Chapter II Systems Described by Ordinary Differential Equations

C H A P T E R I1

Systems Described Ly Ordinmy Dzfferentiul Eputions

In this chapter we consider the qualitative analysis of large scale systems described by ordinary differential equations. Our exposition will intentionally be detailed because (a) most of the existing results on large dynamical systems involve systems described by such equations, (b) ordinary differential equations play a crucial role in technology, and (c) many extensions and results not explicitly stated in the subsequent chapters (for systems described by other types of equations) will become apparent from results of the present chapter. The necessary notation is established in the first section while in the second section we present selected results from the Lyapunov theory which serve as essential background material. In the third section we introduce several classes of large scale systems which are analyzed in the fourth section by the use of scalar Lyapunov functions. This is followed by a presentation of several results from the theory of Minkowski matrices (M-matrices) which constitute background material used throughout this book. We then apply M-matrices in the fifth section to the qualitative analysis of large systems. We begin the sixth section with a summary of several standard comparison theorems which usually go under the heading of “comparison principle.” Next, we show how I2

2.1 NOTATION

13

this principle can be applied to vector Lyapunov functions in the analysis of large scale systems. In the seventh section we present a method of determining estimates of trajectory behavior and trajectory bounds. To illustrate how the results can be applied and to demonstrate the usefulness of the method of analysis advanced herein, we consider several specific examples in the eighth section. To point out relative advantages and disadvantages of the present procedure, examples from diverse areas are included which were treated by previous workers using methods that differ significantly from the present approach. In the last section a brief discussion of the literature cited is presented. 2.1

Notation

Let V and W be arbitrary sets. Then V u W, V n W , V - W , and V x W denote the union, intersection, difference, and Cartesian product of V and W , respectively. If V is a subset of W we write V c W and if x is an element of V we write x E V . Iff is a function or mapping of V into W we writefi V - + W and we identify the domain o f f and the range o f f by D ( f ) and Ra(f), respectively. Let @ denote the empty set, let R denote the real numbers, let R + = [0, a), and let J = [to,a)where to 2 0. We let R" be the Euclidean n-space and we let I . I represent the Euclidean norm. If x E R",then xT= (xl, .. ., x,,) denotes the transpose of x. If x, y E R", then x 5 y signifies xi< yi,x < y signifies xi < y,, and x > 0 signifies xi > 0 for all i = 1, ...,n. Let Y c R". Then Yand dY represent the closure and the boundary of Y , respectively. Also, B ( r ) = {x E R" : 1x1< r } and B(r) = {x E R" : 1x1 r } for some r > 0. Unless otherwise specified, matrices are usually assumed to be real. If A = [aii] is an arbitrary matrix, then A T denotes the transpose of A , A > 0 indicates that aii > 0, and A 2 0 signifies that a y 2 0 for all i , j . Now let A be a square matrix. If A is nonsingular, then A - ' denotes the inverse of A . An eigenvalue of A is identified as A(A) and ReA(A) denotes the'real part of I ( A ) . If all eigenvalues of A happen to be real we write I , ( A ) and I , " ( A ) to denote the largest and smallest eigenvalues of A , respectively. Matrix A is said to be stable if all its eigenvalues have negative real parts and unstable if at least one of its eigenvalues has positive real part. The determinant of an n x n matrix A is denoted by

detA

=

a,,

.'.

.

...

.

14

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

If A is a diagonal matrix we write A = diagca,, ...,a,,]. The identity matrix is denoted by I. Finally, the norm of an arbitrary matrix A , induced by the Euclidean norm, is given by

/ / ~ j j= m i n { c c E ~ +: c c ~ x ~ r ~X~E xR l" ,) = J L , ( A ~ A ) . 2.2

Lyapunov Stability and Related Results

In this section we present essential background material concerned with the stability analysis of dynamical systems described by ordinary differential equations. Since this material can be found in several standard texts dealing with the direct method of Lyapunov (also called the second method of Lyapunov) and related topics, we will not present proofs for any of the results presented in this section. We consider systems which can appropriately be described by ordinary differential equations of the form

P = g(x,t)

(1)

where x E R", t E J , P = dxldt, and g : B ( r ) x J + R" for some r > 0. Henceforth we assume that g is sufficiently smooth so that Eq. (I) possesses for every xo E B ( r ) and for every to E R + one and only one solution x ( t ; x o , t o ) for all t E J , where xo = x ( t o ;xo,to). We call xo an initial point, we refer to t as "time," and we call to initial time. Henceforth we also assume that Eq. (I) admits the trivial solution x = 0 so that g(0, t ) = 0 for all t E J . This solution is also called an equilibrium point or a singular point of (I). In addition, we also assume that x = 0 is an isolated equilibrium, i.e., there exists r' > 0 so that g(x', t ) = 0 for all t E J holds for no nonzero x' E B(r'). The preceding formulation pertains to local results. When discussing global results, we always assume that g : R" x J R" and that g is sufficiently smooth so that Eq. ( I ) possesses for every xo E R" and for every to E R+ a unique solution x ( t ; xo,t o ) for all t E J . In this case we also assume that x = 0 is the only equilibrium of Eq. (I). Since Eq. (I) can generally not be solved analytically in closed form, the qualitative properties of the equilibrium are of great practical interest. This motivates the following stability definitions in the sense of Lyapunov. --f

2.2.1. Definition. The equilibrium x = 0 of Eq. (I) is stable if for every 8 > 0 and any to E R + there exists a d ( 8 , t o ) > 0 such that / x ( t ;xo,r o ) /

whenever

/xol< d ( 8 , to).

< 8 for all t 2 to

2.2

15

LYAPUNOV STABILITY AND RELATED RESULTS

In the above definition, 6 depends on & and to. If 6 is independent of to, i.e., 6 = a(&), then the equilibrium x = 0 of Eq. (I) is said to be uniformly stable. 2.2.2. Definition. The equilibrium x = 0 of Eq. (I) is asymptotically stable if (i) it is stable, and (ii) there exists an ? ( t o )> 0 such that lim,+mx(t; x,, to) = 0 whenever IxoI < q . The set of all x, E R" such that condition (ii) of Definition 2.2.2 is satisfied is called the domain of attraction of the equilibrium x = 0 of Eq. (I). 2.2.3. Definition. The equilibrium x = 0 of Eq. (I) is uniformly asymptotically stable if (i) it is uniformly stable, and (ii) for every & > 0 and any to E R + there exists a 6, > 0, independent of to and &, and a T ( € )> 0, independent of to, such that Ix(t; xo, ?,)I < & for all r 2 to+ T(&)whenever IxoJ< 6,. Of special interest in applications is the following special case of uniform asymptotic stability. 2.2.4. Definition. The equilibrium x = 0 of Eq. (I) is exponentially stable if there exists an ci > 0, and for every & > 0 there exists a 6(&) > 0 such that whenever Ix, 1 < 6 (E). 2.2.5. Definition. The equilibrium x = 0 of Eq. (I) is unstable if it is not stable. (In this case there exists a sequence {x,,} of initial points and a sequence { t , } such that Ix(t,+ t,;xOn,t,)l> & for all n.) When g : R" x J + R" and Eq. (I) possesses unique solutions for all x, and every to E R + , the following global characterizations are of interest.

E

R"

2.2.6. Definition. A solution x(t;x,, to) of Eq. (I) is bounded if there exists a I x ( t ;x,, to)l < p for all t 2 to, where fi may depend on each solution.

p > 0 such that

2.2.7. Definition. The solutions of Eq. (I) are uniformly bounded if for any ci > 0 and to E R', there exists a p = P(ci) > 0 (independent of t o ) such that if Ixol < ci, then Ix(t;x,, to)(< p for all t 2 to. 2.2.8. Definition. The solutions of Eq. (I) are uniformly ultimately bounded (with bound B ) if there exists a B > 0 and if corresponding to any CI > 0 and to E R', there exists a T = T(a) > 0 (independent of to) such that lx,l to+T. 2.2.9. Definition. The equilibrium x = 0 of Eq. (I) is asymptotically stable in the large if it is stable and if every solution of Eq. (I) tends to zero as t + co. (In this case the domain of attraction of the equilibrium of Eq. (I) is all of R".)

16

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

2.2.10. Definition. The equilibrium x = 0 of Eq. (I) is uniformly asymptotically stable in the large if (i) it is uniformly stable, (ii) the solutions of Eq. (I) are uniformly bounded, and (iii) for any a > 0, any d > 0, and to E R + , there exists T(F,a) > 0, independent of to,such that if lxol< a then Ix(t;xo,to)l< & for all t 2 to+ T(b,a). 2.2.11. Definition. The equilibrium x = 0 of Eq. (I) is exponentially stable in the large if there exists ci > 0 and for any p > 0, there exists k(') > 0 such that lx(t;x,, to)l I k ( a ) Ixo(e-"('-'O) for all t 2 to

whenever

IxoJ< p.

Results which yield conditions for stability, instability and boundedness in the sense of the above definitions involve the existence of functions u : D + R, where in the case of local results D = B(r) x J for some r > 0, while in the case of global results D = R" x J . Henceforth we always assume that such v-functions are continuous on their respective domains of deJinition and that they satisfy locally a Lipschitz condition with respect to x. Also, unless otherwise stated, we assume henceforth that u(0, t ) = 0 for all t E J . The upper right-hand derivative of v with respect t o t along solutions of Eq. (I) is given by Dv(,)(x,t)= lim s u p ( l / h ) { u [ x ( t + h ; x , t )t+h] , - u(x,t)} h+O+

=

lim s u p ( l / h ) { u [ x + h . g ( x , t ) ,t+h] - u ( x , t ) } .

(2.2.12)

h+0+

If u is continuously differentiable with respect to all of its arguments, then the total derivative of u with respect t o t along solutions of Eq. (I) is given by DU(,)(X,t ) = V u ( x , t)'g(x, t )

+ h ( x ,tyat,

(2.2.13)

where V v ( x ,t ) denotes the gradient vector of the scalar function v and &/at represents the partial derivative of u with respect to t. Whether u is continuous or continuously differentiable will either be clear from context or it will be specified. In the former case Du(,, is specified by Eq. (2.2.12) while in the latter case Do(,) is given by Eq. (2.2.13). We now characterize several properties of u-functions in terms of special types of comparison functions. 2.2.14. Definition. A continuous function cp: [0, r l ] -+ R' (or a continuous function c p : [0, co) 4 R') is said to belong t o class K , i.e., cp E K , if cp(0) = 0 and if cp is strictly increasing on [O,r,] (or on [0, m)). If cp: R + -+ R', if cp E K , and if lim,+m cp(r) = co,then cp is said t o belong t o class KR. 2.2.15. Definition. Two functions cpl, cp2 E K defined on [0, r l ] (or on [0,a)) are said to be of the same order of magnitude if there exist positive constants

2.2

LYAPUNOV STABILITY AND RELATED RESULTS

17

k , and k , such that k , rpl(r) 2 rp2(r) I k,rp,(r) for all r E [O,r,] (or for all r E co, a>>. 2.2.16. Definition. A function u is said to be positive definite if there exists rp E K such that u(x, t ) 2 (~(1x1)for all t E J and for all x E B ( r ) for some r > 0. (Recall the assumption u(0, t ) = 0 for all t E J.) 2.2.17. Definition. A function v is said to be negative definite if - u is positive definite. 2.2.18. Definition. A function u, defined on R ” x J , is said to be radially unbounded if there exists cp E K R such that u(x, t ) 2 cp(lx1) for all x E R” and t E J . (Recall the assumption u(0, t ) = 0 for all t E J . ) 2.2.19. Definition. A function u is said to be decrescent if there exists cp E K such that Iu(x, t)l I cp(lxl) for all t E J and for all x E B ( r ) for some r > 0. In this case u is aiso said “to admit an infinitely small upper bound” or “to become uniformly small.” 2.2.20. Definition. A function u is said to be positive (negative) semidefinite if u(x, t ) 2 0 ( u ( x , t ) 5 0) for all t E J and x E B ( r ) for some r > 0. (Recall the assumption v(0, t ) = 0 for a11 t E J . ) For alternate equivalent definitions of the above concepts (positive definite, negative definite, etc.) the reader is referred to several standard texts dealing with the Lyapunov theory cited in Section 2.9. We are now in a position to summarize several well-known stability and instability results. In the first four of these, which are local results, we assume that u is defined and continuous on B ( r ) x J for some r > 0. 2.2.21. Theorem. If there exists a positive definite function u with a negative semidefinite derivative Do(,,, then the equilibrium x = 0 of Eq. (1) is stable. 2.2.22. Theorem. If there exists a positive definite, decrescent function u with a negative semidefinite derivative Du(,,, then the equilibrium x = 0 of Eq. (I) is uniformly stable. 2.2.23. Theorem. If there exists a positive definite, decrescent function u with a negative definite derivative Du(,,, then the equilibrium x = 0 of Eq. (I) is uniformly asymptotically stable. In this case there exist cpl, cpz, (p3 E K such that cpt(lxl>

for all x E B ( r ) and t

u(x,t> I cpz(lxl), E

Du(l,(x,t)5

-cp3(IxI)

J.

2.2.24, Theorem, If in Theorem 2.2.23 cpl, cpz, cp3 E K are of the same order of magnitude, then the equilibrium x = 0 of Eq. (1) is exponentially stable.

18

11 SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Theorem 2.2.24 is true if in particular there exist three constants c1 > 0,

c2 > 0, and c3 > 0 such that c1

lxI2 I u ( x , t ) I c2lxl

2

,

DU(,)(X,t) 5 -c31x12

for all x E B ( r ) and t E J. In the next three theorems, which are of great practical importance, we assume that u is defined and continuous over R" x J and that Eq. (I) possesses unique solutions for all xo E R" and all to E R'. 2.2.25. Theorem. If there exists a positive definite, decrescent, and radially unbounded function u with a negative definite derivative D U ( ~ then ), the equilibrium x = 0 of Eq. (I) is uniformly asymptotically stable in the large. In this case there exist 'pl, q 2 E K R and q3 E K such that CPi(IXI)

5 u(x,t) 5

q2(IXI),

D ~ c I5) -4O3(IXI)

for all x E R" and for all f E J . 2.2.26. Theorem. If in Theorem 2.2.25 q 1 , q 2 , q E3 K P and if cp1,q2,q3 are of the same order of magnitude, then the equilibrium x = 0 of Eq. (I) is exponentially stable in the large. Theorem 2.2.26 is true if in particular there exist three constants c1 > 0, c2 > 0, and c3 > 0 such that C]

for all x

E

lx12 I u(x,t) I c21xI2,

DU(,)(X,t)I -c3lxI2

R" and t E 1.

2.2.27. Theorem. If there exists a function u defined on 1x1 2 R (where R may be large) and 0 I t < m, and if there exist J/,,I,!J2 E K R such that u ( x , t ) I I,!Jz(lxl), (i) $ 1 (Ixl) I (ii) D q , , ( x ,t ) I 0, for all 1x12 R and 0 I t < co, then the solutions of Eq. (I) are uniformly bounded. If i n addition there exists q3 E K (defined on Rf)and if (ii) is replaced by (iii) Du(,,(x,t) I -J/3(IxI) then the solutions of Eq. ( I ) are uniformly ultimately bounded. The next theorem, which yields conditions for instability, is a local result. We assume once more that for some r > 0, u is defined and continuous on B(r) x J . 2.2.28. Theorem. Assume there exists a function u having the following properties. (i) For every W > 0 and for every t 2 to there exist points x' such that u ( x ' , t ) < 0 and such that Ix'/ < 8.The set of all points (x,t ) such that 1x1 < r

2.2

LYAPUNOV STABILITY AND RELATED RESULTS

19

and v(x, t ) < 0 will be called the "domain v < 0." It is bounded by the hypersurfaces 1x1 = r and u = 0 and may consist of several component domains. (ii) In at least one of the component domains D of the domain u < 0, u is bounded from below and 0 E BD. (iii) In the domain D,Du(,)I - cp (v) where q E K . Then the equilibrium x = 0 of Eq. (I) is unstable. If in particular there exists a positive definite function u (a negative definite function v) such that Du(,,(x,t ) is positive definite (negative definite), then the equilibrium x = 0 of Eq. (I) is unstable. In fact, in this case the equilibrium is said to be completely unstable. Equation (I) is called a nonautonomous ordinary differential equation. Many systems of practical interest can appropriately be described by autonomous ordinary differential equations given by f = g(x)

(1')

where x E R", f = dxldt, and g: B ( r ) -+ R" for some r > 0, or in the case of a global setting, g: R"-+ R". For the unique solutions of Eq. (1') we have x ( t ; xo, to) = x ( t - to;xo,0) which allows us to assume to = 0 without loss of generality. Furthermore, since Eq. (1') is a special case of Eq. (I), all preceding statements made for (I) hold equally as well for (I'), with obvious modifications. In particular, since Eq. (1') is invariant under translations of time, it makes sense to consider only uniform stability, uniform asymptotic stability, uniform asymptotic stability in the large, and so forth. Conditions for stability, instability, boundedness, and the like, involve in this case the existence of functions v : B ( r ) -+ R for some r > 0 or u : R" -+ R . Such functions are characterized as being positive definite, negative definite, or radially unbounded as was done before using functions of class K and class K R and deleting all reference to t E J . For Eq. (If), the preceding theorems are modified by replacing u ( x , t ) by u(x). In addition, in Theorems 2.2.22, 2.2.23, and 2.2.25, all references to the word "decrescent" are deleted. For system (1') there is a significant extension for asymptotic stability, which we want to consider. First we require the following concept, given here in a global setting. 2.2.29. Definition. A set I-' of points in R" is invariant with respect to Eq. (1') if every solution of Eq. (1') starting in remains in I' for all time, i.e., if xo E r then x ( t ; x , , O ) E r for all t E R .

This concept has been used to prove the results summarized in the next theorem. 2.2.30. Theorem. If there exists a continuously differentiable function u : R" -+ R (recall the assumption u(0) = 0) and $ E K R such that

20

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

(i) u(x) 2 $(lxl) for all x E R", (ii) Du(,.,(x)I 0 for all x E R", (iii) the origin x = 0 is the only invariant subset of the set E

= {X E

R" : Du(,.,(x)= 0},

then the equilibrium x = 0 is asymptotically stable in the large. If hypothesis (iii) is deleted then all solutions x ( t ; x,,O) of Eq. (1') are bounded for t 2 to. The preceding results are phrased in terms of the Euclidean norm 1.1. They are also true with respect to a n y other equivalent norm defined on R". In this case, convergence needs to be interpreted relative to the particular norm that is used. Henceforth we refer to results such as those given in this section as Lyapunov-type theorems and we call a function u satisfying any theorem of this type a Lyapunov function. The Lyapunov theorems are very powerful. However, in general, great difficulties arise in applying these results to high-dimensional systems with complicated structure. The reason for this lies in the fact that there is no universal and systematic procedure available which tells us how to find the required Lyapunov functions. Although converse Lyapunov theorems have been established, these results provide no clue (except in the case of linear equations) for the construction of Lyapunov functions. For this reason we will pursue an approach which allows us to analyze the stability of highdimensional systems with intricate structure in terms of simpler system components, which we shall call subsystems and interconnecting structure. This viewpoint makes it often possible to circumvent many of the difficulties associated with the Lyapunov method. First we need to consider the description of large scale systems, also called interconnected systems or composite systems. This is the topic of the next section. 2.3

Large Scale Systems

Since it simplifies matters considerably, our exposition in the present section is in a global setting. A development in a local setting involves obvious modifications. To fix some of the subsequent ideas, we begin with a specific example. In particular, we consider systems described by the set of equations ii = AiZi

y. =

+ DiU,

H.Z. 1 1

(2.3. I ) (2.3.2)

2.3

LARGE SCALE SYSTEMS

21

where zi E Rni,ii = dzi/dt,Ai is an n,x ni matrix, Di is an ni x mimatrix, ui E R"1, Hi is a p i x ni matrix, and y i E RPL.Equations (2.3.1) and (2.3.2) describe the input-output characteristics of a linear time-invariant system described by ordinary differential equations. We call this system the ith transfer system, where ui is interpreted as the input and y i as the output. Associated with Eq. (2.3.1) is the system described by the linear equation 2.

