CONTROL OF A DC MOTOR: COMPARING CLASSICAL AND ADAPTIVE
TECHNIQUES
By
Jay L. Adams
Submitted in Partial Fulfillment ofthe Requirements
For the Degree of
Master ofScience ofEngineering
in the
Electrical Engineering
Program.
YOUNGSTOWN STATE UNIVERSITY
August 2004
CONTROL OF A DC MOTOR: COMPARING CLASSICAL AND ADAPTIVE
TECHNIQUES
Jay L. Adams
I hereby release this thesis to the public. I understand that this thesis will be made
available from the OhioLINK ETD Center and the Maag Library Circulation Desk for
public access. I also authorize the University or other individuals to make copies ofthis
thesis as needed for scholarly research.
Signature:
Approvals:
JaYAdamS'~
ossayebi, Thesis Advisor
gj121100~
Date
8112.. OLf
Date
8)12/2..00'-!-
Date
Date
Date
111
ABSTRACT
The intent of this project is to compare and contrast the advantages and
disadvantages ofusing adaptive techniques as opposed to classical ones in the control of
a DC motor. To that end Root Locus and the adaptive techniques of self-tuning
regulation, one-step ahead adaptive control, and model-reference adaptive control are
presented. The controllers are designed based on a model that is commonly used for a
DC motor. To investigate the effect of a non-ideal model on the controlled system,
several variations are considered. The effect of model mismatch is investigated by
making the order ofthe plant and the type ofthe plant higher than the assumed model for
controller design. Furthermore, the effect ofload on the motor as well as the presence of
additive noise is considered. The Root-Locus designed controller does not meet the
criteria in most non-ideal situations. The self-tuning regulator meets the design criteria in
all the non-ideal cases, but has a high control effort before the parameter estimates
converge, after which the control effort is more reasonable. The weighted model 
reference adaptive system best meets the design criteria with the least control effort in all
the non-ideal cases if the model used to design the controller has been over 
parameterized. This thesis concludes with a summary ofthe project results and ideas for
future research topics.
IV
ACKNOWLEDGEMENTS
It pleases me to express gratitude to several people who have been instrumental in
my success with this work and the master's degree. First and foremost, I must thank the
God who created me with the capacities and abilities for reasoning that I have, without
whom all my efforts would have been and be in vain. Dr. Faramarz Mossayebi has
contributed greatly to this work and my future education with his expertise, experience,
and encouragement. Our weekly meetings have kept me on track and given me
motivation to do what needed to be done. Further, I must thank Dr. Robert H. Foulkes, Jr
and Dr. Salvatore Pansino for serving on my thesis committee. I am also compelled to
thank Dr. Thomas J. Cavicchi who served as inspiration and introduced me to systems
and controls. My parents, J. Lawrence and Beth Ann, have ever been ardent supporters
ofmy academic endeavors, believing in me, and spurring me on toward greatness, even
when future prospects looked bleak; no greater gift could I have been given. The support
and patience exhibited by Lynn Irene Schwartz, listening to my defense, providing
technical help, and encouraging me through out my work at Youngstown State University
has been exceedingly helpful as well. Finally, I thank everyone who has been thinking
about and praying for me in this and future academic endeavors.
v
TABLE OF CONTENTS
MODELLING THE DC MOTOR.................................... 3
1.1 Determining the Transfer Function. .. ... ........ ... ...... 3
1.2 Numerical Transfer Function 6
1.3 Determining the Pulse-Transfer Function. ........... ... . 7
1.4 Characteristics ofthe Open-Loop Responses 13
1.5 Design Criteria ... ............................................ 14
1.6 Varying the Motor Load... .. ...... ... ...... .... ............ 16
PARAMETER ESTIMATION 19
2.1 Least Squares Estimation 19
2.2 Recursive Least Squares 21
CLASSICAL CONTROL TECHNIQUES 25
3.1 Introduction to Root Locus 25
3.2 Designing the Compensator........................ ........ 26
SELF-TUNING REGULATORS 33
4.1 Introduction to Self-Tuning Regulation............ 33
III.
II.
IV.
PAGE
ABSTRACT 111
ACKNOWLEDGEMENTS IV
TABLE OF CONTENTS............................................................... V
LIST OF FIGURES..................................................................... V111
INTRODUCTION 1
CHAPTER
I.
4.2 Controller Design .
4.3 Self-Tuning Regulator Algorithm .
4.4 Self-Tuning Regulator Design Specifications .
4.5 Self-Tuning Regulator with Pole-Zero Cancellation .
4.6 Self-Tuning Regulator without Pole-Zero Cancellation ..
4.7 Conclusions .
V. ONE-STEP AHEAD AND MRAS ..
5.1 Introduction to One-Step Ahead Adaptive Control .
5.2 Unweighted and Weighted One-Step Ahead Controllers.
5.3 Model Reference Adaptive System .
5.4 Conclusions .
VI. SYSTEM AND LOAD ERROR RESULTS .
6.1 Comparison ofTechniques ..
6.2 Disturbance Rejection " .
6.3 Non-Ideal Model Situation .
6.4 Load Error .
VII. SUMARY AND CONCLUSIONS ..
7.1 Summary .
7.2 Conclusions .
REFERENCES .
APPENDIX A MATRIX DERIVATIVE AND INVERSION LEMMA .
Al Matrix Derivative '" .
A2 Matrix Inversion Lemma " .
vi
35
39
39
40
45
47
51
51
52
62
68
69
69
71
73
80
85
85
88
90
92
92
94
APPENDIX B LINEAR PREDICTION MODEL................................... 95
APPENDIX C~lrLi\BCODE 97
Classical Controller Simulation 97
Root Locus..................... ... ... .. .... .. ... . .. .... 97
Step Response 97
Self-lruning Regulator Simulation... .. .......... ..... ... ... .......... ....... 98
Model-Reference i\daptive System Simulation 101
Vll
~lrLi\B
LIST OF SYMBOLS
SYMBOL DEFINITION
A Denominator ofthe transfer function or pulse-transfer function
A
CL
Closed-loop denominator
Am Desired closed-loop denominator
A
p
Denominator ofthe pulse-transfer function
A(s) Denominator ofthe transfer function
B Damping constant ofthe motor
B Numerator ofthe transfer function or pulse-transfer function
B
m
Desired closed-loop numerator
B
p
Numerator ofthe pulse-transfer function
B(s) Numerator ofthe transfer function
D(z) Discrete-time compensator
d Prediction horizon
do Pole excess
d
- ( ) Derivative with respect to time
dt
G
m
(s) Desired transfer function
Gp (s) Transfer function ofthe plant
H Linear function
H m (z) Desired pulse-transfer function
pq
viii
ix
Hpq(z) Pulse-transfer function ofthe plant
i Current through the motor armature
J Moment ofinertia ofthe rotor
K Motor constant
K System gain
K
e
Electrical constant ofthe motor
K
1
Torque constant ofthe motor
L Inductance
m Order ofnumerator ofpulse-transfer function
n Order ofdenominator ofpulse-transfer function
p
Confidence matrix
q Forward shift operator
-1
Reverse shift operatorq
R Self-tuning regulator control law denominator
R Resistance
S Self-tuning regulator feedback numerator
s Laplace variable
T Self-tuning regulator feedforward numerator
r:
Sampling time
T
Sel
Settling time
u(k) Input into the plant
u(t) Input into the plant
x
uc(k) Reference signal
uc(t)
Reference signal
v(k) Noise into the plant
v(t) Noise into the plant
v(t) Source voltage
x Parameters to be estimated
y(k) Output ofthe plant
y(t) Desired output ofthe plant
YIII(k)
Desired output ofthe plant
Ym(t)
Output ofthe plant
Z Noisy observation
ZT{} Z-transform
e Estimation error
S
Damping ratio
A Weighting factor
¢(k) Regression vector
T Torque ofthe rotor
e Position ofthe motor shaft
B Speed ofthe motor shaft
B(k) Parameter vector
OJ
n
Natural frequency
%08 Percent overshoot
FIGURE
LIST OF FIGURES
PAGE
xi
1.1 Model ofthe DC Motor 4
1.2 Open-Loop Response ofthe DC Motor 8
1.3 Open-Loop Response for the Continuous and Pulse Transfer Functions. 11
1.4 Desired Step Response ofthe DC Motor..... ........ .......................... 14
1.5 Desired Step Response for the Continuous and Pulse-Transfer Functions 16
3.1 Block Diagram ofClassical Controller '" 26
3.2 Root Locus ofthe Uncompensated System 27
3.3 Root Locus ofIntegrator Compensated System 28
3.4 Root Locus ofthe Classically Compensated System 29
3.5 Step Response ofthe System with Controller Exactly Meeting the
Damping Ratio Requirement 31
3.6 Step Response ofthe System with Controller Exactly Meeting the
Settling Time Requirement 32
4.1 Block Diagram for Self-Tuning Regulator. .. ......... ... ... .................. 34
4.2 Reference and Output ofa Self-Tuning Regulator-Controlled System... 43
4.3 Control Signal for a Self-Tuning Regulator with Pole-Zero Cancellation 44
4.4 Parameter Estimates for a Self-Tuning Regulator with Pole-Zero
Cancellation ,..... . 45
4.5 Reference and Output ofa Self-Tuning Regulator-Controlled System
without Pole-Zero Cancellation 47
4.6 Control Signal for a Self-Tuning Regulator without Pole-Zero
Cancellation. ... ....... .... .. .... ..... ....... .. ... .... ........................ .... 48
4.7 Parameter Estimates for a Self-Tuning Regulator without Pole-Zero
Cancellation " ... . .. .... 49
5.1 System Response ofa One-Step Ahdead Adaptive Controller 55
5.2 Control Signal for a One-Step Ahdead Adaptive Controller 56
5.3 Response and Control Signal for a Weighted One-Step Ahdead Adaptive
Controlled System 58
5.4 Response and Control Signal for a Weighted One-Step Ahdead Adaptive
Controlled System.. .... .. .... ...... ... ... ... .. ... ... ...... ....... .............. .. 59
5.5 Response and Control Signal for a Weighted One-Step Ahdead Adaptive
Controlled System 60
5.6 Response and Control Signal for a Weighted One-Step Ahdead Adaptive
Controlled System 61
5.7 Response and Control Signal for a Model-Reference Adaptive System ... 63
5.8 Response and Control Signal for a Weighted Model-Reference
Adaptive System " ... ........ ....... ..... 64
5.9 Response and Control Signal for a Weighted Model-Reference
Adaptive System......................................................... ......... 65
5.10 Response and Control Signal for a Weighted Model-Reference
Adaptive System '" ..... .... .. ....... ...... 66
5.11 Response and Control Signal for a Weighted Model-Reference
Adaptive System........... ................... .. .......... ............. ........... 67
6.1 Comparison ofResponses and Control Signals with Ideal Model......... 70
6.2 Comparison ofDisturbance Rejection............................... 72
6.3 Responses and Control Signals for a Plant ofHigher Type 74
6.4 Responses and Control Signals for a Plant ofHigher Type 76
6.5 Responses and Control Signals for an Unstable Plant 77
6.6 Responses and Control Signals for a Plant ofHigher Order 79
6.7 Responses and Control Signals for a Loaded Plant 82
xii
6.8 Responses and Control Signals for a Maximally Loaded Plant 84
XUl
1
INTRODUCTION
Control theory is concerned with obtaining a desired response from a system, or
plant. Classical control theory, in the form ofroot locus, pole placement, etc, can be used
when the designer has a priori knowledge of the plant. Control systems allow the
management of large and small equipment with a degree of precision that would
otherwise be impossible. Further, the use of control systems incorporating feedback
allows compensation for disturbances, such as mechanical or electrical noise
[3],[8],[11],[14].
