mse 490 final project paper

A Review of Project Paper #24 A simple method of tuning PID controllers for stable and unstable FOPTD systems (R. Padma Sree, et al.)

MSE 490 – Advanced PID Control

Kjell Sadowski

12/04/2014

Abstract

This paper reviews the PI/PID controller design method proposed by R.Padma Sree, M. N. Srinivas and M.

Chidambaram in their paper titled: A simple method of tuning PID controllers for stable and unstable FOPTD systems.

A derivation of the theory for the proposed method is displayed and further analyzed. Results of the relevant

sections are replicated using simulations in MATLAB to check the correspondence of them to the results

they are reporting. Further analysis will show the discrepancies between results from the simulations and the

ones reported in the paper. A discussion of the results will be presented along with a brief literature review.

Page i

CONTENTS

List of Figures ...................................................................................................................................................................... ii

List of Tables ....................................................................................................................................................................... ii

Glossary ................................................................................................................................................................................ ii

1 Introduction ................................................................................................................................................................ 1

2 Problem Statement .................................................................................................................................................... 1

3 Theory.......................................................................................................................................................................... 2

3.1 Derivation of Proposed Method .................................................................................................................... 2

3.2 Derivation of Two-Parameter PID Tuning Method .................................................................................. 5

3.3 Analysis of Proposed Method ........................................................................................................................ 6

4 Simulation Studies...................................................................................................................................................... 7

4.1 PI Controller ..................................................................................................................................................... 7

4.2 PID Controller ................................................................................................................................................ 10

5 Analysis of Results ................................................................................................................................................... 14

6 Discussion ................................................................................................................................................................. 16

7 Literature survey ...................................................................................................................................................... 16

8 Conclusion ................................................................................................................................................................ 16

9 Bibliography .............................................................................................................................................................. 17

Appendix A: MATLAB Error Calculation Code ......................................................................................................... 18

Page ii

LIST OF FIGURES

Figure 1 - PI Controller Set Point Response .................................................................................................................. 8 Figure 2 - PI Controller Regulatory Response ............................................................................................................... 8 Figure 3 - PI Controller Uncertainty in Parameters Regulatory Response ................................................................ 9 Figure 4 - PI Controller Uncertainty in Parameters Set Point Response ................................................................. 10 Figure 5 - PID Controller Set Point Response ............................................................................................................ 11 Figure 6 - PID Controller Regulatory Response.......................................................................................................... 12 Figure 7 - PID Controller Parameter Uncertainty Regulatory Response ................................................................ 13 Figure 8 - PID Controller Parameter Uncertainty Set Point Response ................................................................... 14

LIST OF TABLES

Table 1 - Analysis of System Types ................................................................................................................................. 6 Table 2 - PI Controller Errors .......................................................................................................................................... 7 Table 3 - PI Controller Parameter Uncertainty Errors ................................................................................................. 9 Table 4 - PID Controller Errors ..................................................................................................................................... 11 Table 5 - PID Controller Parameter Uncertainty Errors ........................................................................................... 12 Table 6 - PI Controller Simulation Results Comparison ............................................................................................ 14 Table 7 - PI Controller Parameter Uncertainty Simulation Results Comparison ................................................... 15 Table 8 - PID Controller Simulation Results Comparison ........................................................................................ 15 Table 9 - PID Controller Parameter Uncertainty Simulation Results Comparison ............................................... 15

GLOSSARY

FOPTD – First Order Plus Time Delay

IAE – Integral Absolute Error

IMC – Internal Model Control

ISE – Integral Square Error

ITAE – Integral Time Absolute Error

PID – Proportional Integral Derivative

SOPTD – Second Order Plus Time Delay

Z-N – Ziegler-Nichols

1 INTRODUCTION

The main idea behind the development of this method was to design a tuning method for a FOPTD model

which was simple and could be applied to both stable and unstable systems, which would be designed for set

point tracking and would provide an overall good response. This method would be able to be used to

calculate tuning parameters for both PI and PID type controllers.

In this report, the proposed tuning method of the analyzed paper [1] and the results for a stable FOPTD PI

and PID controller will be reproduced and analyzed along with the results from the related simulations.

Although this paper provides an analysis and results for unstable systems, this data will not be presented

because it is not within the scope of this analysis.