=

A I. z .

17

(2.3.3)

which we call the ith isolated subsystem or the ith free subsystem. Next, let us consider I transfer systems (i.e., i = 1 , ...,1) and let us interconnect these by means of the equations 1

ui =

C BYyj + G i u ,

j= 1

(2.3.4)

i = 1, ...,I, to form a composite system or interconnected system. Here B, is an mi x p j matrix, Giis an m i x q matrix and u E Rq. In this case B, represents a linear, time-invariant connection from the output of the j t h transfer system to the input of the ith transfer system. Frequently we may assume Bii to be zero, since feedback around any transfer system can often (for purposes of analysis) be combined with the matrix A i . Combining Eqs. (2.3.1), (2.3.2), and (2.3.4), we obtain ii = A i z i +

I

1C , z j + K i u ,

j= 1

(2.3.5)

i = I , ...,I, where C , = Di B, H j and Ki = Di Gi are n, x n and ni x q matrices, respectively. Letting ni = n, xT = ( z I T..., , zrT)E R",

cf=

we can rewrite Eq. (2.3.5) as f = AX

+ CX + Ku.

(2.3.6)

Finally, let the output of this interconnected system be represented by y = HX

(2.3.7)

where y E RP and H is a p x n matrix. Equations (2.3.6) and (2.3.7) describe the input-output characteristics of the composite system considered, where u denotes the input and y the output. This system may be viewed as a linear interconnection of I isolated subsystems

22

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

described by Eq. (2.3.3) with interconnecting structure specified by Eqs. (2.3.1), (2.3.2), and (2.3.4) and output characterized by Eq. (2.3.7). Systems described by equations of the form (2.3.6) and (2.3.7) are examples of large scale systems. For obvious reasons, the terms interconnected system and composite system are more descriptive. In control theory, such systems are sometimes also called multi-loop feedback systems, decentralized systems, and the like. In the Lyapunov stability analysis of the equilibrium x = 0 of the above system, the output equation (2.3.7) is of no concern whatsoever. Furthermore, in such an analysis it is the free or unforced dynamical system that is of interest. Thus, when investigating the Lyapunov stability of the above system, we consider the set of differential equations ii = Aizi+

I

C C,zj,

j= 1

(2.3.8)

i = 1, . ..,I, or equivalently, the differential equation i= AX

+ CX A Fx,

(2.3.9)

which is a special case of Eq. (1) of Section 2.2. On the other hand, when studying the qualitative input-output properties of the above system, Eqs. (2.3.6) and (2.3.7) are of interest. We will devote Chapters VI and VII in their entirety to such investigations. Meanwhile, we primarily concern ourselves with Lyapunov stability and related concepts for large scale systems. Many systems of practical interest (e.g., power systems, aerospace systems, circuits, economic systems, etc.) are appropriately described by nonlinear time varying ordinary differential equations and may often be viewed as interconnected or composite systems. In the following we consider several classes of ordinary differential equations which may be used to characterize such systems. We consider systems described by equations of the form 5, = f i ( z i , t )+gi(zl,..',z*,f),

(xi)

(2.3.10)

i = l , ..., I, w h e r ez iERni ,t E J , f i : R n i x J + R n i ,a n d g , : R " ' x . . . x R " ' x J - , R"'. Henceforth we assume thatfi(zi, t ) = 0 for all t E J if and only if z, = 0. Letting Cf= n j = n,

xT = (zlT,...,)z:

f(x,t IT

=

g(X>t)T=

E R"

CfI(Zl? t IT, ...,f;(z1, tY1 CS1(Z1,...,Z~,t)T,...,9*(Z1,...,ZI,t)T1

and gi(z,,

...,zl, t )

g i ( x ,t ) ,

i = 1, ..., I,

2.3

LARGE SCALE SYSTEMS

23

so that g ( x , t)T = [g,(x, t ) T ,...,gl(x,t)T],we can represent Eq. (2.3.10) equivalently as (9) f = f ( x , t ) + g ( x , t ) Li h ( x , t ) . (2.3.1 1) Clearly, f : R" x J - + R",g: R" x J + R", and h : R" x J - + R". We always assume that h ( x , t ) = 0 for all t E J if and only if x = 0. A system described by Eq. (2.3.1 1) may be viewed as a nonlinear and time varying interconnection of I systems represented by equations of the form

(m

(2.3.12)

ii = f i ( Z i , t ) .

Henceforth we assume that for every to E R + and every xo E R",Eq. (2.3.11) possesses a unique solution x ( t ; xo, to) for t 2 to with xo = x ( t o ; xo,to). Also, we assume that for every to E R + and every zio E Rni,Eq. (2.3.12) has a unique solution z,(t;zio,to) for t 2 to with zi(to;zio,to) = zio. Subsequently we refer to Eq. (2.3.11) as composite system ( Y ) ,or interconnected system (Y), or large scale system (9')with decomposition (&) (described by Eq. (2.3.10)). We refer to Eq. (2.3.12) as the ith isolated subsystem (q)or as the ith free subsystem (q.) or as the ith unforced subsystem (8). We call x in Eq. (2.3.11) a hyper vector. Finally note that Eqs. (2.3.1 1) and (2.3.12) are of the same form as Eq. (I) and as such, all results of Section 2.2 are applicable to composite system (9) and to isolated subsystem (q). In specific applications, more information concerning system structure is usually available than indicated in Eq. (2.3.10). The following two classes of systems are examples of special cases of Eq. (2.3.10) encountered in practice. Let (2.3.13) where C, is a constant n i x n j matrix. Then ( X i ) assumes the form

ii = f i ( z i , t )

+ 2

j=l.i#j

cqzj,

(2.3.14)

i = 1, ..., I. This equation represents a system consisting of 1 isolated subsystems ($) which are linearly interconnected. Its structural properties are depicted in the block diagram of Fig. 2.1. Next let (2.3.15) where gl:R"j x J - + Rni.Then ( C i ) assumes the form (2.3.16)

i = 1, ..., 1.

24

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Figure 2.1 Interconnected system (2.3.14) where i = 1 , ..., 1.

Equations (2.3. lo), (2.3.1 I ) , and (2.3.12) characterize a class of composite systems which can appropriately be described by nonautonomous ordinary differential equations. Deleting t E J in these equations, we also consider time-invariant interconnected systems described by the corresponding autonomous ordinary differential equations given by

(xi')

ii = ,fi(zi)+ gi(zl, . .., z,),

(Y')

.i =f(x)

(XI)

i i

+ g(x) A h(x),

i

=

1 , ..., I,

(2.3.17) (2.3.18) (2.3.19)

= fi(zi>,

respectively. Note that in all cases considered thus far, the interconnecting structure enters additatively into the system description. As such, composite system (9) with decomposition ( X i ) is actually a special case of systems described by equations of the form (C:C)

ii = .fi(zi,gi(zl, ..., z I , t ) , t ) = f i ( z i ,gi(x,t ) , t )

A

h i ( x , t),

(2.3.20)

i = I , ..., I , where zi E R"', t E J , I:.=, n j = n, xT= (zIT, ..., z,') E R", g i :R" x J + R r i , f , : Rntx Rrcx J + R"', and hi:R" x J + R"'. Letting hT(x,t ) = [ h , ( x , t ) T ,... , h l ( x ,r)T], Eq. (2.3.20) can equivalently be represented by

(9")

i= h ( x , r ) .

(2.3.21)

2.3

LARGE SCALE SYSTEMS

25

We assume that Eq. (2.3.21) possesses only one equilibrium, x = 0, and that for every to E R+ and every xo E R”, Eq. (2.3.21) has a unique solution x ( t ; x o ,1 , ) for all t 2 to. In the present case we speak of composite system (9”) with decomposition (C;) and leave the notion of isolated subsystem undefined. In the present chapter, as well as in subsequent ones, we advance a method of qualitative analysis at different hierarchical levels, involving the following general steps. Step 1. A large scale system (9) (see Eq. (2.3.11)) is decomposed into I isolated subsystems (q) (see Eq. (2.3.12)) which when interconnected in an appropriate fashion (see Eq. (2.3.10)) yield the original composite or interconnected system. Step 2. The qualitative properties of the lower order (and hopefully simpler) free subsystems (3)are characterized in terms of Lyapunov functions u i ,using standard and well-established techniques involving the results of Section 2.2.

Step 3. Qualitative properties of the overall system (9) are deduced from the qualitative properties of the interconnecting structure and the individual free subsystems.

For composite system (9”) with decomposition (C;), the above procedure has to be modified somewhat, for in this case free subsystems (3) are not defined. Nevertheless, as will be demonstrated, even i n this case a method of analysis at different hierarchical levels is still possible. The process of decomposing a large system (Step 1) into an appropriate form is by no means a trivial task. We will not address ourselves explicitly to the problem of “tearing,” pioneered by Kron [I]. However, we would like to point to two general approaches. In the first of these, the structural properties of the process being modeled usually dictate a natural decomposition. In the second approach, the decomposition is usually influenced by mathematical convenience to overcome technical difficulties. In connection with Step 2 we note that for isolated subsystems of sufficiently low order and simplicity, many well-known results from the Lyapunov theory are available. Note also that in this step the converse Lyapunov theorems can play a crucial role. Thus, if the stability properties of the free subsystems are known a priori, we are usually in a position to search for Lyapunov functions with certain general properties. We will use two methods of implementing Step 3. In the first approach, which is developed in Sections 2.4 and 2.5, scalar Lyapunov functions (consisting of weighted sums of Lyapunov functions for the isolated subsystems) are constructed and applied to the results of Section 2.2. In the second approach, vector Lyapunov functions (i.e., I-vectors whose components are

26

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Lyapunov functions) are employed and stability is deduced from an Zth-order differential inequality involving these vector Lyapunov functions by invoking an appropriate comparison principle. The method outlined above frequently enables us to circumvent difficulties which arise when the Lyapunov approach is applied to high-dimensional systems with complex structure. 2.4 Analysis by Scalar Lyapunov Functions

I n this section, which consists of five parts, we develop qualitative results for the systems considered in the previous section, making use of scalar Lyapunov functions. First we concern ourselves with uniform stability and uniform asymptotic stability. Next, we consider exponential stability. This is followed by instability and complete instability results. Uniform boundedness and uniform ultimate boundedness are treated in the fourth part. The section is concluded with a discussion of the results. A. Uniform Stability and Uniform Asymptotic Stability

In characterizing the qualitative properties of the free subsystem (Yi), we will find it convenient to use the following convention. 2.4.1. Definition. Isolated subsystem (3.) possesses Property A if there exists a continuously differentiable function u i : R”, x J + R,functions $ i , , $i2 E K R , $i3 E K , and a constant oi E R such that the inequalities iil(Izi1)

5

ci(zi, t >

5

$i2(lzil)>

hold for all zi E Rniand for all t E J .

DUi(yi)(Zi,t)

ci$i3(/zil)

(x)

Clearly, if oi < 0, the equilibrium zi = 0 of is uniformly asymptotically stable in the large. If oi = 0, the equilibrium is uniformly stable. If oi> 0, the equilibrium of (Sq) may be unstable.

2.4.2. Theorem. The equilibrium x = 0 of composite system (9) with decomposition ( C i ) is uniformly asymptotically stable in the large if the following conditions are satisfied. possesses Property A ; (i) Each isolated subsystem (ii) given ui and $ i 3 of hypothesis (i), there exist constants aUE R such that

(z)

I

Vui(zi, tITgi(ZI,. . . , z [ , t > 5

C

C$i3(I~il>I”~

for all ziE R“<,i = 1 , ..., I, and t E J ; and

j =1

~ijC~+j3(Izjl)I”~

2.4

27

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

(iii) given oi of hypothesis (i) there exists an h e c t o r aT = (a1,..., al) > 0 such that the test matrix S = [s,] specified by

is negative definite. Proof. For composite system (9) we choose a Lyapunov function

c CliVi(Zi,t) 1

u(x,t) =

(2.4.3)

i= 1

where the functions vi are given in hypothesis (i) and where ai > 0, i = 1, . .., I are constants given in hypothesis (iii). Clearly, v ( x , t ) is continuously differentiable and u(0, t ) = 0 for all t E J , since each vi(zi, t ) satisfies these conditions. Since each isolated subsystem (q)possesses Property A, it follows that

for all x E R",t E J, and zi E R"', i = 1, . ..,1. Since by assumption $il, t+hi2 E KR, it follows that u(x, t ) is positive definite, decrescent, and radially unbounded. Indeed, there exist $1, 11/2 E KR such that

so that

C ai$il(IziI)

i= 1

c mi+iz(JziJ), 1

1

$l(lXl) 5

and

~lrz(ixI)2

i= 1

28

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

We now have Dv(,,(x,t) I wTRw

=

wT((R+RT)/2)w = wTSw,

where S = [sJ is the test matrix given in hypothesis (iii). Since S i s symmetric, all its eigenvalues are real. Also since by hypothesis (iii), S is negative definite, all its eigenvalues are negative so that A,(S) < 0. We thus have

Therefore, Dv(,)(x, t ) is negative definite for all x E R" and t E J. In fact, there exists a function tj3 E K such that ~ , h ~ ( l x5l ) so that

x:=l

Dv(,)(x,t)

' M ( S ) $ ~ ( I X I ) ~ AM(^> < 0,

(2.4.8)

for all x E R" and t E J. Inequalities (2.4.4) and (2.4.8) show that all hypotheses of Theorem 2.2.25 are satisfied. Therefore, the equilibrium x = 0 of composite system (9) is uniformly asymptotically stable in the large. Before proceeding further, we note that the test matrix S above theorem is negative definite if and only if

(-

1)k

.

...

.

Sk'

...

'kk

>0,

k = 1 , 2 ,..., 1.

=

[sJ of the

(2.4.9)

2.4

29

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

We can generalize Therorem 2.4.2 somewhat.

2.4.11. Theorem. The equilibrium x = 0 of composite system (9) with decomposition (Ci) is uniformly asymptotically stable in the large if the following conditions hold. (i) Each isolated subsystem possesses Property A ; (ii) given ui and t+bi3 of hypothesis (i), there exist continuous functions a y :R” x J - t R such that

(z)

I

Vui(Zi,t)Tgi(Z1,

C alj.(x,t)Ct,bj3(I~jI)I”’

. . . , ~ i , t )I C $ i 3 ( I ~ i l ) I ~ / ~

j= 1

for all zi E R”’,i = 1 , ...,I , x E R”, and t E J ; (iii) there exists an I-vector aT = ( a l , ...,aI) > 0 and d > 0, such that for all x E R“ and t E J , the test matrix S(x, [ ) + & I is negative definite, where I denotes the I x I identity matrix and S(x, t ) = [slj.(x,t)] is defined by

Proof. Choose the positive definite, decrescent, and radially unbounded Lyapunov function given in Eq. (2.4.3). Let R ( x , t ) = [ r l j . ( x ,t ) ] be the I x I matrix specified by . . YG(X,t

)

=/

ai[ai+aii(X,t)],

I

aiaij(x, f ) ,

i # j

=

and let w be defined as in Eq. (2.4.6). Using Eq. (2.4.5) and hypotheses (i), (ii), and (iii), we obtain Dv(,,(x, t ) 2 wTR(x,t ) w

=

w T [ ( R ( x ,t ) + R ( x , t ) T ) / 2 ] ~

= WTS(X,t)W

I -&WTW

= -8

c I

Iji3(IZil).

i= 1

Therefore, D v ( , ) ( x , t ) is negative definite for all x E R” and t E J . Indeed there exists a function $3 E K such that $3(IxI) i t,bi3(Izil),so that

xi=,

D v ( , , ( x , t ) 5 -8$3(IxI). Thus, the hypotheses of Theorem 2.2.25 are satisfied and the equilibrium of system (9) is uniformly asymptotically stable in the large. W Deleting all references to t E J , Theorems 2.4.2, 2.4.1 1, and Corollary with de2.4.10 are also applicable to autonomous composite system (9’) composition (Xi’).For system (9’) an extension involving invariant sets is possible.

30

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

2.4.12. Definition. Isolated subsystem (8’) possesses Property A‘ if there exists a continuously differentiable function vi: R”’ -+ R, t+hil E KR, $i3 E K , and a constant rsi E R such that $il(Izil)

2 ui(zi),

vi(0) = 0,

DUi(yi*)(Zi)

~i$i3(I~il)

for all zi E R”‘. If oi< 0, the equilibrium zi = 0 of system (8’) is asymptotically stable in the large and if oi> 0, the equilibrium may be unstable. If oi = 0, all solutions of (q.‘) are bounded for all t 2 to = 0, and also the equilibrium of (8’) is stable. 2.4.13. Theorem. Assume that for composite system (9’) with decomposition (Xi’) the following conditions hold. (i) Each isolated subsystem (Yi’)possesses Property A ’ ; (ii) given ui and $ i 3 of hypothesis (i), there exist constants aii E R such that Vvi ( Z J T g i (

. .., ZJ 5

~ 1 3

I

(Izil)I

C$i3

”’ C

j= 1

C$ j 3 (Izj1)I ’”

for all Z ~ Rni, E i = 1, ..., 1; (iii) given rsi of hypothesis (i), there exists an I-vector aT = (a1, ..., al) > 0 such that the matrix S = [se] specified by . . s.. =

(ai (ci

+4,

j(“iaq+ajaji),2,

1 =J

i#j

is negative semidefinite. Then the following statements are true. (a) All solutions of (9’) are bounded; (b) the equilibrium x = 0 of (9”) is stable; and (c) if the origin x = 0 is the only invariant subset of the set E = {x E R” : Du(,.,(x) = 0}, where u ( x ) = aivi(zi), then the equilibrium x = 0 of composite system (9”) is asymptotically stable in the large.

xi=,

Proof. For composite system (9”)we choose the Lyapunov function

v(x) =

c I

UiUi(Zi)

i= 1

where the functions vi are given in hypothesis (i) and where mi > 0, i = 1, ..., I are constants given in hypothesis (iii). Clearly v ( x ) is continuously differentiable, v ( 0 ) = 0, and u ( x ) is positive definite and radially unbounded. Thus, there exists E K R such that 2

*l(lXl>

(2.4.14)

2.4

31

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

for all x E R”. Modifying the proof of Theorem 2.4.2 in an obvious way, we see that 1

Ct + i s ( l Z i l )

Dv(,,)(x) I J+,(s) i=

for all x E R”. Since S is by assumption negative semidefinite, we have J+,(S) I 0, so that DV(,,)(X) I 0 (2.4.15) for all x E R”. Clearly, the equilibrium x = 0 of system (9’) is stable. Furthermore, it follows from Theorem 2.2.30 that all solutions of (9’) are bounded for t 2 to = 0. If in addition the origin x = 0 is the only invariant subset of the set E = {x E R” : Du(,,,(x) = 0}, then it follows from Theorem 2.2.30 that the equilibrium of (9’) is asymptotically stable in the large. W Next we consider composite system (9”’) with decomposition (Cy). 2.4.16. Theorem. Assume that for composite system (9”)with decomposition (C;) the following conditions hold. (i) There exist continuously differentiable functions u i : R”‘ x J + R and + i l , J/iz E K R , i = 1, ..., I, such that

+i1(lzil) 5 ui(zi,t) I +iz(Izil) for all zi E R“( and t E J ; (ii) given vi in hypothesis (i), there exist constants ug E R and i, j = 1, ..., I, such that V~i(zi,tIThi(-~, t ) + aui(zi, t > / a t I +i4(Izil)

Ic/i4

E

K,

1

C1 a”+j4(I~jI)

j=

for all zi E R”’, x E R”, t E J , and i = 1 , ..., I ; and ...,a,) > 0 such that the test matrix (iii) there exists an I-vector aT = (al, S = [sg] specified by ai aii > i=j s.. = ((aiui+ajuji)/2, i#j is either negative semidefinite or negative definite. (a) If S is negative semidefinite, the equilibrium x = 0 of composite system (9”’) is uniformly stable. (b) If S is negative definite, the equilibrium of (9”’)is uniformly asymptotically stable in the large. Proof. Given the functions vi of hypothesis (i), we choose for composite system (9”’) the Lyapunov function 1

v(x,t )

=

C CliVi(Zi, i= 1

t)

(2.4.17)

32

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

where aT = (aI, ..., a,) > 0 is given in hypothesis (iii). Clearly u ( x , t ) is continuously differentiable and 1

1

i= 1

1

Mi$il(lzil)

5 v(x,t>5

1

i= 1

ai$iz(lZil>

for all zi E R"', i = 1 , . . .,I, x E R",and t E J . Since by assumption $il, E KR, it follows that v ( x , t ) is positive definite, decrescent, and radially unbounded, and there exist $z E K R such that 1

$1

(1x1) 5

Hence,

C Mi $i1

i= I

and

(Izil)

$l(lXl>

5 u(x, t ) 5

I

$Z(IxI) 2

i= 1

ai $iz(IziI). (2.4.18)

$z(lXl>

for all x E R" and t E J . Along solutions of (9") we have I

DV,.,,,(X,

t) =

1

i= I

cli

(vui(zi,t)Thi(~, t)+(dvi(zi, t ) / d t ) )

= W T R W = W T ( ( R + R T ) / 2 ) W = WTSW

I

c $i4(1zil)2, 1

i= 1

where wT = [$14(Izll), ..., $,4(\zJ)], R = [ r V ] [aia,], and S is given in hypothesis (iii). Therefore, if S is negative semidefinite, then so is Du(,.,,(x, t ) and the equilibrium x = 0 of composite system (9") is uniformly stable. Similarly, if S is negative definite then D u ( , . , ) ( x , t ) is negative definite for all x E R" and t E J and the equilibrium of (9") is uniformly asymptotically stable in the large. w

B. Exponential Stability In studying the exponential stability of composite system (9) we will find

it useful to employ the following convention.

