adaptive control for a first-order system
TRANSCRIPT
ADAPTIVE AND OPTIMAL TRACKING CONTROL OF
ELECTROMECHANICAL SERVOSYSTEMS
by
ADITYA GARG
A thesis submirted ro the Depi~rtmenr of Mechanical Engineering in conjonnity with the requiremenrsfor the degrw of
Master of Science (Engineering)
Queen's University Kingston, Ontario Canada K X 3N6
October 200 1
copyright O Aditya Garg, 200 1
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Adiya Garg: Ahplive ami Optimal Tracking Cvritrol of Electromechanical Servosystems. M. Sc. Thesis, Queeti 's Utiiversiiy a Kirigstot i. October, 2 O0 1
Adaptive control is an appealing control approach when a system requires variable
controller gains to compensate for unknown operating parameters. Similarly, optimal
control has drawn a fair degree of attention for trajectory tracking tasks. Optimal control
is an example of a model-based linear controller that provides a tigorous anaiyticai
approach to obtaining a set of fixed gains for a state feedback controller. The objective
of this thesis is to compare the performance of these approaches for trajectory tracking.
This thesis evaluates Model Reference Adaptive Control (MRAC) as compared with 1)
conventional Proportional-Velocity (PV) control and 2) Linear Quadratic Regulator
Tracking (LQRT) control. Simulation studies were conducted on two electromechanical
servosystems, a DC servomotor and a two-link robotic manipulator. Experimentai
studies were conducted with a single DC servomotor apparatus. Tests evaluated the
ability of each system to track step, ramp, sinusoidal and circular trajectories. The effect
of parameter changes on systern tracking performance was also exarnined.
MRAC and LQRT were first validated on an ideal second order system. Simulation
results for the robotic manipuiator showed that MRAC gave robust performance while
PV control failed to keep tracking errors within desired limits when the system was
subjected to inertia changes. Experimentd P-only and PV control results for the DC
servomotor were obtained to validate the control and data acquisition system. The LQRT
aigorithm achieved good tracking performance for al1 trajettories under conditions of
constant parameters. Simulation results for the DC servomotor showed that MRAC
could satisb design requirements with only partid knowledge of the system and under
conditions of changing systeni parameters. Implementation difficulties prcvented
experimental evaluztion of the MRAC aigorithm. It was apparent from inertia change
tests that LQRT is not suitable for systems which undergo significant parameter
variations. On the other hand, LQRT is simpler to implement than MRAC, which makes
it a good choice for those applications with comant and nearly constant parameters.
ACKNOWLEDGEMENTS
1 would like to extend my gratitude to rny supervisor Dr. Bnan Surgenor for his
continuous guidance and support throughout my program at the University.
The financial support provided by the School of Graduate Studies and the Department of
Mechanical Engineering is greatly acknowledged.
1 appreciate my lab mate Yang Xia, who through his sense of humour helped me get
through occasional hard tirnes.
1 would also like to thank my fiiend Abhinandan Jain at University of British Columbia
for his kind words and encouragement every now and then.
1 must express my gratitude for rny parents, my sisters and my girlfriend Elizabeth (soon
to be my fiance), without whose suppon I would not have been able to reach this
important moment in my life. Finally, 1 thank Queen's on the whole for making my two-
year stay here a most enjoyable and fruithl t h e of my life - not just acadernically, but
also in the cordial, carefiee environment and opponunities it provided for doing things,
which 1 had never done before.
TABLE OF CONTENTS
AB STRACT ACKNOWLEDGEMENTS TABLE OF CONTENTS LIST OF FIGUES LIST OF NOMENCLATURE
I iNTRODUCTION 1. I Problem Overview I .2 Thesis Objectives 1.3 Thesis Outline
2 LITERATIRE REVIEW 2.1 Adaptive Controt
2.1.1 Mode[ Reference Adaptive Control (MRAC) 2.1.2 Self Tuning Control (STC) 2.1.3 Design of MIWC Systems
2.2 Adaptive Control of Robotic Servosystems 2.3 Optimal Tracking Control
2.3.1 Integrator Exampie 2.4 Summary
3 SERVOSYSTLM S W A T I O N STUDES 3 . 1 Adaptive Control of a Second-Order System
3.1. i Modified MIT RuIe 3.1.2 MRAC for Generd Linear Systems
3.2 MRAC and PV for a 2-Iink Robotic Manipulator 3 2.1 Cosine Trajectory 2-22 Response to Parameter Changes 3 .Z.3 Straight Line Trajectory with MRAC
3.3 P and PV ControI for DC Servomotor 3 -3.1 Step and Trapezoidal Trajectories 3 -3.2 Response to Parameter Changes 3.3.3 Cucle Trajectory with PV
3.4 Sumrnary
4 SERVOSYSTEM EXPERIMENTAL STUDES 4.1 Background
4.1.1 Apparatus 4.1.3 Control Software 4.1.3 Open Loop Validation 4.1.4 Closed Loop Validation
4.3 LQRT for DC Servomotor 4.2.1 Optimal Replator Design 4.2.3 Ramp and Sine Tracking Design 4.2.3 Ramp and Sine Trajectories with LQRT 4.2.4 Circle Trajectory with LQRT
4.3 MRAC for DC Servomotor 4.3.1 Discrete Model for MRAC 4.3.2 Controller and Observer Design 4.3.3 Recursive Least Squares (RLS) Estimation 4.3.4 Square Wave Trajectories with MRAC 4.3.5 Circle Trajectory with MRAC
4.4 Cornparison of LQRT and MRAC 4.5 Summary
5 CONCLUStONS AND RECOMMENDATIONS 5.1 Conclusions 5.2 Recommendations
REFERENCES 1 O5
APPENDIX A: SIMULATION SOURCE CODE
APPENDIX B: EXPERlEvlENïATiON SOüRCE CODE 117
APPENDIX C: ADAPTIVE CONTROL FOR A FIRST-ORDER 127 SYSTEM
APPENDIX D: FIRST-ORDER SYSTEM SOLUTION FOR O P T W 130 TRACKMG PROBLEM
VITA 134
LlST OF FIGURES
Figure 2.1 :
Figure 2.2:
Essential components of an adaptive control system
Model Reference Adaptive Control system (parameter adaptation)
Figure 2.3: Model Reference Adaptive Control system (signal synthesis)
Figure 2.4:
Figure 2.5:
Figure 2.6:
Self Tuning Control system
Block diagram of a simple feedback loop
Integrator tracking example, cornparison of approximate Solution and exact solution for a ramp reference signal
Simulation of adaptive contrai for a 2nd -order system with MIT nile, showing trajectory and process output for 3 different adaptation gains and comparison of response to square wave command with three different amplitudes
Figure 3.1 :
Simulation of adaptive controi for a 2" -order system with modified M T nile, showing trajectory and process output and comparison of response to square wave cornmand with three different amplitudes
Figure 3.2:
Figure 3.3: W C for a second-order system using Lyapunov stability theory showing 2"d-order reference trajectory and process output
Figure 3.4:
Figure 3.5:
Schematic of a two-Iink manipulator carrying an unknown load
Cornparison of simulation of PV control and MRAC of a 2-link robotic manipulator for a desired cosine trajectory
Figure 3.6:
Figure 3.7:
Figure 3.8:
Figure 3.9:
Parameter estimates for MRAC of a 2-link robotic manipulator for a desired cosine trajectory
Cornparison of simulation of PV control and MRAC of a 2-link manipulator for a desired cosine trajectory with inertia change
Parameter estimates for MRAC of a 2-link robotic manipulator for a desired cosine trajectoty with inertia change
Cornparison of simulation results for MRAC of a t-link robotic manipuIator for a desired straight-line trajectory for two sets of gains
Figure 3. IO:
Figure 3.1 1:
Figure 3.12:
Position and velocity open loop remit for DC motor
Trapezoidai velocity profile
Cornparison of simulation resu1ts for P-only control of step test for DC motor for two different percent overshoots
Figure 3.13:
Figure 3.14:
Figure 3.15:
Figure 3.16:
Figure 3.1 7:
Figure 3.18:
Figure 3.19:
Figure 4.1 :
Figure 4.2:
Figure 4.3:
Figure 4.4:
Figure 4.5:
Figure 4.6:
Figure 4.7:
Figure 4.8:
Figure 4.9:
Figure 4.10:
Figure 4.1 1 :
Figure 4.12:
Comparison of simulation results for P-only control of ramp ifiput test for DC motor for two different percent overshoots
Comparison of simulation results for PV control of step test for DC motor for two dEerent percent overshoots and a fixed settling time
Comparison of simulation results for PV control of ramp input test for DC motor for two different percent overshoots and a fixed settling time
Cornpanson of performance degradation for P-only control of step test for DC motor for two different percent overshoots
Comparison of performance degradation for PV control of step test for DC motor for two different percent overshoots
Camparison of performance degradation for PV trajectory control of DC motor for two different velocity gains
Simulation oPPV control for two DC motors in response to sinusoidal reference signal to track a circuiar erajectory
Experimental Apparatus
PMDI MotionTools Main GUI
Simulation vs. experimental open loop test showing comparison of responses tvith 5 volt and 8 volt step inputs
Simulation vs. experimental open loop test showing comparison of responses with change in inertia (5 wfr step input)
Simulation vs. experimental P-only closed loop test for a step reference signal
Simulation vs. experimental PV closed Ioop step test for short response time
Simulation vs. experimental PV closed Ioop step test for long response time
Comparison of simulation results for closed loop step test with 3 sets of gains
Comparison of experimental results for DC motoi in response to a step input of 1 rot for j sec for three sets of gains
Simulation response for approximate vs. exact tracking solution for 5 rot'sec ramp reference
Simulation response for approximate vs. exact tracking solution for 0.5 Hz sùiusoidai reference
Simdation response for approximate vs. exact tracking solution for 0.5 Hz sinusoidai reference with inertia change
Figure 1.13 : Optimal tracking test showi.ng comparison of simulation and experimental result of approximate solution for a 5 rotj'sec ramp reference signal
Figure 4.14: Optimal tracking test, comparison of simulation and experimental result of exact solution for a 5 rok'sec ramp reference signal
Figure 4.15: Optimal tracking test showing comparison of simulation and experimental result of approximate solution for a 0.5 Hz sinusoidai reference signal
Figure 4.16: Optimal tracking test showing comparison of simulation and experimentai result of exact solution for a 0.5 Hz sinusoidal reference signal
Figure 4.17:
Figure 4.18:
Simulation of LQRT control for two DC motors in response to sinusoidal reference signal to track a circular trajectory
Simulation result for MRAC controller showing the actual response adapting after a few transients of square wave reference signal with initiai low estimate of initial parameters
Figure 4.19: Simulation result for MRAC controller showing the actual response adapting after a few transients of square wave reference signal with initiai high estimate of initial parameters
Figure 4.20:
Figure 4.2 1 :
Figure 4.22:
Figure B. 1 :
Simulation result for MRAC controller vs. PV controller for a square wave reference signal with inertia change
Simulation of MRAC for two DC motors in response to sinusoidal reference signal to track a circular trajectory
Comparison of simulation of LQRT and MRAC for two motors for tracking a circular trajectory with inertia change
Optimal tracking experimental test for exact solution for a 5 roosec ramp reference signal, comparison of results for different sampling times
Figure B.2: Optimal tracking experimental test with expanded scaies for exact solution for a 5 rorsec ramp reference signal, comparison of results for different sampling times
Figure C. 1 :
Figure D. 1 :
Comparison of adaptive control for l'order system with trajectory and process output for 2 different adaptation gains
First-order system optimal tracking example with cornparison of simulation of approximate and exact solution in response to a rarnp reference signal
Figure D.2: First-order system optimal tracking example with cornparison of simulation of approximate and exact solution in response to a sinusoidal reference signal 133
vii
LIST OF NOMENCLATURE
unknown parameter vector of 2-link manipulator
mode1 coefficients
siope of ramp or amplitude of sine reference input
acceleration of trapezoida1 move profile
state space matrices of the DC motor
observer polynomial
unknown parameters of discrete-time motor model
polynomial gains of mode1
extemal optimal control vector
instantaneous gain
steady-state approxirnate solution for LQR
coefficient for 1' order and 2* order system
cosines of joint angles
coriolis matrix
constants in optimal control Iaw
coefficient for 1" order and 2"d order system
components of parameter vector for RLS algorithm
total move t h e of trapezoidal move profile
error between model and process outputs
filtered error
output matrix of filtered error equation of manipulator
uansfer hnction for a general system
state-space matnx for a general system
gravitational constant
state-space matrix for a general system
transfer fhction for a ~eneral system
I l , , Ir2
N
N r * Nu
0 8
P
mode! transfer hnction
gravity m a t e
sampling time
state-space matrix for a general system
a combination matrix of system matrices and controller gains
identity matrix
unknown parameter - moment of inenia
minimum-error criterion
controller gain vector
optimal controller gain vector
system gain
manipuiator control gain
motor gain
proportional gain
velocity gain
lengths of manipulator links
unknown distance of 2"d link mass center fiom 2& joint
ratio of sampling time, Tm and motor cime constant, r,
manipuiator mass math
percent overshoot for dc rnotor
point masses of 2 links of manipulator
unknown parameter - mas
components of parameter vector for RLS algorithm
combined manipulator gravity and coriolis matrix
variables used in controller design of MRAC for motor
observer gain
differential operator
polynomid or matrix for filtered error
factors of polynomial P
vector of n-joint manipulator system parameters
solution of Riccati equation
joint angles
desired joint angle trajectory
reference velocity
polynomiai or matrix for filtered enor
nonnegative definite weighting matrix
step reference signal
controller polynomiai gain
positive detinite weighting matrix
controller parameter
sine of joint angles
reference velocity error
controller polynomiai gain
time
controller parameters
acceleration time of trapezoidal move prof le
deceIeration time of trapezoidal move profile
total move time of trapezoidai move profle
constant velocity time of trapezoidai move profile
controller polynomiai gain
extemal disturbance vector
settling tirne
control input signal
optirnai control
Greek letters
command signai
additionai control signal term in LQR tracking algorithm
open loop step reference input voltage
average velocity of trapezoidal move profile
maximum velocity of trapezoidal move profile
speed of the rnotor
open Ioop steady state velocity of rnotor
matrix of tùnctions ofjoint variables
controllability rnatrix
state vector
reference or desired trajectory vector
process output or position
trajectory output
known rnatrix, fùnction of q, q, q,, q,
a vector used in RLS algorithm
parameter introduced to avoid division by zero
saturation tùnction parameter
unknown parameter - angle relative to original 2"d link
adaptation gain
error augmentation
controI torques
6rst order system constant
motor t h e constant
discrete state space matrices
Acronyms
DC
DOF
DSP
GUI
P
LQ
LQR
LQRT
MRAC
P-only
P D
PV
RLS
STC
non-negative rnatrix
symrnetric positive definite matrix
parameter vector
vector of negative value of gradient of loss hnction
vector of joint positions
natural Frequency
damping factor
polynomial used in controller design of MRAC
parameter vector for estimation in RLS algorithm
vector of rnotor output and control signal
forgetting factor
symrnetric positive detinite rnatrix
Direct Current
Degree of Freedorn
Digital Signal Processing
Graphical User Interface
Index of Performance
Linear Quadratic
Linear Quadratic Reguiator
Linear Quadratic Regulator Tracking
Model Reference Adaptive ControI
Proportional only feedback controt
Proponionai-lntegrai-Derivative
Proponional-Velocity
Recursive Least Squares
Self Tuning Control
CHAPTER 1
INTRODUCTION
Automatic control systems exist in a virtually idnite variety, both in type of application
and level of sophistication. Control engineering can be surnrned up as the design and
irnplementation of control systems to achieve specified objectives under given
constraints. For a complex system, the overall objectives and constraints need to be
translated into performance specifications for the various individual subsystems that
make up the complete system. Control engineering practice includes the use of better and
fister control design strategies for increasing quality levels in manufacturing processes,
improving the energy efficiency of power production facilities, advancing the state of
flight control for aerospace applications, among others. System models are becoming
more and more important for the design of modem control system. The challenge to
control engineers can be stated as the need to model and control increasingly complex
and highly interrelated systems.
The importance of controllers in al1 technological areas is well known and documented.
For exarnple, without good process control, as in the case of a manufacturing plant. issues
such as product quality, production rates, material management, reliability and even
personnel safety would be in jeopardy. Controllers are there to maintain the process or
plant parameters within their prescribed limits, to track pre-designed trajectones to
perform certain tasks. And they rnust do so despite system disturbances, plant
uncenainties, model errors and so forth. Therefore, they have to be robust and able to '
ensure stability and maintain good system performance under adverse operating
conditions. In the context of this thesis, a robust controller is one that can maintain
performance in the face of changing operating conditions and system parameters, and yet
require a minimum of controller gain tuning.
P D is the acronym for the classicaI and most heavily used controI algorithm.
Proportional-Integral-Derivative (PD) controllers are sufficient for many control
problems, pmicularIy where there are benign process dynamics and modest performance
requirements. However, there are numerous control situations in which PID control with
constant gains fails to meet the requirements. For example, systems with large parameter
variations are candidates for more sophisticated contra[ structures.
When knowledge of the system is limited, the issue of adaptive control becomes
important. Adaptive control can rapidly modiQ an initially "imprecise" set of controller
gains as the controller seeks to adjust :O an unknown system state. Adaptive control with
its variabIe gain structure is also of value for those systems where operating parameters
Vary with time. Adaptive techniques are increasingiy being used in industrial controi
systems. Adaptive control is not a new subject. In the early 1950s extensive research
was done on adaptive control for the design of autopilots for high performance aircraft.