2 PROBLEM STATEMENT

The tuning method outlined in this paper was design for closed loop control of a set point tracking problem

or formally known as the Servo problem. The authors report that the “simulation results show that the

method gives a similar response as that of Ziegler-Nichols (Z-N) method and better response than that of

IMC method” [1].

The assumptions that the authors make when deriving the method for determining the controller parameters

are:

The system can be forced to reach the set point: 𝑦(𝑞) 𝑦𝑟(𝑞)⁄ = 1, and

The transfer function is open-loop stable.

The approach which the authors of this paper have taken to calculate the PID parameters is very different

than what many other control systems researchers have derived in the past. Compared to a method like the

Ziegler Nichols Continuous Cycling Method [2] which calculates the parameters from the actual system, these

parameters rely solely on the FOPTD model being as accurate as possible. The controller parameters

calculated from this method will be very sensitive to small changes in the system, which would make this

method difficult to implement in systems that have very dynamic system states.

3 THEORY

This section shows a complete derivation of the PID parameter equations presented in the analyzed paper [1],

a derivation of the two-parameter tuning equations and an analysis of the equations for determining the PID

parameters.

3.1 DERIVATION OF PROPOSED METHOD This section shows the derivation of the proposed method for the PID Controller parameter equations which

are presented in the analyzed paper [1]. Equations (1) to (14) in the analyzed paper are shown here and do not

correspond to the numbers attached to the equations in this section.

𝐹𝑂𝑃𝑇𝐷 𝑠𝑦𝑠𝑡𝑒𝑚:𝐹(𝑠) =𝑘𝑝𝑒−𝜏𝑑𝑠

𝜏𝑠+1

𝑃𝐼𝐷:𝑢(𝑠)

𝑒(𝑠)= 𝑘𝑐 [1 +

1

𝜏1𝑠+ 𝜏𝐷𝑠]

Closed Loop Transfer Function:

𝐹(𝑠) =𝑘𝑐𝑘𝑝𝑒−𝜏𝑑𝑠(1 +

1𝜏1𝑠

+ 𝜏𝐷𝑠)

[𝑘𝑐𝑘𝑝𝑒

−𝜏𝑑𝑠 (1 +1

𝜏1𝑠+ 𝜏𝐷𝑠)

(𝜏𝑠 + 1)+ 1] (𝜏𝑠 + 1)

(1)

= (𝑘𝑐𝑘𝑝𝑒−𝜏𝑑𝑠 (1 +

1𝜏1𝑠

+ 𝜏𝐷𝑠)

𝑘𝑐𝑘𝑝𝑒−𝜏𝑑𝑠 (1 +1

𝜏1𝑠+ 𝜏𝐷𝑠) + (𝜏𝑠 + 1)

) ∗ (𝑠

𝑠) (2)

= 𝑒−𝜏𝑑𝑠[𝑘𝑐𝑘𝑝𝑠 +

𝑘𝑐𝑘𝑝

𝜏𝐼+ 𝑘𝑐𝑘𝑝𝜏𝐷𝑠2

𝑒−𝜏𝑑𝑠 (𝑘𝑐𝑘𝑝𝑠 +𝑘𝑐𝑘𝑝

𝜏𝐼+ 𝑘𝑐𝑘𝑝𝜏𝐷𝑠2) + 𝜏𝑠2 + 𝑠

(3)

Setting a change of variables: 𝑞 = 𝑠𝜏 → 𝑠 =𝑞

𝜏

= 𝑒−𝜏𝑑𝑞𝜏 [

𝑘𝑐𝑘𝑝𝑞𝜏 +

𝑘𝑐𝑘𝑝

𝜏𝐼+ 𝑘𝑐𝑘𝑝𝜏𝐷

𝑞2

𝜏2

𝑒−𝜏𝑑𝑞𝜏 (

𝑘𝑐𝑘𝑝𝑞𝜏 +

𝑘𝑐𝑘𝑝

𝜏𝐼+ 𝑘𝑐𝑘𝑝𝜏𝐷

𝑞𝜏2

2) +

𝑞𝜏

2+

𝑞𝜏

] ∗ (𝜏𝑒0.5𝜖𝑞

𝜏𝑒0.5𝜖𝑞) (4)