2.4.19. Definition. Isolated subsystem (q)possesses Property B if there exists a continuously differentiable function ui: Rnix J + R, $ i l , Icli2, $i3 E K R u.hich are of the same order of magnitude, and a constant oi E R , such that 5 ui(zi,t) $i2(Izil)> D u i ( y i ) ( Z i r t ) 5 oi$i3(IziO for all zi E R". and t E J . Isolated subsystem (Sq.) possesses Property B' if it possesses Property B with $ i l (IZiI) = C ~ Izi12, I $iz(lzil) = ci2 IziI2, and $i3(Izil) = JziI2, where ci2 2 c i I > 0 are constants. $il(lzil>

2.4

33

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

If isolated subsystem (8) possesses either Property B or Property B', and if oi < 0, then the equilibrium zi = 0 is exponentially stable in the large. If cri = 0, the equilibrium of (q) is uniformly stable and if oi > 0, the equilibrium of may be unstable.

(x)

2.4.20. Theorem. The equilibrium x = 0 of composite system (9') with decomposition (Xi) is exponentially stable in the large if the following conditions are satisfied. possesses Property B ; (i) Each isolated subsystem (q) all comparison functions t,hG, i = 1, . .., I, j = 1,2,3 of hypothesis (i) (ii) are of the same order of magnitude; (iii) given vi and ~ i of3 hypothesis (i), there exist constants ag E R such that

for all zi E R"', zj E R"j, i, j = 1, ..., I, and t E J ; and (iv) given oi in hypothesis (i), there exists an I-vector such that the matrix S = [sG] specified by (ai
is negative definite.

j(ctiag+ajaji),2,

ctT =

(ul, ..., a,)

>0

i #j

Proof. First we note that if the above hypotheses are satisfied, then all hypotheses of Theorem 2.4.2 are also satisfied. It follows that the equilibrium x = 0 of composite system (9') is uniformly asymptotically stable in the large. As in the proof of Theorem 2.4.2, we choose a Lyapunov function I

v(x,t) =

and conclude that

1 a,u,(z,,t)

i= 1

and

for all x E R", t E J , and zi E R"', i = I , ..., I, where , l M ( S )< 0 is the largest eigenvalue of test matrix S. $ 2 , $3 E K R which To complete the proof we first show that there are are of the same order of magnitude (see Definition 2.2.15) such that and

11

34

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

for all x E R“ and t

E

J , where

llxlla

=

maxIzil, i

i = 1, ...,I.

By hypothesis (ii), there is a function $ E KR, e.g., $ = +hIi, and positive constants k, such that

and

1

1

Thus, inequalities (2.4.21) and (2.4.22) are true. Since the Lyapunov theorems of Section 2.2 are valid for any norm on R”, such as 11. /Irn, it follows from Theorem 2.2.26 (with 1 . I replaced by 11. lim) that the equilibrium x = 0 of composite system ( Y )is exponentially stable in the large in the norm 11 . /Im. Furthermore, since f E J, /x(t;xo,to)/ I [“zllx(t;xo,to)IIm, it follows that the equilibrium x = 0 of composite system (9’is ) also exponentially stable in the large in the norm 1 . (.

=

2.4

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

35

In practice it is often convenient to use quadratic comparison functions $ij(r), i = 1, ...,I, j = 1,2,3. In this case, the above theorem assumes the following form. 2.4.23. Corollary. The equilibrium x = 0 of composite system (9') with decomposition (Xi) is exponentially stable in the large if the following conditions are satisfied. (i) Each isolated subsystem possesses Property B'; (ii) given ui of hypothesis (i), there exist constants a, E R such that

(z)

for all zi E Rni,z j E R"J,i, j = 1, ..., I, f E J ; and >0 (iii) given oiin hypothesis (i), there exists an I-vector aT = (ctl, ...,al) such that the matrix S = [s,] specified by s.. =

ai("i+

4,

i=j

+

y [ (ai uij ctj aji)/2, i # j is negative definite. Proof. Since each isolated subsystem (8) possesses Property B', we have $ i l ( l z i l ) =cilIziI2,t,bi2(lzil)= ci21zi12,and $i3(lzil)=/zi12,i = 1, ..., 1. Each of these comparison functions is of the same order of magnitude. Thus, all hypotheses of Theorem 2.4.20 are satisfied. Hence, the equilibrium x = 0 of composite system (9) is exponentially stable in the large. If composite system ( Y )is decomposed into n scalar subsystems (q), it is possible to reduce the conservatism of Corollary 2.4.23 by eliminating norms in hypothesis (ii). The price paid for this improvement is an increase of the order of the test matrix S . We have 2.4.24. Corollary. The equilibrium x = 0 of composite system (9) with decomposition (Xi)is exponentially stable in the large if the following conditions are satisfied. (i) Each isolated subsystem (X) possesses Property B' with n, = 1 so that zi = x i ; (ii) given ui of hypothesis (i), there exist constants ay E R such that

n

C

Ca~i(~i,t)/axiICgi(~~,...,~,,f)I 5 xi aexj j =1

for all xi E R,i = 1, ..., n, and t E J ; and (iii) hypothesis (iii) of Corollary 2.4.23 holds with 1 = n. Proof. Choose as a Lyapunov function u(x, t ) =

c n

i= 1

UiUi(Xi,t).

36

IT

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Following along the lines of the proof of Theorem 2.4.2, we see that c j lxI2 I u(x, t ) I c2 1x12 and

Du(,)(x, t ) 5

c3

1xI2

for all

X E R" and f E J, where el = mini{aicil}, c2 = maxi(aici2], and c3 = A,(S) < 0. The conclusion of the corollary follows from the remark following

Theorem 2.2.26.

For composite system (9'with ) decomposition specified by Eq. (2.3.16), we have the following result.

2.4.25. Corollary. The equilibrium x = 0 of composite system (9) with decomposition (2.3.16) is exponentially stable in the large if the following conditions are satisfied. possesses Property B'; (i) Each isolated subsystem (3) (ii) given ui of hypothesis (i), there exists a positive constant ci4 such that IVui(zi, t>l I ci4 lzi( for all zi E R"I; (iii) for each i, j

=

1, . . . ,I, i # j , there exists a constant k, > 0 such that

Igi(zj,t>lI kijlzjl for all z j E R"' and t E J ; and (iv) given ci in hypothesis (i), there exists an !-vector such that the test matrix S = [sJ specified by s.. =

(L:ri:4

tlT =

..., aJT > 0

i=j k , + a j cj4 kji)/2,

i #j

is negative definite. Proof. Given zii in hypothesis (i) and for system (9' the ) Lyapunov function

CI

> 0 in hypothesis (iv), we choose

c CI,v,(z,, t ) . 1

u(x,t )

=

i= 1

From hypothesis (i) it follows that c1 /XI2

I e(x, t ) I

c2

1x12

(2.4.26)

for all x E R" and t E J , where c, = mini{aici,} and c2 = maxi{ccici,}. we have, taking hypotheses (i)-(iv) into Along solutions of system (9')

2.4

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

37

account

=

wTRw = wT((R+RT)/2)w = wTSw 2 1M(S)1x12 (2.4.27)

for all x E R” and t E J. Here, wT = ( 1 ~ ~ 1 ..., , lzll), A,(S) < 0 is the largest eigenvalue of test matrix S given in hypothesis (iv), and R = [re] is specified by r.. = y

[

Ui cj 9

i=j

aici4k , ,

i #j.

The conclusion of the theorem follows now from Theorem 2.2.26 and inequalities (2.4.26) and (2.4.27).

=

For composite system (9’with ) decomposition specified by Eq. (2.3.14), the above result assumes the following form. 2.4.28. Corollary. The equilibrium x = 0 of composite system (9) with decomposition (2.3.14) is exponentially stable in the large if the following conditions are satisfied. (i) Hypotheses (i) and (ii) of Corollary 2.4.25 hold; (ii) given ai of hypothesis (i), there exists an I-vector aT = ( a l , ...,a!) > 0 such that the test matrix S = [s,] specified by

is negative definite.

To prove Corollary 2.4.28, replace in hypothesis (iii) of Corollary 2.4.25 g , ( z j , t ) by C e z j and k, by IIC,II. We now consider system (9”’). 2.4.29. Theorem. The equilibrium x = 0 of composite system (9”) with decomposition (C;) is exponentially stable in the large if the following conditions hold. (i) There exist continuously differentiable functions ui : R“‘ x J + R and constants ci2 2 cil > 0,i = 1, ..., I, such that Cil IZiI2 I Ui(Zi,t) I ci2 lZil 2

for all zi E Rniand t

E

J;

38

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

(ii) given ui of hypothesis (i), there exist constants uii E R, i, j = 1, ..., I, such that I

+

v ~ , ( z , , r ) ~ h , ( x , ta)u i ( z i , t ) / a tI lzil

C aGlzj(

j= 1

for all zi E R“’,i = 1, . .., 1, x E R”, and t E J ; ...,a[) > 0 such that the matrix S = [se] (iii) there exists a vector aT = (al, specified by ai aii i=j s.. = ((aiuG+ajuji),2, i#j 3

is negative definite. The proof of this theorem follows along similar lines as the proof of Theorem 2.4.16.

C. Instability and Complete Instability In our next definition we let Bi(ri)= { z i E Rni : lzil < r i } for some ri > 0.

(z),

2.4.30. Definition. For isolated subsystem let there exist a continuously differentiable function ui:Bi(ri)x J + R, three functions t+ki2,$ i 3 E K , and constants ail,6 i z ,ci E R such that dil$il(tzil)

u i ( z i , t ) 5 6iz$iz(Izil)~

ouicyi,(zitf)

ei+ij(lzil>

for all zi E Bi(ri)and r E J . If hi, = diz = - 1 we say that (q.) possesses Property C. If di, = di2 = 1 we say that (Sq.)possesses Property A”.

If (Sq) has Property C with o i < 0, then the equilibrium zi = 0 is completely unstable. If (Sq) has Property A“ and ci < 0, the equilibrium of (Yi) is uniformly asymptotically stable. 2.4.31. Theorem. Let N # @, N c L = { I , ..., E ) . Assume that for composite system (9’) with decomposition ( X i ) the following conditions hold. (i) If ic N , then isolated subsystem (q) has Property C and if iQ N , i E L , then (Yi)has Property A“; (ii) given ui of hypothesis (i), there exist constants uGE R such that

for all zi E Bi(ri),z j E B j ( r j ) ,i, j E L , and t E J ; (iii) given ciof hypothesis (i), there exists an I-vector aT = (a,, . ..,a,) > 0

2.4

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

39

such that the test matrix S = [sg] specified by

is negative definite. (a) If N # L, then the equilibrium x = 0 of composite system (9’is) unstable. (b) If N = L, the equilibrium of ( Y )is completely unstable. Proof. Given ui of hypothesis (i) and CI of hypothesis (iii) we choose u(x,t) =

C aiui(zi,t). I

i= 1

Proceeding as in the proof of Theorem 2.4.2, we obtain 1

Du(,)(x,t) 5

C $i3(IziI) i= 1

for all zi E Bi(ri), i E L, and t E J . Since test matrix S is negative definite, we have A M ( S )< 0 and Du(,,(x, t ) is negative definite. Now consider the set D

R” x J : zi E B i ( r ) whenever i E N , where r < min r i , and zi = 0 whenever i .$ N , and t E J } .

= { ( x ,t ) E

For (x, t ) E D we have

i.e., in every neighborhood of the origin x = 0, there is at least one point x’ # 0 for which u(x’, t ) < 0 for all t E J . Furthermore, on the set D, u ( x , t ) is bounded from below. Thus, all conditions of Theorem 2.2.28 are satisfied. If N # L , then the equilibrium x = 0 of composite system (9) is unstable. If N = L, then u(x, t ) is negative definite and the equilibrium of (9’) is completely unstable. In the next result we consider composite system (9”’). 2.4.32. Theorem. Let L = { I , ..., r } , N c L , and N # 0. Assume that for composite system (9”’) with decomposition (C;) the following conditions hold. (i) There exist continuously differentiable functions u i : Bi(ri)x J - , R and $il, t+kiZ E K , i E L, such that $il

(Izil) 5 ui(zi, t ) 5 $i~(lzil)~ i .$ N ,

i E L,

40

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

and

(lzil) 5 Ui(zi,t) -+iz(lzil)~ i E N, for all zi E Bi(ri) for some ri > 0, i E L, and all t E J ; (ii) given ui of hypothesis (i), there exist a# E R and t,bi4 E K , i , j E L, such that -$ij

/

for all zi E Bi(ri),zj E B j ( r j ) ,i, j E L , and t E J ; and (iii) there exists a vector aT = (al,..., q) > 0 such that the matrix S = [sJ given by mi aii i= j s.. ‘I == [ ( a i u G + a j a j i ) / 2 , i j 9

+

is negative definite. (a) If N # L , then the equilibrium x = 0 of composite system ( Y )is unstable. is completely unstable. (b) If N = L, the equilibrium of (9”) The proof of this theorem involves obvious modifications of the proofs of Theorems 2.4.16 and 2.4.31.

D. Uniform Boundedness and Uniform Ultimate Boundedness And now we consider the boundedness of solutions of system (Y), 2.4.33. Definition. Isolated subsystem (Sq) possesses Property D if there exists a continuously differentiable function u j : R”‘x J + R, $ i l , t,bjZ, 4hi3 E K R and a constant ci E R such that $ii(Izil)

5

ui(zi,f)

5 $iz(lziI)j

Dui(Y,)(zi,t ) 5

~i+i3(lzil)

for all t E J and for all (zi12Ri (where Ri may be large) and such that ( z i ,t ) are bounded on Bj(Ri) x J . ui (zi ,t ) and If ci 5 0, then the solutions of subsystem (Sq) are uniformly bounded and if ci < 0, then the solutions of (Sq) are uniformly ultimately bounded.

2.4.34. Theorem. The solutions of composite system (9) with decomposition ( X i ) are uniformly bounded, in fact, uniformly ultimately bounded if the following conditions are satisfied. (i) Each isolated subsystem (Yi)possesses Property D; (ii) given ui and t,hi3 of hypothesis (i), there exist constants a# E R such that

2.4

41

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

for all zi E Rni,z j E R"], i, j = 1, ..., I, and t E J ; and (iii) given vi of hypothesis (i), there exists an I-vector aT = (a1, ..., a,) > 0 such that the matrix S = [SJ specified by

is negative definite. ProoJ Given the functions ui of hypothesis (i) and the vector hypothesis (iii), we choose

tl

of

c tliUi(Z,,t). I

o(x,t) =

i= 1

Following the proof of Theorem 2.4.2 it is clear that there exist E K R such that

$1,$z,$3

$1

(1x1) 5 o(x, t ) 5 *z(lxI),

Dq,,(x, t ) I n,(s)

(1x1)

$3

where A,(S) < 0, whenever x E R"- B, ( R , ) x ... x B,(R,) and t E J. To complete the proof we need to consider the situation where some of the ziare such that (zi( < Ri. First consider the case where lzil 2 Ri, i = 1, ..., r , and lzil < Ri, i = r + l , ..., 1. Then I

1

i=r+ 1

aioi(zi,t) +

,

r

1 ai$il(lZil) 5 u ( x , t > 5 C i= 1

i= 1

1

.i$i2(l~il)

+ 1 1 aiui(zi,t). ,=r+

Since ui is continuous on R", x J and bounded on Bi(Ri)x J , i = 1, ..., 1, it follows that there are q l ,q 2 E K R such that q 1(1x1) I o(x, t ) 5 (p2(lxl) f o r a l l t E J a n d forall x ~ R " s u c h t h a tl z i l < R i , i = r + l , ..., 1, and such that Izil, i = 1 , ..., r , are sufficiently large. Along solutions of (9) we have r

Do(,,(x, t ) =

1 ai{Dui(y,)(Zi,t)+Vui(zi, t)Tgi(Z1,...,zl, t > ) i= 1

11

42

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Now for all Izi(< Ri, i = r + l , ..., I , there exist constants K,, K2, K3, and Mi such that I

K1 2

Let

M,’ =

.