In sumrnary, adaptive controI can cope with increasingiy cornplex systems in the
presence ofextreme changes in systern pararneters and input sigals.
There are two main approaches for constructing adaptive controllers. One is the Mode1
Reference Adaptive Control (MRAC) rnethod and the other is the so-ca1Ied Self Tuning
Control (STC) method. This thesis will only apply the MRAC method. In a MRAC
system, the desired performance is expressed in ternis of a reference model, which gives
the desired response to a command signai. The regulator's pararneters are changed based
on the error as given by the difference between the outputs of the systern and the
reference model. In the context of widely u s a i comrol algorithms. an alternative to
adaptive control is optimai control.
Optimal control has drawn a fair degree of attention in particular for trajectory tracking
tasks. Optimal controt is a particdar branch of modem control, as opposed to classical
control, that aims at providing especiaily appeding adyticai designs. The system that
results from applying optimal control design methodology is not merely stable or satisfies
classicd requirements, but is also supposed to be the best possible or "optimal" system.
Linear optimal control is a special type of optimal control. Linear optimal control is an
example of a model-based linear controller that provides a ngorous analyticai approach to
obtaining a set of fixed gains for a state feedback cantroller. There are several
justifications for applying linear optimal control to a large number of systems as opposed
simply to optimal control. For example, a linear controller is simple to implement and
will often be sufficient for plants which are linear prior to addition of a controller. Most
linear optimal control problems are computationally inexpensive. Linear optima1 control
results may also be applied to nonlinear systems on a small signal basis. Through linear
optimal control, the class of systems for which control designs may be achieved can be
vastly extended.
Optimal control using linear quadratic methods aims at providing a system input that will
either 1) take the plant from a nonzero state to the zero state, or 2) cause the system
output to track some prescnbed function. This thesis will concentrate on the second
objective, namely tracking of a desired trajectory, by studying a Linear Quadratic
Regulator Tracking (LQRT) algorithm. A common situation where this objective is to be
achieved is trajectory-controllable systems such as electromechanical servosystems that
have to track a continuous path to do certain tasks like welding, cutting or painting in an
industrial environment.
1.2 Thesis Objectives
The objective of this thesis is to compare the performance of MRAC and LQRT
approaches for trajectory tracking as applied to representative elecuomechanical
servosystems. Servosystems in generaI are required to maintain a prescribed motion
dong a desired time-based trajectory with the controller applying corrective
compensating torques to adjust for any deviations fiom the trajectory and achieve fastest
response tirnes with minimal errors.
Specifically, the following issues wilI be addressed in this thesis:
What is the relative performance of Model Reference Adaptive Control (MRAC)
and Linear Quadratic ReguIator Tracking (LQRT) as applied to representative
electromechanical servosystems ?
Under what conditions can MRAC and LQRT be applied and when is their
application not advisable ?
1s MRAC harder or easier to implement than LQRT? What are the difficulties
encountered in their irnplementation ?
Are there ways that MRAC and LQRT cm be improved in the context of the
algorithms as irnplernented for this thesis ?
1.3 Thesis Outline
The organization of this thesis is as follows:
Chapter 2 is a literature review. It gives background theory on adaptive controi,
specifically on m C , and linear quadratic (LQ) control theory. A review of
difftrent adaptive control aiprirhmr For robotic rervosystems and optimal
tracking algonthms is aIso presented.
Chapter 3 presents simulation results for representative electromechanicai
servosysterns. Application of MRAC is made to three systerns: an ided second
order system, a 2-link robotic manipulator and a DC servomotor. For
comparative purposes, application of P and PV control is aiso made.
Chapter 4 contains experimental results of open and closed loop tests conducted
on a DC servomotor. MRAC is applied to the servomotor and verification of
LQRT is made for various trajectories. The performance of LQRT is compared
directly with that of MRAC, includins the case of an inertia change.
Chapter 5 summaizes findings and conclusions on the adaptive and optimal
tracking control of electromechanical servosysterns. Recommendations for hmre
work are aiso given.
CHAPTER 2
LITERATURE REVIEW
Chapter 2 gives a review of the literature and theory on the subject of adaptive and
optimal control in the context of tracking control of electromechanical servosystems.
Particular attention is paid to MRAC as an adaptive algorithm, application to robotic
manipulator servosystems and LQ as an optimal control algonthm.
2.1 Adaptive Control
M y dynamic systems to be controlled have constant or slowly varying wlcertatn
paramefers. For instance, robot manipulators may carry large objects with unknown
inertial parameters. Power systems may be subjected to large variations in loading
conditions. Adaptive control is a popular approach to the control of such systems.
There does not seem to be a universally accepted definition of what is an adaptive
controller. However, according to Landau (1979), an adaptive system measures a certain
index OF performance (IP) using the inputs, the States and the outputs of the adjustable
system. From the companson of the measured IP values and a set of given iP values, the
adaptation mechanism modifies the parameters of the adjustable system or generates an
auxiliay input in order to maintain the measured iP values close to the set of given ones.
This definition is illustrated in Figure 2.1.
The basic idea in adaptive control is to estimate the uncertain plant parameters (or
equivalently, the corresponding controller parameters) on-line as based on the measured
system sipais, and use the estimated parameters in the control input computation. An
adaptive system c m thus be regarded as a control system with on-line parameter
estimation (Slotine and Li, 1991). Pragmatically speaking, it is a specid type of
nonlinear feedback control in which the states of the process cm be separated into two
categories, which change at différent rates. The sIowly changing states are viewed as
parameters. An adaptive controller has two loops: one for ordinary feedback and the
second For updating regulator parameters. Adaptive control systems, whether developed
for Iinear or for nonlinear plants, are inherently nodinear, and their analysis and design is
intimately connected with the theory of nonlinear control. and in particular with
Lyapunov theory.
Unknown Known perturbations , perturbations
Figure 2. I: Essential cornponents of an adaptive controt system (Chalam, 1987)
---r--- *
An adaptive controller differs fiom an ordinary controller in that the controller
parameters are variable, and there is a mechanisrn for adjusting these parameters on-line
based on signais in the system. Adaptive controt systems may be classified by rnany
cnteria. For example, they can be parameter-adaptive or signai synthesis adaptive
systems, direct or indirect adaptive systems, or determïnistic, stochastic or leamhg
systems. Landau (1993) diicusses the evolution of adaptive control algorithms driven by
the results obtained in the application of fim generation of adaptive controllers.
t
ADJUSTABLE SY STEM
There are two main approaches For consuucting adaptive controllers. One is Model
Reference Adaptive Control (MRAC) method, and the other is Self-Tuning Control
(STC). This thesis shall deal ody with b M C .
2.1.1 Model Reference Adaptive Control (MRAC)
The problem of self-adjusting the parameters of a controller in order to stabilize the
dynamic characteristics of a feedback controi system when drift variations in the plant
parameter occur, was the origin of MRAC systems (Whitaker, Yamon and Kezer, 1958).
The two basic schemes of MRAC are given in Figure 2.2 and Figure 2.3. A MRAC
system is composed of four parts: a plant containing unknown parameters, a refirence
model for compactly specifjmg the desired output of the control system, a feedback
cot~trol l m containing adjunable parameters, and an adapm[otr mechanism for updating
the adjustable parameters (Slotine and Li, 1991). The reference modei gives the desired
response of the adjustable system and the task of the adaptation is to minimize a hnction
of the difference between the outputs, or the states, of the adjustable system and those of
the reference model. This is done &y the adaptation mechanism that modifies the
parameters of the adjustable systems or generates an auxiliary input signai. The
disturbances represented in Figure 2.2 are parameter disturbances modifying either the
reference model or the adjustable system. One of the most important advantages of this
type of adaptive system is its high speed of adaptation. This is because a measure of the
difference between the given iP specified by the reference model and the iP of the
adjustable system, is obtained directly tiy the cornparison of the outputs, or the states, of
the model with those of the adjustable system.
The plant is assurned to have a known structure, although the parameters are unknown.
This means that either the number of poies and zeros or the structure of noniinear
dynamic equations is known.
A refirence mode1 is used to specifj the ide* response of the adaptive control system.
The choice of reference model is part of adaptive system design. This choice should
reflect the performance specification in control tasks, and also this ideai behavior should
be achievable for the adaptive control system, Le., there are some inherent constraints on
the structure of the reference model given the assumed structure of the plant model.
The cottrrofler is usually parameeerized by a number of adjustable parameters. The
controller should have perfect tracking capacity in order to allow the possibility of
tracking convergence. if the control law is linear in terms of the adjustable parameters, it
is said to be li~iearly parumeferized. Existing adaptive control designs nonally require
linear parameterization of the controller in order to obtain adaptation mechanisms with
guaranteed stabiiity and tracking convergence.
The objective of adapmtioti law is to make the tracking error converge to zero. The main
issue in adaptation desi~n is io synthesize an adaptation mechanism that wiil guarantee
that the control system rernains stable and the cracking error converges to zero as the
parameters are varied. Many formalisms can be used to this end. such as Lyapunov
theory, hyperstability theory, and passivity theory.
CONTROLLER
Figure 2.2: Mode1 Reference Adaptive Controi system (parameter adaptation)
9
uc e + ADAPTARON - MECHAMSM
* P L M b
Figure 2.3: Mode1 Reference Adaptive Control system (signal synthesis)
2.1.2 Self Tuning Control (STC)
The self-tuning controller was originally introduced by Astrom and Wittenrnark in 1973.
In STC, a design procedure for known plant parameters is chosen first. This is applied to
the unknown plant parameten using recursively estimated values of the parameters. A
STC system is shown in Figure 2.4.
- REGLJLATOR DESIGN RECURSIVE
P L u b u E E R ESTMATOR
I Adjustable @-stem I
Figure 2.4: Self Tuning Conuol systern
There are strong relations between W C and STC design methodologies. Both kinds of
systems have an inner loop for control and an outer loop for parameter estimation. es rom a theoretical point of view, MRAC and STC can be put under a unified h e w o r k .
However, two methods can be quite different in tems of analysis and implementation.
2.1.3 Design of MRAC Systems
There are essentially three basic approaches to the analysis and design of a MRAC:
1. The gradient approach
2. Lyapunov functions
3. Passivity theory
The gradient method was used by Whitaker et al in 1958 in the original work on the
MRAC. This approach is based on the assumption that the parameters change more
slowly than the other variables in the system. Lyapunov's stability theory (Parks 1966,
Popov 1973, Vidyasagar 1986) and passivity theory (Vidyasagar, 1978) have been aIso
used to modify the adaptation mechanism to yield stable adaptive systems
The gradient approach is a fundamental idea in MRAC approach. The parameter
adjusment scheme is usually called the MIT nile. Assume that an attempt is made to
change the parameters of the regulator so that the error e between the output of the
process and the reference model is driven to zero. The following criterion is introduced:
where J = minimum error criterion or index
9 = parameter vector
e = error between mode1 and process outputs
To make J smail, it is rettsonable ro change the parameters in the direction of the negative
gradient ofJ, that is,
where Y = adaptation gain
1 - - time
The derivative &/28 is the sensitivity derivative of the system. The adjustrnent mie of
Equation 2.2 is cornmonly referred to as the MIT rule. It also applies to the case of many
adjustable parameters. The variable B should then be interpreted as a vector and &/a@ as the gradient of the error with respect to the parameters. The parameter adjustrnent
mechanism described in Equation 2.7 can be regarded as composed of an element for
computinç the sensitivity denvatives from process inputs and outputs, a multiplier, and
an integrator. The parameters are then introduced in the control law using a multiplier.
This is a generic part of many adaptive schemes.
Stability is a basic requirement in a control system. Much effort has been devoted to
analysis of stability of adaptive systems. Stability theory has been the major source of
inspiration for the development of MRAC systems. Assuming that ail the state variables
of a system are measured, the Lyapunov stabiiity theory can be used to design adaptive
control laws that guarantee the stability ofclosed-loop system. This approach is based on
Ftnding suitable candidates for Lyapunov ttnctions.
If we divide the system into two blocks, as s h o w below in Figure 2.5, where H and G
are general noniinear tirne-varyin3 systerns, we can establish properties for each bIock
and then use thern to conclude the stabiiity of the closed-loop system. By using the enor
rnodel below, it is possible to use passivity theorem if system H is passive (Astrom and
Wittenmark, 1989). The strictty positive reai condition of G is an important condition in
the design.
Augmented error cm be introduced to avoid hrther problems. The augmented error
method gives conversence.
Figure 2.5: Block dia- of a sim.$e téedback loop
The method for a direct MRAC can be described in the following steps:
1. Find a regulator structure that admits perfect model-following for the plant.
2. Derive an enar mode1 of the form
where G = transfer function
P = differential operator
P = vector of negative value of gradient of loss ftnction
3. Use the parameter adjustrnent law (a is a parameter to avoiding division by zero)
The error model of Equation 2.3 is linear in the parameters. The mathematical
manipulations used to derive the e m r model include filtering and error augmentation.
The details of complete derivation wiil not be discussed here. The equations needed to
implement the general MRAC are:
where
B m - I I , Y m -- A m
Filtered error: O O e,. = = e = = ( y - y,) P P
Errer augmentation: q = - -ii + Ii 1 b,Q Augmented error: e = t., + - rl
A d ,
Adjustment rate: $ = y@
Control input: tr = -0' (4 p)
Y = process output
Y , = trajectory (ideai output fory)
A,, B, = polynomiai gains of model
II = controi input signal
'1, = command signai
e,. - - tiItered error
P, 0 = poiynornids
rl = error augmentation
P, = factor of polynomid P
4 = observer polynomiai
4 = instantaneous gain
A. and A, are related by Diophantine's equation. Q is a polynomial wbose degree is not
greater than deg Ad,,,. The controller is given by
ir = R-' ( T i r , - Sy) (2.12)
where R.S and Tare polynomials. P=PIPI where P? is a stable monic polynomial of the
sarne degree as R. P,p and p are generated by filter from signals u , I I . and y. For
complete derivation of this set of equations, see Astrom and Wittenmark (1989).
2.2 Adaptive Control of Robotic Servosystems
Robotic servosystems are familiar examples of trajectory-controllable mechanical
systems. With increased demands on manipulator performance comes the need for
improved servo control techniques. However, their nonlinear dynamics present a
challenging control problem, since traditional Iinear control approaches do not apply. The
dynamic equations of a manipulator are of great importance in the design of its control
scheme. One of the major problems in applying sophisticated control atgorithms is that
of imprecise dynamic models. Extensive research has b e n done on control of
mechanical manipulators in cases where the physicai modeis that describe the
maniputators are not well known. Incorrectness or uncertainty in a dynamic mode! can
be spiit into two portions (Craig, 1988, Chapter 1). S~ructured uncerrainry is the case of
correct structurai mode1 with al1 uncertainty due to incorrect parameter vdues, Le. there
exists a correct (but unknown) set of vaiues for the parameters such that the mode1 will
match the actual system. Utrstr~ict~rred rrticertainty means unrnodeled effects, some of
which may be state-dependent and others are external disturbances. This thesis addresses
the case where modeling error is largely due to structured uncertainty.
The dynamic equation of a generaI serial manipulator rnodeted as a set of ri moving
jointed rigid bodies can be wiitten in the form:
where = 11 x 1 vector of joint torques
= ti x I , manipulator mass matrix
= vector of torques from coriolis & centrifbgal forces
= vector of torques due to gravity
= vector of unmodeled dynamics and extemal disturbances
= vector of joint positions
Using Lyapunov theory, following adaptation law can be denved for the manipulator
(Craig, 1988):
where Pr = vector of systern parameters
@, = estimate of P,
A = a non-negative matnx
CV - - matrix of functions of O, 0, O
EI = output matrix of filtered error equation
The stability, parameter error convergence and robustness to bounded disturbances for
the adaptive algorithm can be proved by using appropriate mathematicid tools. Any
adaptive aigorithm must possess these properties before it can quaiijl for
implementation.
Considerable research has been done on adaptive control of robotic manipulators in
recent years. Researchers have studied a variety of robots, a major proportion of them
being simple two- or three-link manipuiators. under different adaptive algoriths. There
has also been some research done on parallel manipulators though for most part, it has
dealt with kinematics and dynaniics.
One of the most notable initiai works on the application of MRAC to robotic
manipulators was done by Dubowsky and DesForges (1979). They justified MRAC by
pointing out that this procedure requires only a relatively simple, linear uncoupled model
of the system instead of cornplex nonlinear system dynamic equations of the actud robot,
leading to less burden on control compter. FolIowing the work of Donalson and
Leondes (1963). they developed the compensation network and parameter adjustment
algorithm for a six degree-of-&dom serial manipulator.
The technique used in the research of Dubowsky and DesForges was not systematized
and needed a difficult stability analysis. Horovitz and Tomizuka (1986) overcame this
shortcoming and presented an adaptive control scheme using the Popov hyperstability
theory. Baiestrino, Maria and Sciavicco (1983) designed an adaptive mode] following
control technique by doing away with assurnption of a constant linear model.
Slotine and Li (1987) did some signifiant research on adaptive control of industrial
robots using Lyapunov Theory for stability anaiysis. Craig (1988) in his book gave a
detailed account of the relative disadvantages of conventional dynamic control of the
manipulators and designed a MRAC aigorithm that maintains the structure of computed
torque servo but in addition has an adaptive elernent. Johansson (1990) presented
algonthms for continuous-time discrete adaptive control of robot manipulators using
detaiied Lyapunov Theory.