In some cases, however, a portion or the entirety ofthe plant is unknown; in other
cases the parameters of the plant change, or disturbances are too great for the classical
controller to rej ect. In these instances different techniques must be applied; one such
class of techniques is known collectively as adaptive control. Adaptive control was first
developed in the early twentieth century to automatically steer a ship. These
developments contributed to proportional-integral-derivative control. Other adaptive
techniques are the self-tuning regulator and the model-reference adaptive controller
[1],[7],[11].
Adaptive control schemes incorporate some sort of on-line parameter estimation
scheme that takes into account any changes in the external or internal factors that
contribute to change in the plant. The most widespread and common parameter
estimation method is that of least squares regression. Least squares regression can be
2
used in nonlinear applications where the system is linear in the parameters, which can
sometimes be selected or manipulated in such a manner as to cause this [1],[3].
The first section of this work, encompassing Chapters I and II, is concerned with
the basic concepts behind classical and adaptive control. Chapter I follows the
development of a transfer function for a DC motor, which is then analyzed for response
and developed into a pulse-transfer function. The establishment of a desired system
response is also Chapter 1. The least squares estimation technique is derived in Chapter
II; both a non-recursive estimation and an online estimation algorithm are given.
Chapters III, IV, and V compose the second section of the thesis, which contains
the design ofthe three different types of controllers. The root locus technique is utilized
in Chapter III to design a classical controller that conformed to the design criteria set
forth in Chapter 1. The self-tuning regulator is developed and designed for the dc motor in
Chapter IV. This chapter contains the design of two self-tuning regulators, one with
process zero cancellation and one without any cancellation. In the final chapter in this
section, Chapter V, development of the one-step ahead and model-reference adaptive
controllers are presented. The one-step ahead algorithm is divided into two different
types, a pure one-step ahead controller and one that takes into account the desire to keep
the control effort low.
The final section of this work includes Chapters VI and VII. Chapter VI
compares the performance of the different controllers when the system is placed under
various changes. The final chapter in the thesis presents a discussion of the results and
the conclusions derived from the work contained in this thesis, as well as
recommendations for future study.
3
CHAPTER I
MODELLING THE DC MOTOR
In this chapter, the required background will be established. First, a mathematic
model of a dc motor will be developed, which will be used to determine the relation
between the voltage input to the motor and the speed at which the motor rotates.
Secondly, because typical control systems are implemented with computers, a discrete 
time representation for the continuous-time plant must be found. This is called the pulse 
transfer function. The open-loop response will be addressed in the chapter, as will the
design criteria. The final topic addressed in this chapter is the effect that varying the load
ofthe dc motor has on the transfer function.
1.1 Determining the Transfer Function
In order to enact any sort ofcontrol on the DC motor under investigation, a model
must be formulated. Figure 1.1 shows a typical model of a DC motor [4],[14]. The
torque produced by the motor is directly proportional to the current through the armature.
In other words,
(1.1)
Fixed
field
1;n(t)
~
ern(t)
Armature
circuit
LaR
a
ia(t) -4
'"
+
ea(t)
Figure 1.1:
Model ofthe
DC Motor.
~
5
holds, where K, is the torque constant. A back electromotive force, or emf, is generated
by the motor, which is directly proportional to the speed ofthe rotor, shown by
em/=K/}, (1.2)
where K
e
is the electric constant. In the system of units is consistant, it can be shown
that
K=K, =K
e
is true [14]. Using Kirrchoff's Voltage Law around the electronic circuit gives
. . diet)
vet) - K . B(t) =let) . R +- .L
dt
(1.3)
(1.4)
where vet) is the source, or input, voltage. Taking the Laplace Transform of Equation
1.4 results in
V(s) - K . sees) =I(s)· R + sI(s)· L =(sL + R)· I(s). (1.5)
The motion of the rotor is opposed by the inertia of the rotor and the friction of rotor.
This is described by Newton's Law and can be written as
K .i(t) - J .O(t) - b .O(t) =0, (1.6)
where B(t) is the angular position of the motor. Taking the Laplace transform of
Equation 1.6 yields
K . I(s) =J. s2e(s) + B· se(s).
Rearranging Equation 1.5 to solve for I(s) yields,
V(s) - K . sees)
I(s) = .
sL+R
(1.7)
(1.8)
6
In order to find a transfer function for the dc motor, with source voltage, v(t) , as the
input and angular velocity, dB as the output, the expression for I(s) in Equation 1.8 is
dt
substituted into Equation 1.7, resulting in
J. s2e(s) + B. sees) =K. V(s) - K· sees) ,
sL+R
which can be rearranged to collect the like terms, giving
((sL + R)(sJ + B) + K
2
). sees) = K . V(s).
This gives the transfer function,
G (s) =sees) = K
P V(s) (sL + R)(sJ + B) + K
2
With some algebraic manipulation, Gp (s) becomes
K
G (s) - JL
p - 2 (B R) RB+K
2
'
s + -+- s+---
J L JL
1.3 Numerical Transfer Function
(1.9)
(1.10)
(1.11)
(1.12)
Based on experiments conducted at Carnegie Mellon University [4], the following
values were selected for the parameters ofthe dc motor:
b =O.lN .m . s ,
N'n
K=K
e
=K
t
=0.01--,
A
(1.13)
(1.14)
(1.15)
and
R=ln,
L=0.5H.
(1.16)
(1.17)
7
Substituting these values into Equation 1.12 results in the transfer function,
2
G~)=,
P S2 +12s + 20.02
(1.18)
with poles at - 2.0025 and - 9.9975. Figure 1.2 shows the step response of this system
as obtained via MATLAB.
1.3 Determining the Pulse-Transfer Function
For computer implementation of control systems, a discrete-time transfer
function, H pq (z) , must be developed. A common technique used to change signals from
digital to analog is by means of the zero-order hold [15],[16]. When forming the
discrete-time transfer function, often called the pulse-transfer function, the characteristics
of the zero-order hold must be taken into account with the sampled response of G/s).
The relation between H pq(z) and G/s) is given in [15],[16], as
(1.19)
where T is the sampling period, and ZT{ GPs(S)} signifies taking the z-transfonn of the
inverse Laplace-transform of Gp (s) , sampled with a sampling period of T
s
' The inverse
s
~)=
8
9
G (s)
Laplace transform of -p- is also called the step response, gsle/f) , of Gp (s). Because
s
the parameters ofthe continuous-time model ofthe motor will be changing, it is useful to
derive the pulse-transfer function in terms ofthe continuous-time transfer function, which
is ofthe form,
G (s)
-p- can be written as
s
Gp(S) c
I
c
2
c
3
--=--+ +-"--
S S - PI S - P2 S - P3
(1.20)
(1.21)
where two of PI' P2' and P3' are the roots of A(s) and the other is 0, and c
I
' c
2
' and
C
3
are the residues of Gp (s). The step response is then,
(1.22)
A sampled-time version ofthe step response is
(1.23)
which can be written as
(1.24)
where
(1.25)
The pulse-transfer function is z -1 times the z-transform of this function, which
z
can be written as
Gp(s)
10
H (z)=bozz+b\z+bz
(1.26)
pq ZZ + a z + a '
\ z
where
a\ =1-d\ -d
z
-d
3
, (1.27)
a
z
=d\d
Z
d
3
,
(1.28)
b
o
=c\ + C
z
+ c
3
'
(1.29)
b\ =-c\(d
z
+d
3
)-c
z
(d\ +d
3
)-c
3
(d
1
+d
z
)' (1.30)
and
b
z
=c\d
Z
d
3
+ c
Z
d\d
3
+ c
3
d\d
z
·
(1.31)
For the transfer function given in Equation 1.18, the pulse-transfer function is
H (z) = 0.961 x 10-
4
Z + 0.9233 X 10-
4
pq ZZ -1.885z + 0.8869
(1.32)
with the sampling period, T
s
' of 0.0Is. Figure 1.3 shows the step response of the pulse-
transfer function on the same axes as the step response of the transfer function. The
pulse-transfer function does a good job ofapproximating the transfer function.
The system can be written in terms ofthe forward shift operator, q, where
qf(k) == f(k + 1).
The system can now be written as
where
and
(1.33)
(1.34)
(1.35)
11
12
(1.36)
It is convenient in determining the response ofthe plant to represent the system in
terms ofthe reverse shift operator, or delay operator, such that the system becomes
A; (q-I)y(k) =B;(q-I)U(k),
(1.37)
where
q-If(k) == f(k -1) , (1.38)
A· ( -I) 1 -I -2
(1.39)P q = +a1q +a
2
q ,
and
B· ( -I) b b -I b -2
(lAO)
P q = 0 + Iq + 2q .
1.5 Characteristics of the Open-Loop Responses
Characteristics of the open-loop system must be analyzed to determine which
specific portions of its performance need to be improved. Because the poles are in the
left-hand plane, the system is stable, so stabilizing the system is not a design criterion,
although it is imperative to maintain the system's stability. Several characteristics about
the open-loop step response can be established by looking at the transfer function in
terms of the natural frequency, OJ
n
, and the damping ratio, ,. These constants can be
determined from the transfer function by noting that for a second order system,
(1.41)
13
holds [8],[14]. For this system, the natural frequency is 4.47 rad , and the damping ratio
s
is 1.34. The damping ratio indicates that this system is overdamped. From Figure 1.2, it
is apparent that the settling time is 2.1s and that the steady-state error is 90.091 %.