C

J=r+l

IaijIC$j3(IzjI>I”2,

[$,3((zll), . . . , $ r 3 ( z) , \ ) ] , l e t t h e r x r m a t r i x R = [ri] bespecifiedby

cci(ai+aii), and let

Mi 2 I ~ ~ i ~ ~ ~ ~ ( z i ~ ~ ) I

s” = ( R + RT)/2. Then

i =j

i#j,

I

Therefore, if the Izi/ for i = 1, ..., r are sufficiently large, the sign of D u ( , , ( x , r ) is determined by L,(s”)C:=,$i3(1zil), and Du(,)(x,t) < 0 for all t E J . Above we have assumed that lzil> R i , i = 1, ..., r, and lzil< Ri, i = r + I , ..., 1. For any other combination of indices, the proof is similar. E. Discussion

At this point, a few remarks concerning the preceding results are in order. In all results for composite systems (9’) and (Y),the analysis is accomplished in terms of the qualitative properties of the lower order isolated sub-

2.4

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

(x’),

43

systems (q) and respectively, and in terms of the properties of the interconnecting structure expressed by gi (x, t ) and gi( x ) , respectively. These results involve generally a reduction in dimension (from n to I ) and are in such a form as to reveal qualitative trade-off effects between subsystems and interconnecting structure. In the case of composite system (Y”), a reduction in dimension is also realized, however, the results do not yield qualitative trade-off information between isolated subsystems and interconnecting structure, for in this case the notion of isolated subsystem is undefined. For systems with a fine decomposition (i.e., a decomposition into many subsystems) it is as a rule easy to find Lyapunov functions for the lower order subsystems; however, the resulting stability (or instability or boundedness) conditions tend to be conservative. (Note however, that in the case of Corollary 2.4.24 involving the finest possible decomposition, the conservatism can be reduced appreciably.) On the other hand, in the case of systems with a coarse decomposition (i.e., a decomposition into few subsystems) it is usually more difficult to find Lyapunov functions for the subsystems (which may no longer be of low order); however, the resulting stability conditions are usually less conservative. This points to the following advantages (items (a) and (b) below) and disadvantages (item (c)) of the present method. (a) It is often possible to circumvent difficulties that arise when the Lyapunov method is applied to systems of high dimension and intricate interconnecting structure. (b) In the case of systems ( Y )and (9’) the analysis is accomplished in terms of system components and system structure. (c) If a system is decomposed into many subsystems, the results may be conservative. Theorems 2.4.16, 2.4.29, and 2.4.32 are of course also applicable to composite system (Y),since this system is a special case of system (9”). This demonstrates that the decomposition of large scale systems into isolated subsystems is a conceptual convenience and not a necessity. Indeed, Theorems 2.4.16, 2.4.29, and 2.4.32 may in principle yield less conservative conditions than corresponding results involving isolated subsystems, when applied to a given problem, for the former set of results involves fewer majorizations (estimates by comparison functions) than the latter set. However, it is usually more difficult to apply the former than the latter. Of all the preceding results, Corollaries 2.4.25 and 2.4.28 are perhaps the easiest to apply. The reason being that these results pertain to systems for which a great deal of information of the interconnecting structure is available. This suggests that the method of analysis advanced herein is in a sense much more important than the specific results presented. That is to say, given a specific system with special structure, it may often be more desirable to arrive at a result tailored to this particular system, using the present method of

44

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

analysis, rather than try to force this system into one of the specific forms considered in Section 2.3. We will demonstrate this further in Section 2.8. In all results presented thus far we require the existence of a vector ciT = (al, ...,ciJ > 0 which arises i n the choice of Lyapunov functions employed. These u-functions are always of the form

The motivation for choosing functions of this type is as follows. We may think of u as representing a measure of energy associated with a given composite system and we may view uias providing a measure of energy for isolated subsystem (9, Once ).the isolated subsystems are interconnected t o form the composite system, it is reasonable to assume that the qualitative effects of different subsystems on the overall system may vary in importance. Therefore it seems reasonable to assign different weights cii > 0 t o different functions u i , thus forming the indicated Mteighted sum of Lyapunov functions which serves as a Lyapunov function for the overall interconnected system. It is of course possible to combine the weighting factors cii with the respective functions ui . In this case the test matrices S are modified, replacing each cii by 1. However, there are the following two good reasons for not doing this. (a) In applications t o specific problems, the presence of a provides us with an added degree of flexibility, and in addition, judicious choice of a enables us frequently to reduce the conservatism of the results. (b) In the special case when the off-diagonal elements of S are nonnegative, the presence of ci will enable us to obtain results which are equivalent t o the present ones, however, are much easier t o apply. This will be accomplished in the next section. In physical large systems, it is usually true that a given subsystem (9, is) i = 1, ..., 1, i # k . Thus, in not connected to every other subsystem applications, matrix S is usually sparse and it may be possible t o apply linear programming methods (see Dantzig [I]) and sparse matrix results (see Tewarson [ 11) t o determine the optimal choice of ci. On the other hand, when the order of S is small, the optimal choice of CI is usually obvious. The concept which gave rise t o the qualitative analysis of dynamical systems at different hierarchical levels is the notion of vector Lyapunov function, introduced by Bellman [2]. In a certain sense, the results of this section may be viewed as analysis via vector Lyapunov functions. Specifically, if VT = ( u l , ..., u l ) E R' and ciT = (ul, ..., CI') E R',ai > 0, i = 1, ...,I, then

(z),

c aiui(.). 1

u(.)

=

ciTV =

i= 1

Nevertheless, u ( . ) is a scalar Lyapunov function. We shall reserve the term

2.4

ANALYSIS BY SCALAR LYAPUNOV FUNCTIONS

45

“vector Lyapunov approach” to cases when an appropriate comparison principle is applied to a differential inequality (of order I ) involving vector Lyapunov functions. This type of approach will be considered in Section 2.6. To simplify matters, we consider in the following discussion only Theorem 2.4.2. However, similar statements apply to the remaining results as well. We call the parameter oi, introduced in Definition 2.4.1, a degree or a margin of stability. In order t o satisfy hypothesis (iii) of Theorem 2.4.2, it is necessary that ( o i + a i i )< 0, i = 1, ..., 1. Thus, if ci > 0, so that (Yi) may be unstable, we require that a,, < 0 and laiil > oi. In other words, this theorem is applicable to composite systems with unstable subsystems, provided that there exists a sufficient amount of stabilizing feedback (i.e., local negative feedback) asStability ). results of the sociated with (YJ,which is not an integral part of (9, type presented herein for composite systems with unstable subsystems and without local stabilizing feedback have apparently not been established at this time. This problem is of great practical importance and needs to be pursued further. (Results for systems with unstable subsystems are given in Thompson [2]. However, they do not involve a reduction in dimension, and as such, this problem may be regarded as being essentially unsolved.) Generally speaking, the greater the margin of stability associated with each subsystem (i.e., the more negative the terms ( c T ~ + Q ~ ~ )i, = I , ..., /), the easier it is to satisfy the negative definiteness requirement of matrix S. Note that as in the case of weights cli, it is of course possible to combine cri with replacing oi by - 1, 0, or 1, when is uniformly asymptotically stable, uniformly stable, or possibly unstable, respectively. We emphasize that the off-diagonal terms of test matrix Scan have arbitrary signs. Tn the next section we consider additional results for which the off-diagonal terms of the test matrix are required to possess the same sign. Note also that in this section the test matrices are always symmetric. This will in general not be required in the results of the next section. Finally note that inequalities of the type encountered in hypothesis (ii) of Theorem 2.4.2, which express restraints on the interaction among the subsystems, are more easily satisfied than may appear at first glance. For example, hypothesis (ii) of Theorem 2.4.2 can be satisfied for appropriately chosen aij if g i ( x ,t ) are linear functions and v i ( z i ,t ) are quadratic, i = 1, .. ., 1. The results of this section can often be utilized in compensation and stabilization procedures at different hierarchical levels. To simplify matters, we consider once more Theorem 2.4.2. Similar statements can be made for the remaining results. We choose in hypothesis ( 5 ) of this theorem cli = 1, i = 1, ..., 1. It can be shown (see, e.g., Taussky [l]) that all eigenvalues of matrix S are negative if the diagonal dominance conditions

(x)

46

I1

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

hold. Now suppose we are able to ascertain by some means (such as Theorem 2.4.31) that a specific system (9) is unstable. Or suppose we are able to determine by Theorem 2.4.2 that (9) is uniformly asymptotically stable, however, the degree of stability of (Y), as expressed by A M ( S ) ,is unsatisfactory. Using appropriate local feedback associated with the subsystems and the interconnecting structure, it is frequently possible to stabilize (Y),or to enhance the stability of (9). In this process, the local compensators are chosen so as to increase the terms -((aii+aji)and decrease the terms l a g + a j i [ .However, it must be noted that this procedure may often yield overly conservative results. Furthermore, it may physically be impossible or undesirable to use this type of an approach.. We conclude our discussion by noting that all preceding results involving test matrices with off-diagonal terms having arbitrary signs are applicable to strongly coupled systems as well as weakly coupled systems.

2.5 Application of M - M a t r i c e s In the special case when the off-diagonal elements of the test matrices S in the results of Section 2.4 are nonnegative, we can utilize the properties of Minkowski matrices, called M-matrices, to establish results which are easier to apply because they do not involve usage of weighting vectors a. However, we are obliged to emphasize at the outset that (a) in the case the off-diagonal elements of S are nonnegative, the results of the preceding section and corresponding results of this section are equivalent, (b) whereas it is always possible to obtain the results of this section from those of the preceding section, the converse is in general not true, and (c) the results of the present section are generally applicable only to weakly coupled systems. This section consists of six parts. In the first of these we give a summary of selected results from the theory of M-matrices. Since this material is well covered in several references, all proofs of M-matrix results are omitted. In the remaining parts we establish several Lyapunov results for composite systems (9’) and (Y”),most of which have corresponding counterparts in Section 2.4. A. M-Matrices We begin with the following definition.

2.5.1. Definition. A real I x l matrix D = Id,] is said to be an M-matrix if d, i 0, i # j (i.e., all off-diagonal elements of D are nonpositive), and if all principal minors of D are positive (i.e., all principal minor determinants of D are positive).

2.5 Let D

= [d,]

APPLICATION OF M-MATRICES

47

be an I x I matrix. If the determinants D,, given by

are all positive, we say that the ‘‘successive principal minors of D are positive,” or, the “leading principal minors of D are positive.”

2.5.2. Theorem. Let D = [d,] be a real I x I matrix such that d, I 0 for all i # j . Then the following statements are equivalent. (i) The principal minors of D are all positive (i.e., D is an M-matrix). (ii) The successive principal minors of D are all positive. (iii) There is a vector u E R‘ such that u > 0 and such that Du > 0. (iv) There is a vector u E R‘ such that u > 0 and such that DTu > 0. (v) D is nonsingular and all elements of D-’are nonnegative (in fact, all diagonal elements of D - l are positive). (vi) The real parts of all eigenvalues of D are positive.

A direct consequence of parts (iii) and (iv) of Theorem 2.5.2 is the following. 2.5.3. Corollary. Let D = [d,] be an I x I matrix with nonpositive off-diagonal elements. Then the following statements are true. (i) D is an M-matrix if and only if there exist positive constants ,Ij, j = 1, ..., I, such that (2.5.4) (ii) D is an M-matrix if and only if there exist positive constants q j , j = 1, .. .,I, such that

C q j d j i > 0, 1

j= 1

i

=

1, ..., I.

(2.5.5)

Additional useful properties of M-matrices which we will require and which are a consequence of Theorem 2.5.2 are the following.

2.5.6. Corollary. Let D = [d,] be an I x I matrix with nonpositive off-diagonal elements. Then D is an M-matrix if and only if there exists a diagonal matrix A = diag[a,, ..., a*], ai > 0, i = I, ..., I, such that the matrix

is positive definite.

B

=

AD

+ DTA

(2.5.7)

2.5.8. Corollary. Let C = diag [el, .. .,el] > 0 be a diagonal I x I matrix and let Q = [q,] 2 0 be an I x I matrix. Then C- Q is an M-matrix if and only if

48

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

there is a diagonal I x I matrix A B

=

= diagca,,

...,cc,]

> 0 such that the matrix

C A C - QTAQ

(2.5.9)

is positive definite. 2.5.10. Corollary. If D is an M-matrix, then D - p I is an M-matrix if and only if p < minitL Re[Ai(D)], L = { I , ..., I}, where I is the 1 x 1 identity matrix and Re[Ai(D)] denotes the real part of the ith eigenvalue of D.

B. Uniform Asymptotic Stability Our first result is as follows. 2.5.11. Theorem. Assume that for composite system (9’ with ) decomposition (C,) the following conditions hold. (i) Each isolated subsystem ($) possesses Property A (see Definition 2.4.1); and (ii) given v, and $ i 3 of hypothesis (i), there exist constants aii E R such that (a) at 2 0 for all i # j , and C:.=~ for all (b) VOi(Zi,t)Tgi(Z1, . . . , ~ l r t ) I C $ ~ ~ ( I Z ~ I ) I ~ ’ ~ac[$j3(I~jI)I”~ z i ~ R n i , z j ~ R “ ~ , i ,..., , j = 11 , a n d t ~ J . is uniformly asymptotically stable in The equilibrium x = 0 of system (9’) the large if any one of the following conditions hold. (iii) given (o, of hypothesis (i), the successive principal minors of the I x I test matrix D = [d,] are all positive, where - ((oi

+a,,),

i

=j

i # j. (iv) The real parts of the eigenvalues of D are all positive. (v) There exist positive constants Ai,i = 1, . . .,I, such that

C 1

-((oi+aii)-

j=l,i#j

(Aj/Ai)uy > 0,

(vi) There exist positive constants qi, i -((oi+aij)-

I j=l,i#j

=

i

= I , ..., 1.

(2.5.12)

1, ...,1, such that

(qj/qi)aji > 0,

i = 1, ..., 1.

(2.5.13)

Proof. Since by assumption ay 2 0 for all i # , j , and since the successive principal minors of matrix D are all positive, it follows from part (ii) of Theorem 2.5.2 that D is an M-matrix. In view of Corollary 2.5.6 there exists a

2.5

49

APPLICATION OF M-MATRICES

diagonal matrix A = diag[a,, ..., 4 > 0 such that the matrix -2s 4 AD

+ DTA

is positive definite. Here, - 2 s = - 2 [SJ is specified by sij =

+ aii),

i=j

(cciai+ajaji)/2,

i # j.

ai(0i

But if - 2 s is positive definite, then the matrix S = [ S J is negative definite. Therefore, if hypotheses (i)-(iii) of the present theorem are satisfied, then all hypotheses of Theorem 2.4.2 are also satisfied. Ijence, the equilibrium x = 0 of system (9’)is uniformly asymptotically stable in the large. The proof is now complete, for hypotheses (iii)-(vi) are equivalent statements (see Corollary 2.5.3 and part (vi) of Theorem 2.5.2). H Henceforth we shall refer to inequalities (2.5.12)and(2.5.13) as rowdominance condition and column dominance condition, respectively. In all results of the previous and present section, we have used Lyapunov functions which are continuously differentiable. In our next theorem, we make use of continuous Lyapunov functions which need not be continuously differentiable. In doing so, we employ the following convention.

(z)

2.5.14. Definition. Isolated subsystem possesses Property & . if there exists a continuous function vi: Rnix J - , R, Iclil, Ic/i2 E KR, t+hi3 E K , constants L i , oi E R , Li > 0, such that Ic/il(lzil) 5 u i ( z i , t ) 5 +iz(Izil),

+

(zi, t ) = lim sup(l/h) {vi [ z i ( t + h; z i , t ) , t h] - v i ( z i , t ) } h+O+ (Ti

and

Ic/i3(lziI),

IUi(Zi’,

for all z i , zi‘, z;

E

t)-vi(z;,t)l

5 LiIzi’-z;I

Rniand I E J.

2.5.15. Theorem. The equilibrium x = 0 of composite system (9’)with decomposition ( X i ) is uniformly asymptotically stable in the large if the following conditions are satisfied. (i) Each isolated subsystem (Sp) possesses Property A ; (ii) given $i3 of hypothesis (i), there exist constants a8 2 0, i, j = 1, ..., I, such that

for all zi E R“’,i = 1 , ...,I, and

tE

J ; and

50

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATlONS

(iii) given ci and Li of hypothesis (i), the successive principal minors of the I x 1 test matrix D = [d,] are all positive, where

- (ui + Li qi),

i

=j

i#j. Proofi Given ui of hypothesis (i), let aT = ( a l , ..., a,)> 0 be a n arbitrary constant vector and choose as a Lyapunov function for system ( Y ) ,

c UiUi(Zi,t). 1

u(x,t) =

i= 1

From hypothesis (i) it follows that u is continuous, positive definite, decrescent, and radially unbounded. Along solutions of system (9’we ) have f k , : , , ( x , r ) = lim sup(l/h) h-O+

lim sup(l/h)

=

h+O+

- ui

C ai[ui(Zi(r+h;zi,t),t+h)-ui(zi,t)]

{ i =l 1

c ai{ui[zi+h.J;(zi,t ) + h . g , ( x , t>+o(h),t+h]

i= 1

(zi > t ) >

lim sup ( I / h )

=

h-O+

C ai {ui [zi+ h.f,(zi,t )+ o(h), t + h] - ui ( z i ,t ) 1

i= 1

+ ui[zi+h.f;.(Zi,t ) + h . g , ( x , r)+o(h), t+h] -

uiCZi+h.fi(Zi, t ) + o ( h ) , t + h ] }

where o(h) denotes higher order terms so that o(h)/h+O as h.0. In view of hypotheses (i) and (ii) we now have I

1

Du(,)(x,t) 5

1 aiDui(y,)(Ziyt)+ iC= a i L i J g i ( x , t ) (

i= 1

1

Now let wT = [t,hl3(Iz,I),..., I + ~ ~ ~ ( / Zand ~ \ ) ]note that w = 0 if and only if

x = 0. We have

D u , ~ , ( x , I~ )-aTDw

=

-yTw,

where aTD = yT and matrix D is defined in hypothesis (iii). Since D has positive successive principal minors and since d,, 5 0 for all i # , j , it follows from parts (ii) and (v) of Theorem 2.5.2 that D is an M-matrix, that D-’

2.5

51

APPLICATION OF M-MATRICES

exists and that D-' 2 0. Thus, c1

= (D-')Ty.

Since each row and column of D-' must contain at least one nonzero element (in fact, the diagonal elements of D-' are positive), we can always choose y in such a fashion that y > 0, so that a > 0. Therefore, DU(,,(X,t)

< - y T w < 0,

x # 0.

Hence, D U ( ~ ) (t)X is , negative definite for all x E R" and t E J . It follows that the equilibrium x = 0 of composite system (9) is uniformly asymptotically stable in the large. By taking advantage of Corollary 2.5.3 and following the proof of Theorem 2.5.15, we can establish our next result. 2.5.16. Corollary. Assume that hypothesis (i) of Theorem 2.5.15 is satisfied with oi E R replaced by a continuous function o i : Rnix J - + R . Assume that hypothesis (ii) of Theorem 2.5.15 is satisfied with ayE R replaced b,y a continuous function ay: R"Jx J R with the property a y ( z j ,t ) 2 0 for all z j E R"J and all t E J . Let D ( x , t) = [d,(zj,t ) ] be determined by -+

- [ai ( z i , t)

d,(zj,t)=

+Lia ii( zi,t)],

- Lia@(zj9 t>,

i

=j

i#j

If there exist constants l i > 0, i = 1, ..., I, and a constant y > 0 such that 1

d j j ( z j t, ) -

C (Ai/Aj)ldy(zj,t)l L y, i=l,i#j

j = 1, ..., 1,

for all z j E R"j, j = 1, ..., I, t E J, then the equilibrium x = 0 of composite system (9) is uniformly asymptotically stable in the large. For system (9"') we have the following result. 2.5.17. Theorem. Assume that hypotheses (i) and (ii) of Theorem 2.4.16 are true. The equilibrium x = 0 of composite system (9'") with decomposition (C:) is uniformly asymptotically stable in the large if ay 2 0 for all i # j and if the successive principal minors of the test matrix D = [Id,], where dy = - a y , are all positive. Proof. Using an argument similar to that of Theorem 2.5.1 I , the proof follows from Theorem 2.4.16 and Corollary 2.5.6. C. Exponential Stability Next, we consider several exponential stability results for composite and (9'"). systems (9')

52

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

2.5.18. Theorem. Assume that hypotheses (i)-(iii) of Theorem 2.4.20 are true. Then the equilibrium x = 0 of composite system ( Y )with decomposition ( X i ) is exponentially stable in the large if a, 2 0 for all i # j and if the successive principal minors of the test matrix D = [d-,] are all positive, where d.. =

La,,

-(oi+aii),

i =j i # j.

