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ALI SABERI

All Rights Reserved

STABILITY AND CONTROL OF NONLINEAR SINGULARLY PERTURBED
SYSTEMS, WITH APPLICATION TO HIGH-GAIN FEEDBACK

By

All Saberi

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering
and Systems Science

l983

ABSTRACT

In part l of this thesis, for a general class of nonlinear
singularly perturbed systems, a two-time-scale analysis and design
procedure for stability, initial-value problem, stabilization and
regulation is presented. It is shown that if the slow and fast
dynamics satisfy an "Interaction Condition", then a decomposition
base on the time scale structure of the system can be successfully
performed for the purpose of analysis (stability, initial-value
problem) or design (stabilization and regulation).

In part 2 of this thesis a problem of designing a robust
decentralized control law, using local state or output feedback,
for a class of large scale interconnected systems is studied. In
the case that local states are available the preposed control law is
static and is based on direct usage of high-gain local state feed-
back. On the other hand if the measurements of the local states
are not available and the local observations are linear in the
local states, a decentralized control law is proposed which is

dynamics and employs high-gain local observer-based controllers.

To my parents and Valeh

11'

ACKNOWLEDGEMENT

The author wishes to thank his advisor Professor H. Khalil
for his invaluable help and encouragement during the preparation of
this thesis. He also would like to thank Professors R.A. Schlueter,
J.B. Kreer, R.O. Barr and J. Mallet-Paret for serving on his disserta-
tion committee. Finally he would like to thank Enid Maitland for
her unlimited help, Pauline Van Dyke, and Ginny Mrazek for their

excellent typing.

TABLE OF CONTENTS

Page
INTRODUCTION ....................... l
Lyapunov Stability .................... 8
l Introduction ..................... 8
2 Stability Criteria for Nonlinear Systems ....... l0
3 Stability Criteria for Nonlinear Systems ....... 30
4 A Synchronous Machine Example ............. 35

Closeness of The Trajectories of the Singularly Perturbed System
to the Trajectories of its Slow and Fast Subsystems . . . 53

1 Introduction ..................... 53
2 Problem Formulation and Main Results ......... 55
Stabilization and Regulation - Compoiste Control ..... 63
l Introduction ..................... 63
2 Composite Control ................... 65
3 Regulator problem . . . ................ 7l
Decentralized Control Using Local High-Gain State

Feedback ......................... 98
l Introduction ..................... 98
2 Problem Statement ................... 99
3 Main Result ...................... l02
4 Exponential Stability . . ............... 108
5 Algorithm . . ..................... llO
6 Examples ....................... lll
Decentralized Control Using Local High-Gain Dynamic Output
Feedback ......................... l20
1 Introduction ..................... l20
2 Problem Statement ................... l2l
3 Case 1 ........................ 123
4 Case 2 ........................ l27
5 Case 3 ........................ l34
Conclusion ........................ l45

iv

Table l

LIST OF TABLES

LIST OF FIGURES

Page

46

CHAPTER I
INTRODUCTION

Singularly perturbed systems often occur naturally due to the pre-
sence of small "parasitic" parameters, typically small time constants,
masses, etc.,multiplying time derivatives or in more disguised forms,
due to presence of high-gain feedback and weak coupling. In the early
l970s the chief purpose of singular perturbation, or more generally, the
two-time-scale approach to analysis and design has been the alleviation
of the high dimensionality and ill conditioning resulting from the inter-
action of slow and fast dynamic modes. However, in view of its rapid
development and its role in control theory in late 70's, the role of
singular perturbation has gone far beyond its early purpose, i.e. order
reduction. For example singular perturbation methods prove extremely
useful for the analysis of high-gain feedback systems, root locus analysis
of multi-input multi-output linear systems, and synthesizing robust
controller. In short,singular perturbation in automatic control had
come of age. For an excellent introduction to the field, the reader
should consult the l976 survey paper [l ].

The main theme of this thesis is the study the singular perturbation
and its application to decentralized control. This study basically is
divided in two parts. In the first part, which consists of Chapters 2,

3 and 4, several problem areas of singularly perturbed nonlinear system

analysis and feedback control design are studied. The second part,

which consists of Chapters 5 and 6,deals with designing decentralized
controllers for interconnected systems using high-gain feedback.
.aa;

In this part the two-time-scale system decomposition provide a
uniform framework within which Lyapunov stability, initial value problem,
and stabilization and regulation of singularly perturbed nonlinear
systems are studied. Each chapter in part l deals with one of these
problem areas, and contains a survey of literature and comparison of
our accomplishment in the discussed problem area with previous works.

A brief abstract of the chapters in part l is given in the
following.

Chapter 2, "Lyapunov Stability?

 

Consider a nonlinear singularly perturbed system;

f(x,y) (l-la)
9(XaYsE) (l-lb)

x

a
where the origin (x=0, y=0) is the unique equilibirium point in the
region of our interest. Assuming that y = h(x) is a unique root of

9(X.Y.O) = O, a reduced order system is obtained by setting 5 ='0 in

(l-l) to get:

k=fum6h9th. (La

A boundary layer system is defined as

-%¥ = 9(x,y(T),0). r = t/e, . (1-3)

where x is treated as a fixed parameter. In chapter 2 we establish
the asymptotic and exponential stability properties of the singularly
perturbed system (1-1) for small a from those of the reduced system

(1-2) and the boundary layer system (l-3). The methodology in this

study employs Lyapunov stability techniques. Estimates of domain of
attractions, of upper bound on perturbation parameter, and of degree
of exponential stability are also obtained. The stability result is
illustrated by studying the stability of a synchronous generator
connected to an infinite bus.

Chapter 3,"Closeness.of the Trajectories of the Singularly Perturbed
System to the Trajectories of its Slow and Fast Subsystems"

 

It is an old engineering practice to approximate the model of a
physical system described by (l-l), which corresponds to a high-frequency
model, by a low-frequency model, described by (l-2). In this chapter
we study the justification of such an engineering simplification. Let
z€(t) = (x€(t), y€(t)) donote the solution of (l-l) defined on infinite

interval I = [to’ m), with initial condition z€(t ) = (x0, yo). More-

0
over let zs(t) = (xs(t), ys(t)) be the solution of (1-2) defined on I,
with initial condition 25(t0) = (x0, h(x0)). A set of sufficient

conditions is given under which z€(t) uniformly converge to 25(t) on

any closed subset of (to, 00). Furthermore a boundary layer correction
is provided to take care of the initial finite jump, which results
from discrepancy of initial conditions z€(t0) and zs(to), in order to
extend the convergence results to the interval I.

Chapter 4, ”Stabilization and Regulation-Composite Control"

Chow and Kokotovic in their pioneer work [2] employed the decomposi-
tion of two-time-scale systems into separate slow and fast subsystems to
promote a separation, with attendant simplification, in the design of
state feedback controllers. The stabilization problem discussed in
this chapter utilize this philOSOphy and the stability result of chapter
2 to perform a two-stage design of a stabilizing controller, so called

composite-control, for a general nonlinear singularly perturbed system;

f(X.y,U) (l-3a)

X

69 9(x,y,u,€) (l-3b)
This design procedure consists of following steps.
Step l: DeSign a stabilizing controller u5 = m(x), so called slow

controller, for the slow subsystem;

x = f(x, h(x,uS), us), (1-4)

where h(x,u) is the unique root of g(x,y,u,0) = 0 in the region of
our interest.
Step 2: Design a stablilizing controller uf = r(x,y), so called fast

controller, for the fast subsystem;

%¥-= 9(x,y(r), uS + uf, 0) (1-5)

§t§p_3: Form a composite control uc = uS + uf.

In this chapter, under a set of conditions, it is established that
the composite control is stabilizing.

For the regulator problem the same two-time-scale analysis and
design procedure is employed. Consider a regulator problem (1-3) with

the cost function

on

J =./~ L(x, y, u)dt, (l-6)
o

and initial condition x(t0) = x0, y(to) = yo.
In step (1) of our design procedure we choose uS = M(x) as an
optimal or near-Optimal solution of the slow regulator (l-4) with the

slow cost function

S S

J =‘/‘ L(x, h(x,us), u ) (l-7)
o

and the initial condition x(to) = x0, Next we proceed no steps 2 and
3 as before. In this chapter we establish a "near performance"
property of the composite control obtained through above procedure.
It is shown that this composite control is stabilizing and produces a
finite cost, Juc’ which tends to the cost of the slow regulator as 6
tends to zero. Furthermore,under different stability requirements on

the fast subsystem,explicit upper and lower bounds on Juc are obtained.

Finally, for each result in this chapter an upper bound on the perturba-
tion parameter c is provided under which the result is valid.
Part II

Synthesizing a robust decentralized control, in the sense that the
closed loop system can tolerate modeling errors and varing interconnections,
involves high-gain feedback. The underlying philosophy of the design
of a robust decentralized control scheme is that each isolated subsystem
should maintain a large margin of stability to maintain the stability
of the overall system at the presence of the interconnections, and
this large margin, of stability requires high-gain feedback. In [13]
we illustrate the role of high-gain feedback in the problem area of
decentralized control. In chapters 4 and 5 we proposed a design procedures
using local high-gain state or output feedback for stabilizing a class
of interconnected systems. A brief summary of these two chapters is

given in the following:

Chapter 5, "Decentralized Control, Using Local High-Gain State Feedback"

Consider a nonlinear interconnected system

)2]. = f1.(x1.) + B1.(u1. + 91(X1XN)), i= i, N (l-8)

we prepare (l-8) for high-gain feedback by performing a local trans-

formation (yI, 2T

1) = Ti x,, where Ti satisfies

l

O
T. = (l-9)

with Gi being a nonsingular matrix. Then (l-8) can be rewritten as

c<.
ll

¢1(yi,zi) (l-lOa)

2. = ni(yi,z.)+ GT(U' + O.(X1 ... XN)). (I'IOb)

l l “I

Now a reduced order isolated subsystem is defined as:

Where Vi is treated as an input vector. We assume that each isolated
subsystem can solve a stabilization problem for the reduced-order system
(l-ll). Based on the local stabilization problem (l-ll) a decentralized
control law is designed, and it is shown that under some, basically
smoothness, requirements the designed decentralized controller is
stabilizing. The method is illustrated by linear and nonlinear examples
and is compared with the existing results.

Chapter 6, "Decentralized Control, Using Local High-Gain Dynamic
Output Feedback"

 

 

Consider an interconnected system

x.
H

A1 x. + Bi(u

1 + gi(x1 ... xN)) + M1.(y1 ... yN) (l-l2a)

.i

yl = CI Xi (I-le)

We assume that each isolated subsystem

X.

A. x. +B. u.
l l l 'I

1 (l-l3a)

y. = c. x. (I-I3b)

is invertible and has all its transmission zeros in the left-half plane.
Furthermore the mappings 91 and Mi are assumed to be Lipschitzian. In
this chapter a local high-gain observer-based controller is designed,
and it is shown that, under the above assumptions, the proposed

decentralized controller is stabilizing.

CHAPTER II
LYAPUNOV STABILITY

1. Introduction

Stability properties of singularly perturbed systems have been
investigated by several authors over the past two decades (see [i] for
a survey). In [4-7] Lyapunov methods have been employed. The main
idea is to consider two lower-order systems, the so-called reduced and
boundary-layer systems. Assuming that each of the two systems is asymp-
totically stable and has a Lyapunov function, conditions are derived
to guarantee that, for sufficiently small perturbation parameter, asymp-
.totic stability of the singularly perturbed system can be established
by means of a Lyapunov function which is composed as a weighted sum
of the Lyapunov functions of the reduced and boundary-layer systems.
The methods available in the literature have different conditions due
to different smoothness assumptions, different classes of Lyapunov
functions and different ways of obtaining a negative upperbound on the
derivative of the composite Lyapunov function. Previous work, that
is relevant to ours, is due to Grujic [5] and Chow [5]. Grujic employed
composite Lyapunov methods (c.f. [8-l0]) with linear-type Lyapunov
functions to derive asymptotic stability conditions. He was concerned
with establishing the existence of a composite Lyapunov function but
did not use that Lyapunov function to investigate the stability pro-

perties of the system, like estimating the domain of attraction. Chow

8

established the existence of a composite Lyapunov function and used
it to obtain estimates of the domain of attraction. His method, however,
is limited to a specialcase where the boundary-layer system is linear.

The new element in this work is the use of quadratic-type Lyapunov
functions, which has been motivated by their successful use in studying
the stability of interconnected systems [H3]. For an asymptotically
stable system x = f(x), the function V(x) is said to be a quadratic-
type Lyapunov function if (va(x))Tf(x).: -aw2(x) andllvaH §_w(x) where
¢(x) is a positive definite function of x and a is a positive constant.
The class of asymptotically stable systems that have quadratic-type
Lyapunov functions is large; several interesting examples are given
'hiElOJ. Khalil [ll]employed quadratic-type Lyapunov functions to study
the special case that was studied by Chow [6], namely, linear boundary-
layer systems. The results of[ll:lwere promising and superior to the
results of [5]. This work extends the results oflfTF]and improves over
them. The extension is in studying a general case in which the boundary-
layer system is nonlinear. 'The improvement comes through exploring
the freedom in forming composite Lyapunov functions and using those
Lyapunov functions to obtain an upper bound on the perturbation parameter,
to estimate the domain of attraction or to estimate the degree of exponential
stability.

In section 2.l, an asymptotic stability criterion for nonlinear
autonomous systems is derived. An interesting feature of this criterion
is that, under mild conditions, any weighted sum of quadratic-type
Lyapunov functions for the reduced and boundary-layer systems is a
quadratic-type Lyapunov function for the singularly perturbed system

when the perturbation parameter is sufficiently small. It is shown

10

that the choice of the weights of the composite Lyapunov function involves
a trade-off between obtaining a large estimate of the domain of attraction
and a large upper bound on the perturbation parameter. That_trade-off
is discussed and illustrated by an example. In section 2.2, the conditions
are sharpened to give exponential stability. The composite Lyapunov
function is used to obtain an estimate of the degree of exponential
stability, which is shown, by means of an example, to be a very tight
estimate. In section 2.3, the stability criterion is extended to non-
autonomous systems. Application of the stability criterion to linear
systems is carried out in section 3 where upper bounds on the perturbation
parameter are obtained and compared with previous bounds due to lien
[12] and Javid [13]. As an illustration of the application of our method
to physical systems in which the perturbation parameter may be fixed
with given value, we apply the method in section 4 to estimate the domain
of attraction of the stable equilibrium point of a synchronous generator
connected to an infinite bus. The generator is represented by a three-
dimensional model in which a field-flux decay is taken into consideration.
The results are compared with previous ones due to Siddiqee [l4].
2. Stability Criteria for Nonlinear Systems
2.l Autonomous Systems: Asymptotic Stability
Consider the nonlinear singularly perturbed system1
x = f(x,y), xéBiiR"

. m (2-l)
5y 3 9(x:.Y9€)9 3/63ch ’ 5 > 0.

 

1The symbol Bx indicates a closed sphere centered at x = 0; By is defined

in the same way.

ll

We assume that, in 8x and B the origin (x = 0, y = 0) is the

y,
unique equilibrium point and (2-l) has a unique solution. A reduced

system is defined by setting a = 0 in (2-l) to obtain

x = f(x,y) (2-2a)

0 = 9(X.y.0) (Z-Zb)
Assuming that in 8x and By, (2-2b) has a unique root y = h(x), the re-
duced system is rewritten as

i = f(x,h(x))e fr(x) (2-3)

A boundary-layer system is defined as

-%¥ g<x.y(.).0) (2-4)

where I ' t/8 is a stretching time scale. In (2-4), the vector xeRn
is treated as a fixed unknown parameter that takes values in Bx‘

Our objective is to establish the stability properties of the
singularly perturbed system (Z-l), for small 5, from those of the reduced
system (2-3) and the boundary-layer system (2-4). We are interested
in cases when both (2-3) and (2-4) have quadratic-type Lyapunov functions.
Our result shows, under mild assumptions, that, for sufficiently small
a, any weighted sum of the Lyapunov functions of the reduced and boundary-
layer systems is a quadratic-type Lyapunov function for the singularly
perturbed system (2-l).

We start by stating our assumptions.

n

(I) The reduced system (2-3) has a Lyapunov function V : R —+ R+

such that for all xeBx

(v,V(x))If,(x) :,-a]wz(x), a, > o.
where w(x) is a scalar-valued function of x that vanishes at

x = 0 and is different from zero for all other x68x.

12

(II) The boundary-layer system (2-4) has a Lyapunov function
N(x,y) : Rn x Rm + R+ such that for all xeBx and yeBy

(vyN(X.y))Tg(X.y.0) _<_-02¢2(y - hm). (12 > 0.

where<b(y - h(x)) is a scalar-valued function of (y - h(x))eRm
that vanishes at y = h(x) and is different from zero for all

other xeBx and yeBy.
(III) The following three inequalities hold VxeBx and yeBy

(a) (vxw<x.y))T f(x,y) :_ C]¢2(.Y - h(x)) + c2v(x)¢(y - h(x)).
(b) (va(X))T[f(X.y) - f(X.h(X))] _<_ ammo - h(x)).

(c) (VyW(X.y))T[g(X.y.e) - 9(x.y.0)] .<_ 6K1¢2(y - h(x))
+eK2w(X)¢(y - h(X))-

The constants c], c2, 8], K1 and K2 will be taken as nonnegative
numbers [see Remark 2 for the case when some of these constants

are negative]. Condition (I) guarantees that x = 0 is an as-
ymptotically stable equilibrium point of the reduced system (2-3).
Condition (III) plays the same role for the boundary-layer system
(2-4) which has an equilibrium point y = h(x); that is why in
Condition (II) the comparison function ¢(-) depends on y - h(x).

It is important to notice that Condition (II) should hold uniformly
in x, i.e., the positive constant 02 should be independent of

x. Inequalities (III-a) and (III-b) determine the permissible
interaction between the slow and fast variables. They are basically
smoothness requirements on f and 9. One way to grasp their meaning
is to consider a special case when the partial derivatives of

V and H are bounded by w and a, respectively, and fr(x) is bounded

l3

by p. In this special case, inequalities (III-a) and (III-b)
follow from the Lipschitzian-like condition

”(M/1) - f(x,yZH 5. My] - yz) (2-5)

which simply says that the rate of growth of f in y cannot be
faster than the rate of growth of ¢(o). Condition (III-a) will
drop out if N is not a function of x. However, choosing N independent
of x may be impossible or, if possible, would make Condition
(II) very restrictive. Finally, inequality (III-c) determines
the permissible dependence of g on a. It will drOp out if g
is independent of e. we believe that Conditions (I)-(III) are
mild. One way to demonstrate that is to show that there is a
wide class of systems for which these conditions hold. Lemma
1 does that.
Lemma 1: Consider a singularly perturbed system in which 9 is in-
dependent of e. Let f,g and h be continuously differentiable in Ex x by.
Suppose that the reduced system is exponentially stable such that H x(tH[g °
c3 ||x(0)|| exp(-c4t) whenever x(0)eBx. Let BXQ'B'x be a set such that
every trajectory starting in 8x remains inside Bk. For every xéBg,
suppose that the equilibrium point 3:: h(x) of the boundary-layer system
is inside 8? and is exponentially stable uniformly in x such that ”y(r)
- h(x)” 1 c5 ”y(0) - h(x)” exp(-c6r) whenever y(0)€B'y. Let Bycg‘B'y
be a set such that every trajectory starting in By remains inside 3}.
Then, conditions (I)-(III) are satisfied in Bx x By.
3592:: See Appendix.

We are now ready to state our stability criterion.
Theorem 1: Suppose that Conditions (I)-(III) hold; let d be a positive

number such that D < d < l, and let 5*(d) be the positive number given

l4

by
* “1“2

ed= ,
( ) GIY + [81(l - a) + 82d12/4d(l - a)

 

where 82 = K2 + C2, y = K1 + C],
then, for all e < e*(d), the origin (x = o, y = 0) is an asymptotically
stable equilibrium point of (Z-l) and
v(X.y) = (l - d)V(X) + dW(X.y) (2-6)
is a Lyapunov function for (Z-l).
£5991: Let 5(x,y) denote the derivative of v along the trajectory of

(2-l). We have

$(x.y) = (i - d)(VxV(X))Tf(X.y) +-§ (VyW(X.y)Tg(X.y.e) + d(va(X.y))Tf(X.y)
= (i - d>(vXV(xi)Tf,(xi + (i - d>(v,v<x))IEf(x.yi - i(x.h<x))l

+ g-(vyH(X.y))Tg(X.y.0) +-§ (vyH(X.y))T[9(X.y.e) - 9(X.y.0)]
+ d(vxw(X.y))Tf(X.y)
i-u-dnnhn+(i-hmunhy-Mni-§%Jo-huh

A

+

“n y-Mni+«guho-hun+dq¥o-huh

+

dc2w(X)¢(y - h(X))

«xi-hfin)+n-amnuno-nun-dE§{}%y-Mni

.4.

d82w(X)¢(y - h(X))

v(X) T _I_ (:W(X) j)
(My-hm) i>(y-h(X))

where

l5

(l—d)a1 -(l-d}B]/2 - d82/2
T - -(l-d)B]/2 - d82/2 d(:_2 - Y)
For asymptotic stability of (Z-l), it is sufficient to require that
The a positive-definite matrix. For any choice of d(0 < d < l),T
will be positive-definite when E is sufficiently small. In particular,
Tis positive-definite for all . < 3(a).
Remark l: If V(x) and W(x,y) are radially unbounded and 8x x By = Ranm,
the origin (x = O, y = 0) will be asymptotically stable in the large.

An interesting point in Theorem 1 is the arbitrariness of d. By
allowing d to take any value on the interval (0,l), the composite Lyapunov
function u(x,y), as given by (2-6), covers all the possible linear
combinations of V and H. According to Theorem l, any one of these linear
combinations is a Lyapunov function of the singularly perturbed system
(Z-l) as e +-0. What is more interesting is that for any given d, Theorem
l provides us with the upper bound e*(d). Fig. (l) shows a sketch of

'k
e*(d) versus d. From that sketch we see that e (d) has maximum value

c* at d = d*. Straightforward calculations show that
oz
8* = ——l-C:2———— (2-7)
“1" * BlBZ
and
t B]
d = ——-—-—-— (2.8)
8i T 82

Corollary l: Suppose that Conditions (I)-(III) hold and e < e*, then
the origin (x = 0, y = 0) is asymptotically stable.

In applying our stability criterion to study the stability of
(2-l) when e is known, we start by comparing a with 5*, If e > 6*,

our criterion is not satisfied which means either that the origin is

l6

not asymptotically stable or that it is asymptotically stable but our
criterion fails to detect that. On the other hand, if e < 6* we conclude
that the origin is asymptotically stable. The next step is to explore
the freedom we have in choosing d. For any given a, there is an interval
(d],d2) (see Fig. (l)) such that any de(d],d2) will be acceptable.

As 6 gets smaller, the interval (d],d2) spreads out, tending eventually
to (0,l) as c + 0. We can say that our stability criterion improves
asymptotically as e + 0 in the sense that as e + 0 the criterion is
always satisfied and there is greater freedom in forming the composite
Lyapunov function.

