m , + T m 2
x i
0.906 0.957
1.008
OS (%)
40 45
50
OS=10%
Model: Method 16;
Underdamped response
Non-dominant time delay
Closed loop overshoot
(OS) defined
Overdamped process;
T„i > Tm2
K (498) _ 2^mTm l
K m ( 2 ^ m T m l + ! „ , ) •
Chapter 4: Tuning Rules for PID Controllers 327
Rule
Viteckova (1999),
Viteckova et al. (2000a) -continued.
Skogestad (2004b).
Model: Method 1
Vitefikova et al. (2000b).
Model: Method 4 Arvanitis et al.
(2000).
Bietal. (2000). Model: Method 8
Chen and Seborg (2002).
Kc
xiSmTmi
KraXm
x, 0.736 1.028 1.162 1.282
OS (%)
0 5 10 15
2^mTml
K m (T C L + x m )
A
0-74Tml
KmTm
45 jj- (499)
l -0128^mTm l
K m x m
46 Tj (500)
Ti
24 m T m l
x i
1.392 1.496 1.602 1.706
OS (%)
20 25 30 35
2 ^ m T m l
Td
Tm l
2 ^ m
* 1
1.812 1.914 2.016
OS (%)
40 45 50
Tm l
2 ^ m
Comment
Underdamped process;
0-5 < ^m < 1
Recommended
TCL = Tm
ra = 3 . 1 4 , <|>m = 6 1 . 4 ° , M m a x =1 .59
2Tm l
2^dm
esTml
l -9747K m x m
T (500)
0-5Tml
Tm l ^edes
0.5064Tm l2
K m t m
x (500)
Tml - Tm2
Model: Method 1
Model: Method 1
45 K (499) 2 ^ s T m
6 K (500) ^ 1 [(Tm| + Tm 2 )xm + Tm |Tm 2](3TC L + T m ) - T C L3 - 3T C L
2 x m
C " K m ( T C L + ^ ) 3
T(500) = [(Tm[ + T m 2 ) x m +T m l T m 2 ] (3T C L + T m ) - T C L3 - 3 T C L
2 t m T m i T m 2 + ( T m i + T m 2 + T m ) T m
T (5oo) = 3T C L2 T m l T r o 2 + T m l T m 2 x m ( 3 T C L + x m ) - ( T m l + T m 2 + x m ) T C L
3
" [(Tm, + T m 2 )x m + Tm ,Tm 2](3TC L + x m ) - T C L3 - 3 T C L
2 x m
.dVdesi
T ( 5 0 0 ) s e - s x „ _ Ti se -
'desired K c ( T C L S + 1 /
328 Handbook of PI and PID Controller Tuning Rules
Rule
Chen and Seborg (2002) -
continued.
Brambilla et al. (1990). Model:
Method 1
Lee etal. (1998). Model: Method 1
Kc
47 K (501)
48 y- (502)
Ti
T (501) i
Td
T (501)
Comment
Robust T (502) ry, (502)
»d
Representative coefficient values (deduced from graph)
^m T m l + T m 2
0.1 0.2 0.5 T ( 5 0 3 )
Km(2A. + Tn
X
0.50 0.47 0.39
,)
1m
Tml + Tm2 1.0 2.0 5.0
49 j (503)
X
0.29 0.22 0.16
•tm
Tm l+Tm 2
10.0
T (503) J d
X
0.14
47 K„ ( 5 °
..) 1 (2^T m , x m + T m , 2 ) [3T C L + T m ) -T C L3 -3T C L
2 T
^ _ K m ( T C L + X J 3
j (5oi) _ j2^mTmlTm+Tml J j3T C L +t m ) - TCL -3TC L
Tm l2+(2f T . + r
/K~"-CL ' v m / J C
+ (^^m^ml + T m) T i + l 2 Sm 'ml + T m / T m
T (501) _ 3 T g L2 T m l
2 + T m l Tm JTcL
+ Tmlz<3TCL + -3TCL^
IdJ, 48
. /desired K c (TCLS + 1 /
K (502) = Tml + T m 2 + ° - 5 T m T (502)
KmTm(2^ + l) ' ' = Tml +Tm
49 T(503) _ 7 ? T 2X -Z
' ~1^'^ ihx + i
,2+0.5xm
j (502) _ Tm |Tm2 +Q-ATml + Tm2/tm Q | < \ d Tm ,+Tm2+0.5Tm ' ' ~ T m ^
<10. Tm ,+Tm2+0.5Tm Tm l+Tm 2
2 ^ ~ T m T (503) _ T (503) _ , , T Tml _
2(2X + T r a ) ' d " ' ^ m l " » +T ] ( 5 0 3 )
x 3 e-
x"s
—,.-,. .m r ; desired closed loop response = — 6Ti
(503)(2>. + Tm) (Xs + l)2
Chapter 4: Tuning Rules for PID Controllers 329
Rule
Lee et al. (1998). Model: Method 1
Huang et al. (1998), Rivera and Jun (2000).
Model: Method I
Marchetti and Scali (2000).
Kc
T (504) i
Km(2A. + Tm)
T; 50 y (504)
* i
Td
T (504) ' d
Comment
X = max imum[0.25Tm ,0.2Tml ] (Panda et al., 2004)
T (505) i
Kra(^ + Tm) rj, (506) 1 i
Kro( + 0 24mTral
Km(xro+A.)
51 T (505)
52 T (506) * i
2^mTml
T (505) *d
T (506) ' d
Tml
2^m
Desired closed loop response =
(X.s +1)
A, specified graphically for different OS and TR values (Huang et al.
(1998)) 53 ^ (507) j (507) x (507)
' d Model:
Method I
50 T ( 5 0 4 ) _ T T 2 ^ 2 - x m2
2(2A + x m ) T (504) _ T (504) T T
T m l T m 2
• (504)
-rrrrr r ; desired closed loop response = — 6T i
(504)(2A + Tm) (Xs + l)2
( T
3 ^
51 x (505) T i ^ ' = 2 ^ m T m l + 2(^ + Tm)
T (505) _ T (505) ») . T
> 'd ~ li lCsm 'ml 6(A + T J
.(505)
52 T (506) : 1 m 1 + * m ~> "*" 2(X-
T (506) _ T (506) ( T , T -v 1 d ~ M I ' m l + ' n i 2 '
6(X + -,(506)
53 K (507) _ 2^mTml +0-5Tm T(507) = 2 £ J + 0 5t T ( 5°7 ) - Tml(Tml + ^ m O
Km(X- 2^ m T m l +0.5T n
330 Handbook of PI and PID Controller Tuning Rules
Rule
Smith (2003). Model: Method 1
Kristiansson (2003).
Wills (1962a). Model: Method 1
Quarter decay ratio tuning. Tmi = Tm 2 ;
^m=0.1Tml
Bohl and McAvoy (1976b).
Landau and Voda (1992).
Model: Method 1
Kc
Tml
Kj^ + O 54 K (508)
Ti
Tml
2^raTral
Td
Tm2
TmI
2^m
Comment
^ e K , . T m 2 ] ,
T m l >T m 2
Model: Method 1
Ultimate cycle x,/Km x2Tu x3Tu
Coefficient values (deduced from graph) x i
10 8 14 33 18
x2
2.9 1.4 2.9 2.9 1.4
x 3
0.036 0.036 0.068 0.143 0.068
55 K (509)
56 £ (510)
3KU
x i
28 6 2
0.6 30
x2
1.4 0.67 0.48 0.28 0.67
T ( 5 0 9 )
4 + (3 1
0.51T„
* 3
0.143 0.068 0.068 0.068 0.143
x i
18 0.9 18 18 13
T (509)
4 1
4 + M,35»
0.13TU
x2
0.48 0.28 0.67 0.48 0.28
X 3
0.143 0.143 0.24 0.24 0.28
Model: Method 1
1 < (3 < 2
(»uTm<0.16
54 v (508) _ 2 ^ m T m l K Kmxm
55 K (509) _ 2 . 2 6 8 9
K „
T:(509)=0.2693T„
Td(509) = 0.0068TU
M „
2.0302+0.9553 In T I IT.
T
. Recommended values of Mraax : 1.4,1.7,2.0.
(V K \1.3135+0.18091n(K„K„)+0.52191np!!.
r \-0.0517-0.0643 In —
T
-4.2613-1.58551n
(K K ) a 5 9 1 4 - 0 0 8 3 2 1 n ( K « K m ) + 0 0 6 8 7 1 n — 1
(K K V11436-°-2658ln(KuKm)-lll001n| ^2-1
56 K (510) 4 + P P 4 2V2|Gp(ja>135„)|
Chapter 4: Tuning Rules for PID Controllers 331
Rule
Landau and Voda (1992)-
continued.
Kc
1.9KU
T;
0.64TU
Td
0.16TU
Comment
0.16<couTm
<0.2
332 Handbook of PI and PID Controller Tuning Rules
4.5.2 Ideal controller in series with a first order lag
Gc(s) = Kc 1 + "Tds Tfs + 1
R(s) ms;
^ 1 i
K c | l + - J - + Tds I T,s J - •
1 T fs + 1
U(s) S O S P D
mode l
V(s)
^w
Table 116: PID controller tuning rules - SOSPD model Gm (s) = K m e ^
(l + sTm,Xl + sTm2)
Gm(s) Kme"
Rule
Umetal. (1985).
Arvanitis et al. (2000).
Model: Method 1
Panda et al. (2004).
Model: Method 1
Huang et al. (2005). Model:
Method 14
Hang et al. (1993a).
Model: Method 8
m v /
K c
T m l2 s 2 + 2 4 r a T
Ti
m l S + 1
Td Comment
Direct synthesis: time domain criteria 1 K (511)
2 K (512)
S m * m l
KmTm
T m l + T m 2
2 ^ % ,
2^mTm l
lmlTm2
Tmi + Tral
Tra, -^tdes
Tm,
2^ m
Direct synthesis: frequency domain cri
KmTm 2^mT r a l
Tm,
24m
A m * 3 , < t , m * 6 0 °
Robust
2Tml + Tm
2{X + xm)Km
Tm, +0.5xm ^ m l T m
2 T m l + x m
Model has a repeated pole ( T m l ) .
Model: Method 1
T ^ %m
T T ml
f " 2N^ m '
N not defined teria
T °-025Tml
^ > 0 . 2 5 x m .
T ^ m
' 2(X + i m )
1 K (5U) =
2 ^ (512)
Tm l+Tm
Km(2^Tc • . T f
T, CL2
24TCL2+tn
KJS
*CL2 T l m
= , ^CL2 ~ ' l r n de%,,+0' f 2^esTr,,+T
2CTm, m lCL2 T '•ra/
= 2 ^ T r i , 2 - i „ 2 1
?m »CL2 T l m
Chapter 4: Tuning Rules for PID Controllers 333
Rule
Rivera and Jun (2000).
Model: Method
1; X not
specified Lee and Edgar
(2002). Model: Method 1
Zhang et al. (2002).
Model: Method 1
5% overshoot: X = 0.5xm
K c
24 m T m l
X
2^mTm l
K m ( 2 T m + X )
3 K (513)
Tmi + Tm2
K m k+2^)
4 K (514)
T ( 5 1 5 )
K m ( 2 ^ + Tm)
T,
2^mT r a l
2^mT r a l
X (513) i
Tmi + T m 2
I ml Im2
Tmi + T m 2
5 T ( 5 1 5 )
Td
2^ m
Tml
2^ m
T (513)
TmiTm2
Tml + Tm2
T m l + T m 2
T (515)
Comment
T f = - c m
T f = ^ 2xm+X
T * ' • 2A. + t m
H ^ method
T f = Xx™ ; X + 2xm
H 2 method
Tf = 0 ;
Maclaurin method
3 K (513) = _ (2f;mTml
0.5x 2 - ^ 2
Km(2X + xm)K S m ml 2X + x
6{2X + xJ 2 ^ T m l +
0-5xm2-X2^
2X + xm
0.5xJ-X2
2X + x„
J (513) . i t T , li - z S m ' m l +
2A. + T m
6(2^+ x m ) 2^mTm l +
0.5x 2 - ^
2X + x
0 . 5 x „ / - ^
m ; 2X + x„
(513)
4 v (514) K
6(2^+ x m )
Ami l m 9
2^mTm l + 0-5xm
2-X2
2A, + xm
0.5xm2 -X2
2A. + T„
K m ( T m l + T m 2 X ^ + 2 i m ) '
5 T ( 5 1 5 ) : T m l + T „ . 2X2-xJ 2(2X + xm)'
t m l ^rtl
• (515) 12A. + 6xm 2 ^ - x „ -(515) 2(2X + x m ) '
334 Handbook of PI and PID Controller Tuning Rules
Rule
Kristiansson (2003).
Model: Method 1
Panda et al. (2004).
Model: Method 1
Kc
6 K (516)
7 K (517)
8 j £ (518)
9 K (519)
Ti
^ S m ^ m l
T ( 5 1 8 )
-p (519)
Td
2^m
T (518)
T (519)
Comment
K (516) 2^mTm
7 K „
Km i r a
2^raTm
a - a - 1 + -M „
1—1 Mn
T T — ml
• f " T ' 1 + 0 . 0 1 9 — - JO.346 + 0.038-!-211- + 0.00036
Suggested M m a x =1.7 ,p = 10.
8 K (518) _24 m X 1 +0-25T m l +0.5^ m T Km(X + 2^mTml)
T(518) _ 4^m Tml +0.STml +4mTm T (518) _ (Tml +2^mTmJTm |^m
' 4^m2Tm, +^ m t m +0.5T m
T f =
2^„
^(Tm l+2^mxm)
44ma. + 44mxm+2Tm
-, X = max imum 0.25 2^n
-0.4^mT,
9 K (519)
0.828 +0.812-2^-+ 0.172Tmle Jm]
-(6.9T„,/Tml) . 0.558Tm2T„,
Tml+1.208Tra2
- + 0 . 5 T „
K „ A, + 0.828 + 0 . 8 1 2 ^ - + 0.172Tmle-(6-9T™;/T"")
Tmi 0.558T ,T
T/ s ' 9 >=0.828+0.812ii^+0.172T . e ^ " " ' ^ ^ u ' 3 3 8 1 ^ 1 " " + 0 .5x n Tml Tml+1.208Tm2
0.828 + 0 . 8 1 2 ^ + 0.172T ,e-(6 '9T-/T">T LU6T^T^ + T Tml »Tm,+1.208Tra2 (519)
1.656 + 1 . 6 2 4 - ^ + 0.344Tm,e-(6-9T"'/T-')+ L 1 1 6 T ^ T " " + T
Tf = 4l-'16Tm,Tm2+Tro(Tm,+1.208Tro2)]
2(^ + xmXTm, +1.208Tm2)+2.232Tm2Tml '
Tml+1.208Tro2
X = max imum °-279Tm2Tml +Q2^ 0 - 1 6 5 6 + 0 - 1 62 4 l !°2. + 0-0344T e-MTm2/Tml)
Tml+1.208Tm2
Chapter 4: Tuning Rules for PID Controllers 335
Rule
Huang et al. (2005). Model:
Method 14
Kc
10 K (520)
T, Td
Ultimate cycle T(520) T (520)
Am«3,<t ,m«60°
Comment
T (520)
T f = d
20
10 K <520) = 0.080-K„ T„
/ 6.283xn
T V u J
T.(520)=0.159K„ Tu 6.283t„
T
Td(520) = 0.159-?-
K„
1 + K„ cos
f \ 6.283xr
T V u J
6.283T„
T
336 Handbook of PI and PID Controller Tuning Rules
4.5.3 Ideal controller in series with a first order filter f - ^
Gc(s) = Kc
V T , s j
b f ls + l
a f ls + l
R( s) E(s)
K, 1+ — + Tds l + bf]s
l + afls
U(s)
H K „ e "
T m l V + 2 ^ T m l s + l
Y(s)
Table 117: PID controller tuning rules - SOSPD model Gm(s) = Kme-
T m l V + 2 ^ m T m l S + l
Rule
Jahanmiri and Fallahi (1997).
Model: Method 6
Kc
2sm'mi
K m ( t r a + ^ )
T i Td
Robust
2smTmi 24m
X = 0.25Tm+0.1^mTml
Comment
b f l =0.5Tm,
^ m
" " 2(Tm+^)
Chapter 4: Tuning Rules for PID Controllers 337
4.5.4 Ideal controller in series with a second order filter (
Gc(s) = Kc 1
1 + — + Tds V T i s )
l + bfls
l + afls + af2s
R(s) E(s),—
T:S
U(s
l + b f l s
l + a f ls + a f 2s
Kme"
Tml2s2 + 2^mTmls + 1
Y(s)
Table 118: PID controller tuning rules - SOSPD model Gm (s) = Kme"
T m l V + 2 4 m T m l s + l
Rule
Kristiansson (2003).
Model: Method 1
Kc Ti Td Comment
Robust 11 K (521) 2^mTmi Tml
2^m
M m a x =1.7 ,
P = 10.
bIf = 0 , a I f =0.08Tm I ,a2 f =0.01Tml2
II ^ (521) _ 2 ^ m T m l 1 + 0 . 0 1 9 ^ - - .0.346 + 0.038-^- + 0.00036-=^
338 Handbook of PI and PID Controller Tuning Rules
r
4.5.5 Controller with filtered derivative Gc(s) = K,
E(s) R(s)
+ & - K
c , 1 T d s
T i s l + ^ s N
, ! T d s
N J
U(s) SOSPD model
Y(s)
Table 119: PID controller tuning rules - SOSPD model G m ( s ) = K m e "
(l + sTmlXl + sTm2) or
Gm(s) = Kme"
Tml2s2+24mTmls + l
Rule
Hang et al. (1994).
Model: Method 1
Hang et al. (1994).
Model: Method 1 Zhong and Li
(2002).
Kc
WpTml
A m K m
0.7854Tml
K m x m
Tml
K m ( T C L + x m )
13 K (523)
Ti Td
Direct synthesis 12 T (522) Tm2
Sample result
Tml Tm2
Robust
Tmi
T (523)
Tm2
T (523)
Comment
N = 20. Tml > T m 2
A m = 2 . 0 ,
<J)m = 45°
N = 20.
T m l > ™m2
Model: Method 1
1
2co„ -4co„ t m i
n v (523) 2 ^ m T m l ( 2 a + T m ) - a (523) 2 a + x„
K m ( 2 a + x m ) ^
j (523) _ rj, (523)
2^raTml(2a + T m ) - a 2
2^mTm laz(2a + T m ) - a 4
(2a + xm)2 N = 2 0 ^ ^ 5 2 3 ,
Chapter 4: Tuning Rules for PID Controllers 339
4.5.6 Controller with filtered derivative in series with a second order f \
filter Gc(s) = Kc 1 + — + — ^ — T i S l + ^ s
V N
l + b f ls + b f 2s A
\ 1 ~r SfiS T Sf^S /
E(s) R(s)+
K, , 1 TdS 1+ + £— T;s
1 + -N
l + bf,s + bf2s
l + afls + af2s
U(s)
K e "
(l + sTml)(l + sTm2)
Y(s)
Table 120: PID controller tuning rules - SOSPD model Gm (s) = Kme"
(l + sTmlXl + sTm2)
Rule
Shi and Lee (2004).
Model: Method 1
Kc | ^ Td Comment
Robust 14 K (524) T (524)
i T (524)
14 j , (524)
br, =" •m 1
4co 2co_ ' > a f 2 _
0.5 } m - 6 d B ,
,b f l =-2co_
2co„6dB(l + co_6dBxm)'
T (524) = T + T 0.5 T (524) = TmlTm2(p_6dB2 - 0.5[(Tml + Tm2 )co_6dB - 0.5]
[(Tml+Tm2)ffl-6dB-0-5]
N = 0.5[(Tml+Tra2)«_6dB-0.5]
Tm,Tm2a_6dB2 -0.5[(Tml +Tm2>o_6dB -0.5] '
340 Handbook of PI and PID Controller Tuning Rules
4.5.7 Ideal controller with set-point weighting 1
U(s) = Kc (FpR(s) - Y(s))+^(F i R(s) - Y(s))+ KcTds(FdR(s) - Y(s)) T:S
Table 121: PID tuning rules - SOSPD model Gm(s) = K„e"
(l + sTmlXl + sTm2)
Rule
Oubrahim and Leonard (1998).
Model: Method I;
Tmi = Tm2 ;
0 . 1 < ^ - < 3
Kc Ti Td Comment
Ultimate cycle
' 0.6KU
20.6KU
0.5TU
Fi= l
0.5TU
F i = l
0.125TU
0.125TU
10% overshoot
20% overshoot
' F p = 1 . 3 1 6 - K " K " , F d = 1 . 6 9 f 1 6 - K " * ° p 17 + K K d
F„ = -38
17 + K m K u y
1.717
(l + 0.121KmKj
Chapter 4: Tuning Rules for PID Controllers 341
4.5.8 Ideal controller with set-point weighting 2 f 1 Af
R(s)
T W + T i S + l
U(s) = Kc
+ _E(s)[—
V T i S J T i V + T i S + l
wx f
\ \ l + — + Tds
V T i s j
K„e~
R ( s ) - Y ( s )
,Y(s)
T m l2 s 2 +2^ m T m l s + l
Table 122: PID controller tuning rules - SOSPD model Gm (s) = K „ e "
Tm l2s2+2^mTm ls + l
Rule
Arvanitis et al. (2000).
Kc T i Td Comment
Direct synthesis 3 ^ (525)
4 K (526)
T ( 5 2 5 )
T (526)
T (525)
x (526) Xd
Model: Method 1
3 K (525) X ^ Kraxm
2^m + T 1 m l , 0 < % < 1, with suggested x = 0.4 ;
,(525) =2m_ a2
1 +
a + 2£' u 2
+ 2 y l 7- ' , • n'l1-! . with
T-^ r + E2 - - 1 >0andwith a = l + j — ^ r: XTm](24mTm+Tral) U J XTml(24mTm+T ra l)
(525) _ 1 T m l T m
4 K (526)
.(526)
X 2 L T » +Tml
1 (2i;TCL2 + Tm p^mxm + Tm, )Tml - TmTCL22
Km*m TCL22+T r a(2^TCL2+Tm)
_ (24TCL2 H^mX^- T m +
' m l m^CL2
^ 2 + ^ m2 T m l ( 2 ^ r a + T m l )
(526) Tml PcL2 + Tm (2^TCL2+xm)]
(2^TCL2 +TraX2£mTm +Tml)Tm, - x m T c
342 Handbook of PI and PID Controller Tuning Rules
4.5.9 Classical controller 1 Gc(s) = Kc
E(s)
1+ v T i s y
1 + Tds
R(s)
+ ®— K,
v T i sy
1 + Tds T
l + - ^ s N
U(s)
SOSPD model
Y(s)
Table 123: PID controller tuning rules - SOSPD model Gm(s) =
Kme"
Kme"
(l + sTmlXl + sTm2)
Rule
Minimum IAE -Shinskey (1988)
-page 149. Model:
Method 1; N not specified
Minimum IAE -Chao et al.
(1989). Model: Method 1
m \ J
Kc
Minim 0.80Tral
KmTm
0.77Tml
Km tm
0.83Tml
KmTm
5 K (527)
Tm l2s2+2^mT
Tj
mlS + 1
Td Comment
um performance index: regulator tuning
l-5(Tm2+Tm)
l-2(Tm2+xm)
0.75(Tra2+xra)
T(527)
0.60(Tm2+xm)
0.70(Tm2+xm)
0.60(Tm2+Tra)
T (527)
T m2 n ""\ T m 2 + t m
T m2 =0.5 Tm2 + xm
T 1m2 - 0 75 T m 2 + t m
N = 8
0.8 < £m < 3; 0.05 < — ^ — < 0.5 2^raTml
5 K (527) -0.01133 + 0.5017^m-0.07813^n
K „ 2smTmi ,
Ti<527) =2^mTm ,(l.235-0.41264m +0.09873^n
Td(527) =2^mTm,(l.214-0.6250^ra +0.1358^
-1.890+0.7902^-0.1554^'
-0.03793+0.3975$„, -0.053545m
2smTm i , 0.5696-0.8484^m +0.05055m
!
^Sm^ml ,
Chapter 4: Tuning Rules for PID Controllers 343
Rule
Minimum IAE -Shinskey (1994)
-page 159. Model: Method 1 N not specified
Minimum IAE -Shinskey (1996)
-page 121. Model: Method 1 N not specified
Minimum IAE Shinskey (1996)
-page 119. Model: Method 1
-^s - - 0.2; Tm,
N not specified
Minimum ISE -McAvoy and
Johnson (1967).
N = 20
N = 1 0
Kc
6 R (528)
2-5Tm, K m T m
0.59KU
0.85TmI
K m ^ m
0.87Tm,
Km tm
l-00Tml
Kmxm
l-25Tml
Kmxm
x i
i
Ti
T ( 5 2 8 )
M
m+0.2Tm2
0.36TU
1.98xm
2.30xm
2.50xm
2.75xm
x 2 T m
Representative coefficient v
^m
1 1 1
1 1 1
2k xm
0.5 4.0 10.0
0.5 4.0 10.0
X l
0.7 7.6 34.3
0.9 8.0 33.5
x2
0.97 3.33 5.00
1.10 4.00 6.25
X 3
0.75 2.03 2.7
0.64 1.83 2.43
Td
T (528)
xm+0.2Tm2
0.26TU
0.86xm
1.65xm
2-00xm
2-75xm
x3 t m
alues (deducec
S-
4 4 7 7 4 4 7 7
0.5 4.0 0.5 4.0 0.5 4.0 0.5 4.0
Comment
T JjH2_<3 xra
i^>3 xm
-^2- = 0.2, Tml
2k = o.2 T m l
2k = o.i Tml
T - ^ . = 0.2 Tml
2k = o.5
2k = i.o Tml
Model: Method 1
[from x i
3.0 22.7 5.4
40.0 3.2
23.9 5.9
42.9
p-aph) x2
1.16 1.89 1.19 1.64 1.33 2.17 1.39 1.89
X 3
0.85 1.28 0.85 1.14 0.78 1.17 0.78 1.04
6 K (528)
,(528) _ ,
48 + 57
100
1 - e x™ K m T m 1 +0.342k _o.2
x m
T„, "\
1.5-e 1 + 0.9 1-e x-
2 ^
,Td<528)=0.56t„
1.2Tml \
1 - e T™ 0-6Tm2.
344 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ISE -Chao et al.
(1989). Model: Method 1
Minimum ITAE - Chao et al.
(1989). Model: Method 1
Kc
7 K (529)
8 K (530)
T,
T(529)
Td
T (529) ld
0.8 < S < 3; 0.05 < — ^ — < 0.5 2i;mTml
T ( 5 3 0 ) T (530)
Comment
N = 8
0 . 8 < 4 m < 3
7 (529) 0.2538 + 0.3875^m -0.04283^,
K „ V Sm 'ml
T;1529* =2^mTn,1(l.8283-0.8083^m +0.1852§m2lf X
-1.652+0.54165m-0.095445m
0.1363+0.23424m -0.01445^ra2
Td<529) =2^raTml(l.077-0.4952^m + 0 . 1 0 6 2 ^ V T
2smTmi j
0.4772+0.04424m+0.0l694^m
2^raTml
8 K (530) _-0 .1033+ 0.53474m -0.07995);, 2 / N-1.798+0.71914m-0.1416^m
2
K „ ^ S n i ' m l J
Ti(530) =2^mTml(o.9096-0.1299^m + 0.04179i;ra
2 J — I
Td<530)=24mTml(l.2226-0.6456^m+0.1373^ 2 ! X
N -0.1277+0.52204m -0.086294m
2^mTmi ,
0.4780-0.03653^m +0.04523^'
2£ T m ml J
Chapter 4: Tuning Rules for PID Controllers 345
Rule
Minimum ITAE - Chao et al.
(1989)-continued.
Minimum ITSE -Chao et al.
(1989). Model: Method I
Kc
9 K (531)
10 K (532)
Ti
T ( 5 3 1 ) i
Td
T (531) 1 d
N = 8 ; 0 . 0 5 < — ^ — <0.5 2smTmi
T (532) i
T (532) 1 d
Comment
0.3<4m<0.8
0 . 8 < ^ m < 3
9 K (531) 1.3292-2.85274m+2.03654„
K „ 2^mTml
-1.6159-0.066t6^m+0.53515m2
T1(53 , )=2^mTmlexp(x1), x, =(3.3368-7.7919^m+4.3556^m
2)
+ (l .4094-4.458 l^m + 3.4857E,m2 )lnj
2^raT, m ml
+ (o.H60-0.8142^m+0.74024m2
; In
Td(53,) = 2^mTml exp(x2), x2 = (2.3054- 6.4328^m + 3.9743^m
2)
+ (l.8243-5.6458^m + 4.6238^m2)ln
2smTm | ,
V SmTml J
(o.2866-1.4727^m+1.2893£m2
; In 24«,T„
10 K (532) 0.1034 + 0.4857i;m - 0.0709^n
K „ 2 ^ m T m i j
Ti(532, =2^mTm ,(l.5144-0.6060^m +0.1414^
-1.713+0.6238^-0.1178!;m
2^mTn
0.1008+0.26255m -0.022035m
Td(532) =2^mTml(l.0783-0.4886^m + 0.1038(;„
ZtuJu
0.4316+0.08163^m +0.009 l73E,m2
346 Handbook of PI and PID Controller Tuning Rules
Rule K „ Comment
Minimum ITSE Chao et al. (1989)-
continued.
X (533) (533) 0.3<^m<0.8
N = 8; 0.05 < 2^mTml
<0.5
Minimum performance index: servo tuning Minimum IAE -
Chao et al. (1989).
Model: Method 1
12 K (534) T; (534) (534) N = 8
0.8 <^m <3; 0.05 < 24mTml
<0.5
K (533) _ 1.8342-3.8984$m+2.7313; 2 / \-1.5641+O.16285n,+O.093375m
!
<2smTml ,
T,'5") =2^mTmlexp(x1), x, =(3.2026-7.4812^m+4.3686^m2)
+ (0.9475 - 3.6318^ra + 2 .9969C W r ^ -
+ (- 0.06927 - 0.3875^m + 0.4220£m2 y
Td(533) =24mTm lexp(x2) , x2 =(l.7393-3.7971^m+1.7608^m
2)
+ (l .0983 -1.5794^m +1.0744£m2 )ln| — -
In 2^mT, v ' S m 1ml J
2$mTn
+ (o.01553-0.01414^ra -0.009083£m2 | In
12 K (534, _ -0 .1779+ 0.67634m-0.1264^
K.
v2smTmi j
\-0.9941+0.16695„,-0.04381£m2
f T „ ^
2smTm i ,
T/534' =2^mTral(o.2857 + 0.4671^ - 0 . 0 8 3 5 3 ^ m2 ( - ^ -
x, = 0.1665 - 0.3324£m + 0.1776^m2 - 0.02897£m
3;
Td(534) =2^mTml exp(x2), x2 = (-0.8776 + 0.4353^ -0.1237^m
2)
+ (- 2.0552 + 2.786^m - 0.5687£m2 )ln
,2^mTml
+ (- 0.5008 + 0.6232^m - 0.1430£ra2
1 In v 2SmT m i ,
Chapter 4: Tuning Rules for PID Controllers 347
Rule
Minimum ISE -Chao et al.
(1989). Model: Method 1
Minimum ITAE - Chao et al.
(1989). Model: Method 1
Kc 13 K (535)
14 K (536)
T,
T (535)
Td
T (535)
0.8 < £m < 3; 0.05 < — ^ — < 0.5
T ( 5 3 6 ) T (536) Xd
Comment
N = 8
0 . 8 < ^ m < 3
13 K (535) _ 0-1446 + 0.4302^ -0 .07501^ f x_ >
K, ^•%m*-m\ /
-1 .214+0.3123? m - 0 . 0 6 8 8 9 5 „ 2
T, (535)=2^mTmlexp(x1), x, =(-0.1256 + 0.1434^m-0.03277^m2)
+ (- 2.2502 + 3.7015£m -1.7699^m2 + 0.2680£,m
3)lnj 2£nJm,
-(-0.4823+ 0.9479^m -0.4913^m2 +0.07792^m
3^ lnf — \ V ' ' S m 1 m l J
Td(535) =2^mTml(0.9105-0.3874^m +0.08662^m
2 J — ^ _
,4 „ (536) - 0.2641+ 0.7736^m-0.1486^2 f
K „ K.
x, = 0.0779 + 0.28554m - 0.01985^m2.
-0.7225-0.04344<m-0.0012474rai
2^mT m i m l J
Ti(536) =2^mTml(o.22257 + 0.7452^ -0.1451$m2] 21 t „
v ^ S m T m l J
x, = 0.03138 -0.04430^m +0.01120^
Td(536) = 2£mTmI exp(x2), x2 =(o.05515-0.7031^ +0.1433^m
2)
+ (-1.2256 +1.8544^m - 0.3366^m2 )ln| — -
2 . m »
24raT, m l >
+ (-0.2315 + 0.3402^ -0 .06757^ 2 ] In %
2smTm i ,
348 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ITAE - Chao et al.
(1989). Model: Method 1
Minimum ITSE -Chao et al.
(1989). Model: Method 1
Kc
15 K (537)
16 K (538)
T,
T (537)
N = 8; 0.05 <
T (538)
Td
T (537)
— ^ — <0.5
x (538)
Comment
0.3<^m<0.8
0 . 8 < ^ m < 3
5 K (537)
• (537)
1.1295-2.8584^m +2.1176£n
K „
-0.9904-2.531 lS„,+2.
TiWJ" = 2^mTml 18.6286-20.70^m +13.2034 2£ T
(537)
x, = 0.7962-2.4809^m +1.7611^
: 2^mTml exp(x2), x2 = (3.9285 - 12.874^m + 8.6434^m2)
+ (3.2228-12.0344m +9.2547^m2)ln
•(o.6366-3.0208^m+2.4603^m2 In
^ S m ^ml j
16 K (538) 0.0820+ 0.4471^m -0.07797£,n
K „ v2smTmi
-0.9246+0.07955^ro-0.02436^„
1™ = 2i;raTral (o.5930 + 0.1457^m -0.02062!;n 2 T „
(538)
x, =-0.2201 + 0 .1576^-0 .03202^
^ m T r a l e x p ( x 2 ) , x2 =(-0.6534-0.1770^m-0.0240^m2)
+ (-1.0018 + 1 .5914^-0.2831^ r a2 ) lnf- l !B_
V^Smiml J
+ (- 0.2291 + 0.3064^m - 0.06344,/ In 24mTra
Chapter 4: Tuning Rules for PID Controllers 349
Rule
Minimum ITSE -Chao et al. (1989)-
continued.
Smith et al. (1975).
Model: Method 1
Smith et al. (1975).
Model: Method 1
Sklaroff(1992).
