PLACE II RETURN BOX to remove this checkout from your record.
TO AVOID FINES Mum on or before date due.

   
  
 

DATE DUE 'DATE DUE DATE DUE

JL__

-___——j
[—1——
L__J
III I
1| II T

MSU I. An AMrmdivo Adm/Equal Opportunity Institution
cwmw

 

 

 

 

 

 

 

 

 

 

 

.JH

 

 

 

 

an91

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

THE DYNAMICS OF THE el—MODIFICATION LAW
AND A NEW ROBUST ADAPTIVE CONTROL
IN THE PRESENCE OF DISTURBANCES

by

Shi Bai

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering
1989

we run?

ABSTRACT

THE DYNAMICS OF THE e 1—MODIFICATION LAW
AND A NEW ROBUST ADAPTIVE CONTROL
IN THE PRESENCE OF DISTURBANCES

by

Shi Bai

This work has focused on stability and robustness of model reference adaptive con-

trol systems. There are two stages in our work.

In the first stage, we investigate the possible dynamics that a prototype model refer-
ence adaptive control system can experience when employing the so-called
e l—modification adaptation law. Limiting attention to first-order plants with a single
unknown pole and external disturbance, we verify analytically the bifurcations that an
example adaptive system has been shown to exhibit based on computer simulation. We
also construct the parameter-space or bifurcation diagram which summarizes all possible
behaviors of the class of these prototype systems. The analysis and the simulation results
demonstrate that in the presence of external disturbances certain instabilities due to
equilibria bifurcation, Hopf bifurcation, and/or complicated dynamics can be generated

in (model reference) adaptive control systems with this adaptation law.

In the second stage, we propose a new adaptation algorithm which is robust in the
sense that all signals in the adaptive loop are bounded, the tracking error converges to a
sufficiently small value and the tracking error will be zero in the absence of disturbances.
For general time-varying disturbances, the proposed new algorithm will adaptively iden-
tify the disturbances and counteract their effect so that the error between the output of the
plant and the output of the reference model is asymptotically reduced. We employ the

same standardized case study examples which were examined using various adaptive

control schemes by K. J. A srb°m, K. S. Narendra, I. D. Landau, G. C. Goodwin, A. S.
Morse, C. D. Johnson and other researchers in the literature. We compare our results with

their results. The comparison reveals various advantages of our algorithm.

ACKNOWLEDGMENTS

The auther would like to express sincere thanks to Fathi M. A. Salam for his
inspiration and guidance. Thanks to H. K. Khalil, R. Schlueter, and C. R. MacCluer for

their comments and suggestions.

‘iv

Table of Contents

LIST OF FIGURES ................................................................................................... vi
Chapter 1 Introduction ............................................................................................... 1
Chapter 2 The o—Modification Law: Simulation Evidence ..................................... 6
2.1 The adaptive system ....................................................................................... 7
2.2 Simulation results ........................................................................................... 9
2.3 Summary ......................................................................................................... 20
Chapter 3 The el—Modification Law: Analysis and Simulation 21
3.1 Adaptive control of a first-order plant ........................................................... 23
3.2 Local stability analysis ................................................................................... 27
3.3 Simulation results ........................................................................................... 30
3.4 Summary ......................................................................................................... 34
Chapter 4 The e l-Modification Law: Parameter Space Approach .......................... 39
4.1 A prototype adaptive system in the physical coordinates ............................. 41
4.2 A prototype adaptive system in the normalized coordinates ........................ 43
4.3 Local stability analysis of equilibria .............................................................. 46
4.4 The generation of limit cycles: proof of the Hopf bifurcation ..................... 49
4.5 Simulation results ........................................................................................... 53
4.6 Summary ......................................................................................................... 57
Chapter 5 A Robust Adaptive Controller For The Plant With Disturbances .......... 63
5.1 Adaptive control of a first-order plant ........................................................... 63
5.2 Adaptive control of a n th first-order plant .................................................. 72
5.3 Examples and simulations results .................................................................. 82
Chapter 6 Conclusions ............................................................................................... 112
List of References ...................................................................................................... 115

Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5
Figure 2.6
Figure 2.7
Figure 2.8
Figure 2.9
Figure 2.10
Figure 2.11
Figure 2.12
Figure 2.13
Figure 2.14
Figure 2.15
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 3.6

Figure 4.1
figure 4.2
Figure 4.3
figure 4.4
Figure 4.5

figure 4.6

LIST OF FIGURES

Three equilibria: one saddle and two stable foci. ................................. 10
A homoclinic orbit. ................................................................................. 11
An unstable limit cycle. .......................................................................... 12
Enlarged scaling of the unstable limit cycle. ........................................ 13
The shrinking of the unstable limit cycle. ............................................. 13
The weakening of the attractivity of the stable equilibrium. ................ 14
The appearance of the unstable focus. ................................................... l4
Confirmation of the unstable focus. ....................................................... 15
Strong repelling ....................................................................................... 15
Stronger repelling focus. ...................................................................... 16
Saddle-node bifurcation. ....................................................................... 16
Invariant curves at the fused saddle-node equilibrium. ...................... 17
single stable equilibrium. ...................................................................... 17
The effect of increasing 0 at d = (1* = 0.263363. ............................. 19
The effect of decreasing 0‘ at d = d* = 0.263363. ............................. 19
Value of d = 0.2. .................................................................................. 32
Value of d = 0.43794904. .................................................................... 32
Value of d = 1.0. .................................................................................. 32
Value of d = 1.5. .................................................................................. 33
Value of d = 2.0. .................................................................................. 33
Value of d = 2.5. .................................................................................. 33
0.4
4% = 1—6 ............................................................................................... 55
E = 03758908. .................................................................................. 55
v 16
{7‘- = T26" .............................................................................................. 55
3
% = 76° ............................................................................................... 56
4
13- = T6" .............................................................................................. 56
5
l:- = 1—6 ................................................................................................. 56

Figure 5.1 - 5.4 Output of the first example in the absence of

disturbance. ..................................................................................
Figure 5.5 - 5.8 Output error e. .............................................................................
figure 5.9 - 5.12 Control pararnenters. ...................................................................

Figure 5.13 - 5.16

Figure 5.17 - 5.20
Figure 5.21 - 5.24
Figure 5.25 - 5.28

Figure 5.29 - 5.32
Figure 5.33 - 5.37
Figure 5.38 - 5.41

Figure 5.42 - 5.45
Figure 5.46 - 5.50

Output of the first example in the presence of
disturbance. ...............................................................................

Output error e. .........................................................................
Control parameters. ..................................................................

Output of the second example in the absence of
disturbance. ...............................................................................

Output error e 1. ........................................................................
Control parameters. ..................................................................

Output of the second example in the presence of
disturbance. ...............................................................................

Output error e 1. ............................ r ...........................................
Control parameters. ..................................................................

vii

85
87
89

91
93
95

99
101
103

105
107
109

CHAPTER 1

INTRODUCTION

Adaptive control has been extensively studied since 1960’s and numerous suc-
cessful applications have been reported. In 1970’s the correct stability and convergence
proofs for several classes of adaptive schemes have been established under certain
ideal conditions. Soon after the stability proof was established, several types of insta-
bilities have been reported in adaptive control systems when one or more ideal condi-
tions are violated. Consequently, much of the research attention in the 1980’s has

focused on robust adaptive control.

In model reference adaptive control the assumptions have been proposed on the
transfer function of the modeled plant in deriving the (ideal) adaptation law. These

assumptions are given as follows:

Let H (s) = K %E% be the transfer function of the modeled plant, and assume

that the following information about H (s) is known.

0 The number of poles of H (s)

The number of zeros of H (s)

The sign of the high frequency gain K

H (s) is minimum phase

No disturbances nor unmodeled dynamics

Perhaps the most unrealistic among the ideal conditions are the assumptions that
there are no disturbances and unmodeled dynamics and that the transfer function of the
modeled part of the plant is minimum phase. Analysis and simulation show that when

an adaptive control system with the original (ideal) adaptation law is perturbed by

2

some disturbances, and/or there are unmodeled dynamics affecting the system, the fol-

lowing types of instabilities may arise in the system

0 Parameter Drift
Egardt [41], 1980; Peterson and Narendra [17], 1982;
Cyr, Riedle and Kokotovic [42], 1983

0 Linear Instability
Rohrs [43,19], 1981, 1982; Ioannou and Kokotovic [21,1], 1983, 1984

0 Fast Adaptation Instability
Ioannou and Kokotovic [21,1], 1983, 1984

0_ High Frequency Instability
Astrom [44], 1983

0 Throughput Instability
Khalil [16], 1981

Robustness of an adaptive system with respect to a class of perturbations has
become a measurement of satisfactory performance of the adaptation law used. To
achieve robust control of an adaptive control system in the presence of certain distur-
bances, many schemes have been proposed, e.g., the dead-zone approach [17], the
o—modification scheme [1], the e l—modification scheme [2], etc.. Using any of these
schemes, a bounded region in the state space can be described such that all of the
states of the adaptive control system will be within this region. However, it is some-
times difficult to quantitatively and practically determine the boundary of such a
region, and as a result, an estimate of the region’s boundary is usually very poor.
Moreover, one does not know the behavior of the overall system within the region and
thus one cannot track the state of the system once it is within the region. Furthermore,
recent reports have demonstrated that some of the available adaptation schemes may

produce multiple equilibria or chaos within the bounded region [3]-[7].

3

One desires to obtain (if possible) a new adaptation law which would

ensure that only one equilibrium point exists in the system

this equilibrium point is globally asymptotically stable when the system is

perturbation-free

when the system is perturbed, it ensures boundedness of all signals in the system

and convergence to zero or small output error .

We have focused our work on the stability and robustness issue of model refer-

ence adaptive control systems with linear time-invariant single-input single-output plant

in the presence of external disturbances.

(1)

(2)

The objectives of our work are:

To study and demonstrate that in the presence of external disturbances certain
instabilities due to equilibria bifurcation and Hopf bifurcation can be generated in
(model reference) adaptive conuol systems with the adaptation schemes used in

the current adaptive control literature.

To develop a new robust adaptive control algorithm for linear time invariant sin-
gle input single output plants of model reference adaptive control systems such
that the overall adaptive control system with this new adaptive control algorithm
will have only one equilibrium point, zero regulation error, and zero or

sufficiently small tracking error under bounded external disturbances.

There are two stages in our work.

In the first stage, we begin by an introduction to the phenomena exhibited by a

prototype adaptive control system employing the so-called o-modification law. We

have simulated the prototype system in the regulation case with the unknown distur-

bance as a varying parameter. The simulation shows that as the disturbance varies over

4

different values, the system may possess multiple equilibrium points. Saddle-node,

Hopf, and saddle-connection bifurcations can b e generated by the prototype system.

After presenting the prototype adaptive control system with the o—modification
adaptation law, we investigate the possible dynamics that a prototype model reference
adaptive control system can experience when employing the so-called e l-modification
adaptation law. Limiting attention to first-order plants with a single unknown pole and
external disturbance, we verify analytically the bifurcations that an example adaptive
system exhibits based on computer simulation. We also construct the parameter-space
or bifurcation diagram which summarizes all possible behaviors of the class of proto-
type systems under consideration. Our study reveals the qualitative similarities of the

o-modification and the e l—modification laws.

In the second stage of our work we propose a new adaptation algorithm which
ensures that the overall adaptive control system has only one asymptotically stable
equilibrium point. This new adaptation algorithm is robust in the sense that all signals
in the adaptation loop are bounded; and moreover the tracking state error will be zero
in the absence of disturbances. For general time-varying disturbances, the proposed
new algorithm will seek to adaptively identify the disturbances and counteract their
effect so that the error between the output of the plant and the output of the reference
model is (asymptotically) reduced We employ the same standarized test examples
which were proposed in [45] as case study examples to measure the effectiveness of
various adaptation laws. The examples were examined using various adaptive conu'ol
schemes by K. J. Asrb'm, K. S. Narendra, I. D. Landau, G. C. Goodwin, A. S. Morse,
C. D. Johnson and other researchers [8] - [13]. We compare our results with their

results. The comparison reveals various advantages of our algorithm [14] - [15].

This dissertation contains six chapters. In Chapter 1, we have introduced the

motivation of this research, current status of the research in Model Reference Adaptive

I! .n A I

5

Control, our objective in this research, and an overview of the accomplishments. In
Chapter 2, we present some computer simulations that completely characterize the pos-
sible dynamical behavior of a simple adaptive control system with the so-called
o—modification law in the presence of unknown bounded disturbances. In Chapter 3,
we present simulations and some local analyses that completely characterize all possi-
ble dynamical behavior of a prototype adaptive control system distinguished by the
so-called e l-modification law. In Chapter 4, we analyze the adaptive control system
given in chapter 3 by using the parameter space approach. In Chapter 5, we propose a
new adaptation law which is robust in the presence of unknown bounded disturbances.
Finally, in Chapter 6, summary and some final remarks are in order. We also identify a

number of research directions suggested by this dissertation.

CHAPTER 2

THE O-MODIFICATION LAW: SIMULATION EVIDENCE

This introductory chapter concerns some computer simulations that completely
characterize the possible dynamical behavior of a simple adaptive control system. The
system is taken from the literature on adaptive control systems, see Riedle, Cyr and
Kokotovic [18]. We have carefully examined the local stability of equilibria and have
identified invariant (stable and unstable) manifolds [34]. Consequently, we have been
able to characterize completely all possible behavior of the adaptive system.

The aim of this chapter is to describe the results without the potential distraction of
technical mathematical analyses. This chapter entails the simulation of the phase plane
portraits; hence, changes in stability or qualitative behavior are clearly displayed. We
explain the behavioral changes based on our knowledge of dynamical systems theory.
Moreover, relevant references are cited for the interested readers to check or verify any
further details.

In essence, a point that we have stressed here is that careful computer simulation,
guided by the understanding of basic concepts from dynamical systems theory, does
prove to be very useful in analyzing engineering problems.

In the system under consideration, three types of bifurcations have been identified:
(1) saddle-node bifurcation,
(2) (subcritical) Hopf bifurcation, and
(3) saddle connection bifurcation.

Prediction of the (local) bifurcations identified here can be achieved by available
mathematical tools and that is contained in [35,38,46]. However, the non-local and global
behavior (i.e. the patching of the local information together) can not be easily predicted
from the available mathematical tools in specific cases such as the system considered
here.

