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Model-reference adaptive control for dynamical systems

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Model-reference adaptive control for dynamical systems
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Dib, Yehia Ahmed
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iv, 73 leaves : illustrations ; 29 cm

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Adaptive control systems -- Mathematical models ( lcsh )
Adaptive control systems -- Mathematical models ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Includes bibliographical references.
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Submitted in partial fulfillment of the requirements for the degree, Master of Science, Department of Engineering and Computer Science.
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by Yehia Ahmed Dib.

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|University of Colorado Denver
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Auraria Library
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Full Text
MODEL-REFERENCE ADAPTIVE CONTROL FOR DYNAMICAL SYSTEMS
by
Yehia Ahmed Dib
B. S., University of Colorado, 1988
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science
Department of Electrical Engineering
and Computer Science
1991


This thesis for the Master of Science
degree by
Yehia Ahmed Dib
has been approved for the
Department of
Electrical Engineering and Computer Science
by
05/07/ /??/
Date
Marvin Anderson


Dib, Yehia Ahmed (M.S., Electrical Engineering)
Model-Reference Adaptive Control For Dynamical Systems
Thesis directed by Professor Edward T. Wall
In this thesis a new approach to adaptive model-
reference control is presented. A design procedure for
single-input/single-output systems has been developed
and the results are verified by computer simulation.
The improved adaptive control algorithm is very simple
and can be followed very easily. In addition, for
implementing the new adaptive mechanism, it is not
required that the very strong perfect model-following
(PMF) conditions are met. Simulation results show that
the improved adaptive control system is highly robust,
to the variations in plant dynamics. In particular,
the controller developed can easily be used to
stabilize a plant with poles in the right half of the
complex plane.
The second part of this thesis presents a high
speed adaptation system for controlled electrical speed
drives. A design procedure for the system has been
developed and the results are verified by computer
simulation.


The algorithms presented in this thesis are based
on the model reference adaptive control technique using
the Popov-Landau method.
Simulation results show the high speed adaptation
of this type of system, and give very good response as
compared to the classical PI controllers. The form and
content of this abstract are approved. I recommend its
publication.
Signed!
Edward T. Wall
iv


CONTENTS
CHAPTER
1. INTRODUCTION.....................................1
2. AN IMPROVED MODEL-REFERENCE ADAPTIVE
CONTROL SYSTEM...................................5
2.1 Introduction................................5
2.2 Model-Reference Intelligent Control.........6
2.3 The Improved Model-Reference Adaptive
Control System..............................9
2.3.1 Error Equations and the Adaptive
Law of the Improved MRAC
System..............................14
2.4 Summary....................................18
2.5 Simulation Results.........................19
2.5.1 Introduction........................19
2.5.2 Numerical Equations of the
Plant and the Reference Model......19
2.5.3 Numerical Examples and
Simulation Results..................28
3. HIGH SPEED ADAPTATION SYSTEMS FOR CONTROLLED
ELECTRICAL DRIVES...............................40
3.1 Introduction...............................40
3.2 Analysis of the D.C. Drive Dynamic
Characteristics............................47
3.3 The Adaptive Model-Following Control
System Equations...........................49


3.4 Analysis and Simulation Results...........55
4. CONCLUSIONS.....................................70
BIBLIOGRAPHY..........................................72
iv


ACKNOWLEDGEMENTS
This study would not have been possible without
the cooperation and support of many people to whom I
will always be grateful.
I especially wish to express my sincere gratitude
to my advisor, Dr. Edward T. Wall for his support and
guidance during the preparation of this thesis. I
would also like to thank Dr. Jan T. Bialasiewicz for
his advice and encouragement, and for his serving on my
committee. Furthermore, I would like to thank
Professor Marvin Anderson for his encouragement and
also for serving on my committee.
I would like to thank Mr. Alexandros John
Ampsefidis who developed the New Model-Reference
Adaptive Control System used in this thesis.
Finally, this thesis is dedicated to my family for
their moral support and encouragement which was very
helpful during my studies in the United States.


CHAPTER 1
INTRODUCTION
The purpose of a model-reference controller is to
adapt the response of a plant to that of a pre-selected
reference model. The plant design specifications are
included in the model such that the step response of
the model will have the specified rise time, overshoot
and settling time. A controller can be designed
according to the requirements of the model input, the
model states, and the error between plant and model
output in order to generate the appropriate control
signals. These control signals, which are the inputs
to the plant, are used to derive the outputs of the
plant so that they will track the output of the model.
An adaptive controller is used to overcome the
difficulties caused by a poor knowledge of the values
of the dynamic plant parameters and their variations.
A properly designed adaptive controller can result in a
superior plant performance given a limited knowledge of
the plant structure and parameters. Model-reference
adaptive control systems can be divided into two
classes. The first is composed of indirect adaptive


controllers in which on-line estimates of the plant
parameters are used for control law adjustment. The
second class, to which the controller designed in this
thesis belongs, contains the direct adaptive
controllers for which no effort is required to identify
the plant parameters. In other words, the control law
is directly adjusted to minimize the error between the
plant and model states.
The first direct model-reference adaptive control
system was designed by the performance index
minimization method proposed by Whitaker [2]. However,
this so-called "MIT design rule" could not ensure the
stability of this adaptive system. Therefore,
subsequent research efforts were emphasized toward the
development of additional stable adaptive algorithms.
One of the first researchers to use the Lyapunov second
method in the design of a stable adaptive controller
for a single input/single output system was Parks [2].
Direct model-reference adaptive control schemes,
designed for multi-input/multi-ouput systems by using
Lyapunov techniques, were developed by Grayson [3], and
Winsor and Roy [4]. However, these approaches require
the satisfaction of the Erzberger perfect model-
following conditions. In other words, these adaptive
controllers function properly only if there exists a
2