The freedom we have in choosing d can be employed to obtain the
largest possible estimate of the domain of attraction. The idea is
illustrated by the synchronous machine example of section 4. In general,
there is no systematic procedure for choosing d in order to obtain the
best domain of attraction; such a choice is problem dependent. There
is, however, a special case in which the choice of d is obvious. Suppose
that the domain of attraction LR, of the reduced systems and LB, of the

boundary-layer system, are given by

LR = {XEBXI V(x) 1 v0}

and
LB = {xeBx, yeB.y |H(x,y) §_wo},

then, an estimate of the domain of attraction of the singularly perturbed
system (Z-l) is given by
L = {xeBx, yEByl (l - d)V(x) + dw(x,y) 5_min((l - d)Vo, dwo)}

(2-9)

17

It is apparent that the choice

d v° (210)
v0 + wo
will result in the largest set L* which is given by
L*={xe8,ye3|1§l‘l+l‘-Q‘-zfl<i} (2-ll)
x y vo wo .—

The trade-off involved in choosing d is illustrated by the following
example.
Example l: Consider the second-order system

x = y2 - 2x3, B

x {xeR| |x| :_l}, (Z—IZa)

x3 - tan y, B {yeRI |y| §_a , where a is (2-l2b)

ey y

a constant less than, but arbitarily close to, n/2.

The reduced system is obtained by setting 6 = 0 in (2-l2b) to get

y = h(x) = tan‘lx3

which, when substituted in (2-l2a), results in
i = -2x3 + (i;an“x~°’)2 é fr(x) (2-l3)

In searching for a Lyapunov function V(x) for (2-l3) we are guided by

two requirements. First, V(X) should satisfy Condition (1). Second,

§_\_l_
3X

not appear explicitly in Conditions (I)-(III), it is helpful because,

should be bounded by h(x). Although the second requirement does

when it is satisfied, inequality (III-b) reduces to the Lipschitzian-

2

like condition (2-5). The Lyapunov function V(x) = 7} x satisfies

Condition (1) with h(x) = x2; however, g; grows faster than ¢(x). So,
it is not the Lyapunov function we are looking for. A Lyapunov function

4

that satisfies both requirements is V(x) = %'x . It satisfies Condition

(I) with ¢(x) = |x|3 and d1 = l; at the same time lg¥l = v(x). The

l8

set LR = {xéBx|V(x).:-%} is included in the domain of attraction of

the reduced system. The boundary-layer system is defined by

-%¥ = x3 - tan y (2'14)
where x is treated as an unknown fixed parameter in Bx' Again in choosing
the Lyapunov function N(x,y) we are guided by two requirements. First,
N(x,y) should satisfy Condition (II). Second, g; should not grow faster
than o. A Lyapunov function that satisfies both requirements is H(x,y) =
%-(y - tan-1x3)2. It satisfies Condition (III) with ¢(y - h(x)) = ly
- tan'1x3l and a2 = l. The set LB = {x68x, yeByl H(x,y) : %(a -‘%)2is
included in the domain of attraction of the boundary-layer system for
every fixed x. Finally, it can be verified that the inequalities of
Condition (III) hold with C1 = 2.64, c2 = 6, B1 = 2.356, K1 = K2 = 0.
So, e*(d) is given by

Jun = ‘ 2 .

2.64 + [2.356(l - d) + 6d] /4d(l - d)
The largest upper bound 5* = 0.0596 is obtained with the choice
d* = 81/(81 + 82) = 0.282 leading to a Lyapunov function

V(x,y) = 0.7l8 V(x) + 0.282 H(x,y)

A sketch of the corresponding estimate of the domain of attraction,
as given by (2-9), is shown in Fig. (2). 0n the other hand, the largest
).

set L* contained in LR and L8 is obtained with the choice d . VO/(Vo+wo
d = 0.448 leading to a Lyapunov function

\)(x,y) = 0.552 V(x) + 0.448 H(x,y).
The corresponding upper bound on e is 2* (0.448) = 0.0534. A sketch
of L* is shown in Fig. (2); of course, L* contains the set obtained
with d* = 0.282. The price paid for enlarging the estimate of the domain

of attraction is a reduction in the upper bound from 0.0596 to 0.0534.

19

As we pointed out in the introduction, there has been previous
work on developing stability criteria for singularly perturbed systems
using Lyapunov's method. Closely related to our work is the work of
Chow [5] and that of Grujic [5]. We pause here to compare our work
with the previous ones. Chow studied the stability of a special class
of singularly perturbed systems in which f and g are linear in y. The
stability criterion derived by Chow is quadratic in nature in the sense
that the derivative of the Lyapunov function is bounded by a quadratic
form. Motivated by the work of Chow, on one hand, and the work of Araki
[10], on the other hand, we thought that a criterion similar to Chow's
could be derived if we start with quadratic-type Lyapunov functions
and perform composite stability analysis in a way similar to [9,l0].
That has been done successfully [ll] for the class of systems in which
f and g are linear in y. This paper extends the results of [11] to systems
in which f and g are nonlinear in y. The examples worked out in [1]]
show that our criterion produces estimates of the domain of attraction
and of the upper bound on the perturbation parameter which are less
conservative compared to Chow's criterion. So, our stability criterion
is an extension and generalization of Chow's criterion and includes
it as a special case. Grujic's criterion, however, is different from
ours. There are two reasons for the difference. First, Grujic uses
linear-type Lyapunov functions while we use quadratic-type Lyapunov
functions. Second, different conditions are imposed on the interaction
between the slow and fast variables. Generally speaking, we require
stronger smoothness properties on f and 9 than those required by Grujic.
As a result, our criterion is satisfied asymptotically as e + 0. This

is not the case in Grujic's criterion. Beside requiring 6 to be less

20

than a certain upper bound e*, Grujic's criterion has a second requirement
on the constants that appear in the inequalities that correspond to

our Conditions (I)-(III). If that second requirement is not satisifed,
the criterion is not satisfied even if e + 0. Since the basic philosonhy
in studying a singularly perturbed system [c.f. l] is to establish its
properties from those of the reduced and boundary-layer systems for suf-

ficiently small a, we think that our criterion better fits the purpose.
To help the reader get a good picture of the difference bEtween the
two criteria, we solve an example that has been solved by Grujic [5]:
We also make the same choice of the Lyapunov functions.

Exggple 2: Consider the absolute stability problem for the singularly

perturbed system

is Aux + Any + q]¢](o]), (2-15)
.5, = Aux + Azzy + 02t2(02). (2-16)
where
0i ‘ CiI" * clgy’

_ T I

Here xER", yéRm, oieR and A1. ., c.

J 13
The nonlinearities o](-) and ¢2(-) are one-to-one, continuous, ¢i(0) = 0

and qi are of appropriate dimensions.

and satisfy the sector conditions

o1(01)

T e[0,ki], i = 1,2, Voi€('°°s°°)t

where kie(0,w). It is assumed that All is a Hurwitz matrix, the pair
(A1],q1) is controllable, the matrices'322(0) and Dé2(k2) are negative

definite, where

21

T
a~ T T T T

022‘“) T [A22 T k“22”“2c221 T [A22 T k(“225’“‘2‘3223
and that

T
i

l

ci Aiiqi 3.0, i = 1,2.

The following numerical values are considered

0 i o \ -0.01
- ’ q z ’ C = ’ A ..
‘1 -l 2 I o.i) 1‘ 0.0 ‘2
l _3 i0"3
,2 - , k1 = 2, A2] = l0 I, c2] = , K2
1 o
-4 l i l
A . q = . c = .
22 I _4 2 I 22 0

Grujic [5] studied the asymptotic stability of the origin for the given.

>
I
I
H
0

n
I
II

...;

numerical values. For the reduced system
x = A x + (c Tx)
ll qi¢i ll
. 1/2
he used a Lyapunov function [V(x)] where

Tx

V(x) = XTHIX + B[C1¢]](o)do, (2-17)
0
. i. 3 ‘
withH1=§fi <1 ]/ and6=l.
For the boundary-layer system
'gf ‘ “22’ T q2¢2(°223)
he used a Lyapunov function [H(y)]1/2 where

wei=yv udm

22

He verified that his stability criterion is satisfied for e < 0.52.

Let us consider applying our stability criterion to the same problem.

In order to have a meaningful comparison with Grujic's solution, we

use his choice of Lyapunov functions. The only difference is that we

use V(x) and N(y) as Lyapunov functions rather than VT/2(x) and HT/2(y)
since we work with quadratic-type Lyapunov functions. It can be verified
that Conditions (I) and (II) of our criterion are satisfied with 01=0.1 ,
¢(x) = Hxll, 42 = 3.838 and ¢(y) = fly" (notice that in this problem
h(x) : 0). To verify Condition (III) we need to impose a stronger smooth-
ness condition on the nonlinearities ¢i(°)' We require that ¢](-) and

o2(-) satisfy a Lipschitz condition, namely,

 

(471(01) ' 431(3-1'”: Lilo-i “SI-i - (2'19)

This condition is not needed in Grujic's work but it is essential in
ours. Hith (2-19), it can be verified that Condition (III) is satisfied
° "' "' " = = -3 V—
With c1 - c2 - o, a] - 0.3416 + 0.02 L], x1 0 and K2 2x10 (1 + 2 L2)-
*
By Corollary 1, the origin is asymptotically stable for all 6 < E ,
where

8* = 561.72
(i + 0.059 L])(l + ‘VE L2)
From (2-20), we see that for values of L1 and L2 of order ten, or even a

 

(2-20)

hundred, the upper bound obtained by our criterion is better than the
0.52 obtained by Grujic's criterion. Although we require a stronger
smoothness condition on the nonlinearities, for a wide class of non-
linearities that satisfy our condition we get a less conservative result.
This, however, is not the important point. The important difference
is illustrated below. Let us resolve this problem for the same numerical

values except for A2] and C2] which will be taken as A2] = 0.1 I and

23

o l . -3 i0‘3 .
C2] = (0’ ) instead of A2] = 10 I and C2] = (0 >, i.e. we allow
for a stronger interaction between the slow and fast variables. Since
changing the values of A2] and C2] does not affect the reduced and
boundary-layer systems, the same choice of the Lyapunov functions will
be used. It can be verified that Grujic's criterion is not satisfied
because, in his notation, the condition :1 + E] < 1 does not hold.
0n the other hand, using our criterion we conclude that the origin is
asymptotically stable for all e < 8* where 8* is given by

8* = 5.61 .

(1+ 0.059 L])(1+ V—Z'Lz)
Once Conditions (I)-(III) hold, our criterion is satisfied for sufficient-
ly small 5 irrespective of the numerical values of the constants appear-
ing in the inequalities.

We conclude our discussion of the asymptotic stability of autonomous
systems by pointing out some ideas that may lead to less conservative
results.

Remark 2: In stating Conditions (I) and (II), the comparison functions
w(.) and<b(-) are not required to be positive-definite functions. It

is merely required that they vanish only at the equilibrium points.

As a result of that, the inequalities of Condition (III) may be satisfied
with some of the constants on the right hand side being negative numbers.
Such a situation will be the exception not the rule because in general
one has to use norm inequalities in order to verify Condition (III).
However, if it happens, it can be used to get less conservative results.
It can be verified that Theorem 1 holds with the exception that we

should say: for all a satisfying

24

.1 >T_1 (2-20
8 e (d)

* 1'
instead of saying Va < c (d) because 6 (d) could be negative, in which
case the inequality (2-21) will be always satisfied since e is positive.
Example 3: Consider the second-order system

x = x - x3 + y,

8&3'X‘y.
With the choice V(X)= -% x4 and M(X.y)= (l/2)(x+y)2, it can be verified
that Conditions (I )t o(u I) hold with o(x) = x3, ¢(y - h(x)) = x + y.

'k
a] = l, 02 = I, C1 = 1, C2 = -l, B.I = l, Kl = K2 = 0. Thus, 6 (d)
is given
1

I+m[(I-d)-d]2

The choice d = %-maximizes* e (d) and yields e* = 1. So, the origin

e*(d)

is aymptotically stable for all e < 1. If we allow only for nonnegative

He will

         

numbers, we can take w(x) = lxl3 and ¢(y - h(x)) =

get the same numbers except c2 = 1. Then,

1
1*i‘nlr‘afl“'d)*d]2

The choice d= E-maximizes e *(d) and yields 2* = 0.5, which is more

5*(d) =

conservative.
The situation when we work with negative numbers is similar to
the work of Michel and Miller [9] on studying stability of interconnected
systems allowing for negative bounds on the interconnections.
Remark 3: The requirement v(x) = 0 in 8x iff x = 0 can be replaced
by requiring that w(0) = 0 and the set S = {xlw(x) = 0} does not contain

a nontrivial trajectory of the reduced system.

25

Remark 4: Instead of using one comparison function for each of the
reduced and boundary-layer systems, more than one comparison function
may be used. This would be similar to the interconnected system stability
criterion of [15]. In the synchronous machine example of section 4
we consider a case when two comparison functions are used with the
boundary-layer system.
2.2 Autonomous Systems: Exponential Stability

Exponential stability result for (2-1) can be obtained by requiring
stronger conditions. Suppose that Conditions (I)-(III) hold with com-
parision functions w(.) and ¢(-) which belong to class at functions
[15], i.e., they are continuous and strictly increasing in 8x and By.
In addition, suppose that V(x) and H(x,y) satisfy the inequalities

(IV) e1v2(X) : V(x) : ezwzh). vxeex.

(V) e3¢2(y - h(X)) :W(X.y) _<_e4i>2(y - h(Xl). VXGBX. W683,-

Then, the conclusion of Theorem 1 holds with exponential stability re-
placing asymptotic stability. This follows from the fact that the
composite Lyapunov function (2-6) and its derivative along the trajecto-
ries of (2-1) will have the same rate of growth. One case of particular
interest is the case when h(x) = H x|| and 0(y - h(x)) = lly - h(x)]l.
For this case, an estimate of the degree of exponential stability is
obtained in Theorem 2.

Theorem 2: Suppose that Conditions (I)-(V) hold with 1(x) 8 H x H and
¢(y - h(x)) = H y - h(x) ", then the origin (x = 0, y = 0) is an ex-
ponentially stable equilibrium point of (2-1). Let a be any positive

number such that 0 < a < 01/2e2, then a is an estimate of the degree

26

of exponential stability for all e < e:(o) where c:(a) is given by
(a - 2&3 )a
e*(o) = T 2 2 (2-22)
1 13, - ZaezTTv + 2ae4T + Bo,

Proof: Consider a composite Lyapunov function v(x,y) as in (2-6).

Similar to the proof of Theorem (1), it can be shown that

T

- ll" 11 ~ ll X II

V _<_ O . '20. V
("Mg") T (Hy-h(x)”)

\

where

1*.

(l-d)(d]-2oe2) --% (l—d)B] --§ dB

1

a
"2'““1’81‘2dB 2 )

2 d(€—- - Y ‘ 20'84

System (2-1) will be exponentially stable with degree a ifT is positive

definite, which is the case for all 8 < €:(d,a) where

(“i ' 2“82)“?
(a1 - 20e2)(Y + Zae4) + [81(1 - d) + 82d]2/4d(l - d).

 

61(d,a) =

Recalling the discussion of section 2.1 on the choice of d, we see that
e:(d,o) is maximum when d = d* = 81/(81 + 82). This choice of d yields
c:(a) as given by (2-22).

In Theorem 2, the number a1/2e2 is the estimate of the degree of
exponential stability of the reduced system using the Lyapunov function
V(x). The restriction a < 01/2e2 means that the estimate of the degree
of exponential stability of the overall singularly perturbed system
has to be less than the estimate for the reduced system, which is natural.
It is interesting that any ae(o, a1/292) is an estimate of the degree
of exponential stability for sufficiently small 6. The upper bound

e:(a) be.;ngs t0 (0,€*) with 8:(o) + E* as a + 0 while €:(a) T 0 as

27

a + al/Zez. When 6 is known, the expression (2-22) can be used to obtain
the estimate a as illustrated by the following example.

Example 4: Consider the stiff network shown in Fig. (3)

 

 

162
151.

1/2F + V1/2F +
_Vc
1v

2

 

lfig. (3)
The maximum time constant of this network can be estimated using Lyapunov
functions as the reciprocal of the estimate of the degree of exponential
stability. This was done in [17] using a general Lyapunov function
that does not take the stiffness into consideration. The estimated
maximum time constant was 0.5 seconds which is within a factor of 2
of the actual maximum time constant (the actual time constants are
T] = 0.256 and T2 = 0.025). This estimate is good taking into considera-
tion that it is based on a general algorithm. However, if stiffness
is taken into consideration and Lyapunov functions are formed from
singular perturbation point of view it is natural to expect better
estimates. Theorem 2 will be employed to demonstrate that this is
actually the case. Let 7;] anch2 be the equilibrium values of v

cl
and Vc2’ respectively. Define

x = vc] - Vcl’ y = vc2 - 7E2 to get state equations in the form
i = -4x + 2y
6y = 0.2x - 4y
where e = 0.1. It can be verified that with the choice V(x) = x2 and

28

W(x,y) = (y - 0.05x)2, Conditions (I)-(V) are satisfied with W(x) = |x|,
¢(y ' h(X)) g ly ' 0.05Xl, a] a 708, a2 a 8, C] 3.0.2, C2 = 0.39,
B] = 4. K] = K2 = 0, e1 = e2 = e3 = e4 = 1. From (2-22) we get

*( ) = 8(7.8 - 2o)
E1 9 (7.3 - 2o)(0.2 + 2g) + 1.56

Since 2 is known to be 0.1, we choose a such that c:(o) is greater than
0.1. For example a = 3.8759 yields a: = 0.1979 which is acceptable,

so a = 3.8759 is an estimate of the degree of exponential stability.

The corresponding estimate of the maximum time constant is 0.258 seconds
which is very close to the actual maximum time constant 0.256.

2.3 Nonautonomous Systems

The results of sections (2.1) and (2.2) for autonomous systems
can be extended to nonautonomous systems with some additional assumptions.
In order to identify the needed additional assumptions, we give the
nonautonomous version of Theorem 1.

Consider the nonautonomous system

x = f(t,x,z), (2-23a)

62 = §(t,x,z,€), (2-23b)
where téR+ and (x = 0, z = 0) is the unique equilibrium point in 8x x 82.
In this case it is more convenient to start by using a transformation

y = z - h(t,x) (2-24)
where h(t,x) is the unique root satisfying

0 = §(t,x,h(t,x),0). (2-25)

It is assumed that h(t,z) is continuously differentiable and that there
is a nondecreasing real-valued function p : R+ + R+ such that

||h(t.X)H :. p(H x H) vtst‘V’xeBx (2-26)

29

Hahn [18] showed that when (2-26) is satisfied h preserves uniform
asyptotic stability. The transformed system is given by
x = f(t,x,y) (2-27a)
69 = 9(t.X.y.s) (2-27b)
where
flnmfléfUJd+hUJH,
and

~ 3
9(t,X,.Y,€) é g(t,x,y + h(t,X),€) ‘5 '5'? h(t,X)

- e<vxn<t.x))Tf(t.x.v + h(t,x))

It is sufficient to study uniform asymptotic stability of the equilibrium
point of (2-27), namely, (x = 0, y = 0). The reduced system is obtained
by setting 8 = 0 in (2-27b) to get y = 0 which, when substituted in
(2-27a), results in

it = f(t,x,0) é fr(t,x). (2-28)

The boundary-layer system is defined as

%¥ = 9(t.x.y(r).0) (2-29)

where téR+ and xeBx are treated as fixed unknown parameters. We assume
2

that there are positive-definite, decrescent Lyapunov functions V(t,x)

and W(t,x,y, and classlar'functions h(x) and ¢(y) such that the

following equalities hold Vt€R+, VxeBx, vyeBy:

 

2V(t t,x) is said to be positive-definite if there is a strictly increasing
continous real- valued function v' (H x H such that v'(0) = 0 and
0 < v ’(fix x“) < V(t x) VXEB - {0}, and is said to be decrescent if
there is a strictly increasIng continous real- valued function v“(”xu)
such that v"(O) = 0,V "(X) > 030d V(t tax) < V (”X“) V’X 68x - {0}

3O

(1') g}- V(t,x) + (va(t.X))Tfr(t.X) _<_ “(1)200 0‘ > 0:

(11') (vyw(t.x.y>iTg(t.x.y.0) :.-a2¢2<y). a, > 0.

(”1"3) .3? W(t.x.y) + (vxw(t.X.y))Tf(t.X.y) :_ c1¢2(y) + CZWIXWY):

(III'-b) (vxv<t.x>>T[f(t.x.y> - f(t.x.0)1 £.B]A(x)¢(y).
and

(III'-C) (vyN(t.X.y))T[9(t.X.y.e) - 9(t.X.y.0)] _<_ EKIAZM + EKZ‘HXR (y).
Theorem 3: Suppose that Conditions (I')-(III‘) hold; let d be any
positive number such that 0 < d < l, and let 8*(d) be
End) g O‘io‘z
ali + [81(1 - d) + 82d12/4d(i - d)

whereB2 = K2+C2 and Y = K1 + C],

then the origin (x = 0, y = 0) is a uniformly asymptotically stable
equilibrium point of (2427) and
0(t,x,y) = (l - d)V(t,x) + dW(t,x,y)
is a Lyapunov function for (2-27)-
3. Stability Criteria for Linear Systems
Consider the linear singularly perturbed system

i(t) = A1](t)x(t) + A]2(t)y(t). (2—30a>

ev(t) = A,2(t)x(t> + A22(t)y(t). (2-30b)

where xER", yeRm and A22 is nonsingular for all t > 0. In particular,
|det(A22(t)| :,C3 > 0, t > 0 (2-31)

It is well known [c.f., 19] that if the reduced system
i(t) = Ao(t)x(t), (2-32)

31

where
Ao A12 A22A21’
is uniformly asymptotically stable and the boundary-layer system

= AZZItly (2-33)

is asymptotically stable uniformly in t, i.e.,

Re 1(A22(t)) :_-c4 < O ‘rt > 0, (2-34)

then the singularly perturbed system (2-30) is uniformly asymptotically
stable for sufficiently small a. A problem of practical significance

is the determination of an upper bound 8* such that the uniform asymptotic
stability of (2-30) is guaranteed for all e < 8*. Zien obtained 8*

for time-invariant systems [12] and Javid obtained it for time-varying
systems [13]. In both cases, 8* was obtained in terms of the transition
matrices ¢o(t,s) and ¢Z(T,0) of the reduced and boundary-layer systems,
respectively. In [12] and [I3], 6* was computed by writing explicit
expressions for the solution of (2-30). The same 5* of_[12, 13] can

be derived using composite Lyapunov stability analysis with linear-type

Lyapunov functions of the form

V(t,x) =J{ ||o0(t,t)x” di

for the reduced system, and a similar Lyapunov function for the boundary-
layer system. In this section we apply the quadratic-type Lyapunov
stability criterion of section 2 to compute 6* and compare our 5* with
the previous ones.
3.1 Time-Invariant Systems

Consider (2-30) with constant matrices. Let Re A(A0).< 0, Re h(AZZ)

< 0 and let PC > 0 and P2 > 0 be the solutions of the Lyapunov equations

32

T

POAO + AOPO = -21n, (2-35)
P A + A TP = -21 (2-36)
2 22 22 2 m'

It can be verified that

                     

V(x) = %~xTPOx (2-37)
and
W(x ) =-l ( +-A'1 x)TP ( + A 1A x) (2-38)
" 2 y 22A2i 2 y 22 Ax21
satisfy the conditions of Theorem 1 withw =H x H and 0(y-h(x))=
It follows that 8* is given by
6* ‘ T (2 39)
Y + 8182
where
Y T '"’2 A22A2iA i2 H. (2'40)
8
1 = (IPOA 12 M (2-41)
and
_ -1
B2 ‘ " P2A22A2iAo N- (2'42)
The upper bound obtained by Zien [12] is given by
* 1
c 3 (2-43)
2"3 T "4
where
M2 g./. ”exp(Aot)A]2"dt, (2-44)
0
W./: ”e‘P(A22t)A22A 21Aolldt’ (2-45)

and

33

M4 =f ||exp(A22t)A£;_A2]A]2|| dt (246)
Ne observeothat computing e* of (2-39) is much easier than computing
5* of (2-43) because it requires merely solving algebraic Lyapunov
equations; there is no need for finding transition matrices or performing
integrations. To compare the two upper bounds, recall that PO and P2

can be expressed as

°'.' 1' -

P0 =- f(exp(Aot)) (exp(Aot))dt, (2-47)
0

p2 = f(exp(A22t))T(exp(Aot))dt. (2-48)
0

Substituting (2-47) and (2-48) in (2-40)-(2-42) shows a great similarity
between 8], 82 and Y on one hand and M2, M3 and M4 on the other hand.
There are two differences. First, the integrals in (2-40)-(2-42) depend
quadratically on the transition matrices while those in (2-44)-(2-46)
depend linearly on them. Second, in (2-40)-(2-42) integration is per-
formed first followed by norm computations, while in (2-44)-(2-46) norms
are computed first followed by integration. while we cannot make a
general statement about the comparison between the two upper bounds,

we expect, in view of the above differences, that 5* of (2-39) will

be less conservative in most cases than that of (2-43), especially in
high-dimensional problems. This is the case in the example given below

which was solved by Zien [12].