7

The saddle connection is traceable by a homoclinic orbit, i.e. an orbit that converges
to the same saddle point both in forward and backward times. This kind of orbit is
responsible for the formation of ‘horseshoe chaos’ when the system, in addition, is
periodically disturbed at the input or periodically excited by a reference signal. We have
established the occurrence of such behavior when either sinusoidal disturbance or
sinusoidal excitation is added, see [4]. Both kinds of sinusoidal signals are commonly
used in adaptive systems: the first models disturbances at the input [19], while the second
corresponds to a persistingly exciting reference signal [19] - [20].

Some implications of our results in the context of developing robust adaptive con-

u'ol systems (see [18] - [27]) are summarized as follows:

The example arises from a specific adaptive scheme and it is illustrated for specific
coefficient values here. However, it is strikingly simple. Thus, although there is no evi-
dence that other schemes exhibit similar exotic behavior, the question remains that
equally there is still no evidence that those other schemes do not generate such behavior
either. As long as the presence of only one globally asymptotic equilibrium point or
periodic orbit can not be ensured in a non-ideal adaptive control system, the existence of
exotic behavior can not be ruled out. Assuring the existence of a single (globally asymp-
totic) equilibrium point (or periodic orbit) in the non-ideal case is not yet likely to be
achievable in many of the existing adaptation laws. Consequently, recent tools of dynam-
ical systems (such as those of bifurcation analysis and the Melnikov method) may prove
to be instrumental in testing proposed schemes and/or in helping the process of devising
new ones. The usefulness of some of the recent methods of dynamical systems in the
design of robust control has been demonsuated. For instance, Melnikov’s method was
used in a cooperative fashion with Liapunov’s direct method to ‘quantify’ stability for
robust feedback stabilization control of the disturbed nonlinear pendulum (see [35] —
[36]). Further remarks are deferred to Section 2.3 at the end of this chapter.

2.1 The prototype adaptive system
Consider the simple adaptive control system:
model: yin: -y,,,+r, (2.1)

plant: 5:, =apyp+u+d, (2-2)

input: u = -kyp + r, (2.3)
adaptive law: k" = w, (y, - ym) - ok, 7 > o, o 2 o. (2.4)
The term (— 6k) in the adaptation law was introduced to counteract the presence of

the disturbance (d ) in the equation of the plant; see Riedle, Cyr and Kokotovic [18]. The
term ( - 0k) is referred to as the o-modification, see Ioannou and Kokotovic [21] - [22].

Now let (the error) e := yp — y", , to obtain the equivalent system:

é =ape-ke+d+(ap+l)ym-kym. (2.5)
k'= —ok +ye2+yey,,,, (2.6)
M=-M+h am

We are interested in the regulation problem, so as in [18], we will consider r = 0 and

0' > 0. The equilibria satisfy
k :0p + 5%, k = %e2, ym =0. (23)
As in [18], a ‘bifurcation’ expression for the number of solutions is

l
2.2
7]

Thus the equilibria are:

do=2

 

sili-

 

(i) for Id 1 > do, there is a single equilibrium,
(ii) for Id I = do, there are three equilibria, two of which are equal;
(iii) for Id l < do, there are three different equilibria.

In all simulations the initial point of the variable y," , i.e. y", (0), was always taken to

be zero; consequently ym (0) = 0 for all t. Thus the system we are actually considering is

even simpler:

é=ape-ke+d, (2.9)

k' = — O'k +ye2. (2.10)

This system is a new prototype system that exhibits fairly rich behavior for certain values

of the parameters ap , o, y, and (the forcing) d .

2.2 Simulation results

A fourth order Runge-Kutta integration routine is used in all the simulation work we
have conducted. The integration stepsize is always kept at less than 0.02 units of time.
We have considered the system (2.5) - (2.7) with y,,. (0) = 0, and have concentrated on the
invariant curves in the state space of e and k . Specifically, we have concentrated on the
stable and unstable manifolds of the saddle point when d < do. We have fixed the param-

eters as follows (recall that r = 0):
(1,, = 1.5, y: 1.0, 6:05.

Consequently, the critical value of d now reads

3

1135:] 7:05.

(1:23

 

Hence for Id I < 0.5, there exist three equilibria. The disturbance variable d has been

used as the varying parameter.

We now discuss the simulation results as the disturbance d increases from 0 to 0.6.
For bifurcation of equilibria, consult [28] - [29]; for Hopf bifurcation, consult [30] - [33];
for saddle connection bifurcation, consult [28] - [29]; and for periodic perturbation of
homoclinic orbits, consult [29], [34], [35].

10

At d = 0.2, Figure 2.1 depicts the phase portrait for ten initial conditions. It displays
three equilibrium points: one saddle and two stable equilibrium points. For six initial
points, the figure shows six trajectories converging to the ‘upper’ stable equilibrium
point. The other four initial points all emanate from very near the saddle point and thus
trace the stable and unstable manifolds of the saddle. The arrows indicate the direction of
the trajectories. Observe that the stable manifold of the saddle point engulfs its own
unstable manifold which in turn converges to the ‘lower’ stable equilibrium point. The
(local) stability properties of the equilibrium points remain the same for 0 S d < 0.35355.
However, the global qualitative behavior persists for 0 < d < 0.263363.

 

 

 

8
K-AXIS

 

Figure 2.1 Three equilibria: one saddle and two stable foci.

11

At d = 0.263363, Figure 2.2 depicts the phase portrait for the stable and unstable
manifolds of the saddle point. Branches of the stable and unstable manifolds are coin-
cident and engulf the lower stable equilibrium point, forming the so-called homoclinic
orbit. The homoclinic orbit defined a structure of considerable interest in dynamical sys-
tems since it precedes the rise of unstable manifolds of the saddle and has the property
that each point on the structure, evolving forward orbackward in time. converges to the
saddle point.

24

 

,i
l
I

l

 

 

Figure 2.2 A homoclinic orbit.

At d = 0.3, Figure 2.3 shows the saddle and the two stable equilibrium points. How-
ever a new a—limit set now appears: an (unstable) limit cycle. We have taken an initial
point inside the limit cycle and integrated the equations backward and forward in time.
Figure 2.4 shows the limit cycle in a blown-up scaling. The integration is taken for a
longer time period to verify the (asymptotic) stability of the equilibrium point.

12

At d = 0.35, the limit cycle decreases in amplitude and the attraction of the equili-
brium point inside it weakens (Figure 2.5). The weakening of the attraction is evidenced
by .the fairly long time a trajectory takes to approach the stable equilibrium point. Figure
2.6, in blown-up scale, shows a trajectory starting at (e (0), k(0)) = ( - 0.8, 1.0) and
evolving over 0 S t S 1400. The trajectory eventually approaches the equilibrium point.
We have observed that the limit cycle keeps on decreasing in amplitude while engulfing
the ’moving’ very weakly stable equilibrium point: this process evolves over the interval
0.263363 < d < 0.35355.

At d = 0.37, the limit cycle and the equilibrium point it is engulfing have fused into
one another forming a locally weakly unstable equilibrium point (which is a source). Fig-
ure 2.7 depicts a trajectory, evolving in reverse time, converging into the source. Figure 8
shows, in blown-up scale, a trajectory when integrated in reverse time eventually con-
verges to the source. The simulation in fact establishes the occurrence of subcritical Hopf
bifurcation very near d = 0.37. (It is analytically shown that at very near d = 0.35355 the

Hopf bifurcation occurs.)

e-AXlS

 

 

GB

 

u K-AXIS

Figure 2.3 An unstable limit cycle.

fifil-Q-I- m fi‘mm

e-RXIS

 

 

 

- ‘2 .-
,6-
-.eL
-l.4*
Figure 2.4 Enlarged scaling of the unstable limitcycle.
24.
10
'5
-‘ -5 2 ‘1 ‘2' 1 2 5 1
.11.

 

Figure 2.5 The shrinking of the unstable limit cycle.

e-RXIS

 

Figure 2.6 The weakening of the attractivity of the stable equilibrium.

 

 

5
e—AXlS
2
1
.g r 3 w 1 2 s 4 s
K-AXIS
-1

 

Figure 2.7 The appearance of the unstable focus.

e-RXIS

-.702+

- .704 h

 

n7“

 

.958 .56 .4132 .454 .556

Figure 2.8 Confirmation of the unstable focus.

e—AXIS

L
v

.968

5t

15

A

.617
K-flXlS

.9178

 

'5

at -:r

Figure 2.9 Strong repelling

-14?

 

 

 

 

 

 

 

 

Figure 2.11 Saddle-node bifurcation.

e—AXIS
2 r
1 «1
-5 e 3 3 '1 VII 4 75
\\ K-AXIS
a‘ 0
Figure 2.10 Stronger repelling focus.
51
e-AXIS
20
l 4,
r? '4 5 Wl 1 ‘ _FS
K-AXIS

l7

e—AXIS

 

 

 

 

 

 

5? 4 .1. ”— 4" 1 "é
K-AXIS
i
-11
I
Figure 2.12 Invariant curves at the fused saddle-node equilibrium
5t
e-AXlS
2i /H \
, \,\\
\»
l
r.— 4 -3 a ‘1‘ m . ‘1

 

 

Figure 2.13 single stable equilibrium.

- "' “’..-‘fl

iu-Sh‘ "- v‘

IF -‘l‘n‘ll‘
I _ .

18

Increasing d to 0.4, then 0.45 strengthens the repelling of the unstable point (the
source) as depicted in Figures 2.9 and 2.10 respectively. The newly created source and
the saddle continue to approach one another until, at the critical value of d = do = 0.5,
they fuse into each other forming saddle-node bifurcation. Figure 11 displays the mani-
folds connected to the now fused equilibrium. The system equations now must possess
a zero eigenvalue of the linearization at this fused equilibrium point [28] - [29]. Figure

2.12 depicts two more trajectories near the fused equilibrium point.

When d > 0.5, the two fused equilibria annihilate one another and the system
now possesses a single stable equilibrium point above the k-axis. This equilibrium
point maintains its asymptotic stability for all values of d > 0.5. Figure 2.13 shows
some trajectories converging to the single stable equilibrium at d = 0.6. However,
notice the peculiar shape of the trajectories. At the beginning, all of the trajectories are
squeezed into one curve. Then, just before crossing the e-axis, the trajectories diverge
from one another quickly. It may seem that, as a consequence, the trajectories violate
the property of the uniqueness of solutions of differential equations. However, that is
not the case; a higher resolution would certainly validate the uniqueness property. We
(remark that a similar peculiar behavior is known to manifest itself in Lerentz’s equa-

tions.

Alternatively, it may be convenient to consider the changes in the phase portraits
as one decreases d from, say, 0.6. At 0.5 bifurcation gives birth to two equilibria: one
a saddle and the other a source. Then, near 0.35355, the source gives birth to a stable
equilibrium surrounded by a (repelling) limit cycle. The limit cycle grows until at
0.263363 the limit cycle acquires an infinite period and touches the saddle point, thus
becoming a homoclinic orbit. For 0 < d < 0.263363, the ‘lower’ stable equilibrium
point has a basin of attraction defined by the stable manifold of the saddle, while the
‘upper’ stable equilibrium point attracts all other points outside this basin of attraction
(and off the stable manifold of the saddle).

Suppose now that one fixes d and varies 0 instead. Figure 14 and 15 show that
changing o has qualitatively the opposite effect of changing d, where d is at
0.263363. That is, increasing (decreasing) o from 0.5 to 0.6 (0.4) is qualitatively the
same as decreasing (increasing) d from 0.263363 to 0.2 (0.35).

e-AXIS

 

 

 

-5 -4 -5 -z -r :j 1 2 5 4
\ K-AXIS

 

Figure 2.14 The effect of increasing a at d = d* = 0.263363.

 

 

5
ms ]
24
‘1
Hal. \1 r r T
-6-4-5-2-1*‘ 2.545
K-AXIS
7'1
—21

Figure 2.15 The effect of decreasing 0‘ at d = d* = 0.263363.

20

2.3 Summary

We summarize further conclusions in the following:

1. The system exhibits rich dynamics and structures. Specifically, as the disturbance
parameter d is decreased from 0.6 to 0, the system consecutively experiences
saddle-node, Hopf, and saddle connection bifurcations. At d = d* = 0.263363, a
homoclinic orbit exists.

2. We observe that over 0 < d < 0.5, one equilibrium is saddle-type while another is
locally stable. The simulation results however illustrate that the ‘lower’ equilibrium
point, i.e. the one with the lowest value of its e-component, changes stability. For
0 S d < 0.35, this equilibrium point is stable; however, for 0.35 < d < 0.5 ( := do),

it is unstable. '

3. Theorem 1 in [22] implies that the solution e (t), k (t) converges exponentially to a
bounded set for all bounded initial conditions. Certainly all the bifurcations shown
in this chapter must occur within this set. It remains to be seen how well the criteria
for characterizing this bounded set can ‘tightly’ estimate the region where the bifur-
cation action takes place. We propose performing calculations for the same numeri-

cal values in our example as a starting step.

4. The presence of the homoclinic orbit implies the possibility that the addition of dis-
turbance, which is a small sinusoidal function of time, could lead to the generation
of a (transversal) homoclinic orbit, see [29], [34]. Such an occurrence implies the
existence of horseshoes in the dynamics of the system, thus giving rise to the so-
called horseshoe chaos [29], [34]. Indeed we have demonstrated that horseshoes do
arise when d = d* + esin(cot ); the results are reported in [35].

The remarks mentioned above, we hope, suggest that non-local considerations in the
design of adaptive schemes are essential. That is, until there are clear ways in construct-
ing adaptive schemes for non-ideal systems (for example, systems under the influence of
disturbances) so that the overall adaptive system possesses a single equilibrium point,
which is guaranteed to be globally asymptotically stable, we perhaps should be inclined

to utilize some useful new tools for nonlinear analysis.

CHAPTER 3

THE e l—MODIFICATION LAW: ANALYSIS AND SIMULATION

The so-called e l-modification adaptation law has recently been pr0posed by N aren-
dra and Annaswamy [2] for model reference adaptive control systems in the presence of
bounded external disturbances and unmodeled dynamics. It has been shown to improve
certain characteristics of model reference adaptive control systems while ensuring the
usual boundedness of all solutions, see [2] and the references therein. The
el-modification law is intuitively similar to the dead-zone approach (see section 3.4),
assuming that the (output) error converges to zero. When external disturbance is present,
however, the (output) error does not converge to zero; consequently the similarity is
deceiving and the dynamics generated in the corresponding overall adaptive systems are
qualitatively different. More details are appropriately deferred to section 3.4 after the

pertinent results are presented.