= 𝑒−𝜏𝑑𝑞𝜏 [

𝑘𝑐𝑘𝑝𝑞 +𝑘𝑐𝑘𝑝𝜏

𝜏𝐼+

𝑘𝑐𝑘𝑝𝜏𝑑𝑞2

𝜏

𝑒−𝜏𝑑𝑞𝜏 (𝑘𝑐𝑘𝑝𝑞 +

𝑘𝑐𝑘𝑝𝜏𝜏𝐼

+𝑘𝑐𝑘𝑝𝜏𝑑𝑞2

𝜏 ) + 𝑞(𝑞 + 1)

(5)

Making the following variable substitutions

𝑘1 = 𝑘𝑐𝑘𝑝

𝑘2 =𝑘1𝜏

𝜏𝐼

𝑘3 =𝑘1𝜏

𝜏𝐷

𝜖 =𝜏𝑑

𝜏

The 𝑒−𝜖𝑞 term is removed from the numerator of equation (5) and separated from the transfer function

because this function only provides a time shift. The closed loop transfer function of an open loop stable

transfer function is analyzed starting with equation (6).

𝑦(𝑞)

𝑦𝑟(𝑞)= 𝑒−𝜖𝑞 [

(𝑘1𝑞 + 𝑘2 + 𝑘3𝑞2)𝑒0.5𝜖𝑞

(𝑘1𝑞 + 𝑘2 + 𝑘3𝑞2)𝑒−0.5𝜖𝑞 + (𝑞 + 1)𝑞𝑒0.5𝜖𝑞

] (6)

Taylor series expansions

𝑒0.5𝜖𝑞 = 1 + 0.5𝜖𝑞 + 0.125𝜖2𝑞2 + 0.0208333𝑞3𝜖^3 (7)

𝑒−0.5𝜖𝑞 = 1 − 0.5𝜖𝑞 + 0.125𝜖2𝑞2 − 0.0208333𝑞3𝜖^3 (8)

Since the objective of the transfer function is to reach the point where the system reaches the set point and

settles there, 𝑦(𝑞) 𝑦𝑟(𝑞)⁄ = 1 will be sets and the analysis will be carried using this result and numerator and

denominator of equation (9) will be equated.

𝑦(𝑞)

𝑦𝑟(𝑞)= 1 = 𝑒−𝜖𝑞 [

(𝑘1𝑞 + 𝑘2 + 𝑘3𝑞2)𝑒0.5𝜖𝑞

(𝑘1𝑞 + 𝑘2 + 𝑘3𝑞2)𝑒−0.5𝜖𝑞 + (𝑞 + 1)𝑞𝑒0.5𝜖𝑞

] (9)

(𝑞2 + 𝑞)𝑒0.5𝜖𝑞 + (𝑘1𝑞 + 𝑘2 + 𝑘3𝑞2)𝑒−0.5𝜖𝑞 = (𝑘1𝑞 + 𝑘2 + 𝑘3𝑞

2)𝑒0.5𝜖𝑞 (10)

𝑞2(𝑒0.5𝜖𝑞 + 𝑘3𝑒−0.5𝜖𝑞) + 𝑞(𝑒0.5𝜖𝑞 + 𝑘1𝑒

−0.5𝜖𝑞) + 𝑘2𝑒−0.5𝜖𝑞

= 𝑞2(𝑘3𝑒−0.5𝜖𝑞) + 𝑞(𝑘1𝑒

−0.5𝜖𝑞) + +𝑘2𝑒−0.5𝜖𝑞

(11)

(0.5𝜖 + 0.125𝜖2 − 0.5𝜖𝑘3 + 0.125𝜖2𝑘1 − 0.0208333𝜖3𝑘2)𝑞3

+ (0.5𝜖 + 𝑘3 − 0.5𝜖𝑘1 + 0.125𝜖2𝑘2 + 1)𝑞2 + (𝑘1 − 0.5𝜖𝑘2 + 1)𝑞 + 𝑘2

= (0.5𝜖𝑘3 + 0.125𝜖2𝑘1 + 0.0208333𝜖3𝑘2)𝑞3

+ (𝑘3 + 0.5𝜖𝑘1 + 0.125𝜖2𝑘2)𝑞2 + (𝑘1 + 0.5𝜖𝑘2)𝑞 + 𝑘2

(12)