Example 5: Let

A -o.2 +0.2j) 0 cf) ' o 0‘
8 ’ A 3 , A = g
1‘ o -o.5 ‘2 0.5 o 2‘ -k 0

34

-l l
A = , ke[0,10]
22 (LA) "A:)

2* of (2-39) is 2* = 0.2935 while that of (2-43) is 5* = 0.0826.
3.2 Time-Varying Systems

Consider the time-varying system (2-30) and suppose that the reduced
system (2-32) is uniformly asymptotically stable and the boundary-layer
system (2-33) is asymptotically stable uniformly in t. Let Po(t) > 0

be the solution of the Lyapunov differential equation
° _ T
-Po(t) - Po(t)Ao(t) + Ao(t)Po(t) + 21, ‘ (2-49)
and P2(t) > 0 be the solution of the algebraic Lyapunov equation
T
P2(t)A2(t) + A2(t)P2(t) = -21m. (2-50)

Condition (2-34) quarantees that inf A (P2(t)) >c5 :>0. Following
:9 min '—
the treatment of nonautonomous systems that has been presented in section

2.3, we start by using a transformation

~

y(t) = m + AggmAumxit) 9 arm + L(t)X(t). (2-51)

The transformed system is

iit) = Ao(t)x(t) + Anumt). (2-52a)

silt) = €th + L(tle(t)]x(t) + [Azzm +eL(t)A,2(tny<t).
(2-52b)
It is assumed that L(t) is uniformly bounded for all t > 0. This
guarantees that (2-26) is satisfied. Therefore, it is sufficient to
study the stability of (2-52). Moreover, it is assumed that A0, A12,
A22, A22 and L are uniformly bounded for all t > 0. This implies that
Po(t), P2(t) and 22(t) are uniformly bounded for all t > 0. Now, it

can be verified that

35

V(t,x) = % xTPo(t)x (2-53)
and
l .1 ~
W(t.y) = g y P2(t)y (2-54)
satisfy the following conditions of Theorem 3 with 0(x) = H x H and

~ ~ * o
0(Y) = H y H. It follows that e is given by

 

* _ l
E ‘ vm+ 8182 (2-55)
where
v= ing nézmu + HP2(t)L(t)A]2(t)IIJ . (2-55)
B] = :up [IIPo(t)A]2(t)II 1. (2-57)
and
82 = sup [iipzmmti + L(t)Ao(t))|]. (2-58)
t

Based on arguments similar to those of the time-invariant case, it is
expected that the upper bound : given by (2-55) will, in most cases,
be less conservative than the one given by Javid [13].
4. Agfiynchronous Machine Example

The stability criterion of section 2 will be employed to study
the stability of a synchronous generator connected to an infinite bus.
There has been quite a bit of literature on the use of Lyapunov's method
to analyze transient stability of power systems (for a survey see
[20] and for recent developments see [21]). In the standard approach,
a generator is represented by a second-order model, usually referred
to as the classical model, where the state variables are 5 and w 8 (g;
5 being the angle between the generated voltage and some reference.

The need for higher models has been recognized [20] and several studies

36

that employ higher order models have been conducted. Most of these
studies are restricted to one machine connected to an infinite bus.
The simplest higher order model that can be considered is a third-order
model, known as the one-axis model, in which a field-flux decay E&
is taken into consideration. Siddiqee [14] (see also [22]) studied
the stability of one generator connected to an infinite bus using the
one-axis model and came up with a Lyapunov function that can be used
to estimate the domain of attraction. More recently, Sasaki [23] criti-
cized Siddiqee's work on the basis that it did not take into considera-
tion that the state variables have different speeds, and proposed an
alternative approach. Between the work of Siddiqee and the criticism
of Sasaki we found an interesting problem where the use of singular
perturbations may be useful.

We consider a synchronous generator connected to an infinite bus

through a transmission line (Fig. (4)).

r X
6 e e E

fig.~(4)

The one-axis model (see [14], [20]<n~ [23]) is given by

- (x. + x.) (x. - x.)
. V cos 6 + E i (2-59a)
' ' = .. -(——————-y + 1———'
s _ w (2-59b)
’ E'V
. . (2-59c)
Mm = -Dm + Pm - xd + e Sln 6,

37

where
130 = direct-axis Open circuit transient time constant;
xd = direct-axis synchronous reactance of the generator;
x8 = direct-axis transient reactance of the generator;
Ed = instantaneous voltage proportional to field-flux linkage;
6 = angle between voltage of infinite bus and Ea;
D = damping coefficients of the generator;

EFD = field voltage (assumed constant);

Pm = mechanical input to the generator (assumed constant);
w = slip velocity of the generator;

xe = reactance of the transmission line;

V = voltage of infinite bus.

In writing equation (2-59) we have employed some standard assumptions,
namely that the transmission line resistance is negligible, a second
order harmonic term on the right hand side of (2-59c), which results

from saliency, is negligible and that x& ==xé (q-axis synchronous re-
actance). Had the field-flux decay been neglected, Ea would have been
constant and equations (2-59b) and (2-59c) would give the classical model.
Siddiqee [14] found a Lyapunov function, which will be given later in
this section, and used it to estimate the domain of attraction. The know-
lege of the domain of attraction can then be used to estimate the critical
clearance time, which we briefly define by means of a simple example.
Consider Fig. (5) in which a single machine is connected to an infinite

bus through two parallel transmission lines A and B. Suppose that a

short circuit fault takes place on line B. Equation (2-59) is written

 

38

for the post-fault system with line B disconnected and an estimate of

the domain of attraction is obtained. Trajectories of the faulted

system are obtained by integrating equations describing the faulted
system starting at the pre-fault conditions. The critical clearance

time is taken as the time when the trajectories of the faulted system

hit the boundary of the estimated domain of attraction. The idea is that
if the fault is cleared any time before the critical clearance time,

the state of the system will be inside the domain of attraction, hence

the system trajectory will approach the stable equilibrium point.

(:::>___. infinflm
' bus

generator ‘ B

 

.short
cucun

fig- (5)

Sasaki's criticism of Siddiqee's work was based on the observation
that the time constant Ida is large relative to the critical clearance
a is greater than five seconds while the critical
clearance time is less than one second). Therefore, EA will move

time (typically, rd

slowly and changes in EA during the fault will be small (typically, less
than 0.3 per unit). He suggested that treating Ea as a state variable

in estimating the domain of attraction might not be appropriate since

extending the domain of attraction in the direction of the Eé-axis,
which would be at the expense of its extension in the direction of the
6-axis, would not be beneficial in computing the critical clearance time.

He has proposed an alternative method in which equations (2-59b) and (2-59c)

39

are used as a model for the machine but with Ea treated as a varying
parameter. He, then, gave a Lyapunov function that should work for

all the values of EA and used it to estimate the domain of attraction.
Sasaki's method, however, is not theoretically justified because he
overlooked that changes in EA would result in changes in the equilibrium
point of (2-59b) and (2-59c) and did not take those changes into considera-
tion. It is interesting, however, that Sasaki's viewpoint of treating

the slow variable as a varying parameter is identical to the way the
boundary-layer system is defined in our singular perturbation method.

We will apply our stability criterion of section 2 to study the
stability of the equilibrium of (2-59). The first task is to bring (2-59)
into the singularly perturbed form. Let EA’ 3 and 5' denote the stable
equilibrium point of (2-59). The state variables are taken as x= ES; :Eé,
y1 = 6 -'3 and y2 = w -'5, so that x will be the slow variable. In

order to have the singularly perturbed form (l), we define a as e = A/Tdo

and change the time scale from t to t/Tdo to obtain

 

x = -ax + b[cos(y1 + '5) - cos ‘6'] 9 f(x,y) (2-60a)
. _ é ,
ey - yz - 91(x.y) (2-60b)
a) = - Ayz - c[(l + x)sin(y1 + 3) - sin 3] e92(x,y) (2-60c)
where
x + x x + x'
e d d d 1 D
x + xd xe + xd M(xe + xd) M

The reduced system is obtained by setting 6 = 0 in (2-60) to get
. -l sin 3 -—
h](X) 51" (w) - (S

y = h(x) = = (2-6l)

40

which when substituted in (2-60a) yields
x = -ax + b[cos(h](x) + §)-cos 3] 9 -ax + bN(x) (2-62)

The boundary-layer system is defined by

Cl5’1 _

—E?'_ y2’ (2-63a)
dy ._ ._

_a% = - Ayz + c[(l + x)sin(y1 + 6) - sin 6] (2-63b)

In order for the reduced and boundary-layer systems to be well defined,
as well as for other reasons that will be apparent later, we restrict

x and y1 to a region defined by the following inequalities
x > - e g , (2-64a)

-(h + h](x) + 23) 5'y1 < n - (h1(x) + 23) (2-64b)

[h-hnhnguh+E)-gMMU)+aiimh-hnhfi
(2-64c)

where the positive numbers n and 6 will be Specified later; 6 should

satisfy ‘_
sin 6
l l<l

l - e
To study the stability of the reduced system (2-62) we choose

x
V(x) =f [a o - bN(o)]do (2-65)
0

as a Lyapunov function. From the definition of N(x) it follows that

N(0) = O, xN(x) > 0 and

 

 

 

 

IAA—Al = A sin'é’
3x \I 2 __2 l + x
(1 + x) - (sin 5)

where

41

2

 

 

KN = 1 Sin E-
\/(I - 6)2 - (sin?)2 A - 6

We assume that the positive number 6 can be chosen such that KN < (%).
If follows that V(x) is positive-definite. The derivative of V(x)
along the trajectory of (2-62) is given by

3V 2
V = x (-ax + bN(x)) = -(ax - bN(x)) .

Therefore, Condition (1) is satisfied with h(x) = lax - bN(x)| and
a] = l. Next, we consider the boundary-layer system (2-63). This system
has equilibrium at y, = h](x) and y, = 0. We choose a Lure-type

Lyapunov function W(x,y) given by

W(x,y) = g— (y1 - hm)2 + y2(.v1 - h1(x))+ ——

where n > l and

Mx(o) = sin(o + 3) - sin(h](x) + 5).

Inequalities (2—64a) and(2-64b) guarantee that W(x,y) is positive-definite.

The derivative of W(x,y) along the trajectories of (2-63) is given by

Q.

—”— = (VYW(x,y))T9(Xa.Y) = -(n - mg

T

- C(l + X)(y1 - h1(x))Mx(yi).

Using inequality (2-64c) we get

42

A comparison function for the boundary-layer system is taken as

lyl-h](X)l
¢(y - h(X)) =
Elyzl
where g is an arbitrary positive number that plays an important role
in improving the upper bound. With the above choice of the comparison
function and with the choice n = l + c(l - e)ng2 , Condition (II) is
satisfied with 02 = c(l - B)n. It can be shown that Conditions (III-a)

and (III-b) hold and we have

 

J“

_ l l = l 2 2 nc 2 =

3h

where K1 is an upper bound on §_l_, which is given by
x

 

 

Thus, all the conditions of Theorem l hold and we conclude that the
equilibrium point of (2-59) is asymptotically stable for all e < 5*(d)

for any d€(0,l) where

3*(d) = c(l - B)n€

 

 

l l ‘l 2 2 22.2]

(2-67)

43

Moreover, 0(x,y), given by

v(XsY) = (1 ' d)Jf[ao ‘ bN(0)]d + d[%‘(y] ' hi(X))2 + y2(y1 ' h](x))
0 y

l
.9. 2 DE. + - -
+ 2A yZ + A (l x)}f Mx(o)do], (2 68)
h1(X)
is a Lyapunov function.

In (2-67) and (2-68), there are four unspecified positive parameters
namely, d, e, n and g. The first three should be chosen to satisfy
certain restrictions that were imposed throughout the derivation, while
the fourth parameter 5 is arbitrary. In making these choices we are

guided by the requirement that 2*(d) (given by (2'57)) Sh0U1d be greater

than a, and by our desire to obtain a large estimate of the domain of
attraction. Let us illustrate that by a numerical example. Consider

the numerical data used by Siddiqee [14]: M = 147 x 10'4, x8 ..0,3,

Xe = 0.4, Xd = l.lS, Tdo = 6.6, Pm = 0.815 and E = 1.22, where M is

FD
in per unit power second squared per radian, T00 is in seconds and all
other parameters are per unit (pu). In addition, we take A = %-=
4 (SEC)-]. Choosing d = 0.01, 9'= 0.42, n = 0.35 and E = 0.1 yields
e*(0.01) = 0.l844 which is acceptable since l/Tao = 0.1515. Hence

0(x,y) is given by
0(x,y) = 0.99 V(x) + 0.01 W(x,y)

An estimate of the domain of attraction can be obtained by means of

closed Lyapunov surfaces as in (2-9). However, a better estimate can
be obtained by means of open Lyapunov surfaces [24]. Let C and D be
two points on the line x = «e such that for any point on the line CD

the derivative x is p051tive and let Cmax = min{v(xc,yc). h(nyD)}.

44

It can be shown that the set
53‘ {X,)’|v(X,Y) i Cmax and x 1 -B}

is included in the domain of attraction. A sketch of the set.2’is
shown in Fig. (6); shown also is a sketch of the domain of attraction
obtained by Siddiqee's Lyapunov function. From Fig. (6), it is
apparent that we succeeded in getting a better domain of attraction in
the direction of the yl-axis at the expense of reducing its size in the
direction of the x-axis. The choice of the parameters d, e, n and 5,
that we used here, was obtained after several trials. Further
improvement is possible.

Let us use this example to explore another angle of using a
singular perturbation decompositions in Lyapunov stability analysis.
In the stability criterion of Theorem 1, singular perturbations have
been employed for two different purposes. First, they were used to
define separate reduced and boundary-layer systems whose Lyapunov
functions were used to form a candidate Lyapunov function for the full
system. This process simplifies the search for Lyapunov functions since
it is done at the level of the lower-order subsystems. Second, the
singularly perturbed form of the system together with inequalities
(I)-(III) were used to show that the derivative of the candidate
Lyapunov function is bounded by a negative upper bound. However, even
if inequalities (I)-(III) do not hold, computing the derivative of the
candidate Lyapunov function, obtained via singular perturbation de:
composition, along the trajectories of the full system might still
show that it is a valid Lyapunov function. Such computations will,

in general, be difficult to carry out but in some special cases that

45

might be possible. In the case of the synchronous machine example
in-hand a candidate Lyapunov function for the boundary-layer system

.could be the energy-type Lyapunov function given by

y1
W(XsY) =12-yg + c(l + M] Mx(o)do. (2-69)
h1(><)

This function, though, does not satisfy the conditions of Theorem 1
since its derivative along the solution of boundary-layer systems is
only negative semi-definite so that Condition II is not satisfied.

However, we may still consider

33(x.y) = (l - d)V(X) + dW(x.y) ' (2-70)

as a candidate Lyapunov function. It is interesting to note that com-
puting the derivative of'? along the trajectories of the full system
(2-60) shows that there is a unique choice of d, namely d = 525’

which makes v negative semidefinite. Moreover, it can be checked that
the trivial solution is the only solution of 0 = O. Thus'U is a
Lyapunov function for all e > 0. What is more interesting is to observe

(9%21 M is the Lyapunov function used by

that C'with the choice d =
Siddiqee [l4]. In comparing the two Lyapunov functions 0 and b’we

note that in 3 there is no freedom in forming the composite Lyapunov
function while in 0(x,y) the parameter d is free and can be used to
generate a family of composite Lyapunov functions; that is why we were
able to pick d that gave us a better shape for the domain of attraction.
The price paid for that freedom in choosing d is the restriction of the
range of e to (O, e*(d)). Getting an acceptable 5*(d) is hinged on
having A = D/H not too small. With the numerical values we used here

we need A.: l.

46

To illustrate the effect of A, Fig. (7) shows estimates of the

domain of attraction for A = l,2 and Table (l) gives corresponding values

of d, e, n, g and e*(d).

 

 

 

 

 

 

 

 

*
A d B n E E (d)
1 0.0072 0.38 0.544 0.05 0.15186
2 0.007 0.4 0.45 0.05 0.169267
Table (1)

As A gets smaller the improvement of the estimate of the domain of

attraction over Siddiqee's result becomes less significant, because

the choices of the parameters d, e, n and g are primarily directed

*
towards improving c (d). The requirement AI: 1 is realistic. Although

for uncontrolled machines, when D accounts only for field damping, a

typical value of A would be as low as 0.2 [25], for a feedback

controlled machine, in which 0 accounts for damping introduced by

power system stabilizers, typical values of A would be as high as l0 [26].

47

Appendix: Proof of Lemma l

 

The proof of Lemma 1 is similar to the proof of the well-known Converse
Theorem [27] except that extra work is needed to verify condition (III).
Letél(t,x) be the trajectory of the reduced system (3) starting at initial
point x at time t = O, and let s(t,y;x) be the trajectory of the boundary-
layer system (4) starting at initial point y at time t = 0. By assumption,

we have

-c t
".fllt,X)” .3 c3” x ”e 4 , vxeBx (A-l)

and

T

, V(x,y)€BxxBy.
(A-Z)

C5
IIS(r.y;X) - h(X)” < c5 Ily - h(x )e”

Consider conceptual Lyapunov functions V(x) and W(x,y) defined by

T

V(x) =j ll-Q”(t.x)u Zdt (A-3)
and 0

T

W(x,y) =f [M an - h(x112ds.(A-4)
It follows frog the Converse Theorem [27] that

(vxvoandx) :. -..," x 112, Vxeex. (A-S)

“Vx V(x) )H :_ Te1 "xll, -Vx£Bx, (A-6)

(v y wx.( yTn sx< .y) _<_ «12 My - h(x ll! 2 v<x.y)eexxsy. 01-7)
and

llvyW(X.y)l| _<_ 92 II)! - h(X)” . V(X.y)€BxxBy. (A-8)

,

Therefore, Conditions (I) and (II) are satisfied. To verify inequality

(IIIa) consider

48

T
(VXW(X.y))Tf(X.y) = 2[[5(Tay;x) - h(X)]T[vx(S(t.y;X) - h(X))]f(X.y)dt.

Using that f, g and h are continuously differentialbe on the closed
bounded set BxxBy, which implies that vx(s - h) is uniformly bounded,

we get that

‘ T
(vxw(x.y)>Tf<x.y> 1 1.x," y - h(x)“ W2 nxn Wham) - h(x)“ a.
0 (A-9)
where f(x,y) was taken as
f(x,y) = f(x,y) - f(x,h(x)) + f(x,h(x)). (A-lO)
Using (A-2) in (A-9) yields

c -c T
(VXH(X.y))Tf(X.y) = 3g (1 - e 6

6 >95 11y - h<x>u 2 +05 IIXII uy - h(leJ

Hence, inequality (III-a) holds. Finally, (III-b) follows form (A-6)

and the continuous differentiability of f w.r.t. y.

49

50') I

 

Fig. (1)

Domain of. attraction
corresponding to e‘

l.

50

 

 

Fig. (2)

.83

51

"A

 

 

 

 

Our domain of attraction

 

------- Siddiquee's domain of attraction

Fig. (6)

52

 

 

 

 

...... _....-x:1
>52

 

(7)

Fig.

CHAPTER III
CLOSENESS OF THE TRAJECTORIES OF THE SINGULARLY PERTURBED
SYSTEM TO THE TRAJECTORIES OF ITS SLOW AND FAST SUBSYSTEMS

l. Introduction
Simplifying a mathematical model of a physical system has been
an old engineering practice. For instance in circuit theory parasitic
reactances are often freely omitted from circuit models to form an
approximate model. The reason for such a simplication is that parasitics
not only increase the complexity of the system but often lead to equations
with widely separated time constants. Such systems, called stiff differ-
ential equations, may introduce severe computational difficulties, [28-29]
requiring the use of an implicit integration scheme. In practice the
justification of this engineering simplification lies in the fact that
"it works“. However, it happens in some cases that it leads to
results not only unprecise but even qualitatively incorrect. Therefore
it becomes necessary to verify the validity of such an approximation.
Consider a physical system described by a nonlinear singularly

perturbed system

ll
X

i = f(x,y) X(to) o (3-la)

ey = 9(X.y.€) y(to) yo (3-lb)

53

54

where c is small positive parameter. An approximate model is defined
by ignoring parasitics, that is setting a = 0 and removing initial con-

dition y(to) = y0 to obtain

x = f(x,y) x(t ) = x (3-2a)
O = 9(X9yao) (3'2b)

The justification for such an approximation on a finite interval

of time was first studied by Tihonov [30], corrected later on by
Hoppenstead [3T], and was extended to the semi-infinite interval of

time [32]. There has been some other work in this area (eg. [33], [34],

[35]) which basically treat the same problem with different settings
and generalizations. However, the result of Tihonov-Hoppenstead,[30] ,
[32] remains as a fundamental result, so called initial value theorem,
in the singular perturbation theory. In [30], [32] a set of sufficient
conditions is given under which, as c + 0+, the solution of (3-l) converges
to the solution of (3-2)., This convergence of course cannot happen
over the whole interval, finite or infinite, due to discrepancy of initial
conditions of (3-l) and (3-2). Therefore, a boundary layer correction
should be provided to take care of the initial finite jump.

In this chapter we show that our stability result in chapter 2
provides, as a by product, an initial value theorem result. That is
our stability requirements guarantee the uniform convergence of the
solutions of (3-l) to the solutions of (3-2) with a boundary layer correct-
ion. Our results unlike the results of [30], [32], [35] are not local,

that is a set of initial conditions is given so that for trajectories

originating from initial points in this set convergence property hold.

55

Therefore, in a sense it generalizes the result of Tihonov-Hoppensteadt,
[30], [32] and Habets [35].
2. Problem Formulation and Main Results

Consider an initial-value problem of the form

x = f(x,y) xeB}:Rn x(to) = x0 (3-3a)
6)" = 9(X.y.€) yéByCRm y(to) = yo (3-3b)

where f and g are assumed to be continuous functions and the origin

(x = O, y = 0) is the unique equilibrium point in BxxBy. Our purpose
is to investigate the behavior of the solution of (3-3), denoted by
(x€(t), y€(t)), as e + 0+ for to §_t < m. In studying the behavior

of the solution of (3-3) two associated systems, so called reduced and
boundary layer systems, are defined. The reduced system is obtained

by formally setting c = 0, and removing the initial condition y(to) = yo~
inj(3-3). This gives

x = f(x,y) x(t ) = x (3-4a)
0 = 9(Xsys0) (3-4b)

Assuming that in Bx and By (3-4b) has a unique continuous root

y = h(x), the reduced system is rewritten as

i = f(x,h(x)) 9 fr(x) x(to) = x0 (3-5)

Moreover we assume (35) has a solution on [t0,w) which we denote by

xs(t). A boundary-layer system is obtained by stretching the time scale

, ' t - t
through transformation 6 = E. o and then setting 2 = 0 in the result.

 

 

AThe smybol Bx indicates a closed sphere centered at x = 0, By is
defined in the same way.

