STABILITY OF 2NTERCONNEC?ED SYSTEMS

Thesis for the Degree of Ph; D.
MICHIGAN STATE DRIVERSBTY
WILEY E. THGMPSGN
i963

- Hid-n:

LIBRARY

Michigan State
University

 

This is to certify that the

thesis entitled

STABILITY OF INTERCONNECTED SYSTEMS

presented by

Wiley E. Thomps on

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Elect. Engr.

7 ,/ ”2/
5."- - rfl/ _////
//’v::J///' Ix / .7 ’ {2.ka

Major professor

Date February 19, 1968

 

0-169

 

ABSTRACT

STABILITY OF INTERCONNECTED SYSTEMS

by Wiley E. Thompson

Virtually all engineering systems and certain classes of socio-
economic systems are realized as interconnections of components or
subsystems. As such, these interconnected systems can be defined
in terms of their two fundamental structural feature s--that is, (1) the
mathematical models of the unconstrained components or subsystems,
and (2) the constraints imposed by the interconnections between the
components.

Even though stability studies for dynamic systems have
reached a remarkably high level of mathematical SOphistication, mo st
stability criteria still have at least two serious limitations: (1) The
actual application to higher-order systems (particularly nonlinear
systems) is impractical or virtually impossible; and (2) the results
fail to adequately relate stability to the structural features of the
system.

This thesis considers the stability of several classes of inter-
connected systems, consisting of multiterminal, nonlinear, time-

varying components. In addition to dynamic components, algebraic

WILEY E . THOMPSON

and algebraic-dynarnic components are allowed. Scalar and vector
Liapunov functions are constructed for the interconnected systems

in terms of Liapunov functions of the individual components, the
system structure, and a parameter vector determined in a prescribed
optimal manner.

Sufficient conditions for various types of stability are obtained
for the identified classes of interconnected systems. The conditions
are given in forms which are particularly well suited for digital
computation and design. A ninth—order, nonlinear system is considered

for illustration.

STABILITY OF INTERCONNECTED SYSTEMS

BY

If

Wiley EiliThompson

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1968

6 4C: a (if
§-aos(-%

ACKNOWLEDGEMENTS

The author wishes to express his sincere appreciation to his
major professor, Dr. H. E. Koenig, whose guidance and constant
inspiration were invaluable. Thanks are also due to Dr. Y. Tokad for
his efforts as the author's academic advisor.

The author also wishes to thank the Division of Engineering
Research and the National Science Foundation, under whose sponsor-
ship this research was performed.

Finally, the author wishes to thank his wife, Judy, for her
patience, understanding, and encouragement throughout his entire

graduate program .

ii

TAB LE OF CONTENTS

Page
ABSTRACT
ACKNOWLEDGEMENTS ........ . . . .......... ii
LIST OF FIGURES o o o o o o o o o o o o o o o o ooooooo iv
I INTRODUCTION . . ............ . ...... . 1
1.1 Notation and Definitions ..... . ........... 6
1.2 System Concepts ................... 8
1.3 Stability Concepts ...... . . . . . . . . . . . . . 10
II EXPONENTIAL STABILITY OF A CLASS OF
INTERCONNECTED SYSTEMS . . . . . . . ........ 15
2.1 Interconnected Systems . . . . . . . . . ....... 15
2.2 ExponentialStability. . . . . . . . . . . . . . . . . . 18
2. 3 Exponential Stability of a Class of Interconnected
Exponentially Stable Components . . . . . . . . . . 24
2. 4 Exponential Stability of a Class of Interconnected
Systems with Unstable Components . . . ....... 40
III VECTOR LIAPUNOV FUNCTIONS FOR A CLASS
OF INTERCONNECTED SYSTEMS. . . . . . . . ...... 43
3.1 Interconnected.Systenis . . . . . .......... . 44
3.2 Vector Liapunov Functions. . . . . . . . . ..... . 44
3. 3 Vector Liapunov Functions and Stability of
the Interconnected System . . . ............ 50
IV APPLICATIONS.... ..... ..... 64
V SUMMARY AND CONCLUSIONS . . . . . ........ . . 76

REFERENCES . . . . ......... . ........ . . . . . 81

LIST OF FIGURES

Figure Page
1 Representation of an n-terminal component ...... . 9
2 Terminal graph for an n-terminal component . . . . . . 9

iv

LIST OF APPENDICES

Page

APPENDIXA...... ..... ...... 87

I INTRODUCTION

Most of the literature on stability analysis of dynamic systems
assumes that a mathematical model of the system is given in state
model form and that the analysis is to be based on that known model.
For the stability studies of nonlinear systems, the most versatile and
widely used method is Liapunov’s D;rect Method [HA- 1, KA-l, KR-l,
LI-l]. Although several procedures have been given for constructing
Liapunov functions for systems of arbitrary order [IN-1, KR—Z, LU-Z,
PE-l, SC-l, SZ-l, WA-l], the complexities involved in actually
applying these procedures to nonlinear systems make their application
impractical or virtually impossible for high-order systems.

Virtually all engineering and certain classes of socio-economic
systems are realized as interconnected systems (subassemblies, or
components) [KO-1, KO-3] . Generally Speaking, the component
models are simpler in form and of lower order than the model of the
interconnected system, and thus lend more readily to analysis at the
component level.

This thesis considers the stabiiity of an interconnected sys-

tem in terms of the two fundamental structural features--namely,

O the mathematical models of the components or

subassemblies and

0 their interconnection pattern.
This approach to the stability problem, in many cases, not only
,helps to circumvent some of the difficulties encountered in high order
systems, but it gives stability in the form most useful for design--in
terms of system structure. Using this approach [WI —1, WI-Z], various
classes of linear and nonlinear systems of multiterm inal components
can be identified for which the stability is relatively independent of
the topology of the system. For given classes of components, condi-
tions on the topology of the system which are necessary and/ or suf-
ficient for stability have been obtained. Moreover, for certain classes
of components and a given system topology, system stability has been
examined in terms of component characteristics.

Except for cases where the system topology is extremely
simple, as in the problems of Aizerman [AI-l] and Lur'e [LU-l]
where a linear, differential system and single, scalar, nonlinear
algebraic component are connected in a feedback configuration,
essentially nothing had been done in this direction until recently.

Williams [WI-1, WI—Z] examined several classes of systems
containing multiterminal, nonlinear and linear components with

models of the forms

d .
a: \II -— F(Zi' t)
Z() = \I/ (1.0.1)
or Z = C(Z.)
o 1
a d it! - P(t)Z + F(t)
n dt ' 1
20 = \11 (1.0.2)
01‘ Z = C(t)Z. + C(t)
o 1

where \II is a state vector for the component and 20 and Zi are the
vectors of complementary terminal variables for the component.

The particular classes of systems considered by Williams
are defined in terms of certain restrictions on both the system inter-
connection pattern and the component equations. Some of these classes
of linear and nonlinear systems are shown to be stable--a few re-
quiring only minor restrictions on the interconnections. It is also
shown that some of the restrictions on the component equations defining
the classes are necessary for stability.

Williams obtained these results by constructing Liapunov
functions from the structural features of the system. Linear graph
theory and matroid theory are utilized in establishing the relevant
properties of the functions so constructed.

Bailey [BA-l] obtained a sufficient condition for the equilibrium
of a system to be asymptotically stable in the large when the component

equations are of the form

|
El
u

F(\II,t) + DU

Y = H\II (1.0. 3)
and are exponentially stable in the large. Vectors U and Y are the in-
put and output vectors, reSpectively, and \I/ is the state vector. The
condition applies to a restricted topology with no loading. The form
(1. O. 3) is further restrictive in that purely algebraic components are
not allowed.

In Bailey's procedure Liapunov functions satisfying certain esti-
mates are found for each of the components. Krasovskii [KR-1]
showed that under certain conditions, exponential stability assures the
existence of these Liapunov functions. A vector Liapunov function is
formed from the component Liapunov functions, and its time derivative
is evaluated along the solutions of the interconnected system. The
estimates on the individual Liapunov functions are used to obtain a
linear comparison system of differential inequalities with the vector
Liapunov function as the dependent variable.

The idea of considering the Liapunov function as the dependent
variable in a differential inequality, in the scalar case, was presented
by Corduneanu [COR-1] . Under certain conditions on the system of
differential inequalities, asymptotic stability of that linear system
obtained by replacing the inequality by equality implies asymptotic

stability of the original interconnected system.

Aggarwal and Bybee [AG-l] considered the stability of higher
order systems obtained by coupling second order systems having the
form

£13 + gob) +hw) = 0. (4» a scalar) (1.0.4)

and gave sufficient conditions for the stability of the interconnected
system. The approach is essentially that of constructing a Liapunov
function for the interconnected system as a sum of the individual Lia-
punov functions for the components and then examining the time
derivative of this function along the solution to the interconnected system.
The Liapunov function used has a well known form for components of
this type.

A body of results known as "frequency domain stability criterion"
has recently been deve10ped out of the work of V. M. Popov [PO-1] .
These results as given in [BRO-l, KA—Z, KU-l, NA-l, PO-l] apply
to systems constructed by connecting a scalar, algebraic, nonlinear
feedback element to a linear time—invariant system. In [AN—l, IB-l]
multiple autonomous nonlinearities are allowed, while in [DE-1, KU-Z,
SA-l] the system is allowed to be forced with Specific inputs, but a
single nonlinearity is required. Jury and Lee [JU-l] allowed multiple
nonlinearities and inputs.

The classes of problems investigated via the frequency domain
approach have simple component interconnected patterns. For example,

Anderson [AN-1] in generalizing the Popov criterion to the case of a

system containing an arbitrary number of memoryless nonlinearities,
limits the topology to a single feedback loop with alternately connected
linear, time—invariant differential systems and nonlinear algebraic
components.

The remainder of this chapter is devoted to some of the notation,
definitions, and fundamental concepts necessary for an efficient develop-

ment in this thesis.

1. 1 Notation and Definitions
Denote by En the n-dimensional Euclidean Space of real n—vectors.
Capital letters in general are used to denote vectors and matrices, with
lower case letters designating scalars--any exceptions are pointed out
unless obvious. The transpose and conjugate tranSpose of a matrix are
T * . .
denoted by A and A , respectively. The norm IX‘ of a n-vector 18

taken to be the Euclidean norm

NIH

lxl = (X*X)

Z

NIH

T l n
or IXI = (X X)‘2 = ( 2 X1
121

),ifXEEn (1.1.1)

The Euclidean norm of a real m x n matrix A is then
[M = min{e. alXI _>_ IAXI, V ern} (1.1.2)
It can be shown [HAL-1] that
M! = «IA (1.1.3)

>1:
where A is the largest eigenvalue of the matrix A A.

Following Hahn [HA-l], let Hh denote the Spherical neighborhood
of the origin
ab = {XGEnz le 5h} (1.1.4)
and let Rh, 1: denote the half-cylindrical neighborhood
0
Him = {(X,t)€En+l: |x| 5 handt_>_to} (1.1.5)
o

of the t-axis in the motion Space, where h is some finite constant.

Definition 1.1. 1 An n-dimensional vector function F011, t) of the n-

 

dimensional vector \II and t is said to belong to class @ in Hh t if for
’ 0

all (\Il,t) E "h t the following conditions hold:
'
o

O F(W.t) is continuous in \II and t, and

O has continuous first partial derivatives with

respect to the components 4‘1 of \II.
It can be shown [ LE-l] that these conditions imply the Lipschitz

condition
IF(\II.t) - F(\It',t)| 5 quz - \II’ (1.1.6)
with respect to \II in H“ t .
., 0

Definition 1.1. 2 A scalar function ¢(r) of the real variable r is said

to belong to ‘class & if

Q ¢(r) is continuous and real valued on O E r E h,

 

0 M0) = Qand

0 Mr) is strictly monotonically increasing with r.

Definition 1.1. 3 A scalar function v(\II, t) is said to be positive (negative)

 

definiteif for some 426 ?<
v(\Ir.t):¢(1\Ir() (<_-¢(|\II|)) (1.1.7)

is satisfied in Hh t . (If h can be taken arbitrarily large, and ¢(r)

increases unboundedly with r, then v(\II, t) is radially unbounded.)