Proof. The proof follows from Theorem 2.4.20 and Corollary 2.5.6, following a similar procedure as in the proof of Theorem 2.5.1 1.

2.5.19. Corollary. Assume that hypotheses (i) and (ii) of Corollary 2.4.23 are true and assume that a, 2 0 for all i # j . The equilibrium x = 0 of system (9) is exponentially stable in the large if the successive principal minors of the test matrix D = [d,] are all positive, where - (Oi

+U i i ) ,

i

=

;

i # .i. f r o q f The proof follows from Corollaries 2.4.23 and 2.5.6.

m

The next two results are concerned with composite system (9’) having decompositions determined by Eqs. (2.3.16) and (2.3.14), respectively. 2.5.20. Corollary. Assume that hypotheses (i)-(iii) of Corollary 2.4.25 are true. The equilibrium x = 0 of system (9’)with decomposition (2.3.16) is exponentially stable in the large if the successive principal minors of the test matrix D = [d,] are all positive, where

pro(^ The proof follows from Corollaries 2.4.25 and 2.5.6.

2.5.21. Corollary. Assume that hypothesis (i) of Corollary 2.4.28 is true. Then the equilibrium x = 0 of system (9’)with decomposition (2.3.14) is exponentially stable in the large if the successive principal minors of the test matrix D = [d,] are all positive, where

Pro($ This proof follows from Corollaries 2.4.28 and 2.5.6.

m

I n the next theorem we consider system (9”’). 2.5.22. Theorem. Assume that hypotheses (i) and (ii) of Theorem 2.4.29 are true. Then the equilibrium x = 0 of composite system (9”) with decompo-

2.5

APPLICATION OF M-MATRICES

53

sition (C;) is exponentially stable in the large if aY 2 0 for all i # j and if the successive principal minors of the test matrix D = [d,] are all positive, where d.. = - a.. . (f ProoJ The proof follows from Theorem 2.4.29 and Corollary 2.5.6.

The next result provides another interesting application of M-matrices. 2.5.23. Corollary. Let D denote the test matrix ofTheorem 2.5.1 1 (or Theorem 2.5.18). Assume that composite system (9’)with decomposition (Ci) has been shown to be uniformly asymptotically stable in the large (or exponentially stable in the large) using one of these results. Then any modification of the isolated subsystems or their feedback as expressed by a,i, i = 1, . .., I, which increases (ai+ aii),i = 1, .. .,I, by an amount less than p = mink Re [Ak(D)], k = 1, . .., I, will leave system (9’)uniformly asymptotically stable in the large (exponentially stable in the large).

(x),

Proof. The proof follows directly from Theorem 2.5.11 (or Theorem 2.5.18) and from Corollary 2.5.10. The parameter p in Corollary 2.5.23 may be interpreted as a margin of stability or as a degree of stability of the overall interconnected system (9) and may be used to judge how sensitive the qualitative properties of system ( Y )are with respect to structural changes. D. Instability and Complete Instability Next, we consider the instability of systems (9) and (9’”). 2.5.24. Theorem. Assume that hypotheses (i) and (ii) of Theorem 2.4.31 are true, that uv 2 0 for all i # J , and that all principal minors of the test matrix D = EdY]are positive, where d.. =

-(ai+aii), [-uY,

i

=j

i # j.

(a) If N # L, then the equilibrium x = 0 of composite system (9’)is unstable. (b) If N = L, the equilibrium of system (9’)is completely unstable. Proof. The proof follows from Theorem 2.4.31 and Corollary 2.5.6.

2.5.25. Theorem. Assume that hypotheses (i) and (ii) of Theorem 2.4.32 hold, that uv 2 0 for all i #j, and that all principal minors of the test matrix D = [dv] are positive, where dii = -as. If N # L, then the equilibrium x = 0 of

54

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

composite system (9”) is unstable and if N completely unstable.

= L,

the equilibrium of (9”) is

ProoJ The proof follows from Theorem 2.4.32 and Corollary 2.5.6.

w

E. Uniform Boundedness and Uniform Ultimate Boundedness Our last result yields sufficient conditions for boundedness of solutions of composite system (9’). 2.5.26. Theorem. Assume that hypotheses (i) and (ii) of Theorem 2.4.34 are true, that ay > 0 for all i # j , and that the successive principal minors of the test matrix D = are all positive, where

[4]

d.. =

Lag,

-(ai+aii),

i =j i # j.

Then the solutions of composite system (9) with decomposition (Xi) are uniformly bounded, in fact, uniformly ultimately bounded. Proof. The proof follows from Theorem 2.4.34 and Corollary 2.5.6.

F. Discussion We conclude this section with a few observations, phrased in the notation of Theorem 2.5.1 1. Similar statements can be made for the remaining results. To satisfy hypotheses (iii)-(vi) of this theorem, it is necessary that (ai+ aii)< 0, i = 1, .. .,1. Thus, if ai > 0, so that subsystem (q)may be unstable, we require that a,, < 0 and laii\> a i . In other words, as in the case of Theorem 2.4.2, Theorem 2.5.11 is applicable to composite systems with unstable subsystems, provided that a sufficient amount of stabilizing feedback is associated with each unstable subsystem, where the feedback is not an integral part of the subsystem structure. Clearly, the greater the degree of stability associated with each subsystem, as expressed by a i + a i i , i = 1, ..., I, the easier it is to satisfy hypotheses (iii)-(vi) of Theorem 2.5.11. Also, the weaker the interconnections, i.e., the smaller the terms ay > 0, i # j , the easier it is to satisfy these hypotheses. Thus, Theorem 2.5.11 constitutes a set of weak coupling conditions for uniform asymptotic stability of system (9). The same is true in the case of Theorem 2.4.2 when ag 2 0, i # j . These statements are of course not surprising, for if ag 2 0, i # j , then Theorems 2.4.2 and 2.5.1 1 are equivalent. However, it is emphasized that the test matrix S in Theorem 2.4.2 has no restrictions in sign for the off-diagonal elements (i.e., a y , i # j , can have any sign) and therefore Theorem 2.4.2 is a more general result than Theorem

2.5

APPLICATION OF M-MATRICES

55

2.5.1 1. Note however that the latter result is easier to apply than the former. Specifically, in Theorem 2.4.2 we utilize a symmetric test matrix S involving a weighting vector a > 0, while in Theorem 2.5.1 1 we employ a test matrix D which in general need not be symmetric and which does not involve a weighting vector. Note that hypothesis (iv) of Theorem 2.5.11 enables us to deduce the stability properties of the nonlinear, nonautonomous n-dimensional composite system (9’)from the linear autonomous system j = Dy, where I I n. Note also that in the row and column dominance conditions (2.5.12), (2.5.131, ‘there appear arbitrary sets of constants {Ai), {qi}, i = 1, ..., 1, respectively. Due to the simple form of these conditions, usage of linear programming methods appear to be attractive to determine the optimal choice of these constants in specific high-dimensional problems (to obtain the least conservative results). In the case of low-dimensional problems, this choice may often be determined by inspection. The row and column dominance conditions are especially well suited for systematic stabilization and compensation procedures of large scale systems. Conceptually, such procedures involve the following steps. (a) Enhance the degree of stability of specific subsystems, using local feedback at the subsystem level structure (i.e., increase -((ai+aii));or (b) weaken coupling effects, using local feedback at the interconnecting structure level (i.e., decrease appropriate choices of ay 2 0, i # j ) ; or (c) combine items (a) and (b). In this procedure Corollary 2.5.23 is useful in providing a measure for the margin of stability p of the overall composite system (9). The constants av 2 0, i # j , in Theorem 2.5.1 1 provide a measure of dynamic interaction among subsystems. If in particular aij = 0, we view (Yj)as not being connected to (YJand if aji = 0, we view as not being connected to (Yj).Suppose now that any one of the equivalent hypotheses (iii)-(vi) of Theorem 2.5.1 1 have been satisfied for system (9) for a given set of constants a” 2 0, i # j . Then it is clear that if any one (or all) of the aij 2 0, i # j , are decreased or set equal to zero, the stability conditions (iii)-(vi) are preserved. (This has motivated some authors to speak of so-called “connective stability.” Since results of the type considered above have this feature automatically built in, we decline to pursue this notion any further.) Notice further that if an increase of say a i k ,i # k , is accompanied by an appropriate decrease of a.IP p = 1, ..., 1, i # p # k , then stability condition (2.5.12) can be preserved. The preceding discussion suggests that in certain cases one may want to pursue the following approach in the planning of reliable large scale dynamical systems. (a) If applicable, a given system is viewed as an interconnection of subsystems, also called “areas”;

(x.)

7

56

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

(b) each area is planned in such a fashion so that it is endowed with a sufficient margin of stability; (c) the dynamic interaction among various areas is limited in such a fashion that either the row or the column dominance conditions are satisfied; (d) the overall system is planned in such a way that it exhibits a sufficient degree of stability; (e) if at a future point in time it is necessary to increase the interaction among certain areas, this will have to be accompanied by an appropriate increase in the margin of stability of the affected areas and/or by a suitable decrease in the strength of coupling elsewhere. Note that a system planned in this fashion is reliable in the sense that subsystems can be disconnected and reconnected (intentionally or by accident) without affecting the basic stability properties of the main system and the disconnected subsystems. 2.6 Application of the Comparison Principle t o Vector Lyapunov Functions

In the present section we first give a brief overview of several comparison theorems which are the basis of the comparison principle. Next, we show how this principle can be applied in the analysis of large scale systems using vector Lyapunov functions. We will show that for most of the specific cases considered thus far in the literature, this method reduces to the scalar Lyapunov function approach considered in Sections 2.4 and 2.5, where usage of the comparison principle is not required. Furthermore, we will demonstrate that the method of the present section, when applied to interconnected systems, involves test matrices which are always required to be M-matrices. Since the comparison principle is treated in detail in several texts, proofs of the comparison theorems are omitted. We begin by considering a scalar ordinary differential equation of the form

where y E R , t E J , and G : B ( r ) x J + R for some r > 0. Assume that G is continuous on B ( r ) x J a n d that G(0, t ) = 0 for all t 2 to. Under these assumptions it is well known that Eq. (C) possesses solutions y ( t ; yo, to) for every y o = y ( t o ;y o , to) E B ( r ) , which are not necessarily unique. These solutions either exist for all t E [to,a)or else must leave the domain of definition of G at some finite time t , > to. Also, under the above assumptions, Eq. ( C ) admits the trivial solution y = 0 for all t 2 to. As before, we assume that y = 0 is an isolated equilibrium. Finally, for the sake of brevity, we frequently write y ( t ) in place of y ( t ; yo, to) to denote solutions, with y ( t o ) = y o .

2.6

APPLICATION OF THE COMPARISON PRINCIPLE

57

2.6.1. Definition. Let p ( f ) be a solution of Eq. (C) in the interval [to,a). Then p ( t ) is called a maximal solution of (C) if for any other solution y ( t ) existing on [to,a), such that p ( t o ) = y(to) = yo, we have y ( t ) I p ( t ) for all t E [ t o , 4. 2.6.2. Definition. Let q ( t ) be a solution of Eq. (C) on the interval [to,a). Then q ( t ) is called a minimal solution of (C) if for any other solution y ( t ) existing on [to,a), such that q(to)= y ( t o )= yo, we have y ( t ) 2 q ( r ) for all t E [ t o , 4.

For maximal and minimal solutions of Eq. (C) we have the following existence theorem. 2.6.3. Theorem. If G is continuous on B(r) x J and if y o E B(r), then Eq. (C) has both a maximal solutionp(t) and a minimal solution q ( t ) for anyp(to) = q (to)= yo. Each of these solutions either exists for all t E [to,a)or else must leave the domain of definition of G at some finite time t 1 > to. The following comparison theorem is fundamental to the theory. 2.6.4. Theorem. Assume that G is continuous on B(r) x J and that p ( t ) is the maximal solution of Eq. (C) on the interval [to,a) with p ( t o ) = yo. If r(t) is a continuous function such that r(to) 5 yo, if Dr(t) and if

=

lim sup[r(t+h) - r(t)]/h

h+O+

almost everywhere on

Dr(t) 5 G(r(t), t )

[to,a),

then r(t) 5 p ( t ) on [ro,a). We also have 2.6.5. Theorem. Assume that G is continuous on B(r) x J and that q ( t ) is the minimal solution of Eq. ( C ) on the interval [ t o ,a) with q(to) = yo. If s ( t ) is a continuous function such that s(to)2 yo, and if almost everywhere on

Ds(t) 2 G ( s ( t ) ,t ) then

s(t) 2

[to,a),

q ( t ) on [to,a).

Theorem 2.6.4 (as well as Theorem 2.6.5), referred to as a comparison principle, is a very important tool in applications because it can be used to reduce the problem of determining the behavior of solutions of Eq. (I),

1

= g(x,t),

(1)

x E R",g: B(r) x J + R", to the solution of a scalar equation (C). To be more

specific, we have in mind the application of Theorem 2.6.4 (as well as Theorem

58

11 SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

2.6.5) to the case where r ( t ) = v ( x ( t ) ,t ) (or s ( t ) = v ( x ( t ) ,t ) ) , where v : B(r) x J + R is a Lyapunov function and x ( t ) is a solution of Eq. (I). (As in Section 2.2, we assume that g(0, t ) = 0 for all t 2 to.) In particular, applying Theorem 2.6.4 to u ( x , t ) , one can easily see that the following results are true. 2.6.6. Theorem. Let g and G be continuous on their respective domains of definition. Let v : B ( r ) x J - , R be a continuous, positive definite function such that DV(,)(X,

t > 5 G ( v ( x ,t ) , t ) .

(2.6.7)

Then the following statements are true. (i) If the trivial solution of Eq. ( C ) is stable, then the trivial solution of Eq. (I) is stable; (ii) if u is decrescent and if the trivial solution of Eq. (C) is uniformly stable, then the trivial solution of Eq. (I) is uniformly stable; (iii) if v is decrescent and if the trivial solution of Eq. (C) is uniformly asymptotically stable, then the trivial solution of Eq. (I) is uniformly asymptotically stable; (iv) if there are constants a > 0 and b > 0 such that a J x J 0 is most commonly used in parts (iii) and (iv) of the above theorem. Applying Theorem 2.6.5 to u( x, t ) , we can also see that the next result is true. 2.6.8. Theorem. Let g and G be continuous on their respective domains of definition. Let u : B ( r ) x J + R be a continuous, positive definite function such that Du,,,(x,t ) 2 G ( u ( x ,r ) , t ) . If the trivial solution of Eq. ( C ) is unstable, then the trivial solution of Eq. (1) is also unstable.

The generality and effectiveness of the preceding comparison technique can be improved by considering vector-valued comparison equations and vector L ~ ~ u ~ l m o z ~ ~ ~ (see n c t iSection o n s 2.4E). Here, the scalar case is included as a special case. Specifically, consider a system of 1 ordinary differential equations, j,

=

H(y,t),

Y ( h >= Yo

iVC)

2.6

APPLICATION OF THE COMPARISON PRINCIPLE

59

where y E R', t E J, H : B(r) x J R' is continuous on B ( r ) x J, and H ( 0 , t ) = 0 for all t 2 to. Under these assumptions Eq. (VC) possesses solutions y ( t ) for every yo = y(to) E B(r) which again are not necessarily unique. These solutions either exist for all t E [ t o , co) or else must leave the domain of definition of H at some finite time t , > to. Furthermore, under the above assumptions Eq. (VC) admits the trivial solution y = 0 for all t 2 to. Once more we assume that y = 0 is an isolated equilibrium. I f w e l e t a s u s u a l y I z d e n o t e y i I z i , i = 1 , ...,I , a n d y < z d e n o t e y i < z i , i = 1, ...,1 then Definition 2.6.1 of maximal solution and Definition 2.6.2 of minimal solution still make sense. In order to extend Theorems 2.6.6 and 2.6.8 to the vector case, we require the following additional concept. --f

2.6.9. Definition. A function H ( y , 2 ) = ( H , ( y , t), ..., H , ( y , t))' is said to be quasimonotone if for each component Hi, j = 1, . ..,I, the inequality Hi ( y ,t ) I Hj(z, t ) is true whenever yi I zi for all i # j and y j = z j . The above property was used by Muller [I] and Kamke [l]. It is sometimes called the Wazewski condition with reference to the work by Wazewski [l]. 2.6.10. Theorem. If H(y, t ) is continuous and quasimonotone and if yo E B(r), then Eq. (VC) has a maximal and minimal solution. Each of these solutions must either be defined for all t E [to,co) or else leave the domain of definition of H at some finite time t , > to. Analogous to Theorem 2.6.4 we have the following comparison theorem. 2.6.11. Theorem. Assume that H is continuous on B ( r ) x J , that H is quasimonotone, and let p ( t ) be the maximal solution of Eq. (VC) on [to,a) with p ( t o ) = yo. If r ( t ) is a continuous function such that r ( t o ) 5 yo, if D r ( t ) = [Eh+ + ( r l ( t + h) - rl ( t))/A, .. ., hm,, + (rl( t + h) - r, (t))/hIT,and if

-

D r ( t ) I H ( r ( t ) ,t )

almost everywhere on

[to,a)

then r ( t ) I p ( t ) on [to,a). We also have the following theorem. 2.6.12. Theorem. Assume that H is continuous on B ( r ) x J, that H is quasimonotone, and let q ( t ) be the minimal solution of Eq. (VC) on [to,a) with q(to) = yo. If s ( t ) is a continuous function such that so,) 2 yo and if Ds(t) 2 H(s(t), t )

almost everywhere on

[to,a)

then s ( t ) 2 q ( t ) on [to,a). Now let ui (x, t ) , i = 1, .. .,I, denote I continuous Lyapunov functions and let w , t >=

(Ul(X,t),

...,u,(x,t))'.

60

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

We call V ( x , t ) a vector Lyapunov function. Such functions were originally introduced by Bellman [2]. Applying Theorem 2.6.1 1 to V ( x ,t ) the following result is easily established.

2.6.13. Theorem. Let g and H be continuous on their respective domains of definition and let H be quasimonotone. Let V(x,t) be a continuous nonnegative vector Lyapunov function (of dimension I ) such that IV(x,t)l is positive definite and such that

ov,,, (x, t ) I H ( V(x,t ) , t ) .