56
This gives

$5 = 9(X. yo), 0) n0) = yo (3-6)

where x is treated as a parameter which takes value in Bx' We also
assume that the solution of (3—6) for x = x0 exists on [t0,oo) and is
denoted by yf(t,xo).

We assume all the conditions of our stability result in chapter
2, namely 1, II and III, holdwith comparison functions v(-) and o(-)
which belong to class or functions. Moreover we suppose that
LR = {xeBx|V(x) 5-V0} is in the domain of attraction of the reduced
system and LB(x) = {yeBy|W(x,y) :_w0} is in the domain of attraction
of the boundary-layer system, where from the Conditions of our stability
theorem the existence of wO independent of x is guaranteed. Then the set

L = {(x,y) Bx xeByl o(x,y);: min((l-d)vo + d wo)} (3-7)

is in the domain of attraction of the full system (3-3) where 0(x,v)
is the composite Lyapunov function defined in chapter l.

Our first result is given by the following theorem.
Theorem 1: Suppose (x0, yo)eL. Then, as e + 0+, x€(t) converges
uniformly to xs(t) on [to,m) and y€(t) converges uniformly to h(xS(t))
on all closed subset of [to,m).
3399:: The proof of this theorem consists of three steps. In the first
step we use the Lyapunov function W to define a "tube" in [to,w) x
BxxBy which is invariant with respect to the solution of (3-3) for
sufficiently small 6. In the next step we show that if (x0, yo)6L than
for sufficiently small a the solution of (3-3) intersects the "tube“

defined in the first step at an arbitrary time t1 > to. Finally in

57

the last step we show convergence of x€(t) and y€(t) to X5(t) and

h(xs(t)) respectively.

Step l;
~ . *
Lemma l: Given a positive number wo < w0 there eXist so such that for

*
any c < co the tube described by;

r, =11x.y1 B x B 1 W(x,y) :11, :woi

o x y
is invarant with respect to the solution of (3-3), that is, any solution
(x€(t), y€(t)) which meets Th6 cannot leave it there after.
firggfi: Since our stability conditions hold, there exist 8* so that
for e < e* we can assume x€(t) is inside a ball, that is there exists

r > 0 such thatl|x€(t)H < r. Next we consider the boundary of the set

3IW' = {(X syoéBx X By 1 ”(xoy) = WO

0
On the boundary BIW' we have
0
1'1= 1vx w1x.y11Tf<x.y1 + my W(X.y))T(-:; 9(X.y.e))
: c] K<1>:(||y-h(x)|1) + c2 11(IIXH) (Hy-h(x)”) - 1:- ¢Z(Hy-h(X)l|)
+ Mlly- X)H) + K2 wUIXH) Uly-h(X)H)
= (c1 + K1 -—§12 (111-1001) + (c2+1<211(11x111(111-1101111

Let us define

a = inf 01y - h(XlH [W(x,y) = W0}

notice that t > 0. Next we define

01

* 2 a (3'8)
+ K1+ T)‘ ((2211) M") + K2 “1.1)

58

 

e: = min (5*, 6:). (3-9)
Now for any 6 < a; we have

. (c2+K2)w(r r) 2

w:- 1n) luwmum1+mgg1mw1mwmnm

f. - (c2+K2)wr 1110" M (W (X KM“) (CZ +K2)¢'(HXH)¢(H.Y'h(x)H) :0

and the result follows Q.E.D.

Step 2:

In this step we want to show that for sufficiently small a the
solution of (3-3) in fact meets the tube rhb. To show this it is suf-
ficient to show that for some arbitrary time t1 close to to, we can
make||y€(t]) - h(x (t]))||arbitrarily small by choosing 8 small enough.
We need the following lemma;

*
Lemma 2: Let n > 0 and t1 > to, then for sufficiently small 6 we have

*
lly€(t]) - h(x (t]))H :_n for some to < t] < t].

 

t-t

Proof: We start by applying a time transformation T = 0 to
—— . 8
(3-3) and we get

dx _ f A~ -

d?'- e (x,y) x€(0) - xo (3-lOa)

d 1A

'fi=90dfi) ygm=yo (me

/\

where Q;( ) and y€(r ) denote the solution of (3-10). First observe
that 2;(T)= x€(t) and y8 (I ) = yE(t). Next we consider the boundary

layer system at x0, namely.

5% = 9(xo. y(T). 0) yf(0. x0) = yo (3-ll)

59

By our assumption in stability theorem,(3-1l) is uniformly asymptotically

stable. Therefore there exist T(n) such that

lyfh, x0) - h(xo)| _<_ 11/3 n > T01). (3-12)

(since ybfLB(xo)). Now we fix I] so that r] > T(n). From continuous
dependence of solutions of differential equations.on parameters [36,
page 24, theorem 3.4], it follows that, for sufficiently small a, XE(T)
and y€(r) exist and are continuous with respect to 5. Thus there exists

* *
52 such that for e < 62 we have:

”96(1'1) ‘ .Yf(T]9 X0)“ 5. Yl/3 (3-13)
tl ' to -
Now observe that T] =-———;———, so given that T] is fixed we can always

choose 8 sufficiently small so that t1 corresponding to T] be arbitrarily

* *
close to to. Namely let t1 be given then there exist 63 such that

' *_ , *
for e <.e3 we have t0 < t1 < t1. From continuity of h, it follows
*
that there exist 84 so that for e < a: we have,

Hh(x€(t])) - h(xo)H _<_ n/3 . (3-14)

*
0’

* *

i *
Let 8; = min(8 82, E3, 54), then for e < 85 we have

11ye1t11 - h(xe1t1111 11111811211 - h(xo111 + llh(x€(t1)) - 1.1110111
5.1114121) - 1,13. x0111 +11yf1n. x01 - h(x0111
+11h1x (t1) - h1x0111
:_”/3 + n/3 + n/3 = n

where the last step followed from (3-l2), (3-l3) and (3-14) and this

conclude the proof of lemma 2.

60

Our conclusion from steps 1 and 2 is summarized in the following

lemma.

* *
Lemma 3: LEtTl>'0 and t1 > to be given. Then there existe5 so that

<
I r E E:
o 5

111,1t1- h(x,u))” _<_ n- for t1. t < ..

Proof: It follows from lemma 1 and lemma 2 by choosing W6 sufficiently

small.

Step 3:

In this step we are at the position to prove convergence result.
First, since the reduced system is uniformly asymptotically stable,

there exist Tin) such that
HXS(t)ll§_n/2 for T(n)< t < m

Also uniform asymptotic stability of (3-3) implies that there exist
Tln) such that

le (t)H : n/2 for T10) < t < w

Let
A ~

T'(n) = max ”(n). Th1}.
then we have

||xE(t) - xs(t)H < n for T'(n) < t < w (3-l5)
Next we observe that x = x€(t) satisfies

i = f(X. y€(t)), X(to + erll = x0 + 5(6) (3-16)

where 6(5) + 0 as e + 0+ and r] is as chosen in step 2. By a well-known

theorem on continuous dependence of solutions on parameters and on

61

initial conditions, [36, page 24, lemma 3.l.] and in view of the result
of step 2 and continuity of f, it follows that x€(t), + xs(t), as

e + 0+ uniformly on [to, T'(n)]. So in view of (3-l5) it follows that
x€(t) + xs(t), as e + 0 uniformly on [to’ w). Having established
uniform convergence of x€(t) to xs(t) on [to’ m), then it follows from
step (2) that y€(t) + h(xs(t)), as E + O+ uniformly on [t], w), and
this conclude the proof of Theorem 1.

Next we state the result which extend the convergence pr0perties
of the solution of (3-3) to the whole interval. '
Theorem 2: Suppose all the conditions of Theorem l hold. Then
y€(t) . h(xs(t)) + yf(., x0) - h(xo), as e + 0*, uniformly on
o’ m)‘

Proof: Let n'> 0 be given, since (3-ll) is uniformly asymptotical stable,

[t

there exist T](n') so that

||yf(r, x0) - h(xo)H : n‘/2 for T > T1(n') (3-1?)

Now we fix I] > T](n‘) and consider
g x
- A
d: - 9 (x€(T). y. 6) y€(0) = yo (3-18)
Notice that'§é(1) satisfies (3-l8). By continuous dependence of the
solution on parameters we have §;(T) + yf(T’ x0), as 8 + 0+, uniformly

on [0, 1]]. Furthermore h(xs(er + to)) + h(xo) as e + 0+ uniformly

on [0, 1]], So we conclude that there exist 6; so that for e < 8;

we have
Hy€(T) - yf(T. xolH :_ n'/2 for 0 :.T E.T] (3-19)

Hh(XS(ET + to)) - h(xo)H :_n'/2 for o : T :,r1 (3-20)

62

From (3-l9) and (3-20) we get

/\
ily€(r) - h(xs(to + all) - yf(r. x0) + h(xo)|L: d for 0 :_T-:.T]
(3-2l)
Having established (3-2l), we proceed by observing that from Theorem

*
l and (3-2l) there exist €10 such that for any 8 < 6:0 we have

Nloo

||§;(T) ‘ h(§;(T)) f yf(Ta X0) + h(xo)ll E_ n' for 0 :_r :_T]

(3-22)
Next from (3-l7) and (3-22) it follows that there exist 6:] such that

for e < 6:] we have
H9;(r]) - h(?;(rll)ll.: 2n'. (3-23)

By choosing n' small enough we can force the point (QE(T]), y;(T]))
to be inside the tube rhb with arbitrary W6. Now using Lemma l we
conclude that for sufficiently small 6, the trajectory (x}(1), §;(T))
never leaves the tube thereafter. That is, given n there exist 6:2 so

that for e < 5:2

’\

Hy€(r) - h(QgfiT))ll : n/Z for T > T] (3-24)

Finally from (3-l7), (3-2l), (3-24), and Theorem l we conclude

that there exist 6:3 such that for e < e:3 we have

lly€(t) - h(xs(t)) - yf(T. x0) + h(xo)H : n for to.: t < m.

and this concludes the proof of Theorem 2.

CHAPTER IV
STABILIZATION AND REGULATION-COMPOSITE CONTROL

l. Introduction -

 

One of the most interesting results in the control literature on
singular perturbations is the two time scale design of linear regulator
problems presented by Chow and Kokotovic [ 2]. That design introduced
the concept of composite control for singularly perturbed systems.
Design of a stabilizing or near optimal controller can be broken down
into the design of feeback controllers for two separate subsystems, the
so called slow and fast subsystems. A composite control is formed by
simply adding these two controllers. The advantages of this technique
are well known [ l]. Stabilizing and near optimal stabilizing feedback
designs for a class of nonlinear systems which are linear in the fast
state and control input has been already studied by Chow and Kokotovic
in [37], [38] and [39]. The contribution of this paper is two fold.
First, we extend the two-stage feedback design of Chow and Kokotovic
[37] to a wider class of nonlinear systems where nonlinearity in the
fast state and control input is permitted. For this result we use a new
stability criterion for nonlinear singularly perturbed systems presented
in chapter 2. Second, we broaden the context of nonlinear regulator
problems for singularly perturbed systems by adopting an approach which
is different from that of [37], [33] and [39]. We don't necessarily re-

quire optimal controllers for slow subsystems as it is the case in [39].

63

64

We start off with a slow controller which can be optimal or near optimal
with respect to the slow regulator cost function. Then we design a
stabilizing controller for the fast subsystem and form the composite
control. The value of the cost function when the composite control is
applied to the full order system is studied and shown to be close to

the slow regulator cost function. This result, which we call "near

performance, is obtained for different cases corresponding to different
stability requirements on the slow and fast subsystems. Our results
generalize Chow and Kokotovic's result [39] and cover it as a special
case. Dropping the optimality requirements of the slow controller is
significant because finding optimal control for nonlinear systems is a
prohibitively difficult task, while there are ways to generate near
optimal controllers. Another feature of our result is giving explicit
expressions for the upper bounds on the singular perturbation parameter,
a, under which the near performance results hold. Thus, the near
performance results are not restricted to c + 0 as is the case with

the near Optimal result of [39].

The organization of this chapter is as follows. In section 2 we
give the composite control design procedure and show its stabilizing
property. In section 3 we apply the composite control to the regulator
problem. It is shown that the composite control yields a bounded cost;
then near performance results are given for three cases. First, we
consider the case when both the closed loop slow and fast subsystems
are asymptotically stable. Second, the stability requirement on the
closed loop fast subsystem is strengthened to exponential stability.
Finally, both the slow and fast closed loop subsystems are required to

be exponentially stable.

65

2. Composite Control

 

Consider the nonlinear singularly perturbed system

r (4-la)1

x.
H

f(x,y,u) xfBiiRn, ueR

all 9(X.y.U.€). yeBycn'". 14-lb1

where c > 0 is a small singular perturbation parameter. We obtain a

slow subsystem by formally setting a = O in (4-l] to get

x.
H

f(x,y,uS) (4-2a)

C
II

9(XsysUSo0) (4'2b)

where the subscript, s, for the control input signifies control for the
slow subsystem which we refer to as the slow control. Assuming that
(4-2b) has a unique root y = h(x,us) in the region of our interest, (4-2)

can be rewritten as

x = f(x,h(x,us),us) (4-3)

To derive the fast subsystem or the boundary layer system we assume

that the control input to the slow subsystem, us, is known as a feed-

back function of x. We define the fast subsystem as

9x -
dT 9(X.y.uS + uf.0) (4 4)

where T - t/e is a stretched time scale. The vector x€Bx is treated

as a fixed unknown parameter and uf is the control input, which we refer

 

1The symbol Bx indicates a closed sphere centered at x = 0, By is

defined in the same way.

66

to as the fast control. The decomposition of (4—l) into two lower
order subsystems is exactly as in [37]. However, due to the non-
linearity of (4-l) in the fast state and control input, the fast sub-

system (4'4) depends on U It is straight forward to show that if g

s'
is linear in y and u, the effect of uS on the fast subsystem cancels
out by shifting equilibrium to y = 0. As it will be seen later, the
dependence of the fast subsystem on uS makes the design procedure

sequential in nature.

Design Procedure:

 

Step l: DeSlgn a slow control uS = M(x) for the slow subsystem such
that the closed loop slow subsystem has a unique asymptotically stable
equilibrium point at x = 0 in Bx’ and there is a Lyapunov function

Van + R+ such that for all xeBx;

(VXV(x))Tf(x,h(x,M(x)),M(x)) _<_ - a1 1121).), a1 > 0
where h(x) is a scalar-valued function of x that vanishes at x = 0 and
is different from zero for all other xeBx.
Stgp_g: With the knowledge of u5 = M(x) design a fast control
uf = TTx,y) such that
(i) r(x,y) vanishes at y = h(x,M(x)), namely F(x,h(x,M(x))) = 0.

(11 9(Xsy.M(x) + le,y),0) = 0 has a unique root, y = H(x), in the
region of our interest Bx X By.
(iii) The closed loop fast subsystem is asymptotically stable

uniformly in x and there is a Lyapunov function Hanme + R+

such that for all xeBX and yeBy

wwuantoamnl+mnwnls-%afi

y .Y ' h(X,M(X)), 0‘2 > 09

67

where ¢(y - h(x,M(x)) is a scalar-valued function of
(y - h(x,M(x)))€ Rm that vanishes at y = h(x,M(x)) and is
different from zero at all other xeBx and yeBy.
Stgp_§: Verify the following conditions on the interaction of the
slow and fast states.
(a) (VXW(X.y))Tf(X.y.M(X) + r1x,y11 : e1 1211 - h(x,M(x))) +
c2 h(X) 1(1' - h(x,M(X)));s
(b1 (vXV(x))T[f(X.y.M(x) +r(x.y)) - f1x.h<x.M1x11,M<x11] s
81 w(x) c1(y - h(x,M(XH);
1c) (VyN(x.y))T[g(X.y.M(x) + r(x.y). e1 - glx.y.M<x1 + f‘(x.y).0)] s
ck, 1211 - h(x,M1x111 + .12 1<x1 111 - h(x,M(x111;

where the constants c1, c2, 81, k1 and k2 are nonnegative and the in-
equalities should hold for all xeBx and yeBy. We refer to the conditions
of this step as the "Interaction Conditions."

§flgpllgz Form the composite control as

uc = M(x) + r(x,y) (4-5)

The composite control uc is stabilizing as shown by the following
theorem.
Theorem l: Let d be a positive number such that D < d < l, and let

*
e (d) be the positive number given by

8*(d) = a] a2 2 (4-6)
a11 + [81(1 - d) + 82d] /4d(l — d1

 

*
where 82 = k2 + c2, and y = k1 + c]. Then, for all e < e (d), the
origin (x = 0, y = 0) is an asymptotically stable equilibrium point of

the closed l00p system

68

i = f(x,y,uc) (4-78)
1:1./ = 9(X.y.uc.e) (4-7b)
Moreover

, v(X.y) = (l - d)V(x) + dN(X.y)

is a Lyapunov function for this closed loop system.

Proof: The slow subsystem of the closed loop system (4-7) is given by

X f(x,y,uc)

C)
II

9(X9Yaucso) (4‘8)

This is the same as the closed loop slow subsystem
12: f x,h , -
( (x us),us). (4 9)

This.followssimply by observing that g(x,y,uc,0) = D has a unique
root y = H(x). But y = h(x,M(x)) is also a root of this equation.
Hence H(x) = h(x,M(x)) and the equivalence of (4-8) and (4-9) follows.
This observation is a slight generalization of the invariance property
of composite control shown by Suzuki [40]. With this observation
Theorem l follows by applying the stability criterion of chapter 2.
The stability requirements of the design procedure have been dis-
cussed in detail in chapter 2. Roughly speaking, the basic task is to
design controllers for the slow and fast subsystems so that the
corresponding closed loop subsystems have quadratic type Lyapunov
functions. For an asymptically stable system x = f(x), the function
V(x) is said to be a quadratic type Lyapunov function, [10], if
T

(v V)

x f(X) : - aw2(x) and IVXV(x)l i h(x) for some scalar comparison

function ¢(.). The gradient requirement, however, does not appear

69

explicitly in our conditions and, as it will be seen later in Example l,
it could be violated. The interaction conditions in step 3 determine
the permissible interaction between the slow and fast states, and they
are basically smoothness requirements. One implicit feature of the
design procedure is that slow and fast controllers should be designed
to preserve the slow and fast nature of the system, or in other words,
‘ to keep what is slow slow and what is fast fast. For example, a high-
gain feedback slow controller that might force some of the slow variables
to be fast should not be permitted. This feature, in a sense, restricts
the design procedure to situations where the design objective is
compatible with the slow—fast nature of the Open-loop system. If this
implicit requirement is not satisfied we will, typically, end up with
designs for which the upper bound 5*(d) is smaller than the physical
value of e which determines the ratio between the slow and fast motion
in the open loop systems.

Finally the freedom in choosing the parameter d can be employed
to improve estimates of the upper bound on the singular perturbation
parameter s, to improve estimates of the domain of attraction, or to
achieve a compromise between the two kinds of estimates. For further dis-
cussions related to the choice of d the readers are referred to chapter 2.

Corollary 1: If we assume that f, g, and h are continuous and w and o

 

belong to class;fi’function, then the closed loop system resulted from
applying composite control, namely (4-7), satisfies the requirements of
Theorems l and 2 of chapter 3. Therefore the uniform convergence property
of the trajectories of the full system (4-7) to those of‘the slow sub-
system (4-3) and the fast subsystem (4-4), as e + 0+, hold.

70

The design procedure is illustrated by the following example.

Example l:
Consider:
x=xy3 xeBx=[-l,l]
.1: y + u yeBy = [- 1/2, l/2]

It is desired to design a feedback control law to stabilize the system
at the equilibrium point x = O, y = 0. Notice that the desired
stabilization cannot be achieved by linearization at the equilibrium
point because of an uncontrollable pole at the origin. We go through
the design procedure as follows.

Step 1: The slow subsystem is given by:

We chose u5 = x4/3. The closed 100p slow subsystem is x = -x5 and with

the Lyapunov function V(x) = (l/6)x6 we get w(x) = |x|5 and a] = 1.
Observe that V(x) satisfies |VXV| = h(x), so it is quadratic-type in
the sense of [l0].

Step 2: The fast subsystem is defined by:

gx=y+“«s+“1==l/+x4/3+u

dT f

4/3).

We choose uf = -3(y + x The closed loop fast subsystem is

.9! = —2(y + x4/3). The choice N = (l/2)(y + X4/3lz 4/3i

dT
and a2 = 2.

yield5<p= Iy + x

Step 3: It is straight forward to verify that the interaction conditions
hold with 81 = 7/4, 82 = 4/3, and y = 7/3.

71

Step 4: The composite control is formed as

4/3

= 4/3
c s Uf X ' 3(y + x

4/3) = -3y _ 2x

Now the choice d= 21/37 yields a (2l/37) = 3/7, and the system con-
sidered is asymptotically stable for all e < 3/7.

In this example we could have chosen V = (l/4)x4 which yields _
w(x) = x4 and a1 = 1. This choice of Lyapunov function doesn't satisfy
the condition lvx V(x )l < h(x x), but the interaction conditions are
satisfied with the same constants, namely 81 = 7/4, 82 = 4/3, and y = 7/3.
This remark shows that our stability criterion may be applicable even
if the requirement lvx V(x )l < w(x x) is not satisfied.

3. Regulator Problem

 

This section is concerned with establishing a "near performance"
property of composite control. Suppose for solving a regulator problem
of a singularly perturbed system we first ignore the fast part, that is
we set a = O, and solve the resulting slow regulator problem to obtain
an optimal solution. Then, a stabilizing fast control is designed and
a composite control is formed. The question can now be posed whether,
as e + O, the cost of applying this composite control to the singularly
perturbed system tends to the cost of the slow regulator problem. To

investigate this question consider the regulator problem

i = ”1 x,y, u), x(0) = x0, ueRr (4-lOa)

cy = g(x x,y, u,c) y(0) = y0 (4-l0b)

J =f L(x x,,y u) (4-l0c)
0

where L is a positive definite function with respect to its arguments.
A slow subproblem is obtained by formally setting 6 = O in (4-10), and

removing the initial condition y(0) = yo. Then (4-10) becomes

x = f(x,y,u), x(0) = x0 (4-lla)

0 = g(x,y,u,0) (4-llb)

J =fL(x,y,u) (4-llc)
0

As in Section 2 we assume that g(x,y,u,0) = O has a unique root

y = h(x,u) in BxXBy' With this assumption (4-ll) can be rewritten as:

x = f(x,h(x M) S), x(0) = x0 (4-l2a)
Js = J: L( (x, h( (x, uS ),us)dt (4-l2b)

Let us assume that there exists a feedback control, us, which yields a
bounded cost as for all initial conditions in a certain permissible
region B; (that is a region that would guarantee that the trajectory
of the closed loop subsystem remains in Bx)“ Such uS could be an
optimal control or a near-Optimal control obtained through some
approximation technique. With the knowledge of us we define the fast

subsystem as in section 2 and consider the stabilization problem of

finding uf to stabilize

91.
(IT g(x x.s.yu +uf.

Our task is to show a near performance property of the composite control,

0). (4-13)

uC = uS + uf, in the sense that as e + O the cost of full order system
(4-lO) with uC converges to JS, the cost when e = 0. We have classified
the results of this section into three parts depending on the stability
requirements on the slow and fast subsystems.

£2111

Assume that:

Al: The slow subproblem has a solution uS = M(x) which yields a

bounded cost Js(xo) for all initial conditions X06 Bx' Moreover Js(x)

satisfies

73

2( J (x))Tf(x,h(x,M(x),M(x)) 5_- a1 w2(x) where do,

x) i (vx s

o
a] are positive constants and w(-) is a scalar-valued positive-definite

function of x that vanishes at x = 0.