Gourdeau and Schwartz (1991) presented a motion control adaptive scheme with
trajectory tracking results of experimentation on a robot having same configuration as the
MIT Direct Drive robot. The MiT robot has been used as a benchmark by a number of
researchers (Asada and Youcef-Toumi, 1987).
In the previous decade to this century, severd researchers have attempted to design more
general and efficient MRAC algorithms for robotic manipulators using a variety of
simulation tools and also experimenting on the actual hardware. They have exploited the
special structure of rnanipulator dynarnics (Irnura, Sugie and Yoshikawa, 1994) and
eliminated the need for any a priori information on the robot uncertainty to design
sirnpler controllers. Self Tuninç Control and Variable Structure Systems methodology
has also been explored in designing more sophisticated controllers for manipulators.
Young (1978) did pioneering research on controller design for a manipulator using theory
of variable structure systems. Yu (1998) proposed a new combined adaptive and variable
structure adaptive control approach for manipulators. Yu. Zhihong, Chonç and Fei (1999)
developed a robust adaptive stiding mode controller (which is a special case of variable
structure systems) for trajectory trackins.
Nguyen, Antrazi, Zhou, and Campbell (1993) presented the irnplementation of a joint-
space MRAC scheme to control the noncompliant motion of a Stewan platform-based
manipulator (SPBM). Numerous adaptive control schemes were developed for open-
kinematic chah maniputators but oniy a few for SPBMs.
Most industrial robots use electric actuators, for exarnple DC servornotors, for actuation
of their joints. The next section Iooks at the tracking control of electromechanical
servosystems from another perspective; that is from the perspective of Optimal Control.
2.3 Optimal Tracking Control
The Optimal control problern can be stated as the need to determine the control u which
minimizes a given petformance tiinctional J. Given the system
and a quadratic performance index
where - X - state vector
F, G, H = state space matrices for a general system
minimùation pmblem is to find an optimal control ri'(t) for al1 t E [tu, T ] that Mnirnizes
J. The solution of this problem is given by following control law:
where r r 0 ( t ) = optimal control
RI = positive definite weighting matriv
QI = positive definite weighting matrix
P, = solution of Riccati equation
is given by the solution of the following Riccati equation, with initial condition P(T) =O
For details of the derivation of the above solution, one can refer to Anderson and Moore
(1990, Chapter 2).
For a conventional regulator, the objective is to arrive at the zero state and the path that
the system takes to reach that state is typicaliy ignored. The path is known as the
trajectory. This optimal regulator problem is in fact a special case of a wider class of
problems where it is required that the system States follow or track a desired trajectory in
some optimal sense. This type of problem is Frequently encountered in case of trajectory-
controllable systems such as robotic manipulators that have to track a continuous path to
do certain tasks. Another example is for altitude control of a terrain following aircrafl,
where there is knowledge of fiiture terrain. The approach to optimal tracking taken in
this thesis follows that of Anderson and Moore (1990, Chapter 4).
For the system given by Equations 2.15 and 2.16, a performance index can be given of
the form:
- where Y = reference signal or desired trajectory vector
the optimal control II* is given by:
where K' is the optimal gain and 60) is the exact solution to the infinite time tracking
problern as given by following:
In order to make the notation consistent with wider practice, K*is replaced with K for the
remainder of this thesis, where K = - KT . The notation for .? is also changed because -
for this particular case, H = 1, and therefore F = y = x, .
The following equations summarize the Optimal Tracking control algorithm:
where 1, = reference trajectory vector
Various approximately optimal trackers can be designed from above algorithm. Setting
Equation 2.27 to zero, and solving for b to get the steady state solution for the
compensator provides:
where H l = (GK - F)'
b, = steady state approximate solution for LQR
A review of the (steady-state) Riccati equation reveaIs that
and Equation 2.26 with b, substituted for 6 reduces tu the standard form of the LQR
control law:
2.3.1 Integrator Example
To illustrate above tracking theory, consider a simple scalar example where the plant to
be controlled is a pure integrator and reference input is a ramp signal (Anderson and
Moore, Chapter 4, 1990).
We have
F = O , G = l and x, = a,[
where a, = siope of the ramp reference input
Substituting the above values in Equations 2.25 to 7.27 gives:
The exact solution for b can be derived using standard differential equation theory, for
exarnple by finding the complementary but not the particular solution to Equation 2.33
Wce and Strange, 1994). In the case of a ramp reference signal, this becomes:
For a sep reference signal withx, = a, th.e exact solution for b becomes:
a a, For a sinusoidal reference signal with x, = 'sin (a f - -) + -, the exact solution for
2 2 2
b becornes:
where O = frequency of sinusoidai reference signai
The "approximate solution" is the terminology used by Anderson and Moore. This
terminology can be considered somewhat misleading. The tracking result for a process
modeled as a pure integrator is given as Figure 2.6, with a companson of the approximate
and exact solutions, in response to a ramp reference signai. According to Anderson and
Moore, the "approximate solution" is obtained by setting b = O in Equation 2.33. In the
case of the integrator example, this will reduce the control law of Equation 2.32 to
21 = - K . It can be shown, for example by using the Final Value Theorem, that for pure
proponional state feedback, there will be a constant and non-zero steady state error for a
ramp input. This is confirmed in Figure 2.6 for the approximate solution result, with a
steady state error on the order of O. 1.
Figure 2.6 also shows that the feedforward nature of the exact solution significantly
reduces the steady state tracking error, with control action starting before the initiation of
the ramp reference sipal at 0.1 sec. This result matches that given in Anderson and
Moore (1990, Ch. 4, Fig. 4.3-1). AIthough Anderson and Moore outline the theory of
optima1 control for tracking applications, they do not provide the exact solutions for
processes beyond that of an integrator. Extension to first order and second order
processes will be given in Appendix A and Chapter 4, respectively.
0.5 1 Tirne (sec)
Figure 2.6: Integrator tracking e~ample. cornparison of approximate solution (dot-dashed line) and exact solution (sotid Iine) for a ramp reference signal (dotted line).
2.3 Summary
The key observations frorn the theory and Iiterature survey docurnented in this chapter
can be surnmarized as follows:
Adaptive control is a very eEective control technique in the case of dynarnic
systerns that have slowly varying uncertain parameters or that are subjected to
unknown but bounded extemal disturbances.
Model Reference Adaptive Control (MRAC) finds a usehl application in robotic
servosystems. as robots are fiequently required to perform a variety of tasks dong
a set of desired trajectories and in an uncertain environment.
Although there is an abundance of literature on optimal control, there is Iittle on
optimal contour tracking control for servosystems.
Optimal Control can be an appealing approach for tracking control systems with
known and constant parameters, in which case a complex algorithm like MRAC
may not be desirable.
CHAPTER 3
SERVOSYSTEIII SIMULATION STUDIES
Modeling inaccuracies cm have strong adverse eRects on nonlinear control systems.
Therefore, a good design must take them into account explicitly. As mentioned in
Chapter 2, adaptive control is a robust control methodology; one that is able to maintain
good performance characteristics despite mode1 enors and parameter uncertainties. In
this chapter, we present simulation of various systems under adaptive control and also
servo system tracking simulation. The simulation code for al1 the applications in this
chapter is given in Appendix A.
3.1 Adaptive Control of a Second-order System
The design of an adaptive controller for an ideal second order system is given in this
section. For reference purposes, the design of a controller for a first-order system is
given in Appendix C.
Consider a second-order system with transfer hnction given by (Astrom and Wittenmark.
1989):
The objective of the MRAC control system is to adjust a feedforward gain 8 to the value
8' for system coefficients c and d. Applying the MIT rule for parameter adjustment
(Equation 2.2), the adaptive system cm be represented by the following set of equations:
d'y dy - + c - + d y = B u , dt ' dt
The fact that the parameter adjustment rate depends on the magnitude of the command
signal u, can also lead to instability.
Figure 3.1 shows the comparison of simulation results for adaptive control of a second
order system with trajectory, y, and process output, y for two different adaptation gains
y and three different square wave command signal u, amplitudes. For the three grids
piotted, the amplitudes of the command signal were: 0.1 (top), 1 (middle) and 3.5
(bottom). The remaining control parameters used in the simulation were c=d =Bo =1.
It is apparent fiom Figure 3.1 that the response can be unstable if the adaptation gain y is
set too large. Specifically, the response is unstable if y is 1.5. It is dso seen that
instability can result if the command signal u, is too large The figure also shows that the
convergence rate depends on the magnitude of the command signal. SpecificaIly, the
response is unstable when the amplitude of u, is 3.5; and the convergence is too slow
when the amplitude of u, is 0.1. The next section will illustrate how to avoid the problem
of instability and to rernove the dependency of the parameter adjusment rate on the
magnitude of the control signai.
fime (sec)
Figure 3.1 : Simulation of adaptive control for a 2" order qstem with MIT rule, trajectory,~, (dotted line) and process output, y for 2 different adaptation gains, y = [O. 1 (dash-dotted line) 1.5 (solid line)]. Response to square wave command, u, with amplitudes OF: O. 1 (top), 1 (rniddle) and 3.5 (bottom).
3.1.1 Modified MIT Rule
As explained in the last section, it would be desirable to modi@ the MIT mle so that the
adjustment rate does not depend on the magnitude of the cornmand signal. One way is to
rnake normalization (Astrom and Wittenmark 1989), invoduce a saturation to Quarantee
that the pararneter adjustment rate is always below a given lirnit, and replace the MIT mle
in Equation 2.2 with
where
x c - p snt (x . p) = [.ip + g
The algorithm of Equation 3.5 is sometimes referred to as the Modified Parameter
Adjustment Rule (Astrom and Wittenrnark, 1989). For the purposes of this thesis, it wilt
be tenned the Modifled MIT mie. The parameter a > O eliminates the possibility of
division by zero. In Figure 3.2 the second-order system was again sirnulated but with the
modified rule of Equation 3.5 in place of the original rute of Equation 2.2. In this
particular case Equation 3.5 simplifies to:
It should be noted that in some situations it rnight be desirable to have adjustnxnt rates
depend on the magnitude of the command signa1 for smail levels, as measurement noise
may be present. Comparing Figure 3.2 with Figure 3.1, it is clear that the Modified MIT
rule performs better, 1) elimination of the instability and 2) improved pararneter
convergence rate for the given range of command signa1 amplitudes. It is in fact possible
to make the Modified MIT rule work very well over a large range of command signai
amplitudes. However, it should be noted that instabïii is ail1 possible with the modified
rute if the adaptation gain is too large.
-2 , I
O 20 40 6 O 80 1 O0 1 20 140
Time (sec)
Figure 3.2: Simulation of adaptive controi for 2&-order system with Modified MIT nile, trajectory, y, (dotted line) and process output, y (solid line). Comparison of
response to square wave cornmand, JI, with amplitudes of O. 1 (top), 1 (middle) and 3.5 (bottom).
3.1.2 MRAC for General Linear Systems
Lyapunov stability theory can be used ta design adaptive control laws that sarantee the
stability of a closed-loop system. Such an approach is based on finding suitable
candidates for the Lyapunov fiinction Equations 2.6 to 2.12 gave the design for one such
approach to the probiem.
The performance of this system is illustrated by simulation of a second-order systern in
Figure 3.3. The systern is given by
and the standard second-order model is given by
where< is the damping factor, G, the model transfer hnction and K, is system gain.
It is apparent fiorn Figure 3.3 chat by using a control design based on Lyapunov Theory,
systerns with stable performance can be obtained. The adopted filter design is seen to
give rapid convergence of y to y,, and the error between the actual and the ideai model
reduces to near zero as the simulation progresses. Spikes in the plot of the control signal,
ri, are seen whenever there is a step change in the input signal. These spikes are due
largely to the proportionai component of the control law of Equation 2.12 in response to
the step change in the error signal. The bottom two grids of Figure 3.3 show the
parameter estirnates, which are seen to converge to near steady values.
3.2 MRAC and PV for a 2-link Robotic Manipulator
As mentioned in Chapter 2. adaptive control has been considered as an effective
mechanism for robot controller design due to the presence of nonlinearities and
uncertainties in robot dynamic models. MRAC was one of the first adaptive aigorithms
in practice and consequently has been more extensiveIy considered for robot rnanipulator
60 80 Time (sec)
Figure 3.3: MRAC for a second-order system using Lyapunov stability theory, with model reference that o f standard second-order model, second-order trajectory, ym , (dotted line) and process output, y (solid Sie).
The robot control probiem can be senerally stated as that of designing the joint torques so
that the joint motion variables track a desired trajectory.
The efficacy of MRAC as applied to robot manipulators will be illustrated by means of a
simulation of a two degrees-of-freedom (DOF) maniputator (Slotine and Li 1991). The
manipulator is modeled as two rigid, revolute links of lengths 11 and k with second link
carrying a large unknown mass. The two joint variables are joint angles (1, (angle of link
1 From horizontal) and q: (angle of link 2 with respect to link 1) as in Figure 3.4. The
control problem lies in determining the joint inputs required to cause the joints execute
the desired trajectory. The secoqd link, with the payload attached, can be regarded as an
augmented link with four unknown parameters, namely, mass nt,, moment of inertia I,,
the distance I,, of its mass center to the second joint, and the angle S relative to the
original second link.
Figure 3.4: Schematic of a two-[ink manipulator c-ng a large unknown load (Slotine and Li, 199 1)
The dynamic equations ofmotion of a 2-link revolute-joint rnanipulator are given by
(vidYasagar; 1989):
where control torques
Iengths of rnanipulator links
point masses of 2 links of manipulator
joint angles
cosines of joint angles
cosine of sum of joint angles
sine ofjoint angles
gravitational constant
friction tems
These equations can be rearranged to arrive at Following standard matrix fom,
where r = controI torque vector
hl = manipufator mass matrix
IV = vector of torques uïsing fiom centrifùgal, corioiis and gravity forces
The dynamics of the manipulator with payload c m then be wntten as
where
Hl, = a, + 20, cosqz + 2a, sin qz
H,z = HII = a, + aj cosq, + a, smq2
H E =al
h, = n3 sin y, - a, cos q.
C I , = I , i m , i c ; +nt/, ' + m , l , "
4- = f, + me ice'
4 , = m,Z, le= cos6,
a3 = m, il f, sin Je
We consider the case when the manipulator is required to follow a desired trajectory. The
adaptive controller design problem is to derive a control Iaw for the actuator torques, and
an estimation law for the unknown parameters. Applying Lyapunov theory gives
following result (for complete derivation, see Slotine (1991, pg. 404 and 405):
Control law: r = Y i - K,s, (3.22)
Parameter adaptation Iaw: G = - RYr s,. (3.23)
where R is a syrnrnernc positive definite mat* K, is gain and Y is a known matrix
such that
The rnatrix Y has been derived by using an additionai physical property of the system,
namely that, given a proper definition of the unknown parameter vector a describing the
rnanipulator's mass properties, the terms H ( q ) , C(q,q) and G(q), al1 depend linearly on
4. S, can be interpreted as a "velocity error" term
where
The "reference velocity" vector q, is fonned by shifting the desired velocities q,
according to the position error r j . 4) is a symrnetric positive definite matrix.
3.2.1 Cosine Trajectory
With reference to the 2-link manipulator in Figure 3.4, the following is obtained:
The components of rnatrix Y as given in Equarion 3.24 can be written as
Figure 3.5 shows a cornparison of simulation of the manipulator with conventional
Proportional-Velocity (PV) type control and Mode1 Reference Adaptive Control with
regard to tracking errors and control [orques. The PV control Iaw in this case is given by:
where ?, is the position error, q, is the velocity, K, and K,. are the controller gains.
It is evident from Figure 3.5 that PV control is not suitable for trajectory control as it has
larger initial position errors and more imponantly, position errors continue to oscillate
between large maximum and minimum values. On the other hand, W C gives much
better performance with srnaller initial position errors and f ier 1 sec the maximum error
settles to within 2 0.5 deg.
In the case of W C , we start without any a priori information ($0) = 0) for the
desired cosine trajectory. Figure 3.5 confirms the results given in Slotine and Li (1991,
Figures 9.4 and 9.5, pg 408 and 410).
MRACManipulatorl : Kp=2000,Kv=100 Cosine destred trajectory
Time (sec) Time (sec)
Figure 5 3 : Cornparison of simulation of PV control (dotted line) and MRAC (soiid line) of a 2-link robotic manipulator for a desired cosine trajectory.
Figure 3.6 shows the parameter estirnates for the W C resuit of Figure 3 5, except with
the time scale increased from 3 to 5 sec. The parameters shown in the top two grids
exhibit a stigfx drift but they are not unstable and show a trend towards convergence CO a
steady value. The parameters shown in the bottom two grids show an oscillation, but
again they are not unstabte as the amplitudes of their osciIlations are not growing.
MRACManipulatorZ: Kp=2001).Ki~c100 Cosine desired trajectory 0.25 0.8
Time (sec) Time (sec)
Figure 3.6: Parameter estimates for MRAC of a 2-link robotic manipulator for a desired cosine trajectory. Note the convergence pattern of ail the parameters which start without any a prion information
3.2.2 Response to Parameter Changes
In order to show the robustness of the MRAC scherne as applied to the robotic
rnanipuIator problem, Figure 3.7 gives a cornparison of the simulation result for PV
control and bIEWC for a desired cosine trajectory with a step doubling of the inertia of
the second link at f = 2 sec. MIWC gives superior performance in the face of this sudden
load change with the system recoverinç dmost immediately afier an initia1 increase of
errors and control toques at r = 2 sec. The two top grids in the figure show that position
errors for PV control are significantly Iarger as compared to those for MRAC. The PV
control performance degrades significantly after the inertia change and never recovers.