Some assertions have been made in [2] regarding some advantages of the
e l—modification law over the o—modification law [18], [22]. The research work con-
tained in this chapter reveals that, in the regulation case, the two laws are in fact qualita-
tively equivalent in the sense that the overall adaptive system under both laws may pos-

sess multiple equilibria and oscillations.1

Suppose that we are interested in a detailed study of the performance of the
el-modification law when applied to a first order plant with an unknown pole and

bounded external disturbance. The disturbance has a constant average value, say d, and

 

1 See Chapter 2 for similar dynamics generated in the prototype adaptive system using the
O-modification law.

21

22
small additive fluctuating term, say 8(t ). That is, t he external disturbance has the form:
External disturbance = d + 8(t ),

where l5(t)| << 1.

In this work, we limit attention to the case when the fluctuation, 5(t), equals zero.
Consequently, we focus on determining the properties of the whole adaptive system in
the presence of the constant disturbance d. Available methods in analysis can subse-
quently be used to determine the corresponding properties of the whole system when the
fluctuating disturbance 8(t) at 0, see [28], [30] - [32], [37]. In what follows we show that
all necessarily bounded solutions can in fact converge to or diverge from every possible
limit set in the two dimensional plane, namely: an equilibrium point, a limit cycle or a
homoclinic orbit! The type of limit set present in the system is dictated by the relative
values of all (constant) parameters, i.e., the constant unknown disturbance, the amplitude
of el-modification term, etc.. Specifically, we present simulations and some local ana-
lyses that completely characterize all possible dynamical behavior of the prototype adap-
tive control system distinguished by the el-modification law. We have determined the
number, the location, and the local stability of the equilibria; we have also derived expli-
cit expressions for saddle-node and Hopf bifurcations. Based on our simulations and the

local analyses, three types of bifurcations have been identified:

(1) saddle-node bifurcation
(2) (subcritical) Hopf bifurcation

(3) saddle-connection bifurcation.

The first bifurcation occurs when a single degenerate equilibrium point "bifurcates" into
two equilibrium points: one is a saddle-type point, and the other is an unstable node. The
second bifurcation subsequently occurs when the unstable node " bifurcates" into a stable

node surrounded by an unstable limit cycle. Finally, the third bifurcation occurs when the

23
"enlarging" limit cycle eventually touches the still present saddle point forming the so-
called homoclinic orbit; such an orbit is characterized by the property that it converges to
a saddle point as time evolves both forward and backward, see [28]. These three bifurca-
tions occur consecutively as the (external disturbance) parameter is varied as will be ela-

borated upon in section 3.3.

3.1 Adaptive Control of a First-Order Plant

In the prototype adaptive control system, the (first-order) plant is described by the
scalar differential equation (see [2], [18] and Chapter 1)

x',, =apxp +u +d, (3.1)

where up is an unknown constant, d is an unknown disturbance, and u is the conu'ol.

Assume the disturbance to be constant. The reference model is described by the equation
Jim = - amxm + r , (3.2)

where a", > 0 and r is a bounded piecewise continuous function of t. The adaptive con-

troller is usually chosen in the standard form as

where the feedback control parameter 0 is adjusted via an adaptation law that uses the
available input-output data.

Now let the state (or output) error be
el = xp - x", ,
then the adaptation law proposed in [2] reads as:
0=-e1xp-yle1|0,y>0. (3.4)

The term ( -'yle1|0 ) has been proposed and referred to as the el-modification scheme,

see Narendra and Annaswamy [2].

"I Q

24

Using equations (3.1) - (3.4), one derives the equivalent system:

é1=ape1+0e1+d+9xm+(am+ap)x,,,, (3.5)
0=—e12-y|e1|9—e1xm, (3.6)
it”, = -a,,,x,,, +r. (3.7)

Now consider the regulation problem, i.e., let r = 0. The equilibria then must satisfy

0=ape1+0e1+d, (3.8i)

O=-e12—yle1|9, (3.8ii)

O =xm. , (3.8iii)
Write

I e1 1 = e lsgn(e1), (39)

where sgn is the sign function. Then using (3.8i), sgn (el) and (3.8ii), one obtains the

quadratic polynomial in el
elz-ysgn(e1)[ape1+d] =0. (3-10)

Thus, one obtains an explicit expression for the e l—component of the equilibria as

 

4438" (e 1)
_2__

(3.11)
Yap

e1=-%-yapsgn(e1) 1:1: 1+
From Eqn.(3.11), one may write a ‘bifurcation’ expression for the number of equilibria as
do = iiyapz (3.12)

Thus,

(1) for Id 1 > do, there is a single equilibrium point, namely,

 

 

 

 

 

_‘Yap 1+ 1+ 4‘12 , d > do
_ 4 " t Yap 1
R”: 1 1 M 1
— ‘—Y(1p 1+ 1- 2 , d < — do
L t Yap J

[fi=-%-Efi.

(2) at Idl = do, there are three equilibria, two of which are equal, namely,

anwmnd=do

13—111=%Yap(1+ ‘5).

_ 1
[81123 = - ‘2'761 1

d
[3?— 1] 1.2.3

 

{-611.23 = — 0p ‘

where [2]],- denotes the el component of the i th equilibrium point.

(2ii) when d = — do

[3111 = — é—vapu + 45).

_ 1
[61123 = 370 .

d
[51111.23

 

[5112.3 = ‘ 0p ‘

(3) for Idl < do, there are three distinct equilibria, namely,
(3i) when 0 S d < do

:1 1
1:11 171

 

 

- 1 4d
[3113: “770p[1+1\/1-W].

(1

[611.23 = — 0p - mo

(3ii) when - do < d _<. 0

 

 

r 1
— =_,1 V_ 4d
[3111 770p. 1+ 1" W J r
— 1 4d
[e]= a 1- \/1+ ,
12 2'ka W
1 F V—_ 4d
1? = a 1+ 1+ ,
[113 77p» W:

‘ d
“’11” = ‘ “P ‘ 171m?

Thus the equilibria are:

(1) for Id l > d 0. there is a single equilibrium point.
(2) at Id | = do, there are three equilibria, two of which are equal.
(3) for Id l < do, there are three distinct equilibria.

There is always one, and only one, equilibrium point with positive (respectively, nega-

tive) e1 component for all positive (respectively, negative) values of d .

Remark 3.1: The o—modification adaptation law is given by
O = - e 1Xp - 09,

where o is a positive constant, i.e., o > 0. The new modified adaptation law, proposed in

[1], basically replaces a by yl e1 | .

27
3.2. Local Stability Analysis

As it is often the case, see [18] and Chapter 1, the initial point of the variable xm,
i.e. x,,, (0), shall be set to zero; consequently xm(t) = 0 for all t. In essence, we are con-

sidering the simpler system:
é1=ape1+6e1+d, (3.13)
0=-e12—y|e1l9. (3.14)

This system captures the essential dynamic features of the adaptive control system (3.1)-
(3.4). The dynamic features are dependent on the relative values that (the system parame-
ter) op , (the adaptive control parameter) 7, and (the external disturbance parameter) (I
will take. Expression (3.12) exhibits the dependency of the number of equilibria on the

relative values of these parameters.

Now we analyze the stability of each equilibrium point. The linearized system at

any equilibrium point reads

éil ' _ a +6 _ E1 8:1
[0] - [ - 251 £758"(?1)9 -’Ysgn(e‘1)e’1] [a] . (3.15)

Remark 3.2: The right-hand side of (3.14) is continuous; however, it is not differentiable
at el = 0 because of the absolute value of e1 term. Thus the linearization (3.15) can be
considered to be defined everywhere except when e1: 0. Observe however that the
linearization is evaluated at the equilibrium points only, and none of the el components
of these points equals zero. This justifies taking derivatives in evaluating the lineariza-

tion.
Let us denote the determinant of the Jacobian matrix by det. Then,

det = 2312 - apysgn(é'1)t?1

28
= 2211?. - ol—vap sgn (2111.
Let us also denote the trace of the Jacobian matrix by Trace. Then,
Trace = 0,, + 5 — ysgn(?1)e'1

= — 21—1-14 + vsgn (6116121.

where we used equality (3.8i). To ease the presentation, the following discussion only
considers the case d > 0; a similar conclusion follows for d < 0.

(i) For d > do (There exists a single equilibrium point.)

det > 0,
Trace < 0,

so the two eigenvalues both have negative real parts. Thus the equilibrium point is
(locally) stable.
(ii) For d = do (There exist two distinct equilibrium points.)

First,
{8-111 = é—Yapu + ‘5).
det > 0,

Trace < 0,

so the two eigenvalues have negative real parts. Thus this equilibrium point is (locally)
stable.

Now consider the coincident two equilibria. One gets
- _ 1
[3112.3 - "' 'Z‘Yap 9
det = 0,

Trace = é-ap (l - 72),

 

29

so there is a zero eigenvalue. Thus the coincident equilibria are degenerate.

(iii) For (1 < do (There are three distinct equilibria.)

First,

det > 0,
Trace < 0,

so that the two eigenvalues have negative real parts. Thus, this equilibrium point is
(locally) stable.

The second equilibrium point gives,

1 — 1 4d
[3112: ‘770p[1‘\/1-W]r

det < 0,
Trace may be zero, positive, or negative ,

so one eigenvalue is positive and the other is negative. Thus ([2 1]2,[§]2) is a saddle

point.

The third equilibrium gives,
16113=—3zvap[1+\/1-7:§,].
det>0,

Trace=0 if
d- 1730 2[1+\/1- 4"12:0.
4 P 7.37

Using the last equality, we write a ‘Hopf bifurcation’ expression for the critical stability

of this equilibrium point as

3O

_ a 7'13

Consequently, we have

Trace < 0, for d < d” , and thus this equilibrium point is stable
Trace = 0, for d = d” , and thus this equilibrium point is critical

Trace > 0, for d > d” , and thus this equilibrium point is unstable

Observethatatd=dH,wealsohave W1 I durfiO.

One immediate conclusion is that the stability has changed at this equilibrium point
when the value of d passes through the value of d”. In addition a Hopf bifurcation takes
place at d = d”. In fact further analysis analogous to [38] reveals that it is a subcritical
Hopf bifurcation. (Recall that at d = do a saddle-node bifurcation occurs also.)

3.3. Simulation Results

We have considered the system (3.5)-(3.7) with x", (O) = 0, in other words, we have
considered the simpler two-dimensional system (3.13)-(3.14), and have concentrated on
the invariant curves in the space of el and 0. Specifically, we have concentrated on the
stable and unstable manifolds of the saddle point when Idl < do. The parameters have

been fixed as follows (recall that r = 0):
ap = 4.0 , y: 0.5

Therefore, the equilibria-bifurcation value of d now reads
do = zl-yap 2 = 2.0

Hence for Idl < 2.0, there exist three distinct equilibria. We have chosen the disturbance
variable d as the varying parameter. For simplicity of the presentation we only discuss

the simulation results as the disturbance d increases from 0 to 2.5.

31

At d = 0.2, Figure 1 displays three equilibrium points: two stable equilibrium points
and one saddle lying near (e1, 9) = (- 0.05132,- 0.10263). The four initial points are
located along the two eigenvectors of the linearization of the two dimensional system at
the saddle point evolving forward or backward in time and thus trace the stable and
unstable manifolds of the saddle. The arrows indicate the direction of the trajectories.
Observe that the unstable manifold of the saddle is engulfed by its own stable manifold
which in turn converges to the ‘left’ stable equilibrium point. The (local) stability proper-
ties of the equilibrium points remain the same for 0 S d < 1.28, however, the global qual-
itative behavior persists for only 0 < d < 0.43794904.

At d = 0.43794904, Figure 2 shows that the branches of the stable and unstable mani-
folds are coincident and engulf the ‘left’ stable equilibrium point, forming the so-called
homoclinic orbit. The homoclinic orbit defines a structure of considerable interest in
dynamical systems since it precedes the rise of ‘horseshoe chaos’ in periodically dis-
turbed systems. This orbit is the coincidence of the stable and unstable manifolds of the
saddle and has the property that each point on the structure would converge to the saddle
point when evolving forward or backward in time.

At d = 1.0, Figure 3 shows that a new a—limit set now appears: an (unstable) limit
cycle. The equilibrium point inside the limit cycle is (locally) asymptotically stable. As d
increases the limit cycle decreases in amplitude and the attraction of the equilibrium
point inside it weakens.

At d = 1.5, Figure 4 shows the limit cycle and the equilibrium point it is engulfing
have fused into one another forming a locally unstable equilibrium point (which is a
source).

The newly created source and the saddle now continue to approach one another and at
d = do = 2.0 they fuse into each other forming saddle-node bifurcation. Figure 5 shows
the manifolds connected to the now fused equilibrium. The linearized system now

possesses a zero eigenvalue at this fused equilibrium point [28].

32

6-1
5.:

4-1

 

 

 

 

 

e1

0‘

 

 

 

-4.

2:1

 

 

Fig. 2 value of d = 0.43794904

 

  

 

Fig. 3 value ofd= 1.0

 

 

 

 

 

 

 

l I
-5 —4 2 3 3. 5 e1
—5
0 -6J
Fig. 4 value ofd = 15
0
1.:
15 I4 :3 r2 I1 I r ‘1 e1
_ ._ 2 I
.44 3 4 s
-3-
-4-
-5;
-61
Fig. 5 value of d = 2.0
6
.1
3.1
2-1
1..
.15 J4 ~13 -12 1 el

-44
-54

-81

Fig. 6 -value of d = 2.5

 

  

  

34
When d > 2.0, the two fused equilibria wipe out one another and the system now
possesses a single stable equilibrium point with positive e1 component. The (global)
asymptotic stability of this equilibrium point will remain the same for all values of d >
2.0. Figure 6 shows some trajectories converging to the single stable equilibrium at d =
2.5. Varying 7 while fixing d qualitatively has the opposite effect. The results are qualita-
tively equivalent to those reported in [3]. The interested reader is referred to [3] for a

qualitatively similar system.

3.4. Summary

The dead-zone approach, see [17],[39], has been shown to be successful in over-
coming external disturbances and unmodeled dynamics. The intuitive idea behind its
operation is that one uses the modified adaptation law when the state of the overall sys-
tem lies outside a prespecified bounded region in the state space. Once the state enters the
prespecified region the adaptation law switches to the nominal adaptation law or stops
the adaptation process. Intuitively, the e l-modification attempts to recreate such a
scenario. Assuming that e1 goes to zero, the contribution of the additive term - y le1|0
would eventually vanish, as it is the case in the dead-zone approach. That is, the
modification or correction factor (—7 lelle) is available when e1¢0; as e1 —) 0 the
contribution of the correction term diminishes. Consequently, it resembles the dead-zone
approach in that respect.