Kc
17 £ (539)
T:
T (539) 1 i
Td
T (539)
Comment
0 .3<^ m <0.8
N = 8; 0.05 < — ^ — <0.5
Direct synthesis ^Tml
Kn.K.+l)
x l *ml
K m T m
Tml
Tral
Tm2
Tm2
TCL
N not specified
D - T m ' Trai + Tm2
N=10 Coefficient values (deduced from graph)
4m 6
1.75 1.75 1.5 1.0
P
>0.02 0.27 0.07 0.11 0.25
x i
0.51
0.46 0.36 0.33 0.46
18 K (540)
ra
3
1.75 1.5 1.5 1.0
P
>0.04 0.13 0.33 0.08 0.17
Tm i+Tm 2
x i
0.50
0.42 0.44 0.28 0.48
4m 2
1.75 1.5 1.0 1.0
TmiTm2
Tmi + Tm2
P
>0.06 0.09 0.17 0.50 0.13
x i
0.45
0.39 0.38 0.40 0.49
Also given by Astrom and Hagglund (1995) -page 250-251. Model: Method 7. N = 8 - Honeywell UDC6000 controller.
„ K ,539, ^ 0.9666-2.3853^m + 2 . 4 0 9 6 ^ f xn
K „
-1.533-0.054425ra+10.9165,,
.24mTm l .
T,(539) =2^mTml(6.3891-15.773^ + 1 0 . 9 1 6 ^ m2 | - ^ -
x, = 0.04175-1.38605m+1.40534n
Td(539) = 24mTml exP(x2), x2 = (s.2620- 10.7894m + 7.6420!;m
2)
+ 2.277-8.94024m +7.7176^,/lln )lnf T- 1 V^SmTml
(o.3465-1.92744m+1.80304ro2, In
f \ In,
24mT, V^^m 'ml J
18 j r (540)
K j l + 3x„
Tml + T m 2 J
350 Handbook of PI and PID Controller Tuning Rules
Rule
Poulin et al. (1996).
Model: Method 1
Schaedel (1997). Model: Method 1
Panda et al. (2004). Model:
Method 7 Panda et al.
(2004). Model: Method 1
Kc
Tml
K m (T m l +x m )
19 K (541)
20 K (542)
2Km tm
Ti
Tml
T(541)
2^mTml
TB,
Td
Tm2
T (541)
2^m
Tm2
Comment
N = 5; Tml ^5T m 2 ;
Minimum
N not defined
N = 8
N not defined
19 jr (541) _ 0 3 7 5
K „
4^ m2 T m l
2 +2^T m l T r o + 0 .5T m2 -T m l
2
(2?mTml + T J 2 +1 i-lj(2?mTml +im)TmlV2(24ra2-l)-x
* = 44m2Tml
2 +2^mTmlxm + 0.5xm2 - T m l
2 ;
• (541) _ 4^ m2 T m l
2 +2^ m T m l T m +0.5T m2 -T m l
2
2 ^ T m , + T m
Td(54" = V t e n J m l +x m ) 2 -2(Tml
2 + 2 ^ T m l x r a +0.5xm2) .
20 K (542)
Km 1 1 ^ „
Chapter 4: Tuning Rules for PID Controllers 351
4.5.10 Classical controller 2 Gc (s) = Kc
E(s)
1+ — v T i s y
l + NTds
1 + T d s y
Kme"
(l + TmlsXl + Tm2s)
Y(s)
Table 124: PID controller tuning rules - SOSPD model Gm (s) = Kme"
(l + sTmlXl + sTm2)
Rule
Hougen(1979)-page 348-349.
Model: Method 1
Hougen (1988). Model: Method 1
Kc Ti Td Comment
Direct synthesis
0.8Trala7Tra2
a3
Km
0.84Tm,°-8Tra20-2
tra
22 K (544)
0-5Tml+Tm2
21 j (543)
0.5Tml+Tra2
0.lV^Tm.Tm2
T (543) J d
T (544)
N=10
N=30
N=10;xm<Tm l
2, Tj(543) = a 5 3 T m i +1.3Tm2,Td(543> =0.08(xmTralTm2)a28 .
K „ -10
X|-x2log, m . Coefficients of Kc<544) deduced from graphs:
T m / T m 2
* 1
x 2
0.1
0.9
0.7
0.3
0.42
0.7
0.5
0.24
0.68
1
-0.10
0.70
2
-0.39
0.69
T d( 5 4 4 ,=(Tm ,+Tm 2>0
0.331ogl0| ^ |-1.40
= 0.1
Td(544) = (Tml+Tra2)l0l m-0.3 l l o g , 0 - J ! ! - -1.63
, 0.2<-2^-<l.
352 Handbook of PI and PID Controller Tuning Rules
4.5.11 Series controller (classical controller 3)
Gc(s) = Kc
U(s) v T i s y
R(s) E(s)
+ K,
v T i sy (l + sTd)
K fi
ll+TmlsXl+Tm2s)
0 + Tds)
Y(s)
Table 125: PID controller tuning rules - SOSPD model Gm (s) = Kme"
(l + sTm,Xl + sTm2)
Rule
Minimum ISE -Haalman(1965).
Bohl and McAvoy (1976b).
Kc Ti Td Comment
Minimum performance index: regulator tuning 2Tml
3xmKm T r a l Tm2 Model: Method I
T r a l >T r a 2 .M s =1 .9 , A m =2.36 , ^ = 5 0 ° .
1 K (545) T (545) M
T (545)
Minimum ITAE; Model: Method 1
T (545) x (545)
- ^ - = 0.1,0.2, 0.3,0.4,0.5; - ^ = 0.12,0.3 ,0.5,0.7,0.9.
K (545) 5.5030
K „
f N-1.1445+0.1100
1 0 -T V 1ml J
iiot)
( ^0.1648+0.17091n| 1 0 - ^ |-0.21421n| 1
V T m l J
T < 5 4 5 ) = 1 . 8 6 8 1 T T r 10
0.6212-0.0760 In 10-
f \0.3144-0.00621nl 10-=^ 1+0.0085 Inl 10-
l02k T
V 'ml J
Tm,
Chapter 4: Tuning Rules for PID Controllers 353
Rule
Yang and Clarke (1996).
Model: Method 8
O'Dwyer
(2001a). Model: Method 1
Skogestad (2003).
Majhi and Litz (2003).
Model: Method 8
Bohl and McAvoy (1976b).
K c
fflpTml
K m A m
x iTm i
K m x m
0.785Tml
K m x m
0-5Tral
K m T m
3 K (547)
4 K (548)
Ti Td
Direct synthesis
2 T ( 5 4 6 )
Tml
Tml
Tm2
Sample result
Tml
min(Tml ,8Tm)
T (547)
Tm2
Tm2
TmI
Ultimate cycle
T (548) T (548)
Comment
Tmi = Tm2
A - % • 2x,
*m =0.57C-X1 .
A m = 2.0;
* m = 45°
Model: Method 18
Tmi = Tm2
T (548) T (548)
Model: Method 1
2 T (546) _
4 f f lP Tm 1 2C0_ +
P 71 T, ml
3 v (547) _ ^ m 2$ + i t ( A m - l ) Tm
2(A m2 - l ) Km
, (547)
1 + K1^)1 T.i . A » - ^ [ 2 ^ t , ( A „ - l ) | l - , A " . 1 ( 2 ^ „ + » ( A J , - 1 ) )
4 K (548) 0.5458
K „
N 1.2264+0.4568 In —
T;<548) = 0.0393T, T v u
t ( A m2 - l )
/ „ „ y.2190-0.09581n(K„K|T,)-0.10071n ~
-1.9314-0.62211n
(KuKm) -0.0703-0.1125ln(KoKm)-0.29441nP!!!
354 Handbook of PI and PID Controller Tuning Rules
4.5.12 Non-interacting controller 1
E(s) %-Y(s)
Table 126: PID controller tuning rules - SOSPD model G ra (s) = Kme-
(l + sTm,Xl + sTm2) or
Gm(s) = K „ e - -
T m l2 s 2 +2^T r a l s + l
Rule
Huang et al. (2000).
Model: Method I
Kc Ti Td Comment
Direct synthesis *KC<*»> T ( 5 4 9 )
i T (549)
1ral ' im\ ' im\ '
Servo tuning
X = 0.6^m<\ A. = 0.7, K ^ r a < 2
^ = 0-8»-z;— £m > 2 > Regulator tuning
1 m l
5Kc(549) = x 2 T m l ^ m + 0 . 4 T m Tj(549) ^ ^ + 0 . 4 T l I
' <549> = Tml +0-8T m l 5 m T l t
0.8T m l ^ m T m
Chapter 4: Tuning Rules for PID Controllers 355
Rule
Minimum IAE -Huang et al.
(1996).
Kc T| Td Comment
Minimum performance index: regulator tuning 6 ^ (550) T (550)
i T (550) Model: Method
7;N=10
6 Note: equations continued into the footnote on page 356; 0 < Tm2 /Tml < 1 ;
0 . 1 < x m / T m l < l .
K (550) 1
1
K„
K„
K„
K„
0.1098 - 8.6290-2- + 76.6760- -3.3397 x m T „
1.1863 IHL. Tm>
-23.1098 , N 0.6642
+ 20.3519 , N 2 . 1 4 8 2
X,
V T m i y
-52.0778
, N 0.8405
T V ml J
9.4709-
•19.4944-
T r a i j
Tm, VTml 7
• 12.1033 M 2 2 -
-13.6581-^2-
-28.2766-
*m
T m i
A T \ 1 1 4 2 7
Xm2
T v 1mi y
C!k -19.1463eT"" + 8.8420eT™' +7.4298e T"" - H . 4 7 5 3 ^ 2 -
,(550) = T - 0.0145 + 2.0555^2- + 0 . 7 4 3 5 ^ _ 4.4805 V^ml
^ 2 _ +1.2069-^-^1 xmTm
+ T„
+ Tm,
+ Tm
+ T„
0.2584 Lm2
r m i j + 7.7916 -6.0330-
ml / Tmi l^Tml J
3.9585-
7.0004-
3.1663
l m l
^-\ -3.0626 / T \3
V ^ml
VTml 7
VTml 7 - 7.0263
-2.7755 V^ml VTml
VTmi y
-1.5769 VTmi y 'mi y
VTmi y + 2.4311 -0.9439
1 mi y
1m2
VTml J
356 Handbook of PI and PID Controller Tuning Rules
+ Tm
+ T „
- 2 . 4 5 0 6 - - 0 . 2 2 2 7
2 /
T I. Jml
1.9228 V ^ml ml J
-0.5494-^S- p 2 2 -Trai V.Tmi
(550) - 0.0206 + 0.9385-^2- + 0 . 7 7 5 9 ^ _ 2.3820 — Tmi Tml vTm l
+ 2.9230 T m T m 2
+ T „ - 3 . 2 3 3 6 | ^ - +7.27741-^5- -9.9017-Tmi vTm l
2.7095-T ( T 'ml V ml
+ 6.1539 V ^ml
l m 2 ' — 11.10181 -^s lml J
+ T m 10.6303-^ml V ml
+ 5.7105 VTml J
+ T „
+ T „
- 7.9490 - i s - p n ? .
Tml V Tml )
- 4 . 4 8 9 7 -
-6 .6597 VTml )
'm2 T
V 'ml J 4
+ 8.0849
- 2.274 V -ml
f N5
V^ml
2.2155
0.519-
'ml V ml J
(T \
• 7.6469 VTml J V^ml
ml / T
V 'ml J
-5.06941-^2-
-1.1295 'm2
VTml J
Lm2 r 'ml 7
+ 4.1225 f-^-l T m l .
•1.6307 T
V 'ml
^m2
VTml J
+ T „ 2.2875
ml /
+ 0.9524 V ml
V ^ \ 3
VTml J - 0 . 9 3 2 1 - T m I T m 2
Chapter 4: Tuning Rules for PID Controllers 357
Rule
Minimum IAE -Huang et al.
(1996).
Kc
7 K (551)
Ti
T ( 5 5 1 )
Td
T (551)
Comment
Model: Method / ; N = 1 0 .
7 Note: equations continued into the footnote on page 358. 0.4 < £,m < 1,
0.05<xm/Tm, < 1 .
K (551)
K „
K „
K „
K „
K „
K „
K „
- 35.7307 + 14 .19^_ + 1.4023^m + 6.86184m -Tmi \ T, ml J
-0.9773 ^ - +55.5898£,n
, \ 0.086 X,
T - 3.3093^n
V^ml
53.8651^=Hm2 +11.491 \\J +0.8778| •
-29.8822
V ml j
-0.6951 / N -0.4762
+ 53.535 ^ j -16.9807^, u 1 9 7
^T m i
-25.4293^m14622 -0.1671^m
58981 +0.0034^n
, N -2.1208
VTml )
. 25.0355^Km3 '0 8 3 6 - 54 .9617 -^Hj ' 2 1 0 3 -0.1398eT-
-8.2721e^ +6.3542e T™' + 1.0479IE-JSL
.(551) 0.2563 + 11.8737-21- - 1.6547^m -16.1913 / A 2
- 9.7061£ 1 ml J
+ T„ 3.5927^m2 + 19.5201 ^ - \ - 14.5581£,n
xn
V T ra l7 + 2.939^
i m l
/ A t m
V ' m l /
+ T. ml -0.4592^m -34.6273 V T m l J
+ 50.5163^n VTml )
+ T „ i.9259CP=-V Xml J
+ 8.6966^-^m3 -6 .9436^m
4
iml
358 Handbook of PI and PID Controller Tuning Rules
+ T „ 27.2386 f "\5
-Tml 8.5019
VTmlV
X
20.0697^n VTml J
-42.2833^„ ^T m i
Em3 -12.2957 ±S- \m* + 8 . 0 6 9 4 ^ - 2 . 7 6 9 1 ^
1ral J V 1ml .
+ Tm ft V
•7.7887 - E -V T m l ,
-2.3012£n Tml
+ 4.6594 - ^ K m5
8.8984 V T m i y
£„/+10.2494 / V ^ U m
3 - 5.4906 V l m l J
f \ 2
V ' m l
(551) -0.021 + 3.3385—2- +-0.185^m -0.5164 ^ M - 0 . 9 6 4 3 4 m ^
V 1ml ) 'ml
+ Tm
+ T „
+ T „
+ Tm
+ T „
-0.8815^m2+0.584 - ^ - 1 2 . 5 1 3 4 m | ^ L | + 1.3468^
V ml .
2.3181^m3-5.2368-^2-J + 15.3014£n
V*ml VTml 7 -3.1988£,n
H-9607^
-0.8219£n
0.484 if ^ Un.y
^=- - 2.041 \ ^ J +3.4675 f ^ x,
V T m i y
T V 1ml J
•15.07184n V^ml J
•1.8859 f \ 2
±m_] f 3
V 'ml J
C + 2 . 2 8 2 1 4 m5 - 0 . 9 3 1 5 | ^
ml J
+ 0 . 5 2 9 ^ m | - ^ 1ml.
-0.6772^,-1.4212 - ^ U r a2 +7.1176 ^ L . ^
-2.3636 ^ b m4 +0.5497 ^ J
V ' m i y 'ml
Chapter 4: Tuning Rules for PID Controllers 359
Rule
Minimum IAE -Huang et al.
(1996).
Kc T; Td Comment
Minimum performance index: servo tuning 8 K (552) T (552) T (552)
1 d Model: Method
1. N=10
' Note: equations continued into the footnote on page 360; 0 < Tm2/Tml < 1,
0 . 1 < x m / T m l < l .
K (552) _ _
K „ 7.0636 + 66.6512—25- -137.8937- -122.7832-
. T m T „
Kn
K „
K „
K „
K „
26.1928 V *ml
' m l
0.0865
+ 33.6578 / N 2.6405
X
-10.9347 / T \2M5
VTml J
34.3156^521 Tmi
( . \
T V 1 m l J
V T m l J
+ 141.51l| — ' m l j
-70.1035-2^ T,
+ 3.0098 (T
V ^ m l
+ 29.4068-
152.6392-T m l V ml
-47.9791-
T ml V ml J
V ^ m l
-57.9370eTml + 10.4002eT«" +6.7646e T""2 +7.3453
f \ -0.4062
V 1 m J
• (552) _ , 0.9923 + 0.2819—52- - 0.2679: Ami
1.4510 f A2
T.
V T m l J
-0.6712-, T X
0.6424p5Sl +2.504^555-VTmlJ VTml,
m2 | ^m + 2.5324-^ m l V ml J
+ T„
+ T„
+ Tm
2.3641-T *m
T m l V ml J
+ 2.0500 v ' m l
•1.8759 ' x ^
0.8519-
-2.4216-
T m i j
V ^ m l
-1.3496-V T m i y
V T m l J
- 3.4972 V T m l J Lml /
•3.1142 - l m l
V^ml J + 0.5862
VTmlV
360 Handbook of PI and PID Controller Tuning Rules
+ Tm
+ T„
0.0797 W ^ M +0.985 i ^ -
1.2892
V^ml
( - ^VT
T V m
2 ^ x „ V
VTral J
V T m i y V '"ml + 1.2108
VTml J
(552) 0.0075 + 0.3449^2- + 0.3924^mL - 0.0793 -T
+ 2.7495 ^ V
+ T„ 0.6485 f-r \2
VTml J + 0.8089^2- -9.7483^21
VTml 1
3 . 4 6 7 9 ^ - f ^ - - 5 . 8 1 9 4 | - ^ - ] - 1 . " " " " ' X
T .0884
+ T„ 12.0049-( \ 3
T V 1ml J
-1.4056 V^ml
T V 'ml J
i a i - J -3.7055^ffl-|in2_ V ml J Tm, ^ Tml j
10.0045 / T \
T V 'ml J
+ 0.3520 f \ 5
-3.2980 VTml 7 V 'ml
T
+ 7.0404
. 6 . 3 6 0 3 ^ ml V ml )
V 'ml V^ml
+ T„
+ T„
+ Tm
1.4294 V 'ml
'm2
T V 'ml J - 6.9064
fry \$
T V 'ml J
-0.0471 VTml J
1.1839-Tm l V ml 7
+ 1.7087 / \ 6
VTml 7
+ 1.7444| JEs
ml J
• 1.28171 2 _ P ^ - l -2.128l| T" + 1.5121^IS-LlmL
Tm l V ml ,
Chapter 4: Tuning Rules for PID Controllers 361
Rule
Minimum IAE -Huang et al.
(1996).
Kc
9 K (553)
Ti
T (553) i
Td
T (553)
Comment
Model: Method 1. N=10 .
Note: equations continued into the footnote on page 362. 0.4 < ^m < 1;
0 . 0 5 < x m / T m l < l .
K (553)
K „
K „
- 8.1727 - 32.9042-2- + 31.9179£m + 38.3405£n l m t V T m l J
+ 0.2079 + 29.3215 , xO.1571
V T m l J
+ 35.9456
^ [ - 2 1 . 4 0 4 5 ^ m0 1 3 U +5.1159^m'-9814 -21.9381^m'-737]
K „
K „
K „
T (553) _ T
+ Tm
+ Tm
-17.7448^m - s u -26.8655£n
ml J
, N 1.2008
V ^ m l
T„ 1
52.9156—s-^m11207 -22.4297-S_^ra°'3<i26 -3.3331eT'
^ m
8.5175e^-1.5312e T™' +0.8906 I T
1.1731 + 6.3082—^- - 0.6937^m +8.5271
[Huang (2002)];
^ - 2 4 . 7 2 9 1 ^ ^ V * ml J lm\
•6.7123^m | -^- +7.9559^m2-32.3937 ^ s - | + 5.5268!;, 6
-27.1372 V^ml - ^ - + 166.9272^ m \ +36.3954^„2
V^ml
f \
V T m l J
-94.8879^n V^ml
-^2_ -22.6065^m3 -1.6084^2_^m
3 + 29.9159^
+ T^ 49.6314' -84.3776^n l m l /
-93.8912^ Lml J ml /
362 Handbook of PI and PID Controller Tuning Rules
+ T „
+ T „
+ T „
(553) _r
( V 110.1706 -^2- £,m
3 -25.1896| KTral J
1™ | ^ m4 -19 .7569^ 5
V 1 ra l
-12.4348 V T m l
•11.75894n
V T m l 7
+ 68.3097 ro U, ml
-17.8663 -^M ^m3 - 22.5926J
;Tmi J
r^r-) C + 9 . 5 0 6 1 ^ m5
V Xml J lm\
0.0904 + 0.8637—— - 0.1301^m + 4.9601 TmJ
+ 14.3899^n
+ Tml 0.7170^m2 -12.53 l l | * " -42.5012^ ^ " 2
V T miy + 17.0952^
+ Tm
+ T „
+ T „
+ Tm
+ T „
-21.4907!;,
102.9447^m
vTmiy -6.95554m -12.3016
' T ^
V T m l 7 + 7.5855^,
V T m i y
VTml
+ 19.i257^-^m3
( - \ 10.8688
V T m i y •17.2130^n
< ' T „ A
V T m l 7 -110.0342£n
V T m l 7
50.6455U=- ^m3 - 1 6 . 7 0 7 3 U ^ I C -16.2013^m
5 +5.4409^n V ml J V ml ,
-0.0979
21.6061
V^ml
^ „ ^
V T m i y
-10.9260^
5m3-24.1917'
V T m i y + 29.4445
TmJ
v T m i y C + 6 . 2 7 9 8 ^ C
Chapter 4: Tuning Rules for PID Controllers 363
4.5.13 Non-interacting controller based on the two degree of freedom structure 1
r \ f \
U(s) = Kc , 1 T d s 1 + — + —
T:S - X
V 1 + ^ s
N ;
E(s)-K ( ot + PTds
1 + ^ - s
N ;
R(s)
a + pTds
T
N
K „
R(s)
-Mg> + T
E(s)
K, 1 THs
1 + + -
N ;
* U ( s )
Y(s)
+
K e"
Tml2s2+2^mTmls + l
Table 127: PID controller tuning rules - SOSPD model K ^ e - 1 -
Tm l2s2+2^mTm ls + l
Rule
Minimum IE -Shen (1999).
Kc T i Td Comment
Minimum performance index: regulator tuning
1 K (554) j (554) T (554) 2 d
Model: Method 1;
K m = l
1 X l = e x p [ a + b | ^ | + c
V T ml7 + d
V^ml + e % ^ + fi;m+g^2
Tm,
+ h S m m +i r \3
T
V T m i y Sm +j
V lm\ )
K (554) _ T m l • (554) (554) :x x., a = 1-x, ; coefficients of x, are
given below (and continued into the footnote on page 364). 1.4 < Ms < 2 ; |3 = 0; N = 0.
a b c
K c ( 5 5 4 ,
1.40 -11.60 14.17
T (554)
2.43 -2.50 -4.16
T (554)
1.32 -0.51 0.70
a
-1.44 1.27 1.34
364 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum IE -Shen (2000).
Kc
2 K (555)
Ti
T (555)
Td
T (555)
Comment
Model: Method 1;
K m = l
(continued)
d e f
g h i
J k
K c ( 5 5 4 ,
-6.44 8.28 -0.07 0.13 -4.66 -0.99 -3.03 2.89
T (554) i
2.85 -0.30 0.1
-1.72 15.51 -7.98 -8.45 4.66
T (554) ' d
-1.06 0.02 -0.09 -7.43 1.77 2.64 4.77 -3.74
a
-0.37 1
-5.97 -0.78
-10.63 18.85 -10.97 6.44
2 x,=exp[a + b f — ^ — ] + c f — ^ — ] + d f — ^ — ) +e^ra+f^ra2
Um+Tm lJ l.Tm+Tm,J Um+Tm lJ
( \ ( V c V + g ^ h — L + h - — ^ — 5m+i—Jbr-Um
ltm+TmlJ ^ m + T m l J {lm +Tm\ J
Kc(555) = _ n i . X | j Xf =TmXi,T d = Tmx,, a = 1-x, ; coefficients of x, are
x m
given below; Ms = 2 ; |3 = 0 ; N = 0; 0 < £m < 1; 0 < -^2- < 1. ml
a b c d e f
g h i
J k 1
K c , 5 5 5 ,
1.73 -17.51 30.73 -24.69 0.15 -0.12 3.30 33.07 -37.64 2.63
-34.57 36.88
T (555)
2.28 0.87
-24.00 15.09 -0.01 -0.02 -8.03 45.21 -22.48 3.89
-22.39 8.32
T (555)
1.43 -1.94 14.82 -21.44 -0.25 0.16 -1.06
-43.39 51.16 -3.25 36.04 -34.95
a
-1.60 4.30
-12.16 26.54 1.33 -1.08
-17.56 38.13 -56.45 14.80
-39.33 51.18
Chapter 4: Tuning Rules for PID Controllers 365
Rule
Minimum IE -Shen (2000).
Model: Method 1
Minimum ITAE - Pecharroman
and Pagola (2000).
Minimum ITAE - Pecharroman
(2000). Model:
Method 11
Kc
3 K (556)
Ti
T (556) i
Td
T (556) *d
Comment
K m = l
Minimum performance index: servo/reguiator tuning
0.7236K, 0.5247TU 0.1650TU
N=10; Model:
Method 11
a = 0.5840, (3 = 1, 4>c =-139.65° ,Km = 1; Tml = 1; ^m = 1
x,Ku x2Tu "3TU
Coefficient values and other data x i
0.803
0.727
x2
0.509
0.524
x 3
0.167
0.165
a
0.585
0.584
4>c
-146°
-140°
x, = exp[a + b| xm+T,
+ g
ml / V T m + T m l + d
V T m + T r a , ^ m + h
V T m + T m l J
^ m + T m i y
^ m + i |
+ e4m+f^„
+ J Tm + T m i j Sm + k Tm + T m ] J Xm + T m ] J
4 m 1 ;
K ( 5 5 6 ) = J J H L X I J T i » o ^ = T m X Tdk"° '=T r ax, , a = l - x , ; coefficientsof x, , (556) (556)
are given below; Ms = 2 ; p
a b c d e f
g h i
j k 1
K (556)
128.9 -560.3 769.2 -338.9 -324.8 194.3 1421
-1981.6 895.0 -848.1 1186.1 -538.1
= 0 ; N = 0; 0.4 < ^m < 1; 1 < -&- < 15 . Tmi
T (556)
92.8 -391 521.6 -225.5 -224.5 137.5 963.9
-1310.6 574.4 -590.8 808.0 -356.7
T (556) x d
-3.1 46.3
-107.4 63.1 5.2 -2.5 -79.6 176.0 -103.6 36.2 -78.4 45.8
a
-119.9 539.5 -785.1 371.8 233.9 -115.7 -1068 1560.5 -741.5 529.2 -774.6 369.2
366 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ITAE - Pecharroman
(2000) -continued.
K m = l ;
Tm, = l ; P = 1;N=10;
0.1<xm <10
Taguchi and Araki (2000).
Model: Method 1
Kc T, Td Comment
Coefficient values and other data (continued) x i
0.672
0.669
0.600
0.578
0.557
0.544
0.537
0.527
0.521
0.515
0.509
0.496
0.480
0.430
x2
0.532
0.486
0.498
0.481
0.467
0.466
0.444
0.450
0.440
0.429
0.399
0.374
0.315
0.242
4 K (557)
x 3
0.161
0.170
0.157
0.154
0.149
0.141
0.144
0.131
0.129
0.126
0.132
0.123
0.112
0.084
T (557)
a
0.577
0.550
0.543
0.528
0.504
0.495
0.484
0.477
0.454
0.445
0.433
0.385
0.286
0.158
T (557)
4>. -134°
-125°
-115°
-105°
-93°
-84°
-73°
-63°
-52°
-41°
-30°
-19°
-10°
-6°
S» = i-0
4 K (557) _ 1
Km 1.389 + -
0.6978
- ^ - + 0.02295]2
T (557) _ T
T (557) _ T LA — ^ r r
0.02453 + 4 . 1 0 4 - a - - 3.434 Tm,
0.03459 + 1.852-a-- 2.741
+ 1.231
+ 2.359 -0.7962
- ] 4 \
a = 0.6726-0.1285-2--0.1371
(3 = 0.8665 - 0.2679-E- + 0.02724 Tml
-0.07345
TmI
Chapter 4: Tuning Rules for PID Controllers 367
Rule
Taguchi and Araki (2000) -
continued.
K, T; Td Comment
T (558) T (558) f = 0.5
Tm/Tml < 1.0; Overshoot (servo step) < 20% ; N not specified
5 R (558)
Direct synthesis
Huang et al. (2000) - servo
tuning. Model: Method 1
6K (559) • (559) (559)
X = 0 . 6 , - 2 K m < l A m = 2 . 8 3 .
X. = 0 . 7 , K - = - $ m £ 2 A m = 2 . 4 3 .
X = 0 .8 , -=^ , . > 2 A m =2.13
2Tml^n - 1 . P = -0.5T
2Tml4m+0.4Tm • • Tml '+0.8Tml4mxm
N = min[20,20Td]
5 K (558) _ _
-(558)
K „
= Tral
0.3363-0.5013
[-J2- + 0.01147]2
T m i
• 0.02337 + 4.858-S- - 5.522 Tm,
+ 2.054 3 "\
(558) _ Tm, 0.03392+ 2.023-^--1.161 + 0.2826
Tm,
a = 0.6678 - 0.05413-^- - 0.5680 Tml
+ 0.1699
P = 0.8646-0.1205^!_-0.1212
6 „ (559) . , 2 T m , ^ m + 0 . 4 T K'3X"=X
m x (559)
K m T m T-1 ' = 2T ,E +0 4x
. (559) _ Tm/+0.8Tml^mxn d 0.8Tml^mTm
368 Handbook of PI and PID Controller Tuning Rules
4.5.14 Non-interacting controller 4
U(s) = K, 1 + E(s)-KcTdsY(s)
R(s) E(s) K 1 + —
+ *g> U(s)
SOSPD model
Y(s)
K cT ds
T Table 128: PID controller tuning rules - SOSPD model Gra(s) =
K m e " ^
(l + sTmlXl + sTra2)
Gm(s)=-K „ e "
T m l V+2^ m T m l s + l
Rule
Minimum IAE -Shinskey(1996)
-page 119. Model:
Method 1;
- ^ - = 0.2 ' m l
Kc Ti Td Comment
Minimum performance index: regulator tuning 1.18Tml
Kmxm
l-25Tml
KmTm
1.67Tml
Kmxm
2-5Tml
^-m^m
2.20xm
2.20xm
2.40xm
2.15xm
0.72xm
1.10xm
1.65xm
2.15xm
T -2^-= 0.1 Tml T - ^ - = 0.2
• ^ = 0.5 ' m l
• ^ = 1 .0
T m i
Chapter 4: Tuning Rules for PID Controllers 369
Rule
Minimum IAE -Shinskey (1994)
-page 159. Model: Method 1
Minimum IAE -Shinskey (1996)
-page 121. Model: Method 1
Kc
1 K (560)
3.33Tml
K m ' C n i
0.85KU
T|
T ( 5 6 0 )
^m+0.2Tm2
0.35TU
Td
T (560) ' d
^m+0.2T ra2
0.17TU
Comment
I E I 2 _ < 3 t m
2k_>3
^ - = 0.2, T ' m l
T - ^ - = 0.2 Tm,
1 R (560) 100
38 + 40 1-e x» /
1 + 0 . 3 4 - ^ - 0 . 2
v
^ 2 ^
T„, "\
0.5 + 1.4[l-e 1 5 t m] 1 + 0.48 1-e T-
J; f 1.2Tm
Td(560)=0.42xn 1-e T- + 0.6Tm
370 Handbook of PI and PID Controller Tuning Rules
4.5.15 Non-interacting controller 5
U(S):=KC b + — V T i S V
E(s)-(c + Tds)Y(s)
R(s)
> ( b - l ) K c
E(s)
+ I K,
v T i s y
+
U(s)
.+ *® Kme""™
(l + sTml)(l + sTra2)
Y(s)
(c + Tds)
I Table 129: PID controller tuning rules - SOSPD model Gm (s) =
Kme" (l + sTm,Xl + sTra2)
Rule
Hansen (1998). Model: Method 1
Kc Ti Td Comment
Direct synthesis
2 K (561)
1.54T.(5«)
1.27T;*561'
1.75T;(561)
T (561)
b = 0.198; nearly zero overshoot b = 0.289;
nearly minimum IAE
b = 0.143; conservative
tuning
2 K (561) 2 k i T m 2 + ( T m l + T m 2 ) r m + 0 . 5 T m2 f
9Km[TmlTra2im +0.5(Tm, + Tm2)xm2 + 0.167tm
3f
T(56.) 3[rm,Tm2Tn, +0.5(Tm, +T m 2 > m2 +0.167Tm
3l
[Tm ,Tm 2+(Tm l+Tm 2)xm +0.5Tm2]
(561) 2k,Tm2+(Tm,+Tm2)rm+0.5Tm2]2
Tr a l + T m 2 + * m
c = K,
3Km[TmlTm2xm +0.5(Tml +T m 2 > m2 +0.167xn
(561) 1
Km '
n — E ;
Chapter 4: Tuning Rules for PID Controllers 371
4.5.16 Non-interacting controller 6 U(s) = —c- E(s) - Kc (1 + Tds) Y(s) T:S
R(s)
>6* + 1 L
E(s) Kc
T i S
U(s)
—4
Kc(l + Tds)
t
K m e-""
TmlV+2^T r a ls + l
Y(s)
^
Table 130: PID controller tuning rules - SOSPD model Gm (s) = Kme-Tml
2s2+2^mTmls + l
Rule
Arvanitis et al. (2000).
Model: Method 1
Kc Ti Td Comment
Direct synthesis
3 K (562) T(562) Tm, 0 < s < l ;
suggested E = 0.5
3 y- (562) _ s T m |
24m + — T
ETr
m J
T, (562)
2^„
[ |Tm l(2^mTm +Tm l)(442(l-S)+ S) ' with
= - L 8(l-e)2k(2^ i nT in + T m l ) fe2(l-e) + e)iHL(2^T m +Tml)+Tn
e + 2ct2(l-s)2[5=L] (24mxm+Tml)2 >0 . w h e r e Jk . (24mxm +Tm l )+-
K V K m /
372 Handbook of PI and PID Controller Tuning Rules
f
4.5.17 Alternative controller 4 U(s) = K sT,
E(s) + R(s )
Table 131: PID controller tuning rules - SOSPD model Gra (s) = Kme"
(l + sTm,Xl + sTm2)
Rule
Bohl and McAvoy (1976a).
Model: Method 1
Kc Td Comment
Direct synthesis: time domain criteria 4 K (563) T (563)
*d N=10
Response reaches a new steady state in minimum time, to a servo input, with no observable overshoot. Upper and lower limits exist on the manipulated variable.
-^2- = 0.1,0.2,0.3,0.4,0.5 ; i s ! = 0.12,0.3,0.5,0.7,0.9 .
K (563) 4.486
K „ 10-
-1.240+0.1409 In 10-
,-0.0252+0.1542 In l o i = i -0.16161n 10-^-. T „ , 1 I T,,, j { Tml ,
(563) ,0.3664-0.0476 In 10-
10-l m l J
0.7797-0.07731n| l O ^ i |+0.02701n| 10 -^ - 1
10Im2 T-T
Chapter 4: Tuning Rules for PID Controllers 373
Rule
Bohl and McAvoy (1976a).