A2: For the fast subproblem there exists a fast feedback control,
uf = r(x,y), which satisfies all the requirements of step 2 of section
2 with ¢(-) being positive-definite.

A3: The interaction conditions as stated in step 3 of section 2
are satisfied, with inequality (b) strengthened by replacing the left-
hand side by its magnitude.

Assumptions Al through A3 are essentially the same as in section 2.
Notice that Js(x) is a Lyapunov function for the system (4-lla), (4-llb),

and it satisfies

(v a <x))Tf(x,n<x.M<x)),M<x))

x s 'L(Xah(X,M(X)),M(x)), (4-14)

and it plays the role of V(x) in section 2.
Now we form the composite control uc = uS + uf. From Theorem l it

follows that uC is stabilizing, that is, the closed l00p system has an
asymptotically stable equilibrium point at the origin.

The asymptotic stability of the equilibrium point is not sufficient
to guarantee that integral cost of the type (4-lOc) will be bounded
(for a counter example refer to [39]). So we need to show that the
application of the composite control, uc, yields a bounded cost. Denote
the cost by Juc(x,y) where x,y are taken as initial conditions. To
show the boundedness of Juc we need the following assumption.

A4: The function L satisfies the following condition for all

(x,y)eB‘y X By
|L<x,y,M<x) + f(x,y)) - L(x,h(x,M(x)),M(x))l : 6, Jo - h(x,M(x)))

+ 62 w(X) ¢(y - h(x,M(X)))

74

where a, and 52 are nonnegative constants. This assumption expresses
the permissible contribution of the fast state to the function L.

Lemma l: Suppose assumptions Al-A4 hold. Let d and e be positive
0.

numbers such that O < d < 1 and e > (1 _ d0

) a . Then there exist
l

 

* * ‘k
e (d,e) < e (d) such that for any 6 < e (e,d) we have

J (x ,yo) :_e((l - d)JS(xo) + dW(x

uc o ))s V(X0,yo)€D, .

o’yo

where D is the set of all initial conditions which give rise to
asymptotically stable trajectories that are restricted to Bx X By. For
example, 0 can be taken as the domain of attraction determined by the
interior of a closed Lyapunov surface in BX X By.

Proof: We consider;

x.
H

f(x,y,uC) (4-lSa)

g(x,y,uc,e) (4-15b)

co

Juc(x,y) =fc L(x,y,uc)do,

a;

where Y,Y’denote the trajectories of (4-l5a), (4-l5b) with initial
point (t,x,y), that is x(t) = x, ylt) = y, and we let 0 be the running

time. First we show that for all (x,y)eBX X By

L(x,y,uc) + e 5(x,y) < 0 (4—16)

where 5 denotes the derivative of composite Lyapunov function v =
(l - d)JS(x) + dW(x,y), along the trajectories of (4-15). In Appendix

l it is shown that

L(X,y,M(X) + ITX.y)) + e 6‘: 0

......J

 

 

 

T
W(X) ' w(x)
- I}
My - h(X.M( )))' My - h(x,M(XH)
where
B B 6
l 2 2
8(1 - d)a].fl- a0 -e(] - d) ——-2— - ed _2. .. _2.
1'l = B a o o
l 2 2 2
-8(1-d)—§-8d—2*-—“2— Ed(“—E‘-{)-5«|
Set
0. (S
. _ o . _ . 2
a 1 - e a] - 1 - d, a 2 - e a2, 8 1 e B] + 1 _ d ,
51
8.2 = e 829 Y' e Y + ——d", and
al a!
e*(e,d) - 1 2 (447)

o'] y' + [e'1(l - d) + e'zd12/4d(i - d)

* *

Notice that e (e,d) < e (d) (compare (4-l7) with (4-6)). It is straight
*

forward to show that. for any a < e (e,d).T1 is positive definite and

therefore (4-l6) holds. Integrating (4-16) from t to w, we obtain;

JUC(X,Y) + G[V(X(®),y(m)) - v(x,y)] :.O

*
Now since 5 < c (d) the trajectories of (4-15) are asymptotically stable,
that is x(w) = O, y(m) = 0 and the result follows.

* 'k *
Remark 1: Notice that e (d,e) + e (d) as e + w and e (d,e) + O as

0. *
e + ll——_37—;— , which shows that we can have s (d,e) arbitrarily
- l

*
close to e (d) and still have bounded cost.
Having established that uc yields a bounded cost we are in a

position to consider the "Near Performance" property of the composite

76

control. Our procedure to show near performance is motivated by [4T].
We start by defining an auxiliary cost function, J], for (4-l5a) and

(4-l5b) as follows

J,<x.y) =ft (L(Yfi'mc) + as} flég—zflido (4-18)

where s} is an arbitrary positive number. It is easy to show that for

'k
any a < e (e,d) we have;

J](x,y) = JUC(X.y) - 631W(x,y) (4-19)

Now, define Q(x,y) as

Q(x,y) é qJS(S) - J1(x,y); (4-20)

where q is an arbitrary positive number such that q > l and as is the
cost of the slow subproblem (4-12) with the initial point (t,x). We
want to show that-Vq > 1, Q(x,y) is positive definite. To do this

first observe that J](x,y) satisfies

9 .
(va])Tfuc + (vle)T —%9 + L + 53 -9E = 0, (4-21)

9 4 Q
where fuc - f(x,y,uc), g - g(x,y,uc,e), L L(x,y,uc), and for

UC UC

the sake of brevity we have dropped the arguments of J1(x,y) and
W(x,y). Substituting (4-20) in (4-2l) we get;

T T TgUC AdW_
q(vas) fuc ' (VXQ) fuc ' (va) _E—'+ Luc + ESl at - 0’
(4-22)
Using (4-l4), (4-22) can be rewritten as
(vaf +(vQ)Tg—UE+L-L Jedi-
x uc y 6 q s uc E 1 dt
q(v J )T(f - f ) = 0, (4-23)

77

where L5 9 L(x,h(x,M(x)), M(x)), and fs 9 f(x,h(x,M(x)),M(x)). Now we
consider another auxiliary cost function, J2, for the system (4-l5a),

(4-l5b), given by;

a)
_ AdN T
J2(x,y) 7.]. {qu ' Luc - ESl'do- ' q(vxds) (fuc 7 175)}do
t (4-24)

*
Lemma 2: For any a < e (e,d), J2 converges. Moreover J2(x,y) = Q(x,y)

V(x,y)€D.

Proof: We have;

J2(x,y) = .f (Luc + e’s‘1 ggio. + of (LS - (vxasflwuc - fs))do
t t
or
J (x y) = -J (x y) + of (L - (v J for - i no. (4-25)
2 ’ l ’ t s x s uc 5

Taking derivative of Js(x) along the trajectories of the full

order system (4-lSa), (4-l5b) gives

d (os<x>) = (v J )Ti

'5? x 5 UC
_ T
- (vxds) (fuc ' fs + fs)
_ T
- -LS + (vxds) (fUC - fs), (4-26)

where in the last step we have used (4-l4). Substituting (4-26) in
(4-25) we get

cn

J2(x.y) = -J,(x,y) - oft-£- (JS('>?))do

Since (4-l5) is asymptotically stable it follows that

78

J2(x,y) -J](X.y) + qu(x) (4-27)

or

J2(X.y) = Q(x,y). Q.E.D.

If we establish that the integrand of J2 is positive definite,
then we have shown that Q(x,y) is positive definite. This is done in
the following lemma.

Lemma 3: For any q > l and e sufficiently small we can always choose
s; such that J2(x,y) be positive definite.

Proof: In Appendix 2 we have shown that for any (x,y)e Bx X B :

y
T /
». dW T J W
ql's 7 Luc 7 ESl at 7 Q(vxds) (fuc 7 fs) 3 ( T2 (>’

where T2 is given by;
(q - l) o1 -(l/2)(o2 + es1 82 + q 8])

A /\
S

'(1/2)(52 + 5?] 82 + q 8]) 1 a2 - 6] - as] v

(4-28)
By inspection we observe that, for sufficiently small a and for q > l,
'3] can always be chosen to make 1% and subsequently J2, positive
definite.
Positive definiteness of Q(x,y) implies that J1 < qJS, or using
(4-19), Juc < qu + 631W(x,y). In order to get the sharpest bound on
Juc we should choose q very close to one. However as q approaches one

the required £3, according to lemma 3, grows and becomes unbounded. To

overcome this problem and get the sharpest bound on Juc’ we set

79

q = l + 50 e1/2 and’s‘1 = c71/251, where So’ and 51 are positive

numbers. To establish our result corresponding to these choices of

50 and 51 we need to define some parameters. Let;

_ 2
a - (l/4) ($081 + 51 82) + s0 s1 a1v,
b = (1/2) (62 + 8]) (s0 8] + 51 82) + 50 a] o]
c = s s o o - (l/4) (o + B )2
0 l l 2 2 l '

Then the upper bound on Juc is summarized in the following lemma.

Lemma 4: Suppose assumptions Al through A4 hold. Let so and 51 be

 

 

2
(62 + 8])
arbitrary positive numbers such that s s > Then there
0 l 4 a] a2
*
exists e (50,51) given by
2
57(5 5 ) = b ‘ U b2 + 46C (4-29)
0’ l 2a
* 'k
such that for any a < min(e (50,51), 5 (e,d)),and for all initial
conditions (x0,y0)€D we have;
J (x y ) < J (x ) + el/2(s J (x > + s w<x y )) (4-30)
uc o’ o —- s o o s o l o’ 0

Now we use the same machinery in order to get a lower bound on Juc'

We start off by defining an auxiliary function, J3, for the system

(4-lSa), (4-l5b) as follows

CD

J3(x.y) =Jf (L(iuy,u ) - es 95§§¥31Jdo. (4-31)
t

where s] is an arbitrary positive number. This cost function plays

the role of J1 in finding upper bound on Juc' For any e < e*(e,d)

we have

80

J3(x.y) = (x,y) + e§}W(x.y). (4-32)

JUC

Define P(x,y) as

P(x,y) 9 J3<x.y> - flush). (4-33)

where’q is a positive number such that q‘< l. As before the next step
would be to show that P(x,y) is positive definite. Observe that

J3(x,y) satisfies

T9

T uc ~ dW _
(VxJB) fuc + (VyJB) 7E. T Luc 7 ESl _t'7 0
Substituting (4-33) in above we get:
T T gUC A T
(pr) fuc + (vyP) _E_ + q(vxds) (fuc 7 fs)
A. ~ dN _
+ Luc - qLS - es1 —E-- 0, (4-34)

where we have dropped the argument of P(x,y). We define another
auxiliary cost function, J4, for the system (4-lSa) and (4-l5b) which
plays the role of J2 in the derivation of the upper bound on Juc’ and
is given by;

(D

J4(x,y) =f {Luc - :51
t
*

Lemma 5: For any e < e (e,d), J4 converges.

(1'0.
0 Z

T
S) (fuc - fs)}do (4-35)

- qLS + q(va
Moreover J4(x;y) = P(x,y) V(x,y)£D.
Proof: Similar to the proof of Lemma 2.

The next step is to show that the integrand of J4 is positive
definite and therefore J4 is positive definite. This is done in the

following lemma.

81

Lemma 6: For any 0 < a‘< l and for sufficiently small a we can always
choose E, such that J4(x,y) is positive definite.

Proof: As in Appendix 2, it can be shown that for any (x,y)e Bx X By

where 13 is given by

(1 - 6M1 ‘ -(i/2) (<52 + as] s, + q 8])

~ A ~
-(T/2) (62 + ES1 82 + q 8]) S1 a2 - 6] - ESTY

It is obvious that for any 0 < fi‘< l and for sufficiently small
e, we can choose 3] so that 13 be positive definite and the result
follows. 7 Q.E.D.

Now for the same reason that was stated in the derivation of

upper bound on Juc we set fi‘= l - 21/250 and?1 = e71/2s1. The lower
bound on Juc is given by the following lemma.

Lemma 7: Suppose assumptions Al through A4 hold. Let so, 51 and

*

*
6 (50,51) be the same as in Lemma 4. Then for any e < min(e (50,51),

5*(e,d)), and for all initial conditions (x0,yo)€D the following holds

J (x .yo);JS(x):e1/2( l).

uc o o Son(X ) T s]W(x

o o’yo

Proof: With the above choices of q and a} the only difference between
l/2
5081/2
. . l/2 .
in T215 replaced by e 5081/21n T3. Thus det (T3) :det (T2)
*

and for all e < 6 (50,51), 15 is positive definite. The rest of the'

1% and 13 is that in the outer diagonal element the term -e

proof is exactly the same as in Lemma 4.

82

Combining Lemmas 4 and 7 gives Theorem 2, the result of this part.

Theorem 2: Suppose assumptions Al through A4 hold. Let d,e,so, and 51

be arbitrary positive numbers such that

a (<3 +8
0 2 l
0 < d < 1,8 > (1 _ d) a , and $051 > 4 a

l

 

 

'k *
Then for any a < min(e (e,d), e (so,s])) and for all initial conditions

(xo,y0)eD the following inequality holds

1/2(

Id (x0.y0) - J (x )1 is H.

s 0 SOJS(x ) + s W(x

uc o l o’yo

Moreover e*(e,d) and e7(so,s]) are given by expressions (4-l7) and (4-29)
respectively.
Remark 2: As we stated before, by appropriately choosing e, 2*(e,d)
can be made arbitrarily close to 27(d). This shows that in Theorem 2
we can ignore e and e7(e,d), and focus on parameters d,so, and 51.
By doing so the result of Theorem 2 holds for any e < min(e7(so,s]),
e (d)).
Remark 3: The choice of the parameter d is usually made based on
stability considerations and does not involve the cost function. The
choice of the parameters so and s], on the other hand, depends on the
cost function since both 50 and 51 appear in the bounds on Juc‘
Generally, the choice of s0 and 51 will be guided by two requirements.
The first one is to get a large 8*(SO,S]) and the second one is to get
a sharp bound on Juc' The two requirements are usually contradicting
and a compromise should be sought.
Part II:

In this part we consider a "Near Performance" property of the

composite control when the closed loop fast subsystem is exponentially

83

stable. It turns out that in this case the closeness of Juc and JS

are of order of e unlike the result in Part I which was order of El/Z.

We start by stating the extra assumptions required for this case.
‘A6: The Lyapunov function for the closed l00p fast subsystem,

W(x,y), satisfies

2(

51¢ y - h(x,M(x))) _<_w(x.y) : €2¢2(y - h(x,M(x)))

where g] and 52 are positive constants.
56:

(i) w(-) is differentiable and satisfies;

|\7x h(x)] :_X < m VxeBX
where X is some positive constant.

(ii) lf(x,y,M(X) + P(x,y))l _<_ C3 ¢(y - h(x,M(X)) + C4 MX).

where C3 and C4 are positive constants.

Assumption A5 implies exponential stability of the fast subsystem.

Assumption A6 restricts f and u further over what has been required

by assumptions Al-A4. However, for a large class of problems assumption

A6 will not add extra restrictions over the interaction conditions A5.
The machinery used to prove the result of this part is similar

to that one used in Part 1, although there are critical changes in

auxiliary cost functions used in the proof. We start by defining

auxiliary cost function 35 for the system (4-lSa) and (4-l5b) as follows

(1)

J5 =ft {L(mmc) + s, Pig—511 + ...-H 3% (W) W(x,y))mndo.

(4-37)
where 53 is an arbitrary positive number and H is a fixed constant to
be chosen later. It is straight forward to show that for any

* .
e < e (e,d) we have:

84

J5(X.y) = JUC

Define D(x,y) as

6(x,y) 9 qu(x) - J5(x.y), q > i

(x,y) - es3W(x,y) - eH v(x) (W(x,y))

1/2. V(x,y)€D

(4-38)

Repeating the derivations of Part I, it can be shown that for any

'k
a < e (e,d) and any (x,y)eD we have:

Q
~ _ dW __d 1/2
Q(x,y) -f {qu - Luc - as, as; - eH do (w )

('1’

T
- q(VXJS) (fuc - fs)}do

Moreover for any (x,y)e‘Bx X By

dW _51 1/2
QLS - LUC - €53 a—t' - 8H dt (ipW )

T
(V J )T(f f ) > w 'T w (4 39)
- q _ ' _
X S UC S - d) 4 q)
and T4 is given by
o(q - l) a] - 5 B2” -o.se
'74 = 2 51
-0.SG $3 0.2 - 6] - ESBY — EH ACBVEZ

where

‘H
G = e s B + —{ + HAV: C
< 3 2 27,52 2 4)

“2
+(Q'T)B]+ 627.781-(1/2)”

‘17:?

85

Notice that if we choose

2 a (o + B )
H = ‘l—:; 2 l (4-40)

(1

 

we get asymptotically, as e + O, sharper estimates on Juc in comparison

with those obtained in Part I. -In fact this has been the motivation

for adding the term eH ag-(o(x)w]/2

(x,y)) to the integrand of J5. So
we set q = l + 252, where 52 is an arbitrary positive number. Taking

H as in (4-40) we define the following parameters:

82H ~

YH
a — s o - —————- B = s B + -—————-+ XHC
l 2 l ’ l 2 l
2\( g] ZVEZ

4 E2

 

52 = 82. '32 = a2 ' “El, T = v + EiE§__E§
S3 S3
82H 5]

Then it can be shown that for any 52 >-—-——-———— and s3 775— there

2 E; a 2
exists e*(sz,s3) given by; 1 1

3 E
8*(sz,s3) =~~ l 2 ~ 2 (4-41)

* *
such that for any e < min(e (52,53), 8 (e,d)) we‘have;

W1/2(X.y)).

V(X.y)€D

Juc(x.y) :.JS(X) + c(szds(X) + s3W(X.y) + H J(X)

We can invoke the same machinery to get the lower bound on Juc’
which We are not going to repeat here. The result of this part can be
summarized in the following theorem.

Theorem 3: Suppose assumptions Al through A6 hold. Let d,e,sz, and 53

be arbitrary positive numbers which satisfy

86

a0 82H 5

T
O < d < T, e > (1 - d) a], $2 > o

>—

_____,S
2Ea13 2

* *
Then there exist a (e,d) and 5 (52,53) such that for any

 

* * .
g < min(e (52,53), e (e,d)) and for all initial conditions (xo,y0)eD

we have;

lduc(xo’yo) 7 ‘Js(xo)I 3-€(52Js(xo) + 53W(xo,yo) + H ¢(X0)w1/2(xo’yo))

Moreover e*(e,d) and e7(sz,s3) are given by expressions (4-l7) and
(4-4l), respectively.

The remarks 2 and 3 of Part I apply to this part as well. More-
over we have the following remark.
Remark 4: The choice of H in (4-40) was made to be independent of e,
however, we can choose H to set off-diagonal terms of matrix '74

equal to zero. This choice of H, denoted by H' is given by

H' = (es2 B] + 52 + B] + 853 82) x

a
(: 2 - FY _ AC4 52:)‘]
2 £2 2 62

which is dependent on 6. It is straight forward to show that for

 

 

82H'

2 and for any 5'3 > (5] + e7(d,e) H'AC3V €2)/(a2 - 6*(d.e)v)
‘51 0‘1

5'2 >

*
the matrix 1% is positive definite. Recalling that e (d,e) can be
*
made arbitrarily close t05:(d), the result of Theorem 3 with the
*
above choice of 5'2, 5'3, and H’ holds for any a < e (d). Notice that

H', 5'2, and 5'3 are greater than H, 52 and 53 as given in (4-40),

87

which implies that with the above choices we have improved the upper
bound on e but for sufficiently small e the choices of H, 52 and 53 used
in Theorem 3 yield a sharper bound on Juc'
Part III:

This part is concerned with the case when both closed l00p slow
and fast subsystems are exponentially stable. The result of this part
is similar to that obtained in Part 11 except that exponential
stability of the closed loop slow subsystem will allow us to drop
assumption A6. The exponential stability requirement on the slow sub-
system is stated by the following assumption.

AZ: The cost function for the slow regulator problem satisfies

p, J26) :Jsm :uz J26)

where p1 and p2 are positive constants.
The procedure to get the result of this part is almost the same
as in Part II. The basic difference is that the term eH a%(w(x)W]/2(x,y))
in the integrand of JS is replaced by eH'ag-(J51/2(x)W1/2(X,y)), where
H will be defined later. Denoting the free parameter s3 in Part II by

55 for this part, we define 5(x,y) as in (4-38). Then it can be shown

that inequality (4-39) is satisfied with 'T4 replaced by 1% which is

given by
68 7|: u
(q - i) a, - ——?—-———2- M2
2 5]
T5 7.7 58"” £2
A/Z 55 a2 - 6] - ESSY - 2 u]
where

Y NTZT- _ “1 VTET
V52 (“727

A = e55 82 + (q - l) B] + (1/2) eH

88

and H is taken as

2(62 + 81%] £2
0‘2 \(“1

By setting q = l + 554, where 54 is an arbitrary positive number, the

fi =

 

upper bound on Juc can be obtained. Using the machinery of Part I the
lower bound on Juc is similarly obtained. To state the result of this

part we define the following parameters

 

 

A - 82 V U2 ~ A - YHVU“ 01H 51
o - s o - -——————-H, B - s B + - ———————
l 4 l 2 l 4 l 2 2
51 E2 “2
B H t o
A~ _ A._ T 2 A. _ ‘_l
82 - 829 Y ‘ Y + 25 9 0'2 - 02 - $5
5 “1
'k
and take e (54,55) as
”o? 6?
* _ l 2
6 (54,55) ' A A (4-42)

/\ A 2
01Y + [B] + 8255] /455

Theorem 4: Suppose assumptions Al through A5 and A7 hold. Let
d,e,s4, and 55 be arbitrary positive numbers which satisfy
on BI‘Tu <5
0 2 “2, and 55 > —1

l - d ’54
( )0‘2 ZaIVg O‘2

* *
Then there exist a (e,d) and 5 (54,55) such that for any 5 <

 

0<d<T,6>

*

min(e (54,55), 5*(e,d)) and for all initial conditons (xo,yo)€D

we have

IJuc(xo.yo ) - Js(xo)| 3 c(s4ds(xo) + 5514(x0.y0)

+ HJ 1”( xo)w‘/2(x0,yo)).

89

'k *
Moreover E (e,d) and 6 (54,55) are given by expressions (4-l7) and

(4-42), respectively.
The result obtained in Parts I, II, and III also satisfy following

corollary.
Corollary 2: Suppose f, g, and h are continuous and w, o belong to class

 

vlffunctions. Then the closed loop system (4-l5) satisfy the conditions
of theorems l and 2 in chapter 3. Therefore the uniform convergence pro-
perty of the trajectories of the full regulator (4-l5) to the trajectories

of the slow regulator (4-l2) under the boundary layer correction holds.
The "Near Performance" results are illustrated by the following

examples.
Example 2: In this example we consider the class of singularly perturbed

systems studied by Chow and Kokotovic [39], namely

ueRr

x.
H

a](x) + A](x)y + B](x)u, x(0) = x0,

ey = a2(x) + A2(x)y + B2(x)u, y(0) y0

J =f PM + sT(X)y + yTM(X)y + uTR(X) Jdt
t

-1 . .
where a], a2, A], A2 , B], 82, P, s, M, and R are continuous with

respect to x in Bx’ Moreover, P + sTy + yTMy is a positive definite
function of its arguments x and y in 8x X By and R(x) and M(x) are
positive definite matrices for all xeBx. The slow subsystem is

given by

x= a (X) + BO(X)uS,

T

0
JS =.l7 [Po(x) + 25$(x)uS + uS R0(x)us]dt
t

90

where

o
B =3 -AA7]B
o l T 2 2
_ T -l T T -l -T
P0 — P - 5 A2 a2 + a2 (A2 ) MA2 a2
_ T T -l -1
SO ' 82 (A2 ) (MAZ az ' (1/2)S)
_ T T -l -l
RO - R + 82 (A2 ) MA2 ‘82

We make the following assumptions.
*
I - There exists a twice differential function JS (x) which satisfies

the Hamilton-Jacobi equation

This assumption implies that

U = -R 71(5 + (T/Z)B T(V J *))
s o o 0 X 5

*
is the minimizing control for the slow subsystem and JS is the
corresponding optimal cost.