In the example we have studied, e1 does not converge to 0; it converges to a nonzero
constant, however. Consequently the effects of (- y Ie 1 I 0) is qualitatively the same as
the effect of (- 0'9). In our opinion, this provides the best intuitive explanation for the
qualitative similarity in the dynamics contributed by the o—modification and
e l—modification, respectively.

We summarize further conclusions in the following:

35
The system exhibits rich dynamics and structures. Specifically, as the disturbance
parameter d is decreased from 2.5 to 0, the system consecutively experiences
saddle-node, Hopf, and saddle connection bifurcations. At d = 0.43794904, a

homoclinic orbit exists.

The simulation results however illustrate that the ‘left’ equilibrium point changes
stability. For 0 S d < 1.28 (= d” ), this equilibrium point is stable; however, for
1.28 < d < 2.0 (= do), it is unstable.

The presence of the homoclinic orbit implies the possibility that the addition of dis-
turbance, which is a small sinusoidal function of time, could lead to the generation
of a (transversal) homoclinic orbit. Such an occurrence implies the existence of hor-
seshoes in the dynamics of the system, thus giving rise to the so-called horseshoe

chaos.

Indeed, the e l—modification scheme exhibits qualitatively the same rich dynamics
as the o—modification scheme, as shown in [3]. So, in as far as dealing with constant

disturbances, both schemes perform similarly.

 

 

 

 

Appendix A
The equilibria are:
(l) for I d l > do, there is a single equilibrium point, namely,
.11., 1.411.344, , 1):,
[311:1 1 : 4d :
1fi=—e d

-T1?T'

(2) at l d l = do, there are three equilibria, two of which are equal, namely,
(2i) when d = do

13111= 3mm +61.

d

12112.3 = - 1312,.
[6112.3 = _ ap '-
(2ii) when d = - do

1211:: - lzyapil +151.

16112.3 = -

(3) for | d l < do, there are three distinct equilibria, namely,

(31) whenOSd < do
- _ 1
[8111- 770p
- _ 1
[3112— " 770p:

[5113 = _ 21-70;):

16112.3 = - a. -

(3ii) when - do < d S 0
— _ 1
[3111- - 770p-
— _ 1
[31h " TWpT

[3113 =

16111.: = -a,. -

ap-

 

 

31.77"?

d

1+

1+

\l—
-—\/—
\j—

”.1—
1__'—\/

d

[5111.23 1

[5111.23 °

1

l-

1—

[3111.23 '

1-

1

[5111.23 °

+

+

+
FJg F Jr

36

F Jr Fula'

‘éJfil

:flal

 

 

37

Appendix B

(i) For (1 > do (There exists a single equilibrium point only.)

 

 

det=-;-y2ap2 1+ 4d [1+ 1+ 4d ] >0.

2 2
Yap Yap

2d

_iyap[1+ 1+ 4‘12] <0,
Yap [1+\/1+ 4d ] 2 Yap

2
Yap

 

Trace = —

 

so the two eigenvalues are both negative. Thus the equilibrium point is (locally)
stable.

(ii) For d = do (There exist two distinct equilibrium points.) First,

[€111 = é—vapu + 12),

det = 72a],2 [1 + :12- ]

 

2
2d 1
Trace=— —- a 1+\/§,
yap(l+\/2) 272” )

so the two eigenvalues are negative. Thus this equilibrium point is (locally) stable.

Now considering the coincident two equilibria, one has
1?] - — iva
1 2.3 2 ’
det = 0,
T - 1 1 72
race - -2-ap( - ),

hence, there is a zero eigenvalue. Thus the coincident equilibria are degenerate.
(iii) For d < do (There are three distinct equilibria.)

First,

 

1 4d
('3' = —- a 1+ 1+ ,

38

det = iyzapzfi I 1 + 4‘12 [1+ 1+ 4‘12 J,
2 7a,, 7a,,
2d 1
_ .572
4d
yap [I + \fl - 2 ]
Yap

so that the two eigenvalues are negative. Thus, this equilibrium point is (locally)

 

 

 

Trace = —

 

stable.

The second equilibrium point gives,

_ _ l ’ 4d

[6112— 113761;,[1— 1- Yapz J,

det= —-;-yzap2r\/1— M2 [l-fi/l- M2 ,

Trace: * 2d -iyzap 1- 1— M2 ],
[1‘\[" 4" 1 2 ‘

a
Y p Yapz

 

 

 

 

 

 

so one eigenvalue is positive and another is negative. Thus ([e‘1]2,[0]2) is a saddle

point.
The third equilibrium gives,

_. 1 I 4d
[e] =——a 1+ 1— ,
l3 2Yp[ 70102]
det=-1—y2ap2*\/1- M2 [1+‘\/1- M2 ],
2 tap tap

Trace =

 

 

 

 

 

CHAPTER 4

THE e l-MODIFICATION LAW: THE PARAMETER SPACE APPROACH

As discussed in chapter 2 and chapter 3, the two laws, namely the so-called e 1— and
o-modification laws [2, 22] are in fact qualitatively equivalent in the sense that the
overall adaptive system under both laws may possess multiple equilibria, oscillations, or

even chaotic dynamics.1

We focus our attention on the so-called e l-modification scheme. We are particu-
larly interested in its performance in the face of external disturbances. We are concerned
with characterizing all its qualitative features-both stabilizing and destabilizing. The rela-
tive values of the constant parameters are very crucial in securing a structurally stable

adaptive control strategy.

The justification of such a view is that (the sufficient) conditions for ensuring the
stability features are often conservative. Moreover, one does not know what combination
of values of constant parameters, which satisfies the sufficient conditions, would result in

the "most" robustdesign.

One knows that the e l-modification scheme is associated with the existence of a
bounded attracting region in the state space, see Narendra and Annaswamy [2]. Some
viable questions are then, "When would this attracting region contain a single limit set,
various limit sets, or infinite limit sets? When would the limit sets entail single or multi-

ple equilibria, periodic orbits, and/or nonperiodic ones?".

 

1 See [3, 38] for similar dynamics generated in the prototype adaptive system using the
6-modification law.

39

 

40
For an adaptive control system with more than two parameters, the stability analysis
is dependent on the variation of parameters. In order to determine the effect on the
overall adaptive control system by the change of one parameter, all other parameters
have to be fixed. The procedure of determining what combination of parameters would
make the system give the desired or the "most" robust performance often becomes tedi-

ous and sometimes impossible.

Parameter space analysis and design reveal its advantages when there are few
parameters in the system. The parameter space approach could separate regions of

parameters which give qualitatively unchanged dynamics.

In this chapter we take a prototype adaptive control system with the
e l-modification law in the presence of bounded external disturbances as an example to
demonstrate the parameter space approach for analysis and design. We show that the
number of parameters can be reduced from four to two essential parameters. We also
present explicit expression for boundaries of qualitatively different regions. In the
parameter space we are able to prove the generation of limit cycles and their stability for

the example adaptive control system.

In particular, we have established that saddle-node, Hopf, and saddle-connection
bifurcations are all exhibited by this system and they are the only kind of bifurcations

present.
We have concentrated on the space of parameters and have characterized the

regions in the parameter space that correspond to qualitatively different phase portraits of

the example system. Curves separating these regions are the bifurcation curves.

From a design point of view, the dividing (or bifurcation) curves in the parameter

space are important for the following reasons.

(1) They form the boundary of the regions of qualitatively different dynamics. Hence,

they characterize and define each region.

'62.

 

41
(2) Knowing the phase portrait on a bifurcation curve identifies the phase poru'aits of

the regions on both sides of the curve.

(3) The bifurcation curves are important in their own right. They represent the location

in the parameter space where new phenomena may arise.

In fact, one can determine the regions in the parameter space where global stability
of a single equilibrium point is ensured. The advantage of using the parameter space is
that one can now select the values of the original parameters so as to ensure the follow-
ing.

(1) a single equilibrium point which is globally asymptotically stable.

(2) the e l-component of the equilibrium point is as small as possible. This amounts to
ensuring the error between the output of the plant and that of the desired model to
be as small as possible. In addition, one has full knowledge of the dynamics of the

system for every parameter range.

More inrportantly, one can easily identify the points for certain design by selecting
those points thatare securely located within a desirable region in the parameter space.
4.1 A Prototype Adaptive System in The Physical Coordinates

A prototype adaptive control system is described by the scalar differential equations
(see [2]. [3] and [18])

16,, =apxp +u +d, (4.1)
in = - amxm + r , (4.2)
where a, is an unknown constant, d is an unknown disturbance, and u is the control.

Assume the disturbance to be constant. a... > 0 and r is a bounded piecewise continuous

function.

-I._L| ‘- .

 

42

The adaptive controller is usually chosen in the standard form as
u = r + 01,, , (4.3)

where the feedback control parameter 0 is adjusted via an adaptation law that uses the

available input-output data.

Now let the state (or output) error be
e 1 = x, — xm .
Then the adaptation law pr0posed in [2] reads as:
0=—aoelxp-yle1|0,y>0. _ (4.4)

The term (-'y| e 1 | 0) has been proposed and referred to as the el-modification scheme,

see Narendra and Annaswamy [2].

_ Using equations (4.1) - (4.4), one derives the equivalent system:

é1=ape1+0e1+d+0xm+(am+ap)x,,,, (4.5)
0=—e12-y|e1|0—e1xm, (46)
in. = -a,,,x,,, +r. (47)

Remark 4.1: The o—modification adaptation law is given by

where o is a positive constant, i.e., o > 0. The new modified adaptation law, proposed in

[2], basically replaces 0‘ by 'yl e; I .

 

”l- V.) (fit ‘71”: I. Vifhf"

43
Since the equation (4.7) is decoupled from (4.5) and (4.6), also we are considering
the regulation problem, i.e., let r = 0. As it is often the case, see [18] and [22], the initial
point of the variable x". , i.e. x... (0), shall be set to zero; consequently x”, (t) = 0 for all t.

In essence, we are considering the simpler system:

é1=ape1+0e1+d, (4.8)

=-096?-Y|€1|e. (4.9)

There are four parameters in the equivalent system equations (4.8) and (4.9).

The analysis of the system (4.8) - (4.9) with d varying and fixed a,, do and 7 has
been done and reported in [5].

4.2 A Prototype Adaptive System in The Normalized Coordinates

The system equations are given by (4.8)«(4.9). We normalize the above system

equations by sealing the states of the system

x;=[“ 11:1, (4.10)

,_ 1

We also reset the time scale
t’:=a,,t. (4.12)

The parameters in the normalized coordinates are defined as

' a0
:= d2, (4.13)
1 [a]

 

v; = ['712L] y, (4.14)

Thus, the system (4.8) - (4.9) in the physical coordinates is transformed into the normal-

ized coordinates to give
x' =1: +xy +1, (4.15i)
1‘ = -1Ix2-v|x Iy. (4.1511)

Now, the system has only the two parameters 11 and v.
We proceed to analyze the system (4.15) by determining the equilibria of the sys-
tem. The equilibria of the system (4.15) must simultaneously satisfy the equalities
0 = x + xy + 1,

0: —pxz-le Iy.

Thus, the equilibria is determined by the intersection of

—_ -_1_
y- l x’ (4.16)

and

y=-%UL MU)

Observe that the first equality, which defines a hyperbola, is independent of the parame-
ters u and v. The second equality, which defines a half line starting from origin in the

third quadrant with slope {f— and a half line starting from origin in the fourth quadrant

with slope - %. The ratio % of these parameters determines the shape of the curves

defined by the second equality.

 

45

Write
lxl =xsgn (x), (4.18)

where sgn is the sign function. Then using (4.16) - (4.18), one obtains the quadratic poly-

nomial in the variable e1

x2 — %xsgn (x) - vasgn (x) = 0. (4.19)

The solutions, i.e. roots, of (4.19), which define the equilibria, can easily be obtained.

The explicit expression for the x-component of the equilibria is given as follows

 

x = -%—%sgn(x)[l 1V1 +4%sgn(x)]. (4.20)

From (4.20), one may write a ‘bifurcation’ expression for the number of equilibria as
147 = 31-, (4.21)

Thus the equilibria are (see Appendix A):

(1) for % > é’, there is a single equilibrium point.
(2) at E- = 4" there are three equilibria, two of which are equal.

(3) for I"? < 31-, there are three distinct equilibria.

In addition to the v and 11 axes, we employ the point (0,0) and the two half lines

with slope '11:"? to partition the (v, tr)—parameter space into six regions.

 

46
Corresponding to each region, or line, in the (v , tr)-parameter space, the curves
divide the whole state space (x , y) into a fixed number of regions within which the signs

of the derivative quantities i andy do not change.

4.3 Local Stability Analysis of Equilibria

The normalized system equations are given as follows

x =1: +xy +1, (4.22)

y: -|.Lx2—V|x|y. (4.23)

This system captures the essential dynamic features of the adaptive control system (4.1)-

(4.4). The dynamic features are dependent on the ratio that (the system parameters) 15

will take. Expression (4.19) exhibits the dependency of the number of equilibria on the
ratio of these two parameters. Now we analyze the stability of each equilibrium point.

The linearization of the normalized system at any equilibrium point reads

' a1: 1 :1:
[j .= [ - 2111* l—tgfign (x*) — vx*§gn (x* )] [$1 ' (4'24)

Remark 4.1: The right-hand side of (4.23) is continuous: however, it is not differentiable
at x = 0 because of the absolute value of 1: term. Thus the linearization (4.23) can be con-
sidered to be defined everywhere except when x = 0. Observe however that the lineariza-
tion is evaluated at the equilibrium points only, and none of the x components of these

points equals zero. This justifies taking derivatives in evaluating the linearization.

Let us denote the determinant of the Jacobian matrix by Det. Then,
Det = px’“ 2 + vsgn (x* ).

Let us also denote the trace of the jacobian matrix by Trace. Then,

 

47

Trace = - Y1" — vx*sgn (X* ).

where we used equality (4.18).

(i) For {71- > 2% (There exists a single equilibrium point)

 

¢_1V \/
x1—7F[1i 1+4l¢—],
Der >0,

Trace < 0,

so the two eigenvalues both have negative real parts. Thus the equilibrium point is

(locally) stable.