certain structural relationship between the plant and
the model.
Another adaptive algorithm for a multi-
input /multi-output continuous systems subject to the
perfect model-following conditions was developed by
Landau [5]. Furthermore, in recent years, the concept
of the command generator tracker (CGT), which was
developed by Broussard and O'Brien [6], has been used
to design adaptive controllers, which do not require
the satisfaction of the perfect model-following
conditions. A paper with this type of controller has
been written by Marbino and Kaufman [7].
In this thesis a new algorithm for direct model-
reference adaptive control of a single-input/single-
output system is presented. This algorithm does not
require the satisfaction of the PMF conditions. In
addition, it is shown that the adaptive algorithm
guarantees that the error will remain bounded under
less restrictive positivity conditions.
Furthermore, this improved model-reference
adaptive control (MRAC) system is extremely simple
compared with other control algorithms. Despite its
simplicity the improved adaptive controller is at least
as effective as the more complicated adaptive systems
mentioned.
3


Finally, this thesis is organized in the following
manner. The second chapter (chapter 2) presents the
design and simulation of the improved MRAC system. In
Chapter 3 the new high speed adaptation system is
analyzed and discussed in detail. Chapter 4 presents
concluding comments, results, and suggestions for
further research.
4


CHAPTER 2
AN IMPROVED MODEL-REFERENCE ADAPTIVE CONTROL SYSTEM
2.1 Introduction
This chapter presents the development and analysis
of an improved model-reference adaptive control (MRAC)
system. The formulation of the design statement
permits a suspension of the conditions for perfect
model-following (PMF) which is required for
hyperstability analysis [15]. In other words, it is
not required that the structure of the model accurately
represent the system. Therefore this improved
controller is more adaptable and hence is more useful
for a wider range of control system applications. The
simplicity and the low order of the adaptive algorithm
proposed are distinct features, which represent a
significant improvement in adaptive design. In
addition, simulation has shown that this improved
design is robust under severe plant parameter changes.
This is a valuable asset. Finally, the organization of
this chapter, as given in section 2.2, deals with a
model-reference intelligent control (MRIC) system [8].
This is the basis of a new and improved approach to


adaptive control. (Section 2.3) presents and develops
the main contribution of this thesis.
2.2 Model-Reference Intelligent Control
The idea of the model-reference intelligent
control (MRIC) was first introduced by Bialasiewiz,
J.T. and Proano, J.C. [8]. The stability of which was
proven by Ampsefidis, A.J. [1]. The MRIC, which is
shown in Fig. 2.1 consists of three parts. The first
Fig. 2.1 Model Reference Intelligent Control System
part the reference model, incorporates the design
characteristics of the plant. The purpose of the
control system is to constrain the plant step response
to track the step response of the reference model. In
the second part of the MRIC system, the theory of the
state estimator is developed. The estimator is
6


described by the differential equation of the reference
model with proper feedback. Because of feedback the
estimator response is faster than that of the reference
model and therefore the adaptation process speed is
improved. The third part of the MRIC system concerns
the adaptive law. This section is designed as a
proportional plus integral (PI) controller which
generates the error signal. This error signal is
defined as the output of the model minus the output of
the plant. In this sense, this new adaptive law could
be considered as continually solving the pole placement
problem. To complete the analysis of the MRIC system
the differential equations, which describes the
improved adaptive system of Fig. 2.1, are considered
next.
In the following equations the A, B and C are n x
n, n x 1, 1 x n matrices respectively. The plant Fig.
2.1 is described by
X ApXp + BpU (2.1)
Yp = cpxp (2.2)
where Xp e Rn, Yp e R and U e R. The reference model
is
xm = Amxm + Bmr (2.3)
xm = cmxm (2.4)
7


where Xm e Rn, Ym e R and r e R. The state estimator
equations are
xe Amxe + mr + Xe2 (2.5)
Ye = Cexe (2.6)
where Xe e Rn and Ye e R, k is the gain matrix and the
error is defined as
e2 = Yp Ye (2.7)
The adaptive control law is given by
uc = eT xe (2.8)
e = ei xe (2.9)
el = Ym Yp (2.10)
where Uc e R, 8 e Rn and e^ e R. Also, the input to
the plant may be expressed as
U Uc + r = 0T Xe + r (2.11)
Finally, without loss of generality, it can be assumed,
as in reference [8], that Cp = Cm = Ce = C, then C =
[l, 0, 0,...0].
From simulation it was observed that the
application of the MRIC system results is an adaptive
system, which is robust under rather extreme plant
parameter variations, even when these variations lead
8


to plant instability. There are however, two main
problems which arise when using the MRIC system. The
first is stability. In reference [8] a development of
the stability of the MRIC system is not presented.
That is, it is not formally determined that e^ (see
equation (2.10)) will converge to zero asymptotically.
Also a limitation of the stability proof given is that
the designer is not provided with guidelines for the
selection of the estimator feedback gains, which force
e to asymptotically approach zero. This problem was
solved by Ampsefidis, A.J. [1], A second set of
problems which evolve, is that the adaptation time of
the unmodified MRIC system is relatively long and the
magnitude of the error signal is large. It follows
from this that this controller can not be used to
control a process in which the adaptation time is a
critical factor. The improved MRIC system developed in
the following section eliminates this problem.
2.3 The Improved Model-Reference Adaptive Control
System
The design of the improved MRAC system is based on
a modification of the MRIC system. This modification
adds to the ideas presented in references [9] and [10]
and results in a useful improvement. In order to
develop the improved adaptive algorithm, it is
9


necessary to formulate a new statement of the design
problem. The idea of the augmented system, shown in
Fig. 2.2, is central to this formulation.
Fig. 2.2 Augmented System
The system shown in Fig. 2.2 is a typical feedback
control system. The state estimator and the matrix of
gains are located in the feedback path. The state
estimator is simply a filter that supplies the gain
controller with the required auxiliary signals.
The necessary concepts have now been established
so that the equation which defines the behavior of the-
augmented system can be developed. The plant shown in
Fig. 2.2 is given by
10