(2.6.14)

Then the following statements are true. (i) If the trivial solution of Eq. (VC) is stable, then the trivial solution of Eq. (I) is also stable; (ii) if J V ( x t, ) / is decrescent and if the trivial solution of Eq. (VC) is uniformly stable, then the trivial solution of Eq. (I) is also uniformly stable; (iii) if lV(x,r)l is decrescent and if the trivial solution of Eq. (VC) is uniformly asymptotically stable, then the trivial solution of Eq. (I) is also uniformly asymptotically stable; 1 IV(x, ~ t ) l , if (iv) if there are constants a > 0 and b > 0 such that ~ 1 x 5 iV(x, t)l is decrescent, and if the trivial solution of Eq. (VC) is exponentially stable, then the trivial solution of Eq. (I) is also exponentially stable; (v) if g : R" x J + R" and H : R' x J + R' and V: R" x J --+ R' and if (2.6.14) is true for all (x, t ) E R" x J , and if the solutions of Eq. (VC) are uniformly bounded (uniformly ultimately bounded), then the solutions of Eq. (I) are also uniformly bounded (uniformly ultimately bounded). Applying Theorem 2.6.12 to V ( x ,t ) we can also prove the following instability result. 2.6.15. Theorem. Let H and g be continuous on their respective domains of definition and let H be quasimonotone. Let V ( x , t ) be a continuous nonnegative vector Lyapunov function (of dimension 1 ) such that IV(x,t)l is positive definite and such that DV(,,(X, l ) 2 H ( V ( x ,t ) , 2 ) .

If the trivial solution of Eq. (VC) is unstable, then the trivial solution of Eq. ( I ) is also unstable. Proofs of the preceding results can be found in several standard references, some o f which we cite in Section 2.9. Let us now see how these results apply i n the qualitative analysis of large scale systems. As Matrosov [ I , 21 has pointed out, from the point of view of applications, the following special case of (VC), j, = Py+m(y,t) (VC')

2.6

APPLICATION OF THE COMPARISON PRINCIPLE

61

is particularly important. Here P = [ps] is a real I x I matrix and the function m : B(r) x J + R’is assumed to consist of second or higher order terms, so that lim Irn(y, t ) l / l yI

lyI-0

=

uniformly in

0,

t 2 to.

Applying the principle of stability in the first approximation (see Hahn [2, p. 1221) to Eq. (VC’), we have the following result. If matrix P has either only eigenvalues with negative real parts or at least one eigenvalue with a positive real part, then the equilibrium of (VC’) shows the same stability behavior as that of the corresponding linearized system, jJ = Py.

Using the principle of stability in the first approximation and Theorems 2.6.13 and 2.6.15, we immediately obtain the following result. 2.6.16. Corollary. Let g be continuous and let V(x, t ) be a nonnegative continuous vector Lyapunov function (of dimension I ) such that IV(x, t ) l is positive definite and decrescent. Suppose there is an I x I matrix P = [ p G ] and a function m (V, t ) such that DV(,,(X, t ) I PV(X, t ) + m ( V ( x , t ) , t )

and

ps 2 0

if

(2.6.17) (2.6.18)

i # j,

and m(V, t ) is quasimonotone in V , and lim Im(y, t ) l / l yl

lyi-0

=

0,

uniformly in

t 2 to.

(2.6.19)

Then the following statements are true. (i) If matrix P has only eigenvalues with negative real part, the trivial solution of Eq. (I) is uniformly asymptotically stable; (ii) if matrix P has only eigenvalues with negative real part and if in addition IV(x, t ) l 2 6 lxI2 for some 6 > 0, then the trivial solution of Eq. (I) is exponentially stable; (iii) if the inequality (2.6.17) is reversed and if P has at least one eigenvalue with positive real part, then the trivial solution of Eq. (I) is unstable; (iv) if the inequality (2.6.17) is reversed and if the real parts of all eigenvalues of P are positive, then the trivial solution of Eq. (I) is completely unstable. To simplify matters, we consider only parts (i) and (ii) of Corollary 2.6.16 in the following discussion. Similar statements can be made for the remaining parts. First we note that in (i) and (ii) of Corollary 2.6.16, the quasimonoticity condition (2.6.18) means that - P is an M-matrix. Therefore, the condition that all eigenvalues of P have negative real parts, i.e., Re[1(P)] < 0, is

62

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

(-

])k

Pll

...

Plk

.

...

.

Pkl

"'

>0,

k = l , ..., 1.

(2.6.20)

Pkk

where V ( x ,t ) = ( u , ( x , t ) , . . ., u,(x, t))'. Under the assumptions of Corollary 2.6.16, this function u(x, t ) is a positive definite and decrescent scalar Lyapunov function such that D o, , , ( x, t )

I uT[PV(x,t)+m(V(x,t),t)] =

(u'P) V ( x ,t ) + crTm(V(x,t ) , t ) .

Since (mTP)V ( x , t ) is negative definite and since u'm(V, t ) consists of terms of second or higher order in V ( x ,t ) , it follows that Do(,,(x,t ) is negative definite in a neighborhood of the origin x = 0. This argument shows that in the important special case of (2.6.17) the vector Lyapunov function V ( x ,t ) can be reduced to a scalar Lyapunov function v ( x , t ) and because of the quasimonoticity requirements, the seemingly more general approach of applying the comparison principle to vector Lyapunov functions is really equivalent to an approach utilizing scalar Lyapunov functions (as discussed, e.g., in Section 2.5). It would be interesting to know whether or not this same equivalence is true for systems of inequalities more general than (2.6.17). Furthermore, we note that whereas the present approach of applying the comparison principle to vector Lyapunov functions appears to require test matrices (e.g., matrix P) whose off-diagonal terms are of the same sign, this is not the case in the approach of Section 2.4. Turning our attention to interconnected systems, let us consider once more composite system (9') with decomposition (Xi) and isolated subsystems (q) (see Eqs. (2.3.10)-(2.3.12)). In particular, recall the equations for (Xi)and

(Xi)

ii = fi(zi,t)+ g i ( x , t ) ,

(sq)

ii = f , ( z i , t ) ,

i

(x),

=

i = 1, ..., I,

(2.6.21)

1, ..., 1.

(2.6.22)

For interconnected system ( Y ) ,the method originally proposed by Bailey

2.7

63

ESTIMATES OF TRAJECTORY BEHAVIOR AND BOUNDS

[1,2] has been the most successful procedure of constructing vector Lyapunov functions (see Matrosov [I, 21). Specifically, we consider isolated subsystems (q)for which we can find positive definite, decrescent, continuously differentiable Lyapunov functions ui such that the derivative of ui with respect to t along solutions of (q) satisfies

(2.6.23)

Dui(yi)(zi,t ) I -Piui(zi, t ) .

Systems for which (2.6.23) can be satisfied are often linear or nearly linear and the corresponding ui is usually a quadratic function in z i . Suppose now that we can find an I x I matrix P = [ p J satisfying pU 2 0,

i # .i,

(2.6.24)

and a quasimonotone function m : R' x J + Rf satisfying lim I m ( y , t ) l / l y l = 0,

uniformly in

(2.6.25)

t 2 to.

lYI+O

Assume also that the following inequalities hold, -Vui(zi,t)Tgi(X,t) +

i

C (pij+6,iPi)uj(zj,t) + m i ( V ( x , t ) , t ) 2 0,

j= 1

(2.6.26)

i = I , ...,I, where 6, is the Kronecker delta, mi(V,t ) is the ith component of m ( V , t ) , and V ( x , t ) = ( u l ( z l , t ) ,..., u f ( z , ,t))'. We now have

D V , , ) ( X ? t ) = CD~,(,,)(z,,t)+V~,(z,,t)'gl(X,t),

+ vuf(ZI

9

'..>D~f(Y,)(ZfJ)

t>'g, ( x , ?)IT

s I:- P I u1 (z1,t )+ vu,( z , ,tl'g, + VV,(Zf t>TgL(X,?)IT. 9

Combining (2.6.26) and (2.6.27) yields D V ( y ) ( x ,t ) 5 P V ( x , t )

( x , t ) , .. ., - Bf Of (Zf 1 ) (2.6.27)

+ m(V(x, t ) , t ) .

7

(2.6.28)

All conditions of Corollary 2.6.16 are now satisfied. I n particular, if matrix P has only eigenvalues with negative real parts, then the equilibrium x = 0 of composite system (9') is asymptotically stable. Parts (ii)-(iv) of Corollary 2.6. I6 apply equally as well, with obvious modifications.

2.7 Estimates of Trajectory Behavior and Trajectory Bounds In this section we obtain estimates of trajectory behavior and trajectory bounds of dynamical systems, using Lyapunov-type results. I n our approach we define stability in terms of subsets of R" which are prespecijied in a given

64

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

problem and which in general may be time varying. The properties of these sets yield information about the transient behavior and about trajectory bounds of such systems. Before considering interconnected systems, we need to develop essential preliminary results. We begin by considering systems described by equations of the form i = g(x,t)

(1)

where x E fz", t E J , and g: R" x J + R". We assume that Eq. (I) possesses for every x o E R" and for every to E R + a unique solution x ( t ;xo, to) for all r E J . However, in the present section we do not insist that x = 0 be an equilibrium of Eq. (1). Subsequently we employ time varying open subsets of R" defined for all t E J and denoted by S ( t ) , S o ( t ) , and D ( t ) , with appropriate subscripts or superscripts if necessary. The boundary of S ( t ) is denoted by a S ( t ) and its closure is denoted by S(t>. Henceforth we assume that all such sets possess the following "well-behaved'' property.

2.7.1. Definition. A time-varying open set S ( t ) is said to possess Property P if, whenever p : J - t R" is continuous and p ( t l ) E S ( t , ) for some t , E J , then either p ( t ) E S ( t ) for all t E [ t i ,a)or there exists a t , E ( t l ,a)for which p ( t z )E dS(t,) and p ( t ) E S ( t ) for all t E [ t l ,t2). For example, the set S , ( t ) = {x E R" : 1x1< keKa', k > 0, a > 0, t possesses Property P. On the other hand, the set

E

R'}

does not possess Property P. In the next definition we let x ( t ; x i , t i ) be the solution of Eq. (I) which satisfies x ( t i ; x i ,ti) = x i .

2.7.2. Definition. System (I) is called stable with respect to {So(to),S ( t ) ,to} ifx, E & , ( t o )implies that x ( t ; xo, to) E S ( t ) for all t E J . System (I) is uniformly stable with respect to { S o ( t ) S, ( t ) } , if for all ti E J , xiE So(ti)implies that x ( t ; x i , t i )E S ( t ) for all t E [ti,co). System (I) is unstable with respect to {S,,(t,),S ( t ) ,t o } , S,(to) c S ( t , ) , if there exists an .yo E So(to) and a t, E J such that x(t,; xo,to) E dS(t,). We emphasize that in Definition 2.7.2 the sets S o ( t ) and S ( t ) are prespecified in a given problem. In the literature the term practical stability is used for the special case S o ( t )= B(u) and S ( t ) = B ( p ) , a p. Also, the term ,finite time stability is used for the special case when J is replaced by the interval [to,to + T ) , T > 0.

2.7

ESTIMATES OF TRAJECTORY BEHAVIOR AND BOUNDS

65

2.7.3. Theorem. System (I) is stable with respect to {So(to), S ( t ) ,t o ) , S(to)3 So(to),if there exists a continuous function u : R" x J + R and a function G which satisfies the assumptions of Theorem 2.6.4 such that (i) Du(,)(x,t ) I G(u(x, t ) , t ) for all x E S ( t ) and t E J ; and ~ to) ( < ~ infxEas(,)u(x, ~ ) ~ ( X t, ) , for all t E J, where (ii) p ( t ; S U ~ ~ ~ ~to), p ( t ; yo, to) is defined in Theorem 2.6.4. System (I) is uniformly stable with respect to { S o ( t ) , S ( t ) }S, ( t ) 3 S o ( t ) for all t E J , if (i) and (ii) are replaced by (iii) Du(,)(x,t ) I G(u(x,r ) , t ) for all x E [S(t)-s,(t>] and t E J ; and (iv) p ( t 2 ;S U P , ~ ~ ~ ~ ( , , ) ~ ( X < ,infx.aS(,2) ~ , ) , ~ , )u(x,t,) for all t , > t,, t i , t 2 E J.

Proof. Since the proofs for stability and uniform stability are very similar, only the proof for stability is given. The proof is by contradiction. Let xo E So(to)and assume there exists t , E (to,00) for which x ( t 2 ;xo,to)$ S(t2). Since S ( t ) possesses Property P, there exists t , E (to,t , ] such that x ( t ; xo,to) E S ( r ) for all t E [to,t , ) and such that x(t,;x,, to) E dS(t,). Define r ( t ) as r ( t ) = u [ x ( t ; x o ,to),t ] (refer to Theorem 2.6.4) and write W t )

=

D v ( , , ( x ( t ;xo, to>,t ) I G [ u ( x ( t ;xo,to),t ) , t ] = G ( r ( t ) ,t ) .

Now r ( t o ) = u(x(to;xo,to),to) = u(xo,to) I S U ~ ~v(x, ~ to), ~ ~and( thus ~ ~ it, follows from Theorem 2.6.4 that u ( x ( t ; xo, to),t ) I p ( t ; S U ~ ~u ( x~, to), ~ to) ~ for all t E (to,t , ] . In view of hypothesis (ii) we can now write + ( t , ; x o , to),t l >

<

inf

xE

~ ( xt ,d .

But the above inequality implies that x(t,;xo,to) $ dS(t,), a contradiction. Hence, no r , as asserted exists and x ( t ; xo, t o ) E S ( t ) for all t E J . Note that from a computational point of view, the u-functions in Theorem 2.7.3 as well as in the remaining results of this section, may sometimes have less stringent requirements than those used in the usual Lyapunov theorems. Thus, in Theorem 2.7.3 there are in general no definiteness conditions on u and we do not require that u ( 0 , t ) = 0. 2.7.4. Theorem. System (I) is unstable with respect to {So(ro),S ( t ) ,t o } , S ( t o )3 So(to),if there exists a continuous function u : R" x J + R and a function G which satisfies the assumptions of Theorem 2.6.5 and a t , E (to,CO) such that

(i) Du(,,(x,t ) 2 G ( u ( x , t ) , t ) for all x E S ( t ) and t E J, (ii) q(t,; infx.SO(,O) u(x, to), t o ) 2 SUPxEas(tl)~ ( xt , A and (iii) v(x,t l ) < t , ) for all x E S ( t , ) where q ( t ;y o , to) is defined in Theorem 2.6.5.

(

~

~

)

66

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Proof, The proof of this theorem is similar to that of Theorem 2.7.3.

It is often convenient to view the right-hand side of Eq. (I) as the sum of two functions f :R" x J + R" and u : R" x J + R", i.e., i= f ( x , t )

+ u ( x , t ) 4k g(x, t ) .

(2.7.5)

Here we view u ( x , t ) as (deterministic) "persistent perturbation terms" and we call (2.7.6)

f =f(x,t)

the "unperturbed system." System (2.7.5) is a special case of systems described by equations of the form f =f(x,u(~,t),t)

h(x,t)

whereu:R"xJ+R', f:R"xR'xJ+R",and

(2.7.7)

h:R"xJ+R".

2.7.8. Definition. System (2.7.7) is totally stable with respect to { S O ( t O ) > S ( t ) ,D ( t > ,t o > ,

if the conditions (a) xo E So(to)and (b) u(x, t ) E D ( t ) whenever x E S ( t ) and t E J imply that x ( t ; xo,to) E S ( t ) for all t E J . System (2.7.7) is uniformly totally stable with respect to { S o ( t )S, ( t ) ,D ( t ) } if for all ti E J the conditions (a) x iE So(ti)and (b) u ( x , t ) E D ( t ) whenever x E S ( t ) and t E [ti,cn) imply that x ( t ; x i , t i ) E S ( t ) for all t E [ti, a). For system (2.7.5) we have the following result. 2.7.9. Theorem. System (2.7.5) is totally stable with respect to {So(to),S(f)?D ( t > ,t o > ,

S(r0) = S O ( t O ) ,

if there exists a continuously differentiable function u : R" x J + R and two integrable functions v : J - . R, q : J + R, such that (i) D v ( ~ , , , ~ )t () x<, v ( t ) for all x E S ( t ) and t E J ; (ii) Vo(x,t ) T 2 ~ q ( t ) for all u E D ( t ) , x E S ( t ) ,t E J; and Ciii) J:,"+~(~)I 5 infx.,,(,,u(x, t ) - s u P x t s o ( t o ) ~ (to). ~, System (2.7.5) is uniformly totally stable with respect to { S o ( t ) S, ( t ) ,D ( t ) } , & ( t ) c S ( t ) for all t E J , if (i)-(iii) are replaced by ~ )all X E [ S ( t ) - S O ( t ) ] and t E J ; (iv) D U ( ~ . , , ~<) v((Xt ),for c

_

2.7

67

ESTIMATES OF TRAJECTORY BEHAVIOR AND BOUNDS

(v) V u ( x ,t)'u I q ( t ) for all X . E [S(t)-s,(t)], u E D ( t ) , t E J ; and (vi) S:: Cv(7) + v (.r)I d7 I inf, as(t2)u(x, t 2 )- supx a s o ( t , ) v ( x , t l ) for t , > t,, t , , t , E J.

all

Proof. Since the proofs for total stability and uniform total stability are similar, only the proof for the former is given. The proof is by contradiction. Let x o E So(to)and assume that u ( x , t ) E D ( t ) whenever x E S ( t ) and t E J . Assume that there exists a t , E (to,m) for which x ( t , ; xo, to) 4 S(t,). Then, since S(t) possesses Property P, there exists t , E (to,t , ] such that x ( t ;x o , t o ) E S ( r ) for all t E [to,1 , ) and such that x ( t , ; x o , to)E a S ( t , ) . Now u(x(t1; xo, to), t l ) = 4 x 0 , to) +

Jr

u(x0,to)

=

+

lr

xo, to), t ) dt

Dq2.,.&(t;

~V~(~(t;xo,to),t)T~(X(t;Xo,to),t)

+ d u ( x ( t ; x 0 ,to),t ) / d t ] dr

In view of hypotheses (i)-(iii) we have u ( x ( t , : x , , to),t , ) <

E

x

E

< =

SUP

x

SO(l0)

l: +

u(x,t o ) +

sup u ( x , t o ) So(t0)

C v ( t > + W l dt

inf

u ( x , t , ) - sup u ( x , t o ) x E SO(t0)

x E as(t1)

inf u ( x , t , ) . asrf,)

But this inequality implies that x ( t l ;x o , to)4 dS(t,) which is a contradiction. Therefore no t , as asserted above exists and x ( t ; x o , to) E S ( t ) for all t E J . For the special case when u ( t ) 3 0, Theorem 2.7.9 yields a stability result with respect to {So(to), S ( t ) ,t o } ,and a uniform stability result with respect to { S o ( t )S, ( t ) } for the unperturbed system (2.7.6). In this case we take q ( t ) = 0 and delete hypothesis (ii) of Theorem 2.7.9 in its entirety. Turning our attention to composite systems, let us consider once more interconnected system (9") with decomposition (Cy), i.e.,

(m

ii

=fi(Zi,gi(zl,

...,zl,t),t)

= fi(zi, g i ( x ,t ) , t ) =

(9"")

2

=

h(x,t)

h i ( x ,t ) ,

i

=

1, ..., I,

(2.7.10) (2.7.11)

68

11 SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

where all symbols i n Eqs. (2.7.10) and (2.7.11) are as defined in Eqs. (2.3.20) and (2.3.21), respectively. However, in the present case we do not require that x = 0 be an equilibrium. In the following, all time varying sets possess Property P. The notation x E Xf= S j ( t ) signifies z j E S j ( t ) , j = 1, ..., I, and t E J . In particular, X;=, B j ( r j ) = {x E R” : lzjl < r j , j = 1, ..., I } . 2.7.12. Definition. Composite system (9”) with decomposition (Cy) is stable with respect to (X:= S o j ( f o )Xj= , S j ( t ) , t o ) if xo E Xi= Soi(to)implies that x ( t ; xo, t o ) E Xi= S j ( t ) for all t E J . Composite system (9”) with decomposition (Cy) is uniformly stable with respect to S,j(t),)(f=, S j ( t ) } if for all ti E J , x i E Soj(ti)implies that x ( t ; x i , ti) E S j ( t ) for all t E [ t i , m).