(ii) For {,1 = 11- (There exists two distinct equilibrium points.)
. First,

1:} = 2(1 + ‘5),

Der > 0,

Trace < 0,

so the two eigenvalues have negative real parts. Thus this equilibrium point is (locally)

stable.

Now consider the coincident two equilibria. One gets

Trace = -5- - 2v,

 

48

so there is a zero eigenvalue. Thus the coincident equilibria are degenerate.
(iii) For %- < 1}- (There are three distinct equilibria.)

First,

 

t_1V \/ E
X1-7F[1+ 1+4v],
Det>0,

Trace < 0,

so the two eigenvalues have negative real parts. Thus, this equilibrium point is (locally)

stable.

The second equilibrium point gives,

x5=—.§-%[1-\/1—41\§)],

Det < 0,

Trace may be zero, positive, or negative ,

so one eigenvalue is positive and another is negative. Thus (x3 , ya) is a saddle point.

The third equilibrium point gives,

x§=--5-—:1-[1+’\j1-4%-],

Der > 0,

Trace = 0 if

 

1 3 2
r-Té—[HVr—fié] =0.

 

49
One may write a ‘Hopf bifurcation’ expression for the critical stability of this equilibrium

point as
u’“ = — v2 + vi. (4.25)
Consequently, we have
Trace < 0, for u < — v2 + v‘}, and thus this equilibrium point is stable
Trace = O, for u = - v2 + v‘}, and thus this equilibrium point is critical

Trace > O, for u > - v2 + V}, and thus this equilibrium point is unstable.

One immediate conclusion is that the stability has changed at this equilibrium point when
the value of it passes through the value of iv“. In addition, a Hopf bifurcation takes place
at it: u" . In fact further analysis analogous to [4] reveals that it is a subcritical Hopf

bifurcation. (Recall that at % = 11- a saddle-node bifurcation occurs also.)

4.4 The Generation of Limit Cycles: Proof of The Hopf Bifurcation

The characteristic polynomial of the linearization of the normalized system is
wt) = 22 - [- 31,,— - vx*sgn 0:" )pt + [rm 2 + vsgn (x* )1. (4.26)

A necessary condition for the birth of the limit cycle is that the eigenvalues of M7»)

become pure imaginary and nonzero. Thus,

71; + vx*sgn (x* ) = 0, (4.27)

and

in" 2 + vsgn (x* ) at 0. (4.28)

From the above two equations one obtain

50

u¢v?
Now recall that equilibria must simultaneously satisfy

y*——1_x11:'v

y* = — %x*sgn(x*).

Thus, one get

,.. = _ +v2

x l1_vi_ (4,29)
* _—_ — 4'

y Iii—yr ‘ 30’

So that

u: -v2+v‘}. (4.31)

The above equation (4.31) is the one for the bifurcation curve for limit cycles.

The real part of the eigenvalues when they are complex is
or(v) = - 114%,,- + vx*sgn (x* )1. (4.32)
In order to prove the existence of limit cycle bifurcation, one has to ensures that

davv ,v. #0. (4.33)

Since we have

x*=—.L:2!_2.’

51

Then,

x* = —v"+. (4.34)
Thus,

atv) =v‘5. (4.35)
And

4%,!l/,. = imi > o. (4.36)

Thus a limit cycle bifurcation necessarily occurs. Now let us determine the stability of

the generated limit cycle. Along the limit cycle bifurcation curve, the system equations

maybeexpressedas
x =v‘}x -v“¥y +xy, (4.37)
y'=2 x-uxZ-va-v‘il(y+vi-1)—J$. (4.38)

V

Linearize equations (4.37)-(4.38) at points where they are differentiable, one gets

. Viki ,

 

 

i‘ = vi _vl
f 21%” +v(v*- 1) -v*

d

The characteristic polynomial of the above linearized system is

w(x)=x2+2{j_-v’t= .
Since

)1: —v2+v‘}.

 

52
Then,

2.2 =v(2-v"5).

For 0 < v < 1%, denote the eigenvalues at criticality by i j a) , we have

2,: 1m: ivaw‘L-z).

 

Now,let
41 9].
vv‘lm
Thus,
1 v‘l‘ —v"l’ 0 -co
T -v%+v —v‘1r T: ‘0 0
Since
2: _ x
[y]‘TH’
Also,
i=vix—v—&y+xy,
'- _ 2_ _ 41 i- _}£
y—Z-fi-x ux le v l(y+v 1) v'
Thus,

. l _l
x =v7x -v 7y +xy,

)3 = [2-‘fi-+v(v3’- 1)]x -v‘1'y

-ux2-le -v“l‘l(y +v‘l1—l)+v“’y —v(v‘l‘— l)x.

53

We can transform above system to

)5 = —(0y +f(x,y), (4.39)

y = mx +g(x,y), (4.40)
where

f(x,y)=vx2+v‘1’cuxy, (4.41)

g(x,y)=vx+vc0y-ttx2—v(v+—1)lx—v‘h (4.42)

—v2l x —v‘%l x -v‘}ml x -v"}| y.

Now applying the formula in ([37], eqn. (6.2.9), [29], eqn. (34.11)), one obtains

160 =fm +fxyy +gny +gyyy +%ny(fn +fyy)"gxy(gxx +8”)
"fngyy +fyygyy]
= 2v"![(1- 2v2 + vi) - 252% -1)] > 0.

Thus the equilibrium at criticality is (weakly) unstable. Consequently, the bifurcation is

subcritical, resulting in the generation of an unstable limit cycle.

4.5 Simulation Results

We have considered the simpler two—dimensional normalized system (4.20)-(4.21),

and have concentrated on the invariant curves in the space of x and y . Specifically, we

have concentrated on the stable and unstable manifolds of the saddle point when % < 31-.

At u = 0.00015625, v = 0.00625 and {j— = 9131—, Figure 4.1 displays three equili-

brium points: two stable equilibrium points and one saddle lying near (1: , y) = (-
1.026334, -' 0.0256584). The four initial points are located along the two eigenvectors of

 

54
the linearization of the two dimensional system at the saddle point evolving forward or
backward in time and thus trace the stable and unstable manifolds of the saddle. The
arrows indicate the direction of the trajectories. Observe that the unstable manifold of the
saddle is engulfed by its own stable manifold which in turn converges to the ‘left’ stable
equilibrium point. The (local) stability properties of the equilibrium points remain the
same for O < E < T’ however, the global qualitative behavior persists for only

021897452
0 < {7‘— < 4 .

At % = O’—21%9—74—7, Figure 4.2 shows that the branches of the stable and unstable

manifolds are coincident and engulf the ‘left’ stable equilibrium point, forming the so-
called homoclinic orbit. The homoclinic orbit defines a structure of considerable interest
in dynamical systems since it precedes the rise of ‘horseshoe chaos’ in periodically dis-
turbed systems. This orbit is the coincidence of the stable and unstable manifolds of the
saddle and has the property that each point on the structure would converge to the saddle

point when evolving forward or backward in time.

At 1i= T’ Figure 4.3 shows that a new or-limit set now appears: an (unstable)
limit cycle. The equilibrium point inside the limit cycle is (locally) asymptotically stable.
As % increases the limit cycle decreases in amplitude and the attraction of the equili-

brium point inside it weakens.

At 15:91:72, Figure 4.4 shows the limit cycle and the equilibrium point it is

engulfing have fused into one another forming a locally unstable equilibrium point

(which is a source).

 

‘r

55

 

_-—-————-————----(

  

 

 

 

 

VJIIa X318

—100

x - axrs
Figure 4.1

 

u_ = 0.8758908

V

-------------------d

  

 

 

 

 

 

‘

Ylaxis

o-.----------_-------_

-1fi

X - 3X18
Figure 4.2

 

 

-------------------d

 

 

VJIIaX.lS

[.0
1
$3
4
.I — I0 Mm
. x.Wa
. F.
.
.
.
.
_
.
. 1m
. _
_
_
.
.
.
.
.
.
. m
_ _ _ _
1|. 0 1w.

56

3 .. . in». r» "sun-rlul..w I“!

 

10

 

 

 

E
v

3
T6

4
T6

  
 

lllllllllll Ila." "n'lvlo

/_

 

 

 

x - axrs
Figure 4 4
I
I
l
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
O
x - axrs
Figure 4 5
I
I
I
I
I
I
I
l
I
I
I
I
I
I
I
I
I
I
I
0

  

-—5

-10

 

 

 

 

 

 

 

 

 

'—10

—1
—II
—I

o-.-----____________
04.---------_---------

1—1
_1_

YIaxis. YIaxis. Yl

57

The newly created source and the saddle now continue to approach one another and
at 13— = 4' they fuse into each other forming saddle-node bifurcation. Figure 4.5 shows

the manifolds connected to the now fused equilibrium. The linearized system now

possesses a zero eigenvalue at this fused equilibrium point [28].

When % > 1L, the two fused equilibria wipe out one another and the system now

possesses a single stable equilibrium point with positive x component. The (global)

asymptotic stability of this equilibrium point will remain the same for all ratios of
4“; > 115 Figure 4.6 shows some trajectories converging to the single stable equilibrium

_ 1.25
at 4;. - T.

4.6 Summary

For many systems, including adaptive control systems, the use of the parameter
space for design is natural. The obstacle lies in constructing the parameter space when
the system is nonlinear, as it is the case for the system considered in this paper. Even

more difficult is the case of high dimensional systems.

We feel that the theory of dynamical systems can be properly utilized to overcome
these difficulties for Model Reference Adaptive Control systems. Our reason is that the
essential ideas and construction of Model Reference Adaptive Control systems are gen-
erated from the example system addressed in this paper. The extension to the higher

dimensional case is done while preserving the same basic structure.

Some remarks and conclusions have been drawn in [5]. We summarize further con-

clusions in the following:

1. The number of parameters has been reduced from four to two for the example sys-

tem by using the parameter space approach.

 

L

tr-

58
2. The saddle-node and the Hopf bifurcation curves divide the parameter space into

regions which have qualitatively different dynamics.

3. The example system demonstrates the advantages of parameter space approach for

analysis, especially for design.

4. The system exhibits rich dynamics and structures. Specifically, as the ratio of the

system parameters 1";— is decreased from 1722 to O, the system consecutively

experiences saddle-node, Hopf, and saddle connection bifurcations. At

1.1. = 0.21897445
4

v , a homoclinic orbit exists.

5. The simulation results however illustrate that the ‘left’ equilibrium point changes
stability. For 0 < P— < T’ this equilibrium point is stable; however, for

0761 < L‘- < 1-, it is unstable.

6. The presence of the homoclinic orbit implies the possibility that the addition of dis-
turbance, which is a small sinusoidal function of time, could lead to the generation
of a (transversal) homoclinic orbit. Such an occurrence implies the existence of hor-
seshoes in the dynamics of the system, thus giving rise to the so-called horseshoe
chaos.

Appendix A

The equilibria are:

(1) for {t— > 21-, there is a single equilibrium point, namely,

x*1=1l—-;’T[li\jl+4%],

59

(2) at % = 711-, there are three equilibria, two of which are equal, namely,

x*1=§1-er-(l+‘/2)

=2(1+\(2),
1*23- “é-fi-
=_2,
y’“ = -1- 1
1 371"

= —1-JZ(1+\/2)-1,

1
a1: __1_
Y23 T 2.3

_1
_ ,2,

(3) for % < -i-, there are three distinct equilibria, namely,

1
* = — l — .
y 1.23 —,-..-—x 113

Appendix B

The determinant and the trace are:

(i) for g > i- (There exist a single equilibrium point only.)

x*1=21--:’T[1+\jl+4%-],

2 2
Der =Zl-VT[1+\’1+4%] +v>0,

 

-1
Trace = —21¢_[1+\]1+413] -21.VT2.[1+\/1+41¢—] <0,

so the two eigenvalues are both negative. Thus the equilibrium point is (locally) stable.
(ii) for «5— = 1; (There exist two distinct equilibrium points.)
First,

x*1 = 2(1 + ‘5.)

Der =8(1+\[2)2+v > 0,

Trace = - ”12‘” (2)-1 - 2v(l + 1(2) < 0,

so the two eigenvalues are negative. Thus this equilibrium point is (locally) stable.

61

Now considering the coincident two equilibria, one has

1*13= —2,
Det=0,

T — 1—
race--2- 2v,

hence, there is a zero eigenvalue. Thus the coincident equilibria are degenerate.
(iii) for g < 71f (There are three distinct equilibria.)

First,

x*1=-5--:’T[l+ \jl+4%—],

2 2
Det=zl—VT[1+V1+4%] +v>0,

 

—1
Trace = —2%[1+\j1+41¢_] -%——‘£13[1+\/1+4%-]<0,

so the two eigenvalues are negative. Thus this equilibrium point is (locally) stable.

The second equilibrium point gives,

 

so one eigenvalue is positive and another is negative. Thus (x5 , yi) is a saddle point.

The third equilibrium point gives,

 

CHAPTER 5

A Robust Adaptive Controller For Linear Plants With External Disturbances

Robustness of an adaptive system with respect to a class of perturbations has
become a measurement of satisfactory performance of the adaptation law used. To
achieve robust control of an adaptive control system in the presence of certain distur-
bances, many schemes have been proposed, e.g., the dead-zone approach, the
o—modification scheme, the el—modification scheme, etc. [1, 2, 17, 26]. Using any of
these schemes, a bounded region in the state space can be described such that eventu-
ally all of the states of the adaptive control system will be within this region. How-
ever, it is sometimes difficult to quantitatively and practically determine the boundary
of such a region, and as a result, an estimate of the region’s boundary is usually very
poor. Moreover, one does not know the behavior of the overall system within the
region and thus one cannot track the state of the system once it is within the region.
Moreover, recent reports have demonstrated that some of the available adaptation

schemes may produce multiple equilibria or chaos within the bounded region [3]-[5].

In this chapter, we propose a new adaptation algorithm which is robust in the
sense that all signals in the adaptive loop are bounded and the tracking error will be
zero in the absence of disturbances. For general time-varying disturbances, the new
proposed algorithm will adaptively identify the disturbances and counteract their effect
so that the error between the output of the plant and the output of the model is asymp-

totically reduced.

5.1 Adaptive Control of a First-Order Plant
The equation of the plant with a disturbance is

JEp-=apxp +u +d(t) (5.1)

63

64

where a is an unknown pole of the plant, d is an unknown bounded

P

external disturbance and u is the control input.
The equation of the reference model is
26 = — amxm + r,

m

where am > O and r is the reference input.