Xp ApXp + BpU
(2.12)
(2.13)
YP CPXP
The state estimator is described by
Xe = AmXe + Bmr + LCpXp LCeXe
(2.14)
(Am LCe) xe bCpXp + Bj^r
ye = CeXe (2.15)
and the input to the plant is assumed to be
U = Uc + r = K(t)
ey
Xe
+ r
Kx(t) ey + K2(t) Xe + r
(2.16)
where K(t) = [Kx(t) K2(t)] (2.17)
The combination of equation (2.12) and (2.16) results
in the following
Xp = Ap Xp + Bp K!(t) ey + Bp K2(t) Xe + Bp r (2.18)
The augmented system, with state vector X = [XpT, XeT]T
may be described by equations (2.18), (2.14) and (2.13)
and can be written in the following state space form
Ap BpK2(t)
LCr
Am LCe
BpK1(t)
0
ey +
(2.19)
11


r
Y = [Cp 0]
(2.20)
Note also that Y = Yp. Consider next equations (2.19)
and (2.20) which can be rewritten as
which follow from the definitions of matrices A, B and
C. The state vector X is a 2n-vector since it is
matrices in the equation for Xp and Xe have appropriate
dimensions. The outputs Yp, Ye, Y and the inputs r and
U belong to the set R. In addition, it is assumed that
the plant (described be equation (2.12) and (2.13)) is
time invariant and linear with unknown parameters.
Since the plant parameters are not known the elements
of the gain matrix (refer to Fig. 2.2) are assumed to
be time variant. This gain matrix, of dimension 1 x n,
is called K(t). Having developed the augmented system
equations the design problem can be stated.
X = AX + Dey + Br
(2.21)
Y = CX
(2.22)
assumed that Xp and Xe are n-vectors
Also, the
12


The output of the augmented system (which is of
order 2n) is required to track the output of an n-th
order reference model of the form
xm = Amxm + Bmr (2.23)
Ym ~ cmxm (2.24)
This model expresses the desired input-output behavior
of the augmented system. since the augmented system
input and the reference model input are the same, the
tracking problem can be expressed in the follow form
Y = Ym (2.25)
It should be observed that it is the function of
the gain matrix K(t) to adjust the dynamics of the
augmented system so that it will perform as a stable
reference model.
For the following adaptive system it is assumed
that there exists an ideal target system of the form
X* = A*X* + D*ey + Br (2.26)
Y* = CX* (2.27)
which satisfies the equation
Y* = Ym (2.28)


It is important to observe that both the augmented
system and the ideal target system have the same input
and output matrices. Also, it is assumed that the
system described by equations (2.26) and (2.27) will
have the same order as the augmented systems. It is
also assumed that the ideal system matrix A* is
A
*
Ap BpK
Lcp Am Lce
(2.29)
where K is an unknown constant gain. Finally, it
should be observed that the value of K is not required
to be known, only its existence must be guaranteed.
The generalized error equations will be developed
next. It is necessary to show how K(t) is to vary so
that the generalized state error converges to zero as t
approaches infinity.
2.3.1 Error Equations and the Adaptive Law of the
Improved MRAC System
The generalized state error is defined as
ex = X* X (2.30)
and the output error as
ey = Ym Y = Y* Y = Cx* Cx = Cex (2.31)
14


The differential equations for the state error may be
obtained by differentiating equation (2.30), that is
ex = X* X = X* A*X + A*X X (2.32)
Substituting the appropriate expression for X* and X in
(2.32) gives
ex = A*X* + D*ey + Br A*X + A*X AX Dey Br
= (X* X)A* + (A* A)X + (D* D)ey (2.33)
where
A A =
LC
BpK2
Am lc
Ap BpK2(t)
LCp Am LCe
Xp
a) X!
O Bp (K2 K2 (t))
0 o
Bp(K2 K2(t))
O
= Bp (K2) K2 (t) Xe
(2.34)
If D D = BpKi BpK!(t) Bp(Ki Kx(t))
0 0 0
ey = Bp [Ki Ki(t)] ey
(2.35)
where
15


B
P
*
(2.36)
Therefore, the generalized error state equation
can be written in the following form
ex = A*ex + Bp* (K2 K2(t))
Xe + Bp (KX Ki(t)) ey = A*ex (2.37)
A*ex = + Bp* [K K(t)]
ey
Xe
Equation (2.37) together with equation (2.31) describes
the dynamics of the error system, which is shown in
Fig. 2.3 and has the follow transfer function
Z(s) = C(SI A*)-1 B* (2.38)
As pointed out in the previous section it is the
purpose of the adaptive controller to make the
16


augmented system approach the ideal target system
asymptotically. In other words, K(t) must be chosen so
that ex approaches zero as t approaches infinity.
In summary, an adaptive control law is developed
using extended principles presented in references [9],
[10], and [11]. The resulting form is
uc = K(t) Xe (2.39)
K(t) = Kp(t) + Kx(t) (2.40)
Kp(t) = eyXeT T (2.41)
Kx(t) = eyXeT T (2.42)
where T and T are selected (constant or time variant)
to be positive definite and symmetric adaptation
coefficient matrices. It was shown in reference [1]
that positive definiteness and symmetry for these
matrices is required for global stability of the
adaptive system. The memory of the adaptive mechanism
is provided by the integral term Kj(t), which is
similar to the adaptive gain proposed in reference [8].
Also, the proportional term Kp(t) and the constant gain
matrices T and T cause the improved MRAC system to have
a faster response than the MRIC system. Furthermore,
the proportional term Kp(t) facilitates direct control
of the input error ey. therefore, as indicated in
reference [12], the output error can be ultimately
17


reduced to zero under the assumption of a disturbance
free environment. A final observation is that the
improved adaptive algorithm, given by equations (2.39)-
(2.42) is considerably simpler when compared with many
other adaptive control designs.
2.4 Summary
In this chapter an improved MRAC system was
developed. The complete new adaptive system is shown
in Fig. 2.4.
Fig. 2.4 Improved Model Adaptive Control System
18