Model: Method 1
Kc Td Comment
Ultimate cycle 5 K (564) x (564)
»d N = 1 0
5 K (564) _ 0.2455
K „
, -.-0.4986-0.7510 In - a
v T u y
,0.3875-0.2667 ln(K„Km)-l .1111 I n p s
Td<564) =0.000726TU
N -6.1523-1.4285 i n - ^
T
(K„Kmy
(K K y1-4494+°-0402 |n(KuKm)-0- ' , 6021n —
374 Handbook of PI and PID Controller Tuning Rules
4.6 I2PD Model Gm(s) = K „ e ~
4.6.1 Controller with filtered derivative with set-point weighting 2
G c ( s ) = K t
A , 1 T
d s 1 + — + %r
Xs . T, V
1 + ^ - s N
l + bfls + bf2s2
l + afls + af2s2 • R ( s ) - Y ( s )
R(s) l + b f ls + b f 2s
l + a f ls + a f 2s
E(s)
Kc 1 + l
+ ^
I N )
U(s)
— •
s2
Y(s)
Table 132: PID tuning rules - I2PD model Gm (s) = — 2 ^
Rule
Liu et al. (2003). Model: Method I
Kc
1 K (565)
T i Td
Robust T(565) x (565)
Comment
1 £ (565) {AX + zm jlX3 + 6X\n + 4Xzm2 + t m
3 ) - XA
Km(AX3+6X2zm+4Xzm2+zjj
(565) _ (4k + T J4X3 + 6X2Tm + 4ATm2 + T 3 ) - X4
\4Xi+6X2xm+4Xzm'+zm 2_ , A\~ 2 , - 3T
T (565, _ (6X2 + 4Xzm + zj )AX3 + 6X2xm + 4Xz2 + zj )
(4X + Tm ^ + 6X2zm + 4Xxm2 + T m
3 ) - A4
4X3+6X2Tm+4Xtm2+Tm
3 af, = 4X + zm ; af2 = 6X2 + 4Xxm + z„
6X2+4Xzm+zm ——. rf-— r—~ ~—; T\ '. 1 » b n = 2X , b f2 = X X3[(4X + tm)(4X3 +6X2tm +4Xxm
2 +Tm3)-X4]
Chapter 4: Tuning Rules for PID Controllers 375
Rule
Liu et al. (2003) - continued.
Kc
X
0.8xm
*m
l-5xm
2xm
Sampi OS
a 33%
« 20%
a 3%
* 0 %
T, Td
e data (deduced from a Rise time
- 4 x m
*5xm
«Vxra
*8xm
X
2.5xra
3tm
3.5xm
4xm
Comment
graph) OS
* 0 %
«0%
* 0 %
* 0 %
Rise time
«10tm
-12.5xm
*14xm
*16xm
376 Handbook of PI and PID Controller Tuning Rules
4.6.2 Controller with filtered derivative with set-point weighting 4 f \
Gc(s) = Kc
R(s)
1 + - U ^ T>s 1 + ^ s
N
1 + b f jS + b f 2s + b f 3 s 3 + b f 4 s >
l + afls + af2s2 +a f3s3 + af4s4 R(s)-Y(s)
J U(s) Y (s )
l + b n s + b f9s2 + b f , s3 + b f 4s4
1 + af ls + a f2s2 + a f3s3 + af4s
Table 133: PID tuning rules - I2PD model Gm(s): Kme"
Rule
Liu et al. (2003). Model: Method 1
Kc
2 j ^ <566>
X
0.8xm
tm
1.5Tm
Sampl OS
= 14%
= 4%
= 0%
T, | Td Comment
Robust - , (566)
i T (566)
e data (deduced from a graph)3
Rise time
= 3rm
= 3.5tm
-4 5x
X
2.5xm
3xm
35rm
OS
= 0%
= 0%
= 0%
Rise time
= 7xm
= 8.5T,„
= 10.5xm
K (566) _ (4X + Tm fr3 + 6X2Tm + 4XTJ + Tm3 ) - X4 ^
Km(4X3+6X2Tm+4Xrm
2+xm3j
_ (4A. + T )(4X3 + 6A2xm + 4Axm2 + x 3)-X4
^ 3 + 6 X 2 x m + 4 X x m2 + x m
3 j
_ (6X2 + 4XT + xm2 &A3 + 6X2xln + 4Ax, 2 + xm
3)
(4X + xm X4X3 + 6A2xm + 4Xxm2 + x m
3 ) - X4
X4
, b f l = 4 A , b f 2 =6A2, b f 3 = 4 A 3 , b t 4 = A 4 ,
.(566)
(566)
4X3+6A2xm+4Xxm2+xm
3
l 2 ' " t m + T m2 6A.2+4Ax +x„
X'[(4X + x m ^ +6X\^+4Xxm2 +xm
3)-X4]
af, = 8A + xm , af2 = 26k2 + SXrm + xm2 , af3 = 4x(lOA2 + 20ktm + 4xm
2),
a f 4=4X2(6X2+4Axm +xm2) .
Chapter 4: Tuning Rules for PID Controllers 377
Rule
Liu et al. (2003) - continued.
Kc
2Tm
A
0.8xm
Tm
l-5xm
2Tm
Ti Td Comment
Sample data (deduced from a graph) - continued «0% *6xm 4xm «0% «13Tm
Sample data (deduced from a graph)4
OS
* 46%
* 34%
* 16%
* 8 %
Rise time
~6xm
«13.5xm
*13xra
«16.5xm
A
2.5xm
3xm
3-5xm
4Tra
OS
«4%
- 2 %
* 1 %
«0%
Rise time
*20.5xm
«24xm
- 2 8 i m
«32xm
4 afl =5A + t m , a f2=10.25A2+5ATm+Tm2, af3 = A(7A2 + 4.25Atm + i m
2 ) ,
a f4=0.25A2(6A2+4Axm+Tm2).
378 Handbook of PI and PID Controller Tuning Rules
4.6.3 Series controller (classical controller 3)
R(s)
~ ^ =
E(s) K,
V T i s 7 (l + sTd)
Gc(s) = Kc
U(s)
l + J-](l + sTd) v T i S
K „ e " Y(s)
Table 134: PID tuning rules - I 2 PD model Gm(s) = K „ e - « -
Rule
Skogestad (2003).
Model: Method 1
Kc T; Td Comment
Direct synthesis 0.0625 K m t m
8xm 8tm
Chapter 4: Tuning Rules for PID Controllers 379
4.6.4 Non-interacting controller based on the two degree of freedom structure 1
f \ ( ^
U(s) = K E(s)-Kc a + PTds
1 + ^ s V N
R(s)
Y(s)
*g)->* + s
U(s)
Table 135: PID tuning rules - I2PD model Gm(s) = ^ ^ -
Rule
Hansen (2000). Model: Method 1
Kc T( Td Comment
Direct synthesis
3.75KmTm2 5.4xm 2-5tm N = 0; p = l ;
a = 0.833
380 Handbook of PI and PID Controller Tuning Rules
4.6.5 Industrial controller U(s) = K(
v Tisy R ( s ) - ^ Y ( s )
1 + -N
Table 136: PID tuning rules - I2PD model Gm{s) = ^ L .
Rule
Skogestad (2004b).
Model: Method 1
Kc Tj Td Comment
Direct synthesis: time domain criteria
0.25
Km(TC L+Tm)2 4^2(TCL+xm) 4(TcL+xm)
'good' robustness -
TcL = T m >
£=0.7 or 1
N e [5,10]; N = 2 if measurement noise is a serious problem
Chapter 4: Tuning Rules for PID Controllers
K e"st"'
381
4.7 SOSIPD Model (repeated pole) Gm(s) sfl + T ^
4.7.1 Non-interacting controller based on the two degree of freedom structure 1
( \ f \
U(s) = K
Table 137: PID controller tuning rules - SOSIPD model (repeated pole) Kme~
s(l+T f f l ls)'
Rule
Minimum ITAE - Pecharroman
and Pagola (2000).
Model: Method 2
K m = i ;
T m = i ;
0.1 < t m <10;
P = l , N = 10.
Kc ^ Td Comment
Minimum performance index: servo/regulator tuning x,Ku x i \ x3Tu
Coefficient values
*! 1.672
1.236
0.994
0.842
0.752
x2
0.366
0.427
0.486
0.538
0.567
x3
0.136
0.149
0.155
0.154
0.157
a
0.601
0.607
0.610
0.616
0.605
*c
-164°
-160°
-155°
-150°
-145°
382 Handbook of PI and PID Controller Tuning Rules
Rule
Pecharroman and Pagola (2000) -
continued.
Taguchi and Araki (2000).
Model: Method 1
Kc Ti Td Comment
Coefficient values (continued)
*| 0.679
0.635
0.590
0.551
0.520
0.509
* 2
0.610
0.637
0.669
0.690
0.776
0.810
1 K (567)
X 3
0.149
0.142
0.133
0.114
0.087
0.068
T (567) *i
a
0.610
0.612
0.610
0.616
0.609
0.611
T (567)
•. -140°
-135°
-130°
-125°
-120°
-118°
^ - < 1 . 0 ; ' m l
Overshoot (servo step) < 20% ; N
fixed but not specified
I K (567) = _
K „ 0.1778 + -
0.5667
T (567) _ T
- + 0.002325 J
0.2011 + 11.16-^--14.98 •13.70 -4.835
n 4 >
' m l
(567) V =Tm l 1.262 + 0.3620-^- a = 0.6666, ml J
P = 0.8206 - 0 .09750-^ + 0.03845
Chapter 4: Tuning Rules for PID Controllers 383
4.8 SOSPD Model with a Positive Zero K m ( l - sT m 3 ) e -^
Gm(s) = Tml
2s2+2^mTmls + l or Gm(s)
_K m ( l - sT m 3 ) e - " -
(l + sTml)(l + sTm2)
4.8.1 Ideal controller Gc(s) = Kc Xs
R(s)
*R\ + Y
T 1
E ( s ) (
Kc f 1 1 1+ — + Tds
T.s v *•" y
U(s)
— • Gm(s) Y(s)
— w
Table 138: PID controller tuning rules - SOSPD model with a positive zero
Gm (S) = M ^ - s T ^ K ^ ^ Kn (1 - sTm3 y^ (l + sTm,Xl + sTra2) Tml¥+2i;mTmls + l
Rule
Minimum IAE -Wang et al.
(1995a). Model: Method 1
Kc Ti Td Comment
Minimum performance index: servo tuning i
i P o q i - P i r Km Po Po
p 0 q 2 -P2 Pi
PoQi-Pi Po
Po : T m l + T m 2 + T r a 3 + T m , p , = T m I T m 2 + 0 . 5 T r a ( T m I + T m 2 ) - 0 . 5 T m T m 3 ,
p 2 = 0.5TmlTm2Tm , qi = Tml + Tm 2 + 0.5tm , q2 = Tm lTm 2 + 0.5xm (Tml + Tm 2 ) ,
r = 8 0 + 8 1 - E - + 5 2 'n,2
VTml J
S7 VTml J
+ 5, VTml J
+ 8
+ 83
'T,
'1113
Tml
'O VTml J
VTml J V T m i y
ml J
f
+ 5B
+ 8C
V^ml
T n , + 8,
fT \
VTml J
S0 = 3.5550 ,8, = -3.6167 ,8 2 = 2.1781,83 = -5.5203 ,5 4 = 1.4704 ,
85 = -0.4918,86 = 2.5356 ,8 7 = -0.4685 ,88 = -0.3318 ,8 9 = 1.4746 .
384 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ISE -Wang et al.
(1995a). Model: Method 1
Minimum ITAE - Wang et al.
(1995a). Model: Method 1
Wang et al. (2000b), (2001a). Model: Method 2
Kc
2
i P o q i - P i r
K m Po2
3
i p0qi - P i r
K m Po2
4 ^ (568)
Ti
Po
Po
Direct s
2smTm i
Td
p 0 q 2 - p 2
PoQi -P i
Poq2 - P 2
PoQi - P i
ynthesis
T 2
"•ml
Po
__P_L Po
Comment
M , = 1 . 5
p0 ,Pi,p2 ,q,,q2 ,r defined as on page 383.
S0 = 3.9395 ,5 , = -3.2164 ,6 2 =1.6185 , 5 3 = -5.8240,84 =1.0933 ,
65 = -0.6679 , 56 = 2.5648 , 57 = -0.2383, 58 = -0.0564, 59 = 1.3508. 3 Po>Pi>P2>qi>q2>r defined as on page 383.
80 = 3.2950 ,5 , = -3.4779 , 5 2 = 2.5336,83 = -5.5929,54 = 1.4407 ,
85 = -0.3268 , 56 = 2.7129 , 87 = -0.5712 , 58 = -0.5790 , 8 9 = 1.5340 .
4 K (568) 1 C0S9 2 ^ r oT m l m
Ms cos[anm - t a n - ' ( - T m 3 c o ) J K m ^
co and 9 are determined by solving the equations:
, V + i
tan cos 9
M, -s in0 tan '(-Tra3co)-coTm-0.57t and
l = tan~ ~ T m3 T mt0 2 ~ l)cOs(cOTm ) - C0Tm sin(cOTm )
rTm3^mco2-l)sin(coTm)+coTmsin(coTm)
Chapter 4: Tuning Rules for PID Controllers 385
4.8.2 Ideal controller in series with a first order lag f
R(s) E ( s )
G c ( s ) = K (
U(s)
1 \
V T i s J
1
/ K r
1 1 + — + Tds
v T i s J T fs + 1 j K m ( l - s T m 3 ) e - ^
(l + sTmlXl + sTm 2)
T fs + 1
Y(s)
— •
Table 139: PID controller tuning rules - SOSPD model with a positive zero
K m ( l - s T r a 3 > - ^ Gm(s) =
(l + sTmlXl + sTm2)
Rule
Li et al. (2004). Model: Method 1
Kc Ti Td Comment
Direct synthesis: time domain criteria 5 K (569) Tm, +Tm2 TmiTm2
Tm l+Tm 2
5 K (569) Tm l+Tm 2 ; . T f Km (2TCL2 + Tm3 + xra ) '
f 2TCL2 + Tm3 + xra '
desired closed loop transfer function = ( l - s T m 3 > -
386 Handbook of PI and PID Controller Tuning Rules
4.8.3 Controller with filtered derivative Gc(s) = Kc
R(s)
, 1 Tds
1 + — + -
Table 140: PID controller tuning rules - SOSPD model with a positive zero
Gm(s): K m ( l - s T m 3 ) e - o r G m ( s ) = JCm(l-_sTm3 )e
(l + sTmlXl + sTm2) Tml2s2+24mTmls + l
Rule
Chien (1988). Model: Method I
Kc T i Td Comment
Robust 6 ^ (570)
7 K (571)
j (570)
x ( 5 7 1 )
x (570) 1 d
T (571)
Tml > Tm2 > Tm3; N=10; X = [TmI,xm J (Chien and Fruehauf (1990)).
T m l + T „ , + -T m l x „
6 K (570) • + T m 3 + T m T (570) _ T T T m 3 T m
X + T m , + T„ Km(^ + T m 3 +x m )
T (570) _ T m 3 T m Xd - ~ - +
T m l T r a 2
^•+Tm3+Tra J + x , Tm3Tn
^ + Tm 3+xm
OE T _j m3 m >m ml ^ x X
7 ^ (571) A + 1 m 3 + X m T (571) _ - j . T 1m3Tm
M - / S m 1 m l + K„ Km(X + T m 3 +x m ) k + T n > 3 + * m '
(571) T m 3 T m
^ + T m 3 + * m oK x , " ^ n . 2^mTml + *- + T m 3 + 1 : m
Chapter 4: Tuning Rules for PID Controllers 387
4.8.4 Classical controller 1 Gc(s) = Kc 1+ — v T i s y
1 + Tds
N
R(s) E(s)
K, v T i sy
1 + Tds
l + ^ s V. N .
U(s), K m ( l - s T m 3 ) e -
(l + sTmlXl + sTm2)
Y(s)
Table 141: PID controller tuning rules - SOSPD model with a positive zero
K m ( l - s T m 3 ) e - ^ Gm(s) =
(l + sTm,Xl + sTm2)
Rule
Poulin et al. (1996).
Model: Method I
O'Dwyer (2001a).
Model: Method I
Chien (1988). Model: Method 1
Kc T; Td Comment
Direct synthesis
8 K (572)
x i T m i
K m T m
Tml
Tm,
Tm2
Tm2
Minimum
Am =0.57t/x, ;
K = 0.57t-x,;
N = Tm2/Tm3
Sample result 0.785Tml
K r a T m Tm, Tm2
A m = 2 . 0 ;
<L=45°
Robust
Tm2
Km(^ + Tm)
Tml
*•*(*•+ 0
Tm2
Tn,
Tml
Tm2
N=10;
^e[Tm l ,Tm] (Chien and
Fruehauf(1990))
Tml > Tm2 > Tm3
8 K (572) _ Tral j ^ _ Tm2 (Tm, + 2Tm3 j K m ( T m l + 2 T m 3 + x J :
= 56° when Tm, = Tm, = xm . m3 1 m l Lm
388 Handbook of PI and PID Controller Tuning Rules
4.8.5 Series controller (classical controller 3)
Gc(s) = Kt 1 + — (1 + Tds)
R(s)
^
E(s)
K, 1 + — Ts
(l + sTd)
U(s)
• K m ( l - s T m 3 ) e
(l + Tm]sXl + Tm2s)
Y(s)
Table 142: PID controller tuning rules - SOSPD model with a positive zero Km(l-sTm3)T
SI™ Gm(s) =
(l + sTmlXl + sTm2)
Rule
Zhang (1994). Model: Method 1
Kc T; Td Comment
Direct synthesis
Kmxm Tml Tm2
"Good" servo response; xm is
"small".
Chapter 4: Tuning Rules for PID Controllers 389
4.8.6 Classical controller 4 Gc (s) = Kc 1 + — v T i s y
1 + - TdS
N ;
R(s) E(s)
-0— K, V T i S /
1 + . ^
V N ;
U(s) Km(l-sTm3)e-
(l + sTralXl + sTm2)
Y(s)
Table 143: PID controller tuning rules - SOSPD model with a positive zero
Km(l-sTm3)e-ST" Gm(s) =
(l + sTmlXl + sTm2)
Rule
Chien (1988). Model: Method 1
Kc T, Td Comment
Robust
Tm2
Km(^ + xm)
Tml
Km(*- + 0
Tm2
Tmi
9 T (573) 1 d
10 T (574)
N=10; X e [T, Tm ] , T = dominant time constant (Chien and Fruehauf
(1990)).
= T „ , + -T m , x „
^ + T m 3 +x m
10 x (574) _ T Tn^Tp. 1^ — AmT -1
^ + T m 3 + Tra
390 Handbook of PI and PID Controller Tuning Rules
4.8.7 Non-interacting controller 1
U(s) = Kc
V T i S / E(s) ^ - Y ( s )
i+3fi N
R ( S ) v, ^ + K
V T . S /
U(s) •
Km(l-sTm3>-
T m l V+2^T m l s + l
Y(s)
Tds
l + ^ s N
Table 144: PID controller tuning rules - SOSPD model with a positive zero
Km(l-sTm3)e-ST" Gm(s) =
Tm l2s2+2^mTm ls + l
Rule
Huang et al. (2000).
Model: Method 1
N not specified
Kc Ti Td Comment
Direct synthesis
..Ke(5"> 2Tml^m+0.4Tm T (575)
' d
^ = 0 . 5 , ^ m < l . X = 0 . 6 , K ^ H m < 2 . 1 m l ' Ami '
T 1 0 7 m £ ~> 9 u - ' > T S n , ^ ^ s e r v o tuning
1 m l
^ = 0 . 6 , ^ m < l.X = 0.7, 1 < - I ^ m < 2 . Tral ' T m I
T T . _ n o l ' m e . 9 *• u - ° > T S m ^ ^ regulator tuning
ml
ii K (575) =-^2T ro l^m +0.4xm T (575) _ Tml + 0,8Tmli;mTn
Km(Tm+2Tm 3) 0-8Tml^mTn
Chapter 4: Tuning Rules for PID Controllers 391
4.8.8 Non-interacting controller based on the two degree of freedom structure 1
U(s) = Kc
f 1 T "\ 1 T d S
1+- + d V " i TjS 1 + Tds/N
E(s)-Kc a + -PTds
1 + Tds/N R(s)
a + PTds
T
N
K „
R(s)
+ E(s)
1 + - U - ^ T>S l + ^ s
N V
U(s)
+
K r a ( l - s T m 3 ) e -
(l + sTml)(l + sTm2)
Y(s)
Table 145: PID controller tuning rules - SOSPD model with a positive zero
Kra(l-sTra3)e-SI™ Gm(s) =
(l + sTmlXl + sTm2)
Rule K „ Comment
Direct synthesis Huang et al.
(2000). Model: Method I
N = min[20,
20Td]
12 K (576) , (576) (576) Servo tuning
X = 0.6,^m<\ A m = 2 . 8 3 . X = 0 . 7 , K ^ m < 2 Tml ' Tml
A m = 2 . 4 3 . X = 0 . 8 , ^ r a > 2 A m = 2 . 1 3 .
a = -2T m , ^
2Tml^m+0.4xn • 1 . P = -
0-5Tml2
Tm /+0.8Tm l^mT n
- - 1
,2 ^ ( 5 7 6 ) = 3 L 2 T m , ^ + 0 . 4 T n , T i ( 5 7 6 ) = 2 T m | ^ + Q 4 i n
K m ( t m +2T m 3 )
(576) Tm l2
+0.8Tm l^ r ax ml^rri "m
0-8Tra]^mTn
392 Handbook of PI and PID Controller Tuning Rules
4.9 SOSPD Model with a Negative Zero
Gm(s) = Km(l + sTm3)e-"-
T m i y + 2 ^ T m l s + l o r G J s ) :
Km(l + sTm3)e-"-
(l+sTmI)(l + sTm2)
/ 4.9.1 Ideal controller Gc (s) = K(
1 \
V T i S J
R(s)
*6 1 ?) E^ »
f-Kcfl + - L + Tds
I T i s J
iU(s; Km(l + sT m 3 >-"-
T m l V + 2 ^ m T m l s + l
V(s)
—v-
Table 146: PID controller tuning rules - SOSPD model with a negative zero
Km(l + sTm3)e-sx™ Gm(s) =
Tm lV+2);mT r a ls + l
Rule
Wang et al. (2000b), (2001a). Model: Method 2
Kc Ti Td Comment
Direct synthesis
1 K (577) 2?mTml T 2
1 m l
M,=1.5
1 K (577) 1 cos 6 2 ^m T ml f f l
Ms cos[mTm-tan-,(Tra3co)] K m ^ T m 3 V + •; co and G are determined by
solving the equations:
tan cos 9
VMS -sinO = tan '(Tm3co) —coxm -0.5TC and
6 = tan"1 T m 3 ^ m m 2 - 1 ) c O s ( ( 0 T m ) - C 0 T m SJn(c0Tm)
l T m 3 ' C m f f l 2 - 1 ) s i n ( a i X m ) + f f > ' C m s i n ( ( a X m ) _
Chapter 4: Tuning Rules for PID Controllers 393
4.9.2 Controller with filtered derivative Gc (s) = Kc
R(s)
+ a
E(s) K „
, 1 Td s
1 + — + • — T 's 1 + ^ s N .
, 1 TdS 1 + — + - d
U(s)
N ; Y(s)
Gm(s)
Table 147: PID controller tuning rules - SOSPD model with a negative zero
Gm(s) = Km(l + sTm 3 )e-^ _^ ^ _ Km(l + sTm3)e-ST-
(l + sTmlXl + sTm2) o rG m ( s ) :
Tml2s2+2^mTmls + l
Rule
Chien (1988). Model: Method 1
Kc Ti Td Comment
Robust 2 K (578)
2smTmi ~ Tm3
T (578) i
2smTmi ~Tm3
T (578) *d
3 T (579)
N=10; *-e[Tml,Tm]
(Chien and Fruehauf(1990)).
2 v (578) _ Tml + Tm2 - Tm3 (578) K
Kra(X + t J . V (578)
: Tml + Tm2 Tm3 .
. TralTm2 (Tmi+Tm2 T^JT^ • > Tm| > Tm2 > Tm3.
3 T (579) Tmi (2i;mTml Tm3jTm3
2smTrai _ T m 3
394 Handbook of PI and PID Controller Tuning Rules
4.9.3 Classical controller 1 Gc(s) - Kc 1 + ^ v T i s y
1 + Tds
I N ;
R(s) E(s)
+ x K, 1 + —
1 + Tds
1 + ^ s N
^ K m ( l + s T m 3 ^
(l + sTralXl + sTm2)
Y(s)
Table 148: PID controller tuning rules - SOSPD model with a negative zero
Km(l + sTm 3>-"-Gm(s) =
(l + sTmlXl + sTra2)
Rule
Zhang (1994). Model: Method I
Poulin et al. (1996).
Model: Method 1
Chien (1988). Model: Method 1
Kc T( Td Comment
Direct synthesis
Trol
K m T m Tmi Tm2 N = Tm2/Tm3
"small" xm ; "good" servo response.
Tm, Km(T r a l+Tm)
Tmi Tm2
N = T r a 2 / T m 3 ;
Minimum
*m=90°;<t)m =
6 0 ° , t m = T m l
Robust
Tm2
Km(^ + t m )
Tml
Km( + 0
Tm2
Tm,
Tml
Tm2
Trai >T ra2;N=10; ^e[Tm l ,xmJ (Chien and Fruehauf( 1990)).
Chapter 4: Tuning Rules for PID Controllers 395
/
4.9.4 Classical controller 4 Gc(s) = K v T iV
l + - T ' s
N
R(s) E(s)
K„
v T i s / 1 + - ^
T
N ;
U(s) Km(l + sTm3)e-
Y(s)
(l + sTmlXl + sTm2)
Table 149: PID controller tuning rules - SOSPD model with a negative zero
K m ( l + s T m 3 ) e - ^ G m ( s ) =
(l + sTmlXl + sTm2)
Rule
Chien(1988). Model: Method 1
Kc Ti Td Comment
Robust
Tm2
K m (* + Tm)
Tm, Km( + 0
Tm2
Tm,
Tmi - Tm3
Tm2 ~~ Tra3
Tm, > T m 2 > T m 3 ; N = 1 0 a 6 [ T r a l , T m ]
(Chien and Fruehauf (1990)).
396 Handbook of PI and PID Controller Tuning Rules
4.9.5 Non-interacting controller 1 r
U(s) = Kc E(s)- TdS Y(s)
N
Km(l + sTm3>-"-
Tm,V+2^mTmls + l
Y(s)
Table 150: PID controller tuning rules - SOSPD model with a negative zero
Km(l + sT m 3 )e-^ Gm(s) =
T r a l V + 2 ^ T r a l s + l
Rule
Huang et al. (2000).
Model: Method 1
N not specified
Kc Ti Td Comment
Direct synthesis
4K C( 5 8 0 ) 2Tral^m+0.4Tm T (580)
servo tuning
l = 0 . 6 , ^ m < U = 0 . 7 , K ^ m < 2 . ^ 0 . 8 , ^ m > 2 .
regulator tuning
4 K (580) = ? 2Tml£m + 0,4tm (580) _ Tm, +0.8Tml£mTm
Km(Tm+2Tm 3) 0-8Tml^mTn
Chapter 4: Tuning Rules for PID Controllers 397
4.9.6 Non-interacting controller based on the two degree of freedom structure 1
f \ f \
U(s) = Kc
N
E(s)-Kc a + (3Tds
1 + ^ s N
R(s)
a + PTds
T 1 + ^ s
N
K „
R(s)
E(s)
K, 1 THs
1 + + -T=s 1 + ^ s
N .
U(s)
*g>> K m ( l + sTm3)e-™
T m i y + 2 ^ m T m l s + l
Y(s)
Table 151: PID controller tuning rules - SOSPD model with a negative zero
Km(l + sTm3)e-"" Gm(s) =
T m l V + 2 ^ m T m I s + l
Rule K, Comment
Direct synthesis
Huang et al. (2000) - servo
tuning. Model: Method 1
sK (581) 2Tral^m+0.4Tn (581) N = min[20,
20Td]
2Tml$n
2Tml^m+0.4Tn • l . p = -
0-5Tra,2
T r a/+0.8Tml^mTn
• 1 .
X = 0 . 6 „ ^ K m < 1 Am = 2.83 . X = 0.7,1 < m < 2 Ami ' ' Ami '
Am =2.43. X = 0.'< -lm>2 A m =2.13
5 K (581) _.| ZTml m +0-4lm ^ (581) _ Tml +0.8Tml^mTn
K m ( i m + 2 T m 3 ) . d 0.8Tml^mim
398 Handbook of PI and PID Controller Tuning Rules
4.10 Third Order System plus Time Delay Model
K e~ST|" K e~STm
G (s) = m or G (s) - m
m W l + a1s + a 2 s 2 + a 3 s 3 m W (l + sTml Xl + sTm2 Xl + sTm3)
/ 4.10.1 Ideal controller Gc(s) = Kc
R(s)
1 A
1+ — + Tds
V T i s
U(s)
K „ e "
(l + STmlXl + sTm2Xl + sTm3)
Y(s)
Table 152: PID controller tuning rules - third order system plus time delay model
K „ e - " -Gm(s) =
(l + sTm,Xl + sTm2Xl + sTm3)
Rule
Standard form optimisation -
binomial -Polonyi (1989). Standard form optimisation -
minimum ITAE -Polonyi (1989).
Kc Ti Td Comment
Minimum performance index
1 K (582)
2 K (583)
T ( 5 8 2 ) i
T (583)
, X m
t m
Model: Method I
Tm, > 10(Tm2+xm)
1 K (582) _
6tm Tm3
T/5821 =4V6T m 2 T r a + t r a .
Chapter 4: Tuning Rules for PID Controllers 399
4.10.2 Ideal controller in series with a first order lag /
Gc(s) = Kc
—JU(s)
1 \
v T i s j
1
K, V T i S
1
T fs + 1
K „ e "
Tfs + 1
Y(s)
l + a,s + a 2 s 2 + a 3 s 3
Table 153: PID controller tuning rules -Third order system plus time delay model
K m e " ^ Gra(s) = -
l + a,s + a2s + a3s
Rule
Schaedel(1997). Model: Method 1
Kc Ti Td Comment
Direct synthesis: frequency domain criteria 3 K (584) T (584) T (584) T f =T d
( 5 S 4 ) /N ;
5 < N < 20
3 K (584) 0.375T (584) , (584) _ a l ~ a 2 + a l T m +0-5'cm
Km U , + T m + -(584)
N - (584) a, + t m - T , (584)
(584) _ a2+a1xm+0.5Tm a3+a2Tm+0.5a,Tm +0.167xn 1 J —
a2+a1Tm+0.5Tn
400 Handbook of PI and PID Controller Tuning Rules
4.10.3 Controller with filtered derivative G c ( s ) = Kc
U(s)
i 1 Td s
1 + + =r
R(s) E(s)
K, 1 THs
1 + — • + -T ' S 1 + ^ s
N
N
K „ e " " "
l + a,s + a 2 s 2 + a 3 s 3
Y(s)
Table 154: PID controller tuning rules - Third order system plus time delay model
Gra(s) = -Kme~
i j a T « 2 a a 3 a
Rule
Schaedel(1997).
Kc Ts Td Comment
Direct synthesis: frequency domain criteria 4 K (585) T (585) T (585) Model: Method 1
4 K (585)
K „
0.375T,- (585)
(585)
N
T(585) = a , 2 - a 2 + a , T m +0.5Tn
• (585) *i + t m - T d (585)
(585) _ a 2 + a,xm+0.5xm a 3 +a 2 x m +0.5a ,x m +0.167Tn
a, +x„ a 2 +a ,x m +0 .5x (585)
N:
a + T m - T j (585) Jki^flt
(585) T (585)
0.375-L ry yjOJfy
K „
0.05 < a < 0.2 , Kv = prescribed overshoot in the manipulated variable for a step
response in the command variable.
Chapter 4: Tuning Rules for PID Controllers 401
4.10.4 Non-interacting controller based on the two degree of freedom structure 1
( \ r \ U(s) = Kc
, 1 TdS 1+ — + d
T i s 1 + ^ s N
E(s)-Kc a + -PTds
1 + ^ s N j
R(s)
a + Fds
T N ;
K,
R(s)
+ n
E(s)
K,
N
U(s)
*g» K „ e - " »
d + sTml)3
Y(s)
Table 155: PID controller tuning rules - third order system plus time delay model -
K m e ^ Gm(s) =
0 + sTml)3
Rule
Taguchi and Araki (2000).
Model: Method 1
Kc Ti Td Comment
Minimum performance index: servo/regulator tuning
5 K (586) ry (586) T (586)
Overshoot (servo step) < 20% ; N
fixed but not specified
' Equations continued into the footnote on page 402; xm /Tml < 1.0 .
K (586)
K „ 0.4020+ -
1.275
- + 0.003273
T (586) rp i " 1 m l 0.3572 + 7.467-a!--12.86
)
^m
T m l .
2
+ 11.77 _ T m i
3
-4.146 . T m ! _
4"l
402 Handbook of PI and PID Controller Tuning Rules
(586) = T„ 0.8335 + 0.2910- - 0.04000
2\
a = 0.6661 - 0.2509-12- + 0.04773 Tm,
, p = 0.8131 - 0.2303—S3- + 0.03621 T
Chapter 4: Tuning Rules for PID Controllers
K „ e " " -4.11 Unstable FOLPD Model Gm(s)
T j - 1
403
f 4.11.1 Ideal controller G (s) = Kc
1 A
V T i s
_RCs)_^E(s)
+
r K,
1 1 + — + Tds
V T i s J
U(s) K e "
T m s - 1
Y(s)
Table 156: PID controller tuning rules - unstable FOLPD model G m (s) = Kme"
T m s-1
Rule
Minimum ISE -Visioli (2001).
Minimum ITSE -Visioli (2001).
Minimum ISTSE -Visioli (2001).
Minimum ISE -Visioli (2001).
Minimum ITSE -Visioli (2001).
Kc Ti Td Comment
Minimum performance index: regulator tuning
1.37
1.37
Km
1.70
f
T
f
T
1
\
I
1
2.42Tra
4.12Tm
\ 1m J
( \ m
T
4-52T m f^ V m j
0.90
1.13
0.60xm
0.55xm
0.50xm
Model: Method 1
Model: Method 1
Model: Method I
Minimum performance index: servo tuning / NO.92
1 . 3 2 ( 0 Km lTmJ 1.38'
T
.90
4 . 0 0 T „ ^ j
4.12Tm
f \
T
0.90
1 T (587)
2 T (588)
Model: Method 1
Model: Method 1
1 T ^ ( 5 8 7 ) = 3 7 8 T ^
2 Td(588) = 3 6 2 T ^
1 - 0.84
1-0.85
/ N0.02
f \09S
T V m
f A 0 9 3
T V m J
404 Handbook of PI and PID Controller Tuning Rules
Rule
Minimum ISTSE -Visioli (2001).