Define the state (output) error
e = xp - xm.
Then, the state error equation is
é = -ame +(ap +am)xp +u -r +d(t).
Define the quantity
9* = -(ap +am),
and the parameter error
(I) = 9 - 9*.
We now choose the control input structure
u =9xp +r —0te +B‘I’: a>0

where the constants 0t and B are to be conveniently defined later.

and define the adaptation law in the following equations

._.__ _ 0, HeII<M
9—¢— Y¢(exp +76): Y-{YO(l-eM—llell)’ M S ||9||

w: -awul+rw—0tle, aw>0,

(5.2)

(5.3)

(5.4)

(5.5)

(5.6)

(5.7)

65

where M is an upper bound of l |6* | l, 7‘, , 70 > O and a1 is to be conveniently

chosen later.

Remark 5.1: The nonlinear term — W in equation (5.5) is introduced to ensure the

eventualboundedness of 9(t) in the overall system [1,22].

The disturbance d(t) is unknown and is in general time-variant. The adaptation
equations (5.5), (5.6) and (5.7) have been introduced here to overcome the affect of the
disturbance d(t ). The adaptation equations (5.5), (5.6) and (5.7) seek to identify the
disturbance d(t ). In the absence of d(t), equations (5.6) and (5.7) together with equa-
tion (5.5) ensure that the tracking error goes to zero asymptotically. When the distur-
bance is time-variant, equations (5.5), (5.6) and (5.7) will closely track the disturbance

so that the output of the plant will closely track the output of the model.

The disturbance d(t) is a bounded signal. We write d (t) as the sum of a dc
(averaged) signal and an ac signal. In the following analysis, we investigate the proper-
ties of the overall system when the disturbance is constant, i.e. dc signal. If the ac part
of the disturbance is "sufficiently small" we know that the conclusions of analysis are
adjusted in a standard ways. Often "sufficiently small" suffices for the relative sizes of
most disturbances considered in practice by virtue of definition. In addition, we will

later discuss general time-variant disturbances.

Let us first determine the equilibrium point(s) of the overall adaptive control sys-
tem for a constant disturbance. The reference input is set to be a constant. Rewrite the

equations of the overall system as

é= -—ame +¢e +¢xm-0re+[3\y+d,

. 0, 9

w: —aww+rw—ale, aw>0,

'lufln x mun—rm “:(9‘

 

66

¢* = a constant such that ||9|| <M,

l l

 

“1* = __d ___r*¢*,
I3 am
aw aW

r * = -—d — r*¢*.

W I3 Ba».
1

at: =__ as.
xm amr

Therefore, there exists one and only one equilibrium point when the disturbance and

the reference input are constants.

Remark 5..2: For any given value of d, the e—component of the equilibrium point is

always zero.

Now we study the stability and the convergence to equilibria of the state of the
overall adaptive control system. First, we want to show that all signals in the overall
system are bounded within a sufficiently small region in the augmented state space.

We choose the positive definite Lyapunov candidate

k
_1_¢2+_

1
We, (I), v. r.,,) = 3c? + 27° 2

(W - -a——r,,,)2 + irz (5.8)
W

67

where k1, k2 > e > O are to be conveniently chosen later.

Then, along trajectories

I}(e,¢,w, rw)= -ame2—aez+ Bwe +de —y¢0

l 1
+ k1(\|1 - 2:er - “111W + rw — ale + a—we) — kzrwe

=_ 2__ 2__ ___1_ 2__
ame oce awklml awrw) 7410

k
+ [('a_1 - kla1+ B)\V+ d]e

v
"1 1 i
+ [-a—(0t1 - T) — kflrwe. (5.9)
v it!

Now choose

Then

I}(e, (I), 11!, rw) = - ame2 — 0th — awklml - '21—’81? - NO + de (5.10)
\V

Complete squares to obtain

 

 

 

' _ 1 d 2 2 l 2

V(er¢9warw)- —(am—a)(e-_2'am+a) -aw(W-:;r\y)
1 a2
— +— .
W0 4am+a
Leta beanupper boundof |d|,then
We 0 v r )s —(a —a)(e--’- )2_02(W__1_r )2
’ ’ ' ‘V m 2am+a V a“, V

 

 

 

1 £22
_ 9+—
Y¢ 4am+0t
By definition (5.5), we have
“$992.0.
Definetheset
S={(e ¢.v;r)|(a +a><e--1- d flaw-in2
’ w m 2 am+a W aw W
1 a2
+ s— ,
W0 4am+a}

Hence,
I}(e, q), 0;, rw) s 0
for (e, 0, ‘1’: rw) e S.
Consequently, one can show that all trajectories converge to the set S.

Within the set S observe that

rat2

 

 

 

 

— 2 920; 5.11'
4 am +01 W ( I)
.1. 5‘2 “ZN—Lr )220; (5.1111)
4am+a w a“, V

and
l 5'2 2(a +a)(e -i d )2. (5.11111)
4am+a m 2 am-t-a

Therefore, the variables e , 0 and (w - '51—'11!) are bounded for a bounded disturbu l 1. t-
v

69

The state of the linear time-invariant reference model, xm, is always bounded
since the reference input r is assumed bounded (in this case constant). Now, we are
going to show that the state trajectories in the set S are bounded. Rewrite the error

equation
é = -ame +¢e +¢xm—ae+Bur+d,

We have shown that within the set S e and 11) are bounded. And we assume that d is

bounded.
Let within the'set s
11 be an upper bound of |e |,
12 be an upper bound of |¢|,
13 be an upper bound of Ix”, I.
Then, let
k: = a,,,11+1112 +1213 + 0113 + a,
and
k; = —(a,,,11+1112 +1213 + (113 + .1).
Then,

é > O, for \y > lice".

B

N1

So, é will remain positive when u; > ik,“ and the value of e will be increased and

I3

eventually become unbounded. However, we have shown that e is bounded within S.

80 111 must be upper bounded by k}. Similarly, for w < - 113-19, , é < 0. That is, \V is

bounded for bounded d. Moreover, r“, is also bounded as a consequence of (5.11ii).

.1 -‘:-Y

70

Thus, all signals of the overall system are bounded.

We have shown that all signals in the adaptive control system are bounded and

 

le | is bounded by . Now, we want to show that after the convergence of all

am
signals to the bounded region, the tracking error will approach zero asymptotically for

a constant disturbance or a time-dependent disturbance which approaches a constant.

r.
To case presentation we show the stability proof of the system for an unknown con- i
stant disturbance only. We will then remark on the necessary changes to include the
time—dependent disturbances.
1
Let us choose the positive definite Lyapunov candidate. i_
(‘0 k0 (‘1 1 k 2
_ _ 2 _ 2 _ _ __ 2 _ 2
V(e:¢’W’r\y)" 2e +27¢¢+ 2(‘V awrw)+ 2(rw+k3d),

where k0, k1, k2 > O and k3 are to be conveniently chosen later.
Then, along trajectories

V(e, 4), W, rw) = — koamez — [(00.62 + kOBWe + kode - kowe

1 1
+ k1(\|1 — a_-r"')( —aw\y+ r“, — ale + z—e)
‘I’ \I’

- k2(rw + k3d)e
1
= — koamez - [‘0an - aukltw - 71,12 — we
\I’

1 l
+ k1(\ll - a—ver - are + Z6)

+k0 we + [Code " kzrwe — k2k3d€

1
= - koamez "" koaez — klathI " any? — kowe

71

+ [1‘05 ' k1(0‘1 " ‘a'l—HWe
w

+[-,-1(0t —-1—)—k ]r
aw 1 av 2 we

+ [k0 — k2k3]de.

For preselected aw, B and 011, we can always select k0, k1, k2 > O and k3 such that

° 1
V(e, 4), w, rw) = — [roame2 — 1(0an - avklmt — -;—rw)2 — kowe
‘4’

50.

We use Barbalatic Lemma [47] to infer that V(e (t), ¢(t), w(t), rw(t)) —> O as t —> oo.
Note that

V(e.¢.w. r..)=0

implies that each term in V(e , o, 111, 'w) is equal to zero. Specifically, it implies that

Since all solutions are bounded, it then follows that as t -—) co, e -—> 0.

Second, we want to show that the tracking error will approach zero asymptoti-
cally in the absence of disturbances. Using the same positive definite Lyapunov candi-

date in eqn (5.8), we rederive equation (5.10) to be

V(e, 0,111. rw) = — ame2 - aez - awklw - 7}"102 - 706 + de
‘1’

72

So, in the absence of disturbances, we have

1
2 — one2 — awkloi; - z—rw)2 — 706 s 0. (5.12)

\V

V(e, 11), w, rw) = — ame

We use Barbalatic Lemma [47] to infer that V(e (t), ¢(t), 1110), rw(t)) —> O as t —> oo.
Note that

V(e,¢,w, r...)=o

implies that each term in (5.12) is equal to zero. Specifically, it implies that

Since all solutions are bounded it then follows that as t —> oo, e —> O.

In summary, in the absence of disturbances, the zero tracking error is guaranteed.
For a general time-varying bounded disturbance, this algorithm ensures that the output
of the plant closely tracks the output of the model such that the tracking error will be
bounded by a arbitrarily small region. The estimation of the error bound becomes
much simpler in this case. When the disturbance is a constant or when it asymptoti-

cally approaches a constant, the tracking error will asymptotically approach zero.

5.2 Adaptive Control of a nth-order Plant

We consider a $180 n th-order linear time-invariant plant with a bounded external

disturbance. The transfer function of the plant without the disturbance is

Z (s)
= L.
Wp(s) KP Pp“) (5.13)

73

where Zp(s) and Pp (s) are monic coprime polynomials of degrees m and n (n > m)
respectively, and Zp(s) is Hurwitz. The degrees m and n and the sign of KP are

known.

A minimum realization of the plant is given as follows

xp = Apxp + bpup,

y, = hgxp, (5.14)

where xpeR” is the state of the plant, ups-3R1 is the input of the plant, yp is the out-

put of the plant.

The transfer function of the reference model is

 

z
Wm(s)=K ”’(S)

,,, Pm(s)’ (5.15)

where Z", (s) and Pm (s) are monic Hurwitz polynomials of degrees fir and it respec-

tively. For ease of presentation and without loss of generality, we assume that

n-m=l.

A minimal realization of the reference model is

Jim = Amxm + bmr,

y... = hfixm, (5.16)

where xmeR” is the state of the model, rERl is the reference input, ymeRl is the

desired output of the model.

74

We use the input up and output yp to generate a (2n — 2)— dimensional auxiliary

vector as

C01=Fw1+qu , (bz=Fc02+qyp, (5.17)

where F is a stable matrix.

Define the vectors

CDT=[r,(1)1T,yp,O)2T]
and

eT=[co,cT,ao,aT].

Also define the output error as

el=yp "ym'

Then, the new adaptive controller structure is given as follows

(bl=F(01+qu ,

d)2=Fc02+qyp, (5.18)
uc=0Tm-ae1+B\V, a>0,
up =uc +v,

where veRl is the external bounded disturbance, uc ER1 is the control input, a and [3

are to be conveniently chosen later.

75

Now we have

. x:P Ap 0 0 xp bp
Xe = (01 = 0 F O (01 + q (uc +v).

P

 

 

Define (b = 9 - 6* , thus the control input can be expressed as

uc =67t0—01e1+[5w

= 07a) + car + c*Tco1+ dahgxodflmz — owl + 13W

Rewrite equation (5.19) as

 

 

"‘ T T T
. AP +1..de hp bp c* bp d* bp ’
Xe = (10th F + qc"‘T qd"‘T Xe + q cor
hT 0 F O
1 4 P .

bp
+ 3(0T01—0te1+[3\y+v).

So,

11, =ACXC +bcc6r +bc(¢T00-0te1+|3w+v)

y = hZXc

and the augmented reference model from (5.6) is given by

(5.19)

(5.20)

(5.21)

(5.22)

(5.23)

76

2,, = Acch + becar (5.24)

y”, = thxmc. (5.25)
Define

e=Xc-ch, e1=yp -ym. (5.26)
Then,

e =Ace +bcq>TC0—abcthe +Bbcty+bcv, (5.27)

el = hce. (528)

Now, we define the adaptation law as composed of the following adaptations

'_ __ __ - 0, l|9|| <M

<1) — Pele) F79. ‘Y- {Y0(1‘ eM - Hell), M S I '9' I (5.29)
11;: -aw+r..,—a1e1. a.,,>0. (5.30)
fu= ~e1. (5.31)

where the matrix F = I“T > O and a1 is to be chosen conveniently later.

Remark 5.3: We assume that the upper bound M of the norm of the desired controller

parameter vector I |9* | | is known.

For the purpose of a stability proof one needs to assume that the transfer function
of the model Wm(s) is Strictly Positive Real (SPR). This entails no loss of generality.
Hence, by the Positive Real Lemma, there exist positive definite matrices PC , QC such

that

.4ch +1321c = - QC,

77

Pcbc = he. (5.32)

First, we are going to show that all signals in the adaptive control system are
bounded within a very small region in the augmented state space. we choose the posi-

tive definite (Lyapunov) candidate

I: k
V(e, 6, w, r“) = eTPce + (Jr-16 + -2—1(\y - -al—r,,,)2 + 753,. (5.33)
‘11

where k, > O and k2 > O are to be conveniently chosen later.

Then,

V(e, 0,111, rw) = - eTch - ZWTG - kaw(\|l - -a—1-r‘|,)2 - 2aeTQoe
‘I’

k
+ [(‘a—l' "' klal ‘1' 2B)“, + 2V]e1

‘1'
[‘1 1
+ [—(a1 — —) — kflrwel, (5.34)
“v “v
where Q0 = 116th 2 0.
Now choose
1 1
k1=aw, (11:1, 0:7(aw—1), k2=1-—>O.
“11!
Then

' 1
We, 8: w, 'v) = - eTece — 2W9 - aiw- 3"er
‘1!

- 20teTQ0e + 2th e. (5.35)

78

That is,
V(e, ¢9 w: rw) = _ [e - (QC + 2aQO)—Ith 1T (QC
+ 201Q0)[e — (Q + 2aQ0)‘1vhc]
— ZWTO - aéoy -- 2!"er
W
+ v2hCT(Qc + ZaQ o)‘1hc. (5.36)
Define the set

S = { (c.1811. r...) | [e — (Q. + 20126)“th We. + 2aQo)te

" (Qc + 2aQ0)-1th] + 0112.101, - a—l'rw)2
\II

+ 27670 5 (Phage + 201Q0)'1hc },
where V is an upper bound of Iv |.