11 - For all xe'Bx the following inequalities hold

. 2 * T—
(1) - do w (x) < (VXJS ) ao < - a] w (x), a0, a] > 0
(ll) |Eb| < a3 M(x) a3 > 0
(iii) )SOI < a4 W(X) a4 > O

91

where 36 = a0(x) + Bo(x)us = a0 - BORO71(so + (i/2)BOT(vaS*)), and
u(-) is a positive-definite scalar-valued function which vanishes at
x = 0
III - The pair (A2,Bz) is stabilizable uniformly in x in the sense that
there exist KF(x) such that min ReX(A22(x) + 82(x)KF(x)) < -0 < 0,
With above assumptions it can be shown that the requirements Al
through A6 of Theorem 3 hold. This example extends the result of [39]
in three directions. First, the fast control could be any stabilizing
control and not necessarily the solution of the fast optimal control
problem as in [39]; although the optimizing control is still the most
natural candidate. Second, upper bound on e and on the performance
criterion are provided. Third, the assumptions are weaker than those
of [39] (see Example 4). This can be seen by observing that h(x)

can be taken as [V We should, however, mention that the require-

xJSI.
ment [sol < o4|vaS| is not explicitly required in [39] although it
is implicit in the Hamilton-Jacobi equation and inequalities (i) and
(ii).

Example 3: This example is borrowed from Chow and Kokotovic [39].

Consider
i = -(3/4)x3 + y Bx = [-l/2, 1/2]
ey = -y + u
J =.j7 (X6 + (3/4)y2 + (1/4)U2)dt
O

The slow subproblem is

X = (3/4)x3 + uS

92

S

m
J = JT (x6 + usz)dt
o

and its Optimal solution is uS = -x3/2, which yields the optimal cost

Js(x) = x4/4. This cost function satisfies assumption A1 with
o(x) = |x|3 and a0 = a1 = 5/4. The next step is to stabilize the
fast subsystem, which is given by:

gx=-y+u +

_ 3
d1 5 ”f ’ 'Y'X (2 T u

f

The open loop fast subsystem is asymptotically stable and uf could
be taken zero but to be able to compare with the result of Chow and
Kokotovic [39], we use their choice of uf, namely, uf = -(y + x3/2).

The composite control is given by

With the choice of W(x,y) = (l/2)(y + x3/2)2, assumption A2 is satisfied
with o = [y + (l/2)x3| and a2 = 2. It is straight forward to show

that the interaction conditions, A3, are also satisfied with e] = l,

32 = l5/32, and v= 3/8. Now we can apply Theorem I. The largest

upper bound e*(d) provided by Theorem l is e*(d) = 8/3 which is

the result of the choice d = 32/47. However, in order to improve the
estimate of the domain of attraction we choose d = l/4 which yields
e*(T/4) = l.699 and Theorem l guarantees that for any e < l.699 the
closed 100p system with composite control uC is asymptotically stable,

and the corresponding composite Lyapunov function is given by

4

v(x,y) = (3/Te)x + (T/8)(y + (T/2)x3)2

93

Taking into consideration that xe-Bx = [-l/2, l/2], an estimate of the

domain of attraction is given by

b = {x4 + (2/3)(y + (i/2)x3)2 < l/l6}

These estimates are less conservative than those obtained in [39]. In
[39] the asymptotic stability is guaranteed for e < 480/88l and the

domain of attraction is obtained by

B = {x4 + (y + (i/2)x3)2 < l/l6}

Now we apply the result of section 2 to obtain bounds on Juc and
determine the range of e for which these bounds hold. Since the fast
subsystem is exponentially stable, bounds can be obtained by applying
the results of either Part I or Part II. We apply both parts.

To apply Theorem 2 of Part I, we observe that the assumption A4
is satisfied with o] = l, 52 = l/2. Choosing s0 = l and s1 = 4, we
get 57(5 ,51) = l.3794. By Theorem 2 we conclude that Ve < l.3794

o
and for all initial conditions (xo,yo)€D we have

|J (Xo,yo) - J (x )t _<_ 61/2904“ + 260 + (1/2)X03)2) (4-43)

UC SO

To apply Theorem 3 of Part II we observe that assumptions A5 and
. . . _ __l _ _ =
A6 are satisfied with g] - g2 - 2, A - 3/4, C3 — l and C4 5/4.
*
We choose 52 = l and s3 = 4 which yields 2 (52,53) = l.1277. From
Theorem 3 we conclude that for any e < 1.1277 and all initial conditions

(x0,yo)€D we have
4 3 2
IJUC(X0.yO) - JS(X0)| _<_ c[xo /4 + 2(yO + (l/2)XO )

+ 0.7SIXOI3 - ly0 + (1/2)X03l] (4-44)

94

We finish up this example by pointing out that, for sufficiently
small a, (4-44) is a sharper bound than (4-43). However, for values of
e not so small the comparison between (4-44) and (4-43) is not definite
and both should be considered.

Example 4: This example is from Chow and Kokotovic [38]. Consider

x = xy 3x = [-l,l]
ey = -y + u
an
J = J[ (x4 + yz/Z + u2/2)dt
0

The slow subproblem is given by

x = xuS
a:
_ 4 2
‘15-]; (x +us)dt

Solving for the optimal control of the slow subproblem we obtain
uS = -x2 and Js(s) = x2. So assumption Al is satisfied with h(x) =

x2 and a1 = 2. The fast subsystem is defined as

We choose the same feedback for the fast subsystem as in [38], namely,
uf = -(\f27- l)(y + x2). With the choice of W = (l/2)(y + x2)2
the assumption A2 is satisfied with o = |y + le and a2 = \f5-

Moreover the interaction conditon, A3, holds with 81 = 82 = v = 2.

Next we choose d = l/2, then from (4-6) we get e*(d) =\[%—u From

 

Theorem l it follows that for all e < i the composite control

95

uc = us + uf = - (\f2 - l)y - \f2'x2

is stabilizing. Moreover the estimate of the domain of attraction

corresponding to this choice of d is given by

2

0 = {x + (l/2)(y + x2)2

1 l}.

The fast subsystem in this example is exponentially stable, so
as in Example 2, we apply the result of both Parts I and II. To apply
the result of Part I, first we observe that assumption A4 holds
with a] = 52 = 2 - \[2T We choose 50 = l and s1 = l which yields
5*(so,s]) = 0.0233. From Theorem 2 we conclude that for any a < 0.0233

and for all initial conditons (xo,y0) ED we have

1J (x .yo) - J (x 11 1 9/2602 + (l/2)(y0 + x 2%)

uc o s o 0

To apply the result of Part II we observe that assumptions A5 and
A6 hold with X = 2, C3 = l, C4 = l and a] = 52 = l/2. We choose

= s3 = 4 and from (4-41) it follows that e*( ) = 0.118. From

52 52,53
Theorem 3 we conclude that Va < D.ll8 and for all initial conditions

(x0,yo)éD we have

2

2
. u

1.1 (x ,yo) -J (x )1 : 5(4x 2)2 + (4 -\/72)x02|y0 + x

+ 2 + x
uc o s 0 (yo

0 0

Finally it is pointed out that the cost function Js(x) is not
quadratic-type but it satisfies all of our requirements with
w(x)=¢ |VXJSI. This problem belongs to the class of problems con-
sidered in [39], but it doesn't satisfy the conditions of [39],
which is mainly due to the fact that u(x) is not equal to lvxdsl

which is an implicit requirement in [39].

96

Appendix l

 

From chapter 2 we have;

T
Q < - w ~ w (A-l)
— (J . D

where

~ (1 - d) a1 -(1 - d) 81/2 - d 32/2

(I

-(l - d) 81/2 - d 82/2 d(—§ - v)
(A-Z)

Also we can write

L(X.y,M(X) + P(x,y)) + ex? = L(X,y.M(X) + P(x,y))

- L(X.h(x,M(X)).M(X)) +
L(x,h(X.M(X)).M(X)) + ex)

Using (A-l) and assumptions Al and A4 we have

P
L(X,y,M(X) + P(x,y)) + ex} 1 a] 02 + 6va - e ()7

II
C“:
V

97

Appendix 2

 

From assumptions Al and (4-l4) it follows that

2
L5 7 al¢

Also taking derivative of W along trajectories of (4-l5) and using

assumptions A2 and A3 we will have

qLS-L -e/§]flé--q(vJ)T(f -f)

UC X 5 UC S
- .« dN T
7 (q 7 ])Ls T L5 7 Luc 7 651 d? 7 q(Vst) (fuc 7 fs)
_>_ (q ' 1) (111112 ‘ 51¢2 ‘ 524W + 9] (12932 ' €€1Y¢2

/\
' ES] 829$ ' q B1¢¢

CHAPTER V
DECENTRALIZED CONTROL, USING LOCAL
HIGH-GAIN STATE FEEDBACK

I. Introduction
Decentralized stabilization of the nonlinear interconnected system

xi = fi(xi) + Bi(gi(x],...,xN) + ui), i = l,2,...,N

is considered. This class of systems includes as a special case the
one studied by Davison [42]. It also includes the linear systems studied
by Ikeda and Siljak [43],Yamakami and Geromel [44], Young [45], Sezer
and Huseyin [46] and Bradshaw [47]. In [3] we showed that the use of
local high-gain state feedback control laws can stabilize this system
provided that N lower-order isolated subsystems are stabilizable. That
result was a conceptual one in the sense that it was shown to hold for
sufficiently large feedback gains. In this chapter we address the
important question of deterimining how large the gains should be in
order for the closed-loop system to possess certain stability properties.
In particular, we determine the gains needed so that the closed-loop
system is asymptotically stable, the gains needed so that it is asymp-
totically stable with a certain domain of attraction or the gains needed
so that it is exponentially stable with a certain degree of exponential
stability.

This chapter is organized as follows. In Section 2 we state the

problem and motivate the structure of high-gain feedback control. In

98

99

Section 3 the main asymptotic stability result is given and the choice
of the feedback gains is discussed. Section 4 contains an exponential
stability result and Section 5 summarizes the stabilization algorithm.
In Section 6 two examples are used to illustrate the procedure and
compare our results with some of the previous ones.
2. Problem Statement

Consider the nonlinear interconnected system

:2]. = fi(xi) + 81(gi(x) + 01.), i = l,2,...,N (5-1)

11 T‘ T

where xiéR i, uicR 1 and x T T

T .

= (x1, x2,...,xN). It 15 assumed that
xi = 0 is an equilibrium point of the ith isolated free subsystem

x. = f.(x ) i = l,2,...,N (5-2)
and that x = 0 is the unique point in the region of interest where
gi(x) vanishes. It is desired to design a decentralized state feedback
control law in the form

ui = hi(xi) (5-3)
such that hi(0) = 0 and x = 0 is an asymptotically stable equilibrium

point of the closed-loop system

A=qu1+mmwn+hwgn. 64)

It is assumed that the nonlinear functions fi’ 9i and hi satisfy the
conditions for existence and uniqueness of the solution of (5-4) and
that the matrix Bi has full rank. In designing the control law (5-3)
we follow the philosophy adopted in [42-47]; namely, ”i is designed to
stabilize the ith isolated subsystem with a large stability margin so
that the interconnected system (5-4) remains asymptotically stable in
spite of arbitrary but bounded interconnections. The main drawback

of this philosophy is neglecting any stabilizing effect of the intere

connections and treating all of them as if they were destabilizing,

100

leading naturally to conservative results. On the other hand, adopting
this philosophy alleviates the need for thorough modeling of the inter-
connections and results in feedback control laws that are robust with
respect to variations in those interconnections. Achieving large stability
margins for the isolated subsystems requires the use of high gain feedback.
We start by assuming high-gain feedback in the form

ui = kihi(xi) (5-5)

where ki is a positive scalar parameter. It is assumed that all the
coefficients of hi(xi) are of order one so that the order of magnitude
of the feedback function is determined by the order of ki which can

be chosen sufficiently large. To see clearly the effect of applying

the control (5-5), we follow [48] in using the state transformation

yl -
2i — Tixi (5-6)

m I“

where yiéR 1, ziéR 1, mi = "i - r, and Ti satisfies

0
TiBi = Gi (5-7)

with Si being an ri x r1 nonsingular matrix. The ith isolated subsystem

is transformed into

91 = ¢l(yl’ 2i)
. (5-8)
zi ‘ “iUi’ 2i) I Giui’
where ¢i and 7i are functions of Ti and Ti' Applying the ith control
law (5-5) to (5-8) and writing ki = l/Ei yields

9T = ¢l(yl’ 7]) (5‘9)

Eizi ‘ 7ifli(yi’ 2i) T Gihi(yi’ 2i)

10]

where h ( z ) = h T71 yi
i yi’ i i i °

2 .
1

Equation (5-9) resembles a singularly perturbed system. Therefore,
stability of (5-9), when Si is sufficiently small, can be investigated
using singular perturbation techniques presented in chapter 2. We de-
fine a reduced-order subsystem and a boundary-layer subsystem. The
reduced-order subsystem is obtained by setting 6i = 0 in (5-9) yielding

y, = 91(yi. 2,)

-)

'l

a. (5-10)
0 = hi(y,. z

where we have used the nonsingularity of Gi‘ It is crucial now to solve
the second equation of (S-lO) to get 21 as a function of yi. We simplify

this process by restricting hi to the form
hl(yl’ 2;) = Ml(zl ' ”1(yi)) (5‘11)

where Mi is an ri x r1 nonsingular matrix. With (5-ll) the reduced-

order subsystem is given by
T] = ¢i(yl’ ”1(yi)) (5-12)

and the boundary-layer subsystem is given by

Since asympotic stability of both the reduced-order and boundary-layer
subsystems is required for the asymptotic stability of the singularly
perturbed system when Ei is sufficiently small (chapter 2), "i(yi) should
be chosen to stabilize (5-12) and Mi should be chosen to satisfy
ReX(GiMi) < 0 (5-14)

We notice that (5-l4) can be always achieved since Si is nonsingular.

The ith control law is given by

=._l
u]. 71' Mi(z‘i - ni(yi)) . (5-15)

102

when (S-lS) is applied to the interconnected system (5-l), the closed-
loop system is given by
y- = My» 2-).
l i i 1 (5-l6)
$1271.: eini(yi, 2].) + 6161.817.” 2) + GlMl(zl - nl('yl))’
where 9i is a function of g1, GT and Ti’ and y, z are given by

T = (yT,,,,,y;), zT = (zT,...,z;). Conditions for the asymptotic

y
stability of (5-l6) will be derived in Section 3. The proof of the
main result in Section 3 utilizes the two-time-scale decomposition
and the composite Lyapunov function techniques [cf. chapter 2,

8 - 9].

3. Main Result

01.

Let 5y c:R 1 be a closed set that contains the origin. Assume
i

*
that the following conditions hold for every i = l,...,N.

(I) There exists a function ni(yi) and a Lyapunov function Vi(yi)
such that ‘VyieSi

(a) [vyiVi(yi)]T¢,-(yi. nib/1)) :-k,,J§<y,>.

(b) llvyiviwi)”: klzwl(yi)’

where wi(-) is a positive-definite function. Let Ci be a positive

number such that

Al
L1.-{y1.|V1.(y1.) : ci}csyi.

 

* ...
We use c1, d1, d., k.. and ktmn to denote positive scalars.

lO3

This condition implies the existence of a stabilizing feedback
control law such that the closed-l00p reduced-order subsystem

(5-l2) is asymptotically stable, has a quadratic-type Lyapunov
function and has L1 as an estimate of its domain of attraction.

(II) The functions o1(y1,z1), n1(y 21) satisfy Lipschitz conditions

1"

in z. i.e.,

i,

(a) (MA 1': 2.1) ' (b.1(y1': E71)”: k13"Z1- " E71" 9

21.11.

= l 3
*ry1, 21651, where S1 {y1eSyi and 215R '"21 - n1(y1)" :15

(b) ““1(.y1°9 2.1) ‘ ”1(3’1: 31')" :— k14))21-
.l.
‘1
(III) The functions 01(y1, n1(y1)), 91(y1, 01(y1)) and ”1(y1) satisfy
the following smoothness requirements 'vy1GS

(a) H"1(y1. n1(y1)|| fi.k15J1(y1),

i

T
(b) IIEVyiTl1-(yi)] 91(3’1: ”1(y1'))”f_k1°6'(’1(y1')9
(C) (IVyin1(y1-)”f_ kj7’

(IV) The functions 6 (y, z) and 0(y) satisfy the following interconnection

constraints ‘Vy, 263 where nT(

S = S1X,...xS

y) = (nT(y),...,n;(y)). and
N.

N
(a) ”Gfi My. n( MH< 2% k
J:

181J1(y1),
N

(b) ”(31-91(1', Z) - (31910. Z)|(:j§1ki9j"2j ‘ 31-”-

From (I-b) and (II-a) it follows that
(II-a)' (vyivH (y 1)) (J1 (y y1. 21) - 9.1(y1. 21)): e11v1(y1) H21 - 21”;

also from (II-a) and (III-c) it follows that

(III-C)' (v “(y ))T(J1(y1. 21)- ¢1(y1.'3

1,1911”: K110 "21° ' 21-”,

104

where e11 and K110 can be taken as K12 K1.3 and K13 K17, respectively.
However, direct evaluation of e11 and K110 by verifying inequalities
(II-a)‘ and (III-c)‘ rather than inequalities (I-b), (II-a) and (III-c)
will lead to less conservative results, since (II-a)‘ and (III-c)‘ are
the inequalities that will be used later. Before we proceed we need
to define some parameters and matrices. Let the symmetric matrix P.

1
be the positive definite solution of the Lyapunov equation

'1 _
P1(G1M1) + (61.111) P1 - -11 (5-17)

where I1 is the r1 x r1 identity matrix and M1 has been chosen to satisfy

(5-l4). We denote "Pi" by p1 and the minimum eigenvalues of P1 by'31.

 

LEt F11 = -2(k15 + ki6 + ki8i)’ tij = -2k18j’
Yii = 2(ki10 + K14 + k191)s Yij = 2k19j’
l
q.. = '1: Qi -Y "’
11 p181 Y111J=1J

and define the matrices Q, T, K and E as
Q = [qu]’ T = [tlJJ’ K = diag(k1])a E = diag(‘911)-

Taking 31, d1, 1 = l,...,N, as arbitrary positive numbers, we define

the matrices

D diag(d.), 5 = diag(d1), 5

gm + 0T0]
E
Q I“: lmE + DT)’

(K
((1
DO liK 1.1
R=2 )TVH/ OD r6

Our stability criterion is stated in the following theorem.

Theorem l: If assumptions (I)-(IV) hold and if the matrices 5 and (5K
- FT’G‘E‘) are positive-definite,then the origin (y = o, z = 0) is an

asymptotically stable equilibrium point of the closed-loop system

105

(5-l6). Moreover, the set L,

d.fi.€.
min ——1 1 1)

Lab, 2v(y, z) < min(min H.c.,
'_ i 1 1 1 pi

q,

where

is included in the domain of attraction.

13599:: Consider the composite Lyapunov function V(y, 2). It is shown
in Appendix A that the derivative of v along the trajectory of (5-16)
is bounded by

 
  

11m) T 11m)
. ¢N(yN) II’N(.YN)
V(y. 2) 1 "2] ? n111,11" R 112] 1 ”111111" vy. zes
"ZN ; nN(yN)" ”ZN ; ”M(yN)"
(5-19)

It is sufficient to show that R is positive definite, which is equivalent
to the positive-definiteness of both '6 and (fix - r5641“). Observing
that LCS completes the proof.

We discuss some aspects of Theorem l. First, it is always possible

to choose 81, €2"°"€N in such a way that the matrices a'and (5k - fmU'II)
are positive-definite. This can be seen by observing that the diagonal

elements of‘fi can be made as large as desired by using sufficiently

small 61's. Second, for the class of systems considered in this chapter
Theorem 1 is more general than our earlier result [3] because it provides
an asymptotic stability rather than exponential stability criterion.

It also allows for a wider class of interconnections. Third, the positive

parametersd1,...,dN,'31,...,dN, which are used as weights in the composite

106

Lyapunov function (5-l8), are arbitrary. The freedom in choosing these
parameters may be used to establish certain stability properties for
the closed-loop system (5-l6). Finally, we notice that the choice

of identity matrix in solving the Lyapunov equation (5-l7) has been
made in order to reduce the upper bound on Q[49].

The design of the feedback control law (5-l5) to meet the conditions
of Theorem l may follow one of two possible approaches. In the first
approach the parameters d1 and 31 are chosen first in such a way that
the estimate of domain of attraction, L, contains a certain desired
set; then the parameters 21 are chosen to make fi'and (5k - TIU'II) positive-
definite. One interesting choice of d1 and 31 is the one that provides
the largest possible estimate of domain of attraction. To find out
dipi/gi

i, Wi+N - pi 1 - 1,000,”, W0 - m1" W1 and

w1 = w1/WO. The set L can be rewritten as

 

that choice, let w1 = 31c

' N N WC p.
L(m = {sz2 (111/c11v1o1) + z (71—"1-1
1‘] i=1 plgl

(21 " ”1(yi)) :1}'

l T
) '1'): (21 ‘ ”1(5’1-HP1-

Since W1.: l for i = l,...,ZN with W3 = l for some j, it is obvious
that L(W) is largest when‘W1 = l‘Vi. In summary we have the following
observation.

Observation 1: For the largest estimate of the domain of attraction
L*, that can be obtained using Theorem l, choose 31 = l/c1 and

* 0
d1 = p1/fi1g1. Then, L is g1ven by

'k
L = :y,z

 

.ltvzz

 

l
_ -—-V.(y.) +
1 c1 1 1

107

Moreover, if conditions 11 and IV hold globally in 2, then choose 51
as large as needed to extend L* in the direction of the zi-axis.

The second approach to meet the conditions of Theorem 1 is to start
by choosing the largest possible parameters 81 for which our results
guarantee that the closed-loop system (5-l6) is asymptotically stable,
and then choose the parameters di and Hi to meet the conditions of the
theorem. This approach is interesting because it provides the smallest
possible feedback gains within the framework of our results. To see

how to achieve that, recall that the matrix R in (5-19) can be written

as
,'5 0 '5 0
- l_ T
R‘z (o 0)V+VK0 o)
where
K E
V:
T Q

Observing that the off-diagonal elements of V are non-negative, it follows
that there exists di and Hi such that R is positive-definite if and
only if V is an M-matrix [10]. Moreover, V is an M-matrix if and only
if both Q and (K - EQ'IT) are M-matrices [50]. Since 61's appear only
on the diagonal elements of O, choosing ei's sufficiently small will
make both Q and (K -EQ']T) M-matrices. Thus, we have the following
observation. I
Observation 2: The largest numbers ei‘s for which our results guarantee
that the closed-loop system (5-l6) is asymptotically stable are those
numbers just enough to make both O and (K - EQ'1T) M-matrices.

We notice that choosing €.'s as in Observation 2 guarantees that

l
the origin will be an asymptotically stable equilibrium point of (5-16),

108

but does not guarantee that L, the estimate of the domain of attraction,
will include any specified set.

We can have explicit expression for ei's in each case by using a
diagonal dominance approach to fulfill the requirement on corresponding
matrices. However this is achieved at the cost of getting conservative
results for 81's and therefore higher gains.

4. Exponential Stability

In this section we give an exponential stability result for the
closed-loop system (5-l6).