 

Definition 1.1.4 A scalar function v(\II,t) is Said to be decrescent if

 

 

for some ¢€ x
|v(xIr.t)l: MW") (1.1.8)

h,t°
O

1.2 Systems Concepts

A component is an entity such as a piece of physical hardware

 

or a part of a socio-economic or biological system for which there can
be defined a finite number of interfaces or terminals (Fig. l) at which
the component significantly interacts with its environment. A com-

ponent can be defined by a set of component equations and a terminal

 

linear graph.

 

 

Fig. 1 Representation of an Fig. 2 Terminal graph for an

n-terminal component n-te rminal component

The essential features of a terminal graph (Fig. 2) for an n
terminal component are that it has n vertices correSponding to the n
terminals of the component and (n—1) directed e_d_g2 (or lines) which
form no circuits. Such a graph is called a _tr_e_e_. Associated with each

edge of the terminal graph is a pair of complementary terminal vari-

 

.a_b_l__e;§--through and across variablesuwhose values are associated
with the terminal measurements of the component.

The component equations as considered here are algebraic
and/ or differential constraint equations interrelating the component

variables. This set of component variables is the minimal set of

 

variables which is necessary to completely characterize the terminal
behavior of the component. The set of component variables then con-

sists of a set of terminal variables and a set of parametric or state

 

10

variables. The set of state variables, which may or may not be dis-
joint from the set of terminal variables, are those variables which are
necessary to completely describe the dynamical behavior of the com-

ponent.

 

An interconnected system is then defined in terms of its structure
which consists of a collection of component equations and a set of

linear constraint equations derived from the systemjraph. The sys—

 

tem graph is constructed operationally and uniquely from the component
terminal graphs by coalescing vertices of these graphs that correspond

to the same terminal of the system.

1 . 3 Stability Concepts

A solution {F(t) = \I/(t; 30, to) of the system
i; = F(\II, t) (1.3.1)

is said to be stable (in the sense of Liapunov) if for every 6 > 0, there
0 A . .
exists a 5(€,to) > 0, such that (\IIO - \I/OI < 5 implies l\I/(t;\I/o, to) ..

A .
\Kt;\II0, to)! < E for all solutions \Il(t, \I/O, to) and all t Z to.

Remarks 1. Geometrically, a stable solution is one which is not
greatly disturbed by a small change in initial position. In fact,
stability of a system iS uniform (with respect to t) continuity of its
solution in \IIO.

2. Stability with respect to parameters other than \IIO

may be defined.

11

The change of variable ‘1)! = \II - (i! (t), (1! (t) a known solution of

(l . 3.1), formally transforms (1. 3.1) into

3 F(\II,t) - F(x’i‘z (t),t)

F(Ez + i (t), t) - F(x’i/ (t),t)

= f‘o'fnt) (1.3.2)

Thus there is a one-to-one correspondence between solutions of equa—
tion (1. 3.1) and (l. 3. 2) with (it (t) corresponding to 3 E 0.
If F(W,t) is linear in ‘11 so that

\II A)? + E(t) (1.3.3)

then

i} F(E’IJ) .-. F(§,t) -.- A’x‘iz’

so that stability of the null solution, CI; = 0, of the unforced system is
equivalent to the stability of any solution of (1. 3. 3).

Following common practice, it is postulated that F(O, t) E O, and
all further discussion of stability will refer to the stability of the null
solution, \II 5 0.

Many types of stability have been defined [ANT-l, HA-l] .
Those definitions important to this deve10pment are given in the
following.

Consider the differential system

i = Wet). F(0.t)Eo (1.3.4)

12

for to _<_ t 5 +00, and let \Il(t; \I/O, to) denote a solution of (l. 3.4) such that
‘Ir(to;\IIo, to) = \IIO; let \II(t) also denote a solution to (l. 3.4). The scalar

components of the n-vector \II are denoted by Li) , 412, . . . , kiln.

Definition 1.3. 1 The solution \II = O of (1.3.4) is stable if for every

 

E > 0, there exists a 6(E;to) > 0, such that (\Ilol < 6 implies l\If(t;\I/o, to” < E

fort>t .
—o

Definition 1. 3. 2 The solution \II = 0 of (1.3.4) is uniformly stable if

 

 

for every 6 > 0, there exists a 6(6) > 0 such that [\IIOI < 6 implies

I\II(t;\IIO,tO)I < E for t2 to.

Definition 1. 3. 3 The solution \II = O of (l. 3. 4) is asymptotically- stable

 

 

if it is stable and, in addition, there exists a 60(t0) > 0 with the property

that if (\II | < 6 then
0 0

11m \I/(t;\IIO, to) = O
t -* 00

Definition 1. 3. 4 The solution \II = 0 of (1. 3. 4) is uniformly asymptoti-

 

 

cally stable if there exists a 60 > O and functions 6(6) and T(E) with the

 

property that (\IIOI < 6(6) implies I\II(t;\I/o,t0)| < E for t_>_ to, and

(\IIO) < 60, t_>_ tO + T(€) implies I\P(t;\I/O,t0)l < 6.

Remark: Halanay [HAL-1] has shown that the functions 6(6) and T(E)
in the definition can be chosen continuous and monotone. It is further
shown that uniform asymptotic stability is equivalent to the existence of

two scalar functions a(r) and b(t), the first monotone-increasing and

13

the second monotone-decreasing, such that
I\II(t;\IIO,tO)l 3 a(|\IIOI)b(t-to) (1.3.5)

In particular, if a(lifol) is linear and b(t-t0) is exponential, the stability

is exponential as defined next.

Definition 1. 3. 5 The solution \I/ = 0 of (l. 3. 4) is exponentially stable

 

 

if there exist two constants c > O and (3 > O which are independent of
the initial values, such that for sufficiently small initial values the

following inequality holds
I‘I’it; \I/O. to)l 5 (3 I \I/O] exp(-(1(t-to)) (1. 3. 6)

If (1. 3.6) is satisfied for all initial values in some region, then (1. 3.4)
will be said to be exponentially stable in this region. If (1. 3.6) is
satisfied for all initial conditions from which Solutions originate, then

(1. 3.4) is Said to be exponentially stable in the large.

 

If the initial conditions \IIO satisfy the relations
liol 3 ho < h/(s, to_>_0 (1.3.7)

where (5 is defined in Definition 1. 3. 5, then denote \IIOE Hh .
0
Let v(\I/, t) be a differentiable function of \I/and t. Then the

total time derivative of v(\If, t) with respect to the system (1. 3. l) is
given by

. d av T -
_. __ + V
v dt v(\I/, t) at ( v) \I/

(1.3. 8)

vt + (V v)TF(\II, t)

14

where

and
v V _ [3.2. .sv_ .31. T
a), on) awn]
A positive definite function v(\I(, t) satisfying any of Liapunov's

stability or instability theorems, or any generalization thereof, for a

system is called a Liapunov function for the system.

 

II EXPONENTIAL STABILITY OF A CLASS OF
INTERCONNECTED SYSTEMS

Sufficient conditions are given for the exponential stability of
the equilibrium of a system of interconnected components taken from
a class of nonlinear, time-varying, multiterminal, differential, alge-
braic, and mixed components. A Liapunov function is constructed for
the interconnected system from Liapunov functions of the individual
components and the interconnection pattern. A parameter vector
appearing in the function is to be selected in a prescribed optimal
manner. The Liapunov function so constructed for the system can be

used to establish a lower bound on the rate of decay of the system.

2.1 Interconnected Systems
Consider a system S for which the direct sum of the N com—

ponent models forms a set of equations of the form
it = F(\II,t) +Dzl

22 = G(\Il,t) + cz1

(2.1.1)

Let the component equations for the i-th component be denoted by

-i i i i i
\It - F(\:1r,1:)+nz1 (2.1.2)
212 = Glorft) +clz‘l

15

16

i . . . i i
where \P is an ni-dimenSional state vector, and Z1 and Z2 are ei-

dimensional vectors of terminal variables representing the through

and across variables associated with the ei edges of the component

terminal graphbi. Each of these 2ei terminal variables must appear

i

1 or 212; however, a through and across

once in either of the vectors Z
variable associated with a particular edge of £91 may both appear in the

same terminal vector, Say 2:. In such case, the terminal variables
. . [b . i
assoc1ated With some other edge of .1 must both appear in Z2.
i i i i i i ,
If F (\II, t), D , and G (If, t) are zero--hence \II is null--then the

i-th component is purely algebraic. If C1 is zero, the i-th component

 

is purely dynamic. Otherwise, the component is said to be a mixed

 

component. Thus the set of components having dynamic parts con-

 

sists of the dynamic and mixed components.

Let in be the number of components in (2. 1. l) with dynamic

N N
parts and let n : 23 ni, e = >3 81' Then F(\II, t) and G(\II, t) are res-
i=l i=1

pectively n- and e-dimensional nonlinear, time-varying vector functions
of the n vector ‘11. The constant matrices D and C have dimensions
n x e and e x e respectively.
The class of interconnections considered is characterized by
the system graphb’ having a tree T1 and a tree T2 such that the com-

bined set of fundamental circuit and cut-set equations [KO-l] may be

written in the form

Z = 432 (2.1.3)

17

When these conditions are satisfied for a system S, the interconnection

is said to satisfy graphic constraint GC. The trees T1 and T2 may or

 

may not be identical. If T1 = Tz then it can be Shown that <I>T = -<I>.

What graphic constraint GC actually implies, in terms of re-
strictions on29’, is that the through variables appearing as components
in Z1 must be included in some tree T1 of band the across variables
appearing as components in Z1 must be included in a co-tree of some
tree T2 of %I. If graphic constraint GC is Satisfied, then following
Frame and Koenig [FR-1], the components of 21 and Z2 are called the
secondary and primary variables, respectively, associated with the trees
T1 and T2 for )b’--the vectors Z1 and Z2 are called secondary and pri-
mary terminal vectors and denoted by ZS and Zp.

If graphic constraint GC is satisfied, then applying the inter-

connection constraint equations (2. 1. 3) to the direct sum of the com-

ponent equations (2. 1 . 1) gives

xi; = F(\Il,t) + Dez
p (2.1.4)
2 = G(\I/,t) + €de
p p

for the system S.

If none of the eigenvalues of the matrix CCID are equal to +1,
then the matrix (U - CCD) is nonsingular, and the interconnected system
S is

ix = F(lII, t) + A0011, t)

(2. 1. 5)
Zp = BG(\II, t)

18

where A D¢(U - (343)-1

and B (U - cm'l

When this condition is satisfied, graphic-component constraint GCC

 

is said to be satisfied. This condition is automatically “satisfied for a

system of purely dynamic components.

2. 2 Exponential Stability

In general, even though a nonlinear system is asymptotically
stable or uniformly asymptotically stable, nothing can be said about the
rate of decay of the solutions or about the asymptotic behavior of the
associated Liapunov functions. Thus in certain engineering design
situations, where a certain rate of decay of the solutions is required,
the assurance of asymptotic stability is inadequate. However if the
system is- exponentially stable, then under certain conditions it is
possible to define a Liapunov function which satisfies certain estimates,
yet to be defined. Further, a system having a Liapunov function
satisfying these estimates is exponentially stable, and these estimates
can be used to establish a lower bound on the rate of decay of the sys-
tem solution. These concepts are stated more precisely in the
following theorem which, along with parts of its proof, are suggested

by the work of Krasovskii [KR-1].

Theorem 2. 2. 1 Consider the n-dimensional system of differential

 

equations

19

\II = F(\Ir,t), F(O,t) = 0 (2.2.1)

(i) If there exists a positive definite scalar function v(\I/, t) and a set

of positive constants c , c , c , and c satisfying (a) and (b) of

123 4

the relations

(a) elli/[Z 5 v(\I/,t) 5 ozlxi/lz
(b) v(\I/,t) _<_ -c3|\II|Z (2.2.2)
(c) IVvI _: c4141)

for all (\II, t) E Hh '1" '7' _>_ 0, then the system (2.2. l) is exponen-

tially stable for all \IIO 6 H Further, the inequality (1. 3. 6) is

h C
o
satisfied with a = c3/2cZ and [3 = N) c2; cl.