The equations that describe the operation of the
system shown in the above figure are given in section
2.3. This makes the improved MRAC system really
versatile. In addition, from simulation results which
are presented next, it is clear that the improved
adaptive controller results in a system, which is
robust under extreme plant parameter changes. Another
advantage is the simplicity of the improved adaptive
algorithm.
2.5 Simulation Results
2.5.1 Introduction
In this section several simulation results are
given to illustrate the effectiveness of the improved
MRAC system. These results will be used to control a
plant consisting of a single link manipulator driven by
a d.c. motor through a gear train. The purpose of the
adaptive controller is to constrain the plant to
respond in a prescribed manner despite the uncertainty
in the plant parameters.
2.5.2 Numerical Equations of the Plant and the
Reference Model
A complete derivation of the dynamical equations
may be found in reference [1], as mentioned, the
dynamical behavior of the single link manipulator
19


changes when the weight of the payload is varied. The
plant equations given below show how the variations of
the weight of the payload effect the plant dynamics.
It is assumed that the d.c. motor, which drives
the single link manipulator, has the following
constants
Jm = 0.041 x 10"3 kg.m2
Km = 0.071 N. m/A
Ra = 1.6 n
Also, the ratio of the gear train used is 20:1.
The numerical equations of the plant are derived
for three different payloads. The distance of the
center of mass of the link to the joint is assumed to
be 1 = 0.15 m for all three cases.
In the first case it is assumed that the total
mass of the link is M = 1 kg. The state space equation
is
*1 0 1 *1 0
p = +
X2 -G -F *2 H
E E E
which describes the dynamical system [1]
20


X
(2.44)
0 1 0
X = -37.69 -32.3 X + 22.82
Since the output of the plant is the angular position
of the link, the output equation is
Y = [1 0] X (2.45)
Using equations (2.44) and (5.2) the transfer function
of the plant is
22.82
H (s) = ------------------ (2.46)
+ 32.3 S + 37.69
The step response of this transfer function is shown in
Fig. 2.5.
In the second case it is assumed that M = 3 kg.
Then equation (2.43) becomes
0 1 0
X +
-52.5 -15 10.6
(2.47)
and, for this case, the transfer function of the system
is
10.6
H(s)
S2 + 15 S + 52.5
(2.48)


The step response of this transfer function is shown in
Fig. 2.6.
Finally, in the third case considered here, the
total mass of the link is assumed to be M = 7 kg.
then, equation (2.43) is
0 1 0
X = -59.14 -17.24 X + 5.11 vs (2.49)
Equation (2.49) together with the output equation
(2.45) can be expressed in transfer function form as
5.11
H(s) = ------------------ (2.50)
S2 + 7.24 S + 59.14
The step response of this transfer function is shown in
Fig. 2.7.
As shown in Figs. 2.5 through 2.7, the dynamical
behavior of the single link manipulator changes
drastically whenever the total mass of the link
changes. However, in practical use it is usually
required that the manipulator link responed in a
specified manner regardless of the changes in payload.
In other words, the manipulator has to have a
predetermined step response, which is independent of
the mass of the payload. The ideal step response of
22


the manipulator can be expressed as the step response
of a well-designed reference model.
In developing the equations of the reference model
the designer should keep in mind that the manipulator
must lift the payload in a relatively short time with
practically no overshoot. Therefore, the transfer
function of the reference model is chosen to be
W2 25
H(s) = ------------------ = ----------- (2.51)
+ 2 ywn S + Wn^ Sz + 105 + 25
where the damping ratio is equal to one (% = 1) and the
natural frequency is equal to 5 rad/s (Wn = 5 rad/sec).
Equation (2.51) can be written in state space form as
0 1 0
X +
-25 -10 25
(2.52)
With an output equation given by (2.45). The step
response of the reference model is given in Fig. 2.8.
In conclusion, it should be stated that the
purpose of the adaptive mechanism is to force the
output of the plant to track the output of the model.
In other words, the adaptive controller will minimize
the error, which is defined as the output of the
reference model minus the output of the plant.
23


Aagolar position (ad)
Fig. 2.5 Plant Step Response (M = 1 kg).


Aagilar position (rad)
0.21
0.22 -
Fig. 2.6 Plant Step Response (M = 3 kg)


(pei) nocxsod .sniSay
Fig. 2.7 Plant Step Response (M = 7 kg).
I


Aagnlarposinon (tad)
l.l -
Fig. 2.8 Model Step Response.