MRACManipulator4: Kp=20.00,K\~100 Cosine desired trajectory
-600 . -200 1 O 1 2 3 4 O 1 2 3 4
Time (sec) i Ïme (sec)
Figure 3.7: Cornparison of simulation of PV control (dotted line) and MRAC (solid h é ) of a 2-Iink robotic manipulator for a desired cosine trajectory with doubling of inertia at t=-l sec.
Figure 3.8 shows the adaptation of the unknown parameters. The parameters show a
sudden increase in values at the time of the inenia change at t = 2 sec but they still
converge to their final values. It should be noted that because none of the four mode1
parameters relate directly to the physical inertia, this approach does not necessarily
estimate the unknown parameters exactly, but simply generates values that allow the
desired task to be achieved.
MRACManipulatorS: Kp=2000,Kv=100 Cosine desired trajectory
0) -.
E - 3 . e UY a l .
Time (sec) O 1 2 3 4
Time (sec)
Figure 3.8: Parameter estimates for MRAC of a 2-link robotic manipulator for a desired cosine trajectory with change in load inertia at t=2 sec.
3.2.3 Straight-Line Trajectory with MRAC
After applying the MRAC scheme to a cosine desired trajectory, the contro! method was
applied to a straight-line trajectory. The end effector of the 2-link robotic manipulator
was required to track a straight-line trajectory in the X-Y plane of the manipulator. Since
the desired trajectory was defined in end effector plane, inverse kinematics was used to
determine the required joint trajectories, which would accomplish the tracking task.
Figure 3.9 gives a simulation result for the manipulator given the task of tracking a
straight line, with the X-coordinate remaining constant and Y-coordinate varying with
time. Initially. the rnanipulator is located to the right of the desired stan point for the
straight line. Note thar the large top grid gives the end effector position in Canesian
coordinates as generated by the angular positions of the joints of the manipulator.
It is apparent from Figure 3.9 that even though MRAC starts from an initially arbitrary
position, it is still able to reach the desired trajectory and maintain a small tracking error
thereafter. This shows that MRAC is capable of giving a satisfactory performance for a
range of trajectories with incornplete knowledge of the systern.
A different set of gains was used to study the effect of a change in gains used for the
control law. The resultant response was plotted as a dash-dotted line in Figure 3.9. It is
apparent that gains needed to be tuned before a good tracking performance can be
achieved. It is also noted that near the full-extension position of the manipulator, small
enors in the joint anges result in a very visible error in the end effector position. The
hard limit to the manipulator extension is r l . 8 on the y-axis (as marked in Figure 3.9).
The trajectory task of Figure 3.9 was repeated for the condition of a step doubling of the
inertia. Due to the reIatively slow tracking speed, there was no detectible effect on the
tracking error as MRAC was able to rapidly correct for the change.
The trajectory task was aIso dane for PV control for the sake of cornparison. However,
there was no significant difference between the PV and MRAC response for this test.
MRACManipulatoiB: MRAC Ibr manipulator. stnight line tnjectory
1 'i j Hard Full extension of the manipulator (small errors in joint 1
i 1.4 i angles result in a visible enor in end eiiector position) 4 bit
End eiiector position, x-axis
Joint 1 Joint 2 - 2 A
v v e l t - p O L - . - g ' j J Z . ' z ', 1 a,/ PI I - -C _ 1 0 - g 0 - 1; U ' U - 2 J
Time (sec) Time (sec)
Figure 3.9: Cornparison of simulation results for MRAC of a 2-link robotic manipulator for a desired straight-line trajectory with two sets of gains: [K, K,] =
il500 601 (dotted line) and [2000 3001 (solid line).
3.3 P and PV Control for DC Servomotors
There are two open-1oop parameters for the SIS0 system (position control of a servo
motor). the motor gain, Km. and the motor time constant, r, . The value of the motor
gain is f ied for a servomotor. The value of the time constant is a fiinction of the load on
the motor. The fügher the rotational inertia of the load, the larger the time constant of the
motor.
The measured speed of the DC motor, v, , can be given as a first order response with the
applied control signal. u, as the input (Kuo, 1991):
The position of the load, y,, can be obtained by integration of the speed signal, v, . Both
v, and ir are measured in volrs. The above mode! c m be convened to a second order
continuous state space form:
where
air - = AX + Bri
A. B = state space matrices of the motor
-r~ = motor position
r . = motor speed
The corresponding state space matrices A and B are given as:
The following values of the two parameters, r, and K m , were obtained by performing
an open loop test (the plot for which will be shown in Chapter 4). The actual response of
the motor to a step change on the control signal will also be given in Chapter 4.
Figure 3. IO shows open loop simulation for the above DC motor for a step reference
input. The figure compares the open loop response for different arbitrary values of motor
time constants and true motor time constant.
Bmotor: Open Loop test: Um=lO, Km=5.6, Ustep=5.00
-10 i f
O 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (sec)
Figure 3.10: Position and velocity open Ioop remit, with Km = 5.6 and rm = L0.0625 (dotted), 0.125 (dashed), and 0.28 (solid)]
3.3.1 Step and Trapezoidal Trajectories
Figure 3. Il shows a trapezoidal move profile with given distance and rnove tirne
parameters. Equations 3.42 to 3.44 give the starting points for determining a trapezoidal
profile.
Figure 3 .1 1 : Trapezoidal velocity profile (Industrial Devices Corporation, 1996)
Assume
where acceleration time
constant velocity tirne
deceleration time
total move time
maximum velocity
average velocity
total move time
acceleration
The average velocity given in Equation 3.43 is defined as:
- Dis tan ce to travel - D - - - Time reqllired t ,
Proportional control of a DC servomotor for a step reference input can be designed for a
given percent overshoot, M, . The tollowing equations can be derived from the classical
analytical design for P-only control of an ideal second order system (Kuo, 199 1):
where
,- - log (lll , , 100) 5 - -
( + ( l o g ( 1 100))')
4' = damping factor
K, = proportional gain (by design)
The P-only control law is given by
where P = error between desired and actual response
Figure 3.12 shows the cornparison of simulation of "Proportional-oniy control" by design
for a step reference input with two different user-defined percent overshoot, M, (5% and
25%). It is apparent from the figure that for Ad, = 25%, the response is faster that that €or
Ad, = 5%. The position signal for smaller overshoot reaches steady state earlier. A h , the
design value of gain does satisf) the desired requirernent of percent overshoot.
Bmotorl: P-only control: Km=5.6,Trn=0.28. Kp=0.978. step input signal
s r : 1 - 1 1C - - - - - - -y - - - - - - - -y-- - -- ',, O;' - --- 1 a -5 1
O 0.5 1 1.5 2 2.5 3 3.5 4 4 0 ,
\
u 1 2 20 i -- ,. . - Y - ----_ I
I y- > O - ---\ --- /
- - - .- 0 -20 - " - - II-
bl -40 &
i - > O 0.5 1 1.5 2 2.5 3 3.5 4
' 0 Y,. z 1 -- I
O 0.5 1 1.5 2 2.5 3 3.5 4 200 .
tn 1 ime (sec) O I
g OL ,,-----. ' ,' -
al r /
U I
-200 a 1
a O 0.5 1 1.5 2 2.5 3 3.5
j 4
Tirne (sec)
Figure 3.12: Cornparison of simulation results for P-only control of DC motor with Km = 5.6, r, = 0.28 sec and percent overshoot hl, = [5 (dotted) 25
(solid)]. Dashed line is the step reference signd.
Figure 3.13 shows the cornparison of simulation of "Proportional-ody control" by design
for a ramp reference input with two different user-defined percent overshoots, M, (5%
and 25%). Proportional-only control wiil result in a non-zero steady-state error both for
step and ramp reference trajectories. Introducing integral control action can eliminate this
steady-state error.
Bmotor2: P-only control: Km=5.6.Tm=0.28, Kp=0.978. ramp input signal
- _./-- \- E O - e - - - -- -----. - * E O 1
! \-
O - 1 0 ~ O 0.5 1 1 .S 2 2.5 3 3.5 4
10or ln y--
1 ime (sec) O .*-.,
--k . '-< - y O? , - - -
- / ,-'
Pi x. b*'- /
C) u -100'
0 0.5 1 1.5 2 2.5 3 3.5 4 Time (sec)
Figure 3.13: Cornparison of simulation results for P-only control of DC motor with Km = 5.6, r, = 0 . 2 8 ~ ~ and percent overshoot M, = 15 (dotted) 25
(solid)]. Dashed line is the rarnp reference signal.
PV control dlows one to design a controllcr fur a specified percent overshoot, M, and
settling time, T,. The following equations cm be derived 6om the classical analytical
design for PV controi of an ideal second order system (Kuo, 1991):
where w = natural frequency
K,. = velocity gain
T, = settling time
The PV control Iaw is given by
Figure 3-14 shows the cornparison of sirnuiation results for PV control applied to the
above DC rnotor for a step reference signal for two different user-defined percent
overshoots and a fixed settiing tirne- it is apparent fiom the figure that the design values
of proportional and velocity gains achieve the required performance in t e m of percent
overshoot and setting tirne. The system settles faster for PV than for P-only control.
Bmotor3: PV control: Km=5.6.Tm=0.28,Ts=0.5,Kp=19.63,K~O.62, step input
-
-20: 1
I
O 0.5 1 1.5 2 2.5 3 3.5 4 ,- 400 Time (sec) - 1 UI
200 1 . --- -- g o c - i/ - - I
1 -'Y. - 1 l /-A
-200 r L K k d 1 O 2 -400
l
O 0.5 1 1.5 2 2.5 3 3.5 4
Time (sec)
Figure 3.14: Cornparison of simulation results for PV control of DC motor with Km = 5.6, rm = 0.38sec, percent overshoot M, = [5 (dotted) 25 (soiid)]
and settling timt T = 0-5 sec. Dashed line is the step reference signal.
Figure 3.15 shows the comparison of simulation of Proportionai-Velocity control for a
ramp reference input with two different user-defined percent overshoots, M , (5% and
25%) and tked settling time. Note that PV control gives much better performance that P-
only control. This is because with PV control. we are allowed to specifj two degrees-of-
freedom - percent overshoot and settling time. This gives us more control over the shape
of the response. This is why the lag between the reference input and the actual output is
significantly Iess in the case of PV control.
Bmotor4: PV control: Km=5.6.Tm=0.28.Ts=0.5,Kp=19.63,K\F0.62, ramp input
O 0.5 1 1.5 2 2.5 3 3.5 4 ;ù 400 , rime (sec) ln 3 200 i- , ,.
' -. A
----- - -.. ? . I
g 0-1 .v : '.- - . ,/ I
5 -200 1 i O 1 2 -400 1
I
O 0.5 1 1.5 2 2.5 3 3.5 4
Time (sec)
Figure 2.15: Cornparison of simulation results for PV control of DC motor with Km = 5 . 5 , r , = 0.28 sec, percent overshoot M, = [ 5 (dotted) 25 (solid)]
and settling time T, = 0.5 sec. Dashed line is the ramp reference signal.
3.3.2 Response to Parameter Changes
The response of the motor can signiticantly degrade if subjected to changes in parameters
Iike motor gain and motor time constant. These changes can be caused by sudden change
in load on the system, for example, applying brakes to reduce the speed of the motor can
actually act as a viscous fiction causing change in both motor gain and time constant.
Figure 3.16 compares the degree of degradation in simulation of Proportional-only for a
step reference input when the system is subjected to tive times change in inertia and
controller is designed for two different percent overshoots.
BmotorS: P-anly control: Krn=5.6.Tm=0.28. Kp=0.978. step input
Time (sec)
Figure 3.16: Comparison of performance degradation for P-only control of DC motor with Km = 5 . 5 , r , = 0.28sec, percent overshoot M , = [5 (doned) 25
(solid)] Dashed fine is the step reference signal.
It is evident from acceleration grid of Figure 3.16 chat the response becomes much more
sluggish after inertia change at i=2 sec. Mso, the overshoot becorne significantly larger
which is clearly a sign of performance degradation. This is tme for both the d u e s of
percent overshoot. Figure 3. L 7 shows that PV control shows a similar trend of degraded
performance.
-20 1
O 1 2 3 4 5 6
200 .; Time (sec] +
l . -200: ,, i
: O 1 2 3 4 5 6 Time (sec)
Figure 3.17: Comparison of performance degradation for PV control of DC motor with Km = 5.5, r, = 0.28sec. percent overshoot M , = [5 (dotted) 25 (solid)]
and settling time T, = 0.5 sec. Dashed Line is the step reference signai.
Figure 3. i8 below illustrates trapezoidal velocity trajectory control using PV controller
when the systern is subjected to 500% change in inertia at f=2 sec. The maximum
velocity is 10 m/s and system has a fixed proponional gain. Figure compares results for
three different values of velocity gain. It is clear that when velocity gain is zero (Le
Proportional-only control), the performance degradation is more severe than when
velocity gain is non-zero (Le. PV control). PV control obviousiy gives more control over
performance in the face of parameter changes as explained before.
TrapTrajl: PV control: Krn=5.6.Tm=0.28, Kp= 2. KF[O 1 21, Trapezoid wlo. traj.
O 0.5 1 1.5 2 2.5 3 3.5 4
Time (sec)
Figure 3.18: Cornparison of performance degradation for PV trajectory control of DC motor with K, = 5.5, r, = 0,28sec, K, = 2, Ky = [O (dotted) 1 (dot-
dashed) 3 (solid)] Dashed line is the reference trajectory.
3.3.3 Circle Trajectory with PV
A planar circular trajectory cm be tracked by using two actuators for each of two
independent axes. We sirnulated cracking of a circular trajectory by using two identical
DC servomotors, one each for two axes in X-Y plane. A PV controller was applied to
control the systeni. The resuit of the simdation is showed in Figure 3.19. The top plot
shows the tracking of circle in 2-D plane. Plots under the top plot show two axes
separately. It's evident chat there is a significant tracking error in case of a simple PV
controller. We can eliminate this error by using the optimal tracking control algorithm
which will be explained in Chapter 4.
CircTrajl: PV Tracking of a circular tnjectory
-1 -0.5 O 0.5 1
Position. Motor 1
l ime (sec) Time (sec)
Figure 3.19: Simulation of PV control for two DC motors in response to sinusoida1 reference signal to track a circular trajectory
In Section 3 , I . a MR4C conuoiler as designed through the gradient appmach and
Lyapunov Theory was tested in simulation on an ideaI second-order system. These
simutations provided an introduction to the nature of adaptive control when dealing with
uncertain systems. In Section 3.2, the MRAC controller was applied to a more
demanding application, namely a two-link serial robotic manipulator. A performance
comparison was made between PV control and MRAC for the tasks of tracking
sinusoidal and straight-Iine trajectories under changing load conditions (doubling of
system inertia). These tests confirmed the superiority of M W C over PV in the case of
changing operating parameters as applied to a variety of trajectories.
In Section 3.3, P-only and PV control oPa DC servornotor was tested in simulation. A
performance comparison was made for the tasks of step, trapezoidal and circular
trajectories. Again, the etTect on pertbrmance of changing load conditions was
examined. These simulations enabled one to contirm a classical analyticd design
procedure For the P-only and PV controllers. This design procedure is used for the
purpose of systern validation for P-only and PV tests in Chapter 4.
The simulations in Section 2 . 3 involved only a single servornotor. On the other band, the
seriai rnanipulator simulated in Section 3.2 involved two servomotors whose cornbined
action moved the robot's end effector whose position was govemed by a set of nonlinear
dynamic equations. Therefore, one cannoe directly compare the results in Section 3.2
with those in Section 3 .3 .
The steady state tracking error seen for the circle trajectory in Section 3.3 highlighted the
need for an explicit tracking aigorithm such as LQRT. This, together with the application
of MRAC to a single DC servomotor for direct cornparison of LQRT and MRAC, is the
subject of the next chapter.
CHAPTER 4
SERVOSYSTEM EXPERIMENTAL STUDIES
Chapter 3 presented simulation results on representative electromechanical servosystems.
Application of bIIWC was made to three systems: an ideai second order system, a 2-link
robotic manipulator and a DC servomotor. This chapter will give experimental results of
tests conducted on a DC servomotor apparatus. Both MRAC and LQRT aigonthms are
tested with a variety of trajectories.
4.1 Background
This section will give a description of the apparatus used for experimentation, together
with open and closed loop results. The results are presented to validate the system mode1
as well as confirm the operation of the control and data acquisition system.
4.1.1 Apparatus
A schematic of the apparatus used for experimental testing is shown in Figure 4.1.
Basically, the experimentai setup consists of a DC servomotor, motion control DSP card
by Precision Micro Dynarnics Inc. (PMDI), a linear power amplifier, power supply and a
Pentium III host PC. The motion controller is used for data acquisition and servo control.