(ii) If F(\II, t) e @ in H , 7 _>_ o, and if (2.2. 1) is exponentially

h,'r

stable for all \IIOE Hh , then there exists a positive definite function
0

v(\II, t) and associated positive constants satisfying (2. 2. 2) for

all (\I/,t) E Hh, '7"

Proof:

 

(i) Suppose that there exists a positive definite scalar function v(\I/, t)
satisfying (2.2. 2). Then from (2. 2.2a) and (2.2. 2b)

c
v < -c |\II)Z < --—3-v (2.2.3)
— 3 — c2

Integrating (2. 2. 3) gives

(ii)

20

C
V(\I'(t: \I’O. to). t) _<_ v(\I/0. to)eXP[ - if (t-to)] (2. 2. 4)

Using (2. 2.2a) to eliminate v from (2.2.4), one obtains the esti-

mate

C
2 2 2
I\II(t;\IIO,t0)| E —C (\IIOI

C3
1 C

exp[- —- (t-to)] (2.2.5)

N

Taking the square root of both sides of the inequality (2. 2. 5) gives

C2 C3
lMt; ‘I’o'toH 5 q IWOI exp[-§-;; (t-to)]

 

or
)‘If(t; ‘Iio,to)| E fil‘llolexp[-o(t-to)]
where c c
(3: i>0anda:23 )0.
C1 C2

This proves the assertion that (2. 2. 1) is exponentially stable.

It is now shown that under the hypothesis of the theorem ex-
ponential stability of (2. 2.1) is sufficient for the existence of a

function v(\II, t) satisfying the estimates (2.2.2). Consider the

function
t +T
° 2
v(\Iro.tO) : S )\II(S, \Ifo,to)l ds (2.2.6)
t
0
where

f f
T : m"‘£{ 2((2‘3). ZnLZ}

 

(2.2. 7)

with L the Lipschitz constant given in (1. l. 6) and a, (‘3 the positive

21

constants given in (1. 3. 6) in the definition of exponential stability. The
point (\IIO, to) is an arbitrary point in Hh T subject to the constraint that
’

\IIOE Hh .
o
(a) Applying the result of Lemma A.l (Appendix) to (2. 2. 6)

 

+
1:o T 2L(s-t ) I\II )2 2L(s-t )to+T
v(\.I(t)<)\II|2 e ods: 0 e o
o' o — o t 2L t
o 0

Then from (2. 2. 7) it follows that

2
v(\I/O,to) _<_ c )Wo)

2

l . . . .
where c > -- > 0. Since (\IIO, to) Is arbitrary subject to \IIO 6 H

2— 2L h’and

0
since (2. 2. l) is exponentially stable, it follows that \Ir(t) does not leave

h,'r° Thus

v(‘\II, t) _<_ G2 )\I/) Z
In a similar manner it follows that

v(\I/,t) Z C [\III2

1
where c1 Z 411: > 0. Thus relation (2. 2.22.) is satisfied by the v(\II, t)
given by (2.2.6) and (2.2.7).

(b) Using Liebnitz's Rule [OL-l] for differentiating v(\II, t)

with respect to t along a solution \II(t;\I/dto) of (2. 2.1) gives

22

t+T

2
dt [5 |\II(S;\I'(t;\IIO,tO),t)| ds]
t

d d
a"? V(‘I’(t,\I’o, to), t) ‘

2 2
- )‘I/(t;\I/(t;\1'0,to),t)l + I@(t+T:\IJ(t;\I/O.to).t)l

t+T d 2
+ _ . 0
St dt |\II(S,\Ir(t,\IIdtO),t)| ds (2.2.8)

By definition,

\II(S;\II(t + Atnlro, to), t + /_\.t)

‘I’(s;‘1'(t;\110.t0). t)

so that

t+T

d 2
5; a? |\II(S;\IJ(t;\IIO,tO),t)| dS = 0

and hence

Eva/,0 = - )\II(t;\IJo,tO))Z + I‘I’(t+T;‘I’O:tO))2 (3-2-9)

However, Since (2.2.1) is exponentially stable it follows that

-c1T
I\II(t+T;\I/o,to)| E (3 e |\II(t;\IIO,tO))

so that (2.2. 9) becomes

d 2 2 ~2aT 2

Using (2.2.7) in (2.2.10) gives

Bit v(\II, t) 5 -c3 I\II(t;\I/O,to)|2

where c3 _>_ % . Thus relation (2.2. 2b) is established.

(c) Denote by LIJO the j-th component of the vector \II . Since

F(\II, t) 6 @r in Hh '1" the partial derivatives
3

a 4’0 exist and

J

23

are continuous [CO-1, SAN-l], so that Vv(\IIO, to) exists and can be

computed by differentiation under the integral [FU-l]

t+T
o

2
VV(‘I/o,to) — St V)\II(t,\I/O,to)| dt
0
Carrying out this operation,
it +T
0
V v(\I/O, to) = 25 J(\I/(t; \IIO, to)) \I/(t; \I/o, t0)dt (2. 2. 11)

t
o

where J(\II(t;\IIO, to)) is the Jacobian matrix of \II(t;\IIO, to) with respect

to \I/o. Taking norms in (2.2.11)

 

tO+T
(v v(\I/o, to)) = 215; J(\II(t;\I/o, to)) \Ir(t;xlro, to)dt|
O
t +T
O
5 2 5; IJ(\II(t;\I/O,to))l |\I/(t;\I/O,to)|dt (2.2.12)
0
But from Lemma A. 2,
aMtnIIono) L(t-to)
< e , l. = 1, 2, o o o 9 n
an)J -
0.
1
so that
L(t-t0)
'J(‘I’(t“1’o' tom .5 e (2.2. 13)

Substituting (2.2. 13) in (2.2. 12) and using the hypothesis that

(2. 2.1) is exponentially stable gives

24

+
to T L(t-to) -<1(t-to)
le(‘I/O,to)l < 2 (3)1110) S; e e dt
0
= c ‘q/l (2.2.14)
4 O
t +T
o
where c4 = 213 X exp[(L-a) (t-to)] dt> 0.
0

But following the same reasoning as in (a), (2.2. 14) implies

|Vv| _<_ c4)\I/|, c4 > 0

and (2.2.2c) is proved.

Remark: Note that in part (iia) of the proof of the theorem, the only
use made of the fact that (2. 2.1) is exponentially stable is to insure

that \Ir(t) remains in the region H T where F(\II, t) 6 ©. Thus if

11.
F(\II, t) e @4 for all )1! 6 En, then (2.2.2a) is satisfied even if (2.2.1)

is not exponentially stable.

2. 3 Exponential Stability of a Class of Interconnected Exponen-

tially Stable Components
Consider now the exponential stability of a class of inter-
connected systems with component models of the form (2. 1. 2) and
interconnection constraints (2. 1. 3). Let the interconnections satisfy
graphic constraint GC and let graphic-component constraint GCC be

satisfied so that the interconnected system model is given by (2. 1. 5).

25

A component or system

i; = F(ir,t,zs) , F(0,t,0) s o
(2. 3. 1)
Z = G(\It,t,Z )
p s
is said to be stable if the null solution of
xi = F(\II, t, 0) (2.3.2)

is stable. If (2.3.2) is exponentially stable then it is said to belong
to class a , denoted F(\Il,t,0) 6 g .

To provide a better understanding of exponential stability
and how it relates to other types of stability, and to give an indication
of the importance of the class 5 , some properties of this class are:

0 The linear, constant coefficient system ‘1'? =F\II + DE(t),
is asymptotically stable, uniformly asymptotically
stable, and exponentially stable, if and only if, the

eigenvalues of F have negative real parts [SAN-1].

o For the continuous, time-varying system \1! = F(t)\II,
uniform asymptotic stability of the solution I! = 0 is
equivalent to both uniform asymptotic stability in
the large and exponential stability in the large of

every solution [SAN-1].

0 If the solution \Il(t) of the system ‘1’ = F(t)\II + E(t),
with F(t) bounded, is bounded for all continuous,
bounded inputs E(t), then the system is exponen-

tially stable [ HAL-l].

26

O The solution )1! 2 C of the system \I/ = F(\II, t), F(O, t) E 0,
is exponentially stable for all \IIO 6 Hho if there exists
some constant c > 0 such that \I/TF(\II, t) _<_ -c 3Z|\IJ| for

3
all (\II, t) E H If h can be taken sufficiently large

h, '1"
so that all \IIOE Hh , then the solution \II = O is exponen-
o
tially stable in the large. (Proof follows from Theorem

2.2.1 with v(\I/, t) = \IIT\II.)
0 Let L(r) be a continuous, scalar function on some
interval, 0 E r _<_ s, with L(O) = 0, L(r) > 0 for

O<r:s, and

dr - + 00
L(r) "
0+

 

If a(r) in (1.3.5) is linear and if |F(\If,t)| E L(r)|\IIl
for |\II| < r, then the system \II = F(\II, t) is exponentially
stable [HAL-l].

0 If F(\If,t) is homogeneous in \I/ of first degree, then
uniform asymptotic stability implies exponential

stability for \I/ 2' F(\II, t) [HAL-l].

Of particular interest here are systems for which AC)(\II, t)

in (2. 1. 5) has one of the three following properties or forms

AG(\I/,t) = M(t)\IJ (2.3.3)
)AG(\Ir,t)| < 711‘1’1 (2.3.4)
AG(\II,t) = M(t)\II + 83611, t) (2.3.5)

with

IEWJH < 72M

27

Clearly the three forms are not disjoint; however, each is important
enough to warrant individual attention.

The new stability results which follow begin with a theorem
giving sufficient conditions on the non-dynamic parts of the component
equations and the interconnection pattern for exponential stability of

a system of interconnected components from Class 5 in terms of

(2.3.4).

Theorem 2. 3. 1 Consider an interconnected system S given by (2. l. 5)

 

with component models of the form (2.1. 2) and interconnection con-
straints (2. l. 3). For each of the m components Si having dynamic

iii
,c,c be

parts, let F101}, t) 6 (® (1 & ) and let vi(\Ill, t), Cl, c2 3 4

Liapunov functions and associated positive constants satisfying

Theorem 2. 2. 1. Then all S for which

 

C2
C4
are exponentially stable, where 71 is related to S by (2. 3.4).
Proof: Let
m i
121

Denote by (my, t) the total derivative of (2. 3. 7) with respect to the

system

28

xi = F(\II, t)

and by 6561/, t) the total derivative of (2. 3. 7) with respect to the

system (2.1. 5). Then from (2. 2. 2a)

where

Similarly,

where

m

From (2.2.2c) one obtains

where

v(\If,t) > 23 C1 I‘I’llz
_ . 1
121
m .
> Z) c |\II1|2 — c IWIZ
— . l 1
i=1
, 1
c1 = minc > 0
i
2
V(‘I’9t) E C2|\I/)
_ i 0
C2 - max C2 >
m i 2i
IVv(\P,t)l = (2: (Viviana! )‘2
i=1
m . . 1
i i 2-
_<_ ( E (c4)\I/|) )‘2
i=1
_<_ 04M
- i > 0
c4 — maxc4
i
. . T
v = v + (Vv) AG(\I/,t)

_<_ v + |Vv| IAG(\II,t)|

(2.3.8)

(2. 3.9)

(2. 3. 10)

(2. 3. 11)

(2. 3. 12)

29

Applying inequalities (2. 2. 2) to (2. 3.12),

m i 12 m i i
{r < E -c (\II| + z c I\II||AG(\I/,t)|
S — . 3 . 4
1:1 1:1
Using (2.3.4),
m i i 12
' - >3 -
VS 5- . (C3 C4 71) ”E"
121
2
h - in(i i ) No if
w ere c3 - m. c3-c4')’:l . w
- i ( i i ) > O 2 3 14)
c3 - min C3-C4 71 (. .

1

then S is exponentially stable by Theorem 2. 2. 1. But (2. 3.14) is
satisfied if (2.3.6) is satisfied.

The next theorem relates the stability in the more general
situation, where AG(\II, t) has the form (2. 3. 5), to that of the reduced

system
if = FOP. t) + Mm)? (2.3.15)
that is, where GOP, t) = 0.