2.5.3 Numerical Examples and simulation Results
Before presenting the simulation results, a brief
discussion is given on how the feedback gains of the
state estimator are to be chosen. The choice of the
feedback gain matrix L (refer to equation (2.14) must
guarantee the existence of a constant gain matrix K,
which will make A* a stable matrix. In the case of a
full order estimator, A can be written as
A
*
Ap BpK
LC Am LC
(2.53)
where C is the identity matrix C = diag [1, l,...l], L
= diag [llf l2...ln] and k = [Klf K2...Kn],
In order to clarify the procedure for choosing the
gain matrix L, a numerical example is given which uses
the equations of the plant and the reference model
presented in this section.
When the total mass of the link is 1 kg the
numerical expression for A can be written by using
equations (2.44), (2.52) and (2.53). This expression
is
28


(2.54)
A
*
0 10 0
-37.69 -32.3 22.82 Kn 22.82 K?
1X 0 -li 1
0 12 -25 -10 -12
First, the values of 1^ and 12 must be such that the
sub-matrix A22* (note A22* = Am LC) is a stable
matrix. If L is chosen to be equal to diag [10, 2] (or
1]_ = 10, 12 = 2) the matrix A22* which is equal to
a22
*
-10 1
-25 -12
(2.55)
is a stable matrix. The eigenvalues of A22 are equal
to -100.00 + j 4.89. Next, the following question has
to be addressed. Given the values for 1^ and 12 do
values for and K2 exist that will make A* a stable
matrix? For example, with 1^ = 10 and 12 = 2 there are
many possible gain matrices K = [K^, K2] that will
ff <|||
result m a stable matrix A With K = [2, 2], A
takes the following form
A
*
0 1
-37.69 -32.3
10 0
0 2
0 0
45.64 45.64
-10 1
-25 -12
(2.56)
29


The A* matrix given above is stable, because the
eigenvalues of A* are located in the left half of the
complex plane. that is the eigenvalues of A* are:
-14.475, -33.849, and -2.988 + j 3.786.
Next, the designer must solve the Lyapunov
equation for P so as to determine the stability
constraints
A*tP + PA* = -Q (2.57)
Q is selected to be a symmetric positive definite
matrix. The matrix P, which satisfies this equation is
the positive definite symmetric matrix
P
5.8673
0.1573
0.4927
0.3036
0.1573
0.0396
0.0099
0.0616
0.4927
0.0099
0.5715
0.1704
0.3036
0.0616
0.1704
0.3035
(2.58)
Finally, the designer must be certain that the selected
gain matrix L guarantees the existence of the required
K for all possible plant matrices Ap and Bp (see
equation (2.53). Therefore, for this case it must be
determined that the gain matrix L is correct when the
total mass of the link has the values 3 kg and 7 kg.
For the 3 kg case (equation (2.47)) and with the
same L and K the gam matrices for the A is found to
be
30


0 1 0 0
= -52.5 -15 21.2 21.2
10 0 -10 1
0 2 -25 -12
(2.59)
The matrix given above is stable because its
eigenvalues are located in the left half plane. The
eigenvalues of A* are -2.635 + j 5.473 and -15.865 + j
5.412. Also, by substituting A* and Q = diag [2, 2, 2,
2] into equation (2.57) it is found that there exists a
symmetric positive definite matrix P that satisfies
equation (2.57). This matrix P is equal to:
P
7.7615
0.2006
0.9533
-0.0594
0.2006
0.0917
0.0179
0.0872
0.9533
0.0179
0.4735
-0.1342
-0.0594
0.0872
-0.1342
0.2262
(2.60)
4?
Finally, it must be verified that the matrix A
satisfies the two conditions when the total mass of the
link is 7 kg. The matrix
0
-59.14
10
0
1
-7.24
0
2
0
10.22
-10
-25
0
10.22
1
-12
(2.61)
was obtained by substituting the appropriate matrices
from equations (2.49) and (2.52) into equation (2.53).
The gain matrices for K and L used are the same as
31


those used for the 3 kg and 1 kg cases. The matrix A*
is a stable matrix because the eigenvalues of A* are:
-1.885 + j 6.842 and -12.736 + j 5.868. In addition,
there exists a positive definite symmetric matrix P
that satisfies equation (2.57) where matrix A is given
by the expression (2.61) and Q is equal to diag [2, 2,
2, 2]. The resultant matrix P is:
P
13.975
0.2358
1.2946
-0.3182
0.2358
0.2007
0.0238
0.1068
1.2946
0.0238
0.3988
-0.1098
-0.3182
0.1086
-0.1098
0.1667
(2.62)
Summarizing, this analysis shows that with L =
diag [3-/ 2] the controlled system will be globally
stable if the constant gain matrices T and the T of the
adaptation mechanism are symmetric and positive
definite. Having shown that the controlled system is
stable, the system is simulated and discussed using the
Simnon program.
The simulation results for the plant when the
total mass of the link is 1 kg are shown in Fig. 2.9
and 2.10. Fig. 2.9 shows the position response of the
plant and the reference model. From this figure it is
seen that the output of the plant tracks the output of
the model perfectly. Fig. 2.10 shows the output error.
From this it is observed that the maximum position
32


error is extremely small and the adaptation time is
0.85 seconds.
The simulation results for the plant when the
total mass of the link is 3 kg, are shown in Fig. 2.11
and Fig. 2.12. From Fig. 2.11 it is seen that the
output of the plant tracks the output of the model
rather closely.
The third and last simulation was performed for a
plant with a link of total mass of 4 kg. The
simulation results for this case are shown in Fig. 2.13
and Fig. 2.14. As seen from Fig. 2.13 the output of
the plant is the same as the output of the model.
The simulation results indicate that the improved
MRAC system performs in a superior manner.
33


Angular posaos (rad)
Time (sec)
----- ----- niAilel
Fig. 2.9 Angular Response (M = 1 kg).


Angular posinoa eaor (rad)
l
Fig. 2.10 Angular Position Error (1 kg).


Angola? position (rad)
Time (sec)
|lnnl ------- tnoilel
Fig. 2.11 Angular Response ( M 3 kg).


Angular position era* (iad)
0.002 -
0.0018 -
0.0016 -
0.0014 -
0.0012 -
0.001 -
0.0008 -
0.0006 -
0.0004 -
0.0002 -
0 -
0.0002 -
0.0004 -
0.0006 -
0.0008 -
0.001 -
l
*K4 * IKS 1 1.2
1.6
'Iliiie (see)
T
X4
~T~
2.8
Fig. 2.12 Angular Position Error (3 kg)


Angular posidon (sad)
l.l -
lime (sec)
|ilnul
moilel
Fig. 2.13 Angular Response (M = 7 kg).