Ail of the interface facilities of the board are available to the host PC. The PMDI card
has digital VO, DIA and A/D converters, programmable timers, and intempt generation
devices. Al1 the hardware on the controller can be programmed using software routines
[to be descnbed in next section). The motion control card resides on the host PC. It is
co~ec ted to the DC servomotor via analog and encoder connectors. The DC servomotor
has a digitai encoder which feedbacks position data to the motion card through the
encoder connector. This encoder data is numericaliy differentiated to obtain an estimate
of motor speed. The host PC bas PMDI MotionTools software, resister access functions
and motion control library. The user can mn MotionTools to control and test the DC
servomotor. Altematively, the user can wnte customized C/C+ programs using register
access functions to implement different control structures and algorithms. The PC's
processor accesses the controller's resources through the PCI bus. From the PCI bus, the
registers are mapped into 32-bit memory. The amplifier amplifies the control signals
Rom a DSP board and supplies them to the motor. The amplifier has multiple current
and voltage mode gain settings. The motor responds to these commands and the position
data is then fed back to the motion card registers. These registers can then be read by
MotionTools or the user's program to complete the control loop.
4.1.2 Control Software
The software used for this research consists of PMDI MotionSuite, which is a
combination of MotionTooIs, motion controller's register access tùnctions and Motion
Control Interface (MCI) Library. A typical user interface is illustrated in Figure 4.2.
MotionTools has a GUI for tuning and testing with a PMDI motion control card. It mns
on a local host computer. It also has the capability of running across a local area network
or even the Intemet to control a remote motion control card and apparatus. MotionTools
was used for several purposes: 1) capture and view motion data, 2) interact with hardware
locally, 3) tune in reai-time by editing controI structure parameters, 4) viewing the
response in a display window and 5) read from and write to memory locations on the
PMDI card. It uses an initialization fiIe format to set various motion control parameters.
The user can also write C/C* prograrns using register access fùnctions and motion
control library. The board communication ftnction group handles basic controller board
communication. These functions deaI with the register map directly and are contained in
the dynamically linked Iibrary (DLL) mcregddll. The experimentation code, as wrïtten
in Visual C*, for al1 tests is siven in Appendix B.
1 PMDI Motion Control/DSP Card 1 m MC8-DSP-ISA
I2 ENCODER
m
f - \ \ Host PC
Linear Power ,- (MotionTools, Register Access fis)
Power ii Digital
Encnd~r
Fisure 4.1 : Experimental Apparatus
Figure 4.2: PMDI MotionTools Main GUI
4.1.3 Open Loop Validation
First logical step in the expenmentation phase is to conduct open loop tests on the system
@C motor in our case) to determine system parameters. Two system parameters as in
Equation 4.1 are sufficient to completely define the characteristics of a DC motor - rnotor
velocity gain, K m , and motor time constant, rm .
These parameters are defined as following:
Motor Velocity Gain, Km : Ratio of open loop steady state velocity to the input
step voltage, K m =v,/II,, , where v, is the open loop steady state velocity of
motor and u , , is the open loop step reference input voltage
Motor Time Constant, rm : Time taken by motor to reach 63% of the steady state
velocity
Inspection of Equation 4.1 shows that the steady state veiocity of the motor is in direct
proportion to the control signal magnitude. Ideally, motor gain should remain constant
with c h g e in the control signal magnitude, since it is calculated as the ratio of v, and
u,, . But in practice, this may not be the case.
Open loop tests were conducted with 2 different control signal magnitudes (5 V and 8 V).
The result is illustrated in Figure 4.3. It was found that the gain slightly changed with the
contrai signal magnitude and the change was nonlinear. The best fit for K, and r , to
the experimental results of Figure 4.3 were: Km = 5.6 and r, = 0.28 sec.
O 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Tïme (sec)
Figure 4.3: Simulation (dotted line) vs. experirnental (solid line) open loop test, cornparison of responses with 5 volt and 8 volt step inputs.
Two approaches were examined to produce a change in ioad for the apparatus in order to
provide a test for controller robustness: different discs (43 % change in r,) and different
magnetic brakes (18 % change in r,). It was found that only the different disc approach
produced a measurable change in controller performance.
Figure 4.4 shows the effect of a change in the inertia for the open loop response. The
inertia was reduced by replacing the origind Iarger and heavier steel disc on the motor
sh& with a srnalier and lighter aluminum d i x . Note that r, was 0.28 sec and 0.4 sec for
the fùll inertia and reduced inertia cases, respectiveIy.
h
O Reduced inertia 2 4 0 L 0 '4 - 2. c U t 0 20- O > Full inenta
Figure 4.4: Simulation (dotted line) vs. experirnental (solid line) open loop test, cornparison of responses with change in inenia (5 volt step inpuij.
4.1.4 Closed Loop Validation
Next step in experirnentation was to apply Proportional control to the motor with a
positive and negative step input cycle. MoeionTools could not be used for this purpose as
it does not allow precise trajectory control. Therefore, a C program was written (see
Appendix B) using DSP r e s t e r access Lunetions to implement the trajectory. This test
also served to validate the motor mode1 we used in simulation and for confirmation of
open loop results.
Figure 4.5 shows a cornparison of simulation and experïmefitai for Proportional control
of the motor. The gain K,= 0.34 was obtained fiom Equations 3.45 and 3.46, with
M, = 5 % and C = 0.69. The experiment (solid line) is seen to be a good match with the
simulation (dotted line). Note that the same gains were used in both experiment and
simulation. This match in the dosed loop tùrther didates the mode1 for the motor as
well as the control and data acquisition system for the apparatus.
Tirne (sec)
Figure 4.5: Simulation (dotted line) vs. experimental (solid line) P-only closed loop test for a step reference signal (with M, = 5%).
As mentioned in Section 3.3.1, PV control ailows the user to speciQ the settling t h e and
the percent overshoit in response to a step command. Figures 4.6 and 4.7 are
cornparisons of simulation and experiment for PV control of the motor with an overshoot
of 5 % and settling times of 0.5 sec and 1.3 sec. The gains K, = 6.72 and K., =0.62 were
obtained from Equations 3.48 to 3.5 1. Figures 4.6 and 4.7 tùrther validate the mode1
parameters obtained fiom open loop test. These results also confirm that PV control bas
been correctly irnplemented with the MotionTools software. In particular, care had to be
taken that the scale factors between experiment and simulation were correct.
Tirne (sec)
Figure 4.6: Simulation (dotted iine) vs. experimentai (sotid line) PV closed loop step test for short response time (with M, = 5 % and T, = 0.5 sec).
O 1 2 3 4 5 6 7 8 9 10 Time (sec)
Figure 4.7: Simulation (dotted line) vs. experimental (solid line) PV closed loop step test for long response time (with M, = 5 % and T, = 1.3 sec).
The foliowing section will deai with the problem of optimal trajectory tracking and the
controller gains will be obtained through an LQ design aigorithm. The LQ approach
tends to generate higher gains than with the analyticai approach used in this section.
In order to provide a comparison with the LQ results, Figures 4.8 and 4.9 illusuate the
step tests with the gains of Section 4.6 (K, = 10, K, = 0.84) for simulation and
experiment, respectively. Three different combinations of gains were used: 1) P-only, 2)
PV with half the velocity gains and 3) PV with hl1 (LQ) gains. The result confirms the
expected behaviour narnely; P only is underdamped, PV (half) is marginally
underdamped and PV (fiil!) is near criticdy damped, with settling times progressively
reduced. It is important to note that this trend is confirmed in both simulation and
experirnent. Recall that the experiment was generated with the controller generated in C-
code and there was some question as to whether the coding was correct.
O 0.5 1 1 .S 2 2.5 3 Erne (sec)
Figure 4.8: Companson of simulation results for closed loop step test with 3 gains: P-only with K=[10 O] (dot-dash line), PV with K=[lO 0.421 (dashed line), and PV with K=[10 0.841 (solid line).
-1 0 O 0.5 t 1.5 2 2.5 3
Time (sec)
Figure 4.9: Comparison of experimental results for DC motor in response to a step input of 1 rot for 3 sec, P-oniy with K=[IO O] (dot-dash line), PV with K=[10 0.421 (dashed line), and PV with K=[10 0.841 (solid line).
4.2 LQRT for DC Servomotor
Linear optimal control is a speciar type of optimal controi. The pIant that i s controlled is
assurned linear, and the controller is constraincd to be linear. Linear control!ers are
achieved by workinç with quadratic performance indices. These indices are quadratic in
the controi and regulatiodtracking error variables. Such methods that achieve linear
optimal control are calIed Linear-Quadratic (LQ) methods.
The design of a LQRT controller for the servomotor as a second order system is ~iven in
this section. For reference purposes, the design of a LQRT controlIer for a first-order
system is given in Appendix D.
4.2. i Op t imd Regula tor Design
Suppose we are given the system (Anderson and Moore, 1990):
i = Fx + Cri
y = H T x
where F, G, H = state-space matrices
Let the quadratic performance index to be minimized is given by:
where 0, = nonnegative definite matrix
R, = positive definite weighting matrix
Then the optimal regulator problem is to find an optimal control rr' that minirnizes the
performance index J. Solving Riccati equation, the solution is given by
where
1,' = - R, .' GP,(t)x(f)
P, = solution of Riccati equation, and is given by
- ~ ( t ) = P(t) F + F F(r) - P(r) G R, -'G'P(I) + Q,
4.2.2 Ramp and Sine Trajectory Design
The regulator theory discussed in previous section deais with the problem of returning a
system to its zero state in some optimal way. This problem is, in fact, a special case of a
wider class of problems where it is required that the outputs of a system follow or crack a
desired trajectory in some optimai sense.
Trajectory followinç problems can be further classified into three categones:
I . Servo problem (plant outputs are to follow a class of desired trajectones)
2. Tracking problem (the desired trajectory is aparricular prescribed
fiinction of time)
3. Model-following problem (plant outputs are to follow the response of
another plant or model)
In this chapter, we will discuss only the tracking problem where the desired trajectory xr
is hown a priori. h example of this type of problem is altitude control of a terrain-
folIowing aircrafl where there is knowledge of tiiture terrain. With a system given by
Equations 4.2 and 4.3, the optimal tracking problem is to find the optimal control u * ,
such that output y cracks the signal x,. The solution of this problem is given by following
three equations (Anderson and Moore, 1990):
where K. = optimal controller gain
Merl = additional control signai term
b = extemai optimal control variable
Changing notation for the control gain (with K = -KI ), the following equations were
adopted for implernentation in MATLAB:
I = Fx + Gu (4. IO)
I I = - K X - R ; ' G ' ~ (4. I l )
With regard to generai tracking task above, various approximately optimal trackers cm
be constructed. Setting Equation 4.12 to zero, and solving for b to get the steady state
solution, b, for the compensator:
A review of the (steady-state) Riccati equation will reveal that
and Equation 4.1 1 with b , substituted for b reduces to the standard form of the LQR
control law:
A state space model of the motor was provided in Chapter 3 (Equations j.38 to 3.40).
Given the sarne motor parameters ( Km = 5.6 and r, = 0.28 sec), then for the case of
RI = 1 and Q, = O], the result foi Equntion 4.14 is calculated with the PC-
MATLAB function LQR as K = [IO 0.841. The initial value for Q, is traditionally
taken as the identi9 rnatrix. The adopted vdue for QI came after a trial and error tuning
exercise where the result was a good balance between speed of response and the
avoidance of over saturation of the control signal.
Given a rarnp reference trôjectory defined as:
The control law uses onIy b2, hence we neglect the solution of b,
Solving Equation 4.12, we get following general solution for b(0:
H22 and C1 =- Hl2
For a sinusoidal reference signai, the exact solution for b(I) becomes:
4.2.3 Ramp and Sine Trajectories with LQRT
Using the optimal control design given in the previous section, simulation tests were
conducted for ramp input signal with a siope of 5 rousec and sine input signal with a
Frequency of 0.5 Hz.
Figures 4.10 and 4.1 1 show the comparison of simulation results for approximate solution
and exact solution for the DC servomotor in response to a ramp and a sinusoidai input
respectively. The top grid shows the tracking enor between the reference signai and the
actual response. It is apparent from the figures that the exact solution succeeds in
reducing the steady state tracking error significantly for both ramp and sinusoidai
reference signal, while the approximate solution fails to do so.
OptimaIMotionSimulal: Q I 1=100. R=1, K=[l O 0.841. ramp input signa1 . 1 : I
Figure 4.10: Simulation response for approximate (dotted line) vs. exact (solid line) tracking sotution for 5 robsec ramp reference (K = [ I O 0.841).
OptimalMotionSimula3: QI 1=100, R = l . K=[10 0.841, sine input signal
Figure 4.11: Simulation response for approxirnate (dotted line) vs. exact (solid line) tracking solution for 0.5 Hz sinusoidd reference (K = [IO 0.841).
Figure 1.12 shows the simulation of deterioration in performance of tracking algorithm in
the event of change of inertia of motor. The inertia of the rnotor was made 5 times of the
initial inertia at K. 1 sec. As cm be seen, this introduces significant tracking errors.
OptimaIMotionSimula3: QI 1=100, R=l, K=[lO 0.841, sine input signal
Figure 4.12: Simulation response for approximate (dash-dotted line) vs. exact (solid line) tracking solution for 0.5 Hz sinusoidal reference and 500% inertia change at t=2.I sec, @=[IO 0.841)
Figure 3.13 shows the comparison of simulation and experimental results for approximate
solution in response to a ramp input. Note that the tracking error for the simulation
reaches a steady state value of 0.5 rot. The cracking error for the experiment is of the
same order of magnitude but doesn't reach a steady state. However, the overall match
between the simulation and the experirnental results is reasonably good. thus vdidating in
part the tracking algorithm. There is considerabte noise in the velocity plot. This is
mainly due to the fact that veiocity data was obtained by numerical direrentiatiun of the
position data from the encoder. The noise in the velocity signal is fiinher arnplified by
virtue of the velocity feedback in the control algorithm.
OptirnaIMotorl: QI t=100, R = l , K=[ lO 0.841. ramp input signal
Time (sec)
Figure 4.13: Optimal tracking test, comparison of simulation (dot-dashed line) and experimental result (solid line) of approximate sotution for a 5 rdsec ramp referme signal (dotted he ) .
Figure 4.14 shows the comparison of simulation and experimental results for exact
solution for the same ramp reference signal. Note that the steady state error is reduced
significantiy and also that simulation resul ts are hrther confirmed by experimental
results, thus vaiidating the algorithm for tracking. There are oscillations in conuol signal,
which as explained berore, are due to noise in velocity signal. Another reason for this
phenomenon is that the motor positian is somewhat osciilatory about the reference ramp
signal, which reflects both in velocity and control input computation. However, despite
the apparently iarge oscillations in the control signal, the algorithm is still abte to achieve
good tracking.
OptirnalMotoR: Q l l=100, R=l , K=[1 O 0.841, ramp input signai
Time (sec)
Figure 4.14 Optimal tracking test, comparison of simulation (dot-dashed line) and experimental remit (soiid Iine) of exact solution for a 5 rocsec ramp reference signai (dotted line).
Figure 4.15 shows the cornparison of simulation and expenmentd results for approximate
solution of DC motor rnodel in response to a sinuddal input. Again there is a steady
state error between desired signal and actud response. Simulation result is a good match
with the experimental result. This is evident fiom the top g i d which shows that the
simulation and experimental tracking error are of comparable magnitudes ( = O 3 ror)
except for the oscillations in experimental plot.
OptimalMotor3: QI 1=100. R=l , K=[lO 0.841, sine input signal
O 0.5 1 1.5 2 2.5 3 3.5 4 2 10 - _- -
7 L - - 0, Y - '- ,de >- +. . - O-/,"- , . '-L - '- 2- 1. - > * L
- I O - -. / .-,--- =
- ----*- 5 - 1 0 :
> O 0.5 1 1.5 2 2.5 3 3.5 4 5 :
2 ; 3 - _ 1-- - -.v*-/-
d'. .: .,-,..)' \ L - .,4 t-- :y, .Y I
; O + . .*,' * -.-, - E ; ; C '- vL'\ 6'.
C ' 1 -:--,.r. ..+ -2..r"u\-J C i
- 5 ; O 0.5 1 1.5 2 2.5 3 3.5 4
Tirne (sec)
Figure 4.15: Optimal tracking test, cornparison of simulation (dot-dashed line) and experimental result (solid line) of approximate solution for a 0.5 Hz sinusoida1 reference signal (dotted line).
Figure 4.16 shows the comparison of simulation and expenmentai results for exact
solution of DC motor mode1 in response to a sinusoidal input. Again, implementing the
tracking aigorithm reduces significantly the steady state enor between desired signal and
actuai response. Again, simulation result is a sufficiently good match to the expenmentai
result.
OptirnalMotor4: Q I 1=100. R = l , K=[lO 0.841, sine input signal
f 1 - . . - . - , - . . . .. m.Y ,ci d O - - - . *. . ,--- e -.. . . ,- - i 'J
\/ j 5 1 1 " -5 ' A
O 0.5 1 1.5 2 2.5 3 3.5 4
Time (sec)
Figure 4.16: Optimal tracking test, cornparison of simulation (dot-dashed line) and expenmentai resuh (solid line) of exact solution for a 0.5 Hz sinusoidal reference signai (dotted line).
42.4 Circle Trajectory with LQRT
In Chapter 3, we explained how a circular trajectory could be tracked by using 2 DC
motors in parallel, one each for rwo axes in X-Y plane. However, Figure 3.16 showed that
there existed a significant steady staie error for the PV controller. The tracking aigorithm
presented in Section 4.2.2 is applied here to both the motors to eliminate this steady state
error.
Figure 4.17 shows the improved cracking obtained by implementing the above algorithm.
Comparing this figure with Figure 3.16 in Chapter 3, it is apparent that the steady state
error with PV control is significantly reduced when LQRT aigorithm is applied to the
circular trajectory.