Theorem 2. 3.2 Let (F(\II, t) + M(t)\I/) E (63' F) E ) and let v(\Ir, t),

, c be a Liapunov function and positive constants

“‘1' “2’ C3 4

satisfying (2.2.2) of Theorem 2.2.1. Then the system (2.1.5) is

exponentially stable if

30

C3
72 < — (2.3.16)
C4

where 72 is given in (2.3.5).

Proof: Let vs be the total derivative of v(\I/, t) with respect to the

system (2. 1.5) and v for (2. 3.15). Then

i = f, + (vV)TE:’(s, t)

<0

_<_ + |vV||E§(\i(,t)|

or using (2.2.2) and (2.3.5)

o 2 ~
VS 5 -c3|\I/) + c4)\P|lG(\If,t)|

2
5- '(°3 " C472) 1‘1"
Let a1 = Cl, a2 = c2, a3 = (c3-c4'yz), and a4 = c4. Then

allxirlz _<_ v(\IJ,t) _<_ azlxir|z

. 2
VS 5 -a3 (\I’I

lVV(‘I’,t)) _<_ a4lxir|2

and by Theorem 2.2. 1, the system (2.1.4) is exponentially stable

ifa3>Oor

O

”I
use)

72<

The next theorem gives sufficient conditions for inter-

connected systems of a class to be exponentially stable, in terms

31

of component Liapunov functions, non-dynamic and interconnection pro-

perties, and an optimally selected parameter vector K.

Theorem 2. 3. 3 Consider an interconnected system (2. 1. 5) with

 

component models of the form (2.1. 2) and interconnection constraints

(2. 1. 3). For each of the m components Si having dynamic parts, let

i

i i i i i i
F(\Il,t) 6 (Q (189 )and let vi(\II,t), c1, c2, 63, c4

be Liapunov func-
tions and positive constants satisfying (2. 2. 2) of Theorem 2. 2. 1.

Suppose that there exists a parameter vector K 2 (k1, k2, . . . , km)T with

positive components such that c > 0, with

3

c3 = max {(min kic:i3 - (M(t)) mJax kjc‘l),
(mirz[1<,ci3 - (1 + “film 13459-1 ).

1
(min[ kc;
i 1 j:1

1 m - . .
-3 z (kicZIMij(t)|+kjei|Mji(t)|)])) (2.3.17)

where M(t) is defined by (2.3. 3) or (2. 3.5) and Mij(t) is the submatrix
of M(t) whose rows correspond to it1 in (2. 1. 5) and whose columns
correspond to I). Then the interconnected system (2. 1. 5) is exponen-

tially stable if

C3

7 < ----—-.- (2. 3.18)

2 k cJ

mix ' 4

J
Proof: Consider the function
m i

v(\II, t) = z: kivi(\II , t) (2. 3. 19)

121

32

for the reduced system (2. 3. 15). Then from (2.2.2a),

m i 12
v(\I/,t) > 2 k.c |xlx|
- . i 1
121
m .
> 7-3 c [\Illl2 = c |x1r|2
— . 1 1
121
where c = min k.c1
l i i 1
In a similar manner, one obtains
2
v(\II,t) 5 czlxir|
i
where c2 — max kic2

i
For the gradient vector Vv of (2. 3. 19), one has
m i 2 l
(va15 t)l = ( >3 IkinibImH )2

i=1

Then applying (2. 2.2b) to the right-hand side gives

m . . l
(was) 5 (.2 (ticpr‘W

i=1
_<_ 64M
1
where 04 - max kic4

l

(2. 3. 20)

(2.3.21)

(2. 3.22)

The total derivative of (2. 3.19) with respect to the system

(2.3. 15) is

33

3

(my, t) = z kivi(\II1,t) +(Vv(\II,t))TM(t)\II
i=1

3

5 2 kix'rimft) + IVv(\II,t)IIM(t)I |w| (2.3.23)
i=1

Using (2.2.2c) to eliminate IVvl and (2.2. 2b) to eliminate viflr: t) from
(2. 3.23) gives
. m i i 2 2
v(\II,t) 5 Z -kic3|\11| + c4lM(t)|)\II|
i=1
2
E, 'C3I‘I’1

, i
where 3 min kic3-c4)M(t)I

0
II

min k.c1 - (M(t)) max k.cJ (2.3.24)
. i 3 j J 4
1

An alternate expression for c3 may be obtained by returning to

(2. 3. 23) and noting that by Lemma A.4

1Vv(\II, t)|2 + [WIZ

 

IVV 01513)) 1‘1" 5

2
It follows then from (2. 2.2) that
m . . m . m .
v(\I’.t) < z ..1<.c1L If) 2 +[ 2 )k.vV.(\lrft)|2 + 2 1.1212] Mal
_ i-l 1 3 i-l 1 1 i-l z

m . . .
E - Z) (kic; - [1 + (kic;)2] WZ—(tfl) )‘I’llz

i=1

5 -c3|\II|2

34
where

c3 = min (Ric:5 - [1 + (kici)z]'J-IVI?(1:-)-l ) (2.3.25)
1

A third expression for c3 is obtained as follows

m 1 m m 1 T °
v(\II,t) = z k.v.(‘II,t) + z z: k.(Vv.(\II,t)) M..(t)\I) (2.3.26)
i=1 ‘ ‘ i=1j=l ‘ 1 ‘5'

Taking norms in (2. 3. 26) and using (2. 2.2) yields

m
_ . 1
i=1

12 m m
\Irl + E 2:
'=1 j=1

i

3| kie;|\r‘| IMij(t)| (all (2. 3.27)

H

Using the inequality

 

2
in (2.3.27) gives
m m m . i 2 ' 2
+
v(\II,t) < z -1. ‘(tllz + 2: 2 1.3 IM..(t)| [ 1‘“ J!” 1
— i 3 . . 4 ij 2
1—1 1:1 le
m i i 2 1 m m i ' i 2
= z -kc (w) +— z (k.c IM..(t)I +k.cJ|M..(t)I)I‘PI
i 3 2 i 4 ij J 4 ji
1:1 121 321
m l m 1 j 12
= .2 {-klc +-2- 2: (kic4IMij(t)| +kjc4|Mji(t)|)] [\III
i=1 J=1
-2
_<_ -C3 1“!)
where
1 1 m 1 1
= . "" + o 0
c3 Iriiln[kic3 2 ji31(kic4|Mij(t)| kjc4IMji(t)|)] (2 3 28)

35

Thus, if any of the values for c given by (2. 3.24), (2. 3.25), or

3
(2. 3.28) is positive, the reduced system (2.3. 15) is exponentially
stable. Further, if (2.3. 18) is satisfied, then by Theorem 2. 3.2 the

system (2. 1.5) is exponentially stable, and the theorem is proved.

Remark 1 If AG(\II, t) has the form (2. 3. 3), then any positive parameter

 

vector K which gives a c > 0 is optimal, in the sense that this is suf-

3

ficient for exponential stability.

Remark 2 If AG(\II, t) has the form (2. 3. 5) then any K for which (2. 3. 18)

 

is satisfied is optimal. However, if (2.3. 18) cannot be satisfied, but

> 0 can be found, then an optimal K is one which gives c > 0 and

3C 3

3

also maximizes the ratio c3/c Under these conditions, (2. 1. 5) is

4.
exponentially stable in some smaller region 1111' t for which
0
o

|G(\Ir,t)| < 'y' (\II,t)€ H 1

h,t

O

29

and

c
fly! < _2.
2 c4

Remark 3 The expressions obtained for c are by no means unique,

3

but they are sufficient to illustrate the general concepts involved.

 

The answer to the question of which expression for c3 is "best" is

clearly a function of the particular system involved.

Remark 4 If no K can be found such that c > 0, then the results

3

 

are inconclusive.

36

The problem of determining the parameter vector K can thus
be stated in terms of a static parameter optimization problem, where
the objective is to maximize the ratio c3] c4, with c3 and c4 given by
(2. 3. 17) and (2.3.22) respectively. This problem may appear to be
an extremely difficult one to solve in terms of analytical approaches;
however, it is a very practical problem for numerical computation
and can be solved on a digital computer by a number of different tech-
niques.

For a more restricted class of components, a more specific
set of sufficient conditions can be obtained. Consider in particular
that class of component models for which there exists Liapunov
functions satisfying (2.2. 2) which have a quadratic form. Denote
this class by 80. The class 890 not only allows a large class of
important components, but the restriction is intuitively reasonable,
considering the conditions (2. 2. 2). The next theorem gives sufficient

conditions for an interconnection of components from class 800 to

be exponentially stable.

Theorem 2. 3.4 Consider an interconnected system (2. 1. 5) with

 

component models of the form (2. 1. 2) and interconnection constraints

(2. 1. 3). For each of the in components Si having dynamic parts, let

F1011: t) 6 (Q n 890) and let

37

. .T . .
Viol}, t) = \Irl P1(t)\111 (2.3.29)

i i i i . . . . .
c1, c2, c3, c4 be Liapunov functions and assoc1ated pOSitive constants
satisfying relations (2.2.2). If there exists a positive parameter

T
vector K — (k1, k2, . . . ,km) such that

, i
c3 - miin kic3 - a” > 0 (2. 3.30)

where s is the maximum value assumed by any of the eigenvalues of

the symmetric part of the matrix

(P(K, t) + P(K,t)T) M(t) (2. 3.31)

where

1
klP (t)
2
kZP (t)

P(K.t) = ° (2. 3. 32)

 

 

_ m J

then the reduced system (2. 3. 15) is exponentially stable. Further, if

min kicg-s
72 < 1 . (2.3.33)

1

max k.c
, i 4
i

 

with 72 defined by (2. 3. 5) then the system (2. 1. S) is exponentially

stable .

38

Proof: For the reduced system (2. 3. 15), choose the positive definite

function

v(\II, t) = pr(K, tn; (2. 3. 34)

where P(K,t) is defined in (2. 3. 32). Then if

1

c1 = min kic1
i
1
c2 — max kic2 (2.3.35)
- kci
c4 “ max 14

i
it follows that

c1|\II|2 5 v(\II,t) 5 c (\IIIZ

2
(2. 3. 36)

)VV(‘I’,t)| 5 c4114

Now the total derivative of (2. 3. 34) with respect to (2. 3. 15) is

MB

v(\Ir,t) kiiioirft) + (Vv(\II,t))T M(t)\Ir

1

H.
II

m . i T T
Z kivi(\If, t) + \II (P(K, t) + P(K, t) )M(t)\II (2. 3. 37)
.:1

pm

But from Lemma A. 3

\IIT(P(K, t) +P(K,t)T)M(t)\II 5 .th = a'|\III2 (2.3.38)

where s is the maximum value assumed by any of the eigenvalues of

the symmetrix matrix,

39

%([P(K,t) + P(K,t)T] M(t) +M(t)T[PT(K,t) + P(K,t)D (2.3.39)

over the interval to S t < 00 so that

where c is given by (2.3.30). Consequently if c > 0, (2.3. 15) is

3 3
exponentially stable by Theorem 2. 2. 1, and if (2.3. 33) is satisfied, then

the system (2.1. 5) is exponentially stable by Theorem 2. 3. 2.

Remarks: The discussion following Theorem 2. 3. 3 regarding its appli-

cation also applies to Theorem 2. 3. 4. It may also be noted that in the

case where the matrix (2.3. 39) is constant, 0' is its maximum eigenvalue.
A special form of Theorem 2. 3.4 results when the Liapunov

functions of the system components satisfying (2.2. 2) can be taken as

.T

i ii
vi(\I/,t) = \II \II

(2. 3.41)

Corollary 2. 3. 4 If in Theorem 2. 3.4, the component Liapunov

 

functions satisfying (2.2.2) are given by (2.3.41), then the conclusions
of Theorem 2. 4.4 hold with 7 taken as the maximum value assumed by

the eigenvalues of
P(K)M(t) + M(t)TP(K) (2. 3.42)

where P(K) is the diagonal matrix

40

P(K) : . (2.3.43)

 

 

and the submatrices kiU are of order ni.