𝑞1: 𝑘1 − 0.5𝜖𝑘2 + 1 = 𝑘1 + 0.5𝜖𝑘2

1 = 𝜖𝑘2

𝑘2 =1

𝜖 (13)

𝑞2: 0.5𝜖 + 𝑘3 − 0.5𝜖𝑘1 + 0.125𝜖2𝑘2 + 1 = 𝑘3 + 0.5𝜖𝑘1 + 0.125𝜖2𝑘2

0.5𝜖 − 0.5𝜖𝑘1 + 1 = 0.5𝑘1

0.5𝜖 + 1 = 𝜖𝑘1

𝑘1 =1

𝜖+ 0.5 (14)

𝑞3: 0.5𝜖 + 0.125𝜖2 − 0.5𝜖𝑘3 + 0.125𝜖2𝑘1 − 0.0208333𝜖3𝑘2

= 0.5𝜖𝑘3 + 0.125𝜖2𝑘1 + 0.0208333𝜖3𝑘2

0.5𝜖 + 0.125𝜖2 − 0.0208333𝜖3 (1

𝜖) = 0.5𝜖𝑘3 + 0.0208333𝜖3 (

1

𝜖)

0.5𝜖 + 0.08333𝜖2 = 𝜖𝑘3

𝑘3 = 0.5 + 0.08333𝜖 (15)

𝑘1 = 𝑘𝑐𝑘𝑝 =1

𝜖+ 0.5

𝑘𝑐𝑘𝑝 =𝜏

𝜏𝑑+ 0.5 (16)

𝑘2 =𝑘1𝜏

𝜏𝑑=

1

𝜖=

𝜏

𝜏𝑑

𝜏𝐼 = 𝜏𝑑 (1

𝜖+ 0.5)

𝜏𝐼 = 𝜏 + 0.5𝜏𝑑 (17)

𝑘3 =𝑘1𝜏𝐷

𝜏= 0.5 + 0.08333𝜖

𝜏𝐷 =𝜏0.5 + 0.08333𝜏 (

𝜏𝑑𝜏 )

𝑘1=

𝜏0.5 + 0.08333𝜏𝑑𝜏𝜏𝑑

+ 0.5(𝜏𝑑

𝜏𝑑) =

𝜏𝜏𝑑0.5 + 0.08333𝜏𝑑2

𝜏 + 0.5𝜏𝑑

𝜏𝐷 =𝜏𝑑0.5(𝜏 + 0.1667𝜏𝑑)

𝜏 + 0.5𝜏𝑑 (18)

Equations (16), (17) and (18) describe the parameters for the system gain, reset time and derivative time. If

the time delay (𝜏𝑑) and time constant (𝜏) are known, all of these parameters can be calculated.

3.2 DERIVATION OF TWO-PARAMETER PID TUNING METHOD This section shows the derivation of the two-parameter controller optimization method presented in the

paper of analysis [1]. This section describes the development of equation (17) in the studied paper.

Starting with equation (12), the left-hand side is multiplied by a constant, α#. q1 is multiplied by α1 and q2 and

q3 are multiplied by α2, which has a linear relation to α1 defined by the relation:

𝛼2 = 𝛽𝛼1 (19)

The expansion of equation (12) follows

(0.5𝜖 + 0.125𝜖2 − 0.5𝜖𝑘3 + 0.125𝜖2𝑘1 − 0.0208333𝜖3𝑘2)𝛼2𝑞3

+ (0.5𝜖 + 𝑘3 − 0.5𝜖𝑘1 + 0.125𝜖2𝑘2 + 1)𝛼2𝑞2 + (𝑘1 − 0.5𝜖𝑘2 + 1)𝛼1𝑞

+ 𝑘2 = (0.5𝜖𝑘3 + 0.125𝜖2𝑘1 + 0.0208333𝜖3𝑘2)𝑞

3

+ (𝑘3 + 0.5𝜖𝑘1 + 0.125𝜖2𝑘2)𝑞2 + (𝑘1 + 0.5𝜖𝑘2)𝑞 + 𝑘2

(20)

𝛼1(𝑘1 − 0.5𝜖𝑘2 + 1) = 𝑘1 + 0.5𝜖𝑘2

𝛼1𝑘1 − 0.5𝜖𝑘2𝛼1 + 𝛼1 = 𝑘1 + 0.5𝜖𝑘2

−𝛼 = 𝑘1 − 𝛼𝑘1 + 0.5𝜖𝑘2 + 0.5𝜖𝑘2𝛼

−𝛼 = (1 − 𝛼1)𝑘1 + (1 + 𝛼1)0.5𝜖𝑘2 (21)