CircTraj2: LQRT of a circulai trajectory
3 .
N L- 0.2 ?j O
= g -0.2 2. a ;!,
-0 .4 ' ! r \ i , .) ! I
-0.61 '... , . Y ,
-. ! -0.8 i
l
-1 -1 4 . 5 O 0.5 1
Position. Motor 1
F m e (sec) Time (sec)
Figure 4.17: Simulation of LQRT control for two DC motors in response to sinusoidal reference signai to track a circular trajectory
4.3 MRAC Tor DC Servomotor
This section gives the application of Mode1 Reference Control (MRAC) to the DC servo
motor (Chen and Naughton, 2001). The technique used for estimation of unknown
parameters is Recursive Least Squares (RLS). The objective is to design an adaptive state
ieedback controller to track a square reference input wave. The controller should meet the
following specifications of the dosed-bop step response:
Percentage overshoot, M, 5 10%
Settlinj tirne, 1: 5 0.8s
4.3.1 Discrete Mode1 for MRAC
We now discretise the system rnodel. Let h be the sarnpling period; the discrete-time,
state-space model is then given by
where 4, i- and H are discrete-time state space system matrices and
i r , = u(kh)
The systern matrices are given by
where A,, AI, i l , and A, are unknown parameters of systern mode1 are given by
following relations:
4.3.2 Con troller and Observer Design
It is assumed that the closed-loop poles are to be located at z = :, and : = zz, for the
purposes of the design of the controller and its associated observer. The following state
feedback control law can be used to track a step reference signai r, (Franklin ec al, 1998):
where K is the gain matrix obtained kom the Ackermann formula (Franklin, Powelt and
Workman, 1998):
where WC = [r # r] is the controllability marrix of the discrete-time model. ' P d ( ) is a
polynomial constmcted from the desired closed Ioop system poles, and N,and Nu are
solutions of the following matnx equation. where 1 is the identity matrix:
Following reduced-order observer can be used to estimate the angular velucity, xzs :
where
- = yi: - - Aj lir-i
and 0, is the observer gain, which can be obtained h m the Ackermann formula:
t
where z = r,, is the desired observer pole.
4.3.3 Recursive Least Squares (RLS) Estimation
The parameters of the system mode1 are given by A, , AS, A, and A, . They can be directly
estimated by using the RLS algorithm. The first step is to obtain the following discrete
transfer fùnction:
where
11, = K , ( h - r , + r,e-L)
11. = K m ( r m - rn, f! -L - h e ' L )
Ci, = - ( 1 + e -L )
ci, = r - L
From Equation 4.4 1 , following input-output relationship is obtained:
-vk = -d,yk-, - +n,uk-, + ntllk-2
Let E = [di d: t 11.1 be the parameter vector for estimation and
L k = [-Y,., -h-: U r - , "k - iP
Then, we can use RLS algorithm to estimate the parameter vector 6 (Astrom and
Wittenmark, 1989):
where O 5 p I 1 is a forgetting factor.
When the estimate of the vector 5 is obtainrd, it is used to determine K m and r , fiom
Equations 4.42 CO 1.45 as:
Subsequently, the values of Km and r, from Equations 4.52 and 4.53 are substituted into
Equations 4.30 to 4.34 to obtain the parameters of the state space model.
4.3.4 Square Wave Trajectories with bIRAC
The MEWC aigorithm as defined by Equations 4.23 to 4.53 was applied to the DC
servomotor model as defined by Equations 3.39 to 3.41 . A design specification for a
percentage overshoot, M, of 10% and settling time, T, of 0.8 sec was applied. The closed
loop poles were determined From these specifications, which were then used to obtain the
initial feedback gain matrix K.
The correct model parameters for the motor as taken fiom Section 4.1.3 are K m = 5.6
and r , = 0.28 sec. Two simulations were done with two different initial (incorrect)
estimates of K m and r,, :
Low estimate Case: Km = 3, r , = 0.1
High estimate Case: Km =IO, r , = 0.5
The simulation for the low and high estimate cases are shown in Figures 4.18 and 4.19,
respectively. It is apparent from both figures that the actual response (shown by solid
line) approaches the ideal response after two transients and satisfies the design
requirements of percent overshoot and settling time. This verifies the W C algorithm
and establishes the fact that MRAC aigorithm is capable of giving desired performance
when there is only a partial knowledge of the system parameters.
Attempts were made to impiement this iMRAC design on the experimental setup for the
DC servomotor. In esperiment under MRAC control, the motor never reached steady
state and consequently there appeared to be a convergence problem with the code as
implemented in Visuai C++. The code for the experimentation of , W C is induded in
Appendix B for reference purposes.
M o t o r M R A C I . M R A C using R L S : initial K m = 3 . Tm =0 .1 , M p = 5 % . T s = 0 . 8 ~ 80 1
1
O
Figure 4.18:
2 4 6 8 10 1 2 1 4
Time ( s e c )
Simulation result for M M C controller, the actual response (solid line) adapts afier a few transients of square wave reference signal (dotted Iine), initial low estimate Km = 3, r , = 0.1 sec, trueK, = 5.6, r , = 0.28 sec.
Motorhi R A C 2 : M R A C using R L S : initial K m =IO. Tm =O.S. M p = S % . Ts=0.8s 6 0 i I
-6 0 O 2 4 6 8 1 O 1 2 1 4
T i m e ( s e c )
Figure4.19: Simulation result for IMRAC controller, the actual response (solid line) adapts after a few transients of square wave reference signal (dotted line), initial high estirnate K, = 10, rm = 0.5 sec, true K m = 5.6, r , = 0.28 sec.
Figure 4.20 shows the cornparison between performance of W C controiler and PV
controiler for the same square wave signal as used in Figures 4-17 and 4.18, with a 500%
inertia change at F I O sec. It is clear fiom the trend in tracking error that MRAC
controller gives better performance than PV controiler in the event of change in motor's
properties.
MotorMRACVsPV: initial Km=3. Tm =0.1. Mp=5%. Ts=0.8s 1 O0 1
Inertia change
-100 -- O 2 4 6 8 10 12 14 16 18
Time (sec)
Figure 4.20: Simulation result for W C controller (solid line) vs. PV controller (dash- dotted line) for a square wave reference signal, 500% inenia change at F I O sec.
4.3.4 Circle Trajectory with iMRAC
Figure 4.21 shows the simulation of MRAC applied to two motors for tracking a circular
trajectory. Note that with each cycle the tracking error decreases and is significantly
reduced at the end of two cycles.
MotorMRAC3: MRAC tncking for a circular trajectory
-1 4 . 5 O O 5 1
Portlion. Motor 1
Time (sec) nrne (sec)
Figure4.21: Simulation of MRAC for two DC motors in response to sinusoidal reference signal (dotted line) to track a circdar trajectory
4.4 Corn parison of LQRT and hlRAC
Figure 4.17 shows simulation of LQRT for two servomotors running in parailel to track a
circular trajectory. [t is apparent from the tigure that LQRT gives good tracking
performance reducing steady state error significantly. Similady, Figure 4.21 shows
simulation for MRAC controller for the same circular trajectory and setup of two rnotors
tracking the desired path. It was noted that after a few transients, MRAC gives a good
match between actual and desired response.
However, it is not clear from Figures 4.17 and 4.21 how do LQRT and MRAC perform in
case where the servomotor inertia changes manifold for the same trajectory. Figure 4.22
shows comparison of simulation for MiUC and LQRT for the circuiar trajectory with
500% inertia change in servomotor at F l sec.
It is apparent from the figure that although both LQRT and M M C reduced steady state
error up until f= L sec, afler inenia change, LQRT gives a poor performance compared to
MRAC. Mer inertia change in servomotor, LQRT produces a much larger steady state
error at the end of the cycle; on the other hand, W C shows a jump in error onIy
initiaily. after which it recovers and brings back the system with much less tracking etror.
Therefore, it can be concluded from this test that MRAC is better than LQRT in cases
where inertia or load on the system changes unpredictably. When the system is not
expected to face unpredictable or sudden variations in loading conditions, LQRT is a
good alternative and it provides satisfactory performance.
A major issue with implementation of iMRAC on reai life systems is the computationai
overheads. Since the design of k W C system is more involved and the aigorithm is
much more complex, greater computationai resources and capabilhies will be required for
satisfactory real-time performance. Although cheap cornp~tation is available now due to
faster microprocessors, the systems too are growing compiex with a large array of
variabIes and parameters to account for. Therefore, MEUC is advisable only when the
knowledge of the system is incomplete, uncertain or varying.
MRACbsLQRf: MRAC VI LQRT b r a circulai tralectory 1 5
Position. Moror 1
-2 ' -5 ' 1
O O S t 1.5 2 O 0.5 1 1.5 2 firne (sec) rime (sec)
Figure 4.22: Cornparison of simulation oFLQRT (dashdotted line) and MRAC (solid Iine) for two motors for tracking a circuIar trajectory with 500% inema change at t= I sec.
Section 4. L gave the description of the apparatus and software that were used for
experimentation on the DC servornotor. A cornparison was given of simulation and
experimental results for open loop tests and closed loop tests with P-only and PV control.
The open loop and closed-Ioop tests confirmed the rnotor constants and validated the
control and data acquisition system for the apparatus.
Section 4.2 highlighted the need for an optimal tracking control algorithm. The
background theory for optimal regdator was provided. An LQRT algarithm for ramp
and sine trajectories was designed, sirnulated and implemented on the DC servornotor. It
was confirmed that the tracking error was reduced significantly by the application of
LQRT. A good match was obtained between the simuLation and experimental results of
the algorithm. LQRT was aIso applied in simulation to circular trajectory tracking with
two servornotors. Again. perfbrmance was cornpared with that obtained with PV control.
There was some concern about the difference in magnitude of the experimental noise in
the velocity signal, when a comparison was made of the ramp and sine trajectory tracking
tests. This difference was seen when one compares Figures 4-12 and 4.14 (for the ramp)
with Figures 4.15 and 4.16 (for the sine), respectively. ft was observed that sine tracking
was srnoother than rarnp tracking. No reason could be advanced for this difference. On
the other hand, it did not appear to affect the tracking performance of LQRT.
In Section 4.3, ;WWC was applied to the DC servomotor, usin3 the RLS technique for
parameter estimation. MIWC was tested on its ability to track a square wave comrnand
signal assuming pmial knowIedoe of the motor gain and time constant. The simulation
resdts showed that was able to obtain the desired performance in spite of these
unknown parameters. A comparison was also made between the performance of MRAC
and PV in the case of an inertid load change part-way through a test. It was found that
MIWC recovered rapidly after the load change. On the other hand, PV control was
unable to maintain the performance afier the sarne load change.
Finalty, a cornparison of LQRT and MRAC as applied to a circular trajectory with Ioad
change was made in Section 4.4. The tracking by the LQRT algoBihm degraded d e r the
load variation. By contrast, W C proved to be a robust technique as &er an initial
increase in tracking error, it was able to restore system performance to its original value.
in conclusion. it was apparent frorn the inertial load change tests that unlike MRAC,
LQRT is not suitabIe for systems which undergo significant parameter variations. On the
other hand, LQRT is simpIer to impIement than MRAC. which makes it a good choice
for those applications with constant and nearly constant parameters.
CHAPTER 5
CONCLUSIONS AND RECOMMENDATIONS
5.1 Conclusions
This thesis examined Model Referencc Adaptive Controi (MRAC) and Linear Quadratic
Regulator Tracking (LQRT) control with application to the tracking control of
electromechanical servosystems. A literature review was given on the theory associateci
with the design of MIWC and LQRT control algorithms.
Simulation of MRAC for an ideal second-order systern was done to validate the
algorithm and illustrate the behavior of ;MRAC when dealing with uncertain systems. A
convincing demonstration was çiven of the ability of MRAC to start with partial
knowledge of the system to be controlled and yet still achieve satisfactory performance.
Next, MIWC was applied to a two-link robotic manipulator and was required to track a
cosine and a straight line with changing inertial load conditions. These results, as
compared with the performance of conventional PV control, funher showed that MRAC
could achieve significantly better positioning performance under conditions of changing
load. The performance of PV control for a trapezoidal trajectory was studied to provide a
benchmark measure of performance. Next, PV control was applied to the task of using
two DC servornotors to track a circular trajectory. The simulation results showed that
there was a steady state error berween the desired and actual trajectories that was inherent
to the system and consequently could not be eliminated.
Moving on to experimentation, open and closed dosed loop P-only and PV control tests
were conducted to validate a systern model and confirm the operation of the control and
data acquisition systern for an experimenta! DC servomotor apparatus. A commercial
software package known as MotionTooIs was used in conjunction with user-writien C-
language routines. The DC servomotor was modeIed as a second order system and an
appropriate LQRT algorithrn was desigfied. The LQRT algorithm was applied to the DC
. servomotor for ramp and sinusoidal trajectories. M e r successful simulations, the
algorithm was validated experimentally.
Simulation of LQRT for two servomotors tracking a circular trajectory was also
perforrned. The circular tracking iesults were compared with those for PV control. It
was found that LQRT significantly reduced the tracking error of the servosystem.
MlWC was also applied to the servomotor for a square wave command signal trajectory
with different sets of initial motor parameters. It was apparent €rom the results that
MRAC could indeed adapt to a lack of aprtort system knowledge and with the aid of its
adaptation mechanism. gave good performance. Finally, both LQRT and MRAC were
sirnulated and compared for a circular trajectory under the condition of a inertial load
change in the system part-way through the test.
In the introduction to this thesis (Section 1.2), a set of 4 issues or questions was raised.
The direct answers to these questions are as follows:
On the issue of relative tracking performance of MlWC and LQRT for
electromechanical servosystem, both MRAC and LQRT gave comparable
pefiorniance under constant gain conditions.
On the issue of the appropriate conditions for their application, MRAC gives a
robust performance when the system to be controlled has unknown or uncenain
parameters or changing environment. On the other hand, although LQRT gives
good tracking performance in the case of invariant systerns, it does not maintain
the desired performance in the face of parameter changes. This is expected as
LQRT is a non-adaptive algorithm and is designed for fixed set of system
parameters. Thus, it cm be observed that the choice between MRAC and LQRT
would be dictated by the context of the application.
On the issue of ease of implementation, it was apparent t?om the expenmental
results that bIRAC is harder to implement than LQRT. The W C aigorithm is
more cornplex because in order to adapt the system, the controller gains need to
be estimated and adjusted at every sampling interval. The MRAC algorithm is
computationally intensive, making real-time performance harder to achieve. On
the other hand, LQRT is computationally straightforward.
The algorithms for MRAC and LQRT perfonned as expected and consequently
no recommendations can be given with respect to improving their performance as
implemented for this thesis.
[n conclusion, the choice between VRAC and LQRT is a tradeoff between ease of
implementation (simplicity) and pertbrmance when there is a risk of changing load
conditions ancilor unknown system parameters. But MRAC would achieve better
performance than LQRT in the case of varying systern parameters. The complexity of
implementing the MRAC algorithm was yaphically illustrated in this thesis by the failure
to get MR4C to work for the experimentai servomotor apparatus.
When using PV and LQRT control in experimentation, it was clear that the quaiity of the
velocity signal was important. The velocity was calculated by numerical differentiation
of the measured position signal from the digital encoder. This resulted in numerical noise
in the velocity signal. It is therefore recommended that a tachometer be incorporated
dong with the existing digital encoder for direct velocity feedback. Since the control
signal depends on the velocity signal, a better quaiity velocity signai wilI improve the
quality of the control signal.
Since PMDI MotionTools software can be used only for fixed-gain control structures iike
P, PV or P D and cannot be expanded to implement MRAC, other control algorithms
have to be written and irnplemented in C/C++ language. However, one of the advantages
of using MotionTools is its user-fiendly GUI. This provides easy access to various
parameters and programrning fiinctions. Using Visual Basic or Visual C*, a sirnilar
GUI for other control algorithms could be developed. It is recommended that such an
interface be developed using a standard Windows Programming language.
Consideration should also be given to procuring at least one more servornotor so that two
servomotors can be run in paraliel. This would allow expenmentation on two-
dimensional trajectories such as a circle (which was simulated in Chapter 3 and 4).
It is also recommended that the MTWC algorithni, as explained in Chapter 4 and
documented in Appendix B. be debugged and implemented successfiiliy on the
servomotor apparatus.
Another area to explore could be the cornbination of different tracking and adaptive
algorithrns to design a novel control technique, which couId ive more robust tracking
performance in the tàce of system parameter variations for a range of trajec~ories.
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APPENDIX A
Simulation Source Code
The following source code is a listing of the routines used in PC MATLAB for
simulation purposes. For al1 simulations, a numerical time step of 0.001 sec with a 1'
Order Euler Method was used. Previous work with the serial manipulator model had
shown that this time step was appropriate from the standpoint of both accuracy and
stability (Garg, 2000). Given that the tirne constant of the servomotor model was on the
order of 0.1 sec, it was evident that a tirne step of 0.001 sec was suficient for this
application as well.