Remark 1 The assumption (2. 3.41) about the component Liapunov

functions is valid, in particular, when the component equations are

such that for some c; > 0, one has

T

\Irl Flor} t) 5 -C

i|i2
3

‘1' | (2. 3.44)

Remark 2 It follows that under the hypothesis of Corollary 2. 3. 4,

 

that a negative semidefinite M(t) is sufficient for c3 > 0.

2.4 Exponential Stability of a Class of Interconnected Systems

with Unstable Components
In the preceding sections, only interconnections of exponen-
tially stable components have been considered. No explicit considera-
tion has been given to the concept that even though acomponent may
be unstable when isolated, it may become stable--even exponentially

stable--when interconnected into a system of other components.

41

This problem is considered in the following theorem.

Theorem 2.4. 1 Consider an interconnected system (2. 1. 5) with

 

component models of the form (2. 1. 2) and interconnection constraints

(2. l. 3). Suppose that:

(1) F101,} t) e (@n 5,0), if- q, for (m-l) of the

m components Si having dynamic parts.

(ii) The Liapunov functions for the components
Si, if. q satisfying (2.2. 2) have thezform
1 1T 1 1
vi(‘II, t) = \II P(t)\II
ith oitdotticiidi
w asscae cnsanscl, 2,cBan c4.

(iii) The q-th dynamic component Sq is unstable.

. _ . i
(iv) o -m:n kic3 a”

3

where s' is the maximum value assumed by any

eigenvalue of the symmetric part of the matrix
(P(K, t) + an t)T)M<t)

1
P t

F(Ka t) = . k Pq(t)
q

' m
ka (t)

— d

 

 

If there exists a positive definite nq x nq matrix

Pq(t) such that for

(2.3.45)

(2.3.46)

(2. 3.47)

(2. 3.48)

42

T
vqpfit) = sq 13‘1“)qu (2.3.49)
one has
\‘Iqbltc,l t) _<_ -c(31|\1r|2, cg aconstant (2.3.50)

and a positive parameter vector K such that c > 0, then the reduced

3
system (2. 3. 15) is exponentially stable. Further, if (2. 3.33) is satis-

fied then the system (2. 1.5) is exponentially stable.
Proof: The proof is similar to that of Theorem 2. 3.4.

Rerrrark 1 In this case, a necessary condition for Theorem 2.4. 1 to

 

be satisfied is that a' be negative, and hence, that the matrix (2. 3.47)

be negative definite.

Rgmark 2 In the actual application of Theorem 2.4. 1 it is opera-

 

tionally expedient to assume a diagonal form for Pq(t) when possible.

III VECTOR LIAPUNOV FUNCTIONS FOR A CLASS
OF INTERCONNECTED SYSTEMS

In this chapter, sufficient conditions are given for various
types of stability for interconnected systems from a subclass of those
systems considered in Chapter II. The approach given here differs
from that of Chapter II in that a vectorLiapunov function is con-
structed from the Liapunov functions for the individual components.
A system of linear differential inequalities, satisfied by the vector
Liapunov function, is constructed from the component Liapunov
functions. This system of inequalities depends upon the component
equations and Liapunov functions, the interconnection pattern, and
also upon a parameter vector whose Optimal selection increases
the domain of systems for which this stability criterion becomes
valuable. The stability properties of the interconnected system
are then examined in terms of those of the linear comparison sys-
tem obtained from the system of linear differential inequalities by
replacing the inequality by an equality.

Under several additional restrictions on the system topology

and component characteristics, and equating all the components of

43

44

the parameter vector to unity, one of the results, Corollary 3.3. 1. 1,

reduces to the main result of Bailey [BA-1].

3. l Interconnected Systems
The class of systems considered here can be defined in terms
of a restriction on the class considered in Chapter II. Specifically,

it is assumed that AG(\II,t) in (2.1. 5) is linear in \II, that is,
AG(\IJ, t) = M(t)\II (3.1.1)

Condition (3. 1. 1) can be satisfied in either of two ways. If C(W, t)
is linear in \It, then (3.1. 1) will be satisfied: or if G(\II, t) is not

linear in \II, then A may be such that (3. l. l) is still satisfied.

3. 2 Vector Liapunov Functions

Corduneanu [COR-l] gave a generalization of Liapunov
stability criteria, viewing the Liapunov function as the dependent
variable in a first order differential inequality. The basic concepts
involved had been used earlier by Conti [CON-1] in connection
with existence and uniqueness studies for ordinary differential
equations. Stability results of this type are also given in [BR-1,
SAN-l].

Bellrnan [BEL-l], Matrosov [MA-l], and Lakshi'nikantham
[LA-l] have given stability results in terms of vector Liapunov
functions and have shown that their use is indeed advantageous for

certain problems.

45

In this section, results of Wazewski [WAZ-l] on differential
inequalities are used to extend the stability results of Corduneanu
[COR-l] to vector Liapunov functions.

Consider the m-dirnensional differential system
R = W(R,t) (3.2.1)

where the components wi of W are defined and continuous in some
open regionfz of the (m+l) dimensional space, and W is such that
there exists a unique solution passing each point (R0, to) E 52. Let
each function wi be non-decreasing with respect to r1, . . . , ri-l'

ri+1' . . . , rm in 52, i. e. , for arbitrary points~

(R,t) = (r1, .. "ri-l’ ri, r,

1+1,o..,rm,t)69

and

O I
(R,t) : (rlgooogr.- ,ri, ri

1
il m,t)€f2

+1.0009r

satisfying

it follows that
1
W(R,t) Z W(R, t)

Further, let the m-dimensional vector function V(t) be continuously
differentiable in the interval [to' 9), and such that V(to) = R0,
(V(t),t) 6 52 when t E (to, 9). Then it follows from the results of

Wazewski [WAZ-l], that if

46

V(t) 5 W(V,t) , t6[to,9) (3.2.2)
then

V(t) 5 R(t: Ro,to) , t e [to,6) (3.2.3)

The point 9 may be taken as 9 = 00, or 9 may be chosen so that

t + 9 as the solution of (3. 2.1) approaches the boundary of 52.

Theorem 3. 2.1 Let F(\II, t)€ @’ in Hh t for the n-dimensional
' o

 

system

i = F(\It,t) , F(0,t) a 0 (3.2.4))
and consider the m-dimensional system

R = W(R,t) , W(O,t) -=- 0 (3.2.5)

where the components wi of W are defined, real, continuous, non-

decreasing with respect to r1, . . . , ri 1, r , . . . , rm in

i+1

52:1(R,t): IRI<R500,t>0}

and are such that there exists a unique solution passing through
each point (R0, to) E 9. Let V(\I/,t) Z 0 be a differentiable m-dimen-
sional vector function, and let the total time derivative of V with

respect to (3.2.4) satisfy

1761/, t) 5 W(V(\Ii,t),t) (3.2.6)
Then:
(1) If the solution R = O of (3.2. 5) is stable and IV(\II,t)| is positive

definite, then the solution )1! = 0 of (3.2.4) is stable. Further,

47

if |V(\II, t)| is decrescent, then the solution \I/ = 0 is uniformly

stable .

(2) If the solution R = 0 of (3. 2.5) is asymptotically stable and
IV(\II,t)| is positive definite, then the solution \II = 0 of (3.2.4)
is asymptotically stable. Further, if |V(\I/,t)| is decrescent,

then the solution \II = 0 is uniformly asymptotically stable.

Proof: (1) Since |V(\II, t)| is positive definite, there exists a func-

tion a E R such that
(v61, t)| _>_ a(lwl) (3.2.7)

Let (3. 2. 5) be stable, and let 6 > 0, to Z. 0 be given. Then from the
stability of (3.2. 5) there exists an n(€, to) > 0 such that 0 < [ROI < 17
implies

IR<t: R0. to)! < a(e) (3.2.8)
From the continuity of V, it follows that there exists a 5(6, to) > 0
such that I\IIO| < 6 implies IV(\IIO,tOI < 17. Since from (3.2.6) one
has

V(Mt: ‘I’o'toM’ g W(V(¢I(t;\Ir0.tO).t).t) , (3.2.9)
it follows from Wazewski's results (3. 2.3) that
V(\II(t;\I/O,to), t) _<_ R(t;V(\I’o,to), to) (3.2.10)
Combining (3. 2. 7), (3. 2. 10), and (3. 2. 8) gives

a(IW(t;)IIO.tO)I)5_ |V(\I'(t;\IIO,tO),t)| 5 lR(t;V(\I/0.to).to)l < a(E)

48

so that

a(|\I((t;\IIO, to)|) < a(E)
and since a E %
|\Il(t;\I/O, to)| < E

for t: to, if [\IIOI < 6 (€,to). Thus (3.2.4) is stable.
If (3. 2. 5) is uniformly stable, then r) in the preceding may be
chosen independent of to. Further, if )V(\II,t)I is decrescent, then

there exists a function b E x such that
|V(\Ir.t)| 5 NM) (3.2.11)

and 6 may also be chosen independent of to. In fact, choose 6 < g,
where b(§) = n. Then proceeding as above, uniform stability of

(3.2.4) is obtained.
(2) If (3.2.5) is asymptotically stable, then

lim R(t;V(\II,t),t)=O
o o o
t->oo

This condition and (3. 2. 10) imply

lim V(\II(t;\Ir ,t ),) = 0 (3.2.12)
t-eoo o 0

lim a(|\II(t;\IIO,tO)I) = o
t->oo

which implies that

49

lim \I/(t;\I/,t) = 0
o o
t->oo

and (3.2.4) is asymptotically stable.
If (3. 2.5) is uniformly asymptotically stable, then there exists

an no > 0, and for E > 0, a quantity T(€) > 0 such that IRO) < no implies

)R(t;R0,tO)| < a(E), for t > tO + T(E) (3. 2.13)
If 6 is chosen such that b(6 ) < 'r) , then )\II ) < 6 implies that

O O O O O

|V(\Iro,to)| < b(l‘I’ol) < b(oo) < no (3. 2.14)
Thus from (3.2.13)

|R(t; V(\II0, to),to)) < a(E), for t_>_to + T(€) (3. 2. 15)
From (3.2.15), (3.2.10), and (3.2.7) it follows that

a()\I/(t;\I/o,to)l) < a(E) , for tzto + T(€) (3.2.16)
Finally from (3. 2. 16) and a e 7")
IxIr(t;xI/0,t0)| < E , for tit0 + T(€), lxlzol < 60

and (3. 2.4) is uniformly asymptotically stable, thus completing this

proof.

Remark This theorem is somewhat related to a theorem on vector
Liapunov functions given in [MA-l] ; however, in that theorem the
requirements on the vector V(\It,t) are too stringent for the theorem

to be applicable here. In Matrosov's theorem it is required that

50

k
‘23 vi(\II,t) , l5k5m
i=1
be positive definite. But if the partial sums v1(\II, t), v1(\II,t) +

v2(\If, t), . . . , are not all functions of all the components of \It, then

clearly they cannot be positive definite functions of \II.

3. 3 Vector Liapunov Functions and Stability of the Interconnected

System
The theory developed in the preceding can now be applied to the
central problem of investigating the stability properties of the inter-
connected system. As before, it is assumed that the interconnection
of components (2. l. 2) given by (2.1. 3) satisfies graphic constraint
GC and that graphic-component constraint GCC is satisfied so that
the system model is given by (2. 1. 5). The following theorem gives

one such result.

Theorem 3. 3. 1 Consider an interconnected system S having the

 

form (2. 1. 5), satisfying (3.1.1), with component models of the
form (2. 1.2) and interconnection constraints (2. 1. 3). For each of

the m components Si having dynamic parts, let F1011: t) 6 (@ (1 83 )
i

3 be Liapunov functions and positive

i
, c

and let v1(\II:t), ci, Cl, C 4

2
constants satisfying (2.2.2).

Define the m-diinensional linear system

it = W(K,t)R (3.3.1)

51

where the matrix W(K, t) is given by

 

 

 

 

i 2N 2
(kic4) miIMi.(t)|
wij = jJ , i/=j (3.3.2)
4km+ikjcl
qi .
. , if q. < O
k c1 1
i 2
wii = 6 0 , if q.1 : 0 (3.3.3)
qi .
. , if q. > 0
k C1 l
i l
K...
with
1 1
qi — km+11 + ki(c4IMii(t)I -c3) (3.3.4)

The submatrix Mij(t) of M(t) is as defined in Theorem 2. 3. 3; $1
is the number of nonvanishing norms lMij(t)| it 0 for j ,E i: and
K is a 2m-dimensional parameter vector.