Thus,
We. 0. v. r...) s o

for (e, 0, w, '11!) at S.
Consequently, one can show that all trajectories converge to the set S.
Within the set S observe that

1:21:3(Qc + 2aQ orlhc 2 21079 2 0;

79

_ 1
1’th (Q. + Zones) 1h. 2 c.1011 - 17,02 2 o;

and
Vzth(Qc + 201Q0)‘1hc 2 [e - (Q. + 201Q0)‘1vhc 1T (QC + 2aQ0)[e

— (Q + 2aQ0)'1vhc] 2 0.

Therefore, the variables e , \V and (w - -al—rw) are bounded for bounded disturbance v.
\v

The state of the linear time-invariant reference model, X”, is always bounded
since the referenceinput r is assumed bounded. Now, we are going to show that the

state trajectories in the set S are bounded. Rewrite the error equation
e = Ace + bchcu — abcthe + Bbcw + bcv,

We have shown that e and 4) are bounded in the set S. And we assume that d is

bounded. Also Xc and r are bounded imply that a) is bounded.
Let
ll be an upper bound of | IAceI |,
12 be an upper bound of I lbccthol |,
13 be an upper bound of | labcthe | I.
Then, let
k¢+=ll+lz+l3+v,
and
kg: -(ll+12+l3+V).

Then,

-——k+.
Bl lbe ll e

é>O, for 01>

80

l

B—l—l-b—l—I-ke‘“ and e will increase and eventually
C

So é will remain positive when u; >

become unbounded. However, we have shown that e is bounded in S. So w must be

1 _

bounded above by Ice”. Similarly, for u; < mice, é < O. The é will remain
C
negative when 1|; < Wig—Wk; and e will decreased and eventually become
C

unbounded. Hence w is bounded for bounded v. Moreover, rw is also bounded as a
consequence of
1
ago]; - -—r\tI)2 2 0.
“it!

Thus, all signals of the overall system are bounded.

We have shown that all signals in the adaptive control system are bounded. Now,
we want to show that after the convergence of all signals to S, the tracking error will
approach zero asymptotically for a constant disturbance or a time-dependent distur-
bance which approaches a constant. To case presentation we show the stability proof
of the system for an unknown constant disturbance only. We will then remark on the

necessary changes to include the time-dependent disturbances.

We choose the positive definite Lyapunov candidate

1‘1 1 k2
We, 0. w. r...) = koeTP.e + ko¢Trl¢ + 7011 - 77‘"): + 70., + k3d)2.

where k0, k1, k2 > O and k3 are to be conveniently chosen later.

Then,

- 1
We. 0. w. r...) = — koeTch - kaTG — 2koaeTQoe - krapov — 712
‘II

1 1 —
+ ((101! - Tr‘Vx - alel + Tel) + 2kOBurel + 2kode1
. w W

81

- kzrwel - k2k3del

= _ kOeTch — ZkowTB - 2k0aeTQ0e — klawml - —1 )2
a
‘11

+ [211013 — k1(a1 - aim/e,
W

k
+ [——1-(a1 - i) — kflrwel

+ [2’50 - k2k3lde1.

For preselected aw, B and 011, we can always select k0, k1, k2 > 0 and k3 such that

. . 1
V(e, 0. v, rw) = - kOeTch ,- 2koy¢Te - 2koaeTQ0e — klaww — T?
W

SO.

We can use Barbalatis Lemma, see pp. 21 in [47], to conclude that

V(ett). 00). 110). rue» —> o as t —+ 2,
Note that

re. 4.11:. r...)=o

implies that each term in We , o, w, ’v) is equal to zero. Specifically, it implies that
e = 0.
Since all solutions are bounded, it then follows that as t —> oo, e —> 0.

Second, we want to show that the tracking error will be zero in the absence of
disturbances. Using the same Lyapunov candidate in (5.33), we again obtain equality

(5.35) which we rewrite here as

82

‘ 1
V(e, (D, W: rw) = - eTch — ZWTG - 03101, " Trw)2

v
- 2aeTQ0e + 2ve1. (5.37)
So, in the absence of disturbances, we have
We. 0.1!. w) = - eche - 2W9 - c.3011 - 71%? (5.380
W
"" ZaeTQoe
S O. (5.38ii)

We use Barbalatis Lemma [47] to infer that V(e (r ), ¢(t), 111(1), rw(r)) —) O as t —-> oo.
Note that

V(e, b, \V. rw) = 0.
implies that all the terms in (38i) must equal zero. Specifically,

e = 0.

Since all solutions are bounded, it then follows that as t —-> oo, e —-> O.

5.3 Examples and Simulation Results

We employ the same examples which were examined using various adaptive con-
trol schemes by K. J. Asrb'm, K. S. Narendra, I. D. Landau, G. C. Goodwin, A. S.

Morse, C. D. Johnson and other researchers, see [8]-[13].

The general structure for the examples to be considered is illustrated in the fol-

lowing diagram.

83
V(t) = Disturbance (Can not be measured)

 

 

r (t ) Proposed
Command Input Adaptive u t + 51??) 120 ) - OUtPlg
Controller

 

 

 

 

 

 

 

 

 

 

 

The reference input and disturbance are given as shown in the figure below

3
2— e V“)

 

 

 

 

 

 

 

I
O 5 10 15 20
Time (sec)

Two examples have been defined. The first example is described by
Plant: 26,, =xp +u +v,

Model: in, = —x,,, +r.

We have simulated this example with and without the disturbance using consecutively

the ideal law, the o—modification law, the el—modification law, and our proposed

method.

For the three laws:
the control structure is
u =9T0); 6=¢+9*

where for

 

'F‘ 'I?

84
The ideal law:
(1): — Fe 1(1).
The o—modification 121w:
b: -I‘(e100+06)

The e l—modification law:

¢= -F(810)+Ye,|e1|9),

For the our promsed law:
The control structure is

u =OTt0—0w1+B\y, ct>0, 9=¢+9*,

where

4.): —l"e1(t)-F'Ygr Y={yo(l_e9f-Hell), MGSIEIEIA/II

til: —aw\u+rw-a1e1, aw> 0,
r", = - e1,

For the simulation we set the design parameters as follows:
I‘ = 10, 0‘ = 0.5, 7,, = 0.5, and

a=10, 13:5, M =5, aw: -5, 611:1.

In the case of no external disturbance, Figures 5.1-5.4 depicts the outputs of the refer-

ence model and the output of the plant when, respectively the ideal, c—modification,

cal-modification, and the proposed algorithms are used; Figures 5.5-5.8 depicts the

respective output errors. Figures 5.9-5.12 depicts the control input of the proposed algo-

rithm. Figures 5.13-5.24, correspondingly repeat the simulations for given external dis-

turbance.

85

 

    
   

 

 

 

 

 

 

 

 

 

 

 

 

 

2
1 _ .....
x .
Output 0 - .............. : .......................
_1_ L ........
'2 I I I
0 5 10 15 20
Time (see)
Fig. 5.1 Ideal adaptation law
2
x»: (t )
xp (‘ )
‘2 1 T I
O 5 10 15 20

Time (sec)

Fig. 5.2 c—modification

86

 

 

 

 

 

 

0 5 10 15 20

Fig. 5.3 e l—modification law

 

 

 

 

 

 

 

0 5 10 15 20

Fig. 5.4 New adaptation law

 

87

 

 

 

 

 

1
10 15
Time (see)

Fig. 5.5 Ideal adaptation law

20

 

 

 

 

 

 

UI-t

I
10 15
Time (see)

Fig. 5.6 c—modification law

 

20

88

 

 

 

 

 

 

 

 

 

 

‘2 l 1 l
0 5 10 15 20
Time (sec)
Fig. 5.7 e1—modification law
2
1 _
Output
error 0 ~(\ ------
e
-1 _
'2 I 1 1
O 5 10 15 20
Time (sec)

Fig. 5.8 New adaptation law

89

 

 

 

 

 

 

 

 

 

 

 

 

0
-1 _
9(t) —2 — \2
-3 _
—4 I I I
0 5 10 15 20
Time (sec)
Fig. 5.9 Control parameter with new adaptation law
1
0.5 .4
111(1) 0 ~1¥ .............. _ - _ _ -2
-0.5 2
-1 I I I
0 5 10 15 20

Time (see)

Fig. 5.10

 

r1410)

90

 

 

 

 

1
0.5 4

0 —[k---- --------------------------------

—O.5 —
"1 I I I
0 5 10 15
Time (sec)
Fig. 5.11

 

 

 

 

 

 

I
5 10 15 20
Time (sec)

Fig. 5.12 Control input with new adaptation law

 

20

91

 

 

 

 

 

 

 

 

Fig. 5.13 Ideal adaptation law

20

 

 

 

 

 

 

 

 

 

Fig. 5.14 o—modification

20

92

 

xp (1)

 

 

 

 

 

 

 

 

X... (t)
‘2 I I I
0 5 10 15 20
Time (sec)
Fig. 5.15 el-modification law
2

 

 

 

 

 

 

 

_
—t
—l

0 5 10 15 20
Time (see)

Fig. 5.16 New adaptation law

93

 

 

 

 

 

 

'2 I I I
0 5 10 15 20
Time (sec)
Fig. 5.17 Ideal adaptation law
2

 

 

 

 

 

 

I
0 5 10 15 20
Time (see)

Fig. 5.18 o—modification law

94

 

 

 

 

 

 

‘2 I I I
0 5 10 15 20
Time (see)
Fig. 5.19 e1—modification law
2
1 2

 

 

 

 

 

 

I
O 5 10 15 20
Time (sec)

Fig. 5.20 New adaptation law

95

 

 

 

 

 

0

-1 _
6(t) —2 —

_3 _

_4 I I I

0 5 10 15 20
Time (see)
Fig. 5121 Control parameter with new adaptation law
2
1 .-

 

 

 

—

I
0 ~ 5 10 15 20
Time (sec)

Fig. 5.22

 

96

 

 

7111(1)

 

 

 

 

 

—‘

I
0 5 10 15 20
Time (sec)

Fig. 5.23

 

 

 

 

 

—

I
0 5 10 15 20
Time (sec)

Fig. 5.24 Control input with new adaptation law

 

97

The second example is described by
Plant: 55,, - 315,, + ZxP = up ,
Model: 515m + 1.425”, +xm = r,

where up = uc + v, uc is the control input and v is the disturbance.

In this case the transfer function of the reference model is not SPR. We proceed as
follows. First, we choose a Hurwitz polynomial L (s) such that M (s) = L (s )M (s) is SPR,
where M (s) is the transfer function of the model. Then, we consider M (s) as the new
reference model and filtes the reference input r (t) through the stable filter L‘1(s). This is
a standard practice in the literature to alter the reference model to a SPR model, see [47].

Note that the altered reference model has the same transfer function as the original.

We have simulated this example with and without the disturbance using consecu-
tively the ideal law, the o—modification law, the e l—modification law, and our proposed

method.

For the three laws:

the control structure is
u =070); 0=¢+0*

where for

The ideal law:

(I): — Fe 100.
The o—modification law:

(I): -l"(e1co+09)
The e :modificaLtion fl:

- e la IO
4): _r('i'+—I§§TE+Y¢1T;1?'E)2

 

98
where
620) = [9T Wm (S )1 - Wm (S )OTICDO)

81(t)=er(t)+82(t), C(t)=Wm(S)1(0(t) §T=le1§Tl

For the our proposed law:

The control structure is
u =6T0)—0w1+[3\y, 01> 0, 9=¢+9*,

where

(p: —Fe10)—F79, ‘Y:{Yo(l_el9f_|lell), Meslllfilh’l!

\il= -—aw\tl+rW—Ot1e1, aw> 0,
rw = - e1,

For the simulation we set the design parameters as follows:
1‘ = diag {10, 10}, o = 0.5, ye, ’= 0.5, and

01:10, |3=5, M =5, aw: -5, (11:1.

In the case of no external disturbance, Figures 5.25-5.28 depicts the outputs of the

reference model and the output of the plant when, respectively the ideal, o—modification,

el-modification, and the proposed algorithms are used; Figures 5.29-5.32 depicts the

respective output errors. Figures 5.33-5.37 depicts the control input of the proposed

algorithm. Figures 5.38-5.50, correspondingly repeat the simulations for given external

disturbance.

-99-

 

 

 

 

 

1000 —
500 -—
Output 0 - x" (I)
_500 -
—1000
I I I
0 5 10 15
Time (see)
Fig. 5.25 Ideal adaptation law
100

 

20

 

 

 

 

l.
8

I
0 5 10 15
Time (sec)

Fig. 5.26 o—modification

[Wm—“m

-100-

 

 

 

 

 

 

100
50 —
Output 0 — .........................
—50 — x,(r)
”100 I I I
O 5 10 15 20
Time (sec)
Fig. 5.27 el-modification law
2

 

 

 

 

 

 

I
0 5 10 15 20
Time (see)

Fig. 5.28 New adaptation law

101

 

1000-

500 —-
Output
error 0 —
81

-500 —

 

—1000

 

 

 

 

I
10 15 20
Time (sec)

Fig. 5.29 Ideal adaptation law

100

 

50—

Output
error 0 — -

—50 —~

 

 

 

I I
0 5 10 15 20
Time (sec)

Fig. 5.30 o—modification law

 

102

100

 

 

 

 

 

 

 

 

 

 

 

 

Output
error 0 - ........................ 2
e1
—50 —
”100 I I I
0 5 10 15 20
Time (sec)
Fig. 5.31 el-modification law
2
1 -
Output
error 0 — - - - ------ v
81
-1 _
'2 I I I
O 5 10 15 20

Time (see)

Fig. 5.32 New adaptation law

103

 

 

 

 

 

 

I
0 5 10 15 20
Time (sec)

Fig. 5.33 Control parameter with new adaptation law

 

 

92(t )

 

 

 

 

I
0 5 10 15 20
Time (sec)

Fig. 5.34 Control parameter with new adaptation law

 

104

 

 

 

 

 

 

 

 

 

 

 

 

I
0.5 —
1110) 0 “KI—J: -------- A ............. 4
—0.5 -
'1 I I I
O 5 10 15 20
Time (see)
Fig. 5.35
l
0.5 -
r,,(t) 0 —I
-O.5 -
'1 I T I
O 5 10 15 20

Time (see)

Fig. 5.36

105

 

 

 

 

 

 

.L
o

I
0 5 10 15 20
Time (see)

Fig. 5.37 Control input with new adaptation law

Now, let us look at the performance of the system in the presence of disturbance. The
simulation results are given as follows:

 

 

 

 

 

 

1000 -
500 4
Output 0 - x” (t)
-500 _.
—1000 -—
I f I
0 5 10 15 20
Time (sec)

Fig. 5.38 Ideal adaptation law

106

 

 

 

 

100
50— In |l I‘HIII 11‘ 1111. 111 , 111 , 111‘ Mi 1 (‘1' I
Output 0— ‘ l l l ‘
‘50“ I" 11111151111 W‘l‘rWlt ”111911111“11111151III‘WJII“ 1:11-1:11 I‘
'100 I I I I
0 5 10 15 20

Time (see)

Fig. 5.39 a—modification

 

 

 

1p“)

 

 

I
10 15 20
Time (sec)

9....