-0.001 -
n
i
T
T
Time (see)
Fig. 2.14 Angular Position Error (7 kg).


CHAPTER 3
HIGH SPEED ADAPTATION SYSTEMS
FOR CONTROLLED ELECTRICAL DRIVES
3.1 Introduction
Direct current machines are generally more
adaptable for adjustable-speed service than alternating
current machines which employ a constant-speed rotating
field. Indeed, the ready susceptibility of direct
current drives to adjustment of their operating speed
over wide ranges and by a variety of methods, is one of
the most important reasons for the strong competitive
position of these drives in modern industrial
applications.
The most common speed-control method employ
(i) flux adjustment, usually by means of field-
current control,
(ii) resistance adjustment associated with
armature circuit, and
(iii) armature terminal voltage adjustment.
The most important and more common is (i) which
employs field current control.


Field Current Control:
Field current control has the outstanding
advantage of the use of a shunt motor. The method may
also make use of compound motors. The adjustment of
the field current and hence the flux and speed
adjustment of the shunt-field circuit resistance (or a
solid-state control when the field is separately
excited) is accomplished simply and inexpensively
without change in motor losses.
The lowest speed obtained corresponds to a maximum
field current. The highest speed is limited
electrically by the effects of armature reaction under
weak-field conditions. Motor instability or poor
commutation may then result. The addition of a
stabilizing winding increase the speed range
appreciably, and the alternative addition of a
compensating winding may still increase that range. A
stabilizing winding insures attainment of a drooping
speed-load characteristic even at weak shunt-field
currents and heavy loads. This stabilizing winding is
sometimes used with adjustable-speed motors which are
intended for operation over a wide-speed range by
shunt-field resistance control. With a compensating
winding, the overall range may be as high as 8 to 1 for
the case of a small integral-horsepower motor.
41


Economic factors limit the feasible range for very
large motors to about 2 to 1, however, a 4 to 1 ratio
(considered in this study) is often regarded as the
limit for an average-sized motor. In order to examine
the approximate limitation on the allowable continuous
motor output, as the speed is changed, the influence of
changing ventilation and changing rotational losses on
the allowable output are neglected. The maximum
armature current Ia is then selected as the nameplate
value so that the motor will not overheat, and the
speed voltage Ea will remain constant (since the effect
of a speed change is compensated by the change of flux
causing it). The Ea Ia product and hence the allowable
motor output will then remain substantially constant
over the speed range selected. For a direct current
motor with a shunt field-rheostat the speed control
system is considered to be a constant-horsepower drive.
Since torque varies directly with the flux the highest
allowable value occurs at the lowest speed. Field-
current control is therefore best suited for a drive
requiring increased torque at low speeds. When a motor
so controlled is used with a load requiring constant
torque over the speed range, the rating and size of the
machine are determined by the product of the torque and
the highest speed. Such a drive in inherently
42


oversized at the lower speeds. this is the principal
economic factor which limits the practical speed range
of large motors.
The need for high performance control systems in
industrial application has created a significant
research effort in modern control theory, and in
particular in non-linear adaptive and optimal control
research.
In applying linear quadratic optimal control, two
main difficulties arise in application. First, it is
very hard to specify a quadratic criterion for
acceptable time domain measures such as: rise time,
over shoot and damping. Also, in addition, with direct
current motors, the non-linear and time-variant
behavior and the inaccurate knowledge of the plant
dynamics parameters create additional problems. These
difficulties must be taken into account when high
dynamic performance is required. Specific problems
associated with these systems are:
(i) The specific form of the main index of
performance required in practice.
(ii) The difficulty, in determining, for some
cases, the values of the dynamic
parameters for processes with known
43


structures, e.g., the squirrel cage
induction motor.
(iii) Large variations of plant parameters may
occur, e.g., changes in the moment of
inertia of the load, changes in the field
current, or changes in the gain of the
thyristor rectifiers.
To overcome the difficulties in applying optimal
quadratic control theory, where the shape of the time
response is pre-specified, use can be made of the
linear model following control approach (LMFC); as a
first approximation to the actual system. A block
diagram is shown in Fig. 3.1.
Fig 3.1 Linear Model Following Control System
44


From analysis it is usually possible, with a
direct current motor control system, to choose a model
corresponding to the optimal closed loop control system
case. The objective would be to use (LMFC) so as to
minimize the error and drive it to zero (if the error
is between the state of the model and the controlled
plant). In this case the (LMFC) would compensate for
the difficulties of the optimal quadratic control. To
overcome the difficulties related to the uncertainty of
the parameter values of the dynamic plant and their
variations, it would probably be necessary to use a
non-linear adaptive control system. The design and
analysis of an adaptive model following control system
for such cases has been a main objective of this
research.
Analysis has shown that an adaptive model
following control system will act as a natural solution
to such cases, and will maintain high performance. In
this thesis a general design method for an adaptive
model following control system (Fig. 3.2) using the
POPOV-LANDAU approach (based on hyperstability), is
employed. The adaptive scheme is synthesized on the
basis of a parameter adaptative approach, and the
implementation is realized for signal synthesis
adaptation. The adaptation mechanism operates by the
45


Fig. 3.2 Parallel AMFC System with Signal Synthesis
introduction of a supplementary signal which modifies
the output signal of a linear model following
regulator. This adaptive speed controller is a
particular application. The same system can also be
used for speed control of an a.c. drive as well as for
the adaptive control of many other parmeters
encountered in electrical drives. for example,
acceleration, armature voltage, and field current or
voltage.
46