𝛼2(0.5𝜖 + 𝑘3 − 0.5𝜖𝑘1 + 0.125𝜖2𝑘2 + 1) = (𝑘3 + 0.5𝜖𝑘1 + 0.125𝜖2𝑘2)

𝛼2(1 − 0.5𝜖) = 𝑘3 − 𝛼2𝑘3 + 0.5𝜖𝑘1 + 0.5𝜖𝑘1𝛼2 + 0.125𝜖2𝑘2 − 0.125𝜖2𝑘2𝛼2

𝛼2(1 − 0.5𝜖) = (1 − 𝛼2)𝑘3 + 0.5𝜖(1 + 𝛼2)𝑘1 + 0.125𝜖2(1 − 𝛼2)𝑘2 (22)

(1

0.5𝜖) (0.5𝜖 + 0.125𝜖2 − 0.5𝜖𝑘3 + 0.125𝜖2𝑘1 −

0.25

12𝜖3𝑘2)𝛼2

= (0.5𝜖𝑘3 + 0.125𝜖2𝑘1 +0.25

12𝜖3𝑘2)(

1

0.5𝜖)

𝛼2 + 0.25𝜖𝛼 − 𝑘3𝛼 + 0.25𝜖𝑘1𝛼2 −0.25

6𝜖2𝑘2𝛼2 = 𝑘3 + 0.25𝜖𝑘1

𝛼(1 + 0.25𝜖) = 𝑘3(1 + 𝛼2) + 0.25𝜖𝑘1(1 − 𝛼2) +0.25

6𝜖2𝑘2𝛼2 (23)

Using equations (21), (22) and (23) to form a matrix of the linear equations:

[

1 − 𝛼1 0.5(1 + 𝛼1) 0

0.5𝜖(1 + 𝛼2) 0.125𝜖2(1 − 𝛼2) 1 − 𝛼2

0.25𝜖(1 − 𝛼2)0.25

6𝜖(1 + 𝛼2) 1 + 𝛼2]

[

𝑘1

𝑘2

𝑘3

] = [

−𝛼1

𝛼2(1 − 0.5𝜖)

𝛼2(1 − 0.25𝜖)] (24)

As stated in the analyzed paper [1], 𝛽 = 0.6 for the PI controller equations and 0.8 for the PID controller

equations. Insufficient parameters are presented in the paper to calculate the equations for the PI controller

so only the equations will be presented.

𝑘𝑐𝑘𝑝 = 0.9719𝜖−0.8915 (25)

𝜏𝐼

𝜏= 0.7719𝜖4 − 3.6608𝜖3 + 6.5791𝜖2 + 5.1652𝜖 + 2.8059 (26)

For the PID controller design, 𝛽 = 0.8 as stated above, 𝛼1 = 1.2 and 𝛼2 is calculated using equation (19).

The 𝜖 value corresponds to the model and is approximately but not exactly 0.5. The equations for the PID

controller developed using this method are:

𝑘𝑐𝑘𝑝 = 1.377𝜖−0.8422 (27)

𝜏𝐼

𝜏= 1.085𝜖0.4777 (28)

𝜏𝐷

𝜏= 0.3899𝜖 + 0.0195 (29)

3.3 ANALYSIS OF PROPOSED METHOD

In order to understand the effect of varying time delay and time constant on equations (16), (17) and (18), the

limits of the gains have been taken and the results are summarized in Table 1.