If there exist positive values for the components of K such
that the linear comparison system (3. 3. 1) is stable (uniformly stable,
asymptotically stable, or uniformly asymptotically stable), then‘the
interconnected system S is stable (uniformly stable, asymptotically

stable, or uniformly asymptotically stable).

Proof: Since F1(\II,1 t) E (@ F) {i ), the existence of the component
Liapunov functions satisfying (2. 2. 2) is assured by Theorem 2. 2. 1.

Consider the m-dimensional vector V whose i-th component is

52

given by

1
vi — kiv (3.3.5)

where v is the Liapunov function corresponding to the i-th dynamic
component and ki > 0 is a constant. The total derivative of V1 with
respect to the system S is

i

o a V iT'i
: —— + V .
vi ki at ki( v) \I/ (3.3 6)
and from (2.1.5) and (3.1.1)
. .i i T
vi = kiv + ki( Vv ) Mi(t)\II (3. 3.7)

where Mi(t) is the row submatrix of M(t) formed by the rows of M(t)
corresponding to iii in the system model S. Thus (3. 3. 7) can be
written as

m
E

1,, = k.v1+k.(Vv1)T
l l. l

Mij(t)\1ri (3. 3. 8)
i=1

Applying the Schwartz inequality and inequalities (2. 2. 2) to the right
side of (3.3.8) gives
1’) < k (c1 |M .(t)| -ci) (xi/HZ + k c:1 [\Ilil I; [M (t)| I‘lfil (3.3.9)
i— i 4 i1 3 i 4 j‘l ij
jfii
Using Lemma A. 7 to eliminate the product terms (\Iill I119) from

(3. 3. 9). one obtains

53

i i 12
v. _<_ 1km,+ki(c4|Mfi(t)I-c3)llwI

 

l i m ° 2
+ (k.c z |M..(t)||‘I'JI)
4k . 14 . 1J
m+1 J31
jfii

Applying Lemma A. 6 to the last term in (3. 3.10) gives

. 1 1 12
V1 5- [km+i + ki(c4‘Mii(t)| ‘C3m‘1’ l
1 2
(k.c ) m .
+ Ji— si. 2: (1)/1..(t)|?‘(19(2
4k . 1 . lJ
m+1 J=l
jfii

Using (2.2.2) to eliminate [\IIIIZ and qujlz from (3.3.11)

 

i 1 m .

g.- v + Z w k,v‘], ifq <0
1 1]]

c 321
2 . .

1141
v15 <

I 1 m .

34V + z w.,k,v3, ifq.>0
i IJJ 1

c 3:1
1 ‘161

L J

Then from (3. 3. 5) the system

if 5 W(K, t)V

results with the entries wij of the matrix W(K, t) as given in the

theorem and V = [v1v2. . . vm]T. Since the v1(\Ir:t) and hence

(3. 3. 10)

(3. 3. 11)

(3. 3. 12)

i . . . . . . i .
vi(\II, t) are pOSitive definite functions of their arguments \II , it can

be shown that IV(\II, t)) is a positive definite function of its argument \II.

54

Further, from (2.2. 2a) the V1 (\II} t) and hence vi(\IIf t) are decrescent,
so that |V(\II, t)| is decrescent. Thus the proof follows from Theorem
3. 2. l .
At various steps throughout the derivation of the matrix
W(K, t) in the preceding theorem, there is a certain amount of choice
as to exactly what inequalities are used and how they are used. Clearly,
the answer to which is the better choice depends upon the particular
system structure. The following corollaries give some additional

relations for W(K, t), which are closely related to those in the theorem.

Corollary 3. 3.1. 1 Theorem 3. 3. 1 holds for W(K, t) with

 

m 2

2: IMi.(t)|

i=1 J

w.. = #1 . . ifij (3.3.13)

1J J
4km +ikjcl

1 2
(kic4)

 

and wii given by (3.3.3) and (3. 3.4).

Proof: The result follows from the proof of the theorem by apply-

 

ing Lemma A. 5 instead of Lemma A. 6 to the last term in (3. 3. 10).

An application of Corollary 3. l. 1. 1 gives the result of
Bailey [BA-l] as a special case. In the notation of this thesis,
Bailey considers component models of the form

it = F(xlet) +Dlz: (13.1)

55

In addition to the hypothesis of Theorem 3. 2. 1, he assumes that no
loading occurs and that the system t0pology is restricted so that in

Theorem 3. 2. l

M.. = 0, i=l,2,...,m (B.2)

In his main theorem, Bailey gives the comparison system (after

 

corrections)
i
°~3
W.. : " —T', izj (Be 3)
ii 2ci
2
. m
1 2 2
(c4) 27 W l
w.. = 1:.1. . 11:1 (3.4)
1‘] 201C]
3 1

If in Corollary 3.3. l. 1, one takes M11 2 0; C = 0 and G(\I/, t) has
. , _ = i . = ,
the form Gil/in (2.1.1), ki— land km+i c3/2, i l,2,...,m,

then (3. 3.13), (3.3.3), and (3. 3.4) reduce to (B. 3) and (B.4).

Coronary 3. 3. 1. 2 Theorem 3.3.1 holds for W(K, t) with ’n‘n’i in

 

(3. 3. 2) replaced by r'ni, where r’r‘ii is the number of nonvanishing

norms IMij(t)| )5 0, all j; and wii is given by (3. 3. 3) with

i 2
1 “$45311 lMiim '
q. = k . -k.c + __ (3.3.14)

1 m+1 i 3 ‘1ka

 

56

Proof: The proof of this result follows from that of the theorem by

 

considering inequality (3. 3.9) in the form

. . . . m .
6. < -k.c1|\Irl|2 +k.cl|\Irl| z |M..(t)||\IIJ| (3.3.15)
1 — 1 3 1 4 j‘l 13

Corollari3. 3. 1. 3 Theorem 3. 3. 1 holds with W(K, t) given by

 

 

i 2 m 2
(kic4) ZS [M..(tH
.=1 13
wij = J J- » ifij (3.3.16)
4km+ikjcl
and wii by (3. 3.3) with
i (kic;)2 m 2
‘11 = km+rkiC3 + _4k . (_2 IMij(t)I ) (3.3.17)
m+1 J21

M: The proof follows from the proof of the theorem with both
of the modifications given in the proofs of Corollary 3. 3. 1. 1.and
Corollary 3. 3. 1. Z.

The next theorem and corollaries give stability criterion in
terms of a comparison system that does not depend on a parameter
vector. A parameter vector is not involved, because in this particular
derivation, the parameters as appearing in (3.3. 5) would have no effect
on the stability of the comparison system. Further, the inequality
which introduces the second m components of K, in the preceding,

is not used.

57

Theorem 3. 3. 2 Consider an interconnected system S having the form

 

(2. 1. 5), satisfying (3. l. 1), with component models of the form (2. 1. 2)

and interconnection constraints (2. 1. 3). For each of the m components

Sihaving dynamic Parts. let F‘s/ft) e (630 E; > and let visit). c1. c2.

i

4 be Liapunov functions and positive constants satisfying (2.2. 2).

i
c

3’°

If the m-dimensional, linear comparison system

I? = W(t)R (3.3.18)
with
i
C4
w.. = -—., ifij ' (3.3.19)
1J ZcJ
1
Ci m
q. = -<=1 +-‘-1-(1 + E IM..(t)|2) (3.3.20)
1 3 2 , 13
F1
and
r.
‘11
-i- , for qi < 0
c2
Wii =< O , for qi = 0 (3.3.21)
qi
-i- , for qi > 0
C

 

L

is stable (uniformly stable, asymptotically stable, or uniformly
asymptotically stable), then the interconnected system S is stable
(uniformly stable, asymptotically stable, or uniformly asymptotically

stable).

58

Proof: Since F1011: t) 6 (@ fl 8) ), the component Liapunov functions

 

vi(\II: t) satisfying (2.2.2) exist by Theorem 2. 2. 1. Consider the m-
dimensional vector V with i-th component vi(\II: t). The total derivative

of vi with re spect to the system S is

 

5V1 T .1
(xi = at + (Vvi) \1/ (3.3.22)
or
61011») = 610111, t) +(v vi(\I/i, t))TMi(t)\II (3.3.23)

where 659, t) is the total derivative of vi with respect to the component
model, Vvi(\I/i, t) is the gradient vector of vi with respect to its argument
\I/i, and Mi(t) is the row submatrix of M(t) defined in the proof of
Theorem 3. 3.1. Rewriting (3. 3. 23) as

. . m .
3 (x11, t) = {rim/1, t) + (Vvi(\II1, t))T z: Mij(t)\PJ (3. 3.24)

j=1

and applying the Schwartz inequality and inequalities (2.2. 2) gives
m

+cl 2 |M..(t)||\1rl||x1r’| (3.3.25)
4 j=1 13

i

12
3|

v. E-c [\II

1

From Lemma A.4 with a = ‘Mij(t)l |\II1| and b = (\IaJi, one has

i

i
. 2 :1
3 2

m
|\Ir1| + E

s
V. < -C
1 .—

((Mij(t)|2(oi(2+ Iqrilz) (3.3.26)
i=1

Rearranging terms in (3. 3. 26) and applying inequalities (2.2. 2) gives
the result. By the same reasoning as in Theorem 3. 3.1, |V(\It,t)I
is positive definite and decrescert, thus the theorem follows from

Theorem 3. 2. 1 .

59

Corollary 3. 3. 2. 1 Theorem 3. 3. 2 holds for W(t) given by

 

1 2
c4 lMi.(t)|
w.. = .J . ifij (3.3.27)
11 2cJ
1

 

and

z i
(m+IMii(t)| )c4

qi = -c3 + 2 (3.3.28)

 

Proof: The proof follows from the proof of the theorem by taking

 

a = (Ill) and b = 'Mij(t)| I\II]| when applying Lemma A.4.

Corollary 3. 3. 2. 2 Theorem 3. 3. 2 holds for W(t) with wij given by

 

(3. 3. l9) and

1 1 1 m 2
qi = -C3 +C4 (|Mii(t)| +2- jfl Nij(t)l ) (3. 3.29)
j£1

Proof: The proof follows from the proof of the theorem with (3.3. 25)

 

replaced by

. 1 12 1 12
1 m 1 °
+ c4( 2: [M..(t)|I\II | WI) (3.3.30)
. 1]
J=1
j£1

Corollary 3. 3. 2. 3 Theorem 3. 3. 2 holds for W(t) with

 

1 2
c4 IMi.(t)|
W.. = .4 . iaéj (3.3.31)
11 2cJ
1

 

60

and

_ i i m-l
c1i — -c3 +c4(|Mfi(t)l + 2 ) (3.3.32)

Proof: The proof follows from the proof of the theorem with both

 

of the modifications given in the proofs of Corollary 3. 3. 2. l and

Corollary 3. 3. 2. 2.

Corollary 3. 3. 2.4 Theorem 3. 3. 2 holds for W(t) with

 

 

c; lMi.(t)|

w.. = .J , ifj (3.3.33)

11 2C1

1

and
1
1 C4 m
2 - —-- + z o o

qi c3 + 2 (lMiiml j_1 IMij(t)|) (3 3 34)

MI The proof follows from the proof of the theorem by taking
a = If) and b = I‘ljl when applying Lemma A. 4.
If it is assumed that the component models belong to class
eg, more specific sets of sufficient conditions can be obtained
corresponding to Theorem 3. 3.1 and Theorem 3. 3. 2 and their
corollaries. One such result given for class 80 in the following

theorem corresponds to Corollary 3. 3. 2.4 for class Eh.