Adaptive Control for a l"-order Syvtcm
Y- Input square signal genention [U,V = gensig('square',70.time. samplel: Z = ones(size(U); UC = (Z-U); for j = L:length(UC); if UCQ) == O; UCQ) = -L; end end
Yi- Model response IO square input sys = tf(am,[l bml); [YM.îMl= Isim(sys.UC.T):
YI- Process for i = 1 :lengtIi(YbO
dy = <*y + d * u v = y + sample*dy Yi- actwl process output é = y - ~h.i( i) ; %- error between actual and mode1 output du = -am*st + UC(i): xt = .ut + wmple*d~t: du = -am*.u + y: .YS = .ss + sample*d~~. dtO = -(gamma*e)*.\;t: 10 = tr) + mple*dtO: %- conuoIIer parameter dsO = (g;imma*e)*.us. sO = sO + samplecds0: O/&- mntmIler parameter u = tO*UC(i) - sO*y Y- corn1 signal Y(i) = y: Ul(i)z u;
End
Adaptive Control for r Zn*-order System
This code represents the impleinentation of Equations 3.1 to 3.1.
Y i - Input square signal genention [U.Tl=gensig('square1. period, time. sample): Z=ones(size(U)); UC=(Z-U): UCI=O. L*UC: UCZ=3.5*UC:
Y- h c s s for i= l:Iength(Yhl)
dy2 = tlieta*UCl(i) - al*dyl - aï*?; dyl = dy 1 7 sample*dy2: y = y + sample'dy 1; %- actual pmess outpu1 e = y - YW); dtlieta = -(gamma*e)*YM(i): Y- panmeter adaptation theta = rhcta + sample*dtheta: Y(i) = 5
end
Aduptivc control fur 2"-order System with Modified MIT Rule
This code represents 11ic ti~ipleineiitation of Equations 3.5 IO j .7.
Y- Process for i= l:lengtli(YM)
dy2 = thetaeUC l ( i ) - a l*dy 1 - aZ*y dy l = dy 1 + saiiiple*dy2: y = y + sainple*dy l; e = y - YM(i): Y M L=Yivl(i j: dtheta = -gamma8SAT(e.YM1.thet30.alpha.bet'd); Y- Saturation function theta = theia + sample*dtlieta; Y(i) = y:
end
function zI = SAT(e.YMl.thetaO.alpIu.bet2l): theta0 l=ilietac)*2; P = (e*Wl*ibetaO)l(alph*thetaOI + YMIY); if P c -beta Y&- Saturation parainetet
zl = -ben: eiseif abs(P) <= beta
21 =P: else
z l = betx end
MRAC of a 2-link manipulrtor
This code represents the implementation of Equations 3.12 to 3.36
'7" Mode1 parameten and c o ~ r s ml = 1; II = 1; Icl = 0.5: lm1 = 0.5: KP = 2000; KI = LOO: KV = 100; %- ControlIer gains lambda = diag([0.03 0.05 O. 1 0.31); g = 9.8; zeta = 2O*eye(2);
Y&-- Simulation IOOP for i = l:(tiine/dc+l)
est-Hl1 = alplu + 2*eps*cZ + 2*ctafs2: Y- Esuinated henia matrix at-H 12 = beta + eps*c2 + cta*s2: est-HZ1 = beta + epsW - eta*s2: est-HZ2 = beta: est-H = [est-Hl 1 est-HF: est-H2I est-MZI;
HI 1 = 3.3 + 2.08*cL + l.Z*sZ: Y i Tme [nenia matrix Hl2 = 0.97 L.O.l*cZ + 0.6*s2: H21 = 0.97 + l.O-i*c2 + 0.6*s2: H22 = 0.97: H = (HI1 H12: H21 H221: H-diff = est-H - H:
C 1 1 = -h*qv(2); C 12 = -ll+(qv( 1 )+qv(2)): C2I = hfqv(l): C22 = O; C = [Cl 1 CI2: C X C22l:
Y- Coriolis mritriu
qv= qv cdc*qiL q = q + dc*qv:
%- MRAC contr~ller m d e i qe =q-qd; %- T ndring error qve = qv - qdv: s = qve + zeta*qe; Y- Velocity enor qw = qdv - zeta*qe; qra = qda - zeta*qve: tq = est-H*qn + est-C*qm - KV*s: O/&- Conuol law
Y11 = qn(l.1); Y&- Y mayt~ Y12 = qra(2.1); Y21 =O; Y22=YlI+Y12: Y 13 = (2*qn( 1.1) + qn(2,1))*c2 - (qv(2.l)*qni 1.1) +
qv( 1.1 )*qni(2, 1) + qv(2.1)*qrv(2. I))*s2: YI4 = (2*qra( 1.1) + qn(2.l))*s2 + (qv(2.L)*qrv(l.l) +
qv( l.l)*qtv(2.1) + qv(2.l)*qrv(2.l))*c2: Y23 = qn(l.l)*cZ + qv(l.l)*qw(l, l)*s2: Y24 = qn( l.l)*sZ - qv(l.l)*qrv(l.l)*cZ: Y = [YI 1 Y 12 Y l3 Y 14: Y21 Y22 Y23 Y2-41;
desta = -(larnbda*Y')*s: Y- Panineter estimation law e s t a = e s t a + dctdcst-a: alplia = esta( 1-11: bcta = esr-a(2.1): eps = est-a(;. 1); eta = est_a(4, 1);
end
%- Km: Motor gain Yo- Tm: Motor tiine constant %- Am = [O 1: O - 1ITinl; Bm = [O; KmTinl; ?& state-space muices
%- Mp: Desired Ovenlioot Yo- zeta = ~10g(~lp/100~)/çq~(pi~3+(log(MplIOU))~2): O/&- Vm'i: Setpoint uelocip %- If Open Loop test. Usrep: Step size Yo" if PV tesr. Ts: Desired Sertling T h e
Kv = ( (S8Tf l s ) - I ) /Km: wn = -I/(Ts*reta): Kp = wn*wn7m/Km;
O/o_ If P-only test. Kv = O: Kp = l/(Km*Tm*(2*ze1a)~S); %- dt: T h e Step t = 0:dt:time: uc = O; 5' = 0: sp = y, dxdt = O; yt = s p dytdt = O: dtydtt = 0:
O/&- Main simulation loop for ii= l:length(t);
W ( i i ) = y: Yref(ii1 = s p üü(ii) = u c W(ii) = dydt: AA(ii) = d2ydt2:
tf P-only or PV test if Vrnas-=O:
if t(ii) > 2 & sp > U: sp = sp - Vinas*dt; Y' tnpezoidal move profile
elseif t(ii) > 0.1 & sp < 10; sp = sp + VinsPdt:
end else
if t(ii) > O. I; sp = 10: end %- step tcst if t(ii) > 2: sp = O: end
end e = (sp -y): uc = Kp*e - KPd~dt : %- contml law
else if t(ii) > 0.1; uc = Ustep: end O/&- open loop test
if t(ii) > 0.4: uc = 0: end if t(ii) > 2.0; uc = 0: end if t(ii) > 2.3; uc = 0: end
end if uc> 10: uc = IO: end if us-10: uc = -10; end
% - motor modei d2ydt2 = (Km*uc - dydt)TTm: YOC d2pdt2 = (Krn*uc - dydt)l(j*Tm): O/&- parameter change dydt = dydt + dt8d2!dt2: y = y + dt'dydt:
end
O/- Initial conditions t = &dl: time; ucl = k y l =O;spl =yl;dydtl =0;yt1 = spl; dytdtl =O; dZydt21 = 0: uc2 = O: y2 = 0: sp2 = y2; dydt2 = 0: ytî = sp2; dytdt2 = 0; dLydt22 = 0;
for ii= l :(time/dt+l): W l ( i i ) = yl; Yrefl(ii) = spl; Wl(ii) = ucl: Wl( i i ) = dydtl; AAl(ii) = dîydt21: WZ(ii) = y2: Yrefl(ii) = sp2; üiJ2(ii) = uc2; WZ(ii) = dydt2; AA2(ü) = d2ydiZ2;
If P-only or PV test if [(if) >= 0:
sp 1 = sin(2*pi4(ii-l )*dtltirne); end e l = (spl -yl): uc 1 = Kp*e l - Kv*dydt 1: if [(ii) >= O:
sp2 = cos(2*pib(ii-1 )*dtltime); end e2 = (sp2 - y?): uc2 = Kp*e2 - KvZdydt2;
end ifucl> 10; ucl = 10: end ifucle-IO: ucl = -10; end if uc2> 10: uc2 = 10: end i l uc2<- 10: uc2 = -IO: end
O/&- mofor mode1 d2ydt2 1 = (Km4uc 1 - dydt 1 )fiin: d2vdt22 = (Km*ucZ - dvdt2)flm. dGt1 = dydt 1 + dtLd2ydtl 1; dydU = dydt2 + dtqd2!dt22: yl = y 1 + dtfdydtl: ' 2 = y2 + dt*d!.dQ:
end
for kk= 1 :lengtli(t) TERRl(kk) = Yrefl(kk) - W l(kk); TERRZ(kk) = YrefZ(kk) - YY?(kk):
end
mu = input('motor tifne constant'): Y&- Motor parameters Km = input('motor gain constant'); F = [O 1; O -1ltauj; G = [O Kmltaul'; [KSI = 1qflF.G.QQ.R): [phi gamma] = cZd(F.G.dt): H=(G*K -F)'; invH = inv(H); C l = Q I lM(I.2) %- calculare parùcular solution C2 = H(Z.Z)rH( 1.2)
for ii= l:lefigth(tt): iJü(ii) = u; Xl(ii) = ~ ( 1 ) ; XZ(ii) = s(2): RXL(ii) = slr. BB2(ii) = b2; if hg&= 1; u = -K(l)*(s( 1) - sir) - K(2)*~(2):
elscif flagP==2 XR =[sir 01'; bss = -invH*QQ*,W:
if flagQ==O; % - step solution b2 = bss(2):
elseif flagQ== 1; % --- exact nmp solution if tt(ii) >=?.O
b2 = bss(2); % - nvitcli to stcady state solution elseif tr(ii) c 3.0 & tt(ii) > 0.1;
b2 = -C 1 *aaf (tt(ii) + C2 - 0.1 }: else
b2 = 0: end
clseif flagQ==2: % - e.ua sinusoid solution if tt(ii) > 0.1:
b2 = -CI*((aalZ)*sin(wn* (tt(ii) + C2 - 2))); end
end u = -K*s - ( l/R)*G'*[O b2j';
end
%- process mode1 (digital implementation) .u = phi% + gamma*^:
end
MRAC Cor DC Motor
This code represents the iinpleinentation of Equations 4.22 to 4.53
T = inputlSampling tirne'); n = 3: T p 4 : TF1 4: t=O:T:TI: Ns=length(t); rad=pi/l80; theta = [j: ref = jO.O*rad: for ii=l :length(t) y01 genention of square wave reference signal
if ((i(ii)>=O & t(ii)<2)) reÎt(ii) = O;
elseif ((t(ii)>=i & t(iiF=-t) i (t(ii)X & i(ii)<=8) 1 (t(ii)>lO & t(ii)c=It) 1 (t(iipl4 & t ( i ip16)) reh(ii) = jO*nd
efseif ((t(iiP4 & t(ii)c=6) 1 (t(iiP8 & t(ii)<=LO) 1 (t(iipl2 & t(ii)<=l4) 1 (t(ii)>l6 & f(ii)<=lS)) refv(ii) = -5O*nd;
end end
po = input(' Ovenhoot. dehuIt 5%?); u2 = input(' Settling tiinc. default 0.8s:"): zeta = -Iog(po)/~qn(log(po)~2 + piA2); wn = 4lts2izeta: J = qn(-1): sd = -zeta*wn + Jtwn*sqR( 1-zetaA2): sd = [sd. conj(sd)l: zdc = esp(sd*T): pdc = polgzdc): O/&- closcd-loop poles
npann = 4; P= IOOO*e~e(J): Q=eyeO); Iainbda = esp(-T'rrp): Kin[ = 5.6; Tm[ = 0.28; O/&- tme panmeten Km = 3: Tm = 0.1; 9 1 ~ initial piininelen E = esp(-Tfim): a l = -Tm*(E-1): a Z = E bl = Km8(T + Tm*E - Tm): b2 = Km*( l - E): der0 = b2*(a2*b 1 - b 1 - al *b2): n l = Km*(T + Tm*€ -Tm): n2 = Km*(Tm - Tm*E - T*E): dl = - ( I + E): dZ = E: parm = [dl d2 nl nt]':
Phi = [l al; O a21; O/&- ?stem matrices Gamma = [b 1 b2l': c = [i 01: M = [ p h i - qe(2). Gainmal: [C 01 1; R=[00 11'; N = Ns = N( 112); Nu = N(3);
oldul = O;old\;k = [O O]';ol&x2h = O;
for k = I : ( W Det l(k) = 0; tlieta(k) = 0;
end
dydt = O; '-0; u=O: for k = (m+I):length(t) % main siindation loop
if t(k)>= IO d2ydt2 = (Km l *u - dydt)/(Tm 1 9 ) ;
else dQdt2 = (Kml*u - dydt)ITml: end
dydt = dydt + T*dZyd12; y = y + T*dydt;
Plu = [ l al; O a2I; Yi-update .stem marices Gamma = [b 1 b21';
Der L(k) = det(cüb(Plii,Camma)); Kp = [O 1 ~*inv(ctrb(Plii.Ga~nm))~poIy~alm(pdc,Plu);
ytilde = y - oldy - b l *oldu: Lr = a2131; s2Ii = al*olds21i+ bPoldu + Lr*(~lildc - a l *olddIl): .sk = [y .dhl':
ref = refv(k): u = Nu*rcf - Kp*(.xk - Ns*ref);
%-- RLS panrnerer estimation psi = [-oldy oldy 1 oldu oldu II'; K = P*psi/(lambda + psi'*P8psi): P = (eye(nparm) - K*psi')*Pll;uiibdi; parm = parm + K.(! - psi'*parni):
theta(k) = y oldxk = &d<: olddli = .dh; oldu 1 = oldu; oldu = u: oldyl = oldy: oldy = y:
end
EXPERIMENTATION SOURCE CODE
The following source code is a listing of the routines used in Visual C++ for experimental
control and data acquisition purpases. For al1 wperiments in this thesis, a sample time of
0.001 sec was used as this was found to be suficient to enable one to assume that contrai
could be considered continuous in nature. Closed loop tests conducted with a sample
time of 0.1 m e c showed no significant change in the results. On the other hand, one
began to see discretization effects when the sample time was increased to 0.0 IO sec. This
is shown in Figure B. 1. The discretization is observed in both velocity and control signal.
It is also apparenr from Figure B. 1 that when a sampIing time of 15 msec was used for
control, the performance gets worse. It was also noted that the control and velocity
signals obcained by using 10 msrc and 25 rnsrc were slightly out of phase with those from
L msec. The PMD[ hardware and software is capable of a sample time of 0.00 1 mec.
OptimalMotor5: Q I 1=100, R = l , K=[1O 0.841. ramp input signal
O 0.5 1 1 .S 2 2.5 3 Time (sec)
Figure B. 1 : Optimal tracking experimental test for exact solution for a 5 rot'sec ramp reference signal. cornparison of results for different sampling times: 1 mec (solid line), 10 msec (dash-dotted line) and 25 msrc (dashed line)
Figure B.2 shows resutts of a cornpanson of results of experimental test for sarnpling
time of 1 msec and 0.25 mec. Expanded scales are used in the figure to highlight that the
tracking errors decay with time and after 2 sec, the actuai trajectory is a sufficiently good
match with the desired trajectory. The initial 0.5 sec is not shown, as the time period up
to about 0.5 sec is a transient. It is apparent Frorn Figure B.2 that use of sampling time
srnaller than l msec is not warranted as there is Little improvement in accuracy by doing
do. It is also noted that the control and velocity signals obtained by using 0.35 m e c were
slightly out of phase with those from 1 msec.