Fig. 5.40 el—modification law

107

 

 

 

 

 

 

 

‘2 I TT I
0 5 10 15 20
Time (sec)

Fig. 5.41 New adaptation law

 

 

 

 

 

 

I
0 5 10 15 20
Time (sec)

Fig. 5.42 Ideal adaptation law

108

 

 

 

 

100
50—
Output
error 0— -
81 . a , .
-50__ i 111“!“ M II Ml 11,1112,th I110. ’11 I‘ll!” WII lll' 1,1111!“ '1;
‘100 I I I fl
0 5 10 15 20

Time (see)

Fig. 5.43 o—modification law

 

_.__-__-—-___-__-___-____-_.

 

 

 

 

I I
0 5 10 15 20
Time (sec)

Fig. 5.44 el—modification law

109

 

 

 

 

 

 

_

I
5 10 15 20
Time (see)

Fig. 5.45 New adaptation law

 

 

 

 

 

I
0 5 10 15
Time (see)

Fig. 5.46 Control parameter with new adaptation law

 

20

 

110

 

920 )

 

 

 

 

I
0 5 10 15 20
Time (see)

Fig. 5.47 Control parameter with new adaptation law

 

 

 

 

 

I
0 5 10 15 20
Time (see)

Fig. 5.48

"If

 

if

 

111

 

 

 

 

 

 

0.5 -
rw(t) 0 -—
—0.5 —
’1 I I I
O 5 10 15 20
Time (see)
Fig. 5.49

 

 

 

 

 

 

I
O 5 10 15 20
Time (sec)

Fig. 5 .50 Control input with new adaptation law

 

CHAPTER 6

CONCLUSIONS

In this research we have attempted to give our view of the complex field of adap-
tive control. We have shown that adaptive controllers are by definition nonlinear and
more complex than linear controllers with constant parameters. When disturbances are
taken into consideration the usual Lyapunov-like stability methods and functions would
at best guarantee the convergence to a region in the state space. One does not know
the behavior of the overall system within this region and thus one cannot track the
state of the system once it is within the region. Moreover, within such a region various

bifurcation phenomena may occur.

In chapter 2, chapter 3 and chapter 4 we have employed a prototype adaptive
control system, with a single unknown pole and an unknown constant disturbance at its
input. We investigate the possible dynamics that this prototype adaptive control system
exhibits when it is associated with the o—modification law or the el-modification law.
We have shown that in the case of regulation the system possesses multiple equilibria
and undergoes various bifurcation. We have also constructed the parameter-space or
bifurcation diagram which summarizes all possible behaviors of the class of prototype

systems under consideration.

For many systems, including adaptive control systems, the use of the parameter
space for design is natural. The obstacle lies in constructing the parameter space when
the system is nonlinear, as in the case for the system considered here. Even more

difficult is the case of high-dimensional systems.

We feel that the theory of dynamical systems can be properly utilized to over-
come some of these difficulties for (model reference) adaptive control systems. Our

reason is that the essential ideas and construction of (model reference) adaptive control

112

" fl '4 _'_‘7

r—-—Ac_. —-

 

113

systems are derived from the prototype system addressed in this research. That is, the
extension to the higher dimensional case is carried out while preserving the same basic
structure used for the prototype first-order plant. One can determine the region in the
parameter space where global stability of a single equilibrium point is ensured. An
advantage of using the parameter space is that one can now select the values of the

original parameters so as to ensure the following:
(1) a single equilibrium point which is globally asymptotically stable,
(2) the e-component of the equilibrium point is as small in amplitude as possible.

This amounts to ensuring the error between the output of the plant and that of the
desired model to be as small as possible. In addition, one has full knowledge of the

dynamics of the system for every parameter range.

Analysis and simulation in chapter 2, 3 and 4 provide some information and
suggestion for the design of adaptive control systems, specifically, the design of adap-

tation algorithms. One desires to obtain (if possible) an adaptation law which
(1) ensures that only one equilibrium point exists in the overall system,

(2) this equilibrium point is globally asymptotically stable when the system is

perturbation-free,

(3) when the system is perturbed, it ensures boundedness of all signals in the overall

system and zero or small amplitude output convergence error.

Based on the analysis and simulation in all previous chapters and objectives (1) -
(3) above, we have proposed an adaptation law in chapter 5. This new adaptation law
ensures one and only one equilibrium point in the overall adaptive control system. It
also possesses an important property that the e—component of the equilibrium point is
always zero irrespective of the presence of disturbances. When disturbance is an unk-

nown constant or one which would asymptotically approach a constant, the adaptive

F_-_-._ " __"—_‘I

 

114

control system associated with this new adaptation law will be stable and the output

error will asymptotically approaches zero.

The results in chapter 5, we think, are an important step to achieve global asymp-

totic stability and convergence of adaptive control systems under disturbances.

The continuation of the research contained in this work is to devise an adaptive

control structure and an adaptation algorithm such that
(1) there exists one and only one equilibrium point.

(2) the equilibrium point is globally asymptotically stable when the system is
perturbation-free.

(3) when the disturbances are taken into consideration, all signals in the adaptation
loop must be bounded and the tracking error asymptotically approaches zero or
has sufficiently small amplitude. The result should follow when, in addition,

unmodeled dynamics are considered.

 

LIST OF REFERENCES

 

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

LIST OF REFERENCES

P. A. Ioannou and P. V. Kokotovic, "Robust redesign of adaptive control," [EEE

Trans. Automat. Contr., Vol. AC-29, No. 3, 1984.

K. S. Narendra and A. M. Annaswamy, "A new adaptive law for robust adapta-
tion without persistent excitation," IEEE Trans. Automat. Contr., vol. AC-32, pp.
134-145, 1987.

Fathi M. A. Salam and Shi Bai, "Disturbance-generated bifurcations in a simple
adaptive system: Simulation evidence," Systems Control Lett. 7 (1986) pp. 269-
280. '

Fathi M. A. Salam and Shi Bai, "Complicated Dynamics of a Prototype
Continuous-Tirne Adaptive Control System," IEEE Trans. on Circuits and Sys-

tems, Vol. CAS-35, July 1, 1988,

Shi Bai and Fathi M. A. Salam, "Disturbance-Generated Bifurcations in a Proto-
type Adaptive System with e l—Modification Law," IEEE Trans. on Automatic
Control, Vol. AC-33, 1988,

Shi Bai and Fathi M. A. Salam, "Analysis of Disturbance-Generated Bifurcation
in an Adaptive System with e l-Modification Law," the 27th IEEE Conference on
Decision and Control (CDC), December, 1988.

Shi Bai and Fathi M. A. Salam, "Local and Hopf Bifurcation Analysis of an
Adaptive System with el-Modification Law," IEEE Trans. on Automatic Control,

submitted for publication.

K. J. Astro'm, "Robust and adaptive pole placement," 1988 American Control
Conference (ACC), pp. 2423-2428.

115

r..- .-

!.

116

[9] K. S. Narendra and M. A. Duarte, "Robust adaptive control using combined direct

methods," 1988 American Control Conference (ACC).

[10] M. M. Saad, I. D. Landau, M. Samaan and M. Duque, "Performance oriented
robust adaptive controller," 1988 American Control Conference (ACC), pp.
2434-2439.

[11] G. C. Goodwin, "Indirct adaptive control: an integrated approach," 1988 Ameri-
can Control Conference (ACC), pp. 2440-2445.

[12] J. Huang and A. S. Morse, "A computer study of adaptive control systems," 1988
American Control Conference (ACC), pp. 2446-2451.

[13] C. D. Johnson, "Applications of a new approach to adaptive control," 1988 Amer-

ican Control Conference.

[14] Shi Bai and Fathi M. A. Salam, "A robust adaptive controller for linear time-

invariant plants with external disturbances," 1989 American Control Conference.

[15] Shi Bai and Fathi M. A. Salam, "A robust adaptive controller for linear time-
invariant plants with external disturbances," Technical Report, Memorandum No.
MSU/EE/S 89/04 Department of Electrical Engineering , Michigan State Univer-
sity.

[16] H. Khalil, "On the robustness of output feedback control methods to modeling
errors," IEEE Trans. Aut. Control AC-26.

[17] B. B. Peterson and K. S. Narendra, "Bounded error adaptive control," IEEE
Trans. Automat. Contr., Vol. AC-27, No. 6, pp. 1161-1168, 1982.

[18] B. Riedle, B. Cyr and P. V. Kokotovic, "Disturbance instability in an adaptive
system," IEEE Trans. Automat. Control 29 (1984) pp. 822-824.

 

117

[19] C. E. Rohrs, L. Valavani, M. Athans and G. Stein, "Robustness of continuous-

time adaptive control algorithms in the presence of unmodeled dynamics, IEEE

Trans. Automat. Control 30 (1985) pp. 881-889.

[20] K. J. Astrom, "Interactions between excitation and unmodeled dynamics," Proc.

of The 23th IEEE Conference on Decision and Control, las Vegas, NV (1984) pp.
1276-1281.

[21] P. A. Ioannou and P. V. Kokotovic, Adaptive Systems with Reduced Models, Lec-

ture Notes in Control and Information Sciences (Springer, Berlin-New York,
1983).

[22] P. A. Ioannou and P. V. Kokotovic, "Instability analysis and robustness of adap-

tive control," Automatica

[23] K. S. Narendra, Y. H. Lin and L. S. Valavani, "Stable adaptive controller design,
Part II: Proof of stability," IEEE Trans. Automat. Control 25 (1980) pp. 440-448.

[24] K. S. Narendra and A. M. Annaswamy, "A general approach to the stability

analysis of adaptive systems," Proc. of The 23th IEEE Conference on Decision
and Control, las Vegas, NV (1984) pp. 1298-1303.

[25] M. Bodson and S. Sastry, "Exponential convergence and robustness margins in
adaptive control," Proc. of The 23th IEEE Conference on Decision and Control,
las Vegas, NV (1984) pp. 1282-1285.

[26] S. Sastry, "Model reference adaptive control - stability, parameter convergence,

and robustness," [MA J. Math. Control Inform. 1 (1984) pp. 27-66.

[27] S. Boyd and S. Sastry, "Necessary and sufficient conditions for parameter conver-
gence in adaptive control," Proc. of the Berkeley-Antes Conference on Nonlinear
Problems in Control and Fluid Dynamics (Math. Sci. Press, Brookline, MA,
1984) pp. 81-101.

 

118

[28] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory (Springerr, Berlin-
New York, 1982).

[29] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and
Bifurcations of Vector Fields, Applied Mathematical Sciences No. 42 (Springer,
Berlin-New York, 1983).

[30] J. Carr, Applications of Centre Manifold Theory (Springerr, Berlin-New York,
1981).

[31] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of
Hopf Bifurcation (Cambridge Univ. Press, Cambridge, 1981).

[32] J. E. Marsden and M. McCracken, The Hopf Bifurcation and [3 Applications
(Springerr, Berlin-New York, 1976).

[33] A. I. Mees and L. O. Chua, "The Hopf bifurcation theorem and its applications to
nonlinear oscillations in circuits and systems," IEEE Trans. Circuits and Systems

26 (1979) pp. 235-254.

[34] F. M. A. Salam, J. E. Marsden and P. P. Varaiya, "Chaos and Arnold diffusion in
dynamical systems," IEEE Trans. Circuits and Systems 30 (1983) pp. 697-708.

[35] F. M. A. Salam, "Feedback stabilization, stability, and chaotic dynamics," 24th
IEEE Corference on Decision and Control, Fort Lauderdale, FL 1985.

[36] F. M. A. Salam, "Feedback stabilization of the nonlinear pendulum under uncer-
tainty: A robustness issue," Systems Control Lett. 7 (1986) pp. 199-206.

[37] A. I. Mees, Dynamics of Feedback Systems. New York: Wiley, 1981.

[38] F. M. A. Salam, "Parameter space analysis and design of an adaptive system,"
25th IEEE Conference on Decision and Control, Dec. 1986, pp. 1155-1160.

'“I" Hal...“ ‘3‘...-

 

119

[39] B. Egard, Stability of Adaptive Controllers. New York: Springer-Verlay, 1979.

[40] P. A. Ioannou and K. S. Tsakalis, "A robust direct adaptive controller," IEEE
Trans. Automat. Contr., vol AC-31, no. 11 pp. 1033-1043, 1986.

[41] B. Egardt,"Stability analysis of adaptive control systems with disturbances," Proc.
JACC, San Francisco, CA.

[42] B. Cyr, B. Riedle and P. Kokotovic, "Hopf bifurcation in an adaptive system with
unmodeled dynamics," Proc. of the [PAC Workshop on Adaptive Systems in Con-

trol and Signal Processing, San Francisco, CA.

[43] C. E. Rohrs, L. Valavani, M. Athans and G. Stein, "Analytical verification of
undesirable properties of direct model reference adaptive control algorithms,"
Proc. of 21th IEEE CDC

[44] K. J. Astrom, "Analysis of Rohrs counterexarnples to adaptive control," Dept. of

Automatic Control , Lund Institute of Technology Report.

[45] M. K. Masten and H. E. Cohen, "Introduction to a showcase of adaptive con-
troller designs," 1988 American Control Conference (ACC), pp. 2418-2422.

[46] Fathi M. A. Salam, S. A. Van Gils and Z. F. Zhang, "Global bifurcation analysis
of an adaptive control system," Proc. of The 27th IEEE Conference on Decision
and Control, December, 1988.

[47] S. Sastry and M. Bodson, Adaptive Control Stability, Convergence, and Robust-
ness. Prentice Hall, 1989.