,

xf=

xi= ,

We are now in a position to prove the following result. 2.7.13. Theorem. Composite system (9”) with decomposition (Cy) is stable with respect to Soj(lo), S j ( t ) ,to>, S j ( t o )2 Soj(to),j = 1 , ..., I, if the following conditions hold. (i) g i ( x ,t ) E D i ( t )c Rri whenever x E Xi=, S j ( t ) and t E J ; (ii) each system described by

{xf=

xi= ,

i i =f i ( Z i , S i ( X ,

t),t )

(2.7.14)

is totally stable with respect to {Soi(to),S’(t),D i ( t ) ,t o } , Soi(to)c S’(to), i=

],...,I.

Composite system (9”‘) is uniformly stable with respect to

{

j = 1

Soj(t),

j= 1

I

Sj(t) ,

S j ( t ) 3 S O j ( t ) , j = 1, ...)I,

for all t E J, if (i) is true and if (ii) is replaced by the following. (iii) Each system described by Eq. (2.7.14) is uniformly totally stable with respect to { S o i ( t )S, i ( t ) ,D i ( t ) } So’(t) c S i ( t ) , i = 1, ...,I, and t E J . Proof. The proofs for stability and uniform stability are similar. Only the proof for stability is given. Let xo E Xi= Soj(to)c Sj(to). For purposes of contradicton, assume that at some t , E J , x ( t 2 ;xo,t o ) $ Sj(t,). Since x ( t ;xo,to) is continuous and since sets S j ( t ) , j = 1, ..., I, possess Property P, there exists a finite first time t , < t , such that

xi.=,

-

x;=

xf=,

Hence, x ( t ; xo,t o ) E S j ( t ) for all t E [to,t , ] . Now by hypothesis (i), gi(x,r ) E D i ( t ) whenever x E S i ( t ) and t E J. Thus, g , ( x ( t ;xo,to), t ) E

2.8 D i ( t ) , i = I , ..., 1, for all t

E

69

APPLICATIONS

[to,tl]. Next, define

i = 1, ..., I,andchooseu,(f)insuchafashionthatu,(t) E Di(t)forallt E (tl,m). Then clearly gi*(t) E D i ( t ) for all t E J . Corresponding to each equation describing (Ci), consider j i = .L(yi, gi*(t),t),

i

= 1,

...)1,

where the mappings fi, i = 1, ...,I, are defined as in (Ci). Let yio = zio. By hypothesis (ii), each system described by Eq. (2.7.14) is totally stable with respect to { S o i ( t o ) , S i ( t ) , D i ( tt)o, } , Soi(to)c S i ( t o ) , i = I , ...,I. Thus, y i ( t ;y i o ,to) E S ' ( t ) for all t E J . By causality (i.e., by uniqueness), z i ( t ;zio,to) = y i ( t ;yio,t o ) for all t E [ t o , t ! ] . Therefore, z i ( t ;zio,to) E S i ( t ) for all f E [ t o , t l ] and x ( t ; x , , t , ) E X ~ = , S ' ( r )for all t e [ t o , t l ] .But above we assumed that

Thus, we have arrived at a contradiction and there exists no t , E J such that x ( t , ; x o , to)E a Sj(tl)}. This in turn implies that there is no t , E J such that x ( t , ; x o , to) # Sj(t,), because each Sj(t) possesses Property P and Si(t) because of the continuity of x ( t ; xo, to). Therefore x ( t ;x o , to) E for all t E J.

{xi=

xf=

xi=

2.8 Applications

In order to demonstrate the usefulness of the method of analysis advanced in the preceding sections, we consider several specific examples. 2.8.1. Example. (Longitudinal Motion of an Aircraft.) The controlled longitudinal motion of an aircraft may be represented by the set of equations (see Piontkovskii and Rutkovskaya [l]) i k

=

& =

-pkxk 4

+

6,

k

=

1,2,3,4,

1 Pkxk- rpa -f ( o )

(2.8.2)

k= 1

where p k > 0, r > 0, p > 0, Pk are constants, where xk E R , o E R , and where + R has the following properties: (a) it is continuous on R , (b) f ( a ) = 0 if and only if a = 0, and (c) of(.) > 0 for all a # 0. We call any function f

f :R

70

11 SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

with these properties an admissible nonlinearity. If the equilibrium xT = ( x , , x,, x 3 , x 4 , 0 ) = 0 is asymptotically stable in the large for all admissible nonlinearities f, we call system (2.8.2) absolutely stable (see Aizerman and Gantmacher [ l ] , Lefschetz [ l ] ) . System (2.8.2) may be viewed as a linear interconnection of isolated subsystems (Y1)and (Y,)described by i k

k

= -fkXk,

=

1,2,3,4,

(91)

and 6 = -rpa -f(o).

Systems (Y,)and (Y2) are interconnected to form system (2.8.2) by means of the matrices CT, = [ l , 1 , 1 , 1 ] and C , , = [fi1,fi2,p3,fi4],where the notation of Eq. (2.3.14) is used. Without loss of generality we assume in the following that p1 I p, I p 3 I p4. Let z I T = ( x , , x 2 , x 3 , x 4 ) and z2 = 0 . For (Y,) and (9, choose ) u,(z,) = c,z:zl and u2(z2)= c 2 z Z 2 where , c1 > 0 and c, > 0 are constants. Then DU2(Y*)(Z2)I -2rpcz lz212,

IVU,(Zz>I

2c2Iz2l

for all z,E R4 and z2 E R . The norms of C , , and C , , induced by the Euclidean norm are 11 C , , 11 = 2 and 1 C , , / = Hypothesis (i) of Corollary 2.4.28 is satisfied. The test matrix S of this corollary is specified by

(xf=

s,, =

-2c(,c,p1,

s,, =

s,1

Choosing a, = 1/(4c,) and form

= c(,

2c(,c,

=

s,,

+

=

(,.

-2a2rpc,, i= I

p,.)"'.

1 / [ 2 c 2 ( x i 2 ,p ~ ) ' ' 2 ] matrix , S assumes the

This matrix is negative definite if and only if

2 ti2 < 4

1 3

ti =

(2Pi>/(p,p r ) .

(2.8.3)

i= 1

I t follows from Corollary 2.4.28 that the equilibrium x = 0 of Eq. (2.8.2) is exponentially stable in the large for any admissible function f if inequality (2.8.3) is satisfied.

2.8

APPLICATIONS

71

We can also apply Corollary 2.5.21. The test matrix D of this corollary is given by

Matrix D has positive successive principal minors if and only if inequality (2.8.3) is satisfied. Thus, with the above choice of a, and a,, Corollaries 2.4.28 and 2.5.21 yield the same result. Utilizing an approach of the type discussed in Section 2.6, Piontkovskii and Rutkovskaya [11 obtain the stability condition

c 45: 4

i= 1

<1

(2.8.4)

using the above Lyapunov functions u1(2,) and u2 (2,) as components of a vector Lyapunov function. Condition (2.8.3) is clearly less conservative than condition (2.8.4). 2.8.5. Example. Let us reconsider system (2.8.2). For (9,) choose u l ( z l ) = lz,l and for (9, choose ) u2(z2)=1z21. Then D U ~ ~ ~I, - ~ p l ( Iz,] Z ~ and ) D Z I , ( ~ ~ ) (5 Z ,-rplal. ) Note that u, and u2 are globally Lipschitzian with L1 = L, = 1. Hypothesis (i) of Theorem 2.5.1 5 is clearly satisfied. Hypothesis (ii) of this theorem is also satisfied with a,, = u2, = 0, a 1 2= 2 and a,, = The test matrix D of this theorem assumes the form

(xf=

This matrix has positive successive principal minors if and only if the inequality 4

(2.8.6)

is satisfied. It follows from Theorem 2.5.15 that the equilibrium x = 0 of system (2.8.2) is uniformly asymptotically stable in the large if inequality (2.8.6) is true. Thus, with the preceding choices of Lyapunov functions and weighting vector, Corollaries 2.4.28 and 2.5.21 and Theorem 2.5.15 yield the same stability condition for system (2.8.2).

2.8.7. Example. Let us alter system (2.8.2) by assuming thatf((a) is given by f(0) = a((a2-a2),

a > 0,

(2.8.8)

72

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

which is not an admissible nonlinearity in the sense of Example 2.8.1. Specifically, in this case we have af(o) > 0 for all 101> a, lim,o,+mof(o) = 03, f ( 0 ) = O, f(a) = O,f( - a ) = 0, and ~ - f ( o<) 0 for all 0 < 101 < a. Utilizing either Theorem 2.4.34 or 2.5.26 and proceeding in an identical fashion as in Example 2.8.1, we conclude that all solutions of system (2.8.2) withf(a) specified by Eq. (2.8.8) are uniformly bounded, and in fact, uniformly ultimately bounded if inequality (2.8.3) is satisfied. 2.8.9. Example, (Indirect Control Problem.) An important class of problems arising in automatic control theory is the indirect control problem (see, e.g., Aizerman and Gantmacher [l], Lefschetz [l]) characterized by the set of equations

+ bf(0)

1

=

Ax

6

=

- p 0 - rf(0)

+ aTx

(2.8.10)

where x E R", (T E R, A is a stable n x n matrix (i.e., all eigenvalues of A have negative real parts), b E R", p > 0, r > 0, a E R", and f : R + R has the following properties: (a) it is continuous on R , (b) f ( a ) = 0 if and only if r~ = 0, and (c) 0 < of(a) < ka2 for all 0 # 0, where k > 0 is a constant. System (2.8.10) is called absolutely stable if its equilibrium (xT,a) = 0 is asymptotically stable in the large for any admissible nonlinearity f with the above properties. System (2.8.10) may be viewed as a nonlinear interconnection of isolated subsystems (9,) and (Y;)described by

1

and

6

=

=

AX

(91

-pa - rf(0).

)

(92)

Using the notation of Eq. (2.3.16), these subsystems are interconnected to form composite system (2.8.10) by means of the relations g , 2 ( 0 )= f ( o ) b and q z l(x) = a'x. In the subsequent analysis, as well as in later examples, we utilize the following well-known result (see, e.g., Hahn [2, p. 1171). 2.8.1 1. Theorem. Let y

E

R", let B be ann x n matrix, and consider the equation j,

=

(2.8.12)

By.

If all eigenvalues of B have negative real parts or if at least one eigenvalue has positive real part, then there exists a Lyapunov function of the form v ( y ) = yTPy,

PT = P

(2.8.13)

-Y'G

(2.8.14)

such that DU(2.&?.12)(Y) =

is definite, where

2.8

APPLICATIONS

73

-C

=

BTP i- PB.

(2.8.1 5)

Thus, if the conditions of Theorem 2.8.1 1 are satisfied, then it is possible to construct a Lyapunov function v( y) by assuming a definite matrix Cand solving the Lyapunov matrix equation (2.8.15). Returning to the problem on hand, since A of (9,) is a stable matrix it follows from Theorem 2.8. I 1 that there exists a function u1 : R" --t R and constants c l i > 0, i = 1,2,3,4, such that C111Xl2

-C13IXl2, D~,(,,,(X) I

I v1(x) 5 c121x12,

IVv,(x)I I c141xI

for all x E R". For (9,we ) choose v,(o) = a2/2. Then D Y , ( ~ ~ ) I ( O- p) and IVu,(o)l= lo1 for all o E R. Hypotheses (i)-(iii) of Corollary 2.4.25 are now satisfied with k,, = k Ibl and k,, = lal. Choosing ctl = l / ( k lbl) and tl, = c14/lal,matrix S of Corollary 2.4.25 assumes the form

This matrix is negative definite if and only if

<

(PCl3>/(bl

Ibl c14).

(2.8.16)

It follows from Corollary 2.4.25 that the equilibrium (xT,o)= 0 of composite system (2.8.10) is asymptotically stable in the large (i.e., system (2.8.10) is absolutely stable) if inequality (2.8.16) is true. We can apply Corollary 2.5.20 as well. The test matrix D of this corollary is given by

Matrix D has positive successive principal minors if and only if inequality (2.8.16) is satisfied. Therefore, with the above choice of CI, and a,, Corollaries 2.4.25 and 2.5.20 yield the same stability result. Using an approach of the type discussed in Section 2.6 involving Lyapunov functions u l ( x ) and v,(a) as components of a vector Lyapunov function, Piontkovskii and Rutkovskaya [1) obtain the stability condition k < C(Pcl3>/(lallbl C14)l ( ~ l l / ~ l z ) 1 ' 2 .

(2.8.17)

~ ~I~, it follows that condition (2.8.17) is in general more Since ( c , , / c , ~ )I conservative than condition (2.8.16).

74

11 SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

To determine the constants ~ 1 and 3 c14 in (2.8.16), let v ( x ) = xTPx with P T = P, so that Dv,(su,) (x) = - xTCx. Given a positive definite matrix C, we can solve for P by solving the Lyapunov matrix equation - C = A T P + PA. In so doing we obtain &(P) 1x1, I u1 (x) I A,(P) [XI,, Dv,(,,,(x) I -A,(C)lx12, and I V v , ( x ) ) 1 2 A M ( P )1x1. Inequality (2.8.16) assumes now the form k < :[Am(c>/AM(P>l[P/(bl lbl>1. (2.8.18) To obtain the least conservative result, matrix C needs to be chosen in such a fashion as to maximize A,(C)/A,(P). 2.8.19. Example. Choosing an appropriate nonsingular linear transformation, we can represent the indirect control problem of Example 2.8.9 equivalently by the set of equations 1, = A , x l f 2

+ b,f(o) + b2f(o) - rf(o) + alTxl+ a2Tx2

(2.8.20)

= A 2 ~ 2

6 = -po

where x, E R"',x 2 E R"', A , is an n , x n , matrix, A , is an n, x n 2 matrix, b , E R ' ,6 , E R"', a , E R"', a , E R"', n , +n, = n, and the remaining symbols are as defined in Example 2.8.9. System (2.8.20) may be viewed as a nonlinear interconnection of three isolated subsystems (Y,),(Y,),(Y3) of the form 2, = A , x , f, = A 2 x Z

6 = - p o - rf(0).

Using the notation of Eq. (2.3.16), these subsystems are interconnected to form composite system (2.8.20) by means of the relations g12(x2) = Y21 ( X I > = 0, g13(a) = f ( a ) b l , g 2 3 ( o ) =f(a)b2> g31(xl) =alTxI, and g32b2)

= a2Tx2.

Assume that all eigenvalues of A , have positive real parts and that A , is stable. In accordance with Theorem 2.8.1 I , there exist 0 , : R"' + R , v,: R 2+ R,and positive constants cij,i = 1,2 and j = I , 2,3,4, such that -c11Ix1l2

I u1(x,) I -c121x12>

DOi,y,j(xi)

5

-Ci31xi12,

IVvi(Xi)l I C i 4 1 x i 1 ,

c211x212

5

u2(x2)

Dv2(Y2)(x2)

1vv2(x2)1

for all x, E R ' and for all x2 E R"'. For (Y3) choose u3(0) = 40,. Then Dv3,,3j(0) I 1 ~ for 1 all o~ R.

5 Czzlxz12,

I -c23

IX2l29

I Cz41x21,

and IVv3(o)l=

~ J O ) ~

2.8

75

APPLICATIONS

Using the notation of Theorems 2.4.31 and 2.5.24, we obtain a 1 3= c14k Ib, 1, = c2,klb21, a 3 L= lal[, a32 =la,/, and a12= a2*= a l l = a 2 2= a33= 0. Hypotheses (i) and (ii) of Theorem 2.4.31 are thus clearly satisfied. The test matrix D of Theorem 2.5.24 assumes the form

a23

0

-Ci4k\bil

0

c23

-c24klb1

-la11

-la21

3

D = [

1

.

P

This matrix has positive successive principal minors if and only if the inequality c13 c 2 3

k < c23 C i 4

la1 I h

P

h

3 C24

1%

(2.8.21)

lb21

is satisfied. It follows from Theorem 2.5.24 that the equilibrium xT = (xlT,x ~cr) = ~ 0, is unstable (for all admissible nonlinearities , f ) if inequality (2.8.21) holds. 2.8.22. Example. (System with Two Nonlinearities.) Consider the system described by the set of equations 5, 0 1

= AIZl

=

+ b,fl(fJl),

i 2 =

T c1 z2,

A,z,

+ b,f2(cr2),

(2.8.23)

6, = C2TZ1,

where for i = 1,2, zi E RnL,Ai is a stable ni x ni matrix, c1 E R"', c2 E R"', and f i : R + R has the following properties: (a) it is continuous on R, (b) fi(ai) = 0 if and only if cri = 0, and (c) 0 < crifi(cri) < kicri2 for all cri # 0, where k i > 0 is a constant. System (2.8.23) may be viewed as a nonlinear interconnection of two linear isolated subsystems (9,) and (Y2), (91)

i, = A , z , , i 2

=

A2~2,

(92)

which are interconnected to form composite system (2.8.23) by means of the relations g12(z2)= b , f l (crl) and gZ1(z1)= b 2 f 2 ( 0 2 )where , the notation of Eq. (2.3.16) is used. Since A, and A, are stable matrices there exist ui: Rni-+ R and constants cij > 0, i = 1 , 2 , j = 1,2,3,4, such that 2 cil /zi/

IUifzi)

s

ci2 IziI2,

2

Dui(yi)(zi)I -ci3 [zi)

3

for all zi E Rni.Thus, hypotheses (i) and (ii) of Corollary 2.4.25 are satisfied.

76

11 SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Hypothesis (iii) is satisfied with k , , = k , JbllIc, 1 and k,, = k , Ib,J Jc,J. Choosing c ( ~= ~24/(k1[clJlb, 1) and c(, = ~,4/(k2Ic21 lb21), matrix S of Corollary 2.4.25 assumes the form

This matrix is negative definite if and only if k J k2

<

( C 1 3 C 2 3 ) / ( C 1 4 C 2 4 / b l / lb21 ('I/ 1'21).

(2.8.24)

It follows from Corollary 2.4.25 that the equilibrium xT= (xIT,x , ~ ) = 0 of system (2.8.23) is exponentially stable in the large if (2.8.24) holds. Following an approach of the type discussed in Section 2.6 and using the above Lyapunov functions vl(zl) and v 2 ( z 2 ) as components of a vector Lyapunov function, Piontkovskii and Rutkovskaya [I] obtain the inequality

as a sufficient condition for exponential stability. Since (cl, C , ~ ) / ( C ~ , c,,) 5 I , condition (2.8.25) is in general more conservative than condition (2.8.24). 2.8.26. Example. The purpose of the present simple example is to provide a case where the equilibrium is asymptotically stable but not necessarily exponentially stable and to present an example where the results of Section 2.5 involving M-matrices do not predict stability while those of Section 2.4 do predict stability, using identical Lyapunov functions for the subsystems. Specifically, consider

(2.8.27) where x , E R and x, E R . This system may be viewed as a nonlinear interconnection of isolated subsystems (Y,),(Y2),

f,

= -x13,

x2

=

-x,

5

(91)

.