Jhunjhunwala and
Chidambaram (2001).
De Paor and 0'Malley(1989). Model: Method 1
Chidambaram (1995b).
Model: Method 1
Kc
, N0 .95
1.35fO K ralTJ
Ti
4 . 5 2 T m f ^ 1.13
Td
3 T (589)
Comment
Model: Method I
Minimum performance index: servo/regulator tuning
4 K (590) T (590) T (590) Model: Method 1
Direct synthesis
5 K (591)
6 K (592)
T (591) i
T 25-27^2-
T (591) 1 d
0.46xm
T m
<1
^2_<0.6 T
m
3 x (589)
4 K (590)
: 3.70T„ 1-0.861
13.0-39.712—2 K m l Tm
• < 0 . 2 :
K (590) 1.397
K „ , 0 . 2 < - ^ - < 1 . 4 ;
T
Ti(590> = 0 . 8 5 6 T e T- , 0 <-^- < 1 ; T: (590) = 0.3444T e T- , 1 < ^ L < 1 . 4 :
T (590) _ T
' d ~ Xrr 0.5643-^2- + 0.0075 , 0 < ^ _ < 1 . 4 . T T
5 K (591)
,(591)
K „ cosJU-^Ll^L+J1 T m / / T m s in j f 1 - - T "
T
l - t „ / T m
Tm/Tm
tan(0.75(f))
f \ (j) = tan
v Tm y
Tm K / \
m \ m V m /
6 v (592) _ _ 1 _ K „ 1.3 + 0.3^2-
Chapter 4: Tuning Rules for PID Controllers 405
Rule
Chidambaram (1997).
Model: Method 1
Valentine and Chidambaram
(1997b)-dominant pole
placement. Model: Method 1
Kc
1.165 Am
, \ 0.245
1.165 l T m J
Ti
7 T (593) 1 i
S T (594) ^i
9 T ( 5 9 5 )
10 j (596)
Td
j ( 5 9 3 )
a
T (594) ^d
Comment
T
0.6 <
0.8 <
<0.6
T m m
m
<0.8
<1
7 (593) =0 .176 + 0.360^2- T m ; a = 0 .179-0 .324^-+ 0.161 \ * m / *• m
f \ 2
2m. I t n vT™
<0.6
a = 0.04, 0 .6<^s .<0 .8 ; a = 0 .12-0 .1 -^ - , 0 . 8 < ^ - < 1 . 0 . T T T
, (594)
9 j (595)
10 j (596)
0.176 + 0.36-m J
0 . 1 7 9 - 0 . 3 2 4 ^ + 0.161 — T T
r - T d (594) 0.176 + 0.36-
o.ne + o^-^psT^ 0.176Tm+0.36T„
0 . 1 2 - 0 . 1 ^ -
406 Handbook of PI and PID Controller Tuning Rules
Rule
Pramod and Chidambaram
(2000). Model: Method I
Gaikward and Chidambaram
(2000). Model: Method 1 Sree et al. (2004) Model: Method 1
Kc
11 K (597)
13 K (599)
14 v (600)
Ti
T (597)
12 j (598)
T (599)
T ( 6 0 0 )
Td
T (597)
rj, (599)
0.4917tm
Comment
T
0.6 <-
- <0.6
T <0.8
0.1<^L<0.6 T
0.01 < ^ L < 0.9 T
11 K (597) = _
K „ 2.121-1.906-2-+ 0.945
T „
f \2
t,
T
, ( 5 9 7 )
0.176 + 0.36-
0 .179-0 .324-^ + 0.161 T
/" \ 2 ' d (597) _
T
0.176 + 0.36-
12 j (598) 0.176 + 0.36- 2.5T„
Kn
0.299 + 0.726- • (599) = T „
( V 4.104^2.
V m J
+ 2 .099-^ + 0.096 T
(599) 0.822 r \2
t
T V m J
+ 0.419-E- +0.019 T „
14 xj (600) _ 1.4183 (' \ K „ T
V m y
y (600) _ T 16.327 VTm/
+ 5.5778^2- + 0.8158 T
, 0.01 <-^< 0.6; T
196 - 247.28 -J2- + 87.72 T
, 0.6 <-^2-< 0.9. T
Chapter 4: Tuning Rules for PID Controllers 407
Rule
Sree et al. (2004) - continued.
Tan et al. (1999b).
Model: Method 1 Lee et al. (2000) Model: Method 1
Kc
15 pr (601)
Ti
y (601)
Td
T (601) x d
Comment
0.01<is-<1.2 T
Robust v (602)
17 K (603)
rp (602) •M
y (603) •M
T (602)
x (603) J d
-^-<1.0 T
0 < ^ 2 - < 2 Tm
m
15 v (601) _ K
1.2824
K „ T
•. (601) _ , 5.5734-iS-- 0.0063 T
, 0.01 < - a - < 0.5; T
. (601) _
16 if (602)
0.483Trae ' T™ , 0.5 < ^ - < 1.2 ; Td(601) = Tn
1
0.507-21-+ 0.0028 T
Kn 0.0373
, (602)
(602)
X
10.5045
+ 0.1862 ^ - +[0.6508-0.1498 log?.]
+ 0.8032 3.2312
- 0.3445
0.0373 + 0.1862 ^ - + [0.6508- 0.1498 log*.]
0.9065X,0-"'"-12--l
Tm 0.2075X + 2.0335
0 0 3 7 3 +0.1862 | ^ - + (0.6508-0.14981og?0 , X not specified.
17 £ (603)
. (603) _
, (603)
K m ( 2 T C L + x m - a ) .<* = Tn i ck + l
T e*.P._l
• T m + a -T C L
2 +aT m -0 .5T m2
2 T C L + t m - a
, (603)
T a Tm2(0.167xm-0.5a)
2TCL + T m - a TC L2+atm-0.5Tm
2
,(603) 2 T C L + x m - a
408 Handbook of PI and PID Controller Tuning Rules
4.11.2 Ideal controller in series with a first order lag f •> \
Gc(s) = Kc
U(s) V T i s
1
Tfs + 1
R(s) / 0 v E ( s )
^ ' v "
+ X
/ K,
1 1 + — + Tds
V T i s j
1
Tfs + 1 Kme" T „ s - 1
Y(s)
Table 157: PID controller tuning rules - unstable FOLPD model Gm (s) = Kme"
• o - i
Rule
Chandrashekar et al. (2002).
Kc Ti Td Comment
Direct synthesis 18 j r (604) ry (604) 0-5Tm Model: Method 1
18 v (604) _ _ J _
K „ . K „ 4282 -1334.6-21-+ 101
T 0<-S-<0.2;
T 1 m
K (604) .1161 f N-0.9427
K m V m J
0.2 < -!=- < 1.0. T
y (604) _ - , 36.842 T
-10.3 X
T V m J
+ 0.8288 , 0 < — < 0 . 8 ; T
T: ( 604 )= 76.241T„ , 0.8 < - 2 - < 1.0 T
Tr = 0 .1233-^ + 0.0033 Tm , 0 <-$-< 1.0.
(TIS + l)e~STm
Desired closed loop transfer function = ^~- r ^ ; TCL2 = 4.5115Tn
/• \ 5.661 x „
(TCL2s + l)2 T
0 . 8 < ^ < 1 . 0 , T C L 2
f \ i \
0.0326+ 0.4958-2L +2.03 T T
V m J
Tm , o < - s - < o.: T
Chapter 4: Tuning Rules for PID Controllers 409
Rule
Chandrashekar et al. (2002).
Model: Method 1
Kc
Kc
101.0000
Kra
10.3662
5.3750
3.2655
2.4516
Kra
2.0217
1.7525
1.5458
1.3723
1.2420
1.1400
T, Td
Alternative tuning rules
Ti
0.0404Tm
0.3874Tm
0.8600Tm
1.8018Tm
3.0400Tm
4.6500Tm
6.7286Tra
10.337Tm
17.693Tra
33.610Tm
80.620Tm
Td
0.0
0.0435Tm
0.0884Tm
0.1375Tm
0.1868Tm
0.2366Tm
0.2866Tm
0.3380Tm
0.3910Tm
0.4440Tm
0.4969Tm
Tf
0.0
0.0134Tm
0.0250Tm
0.0435Tm
0.0581Tm
0.0696Tm
0.0784Tm
0.0885Tm
0.1005Tm
0.1124Tm
0.1247Tm
Comment
TcL2
0.02Tm
0.10Tm
0.20Tm
0.40Tm
0.60Tm
0.80Tm
1.00Tm
1.30Tm
1.80Tm
2.60Tm
4.20Tm
T m / T m
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
410 Handbook of PI and PID Controller Tuning Rules
4.11.3 Ideal controller with set-point weighting 1
U(s) = Kc (FpR(s) - Y(s))+^(F iR(s) - Y(s))+ KcTds(FdR(s) - Y(s)) T:S
Table 158: PID controller tuning rules - unstable FOLPD model Gm (s) = T m s-1
Rule
Prashanti and Chidambaram
(2000). Model: Method 1
Kc Ti Td Comment
Direct synthesis 1 K (605)
Fp=0.17
T (605) *i
Fi= l
T (605)
Fd = 0.654
Modified method of De Paor and
O'Malley (1989);
h./Tmj<i
1 K (605) _ _J_
x (605) _
COS J 1 — + . ) T m / / T m sin T V x /T T
v 'my (605)
T n , / T i >
tan(0.75(|») V 1 V / 'm
(j) = tan Tm xra
V m / m \ m \ m /
Chapter 4: Tuning Rules for PID Controllers 411
Rule
Prashanti and Chidambaram
(2000)-Alternative
tuning 1. Model:
Method I;
K (605) T ( 6 0 5 )
Td specified
on page 410.
Prashanti and Chidambaram
(2000) -Alternative
tuning 2. Model:
Method T, K (605) T ( 6 0 5 )
Td(605> specified
on page 410.
Kc
K (605)
T;
T (605)
Td
T (605) *d
Comment
0 - 0 5 < x m / T m
<0.90
Representative results 2
Fp = 0.309
Fp =0.281
Fp = 0.244
Fp =0.213
Fp =0.182
Fp =0.150
Fp =0.116
Fp = 0.080
Fp = 0.040
Fp =0.0091
K (605)
F i = l
F , = l
F i = l
F; = 1
F i = l
Fj = 1
F , = l
F i = l
F ; = l
F | = l
y {605)
Fd = 2.834
Fd = 2.605
Fd =2.413
Fd = 2.346
Fd = 2.336
Fd = 2.207
Fd = 2.093
Fd =1.884
Fd =1.508
Fd =1.159
T (605) Xd
x m / T m = 0 . 0 5
T m / T m = 0 . 1 0
t m / T m = 0 . 2 0
x m / T m = 0 . 3 0
t r a / T m = 0 . 4 0
x m / T m = 0 . 5 0
T m / T m = 0 . 6 0
i m / T r a = 0 . 7 0
x r a / T m = 0 . 8 0
t m / T m = 0 . 9 0
-6 .405T„ ,
Fp = 0.9726e Tm , F; = 1, Fd = 0 ; 0.05 < ^ - < 0.90 .
Representative results
FP = 0 . 7 7 7 , xjTm =0 .05
Fp = 0 . 4 5 7 , t m / T m =0 .10
Fp = 0 .227, x m / T m =0 .20
Fp = 0 . 1 2 1 , t m / T r a =0 .30
Fp = 0 . 0 7 8 , x m / T m =0 .40
Fp =0.046 , T m / T m =0 .50
Fp = 0 .026, x m / T m =0 .60
Fp = 0 . 0 1 3 , T m /T m =0 .70
Fp = 0.0057, x m / T m =0 .80
Fp = 0.0022, x m / T m =0.90
!Fn = 0.3127-0.3369-^- + 0.0378 ^ _ m \ *m J
2 f V - 0 . 0 4 5 ^
V m j
• F= = 1 .
FH =2.779-2.44-2!- + 5
/ ^2 x„
T V m J
-4.81 T
412 Handbook of PI and PID Controller Tuning Rules
Rule
Prashanti and Chidambaram
(2000). Model: Method 1
Prashanti and Chidambaram
(2000) -alternative
tuning. Model: Method 1
Kc
3 K (606)
Fp = 0.542
Fp = 0.437
Fp = 0.366
Fp = 0.303
Fp = 0.263
Fp =0.214
Fp =0.164
F„ =0.151
Fp =0.139
Fp =0.113
Fp = 0.775
Fp = 0.605 , T.
T i
rj, (606) i
Td
^ ( 6 0 6 )
Representative results 3=1 F i = l
F | = l
F j = l
Fi =1
F |= l
F;=l
F ,= l
F ,=l
F i = l
Fd =2.916
Fd = 2.878
Fd = 2.525
Fd = 2.307
Fd =1.948
Fd =1.729
Fd =1.530
Fd =1.297
Fd =1.093
Fd = 0.962
7-1.289^2-+ 0.6267 T
m
0.05 < ^ < T
m
T V xm J 0.90
2
Representative results
n l /Tm=0.05 Fp = 0.322 , T
Comment
xm /Tm=0.05
t m /T m =0 .10
^m /Tm=0.20
^m/Tm=0.30
xm /Tm=0.40
x r o /Tm=0.50
t r a /T m =0.60
xm /Tm=0.70
t m /T m =0.80
xm /Tm=0.90
1, Fd = 0 ;
m /T m =0.50
3 K C( 6 0 6 ) =—[2.121-1 .906^2- +0.945
Km T
f \ 2 t m I , - , (606)
T ' " •
T a
0.176 + 0.36-
a = 0.179-0.324-^2. +0.161 \ l^-\ , — < 0 . 6 a = 0.04 ,0.6 < 1^- < 0.8 ; T T T T
a = 0.12 - 0.1-^2-, 0.8 < - < 1.0; T T
Fn =0.5165-0.8482^2L +0.5133f T" -0.071 f ^
1M. T
; F i = l :
F d =3.206-3 .461^L+0.984p2- | +0.089 m V ' ,
f A3
V m J
Chapter 4: Tuning Rules for PID Controllers 413
Rule
Prashanti and Chidambaram
(2000) -continued.
Jhunjhunwala and
Chidambaram (2001).
Lee et al. (2000). Model: Method 1
Kc T i Td Comment
Representative results (continued) Fp =0.664, Tm/Tm =0.10
Fp = 0.550, Tra/Tm =0.20
FP =0.456, xra/Tra =0.30
Fp =0.395, i m /T m =0.40
4 K (607) T ( 6 0 7 ) * i
Fp = 0.248, xm/Tm =0.60
Fp =0.228, xm/Tra =0.70
Fp = 0.209, xm/Tm =0.80
Fp = 0.169, xra/Tm =0.90
T (607) Model: Method I
Robust 5 K (608)
F p = F i =
y (608)
1 as + 1
ry, (608) x d
T _ l T
m J
0<-^-<2 T
2
e t m / T m ,
4 K (607) = _J_[ 13.0-39.712-T" K „
-^-<0.2: T
K (607) 1.397
K „
2.044tm
, 0.2<-!2-<1.4; T
T(607) = 0.856Tme T- , 0 < ^ - < 1 ; T,'607' = 0.3444Tme T- , 1 < - ^ - < 1.4 ; | III ' rji ' I 111 ' rji
(607) _ . 0.5643— + 0.0075 T
, 0 < - 2 - < 1 . 4 : T
F i = 0 ; Fd = 0 ; F p =0 .6863 , - ^ - = 0.1 ;Fp =0.4772e T™ , 0.2 < - < 1. 1 m * TTI
T; (608)
K m ( 2 T C L + T m - a ) ". a = T n
(T XCL
T + 1 e ^ - 1
,(608) _ -T„ + a •
- T „ a - -, (608) _
TC L2+aT:m-0.5Tm
2
2TCL+Tm-CX
m2(0.167Tm-0.5a) 2 T a + T m - a TCL
2+aTm-0.5Tm2
T(608)
414 Handbook of PI and PID Controller Tuning Rules
4.11.4 Classical controller 1 Gc(s) = K
E(s) R(s)
®—K, + T
Table 159: PID controller tuning rules - unstable FOLPD model G m (s) = K „ e '
Tms-1
Rule
Shinskey(1988) - minimum IAE
-page 381. Method: Model I
N not specified
Kc
Minim 0.91Tra
Kmt r a
1.00Tm
K m x m
0.89Tm
Kmxm
0.86Tm
KmTm m m
0.83Tra
Km tm
Ti um performance
1.70Tm
1.90xm
2.00xra
2.25xm
2-40xm
Td
index: regulator 0.60xm
0.60im
0.80xm
0.90xm
1.00xm
Comment
tuning Kn/Tra| = 0.1
k /T m | = 0.2
k /T m | = 0.5
k /T m | = 0.67
k/T r a | = 0.8
Chapter 4: Tuning Rules for PID Controllers
4.11.5 Series controller (classical controller 3)
Gc(s) = K,
415
\+A"
Table 160: PID controller tuning rules - unstable FOLPD model Gm (s) = Tms-1
Rule
Huang and Chen (1997), (1999).
Kc Ti Td Comment
Direct synthesis: time domain criteria 6 ^ (609) y (609) T (609)
Xd Model: Method 4
0 < —— < 1; 'Good' servo and regulator response.
K (609) 0.5
Km
1 + 1.61519 T
(609),
. ^ (609) = K c KroTm [ 2 x l + 2 m .
V <609>K- 1 I T T ^-,- J^-m ~ 1 \ 1 r a ' m y
with recommended x, =Tm -1.3735.10"3 + 0.22091^2-+ 1.6653PS-T T
T (609) _ T 4 , n O'}c/ in m , ^ 1 £ m r \ _ n
1.5006.10"4 +0.23549—E2- + 0.16970I T V 1 m J
416 Handbook of PI and PID Controller Tuning Rules
4.11.6 Non-interacting controller based on the two degree of freedom structure 1
U(s) = Kc ' , 1 Tds ^
1 + — + - -v
TiS l + (Td/N>
f E(s)-K t
J a +
PTds
l + (Td/N>_ R(s)
a + -(3Tds
T
N j
K_
R(s) E(s)
* K , 1 + ^ + - ^ T ' S 1 + ^ s
N
U(s)
+
K e"
T s - 1
Y(s)
Table 161: PID controller tuning rules - unstable FOLPD model Gm (s) = Kme"
T m s - 1
Rule
Chidambaram (2000c).
Model: Method I
4 = 0.333; N = oo;(3 = l
Kc Ti Td Comment
Direct synthesis
1.165 (T ~\ 0.245
K m l T m J
7 j (610)
4.4Tm+9xra
8 T ( 6 U ) i
T (610) T
0.6 <
0.8 <
- <0.6
T
T m
<0.8
<0.9
7 j (610)
0.176 + 0.36-
0.179-0.324^2-+ 0.161 T
f A 2 ' d (610)
, T
0.176 + 0 . 3 6 - ^ T
T .
8 Ti(61,) = 0.176Tm+0.36xm; a = 0 7 4 + 0 4 2 I ^ _ 0 2 3
0.12-0.1^2- T™
/ \2
T V ' n /
Chapter 4: Tuning Rules for PID Controllers 417
Rule
Srinivas and Chidambaram
(2001). Model: Method 1 Chandrashekar et
al. (2002). Model: Method 1
Kc
9 K (612)
10 K (613)
T,
T ( 6 1 2 ) i
T ( 6 1 3 )
Td
T (612)
0.5Tra
Comment
N = oo
N = oo;|3 = 0
'Sample Kc(612), Tj(612) ,Td
(612) taken by the authors correspond to the dominant pole
placement tuning rules ofValentine and Chidambaram (1997b) (see page 405);
•r 1
• ; P = ,(612) K ( 6 , 2 , K (612) K (612)K 2X ( 6 1 2 )
10 ^ (613)
K „ 4282 - r -1334.6-^- +101
T , 0<-!S-<0.2;
T
K (613) 1.1161
K „ , T , V m J
, 0 .2<-S-<1 .0 . T x m
36.842 f N 2
, T •10.3
t T + 0.8288 0<JU2-<o.8;
T
T:(613) = 76.24IT ' Tn 0 . 8 < - ^ < 1 . 0 . T
a = 0.83, - ^ - < 0 . 1 ; a = 0.8053-0.1935-^-, 0.1<i2_<0.7 T T T
Desired closed loop transfer function =
0.0326 + 0.4958- -2.03
( r j s+_ l )e^ .
(TCL2S + 1)2 '
\2^ Tm , 0 < - ^ < 0.8 ;
T r l , =4.5115T T
, 0.8 < - 2 - < 1.0 T
418 Handbook of PI and PID Controller Tuning Rules
Rule
Chandrashekar et al. (2002).
Model: Method I
Wang and Cai (2002).
Model: Method I
Kc Ti Td 1 Comment
Alternative tuning rules
Kc = — , Tj = x2Tm , Td = x3Tm , Tf = x4Tm , TCL2 = x5Tm K m
Coefficient values x i
101.000 10.3662 5.3750 3.2655 2.4516 2.0217 1.7525 1.5458 1.3723 1.2420 1.1400
x2
0.0404 0.3874 0.8600 1.8018 3.0400 4.6500 6.7286 10.337 17.693 33.610 80.620
11 K (614)
x3
0.0 0.0435 0.0884 0.1375 0.1868 0.2366 0.2866 0.3380 0.3910 0.4440 0.4969
x4
0.0 0.0134 0.0250 0.0435 0.0581 0.0696 0.0784 0.0885 0.1005 0.1124 0.1247
T (614) M
X 5
0.02 0.10 0.20 0.40 0.60 0.80 1.00 1.30 1.80 2.60 4.20
a
0.505 0.742 0.767 0.778 0.803 0.828 0.851 0.874 0.898 0.923 0.948
T (614) 1 d
* m / T m
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
N=0; p = 0
(614) 0.5236Tm 0.4764 Pf^ T (6I4) _ 0.5236Tm - 0 . 4 7 6 4 ^ - I ) A; , -
Chapter 4: Tuning Rules for PID Controllers 419
Rule
Xu and Shao (2003b).
Model: Method 1 Sree and
Chidambaram (2005a).
Model: Method 1
Kc
12 K (615)
13 K (616)
Ti
T (615)
j ( 6 1 6 ) i
Td
T (615) *d
T (616) ' d
Comment
N=0; P = 0
N=oo; p = l
12 K (615) _ 0-5
Km
T T 1 m I ' i f
/
T (615)
T IT
(615) 0.5/L
Tm |T„ A ' a =
2 ^
Tm+VT;
mTm ( A r a > 2 , 4 > m > 6 0 ° ) .
13 Sample Kc(616), Tj*6'6' ,Td
<616) taken by the authors correspond to the tuning rules of
Chandrashekar et al. (2002) and Huang and Chen (1999); the latter tuning rule is defined for an unstable SOSPD process model.
iQ-qfc^s + ilr"-Desired closed loop transfer function =
TCL2s2+2£TCLs + l
The optimum value of a is chosen by minimizing the ISE (servo response). Then,
a = l - -
a = l-
2TCL .(616)
, £,= 1; a = l-2TCL5 T (616) , S < 1 ;
.(616)
^+^~i)(-^-^~i)2 J-z+fiT^ + x,
^+Vi^T -^-V^T +x 4>i
with x, =-(-$-fi*~Tj + ^ + A /^T7j ^ _ ^ T 7 j and
x^^-^T^-^V^J^^+V^J^-V^ If it is desired to minimize the overshoot (servo response), for £, > 1, then a = 1.
420 Handbook of PI and PID Controller Tuning Rules
4.11.7 Non-interacting controller 3
U(s) = Kc
V T i s / E ( S ) - ^ £ Y ( S )
N R(s)
E(s) K „
v T i s y
U(s)
- * R > >
KcTds
1 + ^ s N
K e"sx™
T „ s - 1
Y(s)
Table 162: PID controller tuning rules - unstable FOLPD model Kme"
T m s -1
Rule
Huang and Lin (1995) -
minimum IAE. Model: Method 2
Kc T: Td Comment
Minimum performance index
1 K (617)
2 K (618)
-p (617)
rj, (618)
T (617)
T (618) Xd
0.01 <
r>
T
[=10 <0.8;
Servo response: K (617)
K „ - 0.433 + 0.2056-2-+ 0.3135
T
,(617) = T • 0.0018 + 0.8193-^- + 7.7853 T
6.6423 \
(617) = T
v
T 7.6983-^
0.0312 + 1.6333-SL + 0.0399e T™ T
m J
• Regulator response: Kc(618) = 0.2675+ 0.1226-=i- +0.8781
T
-1.004 A
. (618) 0.0005 + 2.4631 -S- + 9.5795
T i(618)
= 0.001 lTm+0.4759x„
Chapter 4: Tuning Rules for PID Controllers
4.11.8 Non-interacting controller 8
421
f U(s) = Kt
1 \
v T ; s j
E(s)-K i( l + Tdls)Y(s)
R(s) E(s)
-*8h* K + f
+ x-x U ( S )
+ Tds) — • ( X ) • K e"
T „ s - 1
KiO + V )
I
Y(s) - •
Table 163: PID controller tuning rules - unstable FOLPD model Gm (s) = Kme~
T m s-1
Rule
Park et al. (1998).
Model: Method 1
Kc Ti Td Comment
Robust
3 K (619) T(619) T (619) K _ i | T 7
' Km Vtm
T d i =0
3 Apply minimum ITAE - regulator tuning rule or minimum ITAE - servo tuning rule defined by Sung et al. (1996) for SOSPD model, ideal PID controller (pages 313, 316). For these formulae, Km , Tml and £m are replaced by the following parameters from the unstable FOLPD model:
K „ • (Unstable FOLPD) -> Km (SOSPD)
- 1
A 7 1 X 0.25 0.75
0-71Tm xm
V T A05 (Unstable FOLPD) -» Tml (SOSPD)
„ 7 , L 0.5 0.5 L 0.25
rT \0-5 • (Unstable FOLPD) -> %m (SOSPD)
- 1
422 Handbook of PI and PID Controller Tuning Rules
Rule
Lee and Edgar (2002).
Model: Method 1
Kc
4 K (620)
T;
y (620)
Td
x (620) l6
Comment
T d i =0
V <620>l<r 1 4 K (620) _ K i K-m _ 1
Km(^ + x J T K ( 6 2 0 ) K - T
X T v (620) „
(620) _ l m - K - i K - m X „
(620)
K i( 6 2 0 ) K m - l 2 ( X + T J
K m ( ^ O ^ K , < 6 2 0 ) K m - l ) 6(X + x J 4(X + T m ) 2
2(K/-»Km-lJ(X + xJ}'
K, (620)
K „ T V m /
1H.+ In f \
T V 1 m J
1 - ^ T ,
V m y
Chapter 4: Tuning Rules for PID Controllers
4.11.9 Non-interacting controller 10
U(s) = K( v T i s y
E(s)-K f
f T ^
VTm J
423
Y(s)
R(s) E(s)
+ X K,
V T i S /
+ xg> U(s)
K„e-"-
Tms-1
Y(s)
Kf -^-S + l VTm J
T Table 164: PID tuning rules - unstable FOLPD model Gm(s) =
Kme"
T m s -1
Rule
Majhi and Atherton (2000)
- minimum ITSE Majhi and
Atherton (2000) -minimum ITSE.
Model: Method 3
Kc T; Td Comment
Minimum performance index: servo tuning
5 K (621)
6 K (622)
T (621) i
T (622)
0.5tm
ln(l + K)
4tanh-'K m
Model: Method 1
0 < ^ - < 0.693 T
f 2
0.1227+ 1.4550^2--1.271 l ^ r -1 2T
_ 0.801 l(l - 0.9358VRT^)) J _ O Z , K = Ap/(Kmh), Kraln(l + K) KraVln(l + K)' p/V
(622) 0.1227 + 1.4550 ln(l + K)-l.271 l[ln(l + K)f
' 2tanh- 'K
424 Handbook of PI and PID Controller Tuning Rules
Rule
Majhi and Atherton (2000)
- continued.
Kc
7 K (623)
T,
T(623)
Td
T (623) x d
Comment
0.693 <-^2-<1 T
7 K (623)
K „
0.801 l(K + 0.9946) 0.7497VK + 0.9946
K +0.0682 VK + 0.0682
, (623) _ 0.0145K3 + 0.5773K2 + 2.6554K + 0.3488 ,
K3 + 5.2158K2 + 4.4816K + 0.2817
, (623) = 0.0237(K +34.5338),
" K + 4.1530 Kf
_1_ 12(K +0.9946)
KV K +0.0682
Chapter 4: Tuning Rules for PID Controllers 425
4.11.10 Non-interacting controller 12 f
U(s) = Kc 1 \
1 + — + Tds v T i s j Tds + 1
E(s)-K,Y(s)
R(s) E(s)
(1+ +Tds) T;S Tds + 1
+ U(s) K ^ e -
T m s - 1
Y(s)
K,
I Table 165: PID controller tuning rules - unstable FOLPD model Gm (s) =
Kme~
T „ s - 1
Rule
Lee and Edgar (2002).
Model: Method I
Kc T i Td
Comment
Robust
8 K (624) T (624) i
T (624)
8 K (624) _ ^ 1 K , | a \ „ - l T - K , « K „ i „ 1 m 1 m ''tn
Km(^ + x J K, ( 6 2 4 ) K m - l 2(X + xm)
K ( 6 2 4 ) K 2M | ^ , (Tm-K, ( 6 2 4 )KmTm)
2(K, ( 6 2 4 , K m - l ) 6(X + xm) 4(X + T J 2 2(K 1( 6 2 4 ,Km- l ) (^ + Tm)
T V ( 6 2 4 ) f T . (624) _ J m ~ ft-1 K m T m , l m
K,<624»K - 1 2(^ + tn
K,(624,K T m T „ ( T m - K , Kmxm)
2(K, ( 6 2 4 )Km - l ) 6(^ + xm) 4(X + Tm)2 2(K, ( 6 2 4 ,Km - l ) (^ + Tm)
(624)
(624), { ( T m - K , ^ K m x m )
2(K1< 6 2 4 )Km-l) 6(^ + xra) 4(^ + Tm)2 2(K1
(624)Km-l)(A. + Tm)
K (624)
K „ cos J 1 m \ m , m
T
f \ 1 —2HL ^m
T T
426 Handbook of PI and PID Controller Tuning Rules
Rule
Lee and Edgar (2002).
Model: Method 1
Kc
9 K (625)
Ti
T (625)
Td
T (625)
Comment
K, =0.25KU
9 K (625) _ l + 0.25KuKm (Tm -0 .25K uKmxm
Km(X + xJ l+0.25KuK ra 2(X + TJ
0-25KuKm _ xm + xj + (T m -0 .25K u K m xJx m2
.(625) _ i n
2(l + 0.25KuKm 6(X + xm) 4(X + xm)2 2(1+ 0.25KuKm)(k + xm)
2
+ Tm -0 .25K uKmxm + x l + 0.25KuKm 2(X + xm)
0.25KuKm (Tm-0.25KuKmxm)xn
2(l + 0.25KuKm 6(X + xm) 4(X + xm)2 2(1+ 0.25KuKm)(^ + xm)
(625)
0.25KuKm (Tm-0.25KuKmxm)T„ 2(l + 0.25KuKm 6(X + xm) 4(X + xm)2 2(1+ 0.25K uKJ(^ + xm)
Chapter 4: Tuning Rules for PID Controllers All
4.12 Unstable SOSPD Model (one unstable pole)
Gm(s) = * * e
(Tmls-l)(l + sTm2)
4.12.1 Ideal controller G c(s) = Kc
R ( s ) > ( C > E ( s )
+ K,
v T i s
1 + ^ - + Tds
U(s) K e~ (Tmls-lXl + sTm2)
Y(s)
Table 166: PID controller tuning rules - unstable SOSPD model (one unstable pole)
Kme"sx» Gm(s) =
(Tmls-lXl + sTm2)
Rule
Wang and Jin (2004).
Kc Ti Td Comment
Direct synthesis 1 K (626) T (626) „ (626) Model: Method 1
1 v (626) _ K m T i ( T m l ~ T m X T m 2 + T m )
(T C L3+t m ) 3 K „
TCL3 + 3xmTCL3 + 3(TmlTm2 + xmTml - xmTm2 JTCL3 + TmTm2 (Tml - xm j .(626) 1 CL3
. (626) _ (Tm2 +Tm ~TmiJTCL3 + 3TmlTm2TCL3(TCL3 + xmJ + TmlTm2xn
428 Handbook ofPIandPID Controller Tuning Rules
Rule
Rotstein and Lewin(1991).
Model: Method 1
Lee et al. (2000). Model: Method I
Kc Ti Td
Robust 2 K (627) T (627)
i T (627)
X determined graphically - sample values provided Tm/Tm=0.2,^e[0.5Tm ,1.9Tm]
T m /T m =0.4Ae[1 .3T m ,1 .9Tj
Tm /Tm=0.2^e[0.4Tm ,4 .3Tm]
xm/Tm=0.4,A.e[l.lT in>4.3Tm]
i m /T m =0.6 ,X6[2 .2T m , 4 .3TJ
3 ^ (628) T (628) i
T (628) ld
Comment
Km uncertainty
= 50%
Km uncertainty
= 30%
0 < ^ - < 2 T
m
(627) _ . — + 2 | + Tm2>
V ' m l
3 £ (628) . (628)
•Km(2X + T m - a )
(627) VTml J
M— + 2 +Tm
, X = desired closed loop dynamic parameter,
a = T . 1 e, - / T - ' - l
T(628)__T _ X1 + axm -0 .5 i m2
, (628) -Tmia + Tm2a-Tm]Tm2
2X + xm - a
xm2(0.167Tm-0.5a)
n(628)
2X + zm-a X2 + axm-0.5xm2
2X + x„ - a
Chapter 4: Tuning Rules for PID Controllers 429
Rule
McMillan (1984).
Model: Method I
Kc T| Td Comment
Ultimate cycle
4 K (629) T(629) ~ (629) Tuning rules
developed from KU,TU
4 ^ (629) 1.111 Tm,Tm
K m T 1 +
lTm| +T m 2 / r m l T a
T < 6 2 9 , = 2 x j l +
Td( 6 2 9 )=0.5xm l +
_ vTml Tm2 ATmi tm/^mj
(Tmi + Tm2JTmlTm2
\Jm\ Tm2XTmi ^m^m.
(Jm! +Tm2jTmlTm2
\}m\ Tm2j(Tml Tm/C„
430 Handbook of PI and PID Controller Tuning Rules
4.12.2 Ideal controller in series with a first order lag r
R(s) - 1 f
Gc(s) = Kc
U(s)
1 \ 1 + — + Tds
V T i s J Tfs + 1
+ <§>* Kr
1 A 1 + — + Tds
V T i s J
1
(Tfs + l) K„e~
(Tmls-lXl + sTm2)
Y(s)
Table 167: PID controller tuning rules - unstable SOSPD model (one unstable pole)
K m e-"" Gm(s) = (Tmls-lXl + sTm2)
Rule
Zhang and Xu (2002).