1.6 Design Criteria
The desired performance of the system involves the steady-state error, overshoot
and the settling time. A reasonable value for the overshoot is 15%. Using the relation,
t; = -In(%OS/1 00)
~1r2+ln
2
(%OS/100)'
(1.42)
t; , the damping ratio, is found to be 0.5169. The system is to be sped up to have a
settling time, T
s
' of Is. This implies, through use ofthe relation,
(1.43)
rad
a natural frequency, OJ
n
, of 7.738-. Using these values in Equation 1.41 gives the
s
desired performance as
G (s) = 59.88
m S2 + 8.0s + 59.88
(1.44)
Figure 1.4 shows the step response of this transfer function. From Figure 1.4, it can be
seen that the overshoot is 5%, and the settling time is about Is, meeting the desired
design criteria.
14
15
Using the procedure that was presented in Section 1.3, the pulse-transfer function
of the desired transfer function can be obtained. Using a sampling time of O.Ols, the
pulse transfer function is
H m (z) =0.001638z + 0.001594 .
pq Z2 -1.92z + 0.9231
(1.45)
It can be observed that the pulse-transfer functions' step response is a reasonable
approximation of the step response of the continuous-time step response; Figure 1.5
shows the step responses of both the pulse-transfer function and transfer function on the
same axis.
1.7 Varying the Load of the Motor
The prior calculations involved the motor in an unloaded state. When a load is
added to the motor, the moment ofinertia, J, ofthe motor will effectively be changed. It
will be assumed the connection of the motor and the load will be perfectly attached, that
is there is no slipping between the motor and the load. This change in J changes the
location ofthe poles ofthe transfer function, which will change the nature of the system.
For example, if J is 0.0471, then r; is 1.0, resulting in a critically damped system. If J
is 0.0501, then r; is 0.995; then the system becomes overdamped. For the situation
where J is 0.0471 , the transfer function is
G' (s) = 0.4246 = 0.4246 (1 46)
P S2 + 4.123s + 4.251 (s + 2.06 + }0.0349Xs + 2.06 - }0.0349) .
and the pulse-transfer function is
16
17
H (z)=20.94x10-
6
z+20.66xlO-
6
= 20.94 X 1O-6(z +0.9866) .(1.47)
pq Z2 -1.959z + 0.9596 (z - 0.98 + jO.0134Xz - 0.98 - jO.0134)
For a controller using classical control techniques, this would result in a poor
response, as will be shown in Chapter 6.
18
CHAPTER II
PARAMETER ESTIMATION
2.1 Least Squares Estimation
Because all of the system parameters may not be known or may vary from the
known values in an adaptive system, some form ofparameter estimation must be enacted.
For this discussion, linear estimation will be emphasized. For this situation, the
measurement equation has the general form
Z=Hx+&", (2.1)
where Z is the noisy observation, x is a vector ofthe parameters to be estimated, &" is the
error, and H is the linear function operating on x to get the measured quantity [3]. The
noise-free measurement is a linear function ofthe parameters to be estimated.
Least squares linear estimation is a common method for determining parameters
in an adaptive control scheme. In the general case, the measurement is of the form of
Equation (2.1), and the distribution of &" is unknown. The principle of least squares is
that errors that are present in measurements tend to be minimized over time, and thus a
logical estimate for the parameters in x are that values for which the sum ofthe squares
ofthe error is minimum. In mathematical terms, the loss function,
(2.2)
19
should be minimized. Because the function, H, is linear, this loss function can be
rewritten as
(2.3)
Because the choice ofthe parameters, x, are to result in a minimal loss function, the loss
function, J, must be differentiated with respect to x, and equating the result with zero.
Solving the equation,
OJ] T T T
... - ;:;::-2Z H+x H H,
Ox
N
(2.4)
for x, yields the least squares estimate, denoted by x[3],
This can be written as [1]
where h
j
is the i
1h
member of H .
2.2 Recursive Least Squares Estimation
(2.5)
(2.6)
Because most ofthe parameter estimation in adaptive controllers must be done in
real time, a recursive parameter estimation is used to save computation time [1],[7]. The
definition
(2.7)
is introduced. Equation 2.6 can now be written as
20
However,
k
P-l(k)= HT(k)H(k) =Lh;h
T
;,
;=1
which follows from the definition in Equation 2.7, and can be expanded into
k-I
P-I(k) =Lh;h
T
;+h(k)hT(k),
;=1
which can be simplified as
p-I (k) =p-I (k -1)+ h(k)h
T
(k).
Equation 2.8 leads to
k-I
L h;h
T
; =p-I (k -l)x(k -1),
;=1
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
which upon using a rearranged Equation 2.11 to substitute for p-I (k -1) results in
k-I
L h;h
T
; =p-I (k )x(k -1)-h(k)h-
I
(k )x(k -1).
;=1
(2.13)
The estimate for time k can now be written in terms ofthe estimate for time k -1 as
x(k) = x(k -1)-P(k )h(k)h
T
(k )x(k -1)+ P(k )h(k)z(k), (2.14)
which can be rearranged into
(2.15)
The error, e(k), associated with the prediction of z(k) based on the estimate x(k -1) is
Equation 2.16 can be written as
i(k) =x(k -1)+ R(k)e(k),
(2.16)
(2.17)
21
where R(k) is defined as
R(k)== P(k)h(k). (2.18)
In order to use these results, a recursive relation for P(k) must be found rather than for
p-
1
(k). This can be done by using the matrix inversion lemma found in Appendix A,
which states
(A + BCDt == (A-
1
- A-IB(C-
I
+ DA-
1
Bt DA-
1
). (2.19)
This lemma is used by substituting p-
1
(k -1) for A, the identity matrix, I , for C , h(k)
for B, and h
T
(k) for D. This choice satisfies the required conditions for the matrix
inversion lemma, as p-
1
(k -1), I, and h(k )Ih
T
(k) are nonsingular matrices. Using
Equation 2.19 in conjunction with Equation 2.11 results in
P(k) =P(k -1)- P(k -1)h(kXr
l
+ h
T
(k )P(k -1)h(k)t h
T
(k )P(k -1). (2.20)
Using
(2.21)
the estimation ofparameters can be summarized as follows:
First find R(k), using Equation 2.21 based on the choice of initial conditions for
parameters and matrix, p(0). Next, determine p(k) using
P(k) =(I - R(k )h
T
(k )P(k -1)), (2.22)
which is found by substituting Equation 2.21 into Equation 2.20. The residual error in
the system can be then calculated as
e(k) = z(k)- h
T
(k )i(k -1).
Finally, the parameters may be estimated using
(2.23)
x(k) == x(k -1)+ R(k)&(k).
22
(2.24)
The recursive least squares estimation technique can be used to estimate the coefficients
in the transfer function ofa system. The transfer function ofthe system is assumed to be
ofthe form
A(q)y(k) == B(q }u(k), (2.25)
where q is the forward shift operator. Equation 2.25 can be rewritten as
any(k - n)+ ... + a1y(k -1)+ y(k) == bmu(k - m)+ .. ·+ b1u(k -1)+ bou(k), (2.24)
which can be solved for y(k) as
y(k) == bmu(k - m)+ ... + b1u(k -1)+bou(k)- any(k - n)-· .. - a1y(k -1). (2.25)
Let
¢(k)==h(k)==(u(k-m) ... u(k) y(k-n) ... y(k-1)Y (2.26)
and
Equation 2.25 can now be written as
y(k) == ¢T (k )o(k),
which is ofthe form ofEquation 2.1.
2.3 MATLAB Code
(2.27)
(2.28)
The MATLAB code for estimating the parameters at each time, m, is as follows.
It is assumed that there is an initial guess, theta(:,1), an initial confidence matrix, P(:,:,1),
an initial K vector, K(:,1), the output at times m-l and ,m-2, that is y(m-1) and y(m-2),
23
and the inputs at times m-l and m-2, which are u(m-l),u(m-2). Note that the algorithm
must be initialized by providing an initial guess for the parameters, theta(:,:). This
example is for a second order system, with m = 2 and n = 1.
% Estimate Parameters
Pkml=P(:,:,m-l);
phikml=[-y(m-l) -y(m-2) u(m-l) u(m-2)]';
thetakm1=theta(:,m-l);
e(m)=y(m)-phikml'*thetakml;
K(:,m)=Pkml *phikml*inv(1ambda+phikml'*Pkml *phikml);
P(:,:,m)=(diag([l 1 1 l])-K(:,m)*phikml')*Pkml/lambda;
theta(:,m)=thetakml+K (:,m)*e(m);
al=theta(l ,m);
a2=theta(2,m);
bO=theta(3,m);
bl=theta(4,m);
24
CHAPTER III
CLASSICAL CONTROL TECHNIQUES
In this chapter classical control techniques are explored for the dc motor modeled
in Chapter One. At first a discrete-time lag-lead controller was designed, but it was
impossible to meet the design requirements with such a controller. Either the settling
time was lengthened or the error was unacceptable. In what follows a classical controller
designed based on the traditional Root Locus technique will be presented.
3.1 Introduction to Root Locus
One technique that can be used to design a controller for a continuous-time
system is the Root Locus method. A Root Locus is a graph that shows the positions of
the closed-loop poles as the system gain, K, is varied [8],[11],[14]. This technique can
be used for discrete-time systems as well because the characteristic equation used is both
discrete-time and continuous-time is of the same form, only in different complex
variables and domains. The stability boundary is the unit circle for discrete-time systems
rather than the j OJ -axis, as it is in continuous time case. Likewise, the curve ofconstant
parameters such as damping ratio and natural frequency, are changed from continuous 
time to discrete-time systems; a cartoid is the curve ofconstant damping ratio in discrete-
25
time instead of a line radiating from the origin and a circle centered at z =1 is the curve
of constant natural frequency in the z-domain [15],[16]. Figure 3.1 depicts the block
diagram ofthe system that is used for Root Locus analysis and design.