3.2 Analysis of the D.C. Drive Dynamic Characteristics
The transfer function of the system of Fig. 3.1 is
Fig. 3.3 D.C. Motor Configuration
Gg Gg
Hp (s) = --------------------
(1 + ST) (1 + S4) T4
1
(1/T4) + [(T + r)/T4] s + s2
(3.1)
where g is the gain of the tachogenerator, G = k/f, ip=
J/f, k is the motor torque constant, J is the moment of
inertia as seen at the motor axle and f is the
friction.
The system is subject to three types of dynamic
parameter variations:
1. Variations of the load
2. Modification of the gain and the time
constant of the thyristor supply at low
f*.
47


H> <4 < $
firing angle, when discontinuous current
flows through the motor.
3. Modification of the field current In the
case of constant working conditions and
assuming that the motor is fed through the
thyristor rectifier by a triangular voltage
pulse of magnitude U, width A and with a
period T, the transfer function of the d.c.
motor, around a mean value [WQ UQ], is
A0)o(s) _k km
AU0(S) km +fR^
1
1-exp (-A./9)
l
l+Ys
(3.2)
where
km = torque and back-e.m. f constant
jli = j/f-mechanical time constant
Ri = armature resistance
4 = T/{W/6) + [ (T- )//*]} global time constant
6 = RiJ/(Km2 + fRI) time constant for the continuous
conduction mode
= motor speed
= motor voltage
= moment of inertia of the load
= friction constant of the load
In fact, when a thyristor bridge is used, the
transfer function between AW and the variations of the
conduction time of thyristor is
Ac(s) =k km________________________X/Q______________1
A^(s) km + fR (V0) + [(X X)/[i] l+\jfs
48


where K is a variable gain depending especially on the
magnitude of the a.c. supply voltage and on the
conduction mode of the thyristors.
3.3 The Adaptive Model-Following control System
Equations
If it is assumed that the PI controller for the
armature current is adjusted in such a way that at each
operating point, the response of the control loop can
be approximated by a first-order transfer function, and
since the parameters of the equivalent transfer
function depend on the operating point, the state
equation of the controlled plant can be written as
(with f = 0 and neglecting (the back e.m.f.)
Xp ApXp + BpUp
(3.4)
let
1 1
(3.5)
0 1/ T
0
(3.6)
Km A/Jr
49


(3.7)
Xp =
Xpl
Xp2
W
a
W is the motor speed
a = acceleration
In this case r, J, Km, A are parameters subject to
variations.
The state equation of the model is chosen as
xm Amxm + BmUm
(3.8)
Am -
0 1 1
o:Km A/Jr -1/r
(3.9)
Bm =
aKm A/Jr
(3.10)
xm -
xml wm
xm2 Am
(3.11)
Wm = desired speed
am = desired acceleration
The control input Up is equal to
Up = -KpXp + KuUm = [Upl Up2] (3.12)
50


In the equations, Xp and xm are n- dimensional
vectors, Um is the reference input (m-dimensional), Up
is the plant input (m^-dimensional) and Ap, Bp, Am, Bjj,
Kp, Ku are matrices of appropriate dimensions. The
pairs (Ap, Bp) and (Am, Bm) are assumed to be
stabilizable, and Am is a Hurwitz matrix.
The gain A is introduced in order to obtain a pre-
specified dynamic characteristic of the reference
model, which is used to represent the desired dynamics
of the speed control loop.
The following complete state equations of the
system are
xm = Amxm + BmUm (313)
xp = Apxp + Bpup
The PMF requirements are imposed on the error,
which is defined as
(D II r X t) (3.14)
and Up2 are defined as
II H Kpxp + KmXm + KuUm (3.15)
II OJ Kp (t,e) Xp + Ku(t, e) Um
51


It must be verified that the linear model
following control system Up2 = 0 meets the perfect
model following conditions that is
(I BpBp+)(Am Ap) =0 (3.16)
(I BpBp+JBjn = 0 (3.17)
where Bp+ is the Penrose pseudo-inverse of Bp:
Bp+ = (Bp+Bp)-1 Bp+ (3.18)
It can be shown that these conditions are
satisfied for the given problem independently of the
values of the coefficients of Am, Ap/ Bm, Bp.
Therefore the theory can be used to synthesize an
adaptive loop.
A linear compensator for the adaptive loop to
generate signal is next required. Let
V = De (3.19)
Then in the case of PI (proportional-integral)
adaptation, the adaptive laws are
Akp(t, e)
Aku(t,e)
LV(Qxp)T
MV(Rum)x
dT + LV (Qxp)x + Akp
dT + MV(Rum)T + Aku
(3.20)
(3.21)
52


The linear compensator is defined as
D = [di d2] (3.22)
and is computed for the condition that the linear part
of the equivalent feedback system corresponding to the
AMFC
Z(S) = D(SI - Am) Bp (3.23)
This should be a strictly positive real transfer
function. For a more effective computation using a
positive real lemma
D = BpTP (3.24)
where p is the solution of the Lypunov equation,
(Am BpKm)T P + P (Am BpKm) = -H (3.25)
and H is an arbitrary positive definite matrix. The
fact that the term Km A/Jt of Bp varies from a nominal
value used for P does not affect the positivity
properties. It can be written for every situation of
Bp = F Bp (3.26)
where F is the positive definite matrix
53