Theorem 3. 3. 3 Consider an interconnected system S having the

 

form (2. l. 5), satisfying (3. 1. 1), with component models of the

form (2. 1. 2) and interconnection constraints (2. 1. 3). For each of

61

the m components Si having dynamic parts, let F1(\I11, t) 6 (Q (l 5Q)

and let
1 1T 1 1
vi(\II,t) = \II P (t)\II (3.3.35)
c1, CE, (3313' c; be Liapunov functions and positive constants satisfy-

ing (2.2.2). Define the m-dimensional linear comparison
11 = W(t)R (3.3.36)

with the matrix W(t) given by

I (Pia) + Pi(t)T)Mi.(t)I
wi. = . J ,115 (3.3.37)
J 2c"l

 

and r-

1
31—, ifqi<0
c2

Wu: 0 , 1fqi=0 (3.3.38)

1
L .

i’ 1fqi>0
c1

 

where
= -ci +¢r +91 I; ((1218) +Pi(t)T)M (t)| (3.3.39)
C11 3 1 2 5:1 15
jf-i

and 0'i the maximum value assumed by an eigenvalue of the symmetric

part of the matrix

62

(121(1) + Pi(t)T)Mii(t) (3. 3.40)

If the linear system (3.3.36) is stable (uniformly stable, asymptotically
stable, or uniformly asymptotically stable), then the interconnected
system S is stable (uniformly stable, asymptotically stable, or uniformly

asymptotic ally stable).

Proof: Since F101}, t) 6 (© 0 EDQ), the Liapunov functions (3.3.35)

 

satisfying (2. 2. 2) exist. The total derivative of V1 with respect to the

system S is

(rim, t) = {7101}, t) + (Vvi(\I/i, t))TMi(t)\II (3. 3.41)

or from (3. 3. 35)

1 1T m
3,011.1) = (vim/,1) + \II )3

1 (pin) + Pi(t)T)Mij(t)\Ifi (3. 3.42)

i=1
From (2.2.2) and Lemma A. 3, (3.3.42) gives

i 2 1T m . . T .
+1,-9|qu +11; 2 (P1(t)+P1(t) )Mij(t)\1r] (3.3.43)
1:1
jfii

. < ( i
vi _ -c3
Then using the Schwartz inequality and applying Lemma A.4 with

a= I\I/1I, b: NJ), one obtains

° < ( 1+. +1 I; |(Pi(t)+Pi(t)T)M (t)|)lxIril2
V1— "°3 1 2 Fl 15
jfi
1 m 1 1,1" ‘1; 2
+ 5— : I(P(t)+P(a-) )Mij(t)ll I (3-3-44)
i=1
jfii

63

Inequality (2. 2. 2a) applied to (3. 3. 44) gives the desired result for

W(t), and the theorem then follows from Theorem 3. 2. 1.

Corollary 3.3. 3 Suppose that the component Liapunov functions

 

vi(\Ill, t) in Theorem 3. 3. 3 can be chosen as

. . 2
vi(‘II1,t) = lm‘l (3.3.45)
Then the theorem holds with
lMi.(t)|
w.. = ——.l-— , 11% (3.3.46)
1J CJ
1
i m
qi = -c3 + o-i + .2 lMij(t)| (3.3.47)
J=1
#1

and 0'i the maximum value assumed by any of the eigenvalues of the

matrix

.1. T
M11“) + Mum (3. 3. 48)

IV APPLICATIONS

The purpose of this chapter is to illustrate, by means of example,

some of the stability theory developed in this thesis.

In so doing, an

attempt is made to indicate the importance of these concepts to design

work.

Consider a hypothetical interconnected system S composed of

the following subsystems

S : a second order, three terminal, nonlinear, mixed component

1

 

 

 

 

 

 

 

 

 

 

64

 

 

. 300

. 400

 

. 086

1
-. 549 1. 943

-.239

.100

 

 

 

 

- L.

 

 

(4.1)

65

$2: a third order, three terminal, linear, dynamic component

I
F- - —

 

 

 

 

 

 

 

 

 

 

 

o (- " " "
. : -. -. . + -0 1 o
04 700 200 500 (04 57 213 (Y4
i5 .400 -.500 -.300 13’ 1.000 -.652
- _ (4.2)
1.000 . 00
x3 - 5 0 4:3
x4 0 -.500 2.000 14
‘4’5
3 4
2:8
S3: a second order, five terminal, linear, mixed component
(- '1
i -.300 .200 0 -.223 -.315 -.156 .071 x
6 _ 6 + 5
i7 -.200 -.100 07 1.000 .461 -.978 .603 y6
Y7
y8

 

 

ys F1.000 .600
x6 -.300 4.000
0 1.000
x7
0
x8 0

 

 

 

 

 

S o

4.
qJ8 _ '3¢8
$9 ¢8 -2¢
_ _ _.0
y13 39
x14 4 g
y15 ‘0071

 

 

 

66

1.9941
-0 378

-.395

 

 

F-2.661 -2.910
.368 .273
0 1.000
.199 ..200

5
6

. 500

-.019

F a
-.001 .298 x13
.197 .397 y14
x15

L .1

1.166 -.8991
-1.375 1.465
O 0

O -l.000

 

 

a second order, four terminal, nonlinear dynamic component

 

 

 

(4.4)

 

67

 

 

14
13 b
4
15
SS: a three terminal, linear, algebraic component
y 0 -1.000 x
9 = 9 (4.5)
x10 1.000 .400 le
9 10
:55
S6: a three terminal, linear, algebraic component
x -1.000 .300 y
11 = 11 (4.6)
x 0 -.400

12 y12

68

11

 

The system graphb for the interconnected system S is

 

 

 

12

it?

 

69

and the linear interconnection constraint equations derived from %/ are

 

 

 

 

 

 

3:1— _0 1 0 00 1-1 0 0 0 0 0 0 0 O__Y1_
yz -l O O 0 l 0 O 0 -l O O O l 0 0 x2
3,3 0 0 0 0 -1 0 0 O O O O 0 -1 O -1 x3
y4 O O 0 O l O O O O O O O l 0 1 x4
x5 0 -1 l -1 O -l l O 0 l l -l 0 O 0 Y5
Y6 -1 O 0 O l O O O O O 0 0 l 0 0 x6
Y7 _ 1 O O 0 -l 0 0 0 l O 0 0 -1 0 0 x7
Y8 — 0 0 O O 0 O O 0 -l 0 O 0 O O 0 x8
x9 0 l 0 O O O -l l 0 -l 0 O O 0 0 y9
le O O 0 O -l 0 O O l O 0 0 -1 0 0 x10
3'11 0 0 0 0 -l O O O O 0 0 0 0 0 0 x11
3'12 O O 0 0 l 0 O O 0 O O O l O 1 x12
x13 0 -l l -1 0 -1 l O O 1 O -l 0 1 0 y13
y14 O 0 O O 0 O 0 O O 0 O O -l 0 -1 x14
x15 0 O 1 -l O O 0 O 0 O O -l 0 l 0 y15
Problem 1 Determine a value for the parameter g in the component
equations for S4 so that the interconnected system S is exponentially
stable. The value of g must satisfy the design constraint

41.485 5. g _<_ .915 (4.8)
Solution: Consider first the matrices F(Z) and PB) corresponding
to the dynamic parts of the linear components S2 and S3 respectively.

(4. 7)

70

(2)

The eigenvalues of F are determined to be

11 = -.1871
12 = -.2065 +j0.9220
13 = -.2065-j0.9220

and those of F(3) are

11 = -.2000 +j0.1732

12 = -.2000 - j0.1732

Since the eigenvalues of S2 and S3 all have negative real parts, these
components are exponentially stable and have Liapunov functions

satisfying (2.2. 2). To determine the component Liapunov functions

v2 and v3 for 52 and S3, respectively, assume they have the quadratic

forms
T
v2 = (1(2) P(ztpm (4.9)
T
3 3
v3 = \I/(3) P( )\II( ) (4.10)
Assume further, that the total time derivatives of v2 and v3 with
respect to the component equations (4. 2) and (4. 3), respectively,
are
(2)T (2) 2 2 2
V2 :-\I/ \I/ = -(LiJ3 +4.14 +¢5) (4'11)
and
(3)T (3) 2 2
v3 = -\II \II = -(L|.I6 +417) (4.12)

71

Frame [FR-2] has given a very elegant method for calculating the
positive definite matrix of a quadratic Liapunov function, for time-
invariant linear systems, from the negative definite matrix in the
derivative of the Liapunov function. Application of this procedure
to (4. 11) and (4. 12) give the following positive definite matrices for

(Z) (3)

13 and P .

_ 2.723 . 326 .002

Pm = .326 2.775) .345 (4.13)

 

 

. 002 . 345 2. 240

2.143 -.714
pm = (4.14)

Consider now the nonlinear component S If a Liapunov

1.
function v1 is assumed as
1 (1)T (1) 1 2 2
v1 = -2- \I/ \I/ = 2W1 +1112) (4.15)
then
Ll11
V v = (4. 16)
1 up
2

The total time derivative of v1 with respect to the component equations

(4. 1) becomes

72

 

T
{)1 2 211(2) F(Z)
_ 2 l . 4 2
— “DZ - 2 ¢zs1n2 412 - ((11 - 4J1 (4.17)
' 0
Since glen > -.218,
2 2 1 .
aLPZ < (1112 +E¢zs1n2¢2) (4.18)
for all a i . 782, so that
. 2 2
v:l 5 4.782102 + 411)
5 “78(4): +03) = -.78|9(2)|2 (4.19)
Thus S1 is exponentially stable by Theorem 2.2.1, with v1 satisfying
(2.2.2).
For the nonlinear system S4, assume a Liapunov function v4 as
(4)T (4) (4) 2
v4 = 11/ 111 = (11/ | (4.20)
Then
(”4 = 24““ (4.21)

and the total time derivative of v4 with respect to (4.4) is given by

{r4 = 434:: +211): +4413)
_<_ ~20): +93) = -2|\Ir‘4’l2 (4.22)

The function v satisfies (2. 2. 2) and thus S

4 is exponentially stable.

4

73

Having determined component Liapunov functions, the positive

constants for which the functions satisfy (2. 2. 2) of Theorem 2. 2. 1 are

51‘ oi: .500 oi: .500 6;: .780 631:1.000
sz: cf=2.038 6:: 3.156 c:=1.000 ci=6.312
S3: c: = 1.847 c: = 3.876 c: = 1.000 c: = 7.735
54: c? = 1.000 c: = 1.000 c: = 2.000 c: = 2.000

Since the component Liapunov functions have the quadratic form
(2. 3. 29). Theorem 2. 3.4 is applicable. For the parameter vector

K = [1 l l 1]T, the following values of c as determined by (2. 3. 30)

3!
in Theorem 2. 3.4,, are computed for several values of g in the allowable

interval

Value of g Computed value of c

 

 

3
-l.485 .18
-1.085 .16
- .685 .09
- .385 - .01
- .285 - .06
.115 - .36
.515 - .82
.915 -l.36

BY Theorem 2. 3.4, S is exponentially stable for any g

giving a positive value for c3.

74

From (2. 3. 35), one also has

cl=.500 c2=3.867 c4=7.735

for the Liapunov function constructed for the interconnected system S.
Referring back to the proof of Theorem 2, 2.1, one can see

that there is more information here than just the assurance that S is

exponentially stable for say, g = -l. 485. Following equation (2. 2. 5)

one has
but; 90.10)! 5 Blwolexpl-au-tofl

for S, where

 

 

 

C
2 3 867
2 -— 2 . : Z
(3 c .500 8
1
and
C
3 .18 .
° ‘ 262 ‘ 7.734 ‘ '025

Clearly this upper bound on the solutions of S is not a least upper bound,
because of various approximations made and because of the non-unique-
ness of the Liapunov functions involved. However, this does give a
lower bound on the rate of decay of the solutions of S. Further, if
F(‘If,t) 663/ only for I! in some th- with h fixed, then since a value

is known for (3, the value ho’ defining Hh , can be determined from

(1. 3.7). Then for all \IIOE Hh , one is aSsured of exponential

o
stability .

75

Problem 2 In the preceding problem, a value of g = -. 285 gave

 

c = -.O6.

3 Does this mean that S is not exponentially stable for this

value of g ?

Solution: The first question one should ask is whether a positive c

3
might not be obtained for g = -. 285 by using values for the parameter
T . .

, as used 1n the precedlng.

vector K other than K = [l 1 l l]

A gradient directed search in the K space, using Theorem 2. 3.4

1]T, yields the results shown below.