3.2 Designing the Compensator
The first step in designing the compensator is to construct the Root Locus for the
uncompensated system,
H (z)=96.1xl0-
6
z+92.33xl0-
6
=96.1xl0-6 z+0.9608 .(3.1)
pq Z2 -1.885z + 0.8869 (z - 0.9800Xz - 0.9050)
Figure 3.2 shows the Root Locus for this system. Initially, with a gain of zero, the
system is stable and remains stable. If the gain is increased to about 1,360, the system
becomes unstable. Three system response parameters are important: no steady-state error
in the step response, an overshoot of less than 15%, and a settling time ofat most Is. In
order to have zero steady-state error, integrating action is needed. This is accomplished
by placing a pole at z =1 [16]. This, however destabilizes the system as can be seen
from the Root Locus depicted in Figure 3.3 which is derived by adding a pole at z =1.
This is because poles tend to repel the branches ofa Root Locus, and there are two poles
in the near vicinity of z = 1. One solution is to cancel the other pole at z = 0.9800 by
placing a zero at the same place. Pole-zero cancellation is not perfect, however, the pole
will be effectively cancelled if the zero is very close to the same location [15]. The
compensator is so far given by
D(z)=Kz-0.9800.
z-1
(3.2)
,-..
rf)
'-'"
0..
r-
e,:l
j ,.....;
...;
~
-
I--
0
~:::>
0
u
ta
(,)
.....
til
~
-
U
t+-<
0
-t- I
§
~
.....
0
.r;
0
~
-
Q:l
..
-
M
~
.....
~
I--
26
~
~
~
~
~
~
27
28
29
30
The Root Locus of the compensated system is shown in Figure 3.4. In this Root Locus,
the curves of constant damping ratio at 0.5169 and constant natural frequency of
0.07738 have been added. The desired natural frequency for a discrete time must be in
terms of
rad ,so the natural frequency must be multiplied by the sampling period. In order
sample
for the compensated system to exactly meet the design requirements, the Root Locus
must intersect at the same point as the two curves ofconstant parameter, which is not the
case. The control design is acceptable ifall the criteria are met or exceeded. Where the
Root Locus and the curve of constant damping ratio intersect, the gain is 42.8. Figure
3.5 shows the step response when the compensator gain is 42.8. The percent overshoot
is about 15% , however the settling time requirement is not met; the settling time is about
1.2s. The other option is to use the gain 30.2, which is the gain where the Root Locus
and the curve of constant natural frequency meet. Because this point is inside the curve
ofconstant damping ratio, the damping ratio is greater, thus the overshoot is less. Figure
3.6 shows the step response when the compensator gain is 30.2. The settling time is met
exactly, and the percent overshoot is 8%, which exceeds the design requirement. The
compensator given by
D(z) =30.2 z- 0.9800
z-1
will be used in the discussions on disturbances and varied loads in Chapter 6.
(3.3)
Figure 3.5: Step Response ofthe System \....ith Controller Exactly Meeting the Damping Ratio Requirement
w
-
Figure 3.6: Step Res.ponse ofSystem \vith Controller Exactly Meeting the Settling Time Requirement
V-l
tv
33
CHAPTER IV
SELF-TUNING REGULATORS
The self-tuning regulator is the first adaptive control scheme that will be used to
control the dc motor. First, the process for designing a self-tuning regulator will be
presented. Then the ramifications ofdesigning a controller with pole-zero cancellation as
well as one that has no cancellation will be observed. Finally, two self-tuning regulators
will be designed and compared with respect to meeting the design criteria and the control
effort.
4.1 Introduction to Self-Tuning Regulation
A self-tuning regulator is a control scheme that is useful when the environment in
which the controller operates changes. The block diagram of a general self-tuning
regulator is shown in Figure 4.1 [1]. The process is the system which is to be controlled.
The estimation block involves the estimation of the process parameters, which are fed
into the controller design block. These parameters, along with the control specification,
determine the control parameters. The controller output, also called the control signal,
which drives the plant changes at each time interval based on the output ofthe system,
..-.- -- - - -- ---- - _ - _._ _ .. ""-"'-' - _."_. _ _.. --- --- I
Self Tuning Regulator i
Process Parameters
I
I
I
;
,
i
l
I
i
I
I
I
Controller
~
Estimation I
I
i
I
I
I
Design
!
l
l
j
Controller
-
_" ___. __.......J
parameters
_.
!
,
l
I
;
..
i
l~
eference
;
Process
i
Controller
Output
i
Input I
I
,
:
!
i
-
..-
._~
._--
!
---- --
-
L-_......
-
------
R
Specification
Figure 4.1: Block Diagram for Self-Tuning Regulator.
w
~
--- !
1-"S~/'f'-T~~.j~g._-R~g~i~·t;~---"-"._- __. -'._.' ---
,
,
I
t
I
i
r-
_ •.
--
_.
I
i
I
!
!
~
L_
------
-"
~---'
35
the control parameters, and the input to the system. For this discussion, only least
squares estimation, as discussed in chapter 2, will be considered.
4.2 Controller Design
The design technique for a self-tuning regulator is gIven by Astrom and
Wittenmark [1], which this development basically follows. In order to design the
controller, pole placement is utilized. The model ofthe system is assumed to be
A(q)yet) =B(q)(u(t) +v(t)), (4.1)
where y is the output, u is the input, and v is a disturbance. The polynomials A(q) and
B(q) have degrees of nand m, respectively. The pole excess of the system, do, is the
difference between nand m. Two assumptions about A and B have been made; the
coefficient of the highest term with highest degree in A is one, that is, A is monic, and
A and B have no common factors, which means that they are relatively prime. These
assumptions serve to make the generalization of the design process easier. In general,
A(q) will be written as A.
The controller can be written as
R . u(t) =T . U
c
(t) - s .y(t) , (4.2)
where R,S , and T are polynomials, and u(t) is the signal fed into the system, and U
c
(t)
is the reference signal that the system is trying to follow. Combining Equation 4.1 and
Equation 4.2 results in
which simplifies to
AR· yet) = B(t. uc(t) - S· yet) + R· vet)), (4.3)
(t) = BT .U
c
(t) + BR . vet) .
Y AR+BS
This gives the closed-loop characteristic polynomial, A
CL
' as
A
CL
= AR+BS.
36
(4.4)
(4.5)
Equation 4.5 is called the Diophantine equation [1], and can be used to determine Rand
S. In order to determine T, the response of the desired system to U
c
(t), which is
described by
(4.6)
is used, where Ym(t) is the desired output, Am and Bm are the numerator and
denominator ofthe desired transfer function. The condition,
BT BT B
m
AR +BS = A
CL
= Am '
must hold for the system to follow the model.
In order for the controller to be causal, the conditions
deg S ::; deg R
and
degT::; deg R,
(4.7)
(4.8)
(4.9)
must be true. Equation 4.5 has numerous solutions, thus it is possible to select the
solution which gives the controller with the lowest degree. Because n is greater than
m , and the degree of S cannot be greater than the degree of R , the relationship
degR =degA
cL
- n (4.10)
can be used to find the degree of R. Because it is undesirable to have extra delays in the
controller, R, S , and T are required to have the same degree.
37
This design allows some cancellation of process zeroes with poles of the
controller. In order for the zeroes to be cancelled, they must be both stable and well 
damped, lest the controller introduce unstable or poorly damped poles to the closed-loop
system. The polynomial B can be factored as
(4.11)
where B+ consists of the monic polynomial whose roots are stable and well-damped,
while B- consists of the polynomial whose roots are either unstable or poorly-damped.
Ifthe zeroes are not cancelled, then they must appear in the desired zeroes. This implies
(4.12)
whereB~is the desired numerator with the desired cancelled zeroes, B-, factored out.
Because B+ is cancelled, it must be a factor of A
cL
' thus
(4.13)
is true. Because B+ is a factor of both A
CL
and B , but it is not a factor of A, B+ must
be a factor of R , which can be written as
R =R'B+,
where R' is the feedback relation with B+ factored out ofit.
Equation 4.5 reduces to
and T can be determined from
(4.14)
(4.15)
(4.16)
This leads to three types of controllers where either some zeroes are cancelled, all zeroes
are cancelled or no zeroes are cancelled.
38
If all the process zeroes are cancelled, then the degree of A
o
is do -1. Choosing
(4.17)
results in the factorization ofEquation 4.12 to be
(4.18)
and
(4.19)
The Diophantine equation becomes
(4.20)
and T is
Ifno zeroes are cancelled, then the degree of A
o
is do -1. Choosing
B =flB = Am (l) B
m B(l)
results in the factorization ofEquation 4.12 to be
and
The Diophantine equation becomes
and T is defined by
(4.21)
(4.22)
(4.23)
(4.24)
(4.25)
T =f1rA = Am (1) A .
o B(1) 0
4.3 Self-Tuning Regulator Algorithm
39
(4.26)
The algorithm for determining a self-tuning regulator, taken directly from Astrom and
Wittenmark [1] is as follows.
Data: Given the specifications of the desired closed-loop pulse-transfer operator and a
desired A
o
.
Step 1: Estimate the polynomials ofthe transfer function with recursive least-squares.
Step 2: Solve for Rand S in Equation 4.55 and T in Equation 4.16.
Step 3: Calculate the control variable, u(t) , from Equation 4.2
Repeat Steps 1, 2, and 3 at each sampling period.
4.4 Self-Tuning Regulator Design Specifications
It is assumed that the model is
2
G (s) = ,
P S2 +12s+20.02
as determined in Chapter 1, with pulse-transfer function
H () = B = 0.961x10-
4
q+0.9233x10-
4
pq q A q2 -1.885q + 0.8869
(4.27)
(4.28)
40
which can be generalized to
because the parameters will vary. The desired pulse-transfer function is
H () _ Bm _ Bm
m
pq
q - Am - q2 -1.6015q + 0.6703 '
which can be generalized to
(4.29)
(4.30)
(4.31 )
The numerator has been left undefined because it is convenient to define the desired
numerator differently depending on zero cancellation. The pole excess for the system,
do is one.
4.5 Self-Tuning Regulator with Zero-Cancellation
For zero cancellation, the degree of A
o
is zero. Thus,
(4.32)
is true because A
o
is monic. The desired pulse-transfer function, obtained by choosing
Bm according to Equation 4.17, is
B is factored into
b
B+ :::: q + _1 :::: q + 0.9608
b
o
(4.33)
(4.34)
41
and
(4.35)
Because n is two, the degrees of R, S , and T are 1. Since the degrees of Rand B+ are
both 1, according to Equation 4.14 the degree of R' must be O. Substituting into the
Diophantine equation gives
(4.36)
Equating like coefficients yields
(4.37)
for the first degree term and
(4.38)
for the constant term. However, R is defined by Equation 4.14 as
(4.39)
The expression for S is obtained from Equations 4.37 and 4.38, i.e.