(92)

Choosing u l ( x , ) = x i 2 and v2(x2)= x22,we have D u l ~ y l ~ ( x=l -2x14 ) and D U , ~ , , , ( ~=, -) 2 ~ Using ~ ~ .the notation of Theorem 2.4.2 we make the identifications $, 1 ( ~ )= +h12(y) = r 2 , $ , 3 ( ~ )= r4, I ) ~ ~ ( Y ) = $ 2 2 ( ~ )= r 2 , and $ 2 3 (Y) = r'. The interconnecting structure of Eq. (2.8.27) is characterized by g1(x,,x2)= - 1 . 5 )x2I3 ~ ~ and g2(x1,x2)= x 1 2 x 2 , ,where the notation of

2.8

APPLICATIONS

77

Eq. (2.3.10) has been used. We now have v ~ , ( x l ) g l ( ~ l , x 2=) (2xl)(- 1.5X,)ix,1~= I X ~ I ~ ( - ~ ) I X ~ I ~ =

$1 3

(1x11)1’2

(- 3) $23 (lxzl)l/z

and ~ ’ ) iX1l2 v ~ z ( X 2 ) ~ 2 ( x , , x=2 )( ~ x ~ ) ( x ~ ~5x1x2I3(2) =

$23(1x21)1’2(2)$13(Ix11)1~2.

In the notation of Theorem 2.4.2 we now have a , , = a Z 2= 0, a,, = -3, uZ1= 2, 0, = - 2, and o2 = -2. Choosing C L ~= u2 = I , matrix S of Theorem 2.4.2 assumes the form

Since S is negative definite it follows from Theorem 2.4.2 that the equilibrium (xl, x , ) ~= x = 0 of Eq. (2.8.27) is asymptotically stable in the large. However, since the comparison functions $,, i = I , 2, j = 1,2,3, are not all of the same order of magnitude, we cannot conclude that the equilibrium is exponentially’ stable (see Theorem 2.4.20). Next, note that - S has positive off-diagonal terms and thus the results of Section 2.5 are not directly applicable. However, using the same Lyapunov functions as above, let us attempt to establish stability, using a test matrix D = [d,] for which d, < 0, i # j . In the notation of Theorem 2.5.1 1 we have the estimates VUl (Xd9, ( X l ? XZ)

$13(lx11)1~2(3)~23(Ixz1)1~2

vv, (xz) 9 2 (XI

$23

3

x2)

(1x2

(2) $ 13 (1x1

Hypotheses (i) and (ii) of Theorem 2.5.1 I are now satisfied. The test matrix of hypothesis (iii) assumes the form 2 .=[-2

-3 2

]

Since the successive principal minors of D are not both positive, Theorem 2.5.1 1 fails to predict asymptotic stability. Thus, using the same Lyapunov functions u1 (z,) and u 2 ( z z )we were able to predict stability for system (2.8.27) using Theorem 2.4.2, but failed to do so using Theorem 2.5.1 1. 2.8.28. Example. (The Hicks Conditions in Economics.) Let p i 2 0, i = I , ...,n, denote the prices of n interrelated commodities supplied from the same or related sources which are demanded by the same or related industries

78

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

and let fi(pl, ...,pn) denote the excess demand function of commodity i. Such a multimarket system may be represented by the equation P =f(p)

(2.8.29)

where pT = ( p l , ...,p,) and where f: R n - t R" is assumed to be continuously differentiable with respect to all of its arguments. Assume that p > 0 is an isolated equilibrium price vector, so that f(p) = 0. Linearizing Eq. (2.8.29) about the equilibrium p , we obtain the set of equations

D~ = C a,(pj-pj), n

i = 1 ,..., n,

j= 1

(2.8.30)

where a,. = [dfi(p)/dpj],=a. Here we need to restrict ourselves to "small" initial deviations from the equilibrium. As well as providing mathematical simplicity, this restriction has the advantage of assuring us that in the presence of asymptotic stability, no price will become negative on its path to equilibrium, since it is never far from the equilibrium which we have already assumed to be positive. Here we are invoking theprinciple of stability in thefirst approximation (see Hahn [ 1, p. 1223) which asserts that we can deduce the stability properties of the equilibrium p of Eq. (2.8.29) from Eq. (2.8.30), provided that A = [aij] is not a critical matrix (i.e., A has only eigenvalues with negative real parts or at least one eigenvalue with positive real part). Letting xi = pi- pi, Eq. (2.8.30) assumes the form 1. = a..x. ,I I +

2

aijxj,

j=l,i#j

i = 1, ..., n.

(2.8.31)

Assuming that all commodities are gross substitutes for one another, we have the conditions ay 2 0, i # j . We also make the realistic assumptions a i i < O , i = l ) ...) n. Noting that the asymptotic stability results of Sections 2.4 and 2.5 can be modified in an obvious way to yield local conditions rather than global ones, we may view system (2.8.31) as a composite system of n isolated subsystems

(Z)>

x.L = a..x. 11 I )

(%)

which are interconnected by means of the relations gi(x) = Cl=l,i # j a i j ~ j , xT= ( x l ,..., x,,), where the notation of Eq. (2.3.10) is used. For each isolated subsystem we choose

(x) &)

=

(Xi(.

Note that vi is Lipschitzian with Li = 1. Along solutions of (q) we have Dui(yi)(xi) = aii IXil. We are now in a position to apply Theorem 2.5.15. The test matrix D

= [d,]

2.8

79

APPLICATIONS

of this theorem is specified by

4

laii[,

d.. =

-/ail,

i = j

i f j.

It follows from Theorem 2.5.15 that the equilibrium x = 0 of Eq. (2.8.31) (and hence, the equilibrium p of Eq. (2.8.29)) is asymptotically stable if the successive principal minors of matrix Da re all positive. To put it an equivalent way, the equilibrium price vector p of Eq. (2.8.29) is asymptotically stable if a11

a12

..'

alk

a21

a22

..'

a2k

.

...

.

> 0,

k = 1, ..., n.

(2.8.32)

Conditions (2.8.32) are called the Hicks conditions in economics (see, e.g., Quirk and Saposnik [ 11, Metzler [l], McKenzie [I]). Since D is an M-matrix, we can express these conditions equivalently by requiring the existence of real constants Ai > 0, i = 1, ...,I, such that the inequalities laiiJ-

C n

j = 1, i + j

(Aj//$) laji[> 0,

i = I , ...,n,

(2.8.33)

are true (see Corollary 2.5.3). 2.8.34. Example. (Linear Resistor-Time Varying Capacitor Circuits.) A large class of time varying capacitor-linear resistor networks (see Sandberg [ 101,

Mitra and So [I]) can be described by equations of the form

+ [ADl(t)+BD,(t)]x= b(t)

(2.8.35)

where x E R", A = [ a @ ]and , B = [bi] are constant n x n matrices with aii > 0 and bii > 0, i = 1, ...,n, D , ( t ) and D , ( t ) are diagonal matrices whose diagonal elements [ D l ( t ) l i i , [D2(r)liiare continuous nonnegative functions on J = [ t o , a),and b ( t ) is an n-vector bounded and continuous on J . Henceforth it is assumed that [ D 1 ( t ) l i j + [ D z ( t ) l i2i 6 > 0, i = I,...,n for all ~ E J . Presently we are interested in the asymptotic stability of the equilibrium x = 0. For this reason we let b ( t ) = 0 for all t E J and consider the free or unforced system f

+ [ A D , ( t ) + B D , ( t ) ] x = 0.

(2.8.36)

This system may be viewed as n isolated subsystems (Sq), f i = - { a i i [ D , ( t ) ] i i + b i i [ D 2 ( t ) ] i i }4 xir n i i ( t ) X i ,

(Sq)

11

80

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

which are interconnected by the relations g i ( x j , t ) (using the notation of Eq. (2.3.16)), g , j . ( x j , t ) = - ( a ~ [ ~ l ( t > I j j + b , j . C D z ( t ) I j j >4 x jm , ( t ) x j ,

i ~

j

,

where r n g ( t ) is defined in the obvious way. For (q) we choose

u i ( x i )= ailxil where Ai > 0 is a constant. Note that ui is Lipschitz continuous for all x i E R with Lipschitz constant Li = A i . Along solutions of (q) we have t ) = &mii(l)Ixil.

Dui(yi)(xi,

It now follows from Corollary 2.5.16 that the equilibrium x = 0 of Eq. (2.8.36) is uniformly asymptotically stable in the large if I1

~ , ( t ) Irnii(t)l -

C

i=l,i#j

(Ai/Aj) Imi(t)l 2 y > 0,

.j = I ,

...,n. (2.8.37)

Now assume that

,,

0.. -

C I1

(Ai/Aj)/ail 2

8 > 0,

bjj

C n

-

(AJAj)/bc/2 6 > 0,

i = 1, i # j

i = 1, i#j

j = 1,

...,Iz.

(2.8.38)

Applying the definition of m G ( t )and the properties of D ,( t ) , D , ( t ) to (2.8.37), we obtain n

ajj-

1

i=l,i#j

(&/Aj)laijI

2 a [ D l ( t ) + D 2 ( t ) l j j2

6 6 > 0.

Therefore, the equilibrium x = 0 of the free system (2.8.36) is uniformly asymptotically stable in the large if inequalities (2.8.38) are true. I t is interesting to note that stability conditions (2.8.38) were obtained by Mitra and So [ I ] (see also Sandberg [lo]) by methods which differ significantly from the present approach.

2.8.39. Example. Using Theorem 2.7.3, we now establish the following estimate for the trajectory behavior of composite system (2.8.35). Assume that for unforced system (2.8.36) there exist constants Ai > 0, i = 1 , ..., n, and 8 > 0, such that the inequalities (2.8.38) hold. There is no loss of generality 2;' = 1. Also assume that for forced system (2.8.35), in assuming that

xy=

2.8

81

APPLICATIONS

x:=

AiIbi(t)l I k for all t E J, where b i ( t ) denotes the ith component of vector b ( t ) . Let c = &'h,where 6 > 0 is defined in Example 2.8.34. If a > k/c and C;= lilxiol I a for the initial vector xo = ( x l 0 ,..., xn0)', then for t 2 to,

+

(2.8.40)

Ix(t; xo,to)l I [a- k / c ] e-c(r-ro) k / c .

To obtain this estimate, we choose again u ( x ) = C:= lilxil. Simple computations yield -cv(x>

oq2.8.35)(x)

+ k,

where the constants c and k are defined above. In Theorem 2.7.3 let G(u,t ) = - cu + k. Solving the equation ij = G ( P , t )

we obtain

p ( t ; p o , t o )= (po-kk/c)e-"(r-'o)+ k/c.

Choosing

i

So(?,) = x

and

=(xl,

..., X,)T

S ( t ) = B([a-k/c]

recalling that

x:= l i 2

=

x

1

i= 1

e-'('-'O)

+k/c),

1 by assumption, and noting that

inf u(x) = (a - k / c )e-'('-'O) E

S(')

+ k/c

it is clear that all hypotheses of Theorem 2.7.3 are satisfied. Thus, estimate (2.8.40) follows directly from the properties of the boundary of S(r), d S ( t ) . It is interesting to note that estimate (2.8.40) was obtained by Mitra and So [I] (refer also to Sandberg [lo]) by quite a different method. Observe that in Example 2.8.34, the Lyapunov function u ( x ) = lilxil and the derivative B z + ~ , ~ . can ~ ~ )be( xestimated ) by comparison functions I,!12,I,!13E K R which are of the same order of magnitude. In accordance with Theorem 2.2.26, we can actually deduce that the unforced system (2.8.36) is exponentially stable in the large. Indeed, letting b ( t ) = 0, we obtain from inequality (2.8.40) the estimate

xy=

Ix(t;xo,to)l I ae-'('-'O),

t 2 to,

n

C lilxiolI a.

i= 1

2.8.41. Example. (Nonlinear Transistor-Linear Resistor Networks.) Consider

nonlinear time-varying systems described by x

+ A f ( x , t )+ B g ( x , t ) = b ( t )

(2.8.42)

82

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

where x E R", A = [ae], and B = [b,] are constant matrices with aii > 0 and bii > 0, where f:R" x J + R" and g : R" x J + R" are continuously differentiable in x and continuous in t , where f ( x , t ) = 0 and g(x,t ) = 0 for all t E J if and only if x = 0, wheref,(x, t ) = f i ( x i , t ) and g i ( x ,t ) = gi(xi, t ) , and where b ( t ) is a bounded continuous n-vector. Henceforth it is assumed that , 2 6 > 0 for all xi # 0, t E J, and that [ A ( x i ,? ) / x i ]2 6 > 0, [ . q i ( x i ?)/xi] [LJJ(xi, t)/d~~]l,,=~ 2 q > 0, [ d g i ( x i ,t)/d~~]l,,=~ 2 v] > 0 for all t E J . Presently, the asymptotic stability of the equilibrium x = 0 is of interest. For this reason we let b ( t ) = 0 for all t E J and consider the free system

1 + Af(x,t )

+ Bg(x, t ) = 0.

(2.8.43)

Equation (2.8.42) can be used t o model a great variety of physical systems. For example, a class of nonlinear transistor-linear resistor networks is described by Eq. (2.8.42), where the functions , J ( x i ,t ) = f i ( x i ) ,g i ( x i ,t ) E gi(xi) are assumed to be monotonically increasing in xi(see Sandberg [lo]). The free system (2.8.43) may be viewed as n isolated subsystems (Yi),

(8)

li = m i i ( x i , t ) x i

which are interconnected by the relations g i i ( x i ,t ) (using the notation of

XI

# 0,

(2.8.44)

XI =

0. (2.8.45)

I t follows from Corollary 2.5.16 that the equilibrium x = 0 of Eq. (2.8.43) is uniformly asymptotically stable in the large if

for all s E R" and t

E

J . Applying the definition of m e ( x j ,t ) and taking the

2.8

83

APPLICATIONS

properties of f i ( x i ,t ) and g i ( x i ,t ) into account, it follows similarly as in Example 2.8.34 that the equilibrium x = 0 of unforced system (2.8.43) is uniformly asymptotically stable in the large if there exist constants Ai > 0, i = 1, ...,n, such that a,.JJ

i= 1, i # j

i = 1, i # j

,j = 1 , ...,n.

(2.8.47)

Following a procedure similar to that of Example 2.8.39, it is also possible to obtain an estimate for I x ( t ; xo,t o ) [ , by invoking Theorem 2.7.3. Interestingly, the stability conditions (2.8.47) were obtained earlier by quite different methods (see Mitra and So [I], Sandberg [IC]). 2.8.48. Example. Consider the composite system described by the set of

equations

I

(2.8.49)

where t E J, zi E R"', and A i ( t ) and C g ( t )are real continuous n i x ni and nix n j matrices, respectively. Let Zf= ni = n, let xT= (zIT,..., zlT), and let g i ( x ,t ) = C:.= C , ( t ) z j , so that ii = Ai(t)Zi+ g i ( X , t ) ,

i

= 1,

...)I.

(2.8.50)

This system is clearly a special case of system (9") with decomposition (C;). Now assume that IICii(t)ll< 1, i , j = I, ..., I, for all r E J . Choose S i ( t )= @(pi) = {zi E Rn1: lzil < p i } , Soi(t)E Bi(ai), 0 < ai< pi,i = 1, ...,I, and 1

gi(x, t ) : l g i ( x ,t ) l <

1 pi

j= 1

for all x E

1

X

j = I

B'(Bj) and t E J

1

.

Since Igi(x,t)l=IC:.=lCg(t)zjIIC:.=lCCij(t)ll lzjl
xf=

xi=

(2.8.51) for all t , < t , , t , , t , E J .

84

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Direct application of Theorem 2.7.13 yields now the following result. Composite system (2.8.49) is uniformly stable with respect to

for all t E J and if inequality (2.8.51) holds for each i. It is possible to generalize the preceding result. Let ai: J + R', pi: J + R + be continuous functions and remove the restriction IICij(t)ll< 1. Following a similar procedure as before, we can show that composite system (2.8.49) is uniformly stable with respect to Bj(aj(t)), B j ( p j ( t ) ) } ,0 < 6 I ccj(t)< p j ( t ) , j = 1 , ...,I, t E J, if for each i the inequality

xi=

{Xi=l

1;

{AM

1

~wi(t>l + ( l / a i ( t > ) 11cij([)11 pj(t> j = 1

is satisfied for all t , < t,, t , , t ,

E

J.

dt < l n ~ ~ i ( t ) / a i i t > l

2.8.52. Example. (Multiple Feedback Systems: Direct Control Case.) We begin by considering I isolated subsystems described by the set of equations

(x)

i = I , ..., I, where zi E R"', A , is a stable n, x ni matrix, 6, E Rni,c, E R",,dii E R , ni E R and,fi: R -+ R is a continuous function such thatfi(a,) = 0 if and only if n, = 0 and 0 < aifi(ai)I k i a i 2 for all ni # 0, where k i > 0 is a constant.

I n this case we call f, an "admissible nonlinearity." System (q) is known as the direct control problem (see, e.g., Aizerman and Gantmacher [I], Lefschetz [ 11). Next we consider the composite system

ii

= A i z i - bifi(ai)

yi

z

CiTZ.

(2.8.53)

where yT = (yl, ..., y , ) E R' and d: = ( d i l ,..., dil)E R'. Let let xT = (zlT, ..., 2:) E R", and assume that x = 0 is the only equilibrium of system (2.8.53). It is important to note that in this example the interconnecting structure of composite system (2.8.53) does not enter additatively into the system description. Given as above, system (2.8.53) is not a special case of composite i = 1 , ..., I ,

zi=,n, = n,

(x)

2.8

APPLICATIONS

85

system ( Y )with decomposition (Xi). However, it is a special case of composite system (9") with decomposition (Z;). Let us first apply Theorem 2.5.22. We choose Ui(Zi)

=

Z?PiZi

where Pi = P? is a positive definite matrix which we will need to determine for some particular choice of a positive definite matrix Ci = C : via the Lyapunov equation

-ci

=

A?Pi

+ PiAi.

(2.8.54)

We have Arn(pi)/zi12 5 ui(zi) _< A,(pi) /zi12> and using the notation of Theorems 2.4.29 and 2.5.22, we have

Hypotheses (i) and (ii) of Theorem 2.4.29 are now clearly satisfied. The matrix D = [d,] of Theorem 2.5.22 is given by

From Theorem 2.5.22 it now follows that the equilibrium x = 0 of composite system (2.8.53) is exponentially stable in the large for every admissible nonlinearity (i.e., system (2.8.53) is absolutely stable) if all principal minors of matrix D are positive. In order to obtain the least conservative results, we need to choose matrix Ci so that the ratio A,(C,)/A,(P,) is maximized. Finally it should be noted that Eq. (2.8.53) can serve as a model of a large class of practical systems endowed with multiple nonlinearities. 2.8.55, Example. We have pointed out before that in a sense the method of

analysis advanced herein is more important than the individual results of the preceding sections. To demonstrate this, we reconsider system (2.8.53) in an attempt to obtain less conservative stability conditions. This time we choose for each a Lyapunov function of the Lur6 form (see, e.g., Aizerman and Gantmacher [11, Lefschetz [l]),

(x)

(2.8.56) where Pi= P? is a positive definite matrix and (3, > 0 is a constant. Clearly, ui(zi)is positive definite and radially unbounded.

II

86

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

For composite system (2.8.53) we choose a Lyapunov function of the form

c I

v(x)

=

i= 1

(2.8.57)

LYiDi(Zi)

where aT = (a1,... ,a,) > 0 is a weighting vector. This function is positive definite for all x E R" and is radially unbounded. Along solutions of Eq. (2.8.53) we have I

1

i= 1

i= 1

+ 1 aibi&(ai) c d,cTAjzj where Ci = -(A:Pi+

I

I

i= 1

j = 1

1

I

PiA i ) .Adding and subtracting the nonnegative quantity