Model: Method 1
Kc T, T, Comment
Robust 5 K (630) T (630) T (630)
1d *m < T m l
5 j , (630) . Tm2Tmi(Tmi - x J + X3 +3X2Tm, +3XTJ+T Tm
K m (^+3^ 2 T m , +3XTm ,xm+xm2Tm l)
T (630) _ T M - 1m2 +
ml v ml * m m * ml j
ml ~ r-^"Tml + T m T m l ^3+3X.2T , +3VTm,2+T T, 2
^mllTmi XraJ
T,
(630) Tm2(X, + 3X Tmt + 3XTml +TroTm, )
TmlTm2(Tml - xm)+ X3 + 3X2Tml + 3XTml2 + xmTra
(630) _ ^ TmlVml ~ T m )
Tm l(^+3X2Tm ,+3XTm lTm+Tm2Tm l r
f o r ^ < 0 . 5 , A = 2 - 5 ( t n + T m 2 ) .
Chapter 4: Tuning Rules for PID Controllers 431
4.12.3 Ideal controller with set-point weighting 1
U(s) = Kc (FpR(s) - Y(s))+^(F,R(s) - Y(s))+ KcTds(FdR(s) - Y(s)) T:S
Table 168: PID controller tuning rules - unstable SOSPD model (one unstable pole)
„ . , K m e ^
Rule
Lee et al. (2000). Model: Method 1
u m V
Kc
y~(Tmls-lXl + sTm2
Ti
)
Td
Robust 6 j r (631)
Fp=Fi =
-p(631)
1 f d - , , ' a
as + 1
T (631)
Tm, V ' m l J
Comment
0 < ^ - < 2 Tm
- / T - , _ !
6 K (631)
.(631)
, (631)
Km(2^ + x m - a ) , X = desired closed loop dynamic parameter,
• i r " = - T m l + T n 2 + a - ; X +a,Tm - 0 . 5 T
m " • " " m
- T m l a + T m 2 a - T m l T m 2 -
2X, + xm - a
xm2(0.167xm-0.5a)
(631) 2>. + x m - a X1 + a t m -0 .5x n2
vm m - (631) 2X + xm - a
432 Handbook of PI and PID Controller Tuning Rules
(
4.12.4 Classical controller 1 Gc (s) = Kc
R(s) E(s) K,
f P 1 + -Xs
1 + Tds
l + ^ - s V N ,
v T i s y
1 + Tds
l + -*-s N
U(s)
H K „ e -
(Tmls-lXl + sTm2)
Y(s)
Table 169: PID controller tuning rules - unstable SOSPD model (one unstable pole)
Gm(s)= K ^ e
(Tmls-lXl + sTra2)
Rule
Minimum ITAE - Poulin and Pomerleau
(1996), (1997).
Process output step load
disturbance
Process input step load
disturbance
Kc ^
Minimum performance 7 K (632)
X m + T m 2
Tml
0.05 0.10 0.15 0.20 0.25 0.05 0.10 0.15 0.20 0.25
T (632)
Td Comment
index: regulator tuning
Tm2 Model:
Method 1 Coefficient values (deduced from graph)
x i
0.9479 1.0799 1.2013 1.3485 1.4905 1.1075 1.2013 1.3132 1.4384 1.5698
x2
2.3546 2.4111 2.4646 2.5318 2.5992 2.4230 2.4646 2.5154 2.5742 2.6381
Tm+T r a 2
Tml
0.30 0.35 0.40 0.45 0.50 0.30 0.35 0.40 0.45 0.50
x i
1.6163 1.7650 1.9139 2.0658 2.2080 1.6943 1.8161 1.9658 2.1022 2.2379
x2
2.6612 2.7368 2.8161 2.9004 2.9826 2.7007 2.7637 2.8445 2.9210 3.0003
X l '-ml
7 K (632)
K r a(x 1Tm l-4[Tm+T, ^\Xm + ^ml} T (632) 4Tm l(xm+Tm 2) _
x , T m l - 4 ( T m + T m 2 ) !
0< (T„/N)
^2;0 .1T m 2 <-^-<0.33T m 2
Chapter 4: Tuning Rules for PID Controllers
4.12.5 Series controller (classical controller 3)
433
Gc(s) = K( 1 + — v T i s y
R(s)
—i E(s)
K 1 + v T i sy
(l + sTd)
U(s) K „ e "
(Tmls-lXl + sTm2)
(1 + Tds)
Y(s)
Table 170: PID controller tuning rules - unstable SOSPD model (one unstable pole)
K m e-"-Gra(s) =
(Tmls-lXl + sTm2)
Rule
Chidambaram and Kalyan
(1996). Huang and Chen (1997), (1999).
Model: Method 4
Ho and Xu (1998).
Model: Method 1
Kc
8 ^ (633)
9 K (634)
Ti Td Comment
Direct synthesis: time domain criteria
T (633) i
T (634) i
Tm2
Tm2
Model: Method I
'Good' servo and regulator response
Direct synthesis: frequency domain criteria fflpTml
AmKm
0.7854Tml
K m T m
10 T (635) Tm2
Representative result
"Tml Tm2 A m = 2 ,
4>m=45°
8 Kc(633) and T;
(633) are the proportional gain and integral time defined by De Paor and O'Malley (1989), Rotstein and Lewin (1991) or Chidambaram (1995), for the ideal PI control of an unstable FOLPD model [see Table 31].
9 K (634) 0.5 1 + 1 . 1 1 4 1 6 ^
(634), T (634) _ Kc KmTro I 2xt T^
' ' ~ K c( 6 3 4 > K m - l l T m T m y
Recommended x, =0.160367Tme T-' ; 0 < - 2 - < 0.8 . Tml
•57cop-cop xm-
434 Handbook of PI and PID Controller Tuning Rules
4.12.6 Non-interacting controller 3
R(s) E(s)
Kr
U(s ) = K
U(s)
v T i s / E(s)-i^Y(s)
i3 N
" ®— K e"
(T r a ls-lXl + sTm 2)
Y(s)
KcTds T
1 + ^ - s N
Table 171: PID controller tuning rules - unstable SOSPD model (one unstable pole)
K „ e - " » Gm(s) =
(Tmls-lXl + sTm2)
Rule
Huang and Lin (1995)-
minimum IAE.
Kc T, Td Comment
Minimum performance index: regulator tuning
1 j r (636) j (636) i
T (636) x d
Model: Method 2;
N not specified
Note: equations continued onto page 435. 0.05 < Tm/Tml < 0.4 ; Tm2 < T ra l .
K (636) 1 Km
K „
Kn
- 174.167 - 3 1 . 3 6 4 - 2 - + 0.4642 Tml VTmiy
-103.069-^-
ST. \ 2 M
-83.916 V T m l )
- 66.962 IEILSL + 59.496 i k i s L T ml /
3n_ i m T m 2 T m
. 7 0 7 9 im2_ + 2 3 121eTml +126.924eT"" +26.944e T""' . T „
. (636) 0.008+ 2.0718-2-+ 6.431 f \ 2
T v 1miy
+ 0.4556-E2- + 0.7503
2.4484 T m 2 T m -18.686 T 1ral
V ^ m l
-2.9978 i m 2
V ^ m l
-21.135
VTml J
1 m 2 x m
V ' m l J
Chapter 4: Tuning Rules for PID Controllers 435
Rule
Huang and Lin (1995)-
continued.
Kc
2 K (637)
Ti
- (637) i
Td
T (637)
Comment
N not specified; Model: Method 2
+ T„
+ T„
12.822 T m ' m 2
T 3
V 1ml J
+ 39.001 ' t n ^
VTml J + 22.848
c T 3^ J m 2 T m
T 4
•4.754 rT 3 A
x m2 T m
T "
-0.527 f T 2 2 ^
' m 2 T m
T " + 1.64
VTml 7
(636) = T / \
1.1766-2!--4.4623 Tml V T m i y
,T, + 0.5284-522--1.4281
T „ VTmi y
+ Tm 6Tm2T +nA7J2js_) + L 0 8 8 6 I s l
-0.0301-0.5039
Tm,
/ T 2
T 3
v 1mi y
l T m l
-9.8564
- 5.0229 fT 2 T
T m I m 2
V Tml x„
V^ml -7.528
/ T 3
T " V 'mi y
+ T„ -2.3542 ' x T 3
+ 9.3804 ( 2 T 2 A
T m ^ 2
T 4
V xmi y
-0.1457 f \ 4
T v 1mi y v 1mi y
: Note: equations continued onto page 436. 0.05 < t r a /Tm l < 0.25, Tml < Tm2 < 10Tml ;
K (637) 1 K „
K „
K „
1750.08 +1637.76-=- + 1533.91 VTmi y
-7.917-
6.187 T
v 1mi y
- 6.451 Tm2T,ro +0.002452 T v L i
L3729-sL-l739.77eTml -0.000296eT"" +2.311e T""
j (637) _ T
+ Tm
51.678 - 57.043-SL- +1337.29 Y T
- ^ H + 0 . 1 7 4 2 - 2 ^ - 0.1524
7.7266-. T m , x „
, - 6 0 1 1 . 5 7 — 2 2 I j
T
+ 0.0213
V ml
-9135.52
436 Handbook of PI and PID Controller Tuning Rules
Rule
Huang and Lin (1995)-
minimum IAE.
Kc T i Td Comment
Minimum performance index: servo tuning
3 K (638) T ( 6 3 8 ) i
x (638) ' d
N not specified; Model: Method 2
-65.283 -0.0645 f T 2 \
T m 1 m 2
V ml J
+ 274.851 V Aml J
T T 1 m 2 '•m
T V ' m l J
+ T„
+ T„
0.003926 • m l "Cr.
I T, 4
ml J
-2.0997 fT 2x 2
' m 2 x m T "
V ml J
-0.001077 Tm2
l T , ml J
-49.007eT"" + 0.000026eT"" + 0.2977e ^
T (637) _ T
' d _ ' m l
+ Tm
- 0.0605 + 4.6998^2- - 29.478 — + 0.0117^ TmI lTm
-0.01291-^=^-] + 0.6874 Tm2\m +140.135 T
/ A3
t,
V T m l J
+ T„
+ Tml
0.002455 fT ^ 3
-22-1 +1.4712 V ml
( 2T A X m ' m 2
T J
V ' m l J
-0.1289 ^x T 2^
l m 1 m 2
T 3
V ' m l J
-238.864-^2- +0.6884 l T m l
i m 2 T m
T 4
V ' m l J
-0.007725 i T i fam m2
T 4
V ' m l J
+ T„ -0.1222 f \ 2T 2
T V ' m l J
-0.000135 -Tm2 V
V ml ,
1 Note: equations continued onto page 437. 0.05 < xm/Tml < 0.4 , Tm2 < Tml: -1.344
K (638) . 1
K „
K „
10.741 - 13 .363— + 0.099 — + 727.914—s^-
-708.4811-^21 T,
0.995 T m , + 9.915 Tm2T,m +84.273
T ml J ' m l
' m l 1.031/ N 0.997
1 ^
V ^ m l
- 90.959-2^+ 9.034eT"" -2.386eTml -16.304e T""2
xm
Chapter 4: Tuning Rules for PID Controllers 437
.(638) -149 .685-141 .418- ,717 ' T I T
17 29 T T
V 'ml
+ Tm
+ T„
-12 .82 T m 2 T ,m +3.611 Tm l
2
/ N0.286
V^ml 7 V 'ml
(638)
20.5181 ^ 3 ^ -
l T m , J T m T „ , ; T m ; T m
0 .000805-2^ +141.702eTml -2.032eT"" +10.006eT""2
• 0.4144 + 1 5 . 8 0 5 ^ - - 142.327( - ^ - I + 0 . 7 2 8 7 ^
+ Tm
+ Tm
+ Tm
+ Tml
+ Tm
0.1123
Tm,
-18.317 T m 2 T ,m +486.95 Tm l
2 1SL) _io.542ps2-T
V 'ml
204.009 ( i 2 T
t m '1112 T 3
V 'ml J
+ 41.26 f T 2
Tm *-ml T 3
V 'ml J
-138.038 fx 3 T >*
xm 'm2 T
V 'mi y
+ 52.155 Tm'm2
+ 396.349
3 ^
VTml J
- 646.848 (^ 2 T 2^ lm 'm2
J m l
-289.476
19.302|^22_] - 4 7 3 1 . 7 2 | ^ - | +425.789
V 'ml J
5
T V 'ml 7
Tm im2
TmTm2
T 5
V 'ml 7
-841.807 ^x 3T 2 ^
'm 'm2 T
V 'ml 7
V ml J
+ 1313.72 f. 2T 3A
T m 'm2
T V 'ml J
-3.7688 1m2 T
V 'ml /
( , \ + 6264.79
VTml ) -161.469
Tm 1m2 ry 6
V Xml J
204.689
+ T„ 648.217
f T 5^ Tm ^m2
V Tml J
Tm ^m2
T V 'ml J
+ 25.706
-5.71:
f . 4_. 2 A Xm 'm2
V ^ i J -791.857
r T 2 T 4 \ Lm 1m2
438 Handbook of PI and PID Controller Tuning Rules
Rule
Huang and Lin ( 1 9 9 5 ) -
minimum IAE -continued.
Kc
4 K (639)
Ti
T (639)
Td
T (639)
Comment
N not specified; Model: Method 2
1 Note: equations continued onto page 439. 0.05 < x m / T m l < 0.25 , Tm l < Tm 2 < 10T m l ;
r
K (639) 1
K„
K„
-130.2 + 8 5 . 9 1 4 - 2 - + 34 .82
s -, -0.3055 X „
Tml
+ 10.442 ^ L
•22.547
(» \°-5174 Xm2
V ^ m l J -14.698 Tm2T
2m + 5 2 . 4 0 8 1 ^ -
'ml VTn
(T \
Tml J
- 5 1 . 4 7 (T
s 2 - +53.378eT"« - 0.00000 leT"" +0.286e T™i
, (639) = T 72.806 - 268.146^-- 4.922l[ ^ 2 . Trai J
+ 2468.191 - a . ml J
+ T„
+ T„
+ Tm,
+ T„
+ T„
0 .6724 {j \ 2
VTml J +151.351 Tm2\m -6914 .46 'O*
V T m l J
- 795.465 fx 2 T ^
l m 1 m 2 •14 .27
V 1ml J
T T i ^m x m2 2\
1417.65 x 3 T ^
+ 0 .4057
V 1ml J
T
V 1ml J
X T , 3 ^
+ 5 5 8 0 . 1 7
- 0 . 0 0 9 2 / T "\3
1m2
VTml J
Tml
+ 55.536
V 1ml J
fx 2 T 2 ^ '•m xm2
T " V 'ml J
-0.001119J ^ina_ J _44.903eT"" + 0.000034e T"" -15 .694e T""'
678778 V T m l , V T m l J
T (639) _ ~ Xd - ' m l
+ T „
175.515-86 .2-^2- +348.727 Tm,
( -,1.1798
- ^ - 1 - 0 . 0 0 8 2 0 7 ^ -V ^ m l
- 5 5 . 6 1 9
f NO.1064
_JH2_ + 0,0418 Tm2\m +78.959 VTmlJ Tm, V T m l J V T m l y
Chapter 4: Tuning Rules for PID Controllers
4.12.7 Non-interacting controller 8
439
/ U(s) = Kt
1 \ 1 + — + Tds
V T i s J
E^-K^ l + VJYCs)
R(s) E(s)
K, 1+-J- + Tds V T i S J
+ U(s)
K^e""-
(Tmls-lXl + sTm2)
Y(s)
Ki( l+T d i s )
Table 172: PID controller tuning rules - unstable SOSPD model (one unstable pole)
Gm(s) = (Tmls-lXl + sTm2)
Rule
Kwak et al. (2000).
Model: Method 3
Kc T i Td
Comment
Direct synthesis 5 j , (640) j , (640) T (640) Optimal gain
margin: regulator response
+ Tm 0.005048 'T . ^
-187.01eT"" + 0.000001eT"" -0.0149e T""2
T T 5 Note: equations continued onto page 440. 0 < —— < 1 — —
K (640)
K „ -0.67 + 0.297
, s-2.001
VTml J + 2.189
, ^-0.766
T -<0.9 or
K (640) _ 1
Kn
- 0.365 + 0.260 - i s - -1 .4 VTml J
+ 2.189 V^ml ^ 5» ' T
•>0.9;
,(640) _ . ( - "\
2.212 T
-0 .3 <0.4 or
440 Handbook of PI and PID Controller Tuning Rules
Rule
Kwak et al. (2000) -
continued.
Kc
6 K (641)
Ti
T (641)
Td
T (641)
Comment
Optimal gain margin: servo
response
T^u, = T m i {_o.975+ 0.910 V ' m l
-1.845 + a} , -J2- > 0.4 with T 1 m l
1-e 5.25-0.: V ^ m l
--2.8
(640)
^ K, 1.45 + 0.969 A T A 1 1 7 1
-1.9 + 1.576 l m l J
x,+x2-^-+x3
2 ^
Tml with
/ \ 2 x
X, =-0.0030 + 0 . 6 4 8 2 - ^ ^ - 2 . 2 8 4 1 ^ + 2 . 6 2 2 1 ^ Tmi ^ Tml J ^ T,
-0.9611 ml y V T m l y
X, = 0.2446 - 1 . 0 4 1 0 - ^ -13.6723 ^
X, = 0.1685+ 0.8289-^2--9.3630
-16.76221-5s2-1 +5.1471 ^=2.
^22-) +2.9855 V ^ m l
f'r- ^ 3
T V ml j
V ml j
+ 7.3803 ^=2. l T m l J
K: = A/|Gm(jcou)(l + jTdlcou)|Kn
T T 6 Note: equations continued onto page 441. 0 < —— < 1 — . T m i
K (641)
Kn
-0.04 + 0.333 + 0.949 T,
•> -0 .983
ml J
lm < 0.9 or
Chapter 4: Tuning Rules for PID Controllers 441
K (641)
K „ -0.544 + 0.308 —
Tmi
+ 1.408 /
. lm > 0-9 .
T^640 =Tm][2.055 + 0.072(Tm/Tml)fem, xm/Tml <1 or
T^641' =Tm l [1.768 + 0.329(xra/Tml )]^m , xm/Tral > 1 ;
T (641) Tml .
1-e
X, + X 2 :
0.870
x.
0.55 + 1.683
2^
Tml
T f T X, =-0.0030 + 0.6482-2^- -2.2841| ^2
Trai
-2.6221 'T , ^
-0.9611
X, = 0.2446-1.0410-^-13.6723 Tml
( -Y ^2
T 16.7622
V^ml
+ 5.1471 V^ml
X, =0.1685 + 0 . 8 2 8 9 ^ - -9.3630 \ — \ + 2 . 9 8 5 5 - ^ - + 7 . 3 8 0 3 - ^ . l T . ml / \ ml j
442 Handbook of PI and PID Controller Tuning Rules
4.13 Unstable SOSPD Model (two unstable poles)
Gra(s) = K „ e "
(Tmls-lXTm2s-l)
f 4.13.1 Ideal controller G (s) = Kc
1 \ 1 + — + Tds
-+6 TS a E ( s ) >
< Kc
f 1 1 1 + — + Tds
I T i s J
U(s) ^ K r a e - »
(Tm,s-lXTm2s-l)
Y(s)
— w
Table 173: PID controller tuning rules - unstable SOSPD model (two unstable poles)
Kme-sx™ Gm(s) =
(T r o ls-lXTm 2s-l)
Rule
Wang and Jin (2004).
Kc Ti Td Comment
Direct synthesis 7 K (642) T (642)
i T (642) Model: Method 1
7 jr (642) _ K m T i (Tml ~ Tro XTm2 ~ Tm )
• (642)
. (642)
( T C L 3 + ^ ) 3
= TCL33 + 3 T m T CL3 2 + 3 ( T m , T m 2 + t m T m l - T m T m 2 ) T C U + T m T m 2 (T m l - T m )
\Tm l ~ ^m ATra ~ T m 2 )
= (Tm2 + Tm ~ Tml )^CLZ + 3TmlTm2TCL3 (TCL3 + Tm ) + TmlTm21:m2
T r ' ^ - t j T ^ - x J
Chapter 4: Tuning Rules for PID Controllers 443
Rule
Lee et al. (2000). Model: Method 1
Kc
8 j r (643)
T( Td
Robust T (643) T (643)
Xd
Comment
0 < ^ - < 2 T
K (643)
.(643)
Km(4X + x m - a , ) X = desired closed loop dynamic parameter.
.(643)
(643)
6X2 - a 2 + a!Tm -0.5xm
4X + tm - a .
a 2 - ( T m l + T m 2 ) a , + T m l T , 4X 3 +a 2 x m +0 .167 t m
3 -0 .5a 1 t m2
4X + T m - a , • (643)
6X2 - a 2 +a,Tm -0.5xm
with cC|,a2 values determined by solving 1 (q2s2 + oc1s + l);
(Xs + l)4 _j L = 0 .
444 Handbook of PI and PID Controller Tuning Rules
4.13.2 Ideal controller with set-point weighting 1
U(s) = Kc (FpR(s) - Y ( s ) ) + ^ f a R f s ) - Y(s)) + KcTds(FdR(s) - Y(s))
Y(s)
Table 174: PID controller tuning rules - unstable SOSPD model (two unstable poles)
Kme-S^ Gm(s) = (Tmls-lXTra2s-l)
Rule
Lee et al. (2000). Model: Method 1
Kc
9 j £ (644)
F P = F .
1
Ti Td Comment
Robust T (644) T (644)
0< — < 2 T
m
a2s +a , s + l
( a 2 s 2 +a , s + l ) e - v i , , 0
frs + l)4 ls=^^7
9 K (644) , (644)
Km(4A, + T m - a , ) X = desired closed loop dynamic parameter.
, (644) = -Tmi - T r a 2 + a , 6X2 - a , + a , t m -0 .5t 2
4X + x m - a ,
« 2 - ( T m l + T m 2 ) a l + T m l T , (644)
4 ^ + a 2 x m + 0 . 1 6 7 T mJ - 0 . 5 a , T n
4A, + T m - a 1
, (644)
6X. 2 -a 2 +a |T m -0 .5T m2
4^ + T r a - a ,
Chapter 4: Tuning Rules for PID Controllers 445
4.14 Unstable SOSPD Model with a Positive Zero
Gm(s) = K m ( l - T m 3 s ) e - " "
T m l V - 2 S m T m l s + l
4.14.1 Ideal controller in series with a first order lag f - \
R(s) E(s)
+ < & * K
Gc(s) = K(
U(s) v T i s J
1
1 \ 1 + — + Tds
v T i s j Tfs + 1 > Km(l-Tm3s)e-
2 „ 2 T m ,V-2^ m T m l s + l
Tfs + 1
Y(s)
Table 175: PID controller tuning rules - unstable SOSPD model with a positive zero
K m ( l - T r a 3 s ) r ^ Gra(s) =
TmlV-24mTm ,s + l
Rule
Sree and Chidambaram
(2004).
Kc
10 K (645)
T i Td Comment
Direct synthesis: time domain criteria
11 T(645) i
T (645) x d
Model: Method 1
1' Desired closed loop transfer function = l-sTm3)(l + T,(645)s)e-^
l + sT^Jl + sTg'Jtl + sx,) '
10 v (645) , (645)
( o ^ =
3.5(T
iT& ) +T& ) +x 2 +T i n 3+0.5T i n -T i (645)
) '
(645) 0.5lT;(645) -0.5T,
• (645) » I f -
nl V
0.5TmQ<2L>x2
f " T„,2(T« + T<2> + X , +Tm , +0.5xm - T / 6 4 5 ' ) ' ml \ACL ^ XCL T A 2 T 1 m 3 m I
11 Equations continued into the footnote on page 446.
j (645) _ x l ~ T m l lTCL + TCL + X 2 + Tm3 + T m ]
0.5Tml2Tm3Tm-Tml
4
x, = T M x 2 +0.5xmTm l2 tr^Tg> + T&>T&> + T ^ T ™ ) + ^ T r a l x m T < ^ x 2
with
446 Handbook of PI and PID Controller Tuning Rules
x _ Tm,4(0.5Tm3Tm - T j f a ' T f f + 0.5xmfc'L> + TC2L> - T m 3 ] r f f + 2 i ; m T m 3 x 3 )
x4
, T m , 4 ( - 2 ^ T m l +T m 3 + 0.5x Jo.5xmTc'L>T,l2L> - T r o 3
2 x 3 ^ x4
x 3 = TCL + TCL + Tm3 + xm ,
x 4 =-Tm l2(o.5Tm 3xm - T m l
2 X r m l2 [ r S + T<2> +0 .5 t m + 24mTj-0.5xmT<1
L)T<2
L>) - ( - 2 £ T r a l + T m 3 + 0 . 5 x m )
fc^M + 0 . 5 x m ( T S +T<2L»)]+4mTmlxmT<'L»T<2) - T m l
4 ) .
Chapter 4: Tuning Rules for PID Controllers 447
4.14.2 Non-interacting controller based on the two degree of freedom structure 1
f \ r \
R(s)
X0-
U(s) = Kc , 1 T d s
1+ — + ^— T*s 1 + ^ - s
N
E(s)-K t a + PTds
1 + ^ s N
R(s)
a + PTds
N
K „
+ x E(s)
K, T ' s i + ^ s
N ,
U(s)
+ K m ( l - s T m 3 ) e -
T m l V - 2 ^ T m l s + l
Y(s)
Table 176: PID controller tuning rules — unstable SOSPD model with a positive zero
K r a ( l -T m 3 s )e -^ G m (s ) :
T m l V - 2 ^ ^ , 8 + 1
Rule
Sree and Chidambaram
(2004). Model: Method 1
Kc Ti Td Comment
Direct synthesis: time domain criteria
12 j r (646) 13 T (646) T (646) P = 0; N = oo
( l - sT Vl + T(646)s)e~STra 13 Desired closed loop transfer function = •) 2 i£ '—-x .
(l + sT&lJll + sT&'Xl + sx,)
. (646)
K ^ T ^ + T ^ + x . + T ^ + O . S t , , , - ^ 6 4 6 1 }' (646) O .SJT^ -O.s t j , Tm3+0.5(T^+Tg)
T ( 6 4 6 ) ' T ( 6 4 6 )
13 Equations continued into the footnote on page 448.
448 Handbook of PI and PID Controller Tuning Rules
(646) _ x l ~ Tml \TCL + TCL + X 2 + Tm3 + Tm j w j ^
"•5T m i i m 3 t m _ l m l
*. = 0 2 + 0.5TmTm,2(T<'L>T<2L> + T&>T&> + T<3
L>T<'>)+ ^ m T m , x m T< '>T< 2 >x 2 ,
v _ Tm,4(o.5Tm3Tm -lj)T&lg +0.5tm[T^ + Tff - T ^ j r g +2^mTm3x3) x2 -
x4
, Tro,4(-2^mTm, +Tm3 + 0.5xmjo.5xmT^T^ -Tm3
2x3)^
x4
X3=T<1L
)+T<2L»+Tm 3+xm ,
x4 =-Tm,2(o.5Tm3Tm -Tml2)(Tml
2[TS + T<2) + 0.5xm +2^mTml]-0.5tmT^T<2L
)) -(-2!;Tml+Tm3+0.5xm)
( ^ [ i M +0.5xrafe'» +T<2»)]HmTm,xmT^T<2L> -Tj).
Chapter 4: Tuning Rules for PID Controllers 449
4.15 Delay Model Gm(s) = Kme"
f 4.15.1 Ideal controller Gc(s) = Kc
1 \ 1+ — + Tds
V T . s J
R ( s > ^ E(s)
+ T K „ ! + — + Tds
U(s) K e-SI™
Y(s)
Table 177: PID controller tuning rules - Delay model Gm (s) = Kme
Rule
Callender et al. (1935/6).
Model: Method 1
Kc Ti Td Comment
Process reaction , 0.749
K m' c m
2 1.24 K r a T m
2.734tm
1.31xra
0.303xm
0.303xm
Representative results deduced from graphs
1 Decay ratio = 0.015; Period of decaying oscillation = 10.5xn
2 Decay ratio = 0.12; Period of decaying oscillation = 6.28tm
450 Handbook of PI and PID Controller Tuning Rules
4.15.2 Ideal controller in series with a first order lag r
j « % ^ / K,
1 \ 1 + — + Tds
V T i s J
Gc(s) = K
U(s)
1 A 1 + — + Tds
V T i s J T fs + 1
1
1 + Tfs K e ~
Y(s)
Table 178: PID controller tuning rules - Delay model Gm(s) = Kme
Rule
Bequette (2003) -page 300.
Model: Method 1
Kc
Km(4X + xm)
T(
0.5xra
Td
Robust
0.167xm
Comment
2^2-0.167xm2
4^ + Tm
Chapter 4: Tuning Rules for PID Controllers 451
4.15.3 Classical controller 1 Gc (s) = K
E(s)
'.+-L^ V T i s /
1 + Tds
^ N
R(s)
+ < g > — K
v T i sy
1 + Tds
l + ^ s N
U(s)
* K „ e - " »
Y(s)
Table 179: PID controller tuning rules - Delay model Gra(s) = Kme~
Rule
Hartree et al. (1937).
Model: Method 1
Kc
3 0.70
K m T m
4 0.78
Kmxra
T; Td
Process reaction
2.66xm
2.97xm
*m
0.5xm
Comment
Representative results
3 N = 2. Decay ratio = 0.15; Period of decaying oscillation = 4.49tm
4 N = 2. Decay ratio = 0.042; Period of decaying oscillation = 5.03xn
452 Handbook of PI and PID Controller Tuning Rules
Kme" 4.16 General Model with a Repeated Pole Gm(s) =
(l + sTra)n
/ 4.16.1 Ideal controller Gc(s) = Kc
1 \ 1 + — + Tds
v T i s J
RQO > ( 0 vE ( s )
K, 1 \
1+ + Tds V T i s J
U(s) K„e"
0 + sTJ-
Y(s)
Table 180: PID controller tuning rules - general model with a repeated pole
Gm(s)= K ^ e
d + sTra)n
Rule
Skoczowski and Tarasiejski
(1996).
Kc Ti Td Comment
Direct synthesis
5 R (647) T T (647) Model: Method I
5 £ (647) < w g * m ( i+» g
2 T m2 r co„Tm 1 + co/T,
T (647) n - 1
Km ^ l + W82[Td
(647>f " n + 2
2n + 4 xm 4n + 2
Tm , n > 2 with
o „ = •
b = Tm
n 2 - 2 n - 2 + - + <t»n It T m 71
2Tm 2 . _ 4n + 2 t r a 2 n - 2
n - 4 n + 3 + —+ -
±b
-=—, and with
7C T m 71
n 2_2 n_2 +2E±l^+4 n + 2
c = 4(n + 2) 1
7t Tm 7t
2 „ „ 4n + 2 ( T n - 4 n + 3 + -
71 T,
+ c , with
2 n - 2 + <
= specified phase margin.
Chapter 4: Tuning Rules for PID Controllers 453
4.16.2 Ideal controller in series with a first order lag f 1 \
Gc(s) = Kc 1
1 + — + Tds V T i s j
1 Tfs + 1
R(s)
+ 7) •
L
Kc f 1 ^ I T i s )
1
T fs + 1
U(sj K m e ^
d + sTJ"
Y(s)
— •
Table 181: PID controller tuning rules - general model with a repeated pole
Kme-"» Gm(s) =
(l + sTm)n
Rule
Schaedel(1997). Model: Method 2
Kc T, Td Comment
Direct synthesis: frequency domain criteria
0.563
Km
"n + f
. n - 2 .
6 j (648) T (648) T f =T d( 6 4 8 ) /N ,
5 < N < 20
6 T (648) _. 3(n + 1) / x T (648) _ n + 1 (_ \
5n - 1 6n
454 Handbook of PI and PID Controller Tuning Rules
4.11 General Stable Non-Oscillating Model with a Time Delay
4.17.1 Ideal controller Gc (s) = K I 1 + — + Tds T:S
R(s) / 0 v E ( s )
+ <&
f K,
1 1 + — + Tds
V T i s j
U(s) General model
Y(s)
Table 182: PID controller tuning rules - general stable non-oscillating model with a time delay
Rule
Gorez and Klan (2000).
Model: Method I
Kc
7 K (649)
Ti Td
Direct synthesis
T ( 6 4 9 ) 1 i
T (649)
T (650)
0 2 5 T j (649)
Comment
alternative tuning
alternative tuning
7 K (649) . (649)
T . ( 6 4 9 ) + T
l + J l + 2 T (649) _ j
T V ar J
(649) _ 1 a r , (649) fr, (SW)+Ta&-
t ! " 2T„ 1 - K < 6 4 9 > ^ 1 +
2 T .
3T, (649)
T (650) _ y (649) 2m_l j _ 2m_
T
\ , T^ = average residence time of the process (which
equals Tm + Tm for a FOLPD process, for example); T^ = additional apparent time
constant; T = 1-K, (649)
1+-2T (649)
Chapter 4: Tuning Rules for PID Controllers 455
4.18 Fifth Order System plus Delay Model ^ _ Km(l + b!S + b2s2 +b3s3 +b4s4 +bss
5)e" m ' S ' ~~ ~ Ti 2 3 4 TV
l + ajS + ajS +a3s +a4s +a5s J
4.18.1 Ideal controller Gc (s) = Kc
E(s) U(s)
1+ — + Tds Xs
R(s) Y(s)
+ K, 1 + - - + Tds
Km (l + b,s + b2s2 + b3s
3 + b4s4 + b5s
5 p (l + a,s + a2s
2 + a3s3 + a4s
4 + a5s5 T
Table 183: PID controller tuning rules - fifth order model with delay
Gm(s) = Km(l + b1s + b2sz +b3s3 +b4s + bss )e
(l + a,s + a2s2 +a3s3 +a 4 s 4 +a 5s 5)
Rule
Vrancic et al. (1999).
Kc Ti Td Comment
Direct synthesis 8 K (651) T (651) T (651) Model: Method I
' Note: equations continued into the footnote on page 456.
K (Ml) = a l 3 T a l 2 b l + a l b 2 - 2 a l a 2 + a 2 b l + a 3 ~ b 3 +Yl w i t h
2 K j ^ a 7 2 b , + a , a 2 + a ; b | 2 - a 3 - b , b 2 + b 3 t y j ]
Yi = tm(a i 2 - a 1 b 1 - a 2 +b 2 )+0 .5 ( a l -b l ) r m2 +0 .167x m
3 and
Y2 =(a, - b , ) 2 T m +(a, - b , ) t m2 +0.333xm
3 - ( a , - b , + xm)2Td<651);
T (65i) = a,3 -ai2bi + zxb2 -2a,a 2 +a2b1 +a3 - b 3 +y, .