(4.40)
defines S. Using Equation 4.21 gives T. The simulation of this adaptive scheme is
depicted in Figures 4.2 and 4.3. The output ofthe plant is shown in Figure 4.2, while the
control signal is shown in Figure 4.3. Figure 4.4 shows that the parameter estimates
converge within ten steps. The system does meet all the requirements, however, from
Figure 4.3 it is seen that there is a great initial control effort. Further, the control signal
has ringing which can cause additional wear on the plant.
42
43
44
45
4.6 Self-Tuning Regulator without Zero-Cancellation
For a self-tuning regulator without zero-cancellation, the degree of A
o
is one.
Hence, A
o
is given by
A
o
= q+a
oo
'
(4.41)
Selecting B
m
according to Equation 4.22 results in the desired pulse-transfer function,
0.0031 (b +b)
b +b oq 1
=_-,-0_-,-I _
q2 -1.92q + 0.9231 .
(4.41 )
Substituting for A and B in the Diophantine equation results in
to be solved for R and S. Collecting coefficients of q, results in
Equating like coefficients of q results in the system ofequations,
Solving for the vector consisting ofthe R and S coefficients gives
or
(4.43)
(4.44)
(4.45)
46
The solutions for '1' So' and SI are
(4.47)
So = bO(a OO a2+a2aml -ala2-aooam2~+bl~a:+am2 +aooaml -aooal -amlal -aJ(4.48)
a
2
b
o
+b
l
- albob
l
and
T is defined by Equation 4.26 to be
T
- 1+ amI + am2 A _ 0.0031 ( )
- 0 - q + aoo .
b
o
+b
t
b
o
+b
t
(4.49)
(4.50)
The simulated system output is shown in Figure 4.5. It can be observed that the
system's output settles to its final values with zero error in about one second. The
overshoot of the system is 4%. Figure 4.6 shows the control signal, which has no
ringing and a lesser control effort than is the case for the self-tuning regulator with zero
cancellation. Figure 4.7 shows the convergence of the parameter estimates, which once
again converge within 10 samples.
4.7 Conclusions
Both self-tuning regulators, with and without zero cancellations, meet the
requirements of the design. However, due to the lesser control effort and the lack of
ringing, the self-tuning regulator without zero cancellation can be considered to be a
-aooam2~+bl~a:
1 1
Figure 4.5: Reference and Output ofa Sdf-Tuning Regulator-Controlled System without Pole-Zero CanceJ1at:ion.
+>.
.....:J
Regulator-Controlled
Figure 4.6: Control Signal for a Self-Tuning Regulator without Pole-Zero Cancellation.
~
00
49
50
better choice. This control technique will be used in the discussion on disturbances and
varying loads on the motor, which is provided in Chapter 6.
51
CHAPTER V
ONE-STEP AHEAD AND MODEL-REFERENCE ADAPTIVE CONTROLLERS
This chapter covers two related types of adaptive controller, the one-step ahead
controller and the model-reference adaptive controller. First the one-step ahead
controller is used to attempt to bring the output of the closed-loop system to the desired
value in one step. This in general requires a great control effort, so a form of weighting
on the control signal is also considered to reduce the control effort. A further step to
reduce the control effort is taken. Rather than attempting to bring the output to the
desired output in a single step, the model-reference adaptive technique endeavors to bring
the output of the system to that of the desired model. Several controllers with different
weightings for model-reference adaptive controllers are designed and compared in this
chapter.
5.1 Introduction to One-Step Ahead Control
The one-step ahead (OSA) controller is another adaptive control technique [7].
The one-step ahead controller brings the predicted output y(k + d) to a desired value,
y" (k + d) in a single step. As will be discovered, the control effort for such a controller
52
is often great, so the controller can be weighted in such a way as to reduce the control
effort.
5.2 The One-Step Ahead and Weighted One-Step Ahead Controllers
Once again, the model is assumed to be
(5.1)
where
and
B
·( -I) -d(b b -I b -m)_ -dB'( -I)
p q =q 0 + Iq + ... + mq - qq.
(5.2)
(5.3)
as described in Equation 1.37. This model can be rewritten in a d-step-ahead predictor
form,
(5.4)
where
and
fJ(
-1)_ p P -I .. fJ -m-d+1 - F( -I )'D'( -I)
q - 0 + Iq +. + m+d-Iq - q V q ,
with the coefficients of F(q-I) and G(q-I) given by
10 =1,
H
/; =- 'L!ja;_j ,
j=O
and
(5.5)
(5.6)
(5.7)
(5.8)
d-I
gj =- 'Lfjaj+d-j .
j=O
53
(5.9)
The time delay, d, is selected in such a manner as to ensure that b
o
in Equation 5.3 is
nonzero, which assures that flo in Equation 5.6 is nonzero as well. In order to bring
y(k + d) to the desired value of y. (k + d), while minimizing the cost function,
(5.10)
the desired value is substituted into Equation 5.4, resulting in
(5.11)
This can be expanded into
which can be solved for u(k)
u(k) = y" (k + d)- a{q-l )y(k)- fl1u(k -1)-'"- /3m+d_I
U
(k - m- d +1). (5.13)
/30
Because the selection of b
o
always results in a nonzero /30' the construction is possible
[6].
For the DC-motor plant as given in Equation 1.45
H( -1)= q-d{bo+b1q-l) = q-l{0.0367+0.0321q-I),
q l+a
1
q-l+a
z
q-Z 1-1.6015q-l+0.6703q-Z
(5.14)
the order of the numerator, m, is one, the order of the denominator, n , is two, and the
pole excess, d, is one. This means that F({q-I) is zeroeth order, and G{q-l) is first
order.
54
Because F(q-l) is zeroeth order, the only coefficient present is 10' which from Equation
5.7 is one. The coefficients of a(q-I) are given by Equation 5.9, which evaluates to
(5.15)
and
(5.16)
The predictor is described by
The control signal, u(k) , is given by
u(k) =y. (k +d)+ a1y(k)+ a
2
y(k -1)-b1u(k -1) .
b
o
(5.17)
(5.18)
For adaptive one-step ahead control, the parameters are unknown and must be estimated,
in this case with a recursive least squares estimation algorithm. Figure 5.1 shows the
response for this closed-loop system, and Figure 5.2 shows the control signal for the DC
motor using one-step ahead control.
A simple modification to Equation 5.12 results in the weighted one-step ahead
controller control signal [7], given by
u(k) = flo(/ (k +d)- a(q-I )y(k)-fllu(~-1)-'" - flm+d-lU(k - m - d+I)J .(5.19)
flo + /l,
This is a generalized case ofone-step ahead control, which minimizes the cost function
(5.20)
When Ii is set to zero, the control signal obtained by one-step ahead adaptive control is
recovered. For the DC motor, the control signal becomes
F{q-l)
a{q-I)
a{q-Ifllu(~
55
56
57
(5.21)
where aI' a
2
, b
o
' and b
l
are estimated recursively. Because b
o
is of the magnitude of
10-
3
, A must be less than lOx 10-
6
lest A dominate the denominator. The closed-loop
responses of the system for several different values of A and the corresponding control
signals are depicted in Figures 5.3 through 5.6. Figure 5.3 shows the results when A is
lOx 10-
6
• This system does reach its final value rather quickly, however, the final value
is 0.4879, which gives an unacceptable steady-state error. The control signal for this
system has a maximum magnitude of 15, which will be the smallest control signal. One
additional benefit is the elimination ofthe ringing in the control signal. Figure 5.4 shows
the system information when A is 10-
6
• This system has a final value of 0.905, which is
better than the first system. This is countered by the fact that this system has a maximal
magnitude of 185 for the control signal. Once again there is no ringing in the control
signal. Figure 5.5 shows the system when A is 500 x 10-
9
• The final value for this
system is 0.9506 with a maximal magnitude of 370 for the control signal. The
overshoot for this system is 25%. The final simulation is in Figure 5.6 when A is
100 x 10-
9
• The final value of this simulation is 1, however, the cost for the perfect
steady state is high as the maximum magnitude of the control signal approaches 1,800,
with an overshoot ofabout 45%. Based on comparing these four simulations, either a A
of 500 x 10-
9
or a A of 10-
6
is the best controller for this plant, depending on how
important the steady state accuracy is.
Figure 5.3: Response and Control Signal for a Weighted One-Step Ahead Adaptive Controned SystelIL
VI
00
59
Figure 5.5: Response and Control Signal foc a \Veighted One-Step Ahead Adaptive Controlled System-
0\
o
Figure 5.6: Response and Control Signal for a Weighted One-Step AheadAdapuve Controlled System.
0\
-
62
5.3 Model-Reference Adaptive System
Rather than trying to follow the reference signal, following a model that has an
acceptable transfer function will serve to lessen the control effort. This is the so-called
model-reference adaptive control, which is somewhat related to the self-tuning regulator
[1],[7]. The model that will be used for reference is given by Equation 1.45,
H
m
(z)= 0.001638z+0.001594.
pq Z2 -1.92z + 0.9231
(5.22)
The mechanism that was used to implement the model-reference adaptive control is one-
step ahead and weighted one-step ahead control. Figure 5.7 shows the simulation for
model-reference one-step ahead control. It can be observed that the steady-state error is
0, with the overshoot of 15%. The maximum magnitude of he control signal is about
700, and ringing is present in the control signal. Once again, the control effort can be
improved by using a weighted one-step ahead approach. The system simulation for a A,
of 10xl0-
6
is shown in Figure 5.8. The final value ofthis system output is 0.4873, that
is the steady state error is large and un acceptable. The magnitude ofthe maximal control
input is 12.5. A lesser weighting of A" 10-
6
is used for the simulation shown in Figure
5.9. This simulation shows an overshoot of 10% and a steady-state error of 9.2%. The
control effort has a maximum magnitude of 60. Weighting A, even less, to 500 x lO-
9
results in the system shown in Figure 5.10. The steady-state error is only 4.56% and the
overshoot is 11 %. The control effort is relatively small with a maximal magnitude of
72. Figure 5.11 shows the final simulation, for the system with A, equal to 100 x 10-
9
•
This system has no steady-state error and an overshoot of 15%. The maximum control
63
Figure 5_8: Response and Control Signal for a \\leighted Model-Reference Adaptive System_
01
-I:::>
Adapmre
Figure 5.9: Response and Control Signal for a "lcighted Model-Reference Adaptive System.