Table 1 - Analysis of System Types

System Type Limit System Gain Integral Time Derivative Time

Very slow lim𝜏𝑑→∞

𝑓(𝜏, 𝜏𝑑) 𝑘𝑐𝑘𝑝 = 0.5 𝜏𝐼 = 0.5𝜏𝑑 𝜏𝑑 = 1

Very fast lim𝜏𝑑→0

𝑓(𝜏, 𝜏𝑑) 𝑘𝑐𝑘𝑝 = ∞ 𝜏𝐼 = ∞ 𝜏𝑑 = 𝜏𝑑/6

Large time delay lim𝜏→∞

𝑓(𝜏, 𝜏𝑑) 𝑘𝑐𝑘𝑝 = 0.5 𝜏𝐼 = ∞ 𝜏𝑑 = ∞

Short time delay lim𝜏→0

𝑓(𝜏, 𝜏𝑑) 𝑘𝑐𝑘𝑝 = ∞ 𝜏𝐼 = 𝜏 𝜏𝑑 = 1

In the case where a system is very slow, the controller uses small gains to control the system. Derivative

action is the same as the system gain and the integral gain is the inverse of the time delay. So a slow system

with a small time delay will be mainly controlled by the integral action and a slow system with a large time

delay will use a very sluggish controller, where the derivative action will be the driving force.

A very fast system will not require a PID controller according to these tuning rules. The integral action

becomes zero and the derivative action becomes very small compared to the system gain so control using the

difference in error, or proportional action, will be enough to control the system.

A system with a very large time delay will be controlled by a PD controller where the derivative action is the

main driving component of the controller.

A system with a very short time delay relies mainly on the system gain and derivative action. Integral action

may also play a major part in the system action if the system has a very small time constant.

It is also interesting to note that the analyzed paper [1] makes no mention of using a filter on the derivative of

the controller; thus, all simulation studies will be formed without using one for the proposed method.

4 SIMULATION STUDIES

This section shows all of the replicated results from the paper of study [1]. All results were replicated using

MATLAB to be as similar as possible to the original figures given that enough information was provided in

the section descriptions. Comments regarding major deviations in the results will be made in Section 5.

4.1 PI CONTROLLER The simulation results of the PI controller development using equations (25) and (26), which were partially

developed in Section 3.2, are shown in this section. Graphs for both the set point response and the regulatory

response are shown. Graphs for parameter uncertainty are also provided where each FOPTD model

parameter is increased by 20% individually.

Table 2 provides the information for the errors and Figure 1 and Figure 2 show the results of the proposed

method compared with the Ziegler-Nichols [3] and Abbas [4] PI controller development methods. Table 3

provides ISE error data for the responses to parameter variations. Figure 3 and Figure 4 are the graphs which

describe these responses.

Table 2 - PI Controller Errors

Method Servo Problem Regulatory Problem

ISE IAE ITAE ISE IAE ITAE

Present 1.4227 2.0192 3.1423 0.3576 0.8125 1.3198 Z-N 1.7699 3.6769 17.7768 0.3070 0.9335 3.4799 Abbas 2.1632 3.5847 10.9859 0.7102 1.9993 7.4831

Figure 1 - PI Controller Set Point Response

Figure 2 - PI Controller Regulatory Response

Table 3 - PI Controller Parameter Uncertainty Errors


ISE values for uncertainty in ISE values for uncertainty in

120% kp 120% τ 120% τd 120% kp 120% τ 120% τd

Present 1.4458 1.4840 1.7610 0.4719 0.3176 0.3918 Z-N 1.5626 1.8189 1.8964 0.4156 0.2621 0.3288 Abbas 1.8911 2.2488 2.2849 0.8911 0.6653 0.7342

Figure 3 - PI Controller Uncertainty in Parameters Regulatory Response

Figure 4 - PI Controller Uncertainty in Parameters Set Point Response

4.2 PID CONTROLLER This section contains the simulation results for the PID controller using no tuning parameters and two tuning

parameters, and comparing them against the results for the Ziegler-Nichols [3] and IMC [5] [6] methods. The

graphs are based off of equations (16), (17) and (18) for the model with no tuning parameters and equations

(27), (28) and (29) for the model with two tuning parameters. Graphs for parameter uncertainty are also

provided where each FOPTD model parameter is increased by 20% individually.

Table 4 provides the errors for the set point and regulatory responses of all four systems and Figure 5 and

Figure 6 show the graphs of the responses. Table 5 provides the ISE error information for the set point

tracking and regulatory responses with parameter variation; Figure 7 and Figure 8 show the graphical

responses.