Theorem 2: Suppose assumptions (I-IV) of Theorem 1 hold. Besides,

let Vi(yi) satisfy

2 2 . _

(V) Ol2w1(yl) fi-Vi(yi) :_oi]wi(yi)9 1 ‘ l,...,N,
and wi's be functions of class 1, then the conclusions of Theorem 1
hold with exponential stability replacing asymptotic stability. More-
over, let ‘

wl(yl) = "yin i = 1900-,” and 2 = diag(°i]),
then any positive number a, such that:

* A .
on < a-(l/2)(m1n (Kn/0n” (5-20)
i
is a degree of exponential stability of the closed-loop system (5-16)
if
A "Lik - 205: FT

1 is positive semi definite.

~

I‘ Q - 2&0

R

Proof: From Theorem l and assumption (V) it follows that the comparison
functions of the Lyapunov function (5-l8) and its derivative along the

trajectory of (5-l6), are of the same order of magnitude which implies

109

exponential stability [ 9]. Furthermore, one can majorize the derivative

of (5-18) along (5-16) as

ninth) ’ "h(yl)
ileN> TN(yN)
R

V’
|/\
I

v(y.z -2av(y. 2).

"il ' ”l(yl)” 1 "il ‘ nMl)"

(5-21)

 

 

 

"2N - any )H uzN - nN(y )u‘

which in fact is a simple manipulationiof (5e19). In order for the closed-
loop system (5-l6) to be exponentially stable with degree of exponential
stability a, (i.e. 1 :_-2aV), it is enough to require that R] be positive
semi-definite and this completes the proof of Theorem 2.

Similar to Theorem 1 we notice that matrix R1 can be always made
positive semi-definite by choosing ei's sufficiently small. Also we
notice that Theorem 2 is more general than our earlier exponential
stability result [3], since it provides an estimate of the degree of
exponential stability. The choice of parameters di,d} and 6:i to meet
the condition of Theorem 2 follows the same guidelines discussed in
Section 3. Without repeating the discussions we mention that Observation
l still holds, while in Observation 2 the matrices o and (K - EQ'IT)
are replaced by (Q - 2aI) and (K - 2a2 - E(Q - 2&1)’]T), respectively.

In fact, the requirement that these matrices be M-matrices can be relaxed
since Theorem 2 requires only the positive-semidefiniteness of R].
However, stating the relaxed conditions requires involved notation and

definitions. Interested readers are referred to [51].

110

5. Alggrithm

The stabilization of system (5-l) by means of decentralized state
feedback control law can be performed according to the following algorithm.
Step (ll: Apply local transformation (5-6) to transform (5-l) into

y1 = ¢1(yl’ 21)

ii - ”i(yi’ 2i) + Giei(y’ z) + Giui i = 1...N.

Step (2!: Find ni(yi) such that y1. = ¢i(yi’ ni(yi)) ls asymptotically
stable. Find also a quadratic-type Lyapunov function Vi(yi)
and a comparison function wi(yi) and compute the numbers.
Ci’ kil’ eii’ ki6’ 3"“ kilO'

Step (3): Choose Mi so that Re AEGiMi] < O, and compute Pi’ pi,’Bi.

Step (42: Find the lnterconnectlon bounds ki4’ ki5’ ki8j and ki9j'

Ste 5 : Choose €l"°"€N to meet the conditions of Theorem l or 2.
This choice may be direct as in Observation 2 or indirect
by first choosing di,'di's and then choosing ei's.

Ste 6 : Use the decentralized state feedback control law

.._1 - _
”i - 8i GiMi(zi - ”i(yi)) l - l...,N.

The decentralized feedback law proposed above is a robust feedback
control law, namely we have infinite gain-increase margins and by choosing
ei's appropriately we can have arbitrary gain reduction. Moreover the
proposed feedback law is robust in the structural sense, namely it main-

tains stability in the presence of changes in the interconnections or

disconnection of any subsystem from the others.

111

Another aspect of the proposed algorithm is that stabilization
of subsystems is done using lower-order models, namely we stabilize
reduced-order subsystems of order (n1 - r1).
6. Examples

Two examples will be considered. One linear and the other nonlinear.
Example l: We consider the linear system studied in [46], where the

overall system consists of two linearly interconnected subystems,

and it is represented by:

S1
y] = (1 l)x]
o o 0 ~
x2= o o l X2T o {ule-zlyl}
o l o l
S2

y2 = (l l l)x2

T

_ _ T
Where xl ‘ (xll’ X12) . X2 ‘ (le’ X22: X23)

9 1112' <_ loll-21' : 1.

Ste l : This interconnected system is already in the appropriate

form and we don't need local transformation. Let us write this example
T
)

in our notation by defining y1 = x1], 21 = x12, y2 = (x2], x22 , 22 = x23.
9l T 2l

3‘ 2l T ‘yl'zl T LlZTT ‘)y2 T lezz T ul
_ o l o

52 ”(0 (9,2. (1)22
22 T (0 ‘)Y2 T L2lYl T LZlZl T u2

112

Ste 2 : We choose ”1(y1) = -y], Tl2(y2) = -(1 1)y2 and V](y]) = 1/2 Y].

3/2 1/2
VZTYZ) = yZ 1/2 1 y2’ then we have ¢](y]) =lTy]H9 WZ(YZ) = HszL
k. = k

1] ,6 = l i = l, 2, e]] = l, e22 =if§, kilo = l i = l, 2.
Ste 3 : We choose Mi = -l i = l, 2, then we have, p1 = p2 = l/2.

Ste 4 : In this step we compute interconnection bounds and they

are:

k14 = l, k24 = O, k15 O, k25 ' l, ki8j = O, l = l,2, J = 1,2, k19] = O,
kl92 T 1’ k29l T 1’ k292 T 0'

Ste 5 : To choose 51's in order to get lowest possible gains, using
Observation (2) we should choose Ei's so that matrices Q and (K - EQ'TT)

are M-matrices where:

-§— - 4 -2
E

l l o
. K=( ),
_2 .3. o l

E -
-l o ‘ -2 o

E = , T .
o -2.236 0 -4

We observe that both matrices Q and (K - EQ'TT) are M-matrice for

m

the choice 6] = .27ll, 62 = .l442.

Step (6]: The feedback law is given by:
U] = -k]X] = ”(3069 3.69)X]

U2 = 'kzXZ - -(6093 6.93 6.93)X2
At this point let us compare this result with those proposed in

[42], [46], [43], and [44]. For this purpose let us use norm infinity, i.e.

H . “m, as a measure of the largeness of the feedback gains.

113

Our method produces a stabilizing feedback law with, “k1”oo = 3.69,
szyym= 6.93.

The method proposed in {42] PTOdUCES gains,

3p), 0 > 486 and ”k1” m: (486)4,

szll. = (486)3.

In [45] the proposed gains are:
1 ‘ (3 4). k2 = (32 45 l3), and ”klnm = 4, "k2"- = 45.

7?
I

The method suggested in [43] produces following gains:

1 (l.46 2.22), k2 = (2.l2 6.78 4.98),Hk]"m = 2.22. "k2". = 6.78.

7?
ll

Finally in [44] the proposed gains are:

7?
ll

1 (.4l .68), k2 = (l.OO 3.36 2.78), llklflco = .68, ”k2"- = 3.36.

The above numerical figures show that our feedback gains are seven
orders of magnitude less than Davison's method [42],and, more or less,
of the same order of magnitude as the methods of Sezer and Huseyin [46],
Ikeda and Siljak [43], and Yamakami and Geromel [441- In fact, our gains
are lower than those of [46] but higher than those of [43] and [44]. It
is not our point here to evaluate our method versus those of [43, 44, 46]
since those methods were developed for linear systems using techniques
limited to linear systems (e.g., Riccati equations), so it would not
be a surprise if they work with linear systems better than our method.
We are, however, pleased with the obvious improvement over Davison's
method since it is the only method that is applicable to a class of

nonlinear systems.

Example 2: We consider the Large Space Telescope, L.S.T. example provided
by Siljak and Vukcevic [52]. L.S.T. is described by a nonlinear inter-

connection of three subsystems and it is represented by:

114

5., x1. = Aixi + b].(u1. + h].(x)) i= l,2,3
where x. = (xil’ x1.2

l
O l .
1 and Ai 0 0 l = l,2,3,

(°) (°) °
bl T 85.62 . b2 T l3.69 ’ b3 T l3.2l .

h](x) = -.0026 x22 x32, h2(x) = .064 x12x32, h3(x) = -.061 x12 x22

)1

Let us apply our algorithm to this example.

Ste 1 : This example is already in appropriate form and we don't
need to use any local transformation. In order to write L.S.T. in our
notation let, y1 = x1], y2 = x2], y3 = x3], 21= x12, 22 = x22, 23 = x32

then L.S.T. can be written as

{ yl T 2l
S1

92 T 22
s2 {

22 = +~a22123 + 13.69 u2
S (Y3 23
3 .
1.23 = - 032122 + l3.2l u3
where a] = .2221, a2 = .8754, o3 = .8l12.
Ste 2 : we choose ni(yi) = -y1 i = l,2,3, Vi(yi) =(l/2)y12 i = l,2,3.

Then we have;

wi(yi) T |J’il T T 1:223: kil T ki6 T kilo T eii T 1°

Ste 3 : We choose Mi = -1 so P] = l/l7l.24, P2 = l/27.38, P3 = l/26.42.

Ste 4 : Let us assume lyil E'UT’ Izil 5—0i i = l,2,3 then we have

k. = k. = 0 i = 1,2,3, k

14 15 182 T alU3’ kl8l T k

115

= k k T k393 T kl92 T “103’

383 T 0’ k283 T a2“l’ 38l T O‘3“2 . kl9l T k292

kl93 T Tl°23 k29l T “203’ k293 T T2Ol’ k39l T 8302’ k392 T O‘3"1'

Ste 5 : We have

I 0 0 -2 -28183 0 \\
K = E = o l o , T = ‘ 0 V -2 ~292uI .
0 0 T, -26382 0 -2
F2x85 62 v '
_ET.___ -2 -2a103 -20.]O'2 *
- M
Q ' '28203 52 '2 ‘2“20l
2xl3.21
-2a 0 -29 o -——-———- - 2
L 3 2 3 l e3 J

 

 

According to Observation 2, the lowest possible gains are achieved by
' *
choosing the largest possible Ei's, referred to as 51's, which makes

both 0 and (K - EQ'TT) M-matrices. But, the off-diagonal elements of

*
T and Q depend on oi's and U1.S, which means that si's will depend on

61's and ui's. It can be verified that the smaller Oi's and Ui's the

* *
larger the ei's. Moreover, for any choice of the 01's and ui's, 61's

must satisfy;

* ‘k ‘k

e] < 42.7115, 62 < 6.8390, 63 < 6.6491.

We have chosen c1 = 02 = 03 = .l892, u] = “2 = u3 = .075 and the Eg‘s

* * *
corresponding to this choice are e] < 40.4789, 82 < 5.1678, 63 < 5.576l.
Ste 6 : Let us choose a] = 40.4, 62 = 5, 63 = 5, then the decentral-

ized feedback law is given by

u2 = -.2000(x2] + x22)
U3 = -.2000(x3] + x32),

116

The estimate of the domain of attraction corresponding to this feedback

law is given by

 

_ 2 2 2 2
L - { x1, x2, x3 Sll.l0 x1] + l77.78 (x21 + x3]) + 505.28 (xH + x12) +
l28 20 (x + x )2 + l38 34 (x + x )2 < l
° 2l 22 ° 3l 32 - °

To get a feeling for the interaction between the choice of the
feedback gains and the estimate of the domain of attraction, let us
fix 01 and “i at their previously chosen values, namelycii = .1892,

“i = .075, and find the gains corresponding to the largest estimate
of the domain of attraction. From Observation l, the largest estimate
of the domain of attraction is given by;

3 2 3 2
151077.78 x“) +15] (76.78 (x1.] + x12) )_<_ l}

 

*-
L - X], X2, X3

and from Theorem l, the corresponding feedback law is given by:

u1 = -.0299 (xH + x12)
U2 = ’0233] (X21 + X22)
u3 = -.2277 (x31 + x32),

As we can see we have achieved this estimate, L*, at the cost of higher
gains.

To be able to compare our result with [52], we explore this example
further with higher value for (p and “i' Suppose 0i = 4, “i = l,
i = l,2,3, then the highest upper bound on ei's are: a: < 23.32l4,

a; < l.l8906, e; < l.23808. Let us choose a] = 23.3, 83 = l.l89,
83 = l.238, then the corresponding decentralized feedback law is given

by:

117

 

u1 = -.04291 (xH + x12)
U3 = -.80775 (x3, + x32).
The estimate of the domain of attraction in this case is given
by
_ 2 2 2 2
L - {x1, x2, x3 1.8361 x]] + x2] + x3] + 1.5684 (x11 + x12) +
4341 (x + x )2 + 4684 (x + x )2 < l
' 21 22 ' , 31 32 —- °

Again from Observation 1 the best estimate of domain of attraction,

*
L , for this choice of Oi and “i is given by:

L* = 5

I 2 2 2 1 2 2
)x1’ *2: X3|xll T le T x3l T‘g (X11 T x12) T ) T

(x21 T x22

«01—:

l 2
‘9 (X3l T X32) i—‘}*

and the decentralized feedback law corresponding to, L*, is

and it shows that we have achieved the largest estimate of the domain
of attraction, L*, at the cost of higher gains.

Finally, we recall that the estimate of the domain of attraction
in [521is
L = {x1, x2, x3|4.8“x]fl + 13.76flx2" + 4.30flx3fl,: 1.34},
and the feedback gains that yield L are given by

u1 = -(2.82 x + .3988 x

11 12)

-(17.64 x2] + 2.5 x

”2 22)

118

Obviously, our method results in smaller gains and a larger estimate

of the domain of attraction compared with the method of [52].

119

Appendix
We have, .
N N o
\3=(y.2) 1{1d1-V[1,1-V1-(y1-T¢)] (1.21)+1£1{p(1-n1())TP1-1~["(y1-.z1-)
+ewxn+itmu1-n n11W)w[7mU M] opgn+
d1

—[li(y.z1)+69(y.2)+;-GM 21-- n1(y1-))-

 

 

 

N _ 2 N _
_<_ -1§1d1.k11l121-(y1) + 51 d1 e111 1121 -' 1 H 111)
N d. 2
“151:131II21- - ill-WM + 5:: was.“ 1 1.)”..1. - M(Yi)”
N N N
+1E1Zd1HZ1. " rL1(‘y1)Hj£1k '18ij 'j(.y ) T121111 d1k16w1(‘y1)||zi ' 91-01)”
N 2 N N .
”1531291- k1-4llz1- - 91(y1)ll +131 2d1-Nz1- - n1(y1.)1|1)_:1k 1.91.112. - n11.(y1.)H
N 2
+1212d.1k1.1d[z1.1.)l|
- 1'” 1
1TJ](.)’]) 1T 1111(5'1)
w§(yn) W§(yN)
- R
(121 -1n](y1)ll 1121 14101»!
sz~ 3 nN(yN)H1 U‘TN : nN(yNNL1

 

 

CHAPTER VI
DECENTRALIZED CONTROL, USING LOCAL HIGH-GAIN
DYNAMIC OUTPUT FEEDBACK

I. Introduction

 

Decentralized control methods for the stabilization of an inter-
connected system using only local feedback have developed in two distinct
directions. In the first approach (e.g., [53]-[56]) a decentralized
control is designed using a complete model of the interconnected system
(i.e., a model which describes the local subsystems as well as the inter-
connections). In the second approach (e.g., [42], [43] and chapter.5) a
decentralized control is designed using only models of the local sub-
systems whereas interconnections are characterized by bounds on their
magnitudes. The latter approach invariably employs local high-gain feed-
back either implicity [42], [43] or explicitly as in chapter 5. Com-
paring the two approaches for the case of linear systems shows that the
first approach, which uses more information, can stabilize a wider class
of interconnected systems compared with the second approach. On the
other hand, the second approach, which uses only detailed models of the
local subsystems, can tolerate modeling errors and nonlinearities in the
interconnections. While the use of local output feedback and dynamic
compensators is typical in methods of the first approach, until recently
methods of the second approach have used local state feedback. Since
it is common that the entire state of each subsystem would not be

accessible for feedback control, there is a definite interest in

l20

121

‘extending the methods of the second approach to the output feedback

case. Recently Huseyin et. al. [57] have devised an interesting de-
centralized feedback control strategy in which each isolated subsystem,
which is assumed to be single input—single output, is stabilized using

a dynamic compensator. While designing the local compensators, the gains
of the loops of the interconneCted system are weakened, by an implicit
use of local high-gain feedback, so that stability is retained in the
presence of arbitrary interconnections satisfying the prescribed bounds.
Motivated by the work of [57] and by an explicit use of local high-gain
feedback as in our earlier state feedback work in chapter 5, we propose

a decentralized output feedback control strategy which applies to in-
vertible multi-input multi-output isolated subsystems. The use of‘
explicit high-gain feedback leads to a clear understanding of the structur-
al properties of the system and yields a simple decentralized control
scheme.

2. Problem Statement

 

We consider the interconnected system 2 composed of N linear time

invariant subsystems 21 described by

~ A~ A /\ ~ /\
Xi; Xi - Aixi + Bi(ui + Hi(x)) + Mi(y) (6-la)
A~ -
yi = Cixi 1=l,...,N (6—lb)
~ "1' "‘1' 2i
where xiéR is the state, uiéR. is the control input yReR is the
measured output of Zi’ Y = (Y1,...,§;)T, and y = (yI,...,y§)T. The
matrices X}, g} and 6} are constant and of appr0priate dimensions, and
A
the mappings fl} and Mi satisfy the following smoothness conditions:

122

A N ». .

1|M1~(y)ll : jg] BU. H yjll 1=l ,...N (5-2)
/\ N fi\ .

||H1.(x)||_<_ jg] Yij ||xj||. 1=l,...N (6-3)

We derive our results for the case of square subsystems, that is
2. = m. (i=l,...,N). However in the case of non-square subsystems,
(i.e., 2i a mi), they can be cast in the former case through a square-
down procedure which we will discuss later. Therefore in the following
we assume 21 = mi (i=l,...,N). Furthermore without loss of generality
it is assumed that E} and‘gi are of full rank.

The only requirement for deriving our results is that each isolated
subsystem be invertible and has all its transmission zeros in the open
left-half plane. We have chosen to derive our result for three different
cases depending on properties of infinite zeros of the triple (6}, 3}, 3}).
The first two cases are special forms of the last case. Developing re-
sults for these special cases is helpful for understanding and clarity.
Before starting our analysis for each case we state the common assumption

for all cases, that is:

Al - The isolated subsystem 3i defined by

~ ~ /\ /\
2.; x. = A.x. + B.u.
1 1 1 1 1 1
Y1 = C171
has all its transmission zeros in the open left-half plane. (x is a
A * A
transmission zero of E, if the rank of 1 is
/\
C. 0

1

strictly less than (n1 + min(ti, mi)), [58].)

 

1':
Ir denotes an rxr identity matrix.

123

3. Case l

 

This case is characterized by the requirement that the infinite zeros
/\ /\ /\ . .
of the triple (Ci’ A1, Bi) are of the first order or equ1valently we
assume;
A1A~ , . .
Ag - CiBi TS nons1ngular for 1=l,...,N.

This assumption obviously implies invertibility of 3,. Following

chapter 51~econsider a local transformation xi = 11;} where Ti satisfies
/\
T.B. = (5-4)

1 ' . . . .
where B{€R 1 15 non51ngu1ar. The freedom in ch0051ng Ti can be ex-

ploited to choose Ti such that

€11. = (o , 1m.). (6-5)
1

Performing the local transformation xi = 1,}, brings 21 into the follow-

ing form

Zi‘ Xil = Ailxil + Ai2xi2 + Mi1(Y)

= A + A1 x. + B (u. + Hi(x)) + M12(y)

Xi2 13‘11 4 12 1 1

Y1- =X12 ‘ 1:1,...N
where all the matrices A1], A12, Ai3’ A1.4 and mappings Mil’ M12, Hi are

obtained in an obvious way. Notice that (6-2) and (6-3) in the new

states can be rewritten as

'Nz

llM,1(y)H sij1|lyjH , i=l,...N (6-6a)

J 1

124

llMi2(y)H Bij21lyjH i=l,...,N (6-6b)

(_J.
2M2

'Mz

111110011 _<_ (Yl'j1 lllell + ijz 11sz1111=1~uN (6-6C)

1

J

where Bijl’ B and YijZ are non-negat1ve constants.

ijl
Assumption Al implies that the matrix A1] is a stable matrix, (i.e.,
all its eigenvalues are in the open left half plane), since the eigenvalues
. _ A A A
of Ail are the transmi551on zeros of the tr1ple (Ci’ A1, Bi) [45]. Now
we consider a local static high gain output feedback;
u. =-—— F.y. (6-7)

1 6- 11
1

where Si is a small positive number to be specified later, and F1 is
chosen so that the matrix BiFi is stable. Since Bi is nonsingular we can
always choose F1 such that the matrix BiFi has desired eigenvalues. The
decentralized control law (6-7) is stabilizing for sufficiently small 51.
This is shown in the following theorem.

Theorem l: Suppose assumptions Al and A2 are satisfied. Then there exist

* *

(i=l,...,N) such that for any 21 5_e.

1 the decentralized control law

(6-7) is stabilizing, that is the closed l00p system 2 under (6-7) is

asymptotically stable.

3399:: The closed loop system is in the form of multi parameter singular-

1y perturbed
X‘ z Ailxil + AiZXiZ + ”11(Y) (6‘83)
= B1.F1.x1.2 + Eihi(x) . (6-8b)

yi = x1.2 (o-8C)

125

where

“1(x) = Ai3xil + A14‘12 + BiHi(x) + Mi2(y)'

To show the stability of (6-8) we make a decomposition based on the
time-scale structure of (6-8). We consider (6-8) as interconnection of 2N

subsystems

Xil = Ailxil 1

Eixiz = BiFiXiZ ‘
Based on this decomposition we can choose a composite Lyapunov function
and verify the stability of (6-8). We start by observing that assumption

Al guarantees the existence of a positive definite matrix Pi satisfying

the Lyapunov equation.

P.A. + AT P. = -21( (6-9)

1 11 11 1 mi - n.)

1
Also in the same manner since BiFi is a stable matrix, there exists a

positive definite matrix Qi which satisfies

T — .
QiBiFi + (BiFi) Qi ‘ 'ZIm. (5'10)

1

Next we observe that from (6-6b) and (6-6c) it follows that:

111110011 _E (51.11 11Xj111 1' 513.2 11Xj211):

J-l

where €1j1 and gijZ are non-negat1ve constants.

Now we form a composite Lyapunov function

T -— T
(1/2)(d x. P.x. + dixiZQixiZ) (6-11)

V(x) = 1 1 11 1 11

Ier

1

126

where di and di, (i=l,...,N), are arbitrary positive numbers. Let

911 = ‘11P1A12H ’ 8111 11P1Ha elj = -Blj1 IIPiH l¢j,
s =—-‘--e no H s =-a no II no

ii 61. iiZ i’ ij ij2 i ’

rm. = mam HQiH’ i,j=l,...,N.

and define the matrices E, S, R, D and 5 as

C
II

diag[di], 5': diag[d€].

In Appendix A it is shown that the derivative of the composite

Lyapunov function v(x) along the trajectories of (6-8) is bounded by

 

 

 

 

 

 

f .1 .
11X1111 11x11”
v(X) : - mell T HXN1“ (6-12)
11x12“ 11X12H
Hxlzn Lm2“
where
_ o o o 11 -
T=1/2 . V+ VT , (543)
O 5' O '- .
1 .
V I” E
R 5

Notice that positive definiteness ofrlrimplies asymptotic

127

stability of (6-8). It can be shown that for sufficiently small 61,
(i=l,...,N) the matrix'][ is positive definite. In fact, we can choose
5:, (i=l,...N), just small enough to make the matrixT positive de-
finite. Then we observe that for any 51 < e:, ['7 will remain positive
definite and this concludes our proof.