0\
VI
Figure 5.10: Response and Control Signal for a Weighted Model-Referenc.e Adaptive System.
0\
0\
Figure 5.11: Response and Control Signal for a Weighted Model-Reference Adaptive System.
0\
-...J
68
effort is 85. Each ofthese closed-loop systems has a settling time of less than Is. This
system has similar performance compared to the unweighted model-reference adaptive
system, but at a greatly reduced control effort.
5.4 Conclusions
Several adaptive control schemes were considered in this chapter. The model 
reference adaptive controller based on weighted one-step ahead adaptive contol provided
the best response. By using a weighting of 100 x 10-
9
, all of the design criteria,
overshoot of 15% , no steady-state error, and a settling time of Is, were met with out an
extremely unreasonable control effort. This controller will be used as the basis for the
discussion in Chapter 6, which deals with the effect of varying the load as well as noise
on the closed-loop system.
69
CHAPTER VI
SYSTEM AND LOAD ERROR RESULTS
In this chapter the effect of model uncertainty and noise on the behavior of the
closed-loop controlled systems are investigated. First the disturbance rejection properties of
the controlled systems designed based on an ideal model are considered. The effect ofmodel
uncertainty is then investigated by imposing errors in the type, order, and the stability of the
system. Finally, the classical and adaptive techniques are compared when the motor has a
nonzero load.
6.1 Comparison of Techniques
The closed-loop response ofthe classical controller, self-tuning regulator, and model-
reference adaptive system derived and shown in Chapter 3, Chapter 4, and Chapter 5,
respectively, are shown in Figure 6.1. The plant has the assumed transfer function,
G (s) =----:--_2__
P S2 +12s+20.02
and assumed pulse transfer function
H (z) = 96.1 0 x 10-
6
Z + 92.3310-
6
pq Z2 -1.885z + 0.8869
(6.1)
(6.2)
70
71
From Figure 6.1, it is apparent that all three controllers meet the design criteria of a
maximum overshoot of 15% and a settling time of Is , with the sole exception that the self 
tuning regulator has a settling time of about 1.04s, which is close enough. The self-tuning
regulator has the best performance in terms of percent overshoot, having only 4.21 %
overshoot. The model-reference adaptive controller has the least desirable overshoot, just
meeting the overshot requirement. The classical controller has an overshoot of 8.04%.
Figure 6.1 also shows the control effort for each ofthe three controllers. Both the self-tuning
regulator and the model-reference adaptive controller have higher control signals as the
parameters are being estimated. However, in the steady state, the model-reference adaptive
controller has the best control signal.
6.2 Disturbance Rejection
The first non-ideal situation that will be explored is the case where the input signal
has noise coupled to it. For this purpose, a random signal was created in MATLAB with the
RAND command. The amplitude of the noise was 10% of that of the input. Figure 6.2
shows the results of that simulation as well as the noise that was added to the input. The
disturbance rejection ofall three ofthe controllers is adequate for this magnitude of noise. It
is intuitive that the classical controller will have a decent amount ofdisturbance rejection due
to the presence ofthe pole at z =1, which provides integrating action. The overshoot on the
classical controlled system remains about 8%, the overshoot on the model-reference
adaptive system remains at about 15%, while the self-tuning regulator is reduced to about
3%. The settling time on each system is about Is. The control effort remains largely
similar to that ofthe noise free case simulation.
72
73
6.3 Non-Ideal Model Situation
This situation encompasses a number of circumstances, the three of interest for this
discussion is an error in the type ofthe system, in the stability ofthe system, and in the order
of system. First, suppose that there is an error in the type of the system, that is, the plant to
be controlled is a type-1 system while the controllers were designed based on a type-O
system. Another popular model for a DC motor is
(6.3)
which is a Type-I system. Suppose that the pole at - 2.0025 remains, but that there is a zero
at - 200, and the dc gain is 0.0049; the resulting transfer function would be
( )
s+ 200
Gp S =0.049 ( ) ,
S s+2.0025
and the pulse-transfer function would be
H (z)= 96.92xlO-
6
z
pq Z2 -1.9802z + 0.9802 .
(6.4)
(6.5)
Typically, the zero at s =-200 could be ignored because it has a magnitude of 100 times
that of the next pole or zero. However, it is mathematically convenient to include it in this
discussion because the zero in the continuous-time transfer function increases the magnitude
ofthe pulse-transfer function. Figure 6.3 shows the simulation of the system with this plant.
Because the closed-loop system using the classical controller has a double pole at 1, it is
expected that the output will oscillate and not settle down to a steady state. The
classically-controlled system exhibits marginally stable behavior rather than explicitly
unstable behavior because the pole at 0.9802 is still cancelled by the controller. However, if
74
75
the pole is moved to 0.983, with the resulting pulse-transfer function
H () 96.92 X 10-
6
z
pq Z == Z2 -1.983z + 0.983 '
(6.6)
the resulting output from the classically controlled is unstable, which is the case ifthe pole is
moved any closer to 1 than 0.9802. The closed-loop responses for this plant is depicted in
Figure 6.4, from which it can be seen that the response from the model-reference adaptive
controller and the self-tuning regulator are unchanged by these variations in the model.
Suppose that the system, rather than being open-loop stable, is open-loop unstable.
This is accomplished if a pole is in the right hand side ofthe s-plane for the continuous-time
plant and if a pole is on or outside the unit disc for the discrete-time model. Suppose that the
continuous-time plant is described by
(6.7)
The resulting pulse-transfer function is
H (z)== 0.0001003z + 0.0001007 == 0.0001003(z+1.004) (6.8)
pq Z2 - 2.01z +1.01 (z -lXz -1.01) ,
which has a pole at z == 1.01, which is outside the unit circle. Because the order of he
numerator and the order of he denominator of the pulse-transfer function are the same as the
assumed model, the self-tuning regulator and the model-reference adaptive controller have no
trouble keeping the system stable and within the design parameters, as shown in Figure 6.5.
The control effort for the adaptive schemes once again have spikes on the order of 60 during
the parameter estimation, but settle down to 20 for the self-tuning regulator and 5 for
76
77
78
the model-reference adaptive controller. After the parameter estimates have converged, the
control effort ofthe model-reference adaptive system and the self-tuning regulator are largely
the same. On the other hand, because the classical system is unstable, the control signal is
unstable as well.
As a final consideration in model-type errors, the assumption will be made that there
is an error in the order ofthe model. Suppose that the transfer function has a pole at s =-5,
resulting in
2
Gp (s) =....,....(s-+-2-.0-02-5..,....,Xs-+-5--,-X.,.-s-+-9.-99-7-""5)
and a pulse-transfer function of
(6.9)
H (z) = 323 x 10-
9
(Z2 + 3.867z + 0.9350) = 323 X 10-
9
(z + 3.608Xz + 0.2591) .(6.10)
pq Z3 -2.874z
2
+2.752z-0.8773 (z-0.9049Xz-0.9802Xz-0.9891)
The simulation for the three different controllers is shown in Figure 6.6. The self-tuning
regulator has no problem keeping the output within the desired criteria. This is because the
self-tuning regulator does not need the exact values of the system parameters, only a proper
ratio of system parameters. The estimator tries to formulate a model that has a second-order
denominator and a first-order numerator that fits the input-output relation. The self-tuning
regulator then uses the ratios between numerator and denominator coefficients to formulate
the control signal. The estimator in the model-reference adaptive controller also attempts to
fit the input and output into a second-order denominator and a first-order numerator.
However, the values that are estimated are used directly in formulating the control signal, so
errors in the model estimation dramatically effect the output; in this case causing the output
ofthe system to be unstable, as can be seen in Figure 6.6. The classically controlled system,
79
80
while stable, has an overshoot of 71 %, and has a very long settling time. For this situation
the self-tuning regulator is the best controller. The control effort for any ofthese controllers
has much to be desired. The self-tuning regulator has the best control effort, but it settles at a
value on the order of 6 x 10
5
, while the control signal of the model-reference adaptive
controller and the classical controller oscillate with an increasing magnitude. From these
results, it is seen that the self-tuning regulator can control a system that has a higher order
than designed for, but that the control effort creates an incentive to have the order correct
when designing the controller.
6.4 Load Error
Recall that the transfer function ofthe plant is described by
K
G (s)- JL
p - 2 (B R) RB+K
2
'
S + -+- S+---
J L JL
(6.11)
as shown in Chapter 1. When using the values given in Equations 1.14, 1.15, 1.16, and 1.17,
but letting the value of J vary, Equation 6.11 becomes
0.02
Gp(S)= 2 (2J +0\) 0.2002'
S + s+-----
J J
(6.12)
Assuming that the value of J for the motor shaft and load is 1kg '2
m2
; the transfer function
S
is then, after plugging the value of J into Equation 6.12,
G (s)= 0.02 = 0.02
P S2 + 2.1s + 0.2002 (s+2.000Xs+0.1001) ,
with corresponding pulse-transfer function as
H (z)= 99.3x10-
6
(z+0.9931).
pq Z2 -1.979z + 0.9792
81
(6.13)
(6.14)
Figure 6.7 shows the simulation of the system controlled by the classical controller, a self-
tuning regulator and a model-reference adaptive controller. The performance of both the
self-tuning regulator and the model-reference adaptive controller are unaffected by the
change is system parameters. However, the classical controller's performance is drastically
reduced. By virtue of the integrator the steady-state error is zero, but the settling time is
greatly increased. At a time of 3s, the output is still only at 93.4% of the final value. The
overshoot requirement is met; however, the controller does not reduce the settling time as
compared to the open-loop system. It is reasonable to assume that as the load increases, the
transfer function coefficients vary more from the unloaded values. It follows that the system
response for the classically controlled system will be poorer as the difference in coefficients
grows. The most extreme case that will be addressed is when the inertial constant of the
motor and the load is 5 kg '2
m2
• Substituting this value of J into Equation 6.12 results in the
s
transfer function,
G (s)- 0.004
P - S2 + 2.02s + 0.04004 '
which has the pulse-transfer function,
H (z)= 19.87x10-
9
(z+0.9929).
pq Z2 -1.98z + 0.98
(6.15)
(6.16)
82
83
The zero of the pulse-transfer function remains in the same vicinity, as do the pole. The
main impact that changing the inertia of the load has on the pulse-transfer function is to
change the value and magnitude of the dc-gain. Because the dc-gain is so small, it greatly
lessens the effect ofthe gain introduced by the classical controller. This decreased effect of
gain tends to keep the closed-loop poles in very close proximity to the open-loop poles. The
controller has little effect on the damping ratio and natural frequency, and thus little effect on
the overshoot and settling time of the system. The presence of the integrator does indicate
that eventually, the system will reach zero steady-state error. The self-tuning regulator does
not remain unaffected by the change in de gain. While the output remains the same, the
control effort is increased by a similar magnitude as the dc-gain ofthe pulse-transfer function
is decreased. Figure 6.8 shows the simulation of the system with J =5 kg ·2
m2
. The
s
classically-controlled system has reached a value that is 5.3% of its final value by 3s. The
output of the self-tuning regulator and the model-reference adaptive controller are both
unchanged by the change in system parameters.