Table 4 - PID Controller Errors



Presenta 0.6781 0.9691 0.7913 0.0833 0.4247 0.7433 Presentb 0.7520 1.1165 0.9582 0.0966 0.4076 0.5650 Z-N 0.6939 1.0176 0.8876 0.1173 0.5112 0.8511 IMC 0.7146 0.9968 0.7675 0.1202 0.4754 0.6868

a No tuning parameters b Two tuning parameters

Figure 5 - PID Controller Set Point Response

Figure 6 - PID Controller Regulatory Response

Table 5 - PID Controller Parameter Uncertainty Errors



120% kp 120% τ 120% τd 120% kp 120% τ 120% τd

Presenta 0.7686 0.6860 0.8961 0.1217 0.0729 0.0897 Presentb 0.8252 0.7800 1.0587 0.1141 0.0946 0.1179 Z-N 0.7750 0.7126 0.9070 0.1504 0.1112 0.1217 IMC 0.7051 0.7591 0.8622 0.1453 0.1132 0.1381


Figure 7 - PID Controller Parameter Uncertainty Regulatory Response

Figure 8 - PID Controller Parameter Uncertainty Set Point Response

5 ANALYSIS OF RESULTS

This section presents a comparison of the results of the simulations against those which were provided in the

analyzed paper [1]. The discrepancies which exist are hypothesized to be due to the fact that the analyzed

paper was written 10 years ago. This time discrepancy may allow for the development of better numerical

methods used to evaluate the mapping of transfer functions.

It is also worth noting that the authors of the paper did not provide any information about which

mathematical function was used to test the disturbance rejection of the controller. A square impulse, ramp,

impulse and sinc function were tried but only the ramp impulse provided a response that was stable as well as

visually similar to those produced by the authors.

Table 6, Table 7, Table 8 and Table 9 provide the results comparisons in relative percentages to the numbers

presented in the analyzed paper.

Table 6 - PI Controller Simulation Results Comparison



Present -2% -1% -4% -51% -47% -71% Z-N 0% 0% -1% -75% -75% -86% Abbas 0% 0% -1% -56% -44% -59%

Table 7 - PI Controller Parameter Uncertainty Simulation Results Comparison



120% kp 120% τ 120% τd 120% kp 120% τ 120% τd

Present -4% -1% -1% -51% -55% -57% Z-N -2% 0% -2% -71% -78% -75% Abbas 0% 0% -1% -54% -59% -58%

Table 8 - PID Controller Simulation Results Comparison



Presenta 0% 1% 6% -24% -15% -15% Presentb -1% 0% 0% 0% 11% 12% Z-N 0% 0% 3% 16% 20% 32% IMC -4% -4% 3% -31% -20% -28%


Table 9 - PID Controller Parameter Uncertainty Simulation Results Comparison



120% kp 120% τ 120% τd 120% kp 120% τ 120% τd

Presenta 2% -1% 0% -8% -31% -35% Presentb 0% -1% 0% 20% -18% 30% Z-N 2% -1% 0% 23% 15% -5% IMC -4% -4% -4% -27% -33% -33%


6 DISCUSSION

Aside from the issue regarding the disturbance rejection modelling described in Section 5, the results of the

system appeared to look very good for PID controllers when compared to the Ziegler-Nichols and IMC

models and for PI controllers when compared to Ziegler-Nichols and Abbas models. The results of the

proposed method do exhibit some high overshoot but fast settling time also.

It should be strongly noted that there are issues with the stability of PID (no tuning and two tuning

parameter models) and PI controllers developed using this method. As seen in Figure 4, Figure 7, and Figure

8, these controllers do not display strong stability characteristics for parameter fluctuation; however, the

disturbance rejection capability of PI controllers does not seem to be affected, as seen in Figure 3.

In order to implement this algorithm, the following conditions should be met:

1. Adaptive control must be used, and

2. The system must not change faster than the adaptive control can recalculated the FOPTD model

Adaptive control will ensure that the values which the controller are using are most appropriate for the

current state of the system and thus ensure the best possible response.

This type of method is suggested only for systems that can tolerate an underdamped response, require a quick

settling time and have the capacity to use adaptive control. This method would not be appropriate for a

system that has highly fluctuating parameters and does not need such a fast settling time. A more

conservative and stable method such as IMC may be a better choice for such a system.

7 LITERATURE SURVEY

A survey of the papers provided for this project produced a paper which described the development of a

simple and analytic controller tuning method. The paper titled Simple analytic rules for model reduction and PID

controller tuning by S. Skogestad [7] describes a method which uses the IMC controller framework to create a

simple set of tuning rules for FOPTD and SOPTD systems. These rules are simple equations where the input

parameters are governed by relations which describe the types of output behaviors; for example, 𝜏𝐼 = 8𝜃 will

produce a system that provides “a good trade-off between disturbance response and robustness” [7].