OptimalMolor6: Q I 1 =100. R=l . K=[ l O 0.841, ramp input signal
Figure B.2: Optimal tracking experimental test with expanded scales for exact solution for a 5 rousec ramp reference signai, cornparison of results for dierent sarnpling tirnes: 1 msec (solid lie) and 0.25 m e c (dash-dotted line)
Open loop and closed 1001) (Piinly and PV) control
O/&- Standard NT phtfonn librarïes #inchde <windo\\.s.li> %- windows atatfonn routines
Y- PMDI Header Files #inchde "mc8pnm.li" #inchde "mc8reg.1in
int main(void)
O/&- &ndard UO mutines %- C console and pon UO routines 44- Math fiinctions libncy
Yi- ,W8-DSP-[SA communication primitives %- MC%-DSP-[SA register access hnctions
% MC8 board data structure
O/&- Opcn board O - force it to stay open on nul1
Y&- input: riIn (Open loop, closed Imp P. ciosed lwp PV) switcli(nln) { case '1': /***** OPEN LOOP TEST *****/ iiic8PITload(gpiiic8Info.inten.ril): %- set up PIT incSPITstan(gpinc8Info): %- swn PIT YOI yncronizc PlT and SYNC ~ C ~ S Y N C C ~ ~ ~ ~ ~ ( ~ ~ ~ C ~ I ~ ~ O . E U ~ C ~ S R C P ~ T . ~ . I C C ~ . SRCNONE); % sct intcmpt mask m~8W~teReg(gpmc8InfoMC8~INTEN~OST.Uxl): wliile (!kbliit()) {
Y- check for Pïï interrupi rnc8ReadReg(gpmc8Inf0~h,lC8~~GHOST.&~flag): if ((INTflag&Osl)==0sl) ( mc8EncRead(gpmc8Info.O.&EncCount); Yi-Read encoder EncArra~[iil=EncCount: ii+. if (ii-2000) ( O/&- new output mcSDAC\Mite(gprncSInfo,(int) l.(shon)((Ustep+ 10.0)*8 19.2));
O/&- updale DAC mc8SYNCsofi(gpmc8lnfo): ) eise { if (ii=f 000) { mc8DACwrit~gpmc8Inlo.(int)I.(shon)8 192); %- wite O output mc8SYNCsoR(gpmc8Ido): J else if (ii-6000) : iîont= 1; break: 1) O/&- cIear die intempt ~ c ~ W ~ ~ ~ R ~ ~ ( ~ ~ ~ ~ ~ I ~ ~ O J M C ~ ~ ~ N T F L G H O S T . I N T ~ ~ ~ ~ ) : ) ) mc8PiTsrop(gpmc8 [nfo): Yb- stop P ï ï timer mc8DACwnte(gpmc8Info,(int) l.(shon)8 192); %--zero output
nic8SYNCsofi(gpmc81nlo); break-
%- updale DAC
casc '2': au: '3': /***** P-ONLY I PV TEST *****l Km = 5.6; Tm = 0.28: if (ah==?') Y- P-Only control ( scarif("?hi'".&Mp): O/P Desired ovenhooi zeta = -(log(Mp/LOO))/sqn(3. lJA2+(log(Mp/100))*(log(Mp/Lc)0})); Kv = 0: O/- calculate gains Kp = l/(Km*Tin*(2*zeia)*(2*zeta)); tiiiie = 3000; ) if (szIn=='3') O/&-- PV control { scanf("%t".&Mp): scanf("W. &Ts): %- settling tirne zeta = -(log(Mp/ 100))/sqn(3.1 JAZ+(log(Mpl lOO))*(log(bIp/ LOO))); Kv = ((8*Tm/IS) - l)/Kin:O/o- calculate gains wn = J/(Ts*zcta); Kp = wn*wn*Tin~Krn: tirne = 2000: ) for W=O:j<=ti1ne-I ;j++) (
errorb]=O; dydibl=O: ucb]=O: ; %- set up and sran PlT, clear and check intempts if ((INTflaggtOs l)==Osl) ( iiic8EncRead(gpincSInfo.O.~ncCount); EncAmy [ii I=EncCount: if (ii>=tirne && ii<(l*time)) ( IniEncCount = EncArn![time-2); FinEncCount = InitEncCount + TotEncInc: 1 if ((ii>=(Z*tiine)) && ii<(j*tirne)) ( InitEncCount = EncArny[(2*tirne)-21; FinEncCount = InitEncCount - TotEncInc: ) if (ii>=time &B iiV3*tiine)) { erroriiil = 0.0OQ9765*(FinEncCount-EncCount); dydt[ii] = (O.O009765*(EncAmy[iil - EncAmy[ii-l1))/0.00I: uc[ii] = Kp*error[ii] - Kv*dydt[ii]; if (uc[ii]>9); uc[iil=9: if (uc[ii lC-9); uc[ii]=-9; inc8DACtvrite(gpmcSInfo.(int)I.(short)((uc[iij+ L0.0)*8 19.2)): inc8SYNCsoft(gpincSInfo); O/p,up&te DAC ) if ((ii>=(3*tirne)) &Br (ii<(S*time))) ( error[ii] = 0.U009765*(I;inEncCount-EncCount); dydt[ii] = (O.O009765*(EncArray[ii] - EncArray[ii-11))10.00 1 : uc[ii] = Kp*error[ii] - Kv*dydt[ii]; if (uc[ii]>9); uc[iil=9: if (uc[iil<-9); uc[ii]=-9: %- clear the internipi inc8WriteReg(gp1nc8Info.bIC8~u\ITR.GHOST,Mflag ii*: ) ) mcSPITstop(gpmc8Info); O/&- SIOP PTT umer mc8DACwrite(gprnc8Info,(int) I.(short)8192): O/&- zero output mc8SYNCsofi(gprnc81nfo); break O/&-- SVP PIT timer. and update DAC) 1) mc8Close(gprnc8Info): 1 %- Clean up board data structures
MRAC Routine Cor DC Motor
/****+ ADAllïVE CONTROL (basPd on Recursive L m t Square Estimation **"*l printf(''In\lirlo. of cycles (default:3). n: "); d ( " ~ 0 d " . &n); printfIWtPeriod of a cycle (default: Jsec). Tp: "1: scanf(""/odn. &Tp); printf("\n\tSerpoint value (default: 1). spt: "1: scanf("%dn. Btspt); printf(WtSamp1ing inten7al (default: lm), interval: "1; d ( " Y C , &intemal): T = 0.00 I*inteml; TOI sampting time Hz = UT: O/&- sampling frequenp time = (Tpll)*(int)Hr: Y,- duration of 1 segment (in no. of intervals) Tf = n4Tp + 2: I* T o d run time (2 scc of initial no-activitp3 cycles of4 sec eacli) *I
/**+ Pan B: Initirilisation for System mode1 and RLS ***/ npann = 4: % number of panmeters for (aa=O: aa<=nparm: aa t t ) { O%- creating rwo J*J Identity matrices
for (bb=il: bb<=npam. tb++) { if(aa==bb) {
P[aal[bbl = LUOU: Q[an] [bbl = 1: ;
Else : P[a;il[bbl =O: Q[aalWl = 0: 1 1 ;
K m = j : T m = O . I : O/&-- [nitid qstem panmeter estimates lambda = mp(-Trip): E = ap(-T/Tml; a l = -Tm*(E-1): d = E : b l = Km*n + Tm*E - Tm): b2 = Km*(l- E): n l = Km*(T + Tm*E - Tm); d = Km*[Tm - TmCE - T*E): dl = 41 + E): d2 = E:
parm[OJ = dl; parm[l] = d2; parm[2] = nl; parm[31 = n2:
I*** Pan C: Solve for N s and Nu **Y Phi[01[0] = 1: Phi[Ol[l] = al ; Phi[l][Ol = 0 Plii[il[lj = ril: Gamma[Ol= bl: Gamma[ 11 = bZ: C[Ol = 1.0: Cil 1 = 0.0:
Fpul = fopen("NsNu.ut"."f); '3'0- Reading h m &la file created by MATLAB if( @tri = NLILL )
printf("\n\tERROR: THE FILE WAS NOT OPENED; PLEASE RUN MATLAB FiRSïb" ); fscanf(fpu1, "%f%f Ni". &N 1. &NZ. &N3): Nx[O] = Nl; Ns[l] = NZ: Nu = N3:
I*** Pan D: Control laop ***/ %- Initialise variables o ldv = 0: o l d v 1 = O: old-u = 0: old-ul = 0: old-,uk[0] = 0: old,xk[ll = 0; old-or2h = 0:
rnc8PITload(gpinc8Info.in[e~~al); %- set up PlT inc8PITstan(gpmc8Info): Yi stan PLT YU" syncronizc PIT and SYNC mcc8SYNCconfig(gpmc8Info.MC8SRCP~.MC8SRCNONE.MC8SRCNONE); mcc8WriteReg(gpmc8Info.MC8~INTENAHOST.Oxl): %- set intemipt mask for PIT timer only printf(Wress a key to stan loggingh?; % pause and \a i t for key hit s d n i = getcli(): %- Conml loop s t m while (!kbhit()) {
mcSReadReg(gpmc81nfo.MC8~MTFLGHOST,BiMTflag): ?& check for PIT intemipt if ((MTflag&Os 1 )==Os 1) {
mcSEncRead(gprncSlnfo.O.&EncCount): O/&- read encoder count *I EncArny[ii] = EncCount: Y- Conyen enc count to [rue position ~ [ i i l = 0.0009765*(EncArray[ii] - EncArny[O]): if ((ii>=(time)) && (ii<(f*time))) {
Phi[Ol[OI = i: Phi[Olf LI = 31: Plli[l]{OI = O: Phi[l][l] = d; Gamma[O] = b t ; Gamma[I] = b2;
Y&- CUCULATION OF DLFFERENT MATRICES FOR CALCULATION OF Kp %- CalcuIation of INV(CTRB(Phl.Gamma))' matrix . detCO = b2*(aZ2bl - bl - al*b2): KsO[iij = detC0 invCO[l][O] = -bZ/detCO: invCO[I][l]= b IIdetCO:
if ((ii>=tirne && ii<(7*iiinc)) I I (ii>=(3*time) && ii<(-l*iimc)) 11 (ii>=Wtirne) && ii<(6*time))) { reqiij = sp. ;
if ((ii>=2*(time) && ii<(3*tiine)) 11 (ii>=(J*time) && ii<(j*time)) 11 (ii>=(6*time) && ii<(7*time))) reqii] = O: :
s ~ l = &[O1 - ref(iij*Ns[Ol; ss2 = &[ll - reqiil*Ns[I 1: uc[iij = Nu*refliil - (K[OI*ssl + K[I]*ssZ): tnc8DACw~te(gpmc8Info.(int) I.(shon)((ucf ii]+ 10.0)*8 19.2)). Yi- new output inc8SYNCsoft(gpn1cY Mo): TOI update DAC
/*** RLS calculation ***/ psi[Oj = -01d-v: psi[l] = s l d j l : psi [2j = old-u: psi[3 1 = old-u 1;
kkI = psi[01*(psi[01*P[OI[O] + psi(ll*P[OI[lj+ psi[21*P[01[2] +psï[jl*P[01[31) + psi[ll*(psi[O]*P[I][O] + psi[ll*P[11[1] + psi[ll*P[lj[Z] + psi[3]*P[1][31) + psi[2j*(psi[OJ*P[21[01+ psi[lj*P[Zj[II +psi[2]*P[21[2j + psi[3I*P[2][3]) + psi[3J*(psi[Ol*P[3I[Oj + psi[l[*P[3I[lI + psi[21*P[3][21 + psi[3j*P[3][3]); W = lambda + kkl; for (dd=0: ddc=;: dd++) (
K2[dd] = O: for (ee=O: ee(=3: e) (
K2[dd] = KZ[ddl + Pfddl[ee]*psi[eej: ; Kl[dd] = (ICZ[ddl)/kk2: >
for (qq=O: qq<=3: qq*) { for(rr=O: r r=3: rr*) {
PPZ[qqj[rrj = O: for (=O ss<=3: S*) {
for (gg=O; g p 3 : gg*) ( for(hli=O; lih<=3: hh++) :
P[gg][hlil= PP2[gg][hh]llambda: 1 1
PR = y[iil - (psi[Oltpann[Ol + psi[ 1 l*parm[ll+ psi[ll*parm[2] + psi[31tparm[3)); for (1t=0; tt<=3; [ t u ) (
parm[rtl = pann[ttl + Kl[ttl*Pk 1
d l = parm[Ol; dl = parm[ll: nl = parm[î]; n2 = parm[31: Tm = -Tnog(dZ); El = eq(-TRm): A = T * ( I - E l ) ; Km = (n 1 + n2)IA: al =-l'm'(El - 1): a2= El; b t = Km*(T + Tm*EI - Tm); bZ=Km*(l -El):
old-.sk[0] = .dc[UI: old-.&[Il = .ski 11: old-u l = old-u. old-u = uc[iil: o l d j l = o l d j : o l d j = ~ [ i i j ; I
if (ii==(7*tiine)) ( Kont = 1; break. )
mc8W~teReg(gpmc8[nfo.MCS-INTFLGHOST.INTflag): Yb- clear the intemipt ii*: ) 1 mc8P1Tstop(gp1nctlInfo ): Yb- stop FR' timer mc8DACwfiteI~mcSInfo.(int) l.(slion)S 192): Yi- zero output mc8SYNCsoft(gptnc8Inlo): %- u@te DAC break:
Optimal Tracking
O/&-- input: ah (Approsimate or e.ua soiution) switch(n1n) {
case '1': /***** APPROXUllATE SOLTION *****/ for (ij=Ojj<=tiine-l&+) ( dydtU]=O; ucU]=O; 1 Yi- set up and stan PIT and synchronise PIT and SYNC Y&- set intempt mask for PIT timer while (!kbhit()) { %-check for PIT intempt */ mc8RadReg(gpm~8InfoSvICS-~GHOST.&ïNTnag if ((INTflag&Os l)==O.u 1) ( inc8EncRcad(~inc8lnfo.0.8rEncCount 1; %- R a d encoder count EncAq[ i i l = EncCount; if (ii== 100) ( inidncCount = EncArnyftirne-21: RefEncCount = iniEncCount: 1 if (iixime && ii<(timc+duntion)) ( RefEncCount = RefEncCount + 3: ermr[iil = 0.001)9765*(RefEncCount-EncCounl); d~dt[ii] = (0.000<)76jL(EncArny[iil - EncArray[ii-1 I))/O.Wl: uc[iil = K(l]*error[ii] - K[ZI*dydt[iil; if (uc[iil>9): uc[iil=9 if (uc[iil<-9): uc[iil=-9: O/&- write ncw output, clear rntempt. stop PIT timer and update output case '2': /***** LYACT SOLTION ***Y Yi- set up and stan PIT. synchronise PIT and SYNC O/&- set interrupt tnask for P ï ï titner while (!kbliit()) : ?&- check for PIT intempt mc8RadReg(gpmcS[nfo.MC8-[NTnGHOST.IYt[NTfIag); if ((lNTflag&Os [)==Os 1) ( tncSEncRead(gprncS~nfo.0.&EncCount): O/- R a d encoder count E n c A n [ i i ] = EncCount: if (ii==100) { InitEncCount = EncArray[time-21; RefEncCount = IniEncCount; if (ii>timne && ii<(tinie+duration)) ( RefEncCount = RefEncCount + 5; tirne-elapsed[ii] = 0.00 1 *ii - 0.2; psition[ii] = 1).011119765*(InitEncCount-EncCount): dydt[ii] = (0.0009765*(EncAmy~ - EncArny[ii-1 D)/O.OO 1 : b2[ii] = -Cl*S*(time-eiapsed[iiI + C?): uc[ii] = K[l]*position[ii] - K[2I*dydt[iil - 19.#*bZ[ii]; iF(uc[ii]>9): uc[ii]=9: if (uc[iil<-Y); uc[ii]=-9; %- wite new output clear intempt. stop PtT timer and update ourput )
ADAPTIVE CONTROL FOR A FIRST-ORDER SYSTEM
Section 3.1 gave design and simulation results for adaptive (MRAC) control of a pure
second-order systern. This appendix presents the adaptive control of a first-order system
for reference purposes.
Consider a system described by the mode1 (Astrom 1989):
where c. J = process coefficients
Assume that it is desirable to obtain a closed-loop system described by
where a, , b, = rnodel coefficients
Perfect rnodel following can be achieved with the controller
M(t) = t,,llC ( t ) - soy(t)
with the parameters
Errur: p z ) ' - Y,
It follows frorn Equation 3.1 and 3.3 that
where y is the differential operator. The sensitivity derivatives obtained by taking partial
denvatives with respect to the regulator parameters t~ and s~ cannot be used because
process parameters c and J are not known. Therefore. some approximations are applied,
and following equations for updatinç the regulator parameters are obtained:
where 7 - adaptation gain. -
Simulation of a first-order system with above MRAS was done using MATLAB with
FI , H . 5 , and b,,,=2. The input signal is a square wave of amplitude 1. Figure C. 1
shows a cornparison of the adaptive conuol of a tirst-order system for two different
adaptation gains ( y =[O. I 21). Two points can be noted fiom the figure: I) cIosed-Ioop
system is close to the desired behavior after onIy a few transients, and 2) the convergence
for y =O. 1 is slower than that for y =2. thus showing that the convergence rate depends
critically on the parameters y and d. The MIT rule will perform well if the adaptation
gain y is small. However, it can also give an unstable cIosed-locp system depending on
the magnitude of reference signal and process gain Modified rules can be obtained using
stability theory. These rules are similar to the MIT d e . The sensitivity derivatives are,
however, replaced by other iünctions.
-5 i O 20 40 60 8 0 100 120
Time (sec)
Figure C. 1: Comparison of adapive controI for 1st order system with trajectocy, y, (dotted line) and process output, y for two different adaptation gains, y = [O. 1 (dash-dotted line) 2 (solid Iine)].
APPENDIX D
FLRST-ORDER SYSTEM SOLUTION FOR OPTIMAL TRACKING PROBLEM
Section 4.2 gave design and simulation results for optimal (LQRT) controi of a
servomotor second-order systern. This appendix gives the optimal tracking law for a first-
order system for reference purpuses.
For a pure first order system the rnodel is defined as
where
F = - and G = K , / r ,
K, = system gain
- rr - system constant
and the ramp sirgnal is
which gives
(D. 2)
and the exact solution for b can be shown to be:
For a step reference signal withx, = a, the exact solution for b becomes:
Figure D. 1 shows the comparison of simulation resdts for approxirr?ate solution and
exact solution for a first order plant in response to a ramp input.
I -1
O O . 5 1 1.5
Tirne (sec)
Figure D. 1 : First-order system optimal tracking example, cornparison of simulation of approxirnate solution (dot-dashed line) and exact solution (solid line) in response to a ramp reference sienal (dotted Iine).
For a sinusoidal reference signal wirh x, = Asin " ( w,t - - i) + :, t h exact solution k r 2
b becornes:
b(1) = -0, rr s i n rr )-:)+II 1+K;K 2 I + K , K
Figure D.2 shows the cornparison of simulation results for approximate solution and
exact solution for a tirst order plant in response to a sinusoidal input.
Tirne (sec)
Figure D.2: First-order system optima1 tracking example, cornparison of simulation of approximate solution (dot-dashed line) and exact solution (solid line) in response to a sinusoidal reference signai (dotted line).