84
85
CHAPTER VII
SUMMARY AND CONCLUSIONS
7.1 Summary
In the first chapter a mathematical model for a dc motor was developed, which
was formulated as a continuous-time transfer function. Subsequently, a pulse-transfer
function was derived as well. The desired characteristics of the output of the plant were
used to determine a desired transfer and pulse-transfer function. It was shown that any
variation on the load would change the poles and dc gain ofthe transfer function.
The classical root locus technique was used to design the controller that was given
by
D(z) =30.2 z- 0.9800 .
z-1
7.1
which exceeds the requirements of overshoot of 8%, a settling time of Is, and zero
steady-state error.
A self-tuning regulator was designed that had total pole-zero cancellation. This
adaptive controller resulted in an output that reached its steady-state in about Is, with an
overshoot of only 6% and zero steady-state error. This controller, however, produced a
control signal whose value greatly changed from sample to sample, a phenomenon
known as ringing. A second self-tuning regulator was designed that did not have pole-
zero cancellation. The output of this system has zero steady-state error, an overshoot of
/
86
4% and a settling time of Is. The control signal produced by this controller did not
have ringing and also had a smaller magnitude.
The third type ofcontroller was a one-step ahead adaptive controller. This type of
controller met the overshoot and settling time specifications of 15% and Is exactly.
Two variants, weighted and unweighted one-step ahead controllers were considered. The
unweighted controller resulted in zero steady-state error, however the control signal
exhibited ringing and had too great a control effort. The ringing and control effort
problems were lessened by weighting the control effort. This introduced some steady 
state error into the system.
The fourth and final type of controller considered , model-reference adaptive
controllers, is related to the one-step ahead controller. Rather than trying to follow the
reference signal, the controller attempts to follow the output of an acceptable model to
the reference signal. This results in the lessening the control effort. In the model 
reference adaptive controller, the overshoot and settling times were met exactly. As in
one-step ahead control, the unweighted output had the greatest control effort as well as
some ringing in the control signal. There was a direct correlation between the amount of
steady-state error and the weight that was given to having a low control effort.
The control efforts of each of the controllers were compared. The lowest control
effort was generated by the classical controller. The self-tuning regulator had the greatest
control effort, while the model-reference adaptive system had a moderate control effort.
Based solely on the design criteria of the unloaded dc motor whose mathematical model
was entirely correct and the control effort, the best controller would be the classical
controller. Each off the three controllers exhibited an acceptable amount of inherent
87
disturbance rejection, with the model-reference adaptive controller displaying the greatest
disturbance rejection, and the classical controller displaying the least amount of
disturbance rejection.
Several situations in which the assumed model was inexact were investigated with
the three different controllers, the classical controller, the self-tuning regulator, and the
model reference adaptive controller. The variations from the ideal model were obtained
by including an error in the type of the system, the stability of the system, and the order
ofthe system. When the plant was of a higher type than the assumed plant, the classical
controller remained in sustained oscillation for closed-loop response. The self-tuning
regulator and the model-reference adaptive controller remained unaffected by this type of
error. The control effort was largely unaffected for all three ofthe controllers.
For the second type of plant variation from the ideal case, one of the poles of the
plant was moved to the right half plane, resulting in an unstable system. When the
classical controller was used, the system remained unstable. Once again, neither the self 
tuning regulator nor the model-reference adaptive controller were effected by this change.
Again, the control effort of the three controllers for the unstable case remained largely
unchanged from that of the stable case. The variation from the ideal model was created
by adding a pole to the open-loop transfer function, increasing the order of the system.
The output of the self-tuning regulator controlled system was unaffected by this error.
The control effort was slightly greater with this variation as compared to the ideal case.
The classically controlled system, while stable, did not perform within the desired
criteria. The overshoot was 71% and the settling time was great. This change had a
catastrophic effect on the model-reference adaptive controlled system, which could be
88
avoided by over-parameterizing the model used in design. Both the control effort and the
output ofthe model-reference adaptive system grew without bound.
The final changes that were made to the system were changes to the load ofthe dc
motor, which changed the location of the poles and zeroes of the transfer function from
the assumed poles and zeroes of the transfer function. For the self-tuning regulator and
the model-reference adaptive controller, the response did not change. However, as the
load was increased, the control effort for both the self-tuning regulator and the model 
reference adaptive controller increased. Increasing the load also increased the control
effort of the classical controller. In addition, the settling time also greatly increased as
the load to the controller increased.
7.2 Conclusions
The classical controller has several disadvantages. The first is that it results in an
unstable closed-loop system if the plant is of higher type. Further, this controller cannot
compensate for an unstable plant. The most likely situation that is disadvantageous for
the classical controller is when the dc motor is loaded, for which the settling time is
greatly increased over the open loop transfer characteristic.
If the order of the system is known, then the model-reference adaptive controller
is the best controller because it can follow a desired output no matter what the load, the
type, or the stability of the plant is. However, because of the catastrophic reaction to a
system of higher order than assumed, the model-reference adaptive is a bad controller if
there is the possibility ofa higher order plant. Designing the controller assuming a higher
89
order for the controller would be a solution for this type of error, but would increase the
number of calculation required at each iteration. The increase in calculations should not
present any problem because inexpensive high-speed processors are readily available.
If the smallest control effort is not absolutely necessary, the self-tuning regulator
is the best choice for a controller. It is able to retain its response even if the plant is of a
different order, type or even unstable. The load on the motor has little effect on the
performance ofthe system controlled by a self-tuning regulator.
The work presented here can be expanded by considering experimental
implementation of developed controllers using a microprocessor and digital signal
processing. An extension ofthe implementation would be to observe how quantization of
the process effects the system, and which controller is most effected by the quantization.
The dependence of the closed-loop system response on the initial guess for the system
parameters could also be observed.
90
REFERENCES
[1] Astrom, Karl J. and Bjorn Wittenmark. Adaptive Control, Second Edition. New
York: Addison-Wesley Publishing Company, 1995.
[2] Chen Jie and Guoxiang Gu. Control-Oriented Sytem Identification, An H'"
Approach. Hew York: John Wiley and Sons, 2000.
[3] Elbert, Theodore F. Estimation and Control ofSystems. New York: Van
Nostrand Reinhold Company, 1984.
[4] Example: DC Motor Speed Modeling. Carnegie Mellon Controls Tutorialsfor
MATLAB. 16 June 2004.
<http://www.engin.umich.edu/group/ctm/examples/motor/motor.html>.
[5] Feuer, A and G.C. Goodwin. Linear Deterministic Adaptive Control:
Fundamental Limitations? Systems and Control Letters 49,2003.
[6] Friedland, Bernard. Control System Design: An Introduction to State-Space
Methods. New York: McGraw-Hill Book Company, 1986.
[7] Goodwin, Graham C and Kwai Sang Sin. Adaptive Filtering Prediction and
Control. Englewood Cliffs, NJ: Prentice Hall, 1984.
[8] Goodwin, Graham C, Stefan F Graebe and Mario Salgado. Control System
Design. Upper Saddle River, NJ: Prentice Hall, 2001.
[9] Grewal Mohinder S and Angus P Andrews. Kalman Filtering, Theory and
Practice. Englewood Cliffs, NJ: Prentice Hall, 1993.
[10] Ilchmann, A. and E.P. Ryan. On Tracking and Disturbance Rejection By
Adaptive Control. Systems and Controls Letters 52, 2004.
[11] Kuo, Benjamin C. Automatic Control Systems, Seventh Edition. Englewood
Cliffs, NJ: Prentice Hall, 1995.
[12] Lindfield, George and John Penny. Numerical Methods Using MATLAB,
Second Edition. Upper Saddle River, NJ: Prentice Hall, 2000.
[13] Mossayebi, Faramarz. Adaptive Control ofChaotic Systems. The University of
Akron: Dissertation, 1994.
91
[14] Nise, Nonnan S. Control System Engineering, Third Edition. New York: John
Wiley and Sons, 2000.
[15] Ogata, Katsuhiko. Discrete-Time Control Systems, Second Edition. Upper
Saddle River: Prentice Hall, 1995.
[16] Phillips, Charles C and H. Troy Nagle. Digital Control System Analysis and
Design, Second Edition. Englewood Cliffs, NJ: Prentice Hall, 1984.
[17] Pratap, Rudra. Getting Started with MATLAB. Orlando: Harcourt Brace College
Publishers, 1996.
92
APPENDIX A:
MATRIX DERIVATIVE AND MATRIX INVERSION LEMMA
A.I Matrix Derivative
If f(x) is a vector function of the vector x, then the formal definition of the partial
derivative of f(x) with respect to x is
aJ; aJ; aJ;
ax)
&2
aX
N
Bf(x)
al
2
al
2
al
2
--=
ax)
&2
aX
N
(A.I)
ax
aiM aiM aiM
ax)
&2
aX
N
Two common vector functions that are encountered are
f(x) =Ax
and
f(x) =x
T
Ax.
For the first function, the definition ofthe partial derivative leads to
(A.2)
(A.3)
a(Ax) a
=
ax ax
a
l1
x) + a)2 x2+... + a)NxN
a
2
)x) +a
22
x
2
+ .. ·+a
2N
x
N
(AA)
which becomes, upon performing the partial derivatives,
For the second function,
=A. (A.S)
93
Differentiating x
T
Ax with respect to Xl yields
(A.6)
Collecting like terms in Xi results in the equation,
Similarly the partial derivative with respect to Xi is
If A is symmetric, then Equation (A.9) becomes
(A.ID)
which is the i
1h
entry in the row vector forming the partial derivative. This quantity is
twice as large as the i
1h
entry in the row vector x
T
A. Thus, as long as A is symmetric,
(A.ll)
holds.