' [a,2 - a . b , ~a2 +b2 +(B, -b , ) r m + 0.5xra2 -(a, - b , + xm)rd
(651)J '
Td(651> is found by solving the following equation:
Td(651)x, + 360x2 -360xmx3 + 180xm
2x4 -60xm3x5 - 15xm
4x6
- 3 x m5 x 7 + 7 x m
6 ( b 1 - a 1 ) - x m7 = 0 , w i t h
x, = 360[a,4b2 - a t (a2b] +a3 + b!b2 +b3)
+ a,2(a22 +a 2 (b 1
2 -2b 2 ) + 3a3b1 + 2a4 H-b^- ,+b 22 -b 4 )
-a , (a 2 (2a 3 -3b 3 ) + a3(2b,2 - b 2 ) + 3a4b, +a5 - b , b 4 +2b2b3 - b 5 )
456 Handbook of PI and PID Controller Tuning Rules
- a 2 b ,b 3 +a32 +a3(b,b2 -2b 3 ) + a4b,2 +a5b, —b,b5 +b 3 ]
-360Tm[a,4b, -a , 3 (a 2 + b,2 +2b2) + 3a,2(a3 +b,b2)
+ a,(3a2b2 -3a 3 b, -3a 4 - b , b 3 - 2 b 2 +2b4)
- a 2 (b ,b 2 +b3) + a3(b, - b 2 ) + 2a4b, +a5 - b , b 4 +2b2b3 - b 5 ]
+ 180xm2[a,4 -4a , 3 b, + 3a,2(b,2 +b2) +a1(3a2b, -3a 3 -5b,b 2 +b3)
- a 2 ( b , 2 +2b2) + a3b, +2a4 +b,b3 +2(b22 - b 4 ) ]
+ 60Tm3[4a,3 -9a , 2 b, -a ,(3a2 -5b , 2 - 4b 2 ) +4a2b, - a 3 -5b,b 2 +b3]
+ 15xm4[ 9a,2 -14a,b, - 4 a 2 +5b,2 +4b2] -42xm
5(b, - a i ) + 7Tm6 ,
x 2 = a , b 3 - a , ( a 2 b 2 +a ,b 3 +a 4 +b ,b 3 )
+ a,2(a22b, +a2(2a3 +b,b2 -3b 3 ) + a3(b,2 +b 2 ) + 2a4b, +a s +b 2b 3 - b 5 )
- a , ( a 23 + a 2
2 ( b , 2 - 2 b 2 ) + a2(2a3b, - 2 b , b 3 + b 22 - b 4 )
+ 2a32 +a3(2b,b2 -3b 3 ) + a4(b,2 +b 2 ) + a5b, - b , b 5 +b 3
2 )
+ a22a3+a2[a3(b, - 2 b 2 ) - a 5 - b , b 4 +b5] + a3 b,
+ a3(a4 — b,b3 +b 2 - b 4 ) + a4(b,b2 - b 3 ) + a5b2 - b 2 b 5 +b 3 b 4 ,
x 3 = a , b 2 - a , ( a 2 b , + a 3 + b , b 2 + b 3 )
+ a,2(a22 +a2(b,2 - 2 b 2 ) + 3a3b, +2a4 +b,b3 +b 2
2 - b 4 )
-a,(a2(2a3-3b3) +a3(2b,2 - b 2 ) + 3a4b, +a5 - b , b 4 +2b2b3 - b 5 )
- a 2 b ,b 3 +a32 + a3(b,b2 - 2b 3 ) + a4b,2 +a5b, - b , b 5 + b 3
2 ,
x 4 = a , b, — a, ( a 2 +b , +2b2)
+ 3a,2(a3 +b,b2) + a1(3a2b2 -3a 3b, -3a 4 — b,b3 - 2 b 22 +2b4)
-a 2(b,b 2 + b3) +a3(b, - b 2 ) + 2a4b1 +a5 - b , b 4 +2b2b3 - b 5 ) ,
x 5 = a , -4a , b, +3a, (b, +b 2) + a , (3a2b , -3a 3 -5b ,b 2 +b 3 )
- a 2 ( b , 2 +2b2) +a3b, +2a4 +b,b3 +2(b22 - b 4 ) ,
x6 =4a,3 -9a , 2 b, -a , (3a 2 -5b , 2 - 4 b 2 ) +4a2b, -a 3 -5b ,b 2 + b 3 ,
x7 = 9a,2 -14a,b, - 4 a 2 + 5b,2 +4b2 .
Chapter 4: Tuning Rules for PID Controllers 457
4.18.2 Controller with filtered derivative
Gc(s) = Kt i+ -U Tds
E(s)
R(s) , 1 TdS
1 + — + - d T.s l + ^ s
N .
U(s)
T,s l + (T d /N>
Y(s)
Km (l + b,s + b2s2 + b3s3 + b4s4 + b5s5 p
(l + a ,s + a2s2 + a3s3 + a4s4 + a5s
5
Table 184: PID controller tuning rules - fifth order model with delay
Gm(s) = Km(l + b1s + b2s2 +b3s3 +b 4s 4 + bss
5)e~ST"'
(l + a,s + a2s2 +a3s3 +a 4 s 4 + -7) Rule
Vrancic(1996)-pages 118-119.
Model: Method 1
Kc
ry (652)
2(A,-T i<652)j
T, Td
Direct synthesis
9 j (652) i
T (652)
Comment
N<10
(652) [T, (652),2
N
(652)
-(A32 - A 5 A , ) + .J(A3
2 - A ^ J - ^ ( A 3 A 2 -A5XASA2 -A4A3
N ( A 3 A 2 - A 5 )
A, = K m ( a , - b , + x m ) , A2 = K m ( b 2 - a 2 + A , a 1 - b I T m + 0 . 5 T m2 ) ,
A3 = K m ( a 3 - b 3 + A 2 a , - A,a 2+b 2xm -0 .5b,xm2+0.167xm
3 ) ,
A4 =Km(b4 - a 4 +A3a, - A 2 a 2 +A,
A5 =Km(a5 - b 5 + A4a, - A3a2 + A2a3 - A,a4 +b4xm -0.5b3xm2)
+ Km(o.l67b2xm3-0.042b1xm
4+0.008xm5)
b3Tra +0.5b2Tm2 + 0.167b,xm
3 +0.042xm4)
458 Handbook of PI and PID Controller Tuning Rules
Rule
Vrancic et al. (1999).
Model: Method 1
Kc
10 K (653)
Ti
T (653) i
Td
T (653)
Comment
8 < N < 20
10 K (653) = _
2K,
a, - a , b, +a,b2 -2a ,a 2 + a2b, +a3 - b 3 +y,
^ - a / b j + a . a j + a ^ , 2 - a 3 - b , b 2 +b3 +(a, - b , ) 2 x m +y2
t, with
Yi = t m (a i 2 -a ,b 1 -a 2 +b 2 )+0 .5(a 1 -b , )T: m2 +0.167T m
3 and
y2 =(a, -b , )x m2 +0.333xra
3 - ( a , - b , +xm)2Td
• (653)
(653) r T <6 5 3>12
SV J-(a,-b I +xB); a, - a , b , + a , b 2 - 2 a , a 2 + a 2 b , + a 3 - b 3 + y .
a,2 - a , b , - a 2 +b2 +(a, -b , ) r m +0.5xm2 -(a, - b , +xm)Td
(653) [T, (653)-,2
Td is found by solving the following equation:
[Td(653)]4N-3Z) + [ T d ( 6 »)] 3 N- 2 z 2 +[Td(653)]2N- ,z3 +
TH<653)x, +360x, -360xmx, + 180x 2x< - 60xm
2x< - 15x 4x m A 4 '•m A 6
- 3 t m5 x 7 + 7 T m
6 ( b , - a 1 ) - x r a7 = 0
with x1 ,x2 ,x3 ,x4 ,x5 ,x6 and x7 as defined in Table 183; z,,z2 and z3 are defined as
follows:
z, = 360[a, - a t2 b , +a,(b2 - 2 a 2 ) + a2b, +a3 - b 3 ]
+ 3 6 0 x m [ a 12 - a 1 b 1 - a 2 + b 2 ] + 180xm
2(a ,-b,) + 6 0 x m \
z2 = 360(a, -b , ) [a , - a , b , + a , ( b 2 - 2 a 2 ) + a 2 b , + a 3 - b 3 ]
+ 360tm[2a,3 -3a , 2 b, -a ,(3a2 - b , 2 - 2 b 2 ) + 2a2b, +a3 - b , b 2 - b 3 ]
+ 180xm2(3a,2 -4a ,b , - 2 a 2 +b, 2 +2b2) + 240xm
3(a, - b , ) + 60xm4,
z3 = z4 + z5xm + z6xm2 + z7xm
3 + 135(a, - b,)xm4 + 27xra
5, with
z4 =-360[a,4b, -a , 3 (a 2 + b , 2 + b 2 ) +a,2(2(a3 +b 1 b 2 ) - a 2 b 1 )
+ a,(a22 +a2(b,2 + b 2 ) - a 3 b , - 2 a 4 — b,b3 - b 2
2 +b 4 )
-a 2 (a 3 +b,b2) + a4b, +a5 +b2b3 - b 5 ] ,
z5 =360[a,4-3a,3b, +a,2(2(b,2 + b 2 ) - a 2 ) +a,(3a2b, - a 3 -3b,b 2)
- a 2 ( b , 2 +b 2 ) + a4 +b,b3 +b 22 - b 4 ] ,
z6=540[a, -2a , b, - a , ( a 2 - b , - b 2 ) + b , ( a 2 - b 2 ) ] and
z7 =180[ 2a , 2 -3a ,b , - a 2 + b , 2 + b 2 ] .
Chapter 4: Tuning Rules for PID Controllers 459
Rule
VranCic et al. (2004a), Vrancic
and Lumbar (2004).
Model: Method 1
Kc
11 K (654)
Ti
T (654)
Td
T (654)
Comment
8 < N < 20
Kc( 4>, Tj and Td
(654) are determined by solving the following equations:
K (654) x (654) .
0.5^(A4A, - A 5 K j - 0 . 5 - ^ ( A , A 2 -A3Km) A l ^ m
A<A,Km + A,A,A, - A,A,2 - A,2K„ M ^ l ^ - m T n l n 2 n 3 • r l 4 r M
(654) K 2 L (654) I2
, - - Km lK° ,. J and J r „ 2(A1+K ra
2Kc(654 ,Td
<654))
K, 1 "" l + 2KmKc(654) +
• (654)
K „ A3Km -A,2KmKc
( 6 5 4 )Td '6 5 4 ) - A 2 A , + A2Km2Kc<
654)Td<654> + x
2A,A9K - A , 3 - A , K m2
with
x2 =[(2A,KmKc<654>Td<«4> + A2)A2 - A , A 3 -A 3 (K m
2 + A,2)Kc'«
4»Td<654»]
[A,+K r a2K c< 6 5 4 ,T/ 5 4>],
and with A i = K m ( a i - b l + x m ) , A 2 = K m ( b 2 - a 2 + A , a 1 - b 1 x m + 0 . 5 x m
2 ) ,
A3 =Km(a3 - b 3 + A2a, - A,a2 +b2xm -0.5b,xm2 +0.167Tm
3),
A4 =Km(b4 - a 4 + A3a, - A2a2 +A,a3 -b 3Tm +0.5b2xm2 + 0.167b,xm
3 + 0.042Tm4)
A5 = Km(a5 - b 5 + A4a, - A3a2 + A2a3 - A,a4 +b4xm -0.5b3xm2)
-Km(0.167b: xm3-0.042b,xm
4+0.008xm5).
460 Handbook of PI and PID Controller Tuning Rules
4.18.3 Non-interacting controller 7
R(s) E(s) K „
Xs
U(s) = - ^ E ( s ) - K ( 1+ T ' S
1 + ^ s N ;
Y(s)
U(s) Km(l + b,s + b2s2 + b3s3 + b4s4 + b5s5)s
(l + a,s + a2s2 + a3s3 + a4s4 + a5s5 J
Y(s)
Table 185: PID controller tuning rules - fifth order model with delay
Gm(s) = Km(l + b,s + b2s + b3s +b4s +bss )e
(l + a^ + a ^ 2 +a3s3 +a 4 s 4 +a5s5J
5 \ „ - s t m
Rule
Vrancic et al. (2004a), Vrancic
and Lumbar (2004).
Kc Ti Td Comment
Direct synthesis
12 K (655) T (655) T (655)
Model: Method 1; 8 < N < 20
12 Note: equations continued into the footnote on page 461. Kc ' , T;<655) and Td
are determined by solving the following equations:
0.s42-(A4A1 - A5Kn)-0.5^-(A1A2 -A3Km) K
(655) T (655) A, K „
A,A,Km + A,A,A, - A.A 2 - A 2 K „
K c< 6 5 5 ) _ l + 2 K m K c
(655) +
, (655)
Km2[KH:
2V (655)^ (655)
| r v 2 j - t 3 n 4 * - i ,
|2
and
K (655) A 3 K m ~ A l K m K c (655) T (655)
•A,A, + A , K mz K 2V ( 6 5 5 ) x (655)
+ X
2A,A,K - A , 3 - A , K 2 , with
x '=U2A,KmK c >)A2 (655) T (655) .
l d - t - / \ 2 i n 2 n ] n j n 3 i i v m A , A , - A 1 ( K m
2 + A 12 ) K l
( 6 5 5 ) T (655)
[A,+Km2K(
(655)y (655)1
Chapter 4: Tuning Rules for PID Controllers 461
and with
A, =Km(a, - b , + x m ) , A2 = K m ( b 2 - a 2 + A , a , - b , t r a +0 .5x m2 ) ,
A3 = Km(a3 - b 3 +A2a, - A,a2 +b 2 t r a - O ^ t , , , 2 +0.167-cm3),
A4 =Km(b4 - a 4 + A3a, - A2a2 + A,a3 - b 3 t r a + 0.5b2xm2 +0.167b,Tm
3 +0.042xm4)
A5 = K.m(a5 - b 5 + A4a, - A3a2 + A2a3 - A,a4 +b4xm -0.5b3Tm2)
+ Km(o.l67b2Tm3 -0.042b,xm
4 +0.008Tm5).
Chapter 5
Performance and Robustness Issues in the Compensation of FOLPD Processes with PI
and PID Controllers
5.1 Introduction
This chapter will discuss the compensation of FOLPD processes using PI and PID controllers whose parameters are specified using appropriate tuning rules. The gain margin, phase margin and maximum sensitivity of the compensated system as the ratio of time delay to time constant of the process varies, are used as ways of judging the performance and robustness of the system. This work follows the method of Ho et al. (1995b), (1996), in which good approximations for the gain and phase margins of the PI or PID controlled system are analytically calculated. The method will be extended to determine analytically the maximum sensitivity of the compensated system. Insight will be obtained into the range of time delay to time constant ratios over which it is sensible to apply various tuning rules, to compensate a FOLPD process.
The chapter is organised as follows. Formulae for calculating analytically the gain margin, phase margin and maximum sensitivity are outlined in Sections 5.2 and 5.3. The performance and robustness of FOLPD processes compensated with a variety of PI and PID tuning rules are evaluated in Section 5.4. In Section 5.5, tuning rules are designed to achieve constant gain and phase margins for all values of delay, for a number of process models and controller structures. Conclusions of the
462
Chapter 5: Performance and Robustness Issues... 463
work are drawn in Section 5.6. This work was previously published by O'Dwyer (1998), (2001a).
5.2 The Analytical Determination of Gain and Phase Margin
5.2.1 PI tuning formulae
The controller and process model are respectively given by
Gc(s) = Kc
K T i s ;
Gm(s) = K_e~
1 + sT
Then
r cu*c rw> KmG~>aZm K c(J»T i + l ) (jm(jco)Gc(jco) =
(5.1)
(5.2)
(5.3) l + j©Tm jooT;
From the definition of gain and phase margin, the following sets of equations are obtained:
<t>m =arg|Gc(j©g)Gm(ja)g)j+7i
1 Am =
|Gc(jcop)Gm(jcop)
(5.4)
(5.5)
where co and co are given by
|Gc(jcog)Gm(jcog)| = l
arg[Gc(ja> )Gm(ja> ) J = - T I
(5.6)
(5.7)
From Equation (5.3),
K m K c J l + G>2Ti2
G m ( j c o ) G c ( ] c o ) - ^ ^ ! -2 ^ 2 coTj -y/l + co2Tn
Z - 0.57T + tan-1 coTj - tan-1 ©Tm - coxm (5.8)
Therefore, from Equation (5.4)
(|>m = 7t-0.5n + tan~1cogTi -tan_1cogTm -cogTm (5.9)
464 Handbook of PI and PID Controller Tuning Rules
with co given by the solution of Equation (5.6) i.e.
K m K c J l + co 2T,2
y s = = i (5.io) cogTiA/l + cog
2Tm2
Also, from Equations (5.5) and (5.8), 1 conT; Jl + co 2T 2
A = . - P ' E EL. (5 11) m |Gm(j(Dp)Gc(jcop)| K ^ ^ l + co/Ti2
with co given by the solution of Equation (5.7) i.e. 0.57t + tan ' c o T . - t a n ' co Tm -co xm =-n (5.12)
p i p m p m
From Equation (5.10), co may be determined analytically to be
T,(Kc2Km
2 - l ) + A / ( K c2 K m
2 - l j V +4K c2 K m
2 T m2
^ 2T iTm2
An analytical solution of Equation (5.12) (to determine cop) is not
possible. An approximate analytical solution may be obtained if the following approximation for the arctan function is made:
-1 ?t I I ., , _i 7T 7X I I - . , r- i ^
tan x « — x , x < l and tan x ~ x > l (5.14) 4 ' ' 2 4x M
This is quite an accurate approximation, as is shown in Figures la and lb (page 465). Considering Equation (5.12), four possibilities present themselves if the approximation in Equation (5.14) is to be used. These possibilities, together with the formula for cop that may be determined analytically for each of these cases, are
n± In2 -4m„ T T
(i) c o ^ > 1, copTm > 1: cop = > - ^ '- (5.15)
n±ln2-~-(0.25n7m+Tm) L _ ! (5.1( (ii) co T j > l , c o p T m < l : co = , , (5.16)
2(0.257iTm+Tm)
Chapter 5: Performance and Robustness Issues... 465
X T „ (iii) copT; < 1, copTm > 1: (op = \\4T_m
nT (5-17)
(iv) copT,<l, c o p T m < l : c o p = - j — — - (5.18)
The gain and phase margin of the compensated system, for each of the
tuning rules, as a function of xm/Tm , may be calculated by applying
Equations (5.9), (5.11), (5.13) and the relevant approximation for co
from Equations (5.15) to (5.18).
Figure la: Arctan x and its approximation (Equation 5.14)
0 2 4 6 8 10
Figure lb: % error in taking approximation (Equation 5.14) to arctan(x)
40 T
20-
-20
T
10
466 Handbook of PI and PID Controller Tuning Rules
5.2.2 PID tuning formulae
The classical PID controller structure is considered, whose transfer function is given as
Gc(s) = K, 1 + 1 Y 1 + sT,
T;s 1 + sceT, (5.19)
and the process model is given by Equation (5.2). Substituting Equations (5.2) and (5.19) into Equations (5.4) and (5.5) gives <j>m = 0.5rc + tan-1 cogTi + tan"1 cogTd - tan-1 cogTm - tan-1 cogocTd - cogxm
(5.20) and
_ CQpTj l(l + wp2Tm
2Xl + w p V T d2 )
m " KcKm y (l + cop2
Ti2)(l + cop
2Td2)
From Equation (5.6), (i)g is given by the solution of
(5.21)
KmKc^/l + (ag2Ti
2>/l + (ag
2Td2 _f
4 ' " co„TJl + a)„2T„2^l + a2w„2T,2
The equation for a)g may be determined to be
a)g6+a1o)g
4+a2cog2+a3 =0
(5.22)
(5.23)
with a j = T i
2(Tm2+a2Td
2)-Kc2Km
2T i2Td
T ^ V T , 2
^ - K . X 2 ^ ^ / ) Kc2Km
2
2 ~ , 2™ 2 ^ 2 T 2 3 2 T , 2 „ 2 T 2 T i \ ' a ' T d T i X ' a ' T d
Following the procedure outlined by Ho et al. (1996), the analytical solution of Equation (5.23) is given as
0), =J^/R+VQ T 7R 2 "+^R-VQ 3 + R 2 -— (5-24)
. . ^ 3a, -a . " , „ 9a,a7 - 27a , -2a , with Q = —2- x— and R = —— - —.
9 54 From Equation (5.7), co is given by the solution of
Chapter 5: Performance and Robustness Issues... 467
0.5n + tan -1 copTd + tan"1 copT; - tan"1 copTm - tan"1 copaTd - coptm = 0
(5.25) As with Equation (5.12), an analytical solution of this equation is not possible. An approximate analytical solution may be obtained if the approximation detailed in Equation (5.14) is made. Looking at Equation (5.25), twelve possibilities present themselves if the approximation in Equation (5.14) is to be used; these possibilities are
(1) copT; > 1, ffiT > 1
(a) copTd < 1, ©paTd < 1
(b)copTd > 1 , copaTd <1
(c) o pTd > 1. ^paTd > 1
(3) cOpT; < 1, copTm > 1
(a) ©pTd < 1, copaTd < 1
(b) copTd > 1, ©paTd < 1
(c) ©pTd > 1, copaTd > 1
2 ) c o p T i > l , <DpTm<l
a) ©pTd < 1, copaTd < 1
b) copTd > 1, copaTd < 1
c) copTd > 1, copaTd > 1
^ c O p T ^ l , copTm<l
a) copTd < 1, copaTd < 1
b ) a>pTd > 1. <opaTd < 1
c) copTd > 1, copaTd > 1
The formulae for cop that may be determined for each of these cases are
as follows:
(1) copT, > 1, copTm > 1
(a) ©pTd < 1, copaTd < 1:
7 r ± j 7 t 2 - 7 i ( 4 T m - 7 i T d ( l - a ) )
C O P = ' 4 T m - 7 i T d ( l - a )
(5.26)
(b) copTd > 1, © aTd < 1:
® p = -
( 1 1 1 ^ 2TC ± 4TI2 - 7i(4xm + 7iTda) — + - - -
4xm+TcTda (5.27)
468 Handbook of PI and PID Controller Tuning Rules
(c) copTd > 1, copaTd > 1 : P "^d
t o p = -
J_ J 1 1_ V ^d Tj a T d Tm j
n ±ln -47ix„
4x „ (5.28)
(2)copT i>l, c o T < l : p m
(a)copTd <1, copaTd <1 : p"- J d
2n± Un2 + ^-(nTd-nTm-naTd - 4 x J
cop=-7tTm+4xm+7caTd-7cTd
(5.29)
(b)copTd>l, copaTd<l p^^d
3TT± J9n-n
con
V T d + T i y
( K T m + 7 i a T d + 4 x m )
n T m + 4 x m + ™ T d (5.30)
(c) copTd > 1, copaTd > 1:
2n±J4n +n 1 1 1
co„ V a T d Td T i y
("Tm+4xm)
7tT +4x (5.31)
(3) copTj < 1, coT > 1:
(a) copTd < 1, copaTd < 1: cop =.
(b)copTd>l, copaTd<l:
'Tm(4xm+7r[aTd-Td-T i]) (5.32)
- 7 t ± j 7 l -71
% = "
(TcT i-7taTd-4xm)
rcTj-4xm-7taTd
(5.33)
7t 7t 71
(c)copTd>l, c o p a T d > l : o D = ^ T d T - a T d
7iT. - 4x '^ l m
(5.34)
Chapter 5: Performance and Robustness Issues... 469
(4) a)pTj<l, Q) p T m <l :
(a) copTd < 1, copaTd < 1: cop =
(b) co Td > 1, oo aTd < 1:
2 71
7i(aTd-Td)+^(Tm-T i)+4xn
(5.35)
2n±Un2 - A ( „ T m -nT, +naTd + 4 x J X
C O P = -7tT„ - TTT, + 4x„ 7 i a T ,
(5.36)
(c) co Td > 1, oo aTd > 1
n±Jn -n
< » P = -
v a T d T ay ( 7 i T i - 7 i T m - 4 x m )
(5.37)
The gain and phase margin of the compensated system, for each of the tuning rules, as a function of tm /Tm may now be calculated by applying Equations (5.20), (5.21), (5.24) and the relevant approximation for oop
from Equations (5.26) to (5.37). As Equation (5.19) shows, the procedure applies to tuning rules defined for the classical PID controller structure only; however, Ho et al. (1996) suggest that the method may be applied to tuning rules defined for the ideal PID controller structure, by using equations that approximately convert these tuning rules into their equivalent in the classical controller structure.
5.3 The Analytical Determination of Maximum Sensitivity
The maximum sensitivity is the reciprocal of the shortest distance from the Nyquist curve to the (-1,0) point on the Rl-Im axis. It is defined as follows:
1 M m a x = M a x
all ra l + Gm(jco)Gc(jco)
For a FOLPD process model controlled by a PI controller,
|Gm(jco)Gc(jco)| = Vl + oo'Tj2 KcKr
(5.38)
(5.39)
470 Handbook of PI and PID Controller Tuning Rules
and
arg[Gm (j©)Gc (jco)] = -0.5TT - tan"1 coTm + tan"1 oaT; - coxm (5.40)
For a FOLPD process model controlled by a PID controller,
V(l + co2Tm 2)(l + a V T d
2 ) »T>
and arg[Gm(jco)Gc(jco)] =
- 0.5TC - tan"1 coTm - tan"1 coaTd + tan-1 coT; + tan"1 coTd - coxm (5.42) The maximum sensitivity may be calculated over an appropriate range of frequencies corresponding to phase lags of 100° to 260°.
5.4 Simulation Results
Space considerations dictate that only representative simulation results may be provided; an extensive set of simulation results covering 103 PI controller tuning rules and 125 PID controller tuning rules (for the ideal and classical controller structures) are available (O'Dwyer, 2000). The MATLAB package has been used in the simulations. Figures 2 to 7 show how gain margin, phase margin and maximum sensitivity vary as the ratio of time delay to time constant varies, if some PI tuning rules are used (Figures 2 to 4) and corresponding PID tuning rules for the classical controller structure (with a = 0.1) are used (Figures 5 to 7). In these results, Z-N refers to the process reaction curve method of Ziegler and Nichols (1942); W-W refers to the process reaction curve method of Witt and Waggoner (1990); IAE reg, ISE reg and ITAE reg refer to the tuning rules for regulator applications that minimise the integral of absolute error criterion, the integral of squared error criterion and the integral of time multiplied by absolute error criterion, respectively, as defined by Murrill (1967) for PI tuning rules and Kaya and Scheib (1988) for PID tuning rules based on the classical controller structure. Figures 8 to 15 show gain and phase margin comparisons between corresponding PI and PID controller tuning rules.
It is clear that the gain margin is generally less when the PID rather than the PI tuning rules are considered, over the ratios of time delay to
Chapter 5: Performance and Robustness Issues... 471
time constant taken; the difference between the phase margins is less clear cut. This suggests that these PID tuning rules should provide a greater degree of performance than the corresponding PI tuning rules, but may be less robust. Comparing the individual tuning rules, it is striking that the ISE based tuning rules have generally the smallest gain margin and have also a small phase margin, suggesting that this is a less robust tuning strategy. The results in Figures 4 and 7 confirm these comments.
Figure 2: Gain margin Figure 3: Phase margin Figure 4: Max. sensitivity
•Z-U | f 7= JAp ijegi : +4ltAJEreg
...
o - -°oo<
4 + , * 00°
' ' ' - - - * " 1 1 1 -
/ : t t f V i •... A// ; ' \ t ; V > ° fM--t°--i—1—r--;---:—
;>'! i \ i i - ! — ! — 1 — ! — t " - - \ - - - ; - - -
0.2 0.4 0.6 OB I 1.2 1.4 1.6 IB 2 0.2 0.4 0.6 OB I 12 1.4 1.6 IB 2 02 04 0.6 06 1 12 14 1.6 IB 2
Ratio of tm to T„ Ratio of tm to T„ Ratio of tm to T„
Figure 5: Gain margin Figure 6: Phase margin Figure 7: Max. sensitivity
1- 4 w-w M M : 4 4 iAri reg ; ; ; + = lTAEreg! i i +++++t
•^-lis^tegr-r^f-r--',i : i >*!*' i j^<\
"oofeooiooooo*"?000: : : i
60.. - : . . : . . j / t ; / ,
40 WA 3 0 ° P 4 ° ° J J
0 : - i - \ - f -: : \ ! '• '• *
0.2 0.4 0.6 0 6 1 1.2 1.4 1.6 18 2 0 0.2 04 0.6 0.8 1 1.2 14 1.6 1.8 02 0.4 06 OB 1 1.2 1.4 1.6 1.8 2
Ratio of xm to Tm Ratio of Tm to Tm Ratio of x_ to T„
472 Handbook of PI and PID Controller Tuning Rules
No general conclusion can be reached as to the best tuning rule (as expected); it is interesting, though, that a full panorama of simulation results (O'Dwyer, 2000) show that many tuning rules may be applied at ratios of time delay to time constant greater than that normally recommended. One example may be seen in Figures 5 to 7, where the gain margin, phase margin and maximum sensitivity (associated with the use of the PID tuning rule for obtaining minimum IAE in the regulator mode) tends to level out when the ratio of time delay to time constant is greater than 1; normally, the tuning rule is used when the ratio is less than 1 (Murrill, 1967). On the other hand, it is clear from Figures 8 and 9 that there is a significant degradation of performance when using the PID tuning rule of Witt and Waggoner (1990) and the PI tuning rule of Ziegler and Nichols (1942) for large ratios of time delay to time constant, which is compatible with application experience.
The decision between the use of a PI and PID controller to compensate the process, depends on the ratio of time delay to time constant in the FOLPD model, together with the desired trade-off between performance and robustness, as expected. It turns out, however, that the analytical method explored allows the calculation of a far wider range of gain and phase margins for PI controllers; it is also true that stability tends to be assured when a PI controller is used (O'Dwyer, 2000). Thus, a cautious design approach is to use a PI controller, particularly at larger ratios of time delay to time constant.
Finally, the volume of tuning rules and data generated means that the use of an expert system to recommend a tuning rule based on user defined requirements is indicated; work is ongoing on such an implementation (Feeney and O'Dwyer, 2002).
Chapter 5: Performance and Robustness Issues... 473
Figure 8: Gain margin comparison Figure 9: Phase margin comparison
120
100
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Ratio of Tm to T^
0 .0.. .
0
o 7
oO - - . , - - - U -
: ov
: o : o : /
o •/ o /
5000 )000( >ooo6ooo(
v m v u * m
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Ratio of tm to Tm w m *•" m
Figure 10: Gain margin comparison Figure 11: Phase margin comparison
I 1 1 1 1 1
: - ± IAE re|g PID i : o =j IAB reg PI: i
. _ _ „ . ; . - - - - : — - - ! : ; ; . . . „
: • : : : i Q o °
\—- — -t~ - : — - : - — :-0Q0-:—-
\ : : : ; 00° :
\ : : 0ov : : \ : j,o" : : :
: N»O° : : : ° O O O O ° ^ > ~ - ~ L _ ; ; ...j
; 1/ : /
/? i
960^
. i x ^ ' / ! ! o° ; | ;
A > ° °
r :
) 0o6^e <
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
R a t i o o f ! _ , t o T _ vm *" m
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Ratio of im to Tm * m m
474 Handbook of PI and PID Controller Tuning Rules
Figure 12: Gain margin comparison Figure 13: Phase margin comparison
2.5, ! 1 1 . 1 1 i 1 1 1 I 0 0 r
1SE reg JPIDJ 6 = ISEregPI
—-i—~i
i 1/ o0oO°0 :
,/ : o
•A~t~ : o : 'o ....
0? i o-°4"4- -
in00<
o?° 0° i
—4—-
- — -:- —
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Ratio of xm to Tm " m *•" m
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Ratio of tm to Tm m * v * m
Figure 14: Gain margin comparison Figure 15: Phase margin comparison
L-.=i.iT^ | 0=1117
%fafif-
LEregJ m E reg PI
! _n°<
„<J 0
, 0 ° j
5U
> " i
c
/
, /
1 1 i i _ o O ° (
•A ! i ! '600<? - - - ; - - - - ! - - - - - . ;
pj* io0o^oofo°0(
0 0.2 0.4 0.6 0.6 1 1.2 1.4 1.6 1.8 2
Ratio of tm to T„ fcm v w m
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Ratio of xm to T„ v m ™ x m
Chapter 5: Performance and Robustness Issues... 475
5.5 Design of Tuning Rules to Achieve Constant Gain and Phase Margins, for all Values of Delay
Normally, the gain and phase margins of the compensated systems tend to increase as the time delay increases, supporting the common view that PI and PID controllers are less suitable for the control of dominant time delay processes. However, for the PI control of a FOLPD process model, O'Dwyer (2000) shows that the tuning rules proposed by Chien et al. (1952), Haalman (1965) and Pemberton (1972a), among others, facilitate the achievement of a constant gain and phase margin as the time delay of the process model varies. All of these tuning rules have the following structure: K c = a T m / K m t m , T i=T m . Following this observation, original approaches to the design of tuning rules for PI, PD and PID controllers are proposed, for a wide variety of process models, which allow constant gain and phase margins for the compensated system.
This work organised as follows. In Section 5.5.1, PI controller tuning rules are specified for processes modelled in FOLPD form and IPD form. In Section 5.5.2, PID controller tuning rules are described for processes modelled in FOLPD form, SOSPD form and SOSPD form with a negative zero. Section 5.5.3 deals with the design of PD controller tuning rules for the control of processes modelled in FOLIPD form.
5.5.1 PI controller design
5.5.1.1 Processes modelled in FOLPD form
For such processes and controllers, Equations (5.1) and (5.2) apply i.e.,
K e"STm
Gm(s) = -
with Gc (s) = K
l + sTm
1 1 +
From Section 5.2.1,
<t>m =71-0.571 +tan -1 WgTj -tan"1 cogTm -u)gTm (Equation 5.9)
with GO given by the solution of
476 Handbook of PI and PID Controller Tuning Rules
K m K c J l + ©„2X2
Q)gTiA/l + cog2T,
and
2m 2 m
= 1 (Equation 5.10)
conT: l + co2T 2
Am = — ^ - , V V (Equation 5.11)
with cop given by the solution of
-0.5u + tan_1 c0pT; - tan - 1 copTm-copTm =-n (Equation 5.12)
If Kc and T; are designed as follows: aT
K c = — ^ - (5.43)
and T ;=Tm (5.44)
Then Equation (5.12) becomes
-0.5n-copxm =-7t (5.45)
i.e. cop = 7i/2xm (5.46)
Substituting Equation (5.46) into Equation (5.11) gives Am=7rTm/2KmKcTm (5.47)
Substituting Equation (5.44) into Equation (5.10) gives
KmK c /co gTm=l (5.48)
i.e. co g =K m K c /T m (5.49)
Substituting Equations (5.44) and (5.49) into Equation (5.9) gives
(|>m=0.57t-KmKcTm/Tm (5.50) Finally, substituting Equation (5.43) into Equation (5.47) gives
Am=7r/2a (5.51) and substituting Equation (5.43) into Equation (5.50) gives
(t>m =0.57t-a (5.52) Some typical tuning rules are shown in Table 186.