The advantage of this type of method is that the results for the model can be accurately predicted based on

the type of system input. The disadvantage is that there are probably few real-life models which can satisfy

such relations on a consistent basis. In comparison with the method in the analyzed paper [1], the method

proposed by Skogestad provides more predictable results for more rigidly defined models, whereas the

proposed method is more flexible and provides good results but the controller is not as robust.

8 CONCLUSION

This paper has reviewed the theory and simulation results depicted in the paper: A simple method of tuning PID

controllers for stable and unstable FOPTD systems by R. Padma Sree, et al. A detailed derivation of the theory has

been presented to show the steps required to reach the proposed equations along with an analysis of those

equations. The models which were tested in the paper were reviewed and replicated using MATLAB

simulation. The results from these simulations are presented with the discrepancies in the data. The set point

simulations were easily replicated with slightly better performance but the regulatory response showed much

better results than what was shown in the paper. This is probably due to the fact that the input disturbance

signal was not disclosed in the paper and it therefore had to be estimated. A discussion section presents an

analysis of the results and a suggestion that this method of PID parameter tuning is best for systems which

can handle overshoot, require a fast settling time and have built-in adaptive control. Finally a literature review

is conducted and the current method is compared against a method provided by S. Skogestad [7] which gives

tuning rules that provide predictable outputs for stable systems. When compared against the proposed

method, the proposed method is more flexible but less robust and the S. Skogestad method is more robust

but less flexible.

9 BIBLIOGRAPHY

[1] R. Padma Sree, M. N. Srinivas and M. Chidambaram, "A simple method of tuning PID controllers for

stable and unstable FOPTD systems," Department of Chemical Engineering, Indian Institute of

Technology, Madras, 204.

[2] J. Bennett, A. Bhasin, J. Grant and W. Chung Lim, "PID Tuning Classical - ControlsWiki," 16 10 2007.

[Online]. Available: https://controls.engin.umich.edu/wiki/index.php/PIDTuningClassical#Ziegler-

Nichols_Method.

[3] J. G. Ziegler and N. B. Nichols, "Optimum settings for automatic controllers," ASME Transactions, vol.

64, no. 759, 1942.

[4] A. Abbas, "A new set of controller tuning relations," ISA Transactions, vol. 36, no. 183, 1997.

[5] D. E. Rivera, M. Morari and S. Skogestad, "IMC-PID controller design," Industrial Engineering and

Chemical Process, Design and Development, vol. 25, no. 252, 1986.

[6] Q.-C. Wang, C. C. Hang and X.-P. Yang, "Single loop controller design via IMC principles," Automatica,

vol. 37, no. 2041, 2001.

[7] S. Skogestad, "Simple analytic rules for model reduction and PID controller tuning," Modeling,

Identification and Control, vol. 24, no. 2, pp. 85-120, 2004.

[8] A. Rad, MSE 490: Advanced PID Controllers - Lecture 4, Simon Fraser University: Surrey, 2014.

[9] A. Rad, MSE 490: Advanced PID Controllers - Lecture 3, Surrey: Simon Fraser University, 2014.

[10] G. H. Cohen and G. A. Coon, "Theoretical investigation of retarded control," Transactions of ASME, vol.

75, no. 827, 1953.

APPENDIX A: MATLAB ERROR CALCULATION CODE

% Calculates the ISE, IAE and ITAE for a curve which

function [ISE, IAE, ITAE] = CurveResults(Curve_t,Curve_y)

if(abs(Curve_y(end)) < 0.1) set_point = 0; %Set point for a disturbance rejection scenario else set_point = 1; %Set point for a servo response scenario end

% Calculate error error = Curve_y - set_point;

% IAE Calculation temp = cumtrapz(Curve_t,abs(error)); IAE = temp(end); clear temp

% ISE Calculation temp = cumtrapz(Curve_t,error.^2); ISE = temp(end); clear temp;

% ITAE Calculation temp = cumtrapz(Curve_t,Curve_t.*abs(error)); ITAE = temp(end); clear temp;

end

mse 490 final project paper

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