 

and beginning at K = [ 1 1 1
k1 k2 k3 k4 Computed c
1.0000 1.0000 1. 0000 1.0000 -.06
1.0139 1.0000 .9999 .9854 -.04
1.0333 1.0000 .9998 .9649 -.01
1.0604 1.0000 .9997 .9361 .03
1.0984 1.0000 .9995 .8960 .08
1.1516 1.0000 .9992 .8397 .14
1.2262 1.0000 .9988 .7609 .17
1,2527 1.0923 1.0062 .7328 .18
1.2527 1.1427 1.0501 .7751 .19
1.3480 1.2309 1.1221 .8060 .20
1.3581 1.2481 1.1404 .8412 .21

It can be seen from these results, since an admis sable K can

be found which results in a c

3 > 0, that the system S is indeed exponen-

tially stable for g = -. 285. This clearly shows the value of the

dependence of the system Liapunov function upon K.

V SUMMARY AND CONCLUSIONS

This thesis considers the stability of several classes of inter-
connected systems, consisting of multiterminal, nonlinear, time-
varying components. The central objective is to obtain original stability
criteria which relate to the two fundamental structural features of the
system and which are practical to apply.

Exponential stability of several classes of interconnected systems
is considered in Chapter II and a number of original results are obtained.
Specifically, Theorem 2. 3.1. gives sufficient conditions for exponential
stability of a class of interconnected, exponentially stable components.
This condition is given in terms of a bound on the norm of a vector in the
interconnected system model. The bound is a function of the stability
properties of the dynamic parts of the component models. Theorem
2. 3. 2 provides a basis for studying exponential stability of certain
classes of systems in terms of the exponential stability of a less compli-
cated or reduced system. Theorem 2. 3. 3 gives a sufficient condition
for exponential stability of a class of interconnected exponentially stable

”components. The condition is in terms of an algebraic relation which

76

 

77

depends upon properties of component Liapunov functions, structural
features of the system, and a parameter vector. The condition on the
algebraic expression is obtained by constructing a Liapunov function for
the interconnected system in terms of the individual component Liapunov
functions. Theorem 2. 3.4 and Corollary 2. 3.4 give more specific con-
ditions for exponential stability of systems constructed from a more
restricted classes of components. Theorem 2.4. 1 extends the results
of Theorem 2. 3.4 to allow for systems with unstable components.

In Chapter III, the study of the stability of interconnected systems
is based upon the construction of vector Liapunov functions for the
systems. Theorem 3. 2. 1 extends some results of Corduneanu [COR-1],
on scalar Liapunov functions, to vector Liapunov functions and establishes
a basis for examining the stability of interconnected systems in terms
of vector Liapunov functions. Theorem 3.3. 1 and its corollaries give
sufficient conditions for stability, (uniform stability, asymptotic
stability, or uniform asymptotic stability) of a class of interconnected
systems in terms of that of a linear comparison system, derived
from a vector Liapunov function for the system. The comparison
system is a function of the prOperties of the component Liapunov func-
tions, the system structure, and a parameter vector. Theorem 3.3.2
and its corollaries give similar results, except that the comparison

system, derived in a different manner, is not a function of a parameter

78

vector. Theorem 3. 3. 3 and Corollary 3. 3.3 give more specific
comparison systems for more restricted classes of components.

In Chapter IV a ninth-order, nonlinear system is considered
for illustration. The example indicates an important design application
and shows the value of the parameter vector as an argument of the
system Liapunov function.

The stability criteria obtained are particularly well suited for
digital computation and computer-aided design. Considering stability
in terms of the system structure, in many cases, helps to circumvent
some of the mathematical complexities involved with higher-order
nonlinear systems.

The eXponential stability results in Chapter II are particularly
valuable in engineering design applications, since a bound on the decay
or response of the interconnected system is obtained as a by-product
of the stability analysis.

As with most sufficiency conditions, there is always the pro-
blem that the conditions may be overly sufficient. This problem can
be alleviated, to a certain extent, by the selection of proper values in
the parameter vector in the Liapunov function. This is clearly shown
in the example considered in Chapter IV.

The results of Chapter III on vector Liapunov functions for
interconnected systems are the most general results of this type

published in that they encompass a larger class of components and

79

a larger class of interconnections. Further, the parameter dependence
increases the usefulness of the criteria.
A number of areas for future research related to the work here

seem rather promising. Some specific problems are:

0 The results here apply mainly to system of com-
ponents from Class E; . Perhaps similar results

could be obtained for other classes of components.

0 The problem of obtaining necessary conditions
for stability or instability results could be investi-

gated.

o The hypothesis of part (ii) of Theorem 2.2. 1 might
be weakened so that instead of a differentiability
condition on the system, one has a less restrictive

condition, such as a Lipschitz condition.

0 The component equation forms and interconnections
allowed cover a very broad class of components.
Perhaps some useful results could be obtained by
applying the concepts here to systems with very
special assumptions on the component models and

the interconnections.

0 An investigation to determine explicit information
as to the manner in which the parameter vector
affects the stability of the comparison systems in
Chapter 1H could prove quite valuable. Some
initial efforts have already been made, by the

author, on this problem.

 

80

0 Concepts developed here could be valuable in in-
vestigating conditional stability for interconnected

systems.

 

[AG-1]

[AI-l]

[AN-l]

[ANT-u

[BA-l]

[BE-H

[BEL-1]

[BEL-2]

[1312123]

[BRel]

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[WI-Z] Williams, H. F. , "Performance Characteristics of a System
as a Function of Structure, " Ph. D. Thesis, Electrical Engr.
Dept. , Michigan State University, 1965.

[YA-1] Yakovlev, M. K. , "A Method for Constructing Liapunov
Functions for Linear Systems with Variable Coefficients, "
Differential Equations, vol. 1, pp. 1169-1172, 1965.

 

[ZU-l] Zubov, V. I. , Methods of A.M. Liaflinov and Their Appli-
cation, P. Noordhoff, Ltd., Groningen, The Netherlands,
1964.

 

APPENDIX A

In the following several essential results used in this thesis

are proved or stated for reference.

Lemma A. 1 For the n-dimensional system of ordinary differential

 

equations

\11 = F0151) , \I/(to;\I/o,to) = 1110 (A.l)

F(0, t) E 0

let F(W, t) 6 @ in Hh '1" 7' Z O, and suppose that (A. 1) is exponentially
9

stable for all \II E H , h < h/p. Then for all \p 6 H
o h o o h
o o
-L(t-to) L(t-to)
i‘I’oi e 5. l)?(t;\I/o,to)l _<_ (\I/ole (A.2)

where L is the Lipschitz constant for F(\II, t) in Hh '1'
'

Proof: Since F(\II, t) E 63/ in H , a Lipschitz condition is satisfied

h, 7'
in this region, and since (A. 1) is exponentially stable, for \IIO E Hh ,
o
\II(t;\I/o,to) will remain in Hh 7' Thus with F(0, t) E 0, one has
(F011. t)! 5 LI?! (A. 3)

87

88

so that

-— (4T4) 2 rTFm. t)

dt
5 ZIWHFbIatH

_ 21.14)"- (A.4)

But (A.4) implies that

2L(t-to)

2
I e .

IA

(\1/

2
\I/ "If t
( (t. 0.0)( o

(A. 5)

\Il(to;\lfo,to) = \IIO

and hence

' L(t-to)
[xi/(t;\ro,to)| 5 (1110M

From (A.4) it follows that
d 2 2
a-t-(|\11|)Z -214qu

which similarly give 8

-L(t-to)
|x1r(t;xIro,t0)| 2. (\IIOIe

Lemma A. 2 Let F(\II, t) 6 ® in H
a W(t: ‘110. to)
a 1110.

1
the solution \Ir(t;\IIO, to) of (A. 1) with respect to the i-th component of

h 7" 7'2 0 for the system (A. l),

 

and denote by , tO _>_ 'r the vector of partial derivatives of

the vector \I/o at time t. Then

a ‘1’“; ‘I’O, to)
a 4’0.

1

L(t-to)
: e , i= l,2,...,n (A.6)

 

 

 

89

in Hh 7' where L is the Lipschitz constant for F(\II, t) in Hh 7"
9 3

Proof: Since F(\II, t) 6 @’ , the partial derivatives exist and are continuous

[CO-l]. Let W6 be a point in Hh such that

(so-r3] < (5 (A.7)

Let W(t) and \II'(t) be solutions of (A. l) passing through We and ‘11:) respectively

at t=to. Then they may be written as

t

W(t) = )Il(t ) + S F(\It(s),s)ds
O t
o
and
t
\Ifl(t) = \It'(to) + 5 F(i/‘(s), s)ds
to
so that

' t
r(t) “r(t) = 980) - Wt.) + 3" [F(ns). s) - F(9'(s).s)]ds

t
0

or
t

(r(t) - wwl 5 IMO) - )Il'(to)| + S (Fm/(s). s) - F()Ifl(s), ends
' t

o
Let
r(t) = (r(t) - \II'(t)|
so that
t
r(t) S. r(to) + L 5 r(s)ds (A. 8)
t

O

90

since F()II, t) 6 ©’ .

Defining
t
R(t) = S. r(s)ds
t

O

and using (A. 7), the relation (A. 8) becomes

R(t) - LR(t) _<_ (5 (A.9)
-L(t-to)
Multiplying both sides of (A. 9) by e and integrating the resulting

expression from to to t, gives

-L(t-t ) -L(t-t )
e °R(t)5%(1-e °)

and hence

L(t-to)

R(t) _<_ (e -1) (A. 10)

I.

Substituting (A. 10) into (A. 8) gives

 

 

 

L(t-to)
r(t) 5 6 e
Finally
5‘1, (t; ‘I’o’ to) . W(t) - \Ir'(t) L(tuto)
q} _<_ 11m 5 i
a o. 6* 0

1

Lemma A. 3 Let A(t) be an arbitrary real, continuous, bounded n x n

 

matrix. Then

0' xTx _<_ XTA(t)X _<_ O'nXTX (11.11)

1

 

91

where cyan are the minimum and maximum values assumed by any

of the eigenvalues of the symmetric part of the matrix A(t).

Proof: Let As(t) denote the symmetric part of A(t), i.e. ,

1 T
A8(t) = E(Au) +A(t) )

then

XTAX = XTAs(t)X

For each t E [to, 00) one has defined a constant matrix As for which it
t
is known that [ BE L-2]

XTA X
St
X1 = min -—T-—
t X X X
XTA X
8t
in = max “T—
t X X X

where 11 and kn are the minimum and maximum eigenvalues of A8 ,
t t 1:

respectively. Thus if

= 1
0'1 n: n Alt
and
on = max kn
t t
then one has

GIXTX f. XTA(t)X _<_ a'nXTX

 

92

Lemma A.4 For any two real numbers a and b

 

2
ab 5 %(a2 +b) (A.12)
Proof: (11.12) follows from (a-b)2 > 0

Lemma A. 5 (Cauchy-Schwartz Inequality) For any two real vectors

 

X and Y one has
2
(XTY) _<_ (XTX) (YTY) (14.13)

Lemma A. 6 For any real vector X with components xi, i = 1, 2, . . . , n,

 

one has

n 2 2
( 2 x.) < n z x. (A. 14)
. 1 — 1
1:1
Proof: Inequality (A. 14) follows from Lemma A. 5 by taking y1 = 1,

i: 1,2,ooopno

Lemma A. 7 For all real constants a, b and all I! 6 En,

 

2
alt/(2 + bl\I/( 5 (k+a) (\IIIZ + a: (A. 15)

ifk'>0.

Proof: If b = 0, then clearly (A. 15) is satisfied. Assume b ,£ 0 and

 

consider the quadratic inequality

.4412 +144) _<_ a(rlz +e
or

(d-a) |xxr|‘2 - b|\II] + e: 0 (A.16)

93

But (A. 16) is satisfied for all \II, if and only if,(d-a) > 0 and

b2 - 4(d-a)e 5 0

Let d = k+a for any k > 0. Then with

b2

4(d-a)

 

e:

(A. 16) is satisfied and the result (A. 15) follows.

(A. 17)

(A. 18)