Chapter 5: Performance and Robustness Issues... 411
Table 186: Typical PI controller tuning rules - FOLPD process model
a
7l/3
7t/4
TC/6
Kc
1.047Tm/KmTm
0.785Tm/Kmxm
0.524Tm/KmTm
T Tm
T m
T
Am
1.5
2.0
3.0
*m
7t/6
7t/4
7t/3
These rules are also provided in Table 3, Chapter 3. It may also be demonstrated that, if Kc and Tj are designed as follows:
K„= m ,— u u (5.53) xm , / T 2 +4n2T
and T ^ T m (5.54)
where Ku and Tu are the ultimate gain and ultimate period, respectively, then the constant gain and phase margins provided in Equations (5.51) and (5.52) are obtained.
5.5.1.2 Processes modelled in IPD form
A similar analysis to that of Section 5.5.1.1 may be done for the design of PI controllers for processes modelled in IPD form. The process is modelled as follows:
Gm(s) = K m e- s ^ /s
Corresponding to Equations (5.4) and (5.9), the phase margin is
* , 71-0.571 +tan ' cOgTj-0.57t-cogxm
with cog given by the solution of
K Kcjl + V ? cog T,
If Kc and T; are designed as follows: a
= 1
K = • Kmxm
(5.55)
(5.56)
(5.57)
(5.58)
and
478 Handbook of PI and PID Controller Tuning Rules
% = bxr
then it may be shown that
co„ = a a f ±——T-V; 2xm 2 b x „
a 2 b 2 + 4
0.5
(5.59)
(5.60)
and, substituting Equations (5.59) and (5.60) into Equation (5.56), -i0.5
<t>m=tan-1 0 . 5 a V +0.5abVa2b2+4
Corresponding to Equations (5.5) and (5.11), the gain margin,
A = -®p T i
KmKc>/l + a)p2Ti
2
0 .5a 2 +0 .5 -Va 2 b 2 +4
(5.61)
(5.62)
with co given by the solution of
-0.57i + tan cOpT;-0.57t-copTm
i.e. co is given by the solution of
tan' COpT; = copxm
(5.63)
(5.64)
An analytical solution to this equation is not possible, though if the - i ft i i
approximation tan a^T, « 0.571 (when coT: > 1) is used, P > 4 ( f l p T j | P . | >
simple calculations show that the following analytical solution for co
may be obtained:
%=" 0.5
0.57i + ,|0.257r — (5.65)
The inequality copTj > 1 may be shown to be equivalent to b > 1.273.
Substituting Equations (5.58), (5.59) and (5.65) into Equation (5.62), calculations show that:
Chapter 5: Performance and Robustness Issues... 479
_b_
4a A m = • (5.66)
1 + -
Some typical tuning rules are shown in Table 187.
Table 187: Typical PI controller tuning rules - IPD process model
Kc
0.558/Kmxm
0.484/Kmxm
0.458/KmTm
0-357/Kmtm
0.305/Kmxm
T;
l-4xm
1.55tra
3.35im
4-3xm
12.15tm
Am
1.5
2.0
3.0
4.0
5.0
*m
46.2°
45.5°
59.9°
60.0°
75.0°
It may also be shown that the maximum sensitivity is a constant if Kc
and Tj are specified according to Equations (5.58) and (5.59). The
maximum sensitivity may be calculated to be:
a2
M, l-2acos(co rxm)-2 2
-0.5
(5.67)
with corxm being a constant obtained numerically from the solution of
the equation (a /cor xm ) + cosa>rxm =sin(corxm)/corxm .
It may also be shown that, if Kc and Tj are designed as follows:
K c = a K u (5.68)
and Ti=bTu (5.69)
then the following constant gain and phase margins are determined:
480 Handbook of PI and PID Controller Tuning Rules
A =
2b na
K i \n n 1 4 4b
2
1+- 2 2
an
7t
and <\>m = tan"
- + J - — 2 V 4 4b
T).5 27i2a2b2 +27tabV47i2a2b2 +4
- J0.1257i2a2 + - ^ - V 4 7 r 2 a 2 b 2 + 4
16b
(5.70)
(5.71)
5.5.2 PID controller design
5.5.2.1 Processes modelled in FOLPD form - classical controller 1
The classical PID controller is given by
Gc(s) = K(
f 1 Y l + sTd ^ 1 +
v T i s y 1 + saT, d J
and the process model is given by Kme-ST™
Gm(s) = -l + sT„
(Equation 5.19)
(Equation 5.2)
From Section 5.2.2,
<t>m = 0.57r + tan-1 co Tj + tan"1 co Td - tan"' co T - tan"1 co aTd - co x g m
with co given by the solution of
KmKcA/l + cog2T,.2A/l + co<)
2TH2
" ^ '% 1 d
cogTiA/l + cog2Tm
2^l + a2cog2T,
• = 1 2 „ 2rp 2
d
Also
A, ^pTj l(l + cop
2Tm2J(l + cop
2a2Td2)
KcKm \ (l + cop2
Ti2)(l + cop
2Td2)
(Equation 5.20)
(Equation 5.22)
(Equation 5.21)
Chapter 5: Performance and Robustness Issues... 481
with cop given by
+ tan_1 (OpTj -tan" ' copTm -tan"1 cop
(Equation 5.25)
0.571 + tan"1 copTd +tan cOpT; -tan"1 copTm -tan"1 copaTd -copxm =0
If K
and
then,
c, Tj and Td are designed
Kc =
T,- =
Td = Equation (5.25) becomes
0.5TI
as follows:
aTm
Kmxm
aTm
T ^m
- ® p ^ m = 0
(5.72)
(5.73)
(5.74)
(5.75)
i.e.
cop = V2xra (5.76)
and Equation (5.21) becomes Am=<xrcTm/2KmKcTm (5.77)
Equation (5.22) becomes K m K c / o g T m = l (5.78)
i.e. » g = K m K c / T m (5.79)
Finally, substituting Equations (5.73), (5.74) and (5.79) into Equation (5.20) gives
(t ,m=0.57i-KmKcxm /aTm (5.80)
Substituting Equation (5.72), (5.73), (5.74) and (5.76) into Equation (5.21) gives
Am=7ra/2a (5.81)
and substituting Equation (5.72) into Equation (5.80) gives <|>m= 0 .5TI-a /a (5.82)
This design reduces to the PI controller design when a = 1.
482 Handbook of PI and PID Controller Tuning Rules
5.5.2.2 Processes modelled in SOSPD form - series controller
For such processes and controllers, K e"STm
Gm(s) = 2 (5.83) (l + sTml)(l + sTm2)
with
Gc(s) = Kc
Therefore, » T i s ,
(l + sTd) (5.84)
G m ( s ) G i ( s ) = i S i £ H ± S l ± ^ > (5.85, T|S(l + sTm,Xl + sTm2)
If Tj and Td are designed as follows:
Ti=Tm l (5.86)
and T d =T m 2 (5.87)
then, from Equation (5.85)
G m ( s ) G c ( s ) = K m ^ e (5.88)
which is equal to Gm(s)Gc(s) in Section 5.5.1.1 when T i=T m . Therefore, designing
aT K c = 2 - (5.89)
K m T m
will allow Am = n/2a and §m = 0.571-a , as before.
5.5.2.3 Processes modelled in SOSPD form with a negative zero classical controller 1
For such processes,
Kme- !T-(l + sTm3) Gm(s) = -^S 1 SLL (5.90)
(l + sTml)(l + sTm2) with Gc (s) given by Equation (5.19). Therefore,
Chapter 5: Performance and Robustness Issues... 483
0„(s)G c (s)=K A e""'( ' + S T^1 +;T 'X1 + ^ (5.9!)
If T, ,Td and a are designed as follows:
(5.92)
(5.93)
(5.94)
(5.95)
will allow Am =n/2a. and <j)m = 0.57t-a, as in Sections 5.5.1.1 and 5.5.2.2.
5.5.3 PD controller design
In this case, the process is modelled in FOLIPD form, i.e.
Kme-ST~
and
therefo ire, designing
Ti=Tm l
T<i = Tm2
« = Tm3/Tm2
K c = a T™ Km"1™
s(l + sTm)
with Gc(s) = Kc(l + Tds)
Therefore,
KmKce"STm(l + sTd) Gm(S)Gc(s)= m ' V 6j
S ( 1 + s T m )
Therefore, designing
K - a T m c —
K m T m
and T d = T m
(5.96)
(5.97)
(5.98)
(5.99)
(5.100)
will allow Am =7i/2aand <\>m = 0.57i-a , as in Sections 5.5.1.1, 5.5.2.2
and 5.5.2.3.
484 Handbook of PI and PID Controller Tuning Rules
5.6 Conclusions
This chapter has considered the performance and robustness of a PI and PID controlled FOLPD process, with the parameters of the controllers determined by a variety of tuning rules. The original contributions of this work are as follows: (a) an expansion of the analytical approach of Ho et al. (1995b; 1996) to determine the approximate gain and phase margin analytically under all operating conditions (b) the analytical determination of the approximate maximum sensitivity of the compensated system and (c) the application of the algorithms using a wide variety of PI and PID tuning rules. The implementation of autotuning algorithms in commercial controllers means that choice of a suitable tuning rule is an important issue; the techniques discussed allow an analytical evaluation to be performed of candidate tuning rules.
Finally, the chapter discusses an original approach to design tuning rules for both PI and PID controllers, for a variety of delayed process models, with the objective of achieving constant gain and phase margins for all values of delay. In one of the cases discussed (PI control of an IPD process model), an analytical approximation is used in the development; this approximation may also be used to determine further tuning rules for other process models with delay, and for other PID controller structures.
Appendix 1
Glossary of Symbols Used in the Book
a f l,a f2,b f] = Parameters of a filter in series with some PID controllers
a,,a2,a3,b[, b2 ,b3 = Parameters of a third order process model
a1,a2,a3,a4,a5,b1,b2,b3,b4,b5 = Parameters of a fifth order process model
A, =y,(oo), A2=y2(co), A3=y3(co), A4=y4(oo), A5=y5(oo)
Am = Gain margin
Ap = Peak output amplitude of limit cycle determined from relay
autotuning
b, c = Weighting factors in some PID controller structures
DR = Desired closed loop damping ratio
d(t) = Disturbance variable (time domain)
du/dt = Time derivative of the manipulated variable (time domain)
e(t) = Desired variable, r(t), minus controlled variable, y(t) (time domain) E(s) = Desired variable, R(s), minus controlled variable, Y(s) FOLPD model = First Order Lag Plus time Delay model FOLIPD model = First Order Lag plus Integral Plus time Delay model Fp,F;,Fd = Weights on the desired variable for one PID controller
structure
Gc (s) = PID controller transfer function
GCL(s) = Closed loop transfer function
GCL(jco) = Desired closed loop frequency response
G (s) = Ideal PID controller transfer function
485
486 Handbook of PI and PID Controller Tuning Rules
Gcs(s) = Series PID controller transfer function
Gm(s) = Process model transfer function
Gp(s) = Process transfer function
Gp(jco) = Process transfer function at frequency co
Gp(jco) = Magnitude of Gp(jco)
G (jco,!,) = Magnitude of Gp(joo), at frequency co, corresponding to
phase lag of §
ZGp(jco)= Phase of Gp(jco)
+ h = Relay amplitude (relay autotuning)
IE = Integral of Error = (e(t)dt
IAE = Integral of Absolute Error = J~|e(t)|dt
IMC = Internal Model Controller IPD model = Integral Plus time Delay model
I2PD model = Integral Squared Plus time Delay model
ISE = Integral of Squared Error = f e2 (t)dt
ISTES = Integral of Squared Time multiplied by Error, all to be Squared
= | V e ( t ) ] 2 d t
ISTSE = Integral of Squared Time multiplied by Squared Error =
ft2e2(t)dt
ITSE = Integral of Time multiplied by Squared Error = \~ te2 (t)dt
ITAE = Integral of Time multiplied by Absolute Error = T t|e(t)|dt
Kc = Proportional gain of the controller
Kcp = Proportional gain of the ideal PID controller
Kcs = Proportional gain of the series PID controller
Appendix J: Glossary of Symbols Used in the Book 487
K f = Feedback gain used in non-interacting PID controller 10
2
" ' ' m l KH = 2 t m Km
m '-p 2 TTi i^ml^m (Hwang, 1995)
K 2Kmxm
xm- xmTm | 1 . 8 8 4 K c K u x m
9co„
2KmTm m m
( i m2 | 49Tm
2 | 7Tmxm 4 . 7 1 K u K c ( x m + 1 . 6 T j 1.775KC
2KU2
1324 324 162 81co„ 81a>/
(Hwang, 1995)
K H2 " 2K x 2
V T 2 ^ m x m T m , 1 . 8 8 4 K U K C T „
9co„ +-2 K m T m
-a
with
a :
and
T 4 , T " l , " ^ m ^ m ml T m * ml . ' ^ m l S mT m , ^ ^ m ' m l S m
i . H 1 1 1 a m' 324 81 9 81 9
0 . 4 7 1 K „ K „ a = •
a „
10TmJ | 4Tm,Tm(8^mTm+9Tml) 1.775Ku
2Kc2rm
2
81co„2 81 81
(Hwang, 1995)
Kj = Feedback gain term used in non-interacting PID controller 8, 11
K, = Integral-only controller gain (Pinnella et ah, 1986)
KL = Gain of a load disturbance process model
Km = Gain of the process model
K0 = Weighting parameter used in one PID controller structure
K = Gain of the process
Ku = Ultimate proportional gain
K^ = Proportional gain when Gc(jco)Gp(jco) has a phase lag of <|>
k u = Ultimate proportional gain estimate determined from relay autotuning
488 Handbook of PI and PID Controller Tuning Rules
Kj = Feedback gain term used in one PI controller structure (labeled controller with proportional term acting on the output 2) and the PID non-interacting controller 12 structure
K25% = Proportional gain required to achieve a quarter decay ratio
Kx% = Proportional gain required to achieve a decay ratio of O.Olx
K)35o = Controller gain when Gc(jco)Gp(jco)has a phase lag of 135°
K15Q0 = Controller gain when Gc(jco)Gp(jco) has a phase lag of 150°
Ms = Closed loop sensitivity
Mmax = Maximum value of closed loop sensitivity, Ms
n = Order of a process model with a repeated pole N = Parameter that determines the amount of filtering on the derivative
term on some PID controller structures OS = Closed loop response overshoot PI controller = Proportional Integral controller PID controller = Proportional Integral Derivative controller r = Desired variable (time domain)
0.7071Mmax2-MmaxA/l-0.5Mmax
2
f ' ~ A * 2 i MmaX - 1 (Chen et al, 1999a)
R(s) = Desired variable (Laplace domain) s = Laplace variable SOSPD model = Second Order System Plus time Delay model SOSIPD model = Second Order System plus Integral Plus time Delay
model Tar = Average residence time = time taken for the open loop process step
response to reach 63% of its final value
Td = Derivative time of the controller
Tdi = Derivative feedback term used in non-interacting PID controller 8
Tdp = Derivative time of the ideal PID controller
Tds = Derivative time of the series PID controller
TCL = Desired closed loop system time constant
Appendix 1: Glossary of Symbols Used in the Book 489
TCL2 = Desired parameter of second order closed loop system response
TCL3 = Desired parameter of third order closed loop system response,
with a repeated pole
Tf = Time constant of the filter in series with some PI or PID controllers
T; = Integral time of the controller
Tip = Integral time of the ideal PID controller
Tis = Integral time of the series PID controller
TL = Time constant of a load disturbance FOLPD process model
Tm = Time constant of the process model
TmijT^.T^ = Time constants of second or third order process models, as
appropriate.
Tmi.Tmi,Tmi,T j = Time constants of a general process model
Tp = Time constant of the process
Tr = Parameter of the filter on the set-point in some PID controller
structures
TR = Closed loop rise time
Ts = Closed loop settling time
Tu = Ultimate period A
TU = Ultimate period estimate determined from relay autotuning
T25% = Period of the quarter decay ratio waveform, when the closed loop
system is under proportional control Tx%= Period of the waveform with a decay ratio of O.Olx, when the
closed loop system is under proportional control TOLPD model = Third Order Lag Plus time Delay model u(t) = Manipulated variable (time domain), u^ = Final value of the manipulated variable (time domain) U(s) = Manipulated variable (Laplace domain) V = Closed loop response overshoot (as a fraction of the controlled
variable final value)
x,, x2 , x3 , x4 , x5 = Coefficient values.
490 Handbook of PI and PID Controller Tuning Rules
y = Controlled variable (time domain) y^ = Final value of the controlled variable (time domain)
Y(s) = Controlled variable (Laplace domain)
y,(t)= J^Km - ^ j d t , y2( t)= J(A, -y,(x))lx,
t t
y3(t)= J(A2-y2(x))iT, y4(0= J[A3-y3(T)]iT,
y5(t)=J[A4-y4(x)}iT
a , P, x = Weighting factors in some PI or PID controller structures 2 2
£ = 6 T m l +4^mTm ,xm+KHKmxm ( R w a n g ; 1 9 9 5 )
2Tml xmcoH
= 2Tmcou+KH1Kmxmcou-1.884KuKc ^
0.471KcKu<BH1xm 2 2
60 = 6 T " ^ + 4 T m 1 ^m+K u K m T m ( H w a n g ; 1 9 9 5 )
2 x m T ml fflu
C2 , 6Tm,2(Du +4Tml^mtm(0u + KH2Kmxm2cou -1.884KuKcxm
(0.471KuKcxm2+2xmTml
2cou)coH2
(Hwang, 1995) X = Parameter that determines robustness of compensated system. £, = Damping factor of the compensated system
^m = Damping factor of an underdamped process model
^ e s = Desired damping factor of an underdamped process model
£,m j , £m ; = Damping factors of a general underdamped process model
K = l/KmKu
<)) = Phase lag
Appendix 1: Glossary of Symbols Used in the Book 491
<j)c - Phase of the plant at the crossover frequency of the compensated
system, with a 'conservative' PI controller (Pecharroman and Pagola, 2000).
cpm = Phase margin
§a = Phase lag at an angular frequency of CO
xL = Time delay of a load disturbance process model
Tm = Time delay of the process model
co = Angular frequency
cobw = -3dB closed loop system bandwidth (Shi and Lee, 2002)
co_6dB= Frequency where closed loop system magnitude is -6dB (Shi and Lee, 2004)
coc = Maximum cut-off frequency
cod = Bandwidth of stochastic disturbance signal (Van der Grinten, 1963)
®CL = 2 " / T C L
cog = Specified gain crossover frequency (Shi and Lee, 2002)
coH = | 1 + K " K m (Hwang, 1995) 2 , 2 T m l x m ^ m | K H K m x I ]
TL i ml m ^ m .
1 + K H l K m (Hwang, 1995) xmTm | KH1KmTm
2 0.942KcKutm
3co„
C 0 „ T = l + KH2Km
[T 2 } 2^mxmTml | KH2Kmxm2 0.942KcKuxr
3 6 3cou
(Hwang, 1995)
coM = Frequency where the sensitivity function is maximized (Rotach,
1994)
492 Handbook of PI and PID Controller Tuning Rules
con = Undamped natural frequency of the compensated system
A rh +0 Sir A (A — l) coD = A , p m '- (Hang e/a/. 1993a, 1993b)
lAm2-l>m
cor = Resonant frequency (Rotach, 1994)
cou = Ultimate frequency
co = Angular frequency at a phase lag of (j)
cog()0 = Angular frequency at a phase lag of 90°
co o = Angular frequency at a phase lag of 135° '135'
co o = Angular frequency at a phase lag of 150°
Appendix 2
Some Further Details on Process Modelling
Processes with time delay may be modelled in a variety of ways. The modelling strategy used will influence the value of the model parameters, which will in turn affect the controller values determined from the tuning rules. The modelling strategy used in association with each tuning rule, as described in the original papers, is indicated in the tables in Chapters 3 and 4. Some outline details of these modelling strategies are provided, together with the references that describe the modelling method in detail. The references are given in the bibliography. For all models, the label "Model: Method 1" indicates that the model method has not been defined or that the model parameters are assumed known. Of the 1134 tuning rules specified (442 PI controller tuning rules, 692 PID controller tuning rules), it is startling to relate that only 386 (or 34%) of tuning rules have been specified based on a defined process modelling method.
A2.1 FOLPD Model G m ( s )= K m C
A2.1.1 Parameters estimated from the open loop process step or impulse response
Method 2: Parameters estimated using a tangent and point method (Ziegler and Nichols, 1942).
Method 3: Km ,xm determined from the tangent and point method of
Ziegler and Nichols (1942); Tm determined at 60% of the
493
494 Handbook of PI and PID Controller Tuning Rules
total process variable change (Fertik and Sharpe, 1979). Method4: Km ,xm assumed known; Tm estimated using a tangent
method (Wolfe, 1951). Method 5: Parameters estimated using a second tangent and point
method (Murrill, 1967). Method 6: Parameters estimated using a third tangent and point method
(Davydov et al, 1995). Method 7: xm and Tm estimated using the two-point method; Km
estimated from the open loop step response (ABB, 2001). Method 8: xm and Tm estimated from the process step response:
Tm=1.4( t 6 7 % - t 3 3 % ) , x m = t 6 7 % - l . l T m ; Km assumed known (Chen and Yang, 2000). '
Method 9: xm and Tm estimated from the process step response:
Tm = 1.245(t70O/o - t 3 3 % ) , xm = 1.498t33% -0.498t70% ; Km
assumed known (Viteckova et al., 2000b). 2
Method 10: Km estimated from the process step response; xm= time for
which the process variable does not change; Tm determined
at 63% of the total process variable change (Gerry, 1999).
Method 11: Km estimated from the process step response; xm = time for which the process variable has to change by 5% of its total value; Tm determined at 63% of the total process variable change (Kristiansson, 2003).
Method 12: Parameters estimated using a least squares method in the time domain (Cheng and Hung, 1985).
Method 13: Parameters estimated from linear regression equations in the time domain (Bi et al., 1999).
Method 14: Parameters estimated using the method of moments (Astrom and Hagglund, 1995).
Method 15: Parameters estimated from the process step response and its first time derivative (Tsang and Rad, 1995).
Method 16: Parameters estimated from the process step response using numerical integration procedures (Nishikawa et al., 1984).
' t67o/„, t33% are the times taken by the process variable to change by 67% and 33%,
respectively, of its total value. 2 170o/o is the time taken by the process variable to change by 70% of its total value.
Appendix 2: Some Further Details on Process Modelling 495
Method 17: Parameters estimated from the process impulse response (Peng and Wu, 2000).
Method 18: Parameters estimated using a "process setter" block (Saito et al, 1990).
Method 19: Parameters estimated from a number of process step response data values (Kraus, 1986).
Method 20: Parameters estimated in the sampled data domain using two process step tests (Pinnella et ah, 1986).
Method 21: A model is obtained using the tangent and point method of Ziegler and Nichols (1942); label the parameters Km , Tm
and xm. Subsequently, xm is estimated from the process
step response; then, a parameter labelled
xam=xm-im(Schaedel, 1997).
Method 22: A model is obtained using the tangent and point method of
Ziegler and Nichols (1942); label the parameters Km , Tm
and xm. Subsequently, xm is estimated from the process step response (Henry and Schaedel, 2005).
A2.1.2 Parameters estimated from the closed loop step response
Method 23: Km estimated from the open loop step response. T90% and
xm estimated from the closed loop step response under
proportional control (Astrom and Hagglund, 1988). Method 24: Parameters estimated using a method based on the closed
loop transient response to a step input under proportional control (Sain and Ozgen, 1992).
Method 25: Parameters estimated using a second method based on the closed loop transient response to a step input under proportional control (Hwang, 1993).
Method 26: Parameters estimated using a third method based on the closed loop transient response to a step input under proportional control (Chen, 1989; Taiwo, 1993).
496 Handbook of PI and PID Controller Tuning Rules
Method 27: Parameters estimated from a step response autotuning experiment - Honeywell UDC 6000 controller (Astrom and Hagglund, 1995).
Method 28: Parameters estimated from the closed loop step response when process is in series with a PID controller (Morilla et al, 2000).
Method 29: Parameters estimated by modelling the closed loop response as a second order system (Ettaleb and Roche, 2000).
A2.1.3 Parameters estimated from frequency domain closed loop information
Method 30: Parameters estimated from two points, determined on process frequency response, using a relay and a relay in series with a delay (Tan et al., 1996).
Method 31: Tm and xm estimated from the ultimate gain and period, determined using a relay in series with the process in closed loop; Km assumed known (Hang and Cao, 1996).
Method 32: Tm and xm estimated from the ultimate gain and period, determined using a relay in series with the process in closed loop; Km estimated from the process step response (Hang et al, 1993b).
Method 33: Tm and xm estimated from the ultimate gain and period, determined using the ultimate cycle method of Ziegler and Nichols (1942); Km estimated from the process step response (Hang et al., 1993b).
Method 34: Tm and xm estimated from relay autotuning method (Lee
and Sung, 1993); Km estimated from the closed loop process step response under proportional control (Chun et al, 1999).
Method 35: Parameters estimated in the frequency domain from the data determined using a relay in series with the closed loop system in a master feedback loop (Hwang, 1995).
Appendix 2: Some Further Details on Process Modelling 497
Method 36: Ku and Tu estimated from relay autotuning method; xm
estimated from the open loop process step response, using a tangent and point method (Wojsznis et al, 1999).
Method 37: Parameters estimated by including a dynamic compensator outside or inside an (ideal) relay feedback loop (Huang and Jeng, 2003).
Method 38: Parameters estimated from measurements performed on the manipulated and controlled variables when a relay with hysteresis is introduced in place of the controller (Zhang et al., 1996).
Method 39: Non-gain parameters estimated using a relay, with hysteresis, in series with the process in closed loop; Km is estimated with the aid of a small step signal added to the reference (Potocnik et al., 2001).
Method 40: Parameters estimated using a relay in series with the process in closed loop (Peric et al, 1997).
Method 41: Parameters estimated using an iterative method, based on data from a relay experiment (Leva and Colombo, 2004).
Method 42: Km estimated from relay autotuning method; xm , Tm
estimated using a least squares algorithm based on the data recorded from the relay autotuning method, with the aid of neural networks (Huang et al., 2005).
A2.1.4 Other methods
Method 43: Parameter estimates back-calculated from a discrete time identification method (Ferretti et al., 1991).
Method 44: Parameter estimates determined graphically from a known higher order process (McMillan, 1984).
Method 45: Parameter estimates determined from a known higher order process.
Method 46: The parameters of a second order model plus time delay are estimated using a system identification approach in discrete time; the parameters of a FOLPD model are subsequently determined using standard equations (Ou and Chen, 1995).
498 Handbook of PI and PID Controller Tuning Rules
Method 47: Parameter estimates back-calculated from a discrete time identification non-linear regression method (Gallier and Otto, 1968).
Table 188 summaries the most common FOLPD process modelling methods.
Table 188: The most common FOLPD process modelling methods
Model Method Method 1 Method 2 Method 5 Method 31
PI 112 18 12 6
PID 147 36 26 6
Total 259 54 38 12
It is interesting that in 57% of tuning rules based on the FOLPD process model (252 out of 443), the modelling method has not been specified or the model parameters are known a priori.
A2.2 FOLPD Model with a Negative Zero Gm (s) = - - - -^+ sTm2 ^ l + sT
ml
Method 2: Non-delay parameters are estimated in the discrete time domain using a least squares approach; time delay assumed known (Chang et al, 1997).
A2.3 Non-Model Specific
Modelling methods have not been specified in the tables in most cases, as typically the tuning rules are based on Ku and Tu. Modelling methods have been specified whenever relevant data have been estimated using a relay method. Method 2: Ku and Tu estimated from an experiment using a relay in
series with the process in closed loop (Jones et al., 1997). Method 3: Ku and Tu estimated with the assistance of a relay (Lloyd,
1994).
Appendix 2: Some Further Details on Process Modelling 499
Method 4: co90„ and Ap estimated using a relay in series with an
integrator (Tang et al., 2002).
A2.4 IPD Model Gm (s) = Km&
s
A2.4.1 Parameters estimated from the open loop process step response
Method 2: xm assumed known; Km estimated from the slope at start of the process step response (Ziegler and Nichols, 1942).
Method 3: Km ,xm estimated from the process step response (Hay, 1998).
Method 4: Km ,xm estimated from the process step response using a
tangent and point method (Bunzemeier, 1998).
A2.4.2 Parameters estimated from the closed loop step response
Method 5: Parameters estimated from the servo or regulator closed loop transient response, under PI control (Rotach, 1995).
Method 6: Parameters estimated from the servo closed loop transient response under proportional control (Srividya and Chidambaram, 1997).
Method 7: Parameters estimated from the closed loop response, under the control of an on-off controller (Zou et al., 1997).
Method 8: Parameters estimated from the closed loop response, under the control of an on-off controller (Zou and Brigham, 1998).
A2.4.3 Parameters estimated from frequency domain closed loop information
Method 9: Parameters estimated from Ku and Tu values, determined from an experiment using a relay in series with the process in closed loop (Tyreus and Luyben, 1992).
Method 10: Parameters estimated from values determined from an
500 Handbook of PI and PID Controller Tuning Rules
experiment using an amplitude dependent gain in series with the process in closed loop (Pecharroman and Pagola, 1999).
The most common IPD modeling method is the one where the model parameters are known a-priori or the modeling method is unspecified (i.e. Model: Method 1); 46 PI controller tuning rules, and 47 PID controller tuning rules have been specified using the method. Altogether 76% of tuning rules based on the IPD process model (93 out of 122) have been specified using the method.
A2.5 FOLIPD Model Gm(s) Kme"
3(l + sTm)
Method 2: Parameters estimated from the open loop process step response and its first and second time derivatives (Tsang andRad, 1995).
Method 3: Parameters estimated using the method of moments (Astrom andHagglund, 1995).
Method 4: Parameters estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharroman and Pagola, 1999).
Method 5: Parameters estimated from the open loop response of the process to a pulse signal (Tachibana, 1984).
Method 6: Parameters estimated from the waveform obtained by introducing a single symmetrical relay in series with the process in closed loop (Majhi and Mahanta, 2001).
Method 7: Km estimated from the response of the process to a square
wave pulse; other process parameters estimated from the wavefrom obtained by introducing a relay in series with the process in closed loop (Wang et ah, 2001a).
Method 8: Parameters estimated using a relay in series with the process in closed loop (Peric et ah, 1997).
The most common FOLIPD modeling method is the one where the model parameters are known a-priori or the modeling method is
Appendix 2: Some Further Details on Process Modelling 501
unspecified (i.e. Model: Method 1); 25 PI controller tuning rules, and 49 PID controller tuning rules have been specified using the method. Altogether 85% of tuning rules based on the FOLIPD process model (74 out of 87) have been specified using the method.
A2.6 SOSPD Model Gm(s) = K e~
Tml2s2
+2^mTmls + l or
K e"STm
(l + Tmls)(l + Tm2s)
A2.6.1 Parameters estimated from the open loop process step response
Method 2:
Method 3:
Method 4:
Method 5:
Method 6:
Km , Tml and xm are determined from the tangent and point method of Ziegler and Nichols (1942) (Shinskey, 1988, page 151); Tm2 assumed known.
FOLPD model parameters estimated using a tangent and point method (Ziegler and Nichols, 1942); corresponding SOSPD parameters subsequently deduced (Auslander et al., 1975). Tml = Tm2. xm, Tml estimated from process step response:
Tm l=0.794(t7 0 %- t3 3 %) , Tm=1.937t33%-0.937t70%. Km
assumed known (Viteckova et ah, 2000b). Tml = Tm2. Km , xm and Tml estimated from the process
step response; xm = time for which the process variable does
not change; Tml determined at 73% of the total process variable change (Pomerleau and Poulin, 2004). Parameters estimated from the underdamped or overdamped transient response in open loop to a step input (Jahanmiri and Fallahi, 1997).
502 Handbook of PI and PID Controller Tuning Rules
A2.6.2 Parameters estimated from the closed loop step response
Method 7: Parameters estimated from a step response autotuning experiment - Honeywell UDC 6000 controller (Astrom and Hagglund, 1995).
A2.6.3 Parameters estimated from frequency domain closed loop information
Method 8: Tm and xm estimated In this method, Tml = Tm2 = T,
from Ku , Tu determined using a relay autotuning method;
Km estimated from the process step response (Hang et al., 1993a).
Method 9: Parameters estimated using a two-stage identification procedure involving (a) placing a relay and (b) placing a proportional controller, in series with the process in closed loop (Sung et al, 1996).
Method 10: Parameters estimated in the frequency domain from the data determined using a relay in series with the closed loop system in a master feedback loop (Hwang, 1995).
Method 11: Parameters estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharroman and Pagola, 1999).
Method 12: Parameters estimated from data obtained when the process phase lag is -90° and -180°, respectively (Wang et al., 1999).
Method 13: Model parameters estimated by including a dynamic compensator outside or inside an (ideal) relay feedback loop (Huang and Jeng, 2003).
Method 14: Km estimated from a relay autotuning method; xm, Tml and
2;m estimated using a least squares algorithm based on the
data recorded from the relay autotuning method, with the aid of neural networks (Huang et al, 2005).
Appendix 2: Some Further Details on Process Modelling 503
A2.6.4 Other methods
Method 15: Parameter estimates back-calculated from a discrete time identification method (Ferretti et al., 1991).
Method 16: Parameter estimates back-calculated from a second discrete time identification method (Wang and Clements, 1995).
Method 17: Parameter estimates back-calculated from a third discrete time identification method (Lopez et al, 1969).
Method 18: Model parameters estimated assuming higher order process parameters are known.
Method 19: Parameter estimates back-calculated from a discrete time identification non-linear regression method (Gallier and Otto, 1968).
The most common SOSPD modeling method is the one where the model parameters are known a-priori or the modeling method is unspecified (i.e. Model: Method 1); 15 PI controller tuning rules, and 93 PID controller tuning rules have been specified using the method. Altogether 68% of tuning rules based on the SOSPD process model (108 out of 160) have been specified using the method.
A2.7 SOSIPD model - Repeated Pole G_(s) = —^ -r s(l + sTm)2
Method 2: Parameters estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharroman and Pagola, 1999).
504 Handbook of PI and PID Controller Tuning Rules
A2.8 SOSPD Model with a Positive Zero G_(s) - — •"-" s T m 3 ^ (l + sTmlXl + sTm2)
Method 2: Parameters estimated from a closed loop step response test using a least squares approach (Wang et al., 2001a).
Method 3: Tml = Tm2. Km , xm, Tml and Tm3 estimated from the process step response; the latter three parameters are estimated using a tabular approach (Pomerleau and Poulin, 2004).
A2.9 SOSPD model with a Negative Zero Gm (s) - K m '* + sTm3 ^ (l + sTmlXl + sTm2)
Method 2: Parameters estimated from a closed loop step response test using a least squares approach (Wang et al, 2001a).