(3.27)
0 1 0 0
Km A/Jt i CM CM O i ^m A/Jt
when F is a positive definite matrix this does not
change the positivity properties. For large variations
of Bp it is possible that the matrix p solution for the
nominal point Bp will not ensure positively that the
matrix H is in the whole parameter space. In this case
Km = o is chosen and the linear part of the control
system is an implicit model-following control system.
However, the reference model must always be implemented
since it is a part of the adaptation mechanism. For
the nominal value of Bp and Ap, denoted by Bp and Ap it
follows that
Ku = (Bp)+ Bin = a (3.28)
and for Km = 0
Kp = (Bp) + (Ap Am) = | a, 01 (3.29)
In the adaptation laws
Q =
1 0
0 e
l>e>0 R = 1
L L > 0, L L > 0, M m > 0,
(3.30)
54



Lvxpl dx| xpl + LVX^ + e
+ ELVXpj +
(my umdx
i

LVxp2 dx
xp2
(3.31)
Um + myu|
The integral in the adaptation law insures that the
memory of the adaptive loop is maintained while the
proportional terms have a significant influence on the
speed of reduction of the state error equation (3.14)
The presence of the integrator in the control loop
also ensures that the difference between Wm and the
real speed is null.
3.4 Analysis and Simulation Results
The performance of an adaptive speed controller
design by the method previously discussed, has been
illustrated by applying the method to a D.C. motor.
The results are shown by a computer simulation example
(using Simnon Software).
The selected parameters of the reference model
are:
T = 10 msu A = 3 A/V Km = 2 Nm/A
a = 0.83 J =0.5 Kgm2
where t represents the mean value of the equivalent
time constant in the armature current control loop. A
55


is the gain of the same control loop. J is the mean
value of the moment of inertia of the load. Km is the
value of the torque constant for a nominal field
current. The experimental results are shown for the
case of moment of inertia variation from 0.21 to 0.78
Kgm2.
In this method, it should be noticed that a
comparison with other types of adaptive systems show
that the implementation of the adaptation laws does not
require a real time solution of a set of linear or non-
linear equations. That is this adaptation mechanism
developed in this thesis requires that only summers,
multipliers, and integrators are needed.
The structure of the plant with its adaptive
control system is shown in Fig. 3.4.
The high speed of the adaptation mechanism and the
insensitivity of the performance response of the
control system for the variation of the moment of
inertia are illustrated by the following cases.
56


57


Example 1
choose r = 8 x 10-3s
A = 2.5 A/V
Km = 2.2 Nm/A
J = 0.55 kgnr
those values for the
plant
fo 1 1 [c 1 1

0 1/T 0 -125
0 0
BD = =
P Km A/Jr 1250
BpT = [0 Km A/Jr]
[0 1250]
for the model
choose r = 10 x 103s
A = 3 A/V
% = 2 Nm/A
J = 0.5 kgnr
a = 0.83 *
*m
B.
m
0 1 0 1
-ttKm A/JT -1/r -996 -100
0 0
aKm A/Jt -996
1250
0
1250
-1
[
0
1250
0.0008
58


Ku = Bp
T Bm = 0 0.0008 J
0
996
= 0.7968
Kr
= BpT (Ap Am) [ 0 0.0008 j
0
1250
0 1
996 -100
= 0.7968 -0.02 j
K = 0.7968
nominal values
Kp = 0.7968 -0.02 j
In order to find the linear compensator matrix, a
matrix (Lyapunov) must be determined
(*m BpKm)T p + p (Au BpKm) = -H
where H is a positive definite matrix
suppose H =
1 0
0 1
0 -996 Pi P12 Pi P12
+
1 -100 P21 P22 P21 P22
0 1
-996 -100
59


-1 0
0 -1
-996 P2i ~996 P22
Pi -100 P2i P12 -100 P22
' -996 P2i Pi -100 P12
+
-996 P22 P21 100 P22

-1 0
0 -1
-1992 PX -100 P12 996 P22
Pi -100 P21 -996 P22 2 p12 "20 p22
-1 0
0 -1
PI P12 5.030 0.0005
P21 P22 0.0005 0.005
D = BpT P
5.030 0.0005
0.0005 0.005
0.0625 0.625 1
= |^ 0 1250 j
60


This example is simulated on a digital computer by
using the Software Simnon. The result is shown in Fig.
3.5 to Fig. 3.12.
61


1.1
Tima (sue)
I'lnnl ------- model
Fig. 3.5 Velocity Before Adaptation.'


Angular position (rad)
0.4
0.3 -
Time (see)
Fig. 3.6 Velocity Error


Angular pcsirion (ad)
Time (sec)
Fig. 3.7 Acceleration Before Adaptation.


Angular position (rad)
Fig. 3.8 Acceleration Error.


Fig. 3.9 Velocity After Adaptation


Angular position (rad)
0.02 -
0.015 -
Time (see)
Fig. 3.10 Velocity Error


sa
3
Fig. 3.11 Acceleration After Adaptation


(pE2) noQBod JzpSoy
69
Fig. 3.13 Acceleration Error


CHAPTER 4
CONCLUSIONS
In this thesis the development and testing of an
improved model reference adaptive control system for
single-input/single-output systems have been presented.
For the application of this controller it is not
required that the perfect model-following conditions be
satisfied. Therefore, the plant does not have to be
structurally similar to the reference model.
Furthermore, the improved adaptive algorithm is simpler
than the already existing adaptive schemes.
Despite its simplicity the improved MRAC system is
robust and it performs very well. Actually, as shown
from the simulation results presented in Chapter 2, the
improved MRAC system is as effective as any other
adaptive controller and can be used for more complex
adaptive algorithm. In addition, the simulation
results indicate that the improved MRAC is highly
robust to the variations of the plant dynamics.
Further research should be considered. First, the
effect of the state estimator dynamic on the overall
system performance has to be studied. Second, the


maximum allowable order-difference should be determined
fc
so that the resulting augmented system will have an A
matrix, which satisfies the required positivity
conditions. Finally, research should be extended to
include application of the improved MRAC system to
multi-input/multi-out systems.
71


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73