Remark l: The arbitrariness of the parameters di and 3,, (i=l,...,N)

d

can be employed to obtain the smallest possible gains, , by simply

‘27
* . 1
choosing 51's small enough to make matrix ‘VV M-matrix [l0].

4. Case 2

 

In this case we assume that isolated subsystems are uniform rank
systems, [59], that is the following assumption holds:
A3 - For each isolated subsystem Si there exists a positive integer qi

such that the Markov parameters of E} satisfy

AAkA _ _

CiAiBi - 0 k - 0,. ,(q1 - 2)
where

A Aqi’1A . .

CiAi Bi 1s nons1ngular.

This assumption is a sufficient condition for invertibility of Ei’
[60]. In [6l] it is shown that the freedom in choosing the local trans-
formation Q} = 1171, where Ti satisfies (6-4), can be exploited to trans-

form the subsystem Xi into the canonical form

Zi‘ Xio ‘ Aioxio + Ailxil + Mio(y) (5'14a)
A _ A ~ -_ ..
xij " X1j+1 + M1j(y) J-1s-°-9(qi " 1) (614b)

g} ”-1 A .. A ..
iq, = _ D..x.. + B (u. + H.(x)) + N. (y) (5'14C)

l28
.y1‘ = 1" 1:19-009N (6'14d)

. A . . ,«
where/x1.o 15 an r1 = qimi dimenS1onal and xi

mi dimensional vectors, and the mappings M

j’ (j=l,...,qi), are each
A}j, (j=0,...,qi) and N} are
obtained in an obvious way. To motivate our design and analysis let us

first ignore the interconnections and assume that successive derivatives
of yi can be measured exactly. Then we have all the measurements of

the states’x11,...,x}q . Now upon availability of these states, and in

view of the canonical1structure of the isolated subsystems we consider

a repeated application of high-gain feedback in a manner similar to

case 1. By doing so we convert all the states £},,...,§,q , (i = l,...N),
into fast states. Now in the presence of interconnections1we observe

that state or output coupling that appears on the R.H.S. of the fast

state equations can be tolerated if we have provided enough gain as in

case 1. Moreover the coupling that appears on the R.H.S. of the slow

state equation would depends solely on the fast states and can be tolerated
as in case 1. So we can retain the stability of the system in the pre-
sence of interconnections. Unlike case 1, the high-gain output feedback

in this case is dynamic since it involves derivatives of the local output
yi' To realize such a control law we use local observer-based controllers.
We start off by preparing the system for high-gain feedback. To do that

we scale the states and control variable according to

x. =x. X..=u.

10 10’ 13 " J=]"'°’qi’ “' = 5'

'J
_ l
”' ‘E‘TVi
1
where Eils are small positive numbers to be chosen later and Vi is the

new control variable.

129

In order to maintain the boundedness of the state coupling N}(x)

we restrict the choices of the small parameters 2],...,sN to those

choices satisfying

 

for some numbers mij‘

After scaling (6-l4) is given by

xio Aioxio + Ailxif + Mio(y)

. a-
- 1- -
“'Xif ‘ “i Aizxio + “iAi3Xif + A

+ B. V. + u-B fi-(Xoaxf) + Uifii'l(Y)

1f 1 1 if 1

yi = Cifxif’

where

I
A
X
_o
_a
0
X
A.
N
U
U
X
V
0
X
II
A

Xif '

i1 -

w
I
A
O
1;
O
m

if ’

if

ifxif

(6-15a)

(6-15b)

(6-l6a)

(6-16b)

(6-16c)

471

and the mappings M50, M31 and Hi are given in Appendix B. In Appendix
B it is also shown that the smoothness conditions 2 and 3, in view of

assumption (6-15), can be rewritten as,

_ N .
HMio(y)ll ,5 e'ij1llxj1H, 1:1, ..,N (6-l7a)
J=1
_ N ' .
“Mum“ : £1 a mum”, 1=l,...,N (5-171))
— N qi‘1 .
(6-l7c)

and y.jf are non negative constants.

whereBij], 81j2,Y 1

ijo
Next notice that the system (6-16) is in a singularly perturbed
form and a decomposition based on the time-scale structure of (6-16)

leads us to consider (6-l6) as interconnection of 2N subsystems

x1.0 = Aioxios i=l,...,N (6-18a)
“ixif = Aifxif + Bifvi (6'18b)
yi = Cifxif’ i =1,...,N (6—18c)

the stability of subsystems (6-18a) are implied by assumption Al since

the eigenvalues of A1.0 are the transmission zeros of the triple

/\/\/\

(Ci’Ai’Bi) [6l]. So our immediate task is to stabilize subsystem (6-18b).
Before dealing with this stabilization problem we need the following lemma.
Lemma l: The pair (Aif’ Bif) is controllable and the pair (Aif’ Cif)
is observable.

Proof: This is a consequence of Aif’ Bif’ Cif having a special canonical

structure and Bi being nonsingular.

131

To stabilize the subsystems (6-l8b) and (6-l8c) we use an observer-
based controller. First a state feedback control Vi = Fixif is designed
such that (Aif + BifFi) has allits eigenvalues in the open left half

plane. Then a local observer is considered which is given by

/\ = /\ - /\ ' -
“ixif Aif¥if + Kif(yi Cifxif) + Bifvi’ (5 ‘9)

where Kif is the observer gain which is chosen so that the matrix
(Aif - Kifcif) is a stable matrix.

Now we can apply vi = F{xif to (6—l8b) and (6-19). The resulting
closed loop local subsystem is asymptotically stable. Having stable
subsystems we can proceed to obtain a composite Lyapunov function and
establish stability of z for sufficiently small ei. The result is stated
in the following theorem.

Theorem 2: Suppose assumptions Al and A3 hold. Then there exist 2:,
(i=l,...,N), such that for any ei :_ a: which satisfies (6-l5) the
decentralized observer-based controller

A.
1

A A /\
'xif ’ Aifxif + Kif1yi ‘ Cifxif) + BifFixif (5'2031

/\
u. = ——- F.x.

1
l c. llf

(6-20b)
1

is stabilizing that is the closed loop system 2 and (6-20) is asymptoti-
cally stable.
Proof: Setting ei = §}f - xif as the observation error, the closed loop

system (6—16) and (6-20) can be rewritten as

x1.0 Aioxio + 21(xf) (6-Zla)

1f f if i if
U1 :-
el // 0 (Alf ‘ Kifcif) el
9,-(x0,xf)‘
+ “l. (6-21b)
-g1(XO,Xf)
yi = Cifxif’ 1=l,...,N (6-21c)
where
21(Xf) ‘ Ailxil + io(y)
_ __ _ _ ' qi'L.
gi(xo’xf) ‘ Biin(Xo’Xf) + ”11(Y) + A13x11“ + “i Ai2xio'

Moreover in view of (6-17) it follows that

N
Q.(x ) < Z: k.. _
ll , , ll H Ullxflll
) N l q] -1 1
1191(X0sxf)l f. jg (Y iijxifH + “1 Y ,jOIIXJ-OH)

A decomposition based on the time-scale structure of (6-21) will

lead us to consider (6-2l) as an interconnection of 2N subsystems

xio Aioxio, i=l,...,N (6-22a)

133

if if .
“i = Agf - , i=l,...,N (6-22b)
e1 e].
where
(Alf + BlfFl) BifFi
Aif '
0 (Aif ‘ KifCif)

Let'Pi and’ai be positive definite matrices which satisfy the

following Lyapunov equations

AA N3 -2 523
P1 i0 + io i ‘ ‘ In..q.m. ( - a)
1 1 1
Q AC + (AC )T0 = -21
i if if i 2q m (6-23b)

i 1

Notice that due to the fact that Aio’ and A? are stable, (the latter

f
o a A

one 15 stable $1nce (Aif + BifFi) and (Aif - Kifcif) are stable), Pi
A

and Qi exist and are unique. Now we are at the position to form a com-

posite Lyapunov function

N T
_ . 11A e xif r~ xif
v](x) — IZ(1/z) dixioPixiO + d1. < ) Q1. < , (6-24)

1=1

where as usual di and Hi, (i=l,...,N) are arbitrary positive numbers.
From this point on the proof proceeds in the same way as in case 1.
Remark: Our result for case 2 required that the choices of 6],...,€N

satisfy (6-15), which wasn't required in case 1.

134

5. Case 3

 

This case is the most general case and covers cases 1 and 2 as
special cases. It is characterized by the following assumption.
53 - Each isolated subsystem 31 is invertible.

Recently Sannuti [62] has shown that under the assumption A4 one
can choose local transformation Ti which satisfies (6-4) and brings the
isolated subsystem 31 to a canonical form which is appropriate for high-
gain feedback analysis. To perform this local transofrmation, from
[62], we observe that for each isolated subsystem there exists an integer

pi :_ni and 1nd1ces oij and ri., 3=0,...,p. o. = m. r.o = 0 and

J 1 1o 1’ 1

o1p = 0, so that the transformed system 21 is in the following canonical
1
form
. x. = A. n. _
Zi’ xio Aioxio + Ailxil + Mio(y) (6 253)
A _ A A ~ ._
xijb - Eijxil + xij+l + Mijb(y)’ J-l,...,pi-l (6-25b)
. N
A = A A .=
xija 2:20 Dijtxit + Bljui + ”11”) 1’ Mijo(y)’ 3 1"'“F’i
(6-25c)
/\
yi — Nixil (6-25d)
where;
3? =(3?T ’x‘TlT’)? =3? 3? =6?T ”N T
ij ija’ ijb ’ ija ijo’ ijb ijl" ’ijpi-j
A AT 1 AT A T
x. = (x. , x. ,...,x. ) ,
1 10 11 1pi
. _ T T .
the matrlces Ni and Bi - (Bil""’Bip ) are non51ngular, and the
~ i
mapp1ngs Mijb’ 3:1,...,pi_1, Mijo’ Hij’ J=l,...,pi, and M10 are obtalned

I

135

/\ A A
x.. , x. and x.. are

. ° , A A
1n an obv1ous way. The varlables x. . X- , 133 1jb 131

10 1pia

of dimension hi],

, 0.. and r1j+2

r. . .=
1pi’ rij 13 respectlvely (J l,...,pi_],

2 = l,...,p1_j). Furthermore
A 1' A '01-]
0.. = 0.. - r.. = Z: r. , d. = o.., r. = o. ,
13 13-1 13 2=j+l 12 1f j=0 13 lpi 1pi-l
(6-26a)
n1] = n1 - dif‘ (6-26b)

For further details on this canonical form and the transformation T1,
interested readers are referred to [62]. Now observe that the terms
oijgfi}, in (6-25c) can be included in the state coupling function

Hij(x) and this inclusion doesn't violate the smoothness requirement

(6-3). So we set

Furthermore in our analysis we also consider the term Eijx}] in (6-25b)
as an interconnection term rather than part of an isolated subsystem,

that is we set

_ /\ “ _ -1
Mijb(y) ’ Eijxil + Mijb(y) ‘ EijNi yi + Mijb(y)
pi
Finally observing that mi = Z? rij and the fact that Bi is non-
i=1

singular we can define a new input vector

and partition the new input vector as

u = (31 31 fix '1T
0 g 11,...,1j,...,1p1 ,

136

where G}j€Rrij. In view of above considerations we can rewrite (6-25)
as follows.

Zi; $10 = Aiogio + AilQil + Mio(y) (6-27a)
éhjo = §}j+] + Mijb[y), j=1,...,pi-1 (6-27b)
€21.” = Cu. + 111%?) + Mijow)’ j=1,...,p1. (6-27c)

yi =‘Qi1, (6-27d)

where, since Ni is nonsingular, without loss of generality, we have
taken Ni as an identity matrix. Now we perform a regrouping of the
states of 2i which is an essential part of our analysis. First we

part1t1on Mijb(y) as

(y) = (M1 (y). MT (y).... M1 (y))T

M 111 112 ’ ijpi-j

ijb
which is compatible with the partitioning of §}jb. Then we start by

picking an arbitrary component of'xi1b, say'inj, and we have

A A
xilj ' xiZj-l + Milj(y) (5'28)

Next we choose the state which appears on the right-hand side of (6-28),

namely'xi2j_1, and we have

A

xi2j-l ‘ X133-2 + ”123-1(Y)

and the next step will be to consider xk3j_2 and so on, until we get
which satisfies

A
to Xij+l o

2. _ A. ~ .A
xij+1 o ‘ Uij + Hij(x) + Mij+l o

137

and we stop. Then we group the states xilj’ xi2j-l""’xij+l o' By
performing this regrouping algorithm for every component of xilb’ we

can rewrite (6-27) as follows

- A = A A -
Zi’ xio Aioxio + Ailxil + Mio(y) (6 29a)
2. wt
xi1j - x12j_1 + M11j(y) (6-29b)
" = 2‘ + M ( ) (6-29 )
xizj-l . i3j-2 i2j-l Y C
/\ _ /\ ~ /\
Xij+l o ' uij+l + Hij+1(x) + Mij+l 0(y) (5'29d)
yij = xi1j, 3:0,...,pi-1, (6-29e)

where yij are components of yi which are partitioned compatible with

partitioning of xi]. To write (6-29) in compact form let us denote

Q‘ = (Q1 gJ’ .QT )T :1
lfj llj-1""’ i2j-2""’ 1J0 a J 9° ,p1
’Q = [QT Q1 )T ‘Q = ( QT ‘QT )T
o 109...,N0 , f ...,1fj9...,ifj,.a.,
,

H
l

1
—l
(_J

 

ii

Am“ = O

-----------—-_-
o—c

 

 

“-9.
o
o

138

0
ll

H
1
—l
(_a

ifj
”M
1J (J’1)r.ij
_ .T T T T
Mifj - (ni]j_](y), M12j_2(y),...,Mij0(y)) ,

Using above definitions we have the desired canonical form for Xi given

by
z - z‘ = A ‘§ + A ’“ + M ( ) (6 30 )
1’ xio io io ilxil io y ' 3
/°\ _ A /\ ~ " A
xifj ' Aifjxifj + Bifj(uij + Hij(xo’xf)) + Mifj(y) (6'30”)

— A .—
yij - Cifjxifj 3-l,...,pi (6-30c)

Now it is obvious that our problem in Case 3 has been casted into a
formulation somewhat similar to the one used in Case 2. To prepare

(6-30) for high-gain we use the same scaling scheme we used in Case

2, that is, we let

' l

ij
2 ‘?

 

 

139

>
-—1
o (_JJ—d

j=l,...,pi

where eij’j=]""’pi’ i=l,...,N are small positive numbers to be chosen
later. As in the Case 2 to maintain the boundedness of the state
coupling Hij(x0,xf), where x0 and xf are the scaled vector x0, xf, we
require that the small parameters Eij’j=]""’pi’ i=l,...,N, satisfy a

condition similar to (6-l5), which is

 

EU. :1 (6'313)
Q:l
J

5. .

1‘] oo '= = ' = ..
2-] 1mm.2 < , J l,...,pi,2 l,...,pr,1,r l,...,N. (6 3lb)
_E_'

Era

After scaling (6-3) is given by

2i; xio = Aioxio + Ailxil + Mio(y) (6'323)
“ijxifj = Aifjxifj + Bifj(vij + “in1j(xo’xf)) + “ijMifj(y)
(6-32b)

yij = Cifjxifj J=l,...,pi (6-32c)

where the mappings fiio’ M.f , and fiij are appropriately defined.

1 j
Similar to the case 2 it can be shown that

N N
llfiio(y)ll _<_ f 2 Aizs‘lytsn i=l,...,N (6-33a)

__ N N .
HMjfjb’)” f. 2 2 (SleSH‘ylLSH 1:]s-o-9N (6'33b)

140

P
N N _i

llHij(xo.xf)Il 1 z nijzllxmjl + )5 2; aijgsungsu (cs—33¢)
2=l Q-l s-l

Next we observe that (6-32) is in a singularly perturbed form and a

decomposition based on the time-scale structure of (6-32) will suggest

to consider (6-32) as interconnection of (pi+l)N subsystems

X].O = HlOXlO i=l,...,N (6-34a)
“ijxifj z Aifjxifj + Bifjvij (6‘34b)
yij = Cifjxifj j=l,...,p1 i=l,...,N (6-34c)

The stability of subsystems(6-34a) is implied by the assumption
Al, since the eigenvalues of Aio are the transmission zeros of the triple

(C},’Ai,’§i), [62]. To stabilize subsystems(6-34b), (6-34c) we need

the following lemma.

Lemma 2: The pair (A. ., B.

lfJ )

.) is controllable and the pair (A.

lfJ ifj’ C

ifj
is observable.

Proof: Follows from the special canonical structure of the triple

, A B. .).

(C ifj’ TfJ

ifj
As in Case 2 to stabilize (6-34b), (6-34c) we use an observer-based

controller. First a state feedback vij = Fijxifj is designed such that

ifj + B F ) has all its eigenvalues in the open left-half plane.

(A ifj ij

Then a local observer is considered which is given by

/\

A /\
“ijxifj ' Aifjxifj + Kifj(yij ' Cifjxifj) + Bifjvij (5‘35)

where Kifj is the observer gain, which is chosen so that the matrix

(A K C ) is a stable matrix. The application of the control

ifj - ifj ifj

l4l

law v.. = F to (6-34b) and (6-35) results in an asymptotically

A
13 ijxifj
stable closed loop subsystem. Having stable subsystems we can proceed

to obtain a composite Lyapunov function and establish stability for

sufficiently small Eij' The main result is stated in the following
theorem.

Theorem 3: Suppose Al and A4 hold then there exists €:j (i=l,...,N)
such that for any Eij :_e:j which satisfy (6-3l) the decentralized

observer-based controller;

/\ = A - /\ /\
“ijxifj Aifjxifj + Kifj(yij Cifjxifj) + Biijifxifj (6-36a)
_ l A ._ ._ _
ulj -€—1_J-Fljxlfj J-l,..-.,pi, l-l,...,N (6 36b)

is stabilizing, that is the closed loop system 2 and (6-36) is asymp-
totically stable.

Erggj; Similar to proof of Theorem 2.

Remark: Notice that for each subsystem, 21, pi observers are designed.
These observers are in different time-scales. This decomposition of
the local observer based on its time scale structure is an interesting
feature of our design. Furthermore the remark in previous cases re-
garding minimizing the gains of each subsystem is applicable to this
case as well.

As we stated earlier our results are based on the assumption that
each isolated subsystem is a square system. However, our result can
be extended to the case of non-square isolated subsystems, by perform-
ing a square-down procedure, [63], [64], [65] for all subsystems.

That is we should find a pre-compensator (Post-Compensator) matrixlai

such that the squared-down subsystem (6}, AN, EAC}), ((C}C}, 3%, fi})),

l42

has all its transmission zeros in the open left half plane. From this

point on we can proceed with our investigation as we did before.

143

Appendix A

 

We have

N
. _ T
V(x) ‘1/2 £1 dixilpi(Ailxil T Ai2xi2 + “11”)“

T T T T T
di(xilAil + xiZAT'Z T Mil(y))Pixil T
— T l — l T T
dixizQNE: BiFixiZ l “1”” + WE]? X12(BTF1) + “ (x))leiZ
N 2
_<_ 15] (11(41me +llP1-A12ll'llxflll°||x121|+

N 1 _ l
,5] 81.11 TIP,” 'IIXHH onszn) + d, (- anxfin +

N N
j2::]~€,-j1||Q,—l|'l|><,-2|l°||xj1|l+ , 1a,.jznoin-Hxizn«51.219

J.
_ _T _
llxnll w Hxnul
= - llxpiill T W”
”X12“ Hx‘lzll
LllXfizllJ Lllxriglh 0.5.0.

 

 

 

 

l44

 

Appendix B
We have
Mio(y) = Mio(y)’
Ti <)=(fiT<) Wm “WM q‘J'fiT T
11y il y ’ “i 125’ "'"“i ij 5’ "°"“1 iqi(y))

Now in view of Condition (2), it follows that (6—l7a) and (6-l7b) hold.

New we observe that

Ilfii(xo,xf)[| =p.‘ E(Q). (3-1)

The smoothness requirement (3) implies that

Y ”\ (3-2)

Hm)” < ,..j erjn.

9r

LM2

3 0

where all y. are non negative constants. Now in terms of scaled state

Trj
variables (B-2) can be written as

 

 

q--l
l|_( )H N qr W H II N qr] II H
H. x ,x :_ ,2 Z? ._ x . z: in v. x
l o f r=l j=l uJrl r3 r=l T iro ro

Now in view of Condition (15), (6-l7c) follows.

Q.E.D.

CHAPTER VII
CONCLUSION

In part l of this thesis the analysis and the design of nonlinear
singularly perturbed systems has been investigated. The results of part
1 consist of stability analysis, initial-value problem, stabilization
and regulation. These results cover the related existing results in
the literature as special cases and generalize them in the following
directions:

l. Generalization to a general class of nonlinear singularly

perturbed systems.

2. Providing an explicit upper bound on the perturbation
parameter c, for each result, under which that result
is valid. This is very significant in engineering
practice.

3. Providing a degree of freedom for the designer to exploit
according to his design objective, such as including a
certain set in the domain of attraction or getting the
largest upper bound on e, and so on.

4. The generalization in the regulator problem has significant
features beside those stated in l, 2 and 3. First, ex-
plicit upper and lower boundstonHJuc - JSH are given.
Second, the solution of slow regulator is not required
to be optimal. These features have great practical and

theoretical implications.

l45

l46

One interesting feature of the results of part l is the uniformity
of the requirements, that is to say, to be able to perform a de-
composition based on the time-scale structure of a system for analysis
(e.g. stability) or design (e.g. stabilization, regulation), we have
only required an "Interaction Condition" which sets permissible inter-
actions between slow and fast dynamics, and the rest of requirements in
each case are the natural consequences of the specific problem area
under study. This "Interaction Condition" basically is dictated by
the strength of the stability of slow and fast subsystems. For
example if the slow and fast subsystems are exponentially stable then
the "Interaction Condition" is satisfied by requiring that f and g
be continuously differentiable. 0n the other hand if the slow and
fast subsystems are asymptotically stable then they set a certain
permissible interaction between slow and fast dynamics through their
comparison functions.

It is the author's speculation that most of the control related
results in singular perturbation can be stated in one or two theorems
and the major requirement would be the "Interaction Condition".

Finally we mention that the results obtained in part l lay the
ground and provide the basic tools for solving a host of control
related problems.

In part 2 we consider the problem of designing a robust de-
centralized control law using high-gain state or output feedback.
Although the results of part 2 are stated in the context of de-
centralized control, they can be extended to the design of robust
control using state or output feedback for systems with modeling

errors and uncertainlies.

I47

Decentralized high-gain state feedback pr0posed in chapter 5 is
the first attempt for the class of large scale systems with nonlinear
multi-input subsystems and nonlinear state coupling through input
matrix, which covers the existing results in the literature as
special cases. We should mention that the capability of high-gain
goes far beyond this result and the author forsees a greater role
for high-gain state feedback in the problem areas of decentralized
control and distrubance rejection, in the sense of broadening the

permissible interconnections or disturbances.

In chapter 6 a decentralized high-gain output feedback is designed
for a large-scale system with multi variable linear isolated sub-
systems and nonlinear state coupling through input matrix and arbitrary
nonlinear output coupling. This result generalize the only existing
work in the literature [57] which was for single input single
output isolated subsystems. The result of chapter 6 indicates the
crucial role of transmission zeros in the design of robust output
feedback. There are open problems to be investigated in high-gain
output feedback. For example, a very natural one would be the
extension of the results of chapter 6 to the class of large scale

systems with nonlinear isolated subsystems.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[ll]

[12]

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