Method 3: Tml = Tm2. Km , xm, Tml and Tm3 estimated from the process step response; the latter three parameters are estimated using a tabular approach (Pomerleau and Poulin, 2004).
A2.10 Unstable FOLPD Model Gm (s) = K™e
Tms-1
Method 2: Parameters estimated by least squares fitting from the open loop frequency response of the unstable process; this is done by determining the closed loop magnitude and phase values of the (stable) closed loop system and using the Nichols chart to determine the open loop response (Huang and Lin, 1995; Deshpande, 1980).
Method 3: Parameters estimated using a relay feedback approach (Majhi and Atherton, 2000).
Appendix 2: Some Further Details on Process Modelling 505
Method 4: Parameters estimated using a biased relay feedback test (Huang and Chen, 1999).
K e~STm
A2.ll Unstable SOSPD Model Gm(s) = -, - ^ r or (TmlS-ljO+sTmJ
K e - S T „
(Tm ls-l)(Tm 2s-l)
Method 2: Parameters estimated by least squares fitting from the open loop frequency response of the unstable process; this is done by determining the closed loop magnitude and phase values of the (stable) closed loop system and using the Nichols chart to determine the open loop response (Huang and Lin, 1995; Deshpande, 1980).
Method 3: Parameters estimated by minimising the difference in the frequency responses between the high order process and the model, up to the ultimate frequency. The gain and delay are estimated from analytical equations, with the other parameters estimated using a least squares method (Kwak et al, 2000).
Method 4: Parameters estimated using a biased relay feedback test (Huang and Chen, 1999).
K e""" A2.12 General Model with a Repeated Pole Gm(s)
0 + sTm)n
Method 2: A FOLPD model is obtained using the tangent and point method of Ziegler and Nichols (1942); label the parameters
Km , Tm and xm . Then, n = 10-rL + 1 . Subsequently, xm is
estimated from the open loop step response, and Tar is estimated using "known methods" (Schaedel (1997)).
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Index
ABB (1996), 237 ABB (2001), 15, 29,42, 71, 224,225 Abbas (1997), 47, 169 Aikman(1950), 72 Alfa-Laval, 12, 73 Allen Bradley, 7, 15 Alvarez-Ramirez et al. (1998), 81 Andersson (2000), 58, 81 Argelaguet et al. (2000), 229 Arvanitis et al. (2000), 327, 332, 341,
371 Arvanitis et al. (2003a), 85, 86, 274,
275 Arvanitis et al. (2003b), 94-100, 298-
302 Astrom (1982), 73,236 Astrom and Hagglund (1984), 236 Astrom and Hagglund (1988), 155, 236 Astrom and Hagglund (1995), 45, 64,
70, 73, 75, 76, 93, 185, 236, 245, 259, 284, 349
Astrom and Hagglund (2000), 150 Astrom and Hagglund (2004), 150,
167-168, 223, 260, 272 Atkinson and Davey (1968), 235 Auslandere/a/. (1975), 309
Bailey, 7, 9, 12, 13 Bailey Fisher and Porter, 13 Bain and Martin (1983), 38, 43 Bateson (2002), 73, 238, 258 Bekkeretal. (1991), 45 Belanger and Luyben (1997), 88, 267 Bequette (2003), 151, 181, 450
Bialkowski (1996), 58, 81 Bi et al. (1999), 47 Bi et al. (2000), 47, 327 Blickley (1990), 235 Boe and Chang (1988), 60 Bohl and McAvoy (1976a), 372, 373 Bohl and McAvoy (1976b), 311, 330,
352,353 Borresen and Grindal (1990), 29, 155 Brambilla et al. (1990), 58, 118, 178,
328 Branica et al. (2002), 176 Bryant et al. (1973), 42, 114-115, 151 Buckley (1964), 151 Bunzemeier(1998), 77, 266
Calcev and Gorez (1995), 71, 236 Callender et al. (1935/6), 26, 149, 154,
449 Camachoefa/. (1997), 169 Carr (1986), 235 Chandrashekar et al. (2002), 133, 136,
408,409,417,418 Changed al. (1997), 69 Chao et al. (1989), 106, 107, 111, 342,
344-349 Chau (2002), 237 Chen and Seborg (2002), 48, 81, 172,
261,280,327-328 Chen and Yang (2000), 56 Chen etal. (1997), 59 Chen et al. (1999a), 51, 80, 261, 280
536 Handbook of PI and PID Controller Tuning Rules
Chen etal. (1999b), 51,324 Chen et al. (2001), 193 Cheng and Hung (1985), 157, 164 Cheng and Yu (2000), 80 Chesmond(1982), 73, 238 Chidambaram (1994), 79 Chidambaram (1995a), 175-176 Chidambaram (1995b), 131,404 Chidambaram (1997), 131, 140, 405 Chidambaram (2000a), 63, 226 Chidambaram (2000b), 84, 271 Chidambaram (2000c), 132, 135, 416 Chidambaram and Kalyan (1996), 433 Chidambaram and Sree (2003), 79,
261,271 Chidambaram et al. (2005), 49 Chien (1988), 58, 81, 190, 206, 211,
264, 267, 269, 285, 291, 294, 386, 387, 389, 393, 394, 395
Chien and Fruehauf (1990), 81, 206, 211, 264, 267, 269, 285, 291, 294, 386, 387, 389, 393, 394, 395
Chien era/. (1952), 27, 155 Chien et al. (1999), 65, 86, 229, 275 Chiue/a/. (1973), 42, 326 Chun etal. (1999), 59 Cluett and Wang (1997), 54-55, 78,
170-171,260 Cohen and Coon (1953), 28, 155 Concept, 9 Connell(1996), 156 ControlSoft Inc. (2005), 156 Coon (1956), 76, 89 Coon (1964), 76, 89 Corripio (1990), 236, 253 Cox etal. (1994), 73 Cox etal. (1997), 55
Davydov et al. (1995), 46, 188 De Paor (1993), 236 De Paor and O'Malley (1989), 131,404 Devanathan(1991), 37, 161 Direct synthesis, 24-25, 42-58, 61, 62,
63-64, 65, 67, 68, 71-72, 75, 78-81, 84, 86, 90-91, 93, 100, 114-118,
123-124, 125, 126-127, 129, 131-133, 135-136, 138-139, 141, 142, 143-144, 146, 147, 148, 151, 152, 153, 168-178, 181, 183-184, 185, 186, 188-189, 193, 203-205, 208, 210, 212, 219-220, 225, 226, 227-229, 230, 242, 245, 246-251, 260-261, 263, 264, 265, 268, 271, 275, 277, 279-280, 284, 285, 291, 292, 293, 296, 302, 303, 304, 306, 307, 308, 321-328, 332, 338, 341, 349-350, 351, 353, 354, 367, 370, 371, 372, 378, 379, 380, 385, 387, 388, 390, 391, 392, 394, 396, 397, 399, 400, 404-407, 408-409, 410-413, 415, 416^119, 427, 433, 439^141, 442, 445^146, 447^148, 452, 453, 454, 455^456, 457^459, 460-461 Frequency domain criteria, 49-58,
64, 79-81, 124, 152, 174-178, 181, 183-184, 193, 208, 268, 292, 321-324, 332, 399, 400, 433, 453
Time domain criteria, 42-49, 61, 62, 63-64, 65, 67, 68, 78-79, 123, 141, 142, 143-144, 147, 148, 168-173, 227-229, 230, 265, 306, 325-328, 332, 372, 380, 385, 415, 433, 445-446, 447-448
Dmonetal. (1997), 240
ECOSSE Team (1996a), 37 ECOSSE Team (1996b), 73, 238 Edgar et al. (1997), 31, 71, 197, 224,
253, 257 Ettaleb and Roche (2000), 47 expert system, 472
Farrington (1950), 235 Fanuc, 7 Fertik(1975), 150 Fertik and Sharpe (1979), 29 Fischer and Porter, 9, 12, 17 Fisher-Rosemount, 17 Ford (1953), 259 Foxboro, 9, 11, 12, 17, 209
Index 537
Frank and Lenz (1969), 30, 33, 34, 35 Friman and Waller (1997), 52, 71, 178 Fruehauf etal. (1993), 46, 78, 169
Gaikward and Chidambaram (2000), 132,406
Gain margin, analytical calculation, 463-469
Gain margin, figures, 471, 473-474 Gain and phase margins, constant, 475-
483 PI controller, FOLPD model, 475^77 PI controller, IPD model, 477-480 PD controller, FOLIPD model, 483 PID controller, FOLPD model,
classical controller 1, 480-481 PID controller, SOSPD model,
series controller, 482 PID controller, SOSPD model with a
negative zero, classical controller 1, 482-483
Gallier and Otto (1968), 38, 108, 162, 315
Garcia and Castelo (2000), 239 Genesis, 15 Geng and Geary (1993), 60, 179 Gerry (1988), 149, 159 Gerry (1999), 58, 178 Gerry (2003), 37 Gerry and Hansen (1987), 149 Gong et al. (1996), 190 Goodwin etal. (2001), 189 Gorecki et al. (1989), 45, 49, 168, 174 Gorez (2003), 48, 173,185 Gorez and Klan (2000), 326,454
Haalman (1965), 33, 77, 149, 279, 352 Haeri (2002), 225 Haeri (2005), 49 Hagglund and Astrom (1991), 73 Hagglund and Astrom (2002), 56, 64,
84, 136 Hang and Astrom (1988a), 220-221 Hang and Astrom (1988b), 236 Hang and Cao (1996), 221
Hanged/. (1991), 45, 221 Hang et al. (1993a), 50, 321, 332 Hang et al. (1993b), 50, 70, 211, 255 Hange/a/. (1994), 338 Hansen (1998), 370 Hansen (2000), 151, 271, 379 Harriott (1964), 238 Harris and Tyreus (1987), 206, 234 Harrold (1999), 195,253 Hartmann and Braun, 9 Hartreeera/. (1937), 451 Hassan (1993), 106,312 Hay (1998), 29, 73, 76-77, 156, 238,
259 Hazebroek and Van der Waerden
(1950), 27, 32-33, 77 Heck (1969), 28-29 Henry and Schaedel (2005), 52-53 Hiroi and Terauchi (1986), 217, 222 Ho and Xu (1998), 132,433 Ho era/. (1994), 321 Ho etal. (1995a), 321-322 Ho etal. (1997), 322 Wo etal. (1998), 158, 163 Wo etal. (1999), 51, 158, 163 no etal. (2001), 234 Honeywell, 7, 11, 15, 16, 18, 45, 349 Horned/. (1996), 181, 184 Hougen (1979), 49, 79, 90, 115, 126,
208,268,292,351 Hougen (1988), 116,268,351 Huang and Chao (1982), 195, 197, 198,
199,201,203 Huang and Chen (1997), 415, 433 Huang and Chen (1999), 415, 433 Huang and Jeng (2002), 38, 150 Huang and Jeng (2003), 167, 320-321 Huang and Lin (1995), 420, 434^138 Huang et al. (1996), 32, 38, 103-104,
108-110,355-362 Huang e^a/. (1998), 58, 329 Huang et al. (2000), 212, 219, 354,
367,390,391,396,397 Huang et al. (2002), 182
538 Handbook of PI and PID Controller Tuning Rules
Huang et al. (2005), 58, 60, 205, 207, 332,335
Huba and Zakova (2003), 79, 91 Hwang (1995), 36, 39, 107, 112-113,
159-160, 165-166, 314-315, 316-319
Hwang and Chang (1987), 70 Hwang and Fang (1995), 37, 41, 160,
166, 167
Intellution, 7 Isaksson and Graebe (1999), 59
Jahanmiri and Fallahi (1997), 336 Jhunjhunwala and Chidambaram
(2001), 135,404,413 Jonese/a/. (1997), 73,239 Juang and Wang (1995), 169
Kamimura et al. (1994), 46, 151 Karaboga and Kalinli (1996), 237 Kasahara e? a/. (1997), 219 Kasahara et al. (1999), 190-191 Kasaharae/o/. (2001), 229 Kaya (2003), 277 Kaya and Atherton (1999), 277 Kaya and Scheib (1988), 196, 197, 198,
199,201,215-216,233 Keane et al. (2000), 320 Keviczky and Csaki (1973), 43, 111,
150, 151 Khan and Lehman (1996), 39-40, 45,
47 Khodabakhshian and Golbon (2004),
124 Kim et al. (2004), 219-220 Kinney (1983), 253 Klein etal. (1992), 29 Kookos etal. (1999), 79 Kraus (1986), 209 Kristiansson (2003), 57, 72, 119, 183-
184,241,244,330,334,337 Kristiansson and Lennartson (1998a),
248
Kristiansson and Lennartson (1998b), 248
Kristiansson and Lennartson (2000), 72, 249-250
Kristiansson and Lennartson (2002), 72,250-251,264,285
Kristiansson et al. (2000), 243 Kuhn(1995),46, 188 Kwake/a/. (2000), 439-141
Landau and Voda (1992), 330-331 Lee and Edgar (2002), 66, 87, 101, 140,
182, 231, 232, 276, 278, 303, 305, 333, 422,425,426
Lee and Shi (2002), 62, 206 Lee et al. (1998), 59, 178, 328, 329 Lee et al. (2000), 407, 413, 428, 431,
443, 444 Leeds and Northup, 7 Lennartson and Kristiansson (1997),
248 Leonard (1994), 260 Leva (1993), 73, 246 Leva (2001), 53, 58, 178, 192 Leva (2005), 220 Leva and Colombo (2000), 191 Leva and Colombo (2004), 220 Leva et al. (1994), 57, 322-323 Leva etal. (2003), 57, 118, 192, 324 Li etal. (1994), 177 Li et al. (2004), 385 Lime/a/. (1985), 61,332 Liptak (1970), 238 Liptak(2001), 30, 156 Liu et al. (2003), 286, 287-288, 374-
375,376-377 Lloyd (1994), 239 Lopez et al. (1969), 106,310 Loron(1997), 153 Luoetal. (1996), 237 Luyben (1996), 267 Luyben(1998), 132 Luyben (2000), 124, 181 Luyben and Luyben (1997), 237
Index 539
MacLellan(1950), 72, 238 Maffezzoni and Rocco (1997), 190 Majhi and Atherton (2000), 130, 423-
424 Majhi and Mahanta (2001), 304 Majhi and Litz (2003), 353 Manned/. (2001), 44, 172 Mantz and Tacconi (1989), 252 Marchetti and Scali (2000), 329 Marlin (1995), 31,38,157, 162 Matausek and Kvascev (2003), 57 Matsuba et al. (1998), 60, 179 Maximum sensitivity, analytical
calculation, 469^170 Maximum sensitivity, figures, 471 Maximum sensitivity, constant, 479-
480 McAnany(1993), 45 McAvoy and Johnson (1967), 105, 235,
343 McMillan (1984), 91, 146, 179, 281,
429 McMillan (1994), 29, 60, 70, 73, 236,
238 Mizutani and Hiroi (1991), 14, 15 Modicon, 9 Modcomp, 15 Mollenkamp et al. (1973), 42, 326 Moore, 13, 17 Morari and Zafiriou (1989), 190 Morilla et al. (2000), 171 Moras (1999), 27, 156 Murrill (1967), 28, 30, 33, 34, 155,
157, 158
NI Labview (2001), 74, 240 Normey-Rico et al. (2000), 186, 263 Normey-Rico et al. (2001), 265
O'Connor and Denn (1972), 161 O'Dwyer (2001a), 51, 80-81, 204, 280,
353, 387 O'Dwyer (2001b), 209-210, 254 Ogawa(1995), 59, 81 Ogawa and Katayama (2001), 230
Omron, 13 Oppelt(1951),27 Other tuning rules, 72-74, 82, 151,
237-240, 258 Oubrahim and Leonard (1998), 289,
340 Ou and Chen (1995), 199, 203 Ozawa et al. (2003), 229
Panda et al. (2004), 119, 329, 332, 334, 350
Paraskevopoulos et al. (2004), 137-139 Parked/ . (1998), 421 Parr (1989), 70, 73, 155, 235, 238 Pecharroman (2000), 84, 92, 121, 122,
365-366 Pecharroman and Pagola (2000), 121,
270, 295, 365, 381-382 Pemberton (1972a), 30, 325 Pemberton (1972b), 325, 326 Peng and Wu (2000), 157 Penner(1988), 82 Performance index minimisation, 24,
30-42, 63, 65, 77-78, 84, 85, 89-90, 92, 94-100, 102-113, 120-121, 122, 128, 130, 137-138, 145, 149-150, 157-168, 187, 195-203, 213-214, 215-216, 218, 223, 224-225, 229, 233, 241, 243-244, 253, 256, 257, 259-260, 266-267, 270-271, 272, 273, 274-275, 279, 290-291, 295-296, 297, 298-302, 310-321, 342-349, 352, 355-362, 363-367, 368-369, 381-382, 383-384, 398, 401-402, 403^104, 414, 420, 423^124, 432,434-438
Regulator tuning, 30-37, 77, 85, 89-90, 94-97, 102-107, 137, 145, 149-150, 157-161, 195-199, 215-216, 224, 233, 253, 257, 259, 266-267, 273, 274, 279, 290-291, 297, 298-299, 310-315, 342-346, 352, 355-358, 363-365, 368-369, 403, 414,420,432,434^35 Minimum error, 37, 149, 159
540 Handbook of PI and PID Controller Tuning Rules
Minimum IAE, 30-32, 77, 89, 102-105, 149-150, 157, 195-197, 215, 224, 233, 253, 257, 266-267, 273, 290, 297, 310, 342-343, 355-358, 368-369, 414, 420, 434^435
Minimum IE, 37, 161, 363-365 Minimum ISE, 32-34, 77, 85, 94,
96, 105, 137, 149, 158, 197,215, 233, 259, 274, 279, 298, 299, 343-344, 352, 403
Minimum ISTES, 36, 159 Minimum ISTSE, 36, 158-159,
259, 403 Minimum ITAE, 34-35, 77, 90,
106, 145, 158, 198-199, 216, 233, 291, 310-313, 344-345, 352,432
Minimum ITSE, 35, 107, 199, 259, 345-346, 403
Minimum IAE or ISE, 37 Minimum performance index -
other, 85, 95, 96-97, 161, 274, 298-299
Modified minimum IAE, 157 Nearly minimum IAE and ITAE,
37, 160 Nearly minimum IAE, ISE and
ITAE, 36, 107, 159-160, 314-315
Servo tuning, 38^11, 85, 97-100, 108-113, 130, 137, 150, 162-166, 187, 199-203, 213-214, 216, 229, 233, 260, 274-275, 300-302, 315-319, 346-349, 359-362, 383-384, 403^104, 420, 423^24, 436^38 Approximately minimum ITAE, 40 Minimum IAE, 38-39, 108-110,
150, 162, 187, 199-200, 216, 233, 315, 346, 359-362, 383, 420, 436^138
Minimum ISE, 39-^10, 85, 97, 99, 111, 137, 150, 162-163, 187, 201, 213, 216, 229, 233, 260, 274,300,301,347,384,403
Minimum ISTES, 40, 165, 214 Minimum ISTSE, 40, 164-165,
213-214,260,404 Minimum ITAE, 40, 111, 163-164,
187, 201-203, 216, 233, 315-316,347-348,384
Minimum ITSE, 111, 130, 203, 260, 348-349, 403, 423^124
Minimum performance index -other, 85, 98, 99-100, 274-275, 300-302
Modified minimum ITAE, 164 Nearly minimum IAE and ITAE,
41, 166 Nearly minimum IAE, ISE and
ITAE, 112-113, 165-166, 316-319
Small IAE, 39 Other tuning, 41^12, 63, 65, 77-78,
84, 92, 120-121, 122, 128, 138, 150, 167-168, 218, 223, 224-225, 241, 243-244, 256, 260, 270-271, 272, 295-296, 319-321, 365-367, 381-382,398,401-402,404
Peric et al. (1997), 60, 91, 179, 281 Pessen (1953), 254 Pessen (1954), 254 Pessen (1994), 70, 157,210 Petitt and Carr (1987), 235 Phase margin, analytical calculation,
463-469 Phase margin, figures, 471, 473^174 PI controller structures
Controller with a double integral term, 6-7, 22, 88
Controller with proportional term acting on the output 1, 6, 22, 65, 85-86,94-100,137-139
Controller with proportional term acting on the output 2, 6, 22, 66, 87, 101, 140
Controller with set-point weighting, 6, 22, 63-64, 75, 84, 92-93, 120-121, 122, 128-129, 135-136, 143-144, 148
541
Ideal controller, 6, 22, 26-60, 67, 70-74, 76-82, 89-91, 102-119, 123-124, 126-127, 130-134, 141, 145-146, 147, 149-151, 152, 153
Ideal controller in series with a first order lag, 6, 22, 61, 68, 69, 83, 125, 142
Ideal controller in series with a second order filter, 6, 22, 62
PID controller structures Alternative controller 1,18, 24, 307 Alternative controller 2, 18, 24, 308 Alternative controller 3, 18, 24 Alternative controller 4, 18, 24, 372-
373 Blending controller, 11, 23, 193 Classical controller 1, 11, 23, 194-
207, 253, 266-267, 290-291, 342-350,387,394,414,432,451
Classical controller 2, 12, 23, 208, 268,292,351
Classical controller 4, 12, 23, 211, 255, 269, 294, 389, 395
Controller with filtered derivative, 8-9, 23, 187-192, 246-251, 264, 285, 338,386,393,400,457-459
Controller with filtered derivative and dynamics on the controlled variable, 10,23,265
Controller with filtered derivative in series with a second order filter, 9, 23, 339
Controller with filtered derivative with setpoint weighting 1, 9, 23, 286
Controller with filtered derivative with setpoint weighting 2, 9-10, 23,374-375
Controller with filtered derivative with setpoint weighting 3, 10, 23, 287-288
Controller with filtered derivative with setpoint weighting 4, 10, 23, 376-377
Ideal controller, 5, 7, 19, 23, 24, 154-180, 235-240, 259-261, 279-281, 309-331, 383-384, 392, 398, 403-407, 427-429, 442^143, 449, 452, 454, 455-456
Ideal controller in series with a first order filter, 8, 23, 336
Ideal controller in series with a first order lag, 8, 23, 181-182, 241-242, 262, 282-283, 332-335, 385, 399, 408^109, 430, 445-446, 450, 453
Ideal controller in series with a second order filter, 8, 23, 183-184, 243-244, 337
Ideal controller with first order filter and dynamics on the controlled variable, 8, 23
Ideal controller with first order filter and setpoint weighting 1, 8, 23, 186
Ideal controller with first order filter and setpoint weighting 2, 8, 23, 263
Ideal controller with setpoint weighting 1,10, 23, 252, 289, 340, 410-413,431,444
Ideal controller with setpoint weighting 2, 10, 23, 341
Ideal controller with weighted proportional term, 8, 23, 185, 245, 284
Industrial controller, 17-18, 24, 233-234, 306, 380
Non-interacting controller 1, 12-13, 23,212,354-362,390,396
Non-interacting controller 2a, 13, 23, 213-214
Non-interacting controller 2b, 13, 23, 215-216
Non-interacting controller 3, 15, 23, 420, 434^138
Non-interacting controller 4, 15, 23, 224-225, 257, 273, 297, 368-369
Non-interacting controller 5, 16, 23, 226, 370
542 Handbook of PI and PID Controller Tuning Rules
Non-interacting controller 6 (I-PD controller), 16, 23, 227-229, 274-275, 298-302, 371
Non-interacting controller 7, 16, 23, 230,460^161
Non-interacting controller 8, 16, 23, 276,303,421-422,439^141
Non-interacting controller 9, 16, 23, 258
Non-interacting controller 10, 16, 24, 277, 423^124
Non-interacting controller 11, 16-17, 24,231,304
Non-interacting controller 12, 17, 24, 232, 278, 305, 425^126
Non-interacting controller 13, 17 Non-interacting controller based on
the two degree of freedom structure 1, 13, 23, 217-221, 256, 270-271, 295-296, 363-367, 379, 381-382, 391, 397, 401^02, 416—419, 447-448
Non-interacting controller based on the two degree of freedom structure 2, 13-14, 23, 222
Non-interacting controller based on the two degree of freedom structure 3, 14, 23, 223, 272
PD-I-PD (only PD is DOF) incomplete 2DOF algorithm, 14
PI-PID (only PI is DOF) incomplete 2DOF algorithm, 14
Series controller (classical controller 3), 12, 19, 23, 209-210, 254, 293, 352-353,378,388,415,433
Super (or complete) 2DOF PID algorithm, 15
Pinnella e« a/. (1986), 44 Polonyi (1989), 398 Pomerleau and Poulin (2004), 118, 123,
125 Potocnik etal. (2001), 53-54 Poulin and Pomerleau (1996), 77, 90,
145,291,432 Poulin and Pomerleau (1997), 291, 432
Poulin and Pomerleau (1999), 78, 90 Poulin et al. (1996), 117, 146, 291,
350, 387, 394 Pramod and Chidambaram (2000), 406 Prashanti and Chidambaram (2000),
410,411,412-413 Process modelling
Delay model, 21, 149-151, 449^51 Fifth order system plus delay model,
21,455-461 FOLIPD model, 20, 22, 23, 24, 89-
101,279-308,500-501 FOLPD model, 20, 22, 23, 24, 26-66,
154-234,493^98 FOLPD model with a negative zero,
20, 69,498 FOLPD model with a positive zero,
20, 67-68 General model with a repeated pole,
21, 152,452-453,505 General model with integrator, 22,
153 General stable non-oscillating model
with a time delay, 22, 454 IPD model, 20, 22, 23, 24, 76-88,
259-278, 499-500 I2PD model, 20, 374-380
Non-model specific, 20, 22, 23, 24, 70-75, 235-258, 498-499
SOSIPD model, 20, 122, 381-382, 503
SOSPD model, 20, 22, 23, 24, 102-121, 309-373, 501-503
SOSPD model with a negative zero, 21,125,392-397,504
SOSPD model with a positive zero, 21, 123-124,383-391,504
Third order system plus time delay model, 21, 126-129, 398^102
Unstable FOLPD model, 21, 130-140, 403^126, 504-505
Unstable FOLPD model with a positive zero, 21, 141-144
Index 543
Unstable SOSPD model (one unstable pole), 21, 145-146, 427-441,505
Unstable SOSPD model with a positive zero, 21, 147-148, 445-448
Unstable SOSPD model (two unstable poles), 21, 442^144, 505
Process reaction curve methods, 24, 26-30, 76-77, 89, 149, 154-156, 194-195, 209-210, 211, 217, 222, 224,259,266,309,449,451
Pulkkinene/a/. (1993), 59
Reswick(1956),28 Rice and Cooper (2002), 262, 267 Rivera etal. (1986), 58, 178 Rivera and Jun (2000), 61, 83, 280,
283, 329, 333 Robbins (2002), 71,237 Robust, 25, 58-59, 61, 62, 66, 69, 81-
82, 83, 86, 87, 101, 118-119, 133-134, 140, 151, 178, 181-182, 184, 190-192, 206, 211, 220, 231, 232, 234, 261, 262, 264, 267, 269, 276, 278, 280, 282-283, 285, 286, 287-288, 291, 294, 303, 305, 328-330, 332-334, 336, 337, 338, 339, 374-375, 376-377, 386, 387, 389, 393, 394, 395, 407, 413, 421-422, 425-426,428,430,431,444,450
Rotach (1994), 238-239 Rotach(1995), 78, 261 Rotstein and Lewin (1991), 133-134,
428 Rovira et al. (1969), 38,40,162, 163 Rutherford (1950), 72,237
Sadeghi and Tych (2003), 162, 163, 164
Sain and Ozgen (1992), 156 Saitoh a/. (1990), 169 Sakaie/a/. (1989), 29 S ATT Instruments, 17
Schaedel (1997), 52, 118, 152, 181, 189,204,350,399,400,453
Schneider (1988), 45 Seki et al. (2000), 236 Setiawan et al. (2000), 65 Siemens, 9 Shen (1999), 363 Shen (2000), 364, 365 Shen (2002), 218, 256 Shi and Lee (2002), 206 Shi and Lee (2004), 339 Shigemasa et al. (1987), 227-228 Shines / . (1997), 240 Shinskey (1988), 31, 32, 77, 150, 196,
197, 224, 266, 273, 310, 342, 414 Shinskey (1994), 32, 77, 89, 102, 150,
197, 225, 266, 273, 290, 297, 343, 369
Shinskey (1996), 31, 32, 77, 102, 105, 150, 197, 224, 267, 273, 343, 368, 369
Shinskey (2000), 29, 195 Shinskey (2001), 29 Sklaroff(1992),45, 349 Skoczowski and Tarasiejski (1996),
452 Skogestad (2001), 77, 78, 150 Skogestad (2003), 48, 78, 79, 150, 151,
293,353,378 Skogestad (2004a), 82 Skogestad (2004b), 79, 306, 327, 380 Smith (1998), 55 Smith (2002), 40, 58, 59, 81,82 Smith (2003), 164, 237, 330 Smith and Corripio (1997), 30, 38, 43,
197, 200, 203 Smith et al. (1975), 43, 326, 349 Somani et al. (1992), 50, 116-117, 174 Square D, 15 Sree and Chidambaram (2003a), 67, 68 Sree and Chidambaram (2003b), 142,
143, 144 Sree and Chidambaram (2003c), 141,
144
544 Handbook of PI and PID Controller Tuning Rules
Sree and Chidambaram (2004), 147, 148, 445^46, 447-448
Sree and Chidambaram (2005a), 419 Sree and Chidambaram (2005b), 261 Sree et al. (2004), 49, 133, 173, 406-
407 Srividya and Chidambaram (1997), 79 Srinivas and Chidambaram (2001), 64,
219,417 St. Clair (1997), 29, 195, 253 Streeter et al. (2003), 245 Sung era/. (1996), 313, 316 Suyama(1992),43, 169,326 Suyama (1993), 227-228 Syrcos and Kookos (2005), 161
Tachibana (1984), 279 Taguchi and Araki (2000), 63, 84, 92,
120-121, 122, 128, 218, 271, 296, 366-367, 382, 401-402
Tan etal. (1996), 58, 117, 177 Tan et al. (1998a), 282-283 Tan et al. (1998b), 83, 182, 282 Tan et al. (1999a), 209, 237, 254 Tan era/. (1999b), 407 Tan era/. (1999c), 134 Tan era/. (2001), 237, 254 Tang et al. (2002), 74, 240 Tavakoli and Fleming (2003), 39 Tavakoli and Tavakoli (2003), 187 Taylor, 17 Thomasson (1997), 58, 59, 81 Tinham (1989), 235 Toshiba, 11, 16 Tsang and Rad (1995), 203, 307 Tsang et al. (1993), 62, 203, 210, 307,
308 Turnbull, 12 Tyreus and Luyben (1992), 78
Ultimate cycle, 25, 60, 70-71, 88, 91, 124, 146, 179-180, 207, 210, 220-221, 235-237, 252, 253, 254, 255, 257, 267, 281, 289, 330-331, 335, 340, 353, 373, 429
Valentine and Chidambaram (1997a), 47, 171
Valentine and Chidambaram (1997b), 405
Valentine and Chidambaram (1998), 131
VanDoren (1998), 224, 257 Van der Grinten (1963), 151, 168, 279,
325 Venkatashankar and Chidambaram
(1994), 131 Visioli (2001), 259, 260, 403, 404 Viteckova (1999), 43, 79, 280, 326-
327 Viteckova and Vitecek (2002), 44, 173 Viteckova and Vitecek (2003), 44, 173 Viteckova et al. (2000a), 43, 79, 280,
326-327 Viteckova et al. (2000b), 43, 327 Voda and Landau (1995), 43, 51-52 Vrancic (1996), 71, 75, 127, 129, 247,
457 Vrancic and Lumbar (2004), 459, 460-
461 Vrancic era/. (1996), 71, 127 Vrancic et al. (1999), 247, 455^56,
458 Vrancic et al. (2004a), 127, 459, 460-
461 Vrancic et al. (2004b), 127
Wade (1994), 73, 238 Wang and Clements (1995), 326 Wang and Cai (2001), 296 Wang and Cai (2002), 296, 418 Wang and Cluett (1997), 78, 260 Wang and Cluett (2000), 54-55, 170-
171 Wang and Jin (2004), 427, 442 Wang and Shao (1999), 323 Wang and Shao (2000a), 56 Wang and Shao (2000b), 324 Wang era/. (1995a), 162, 164, 383, 384 Wang era/. (1995b), 71,242 Wang era/. (1999), 323
Index 545
Wang et al. (2000a), 55 Wang et al. (2000b), 384, 392 Wang et al. (2001a), 384, 392 Wang et al. (2001b), 303 Wang et al. (2002), 204-205 Wills (1962a), 315, 325, 330 Wills (1962b), 310, 315 Wilton (1999), 41-42, 319-320 Witt and Waggoner (1990), 194-195,
196, 198,200,202 Wojsznis and Blevins (1995), 179 Wojsznis et al. (1999), 179-180, 237 Wolfe (1951), 28, 76, 149
Xu and Shao (2003a), 296 Xu and Shao (2003b), 419 Xu and Shao (2003c), 296 Xu et al (2004), 58
Yokogawa, 8, 11, 13 Young (1955), 72 Yu (1988), 30-31, 33-34, 35 Yu (1999), 237
Zhang (1994), 46, 388, 394 Zhang and Xu (2002), 139, 430 Zhang era/. (1996), 206 Zhang etal. (1999), 81,282 Zhang et al. (2002), 182, 192, 206, 333 Zhong and Li (2002), 59, 81, 338 Zhuang and Atherton (1993), 33, 36,
39,40, 158, 159, 163, 164-165, 176-177,213-214
Ziegler and Nichols (1942), 27, 70, 76, 154, 235
Zou and Brigham (1998), 261 Zou etal. (1997), 178,261
Yang and Clarke (1996), 321, 353 Yi and De Moor (1994), 246
Handbook of
Controller Timing Rules
2nd Edition
. .ic vast majority ol automatic controllers used to compensate industrial processes arc ol I'l or I'll) type. This hook comprehensively compiles, using a unified notation, tuning rules for these controllers proposed over the last sewn decades i 1935-2005). The tuning rides are carefully categorized and application information about each rule is given. The book ''•"•"sses controller architecture and process modeling
;. as well as the performance and robustness of loops compensated with l'l or I'll) controllers. This unique publication brings together in an easv-to-u.se format material previously published in a large number ol papers and hooks.
This wholly revised second edition extends the presentation ol l'l and I ' l l ) controller tuning rules, lor single variable processes with time delays, to include additional rules compiled since the first edition was published in 2008.
Key Features • Addresses the needs of a niche market where no
comparable book is available
A comprehensive compilation of PI and FID controller tuning rules
Makes t h e tun ing rules easily accessible to researchers and practi t ioners through unified notation
Highlights the marked increase in the number of tuning rules compiled, from 600 in the first edition to 1,134 in this second edition
Imperial College Press
P424 he ISBN 1-86094 <>?'/ 1
www.icpress.co.uk