94
A.2 Matrix Inversion Lemma
If the matrices A, C, and C-
I
+ DA-
I
B are nonsingular, then the matrix A + BCD is
nonsingular, and
(A. 12)
is true. This is shown by multiplication of A + BCD by its inverse. This yields
(A+BCDXA-I_A-IB(C-I+DA-IBtDA-I). The product of the first term of each
multiplicand, which is the first term, is the identity matrix, 1. The second term, which is
the product of the first term of the first multiplicand and the second term of the second
multiplicand, is - B(C-
I
+ DA-
I
Bt DA-
I
. The product of the second term of the first
multiplicand and the first term of the second multiplicand is BCDA-
I
, which is the third
term. The fourth term, which is product of the second term of each multiplicand, is
- BCDA-
I
B(C-
I
+ DA-
I
Bt DA-
I
. The sum ofthe second and fourth term is
- B(C-
I
+ DA-
I
Bt DA-
I
- BCDA-
I
B(C-
I
+ DA-
I
Bt DA-
I
=-B((C-
I
+DA-IBt +CDA-IB(C-
I
+DA-IBt)DA-
I
=- BC(C-
I
(C-
I
+ DA-
I
Bt + DA-
I
B(C-
I
+ DA-
I
Bt )DA-
I
=-BC((C-
I
+DA-IBXC-
I
+DA-IBt )DA-
I
=-BCDA-
I
The sum ofthe second and fourth term, thus cancels the third term, leaving only the first
term, the identity matrix. Thus, A +BCD and A-I - A-IB(C-
1
+DA-
I
Bt DA-
I
are
inverses.
95
APPENDIXB:
LINEAR PREDICTION MODEL
A linear discrete-time system can be written as
(B.l)
where
(B.2)
and
as described in Equation 1.37. The output of the system at time k + d can be written in
the predictor form,
where
( -I) -I -n+1 G( -I)
a\q =a
o
+ a1q +... + an_1q = q
and
(B.4)
(B.5)
F(q-I) and G(q-I) are the unique polynomials that satisfy
with
F(
-I) {' {' -I {' -d+1
q =Jo + Jlq +... + Jd-Iq ,
and
(B.7)
(B.8)
The coefficients are computed in the following manner from Equation B.7.
10 =1,
i-1
J; =-"Ilja
i
- j ,
j=O
and
d-l
gj =-"Iljai+d-j .
j=O
(B.9)
(B.IO)
(B.lI)
(B.12)
96
97
APPENDIXC
MATLAB CODE
Code for Classical Controller Simulation
Root Locus
Bq=[0.96Ie-4 0.9233e-4]; % Set uncompensated numerator
Aq=[1 -1.8850.8869]; % Set uncompensated denominator
Ts=O.OI; % Set sampling time
sysol=tf(Bq,Aq,Ts); % Formulate uncompensated open-loop system
Aqint=conv(Aq,[I -1]); % Determine compensated denominator
Bqint=conv(Bq,[I -0.98]); % Determine compensated numerator
sysint=tf(Bqint,Aqint,Ts); %Formulate compensated open-loop system
rlocus(sysint) % Perform root locus
zgrid(0.5I69,0.07738) % Add contours ofconstant damping ratio and natural frequency
axis([0.8 1.2 -.2 .2]) % Specify graph size
Step Response
Kos=42.8;
Bq=[0.96Ie-4 0.9233e-4]; % Set uncompensated numerator
Aq=[1 -1.8850.8869]; % Set uncompensated denominator
Ts=O.OI; % Set sampling time
sysol=tf(Bq,Aq,Ts); % Formulate uncompensated open-loop system
Aqint=conv(Aq,[I -1]); % Determine compensated denominator
Bqint=conv(Bq,[l -0.98]); % Determine compensated numerator
sysint=tf(Bqint,Aqint,Ts); % Formulate compensatted open-loop system
syscl=feedback(Kos*sysint,1); % Formulate closed-loop system
step(syscl) % Find the closed-loop step response
title('Matching the Overshoot Criteria')
Self-Tuning Regulator Simulation
clear
clf
lambda=O.l; % Initialize forgetting factor
T=O.Ol; % Set sampling time
k=0:300; %Initialize sample index
N=length(k); % Determine the number ofsamples
uc=l *sign(sin((2*pi*k)/400)); % Formulate the reference signal
% Designate Plant
B=[96.le-6 92.33e-6]; % Set uncompensated numerator
A=[l -1.8850.8869]; % Set uncompensated denominator
% Extract transffer function coefficients
brO=B(l);
brl=B(2);
arl=A(2);
ar2=A(3);
% Designate Desired Transfer Function
Bm=[0.1761 0]; % Set desired numerator
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Am=[1 -1.920.9231]; %Set desired denominator
% Extract desired coefficients
bmO=Bm(1);
bml=Bm(2);
aml=Am(2);
am2=Am(3);
aOO=O;
% Initialize estimation variables
e=zeros(1,N);
K=zeros(4,N);
theta=zeros(4,N);
den=zeros(1,N);
% Set initial parameter estimate
theta(:,1)=[0;0;0.01;0.2];
% Perform first two iterations ofestimation and output
theta(:,2)=theta(:,1);
P(:,:,2)=diag([100 100 100 100]);
u(1 )=uc(I);
u(2)=uc(2);
y(1)=O;
y(2)=brO*u(1 );
% Perform ramaining iterations ofestimation, conttrol signal formulation
% and output determination
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form=3:N
% Determine Output
y(m)=-ar1 *y(m-1)-ar2*y(m-2)+brO*u(m-1)+br1*u(m-2);
% Estimate Parameters
Pkm1=P(:,:,m-1);
phikm1=[-y(m-1) -y(m-2) u(m-1) u(m-2)]';
thetakm1=theta(:,m-1 );
e(m)=y(m)-phikml'*thetakm1;
K(:,m)=Pkm1 *phikm1 *inv(lambda+phikm1'*Pkm1 *phikm1);
P(:,:,m)=(diag([l 1 1 1])-K(:,m)*phikm1')*Pkm1/lambda;
theta(:,m)=thetakm1+K (:,m)*e(m);
a1=theta(1 ,m);
a2=theta(2,m);
bO=theta(3,m);
b1=theta(4,m);
% Formulate Control Signal
den(m)=a2*bO*bO+b1 *b1-a1 *bO*b1;
%Determine the S polynomial
sO=(bO*(aOO*a2+a2*am1-a1 *a2-aOO*am2)+bl *(al*al+am2+aOO*aml-aml*al 
a2))/den(m);
sl=(bO*(aOO*a2*aml +am2*a2-aOO*a1 *am2-a2*a2)+b1*(al*a2+aOO*am2-aOO*a2 
a2*aml))/den(m);
S(:,m)=[sO;sl];
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%Determine the R polynomial
rO=l;
r1 =(bO*bO*am2*aOO-bO*b1 *(am2-a2+am1 *aOO)+b1 *b1 *(am1-a1 +aOO))/den(m);
R(:,m)=[rO;r1];
%Determine the T polynomial
tt=((1 +am1 +am2)/(bO+b1));
to=tt;
t1=aOO*tt;
Tmat(:,m)=[tO;t1];
u(m)=-rl*u(m-l)+to*uc(m)+t1 *uc(m-l)-sO*y(m)-s1*y(m-l);
end
%Plot the system response
t=k*T;
stairs(t,uc,'r')
hold on
stairs(t,y,'b')
title('Output (blue) and Input (red) ofa Self-Tuning Regulator without Zero Cancelation')
xlabel('Time (s)')
ylabel('Output')
Model-Reference Adaptive System Simulation Code
% MRAS Weighted
clear
elf
lambda=O.I; % Determine forgetting factor for estimation
lamc=O.OOOOOOI; % Determine small control signal weighting
T=O.OI; % Set sampling time
k=0:300; % Create sampling index
N=length(k);
uc=sign(sin((2 *pi*k)/400));
% Designate Plant
B=[0.961e-4 0.9233e-4]; % Set uncompensated numerator
A=[1 -1.8850.8869]; % Set uncompensated denominator
% Extract transfer function coefficients
brO=B(l);
brl=B(2);
arl=A(2);
ar2=A(3);
% Designate Desired Output
Bm=[0.00291428474 0.0028375860]; % Set desiered numerator
Am=[1 -1.91736448 0.923111635]; % Set desired denominator
yml=filter(Bm,Am,uc); % Determine the desired system response
ym=[yml yml(N)]; % Increase the size ofthe desired output by one
%Initialize estimation variables
e=zeros(1,N);
K=zeros(4,N);
theta=zeros(4,N);
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theta(' 1)=[-2'1'0'0]'
-, , " ,
% Perfonn the first two iterations ofestimation, control signal
% fonnulation, and output detennination
theta(:,2)=theta(:,1);
P=diag([100 100 100 100]);
u(l)=uc(1);
u(2)=uc(2);
y(l)=O;
y(2)=brO*u(1 );
% At each iteration after the first
form=3:N
% Detennine Output
y(m)=-ar1*y(m-1)-ar2*y(m-2)+brO*u(m-1 )+brl*u(m-2);
% Estimate Parameters
Pkm1=P;
phikm1=[-y(m-1) -y(m-2) u(m-l) u(m-2)]';
thetakm1=theta(:,m-1 );
e(m)=y(m)-phikm1'*thetakm1 ;
K(:,m)=Pkm1 *phikm1 *inv(1ambda+phikm1'*Pkm1 *phikm1);
P=(diag([l 1 1 1])-K(:,m)*phikm1')*Pkml/lambda;
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theta(:,m)=thetakm1+K(:,m)*e(m);
a1 =theta(l,m);
a2=theta(2,m);
bO=theta(3,m);
b1=theta(4,m);
% Formulate Control Signal
u(m)=bO*(ym(m+l)+al*y(m)+a2*y(m-l)-bl *u(m-l))/(bO*bO+lamc);
end
% Plot data
t=k*T;
% Plot input and controlled output
subplot(211)
plot(t,uc,'r')
hold on
stairs(t,y,'b')
title('Output (blue) and Input (red) of a Weighted MRAS GSA-Controlled DC Motor for
\lambda = 100* 1Q/\_1\9')
xlabel('Sample')
ylabel('Output')
subplot(212)
% Plot control signal
stairs(t,u)
title('Control Signal')
xlabel('Sample')
ylabel('uc(t)')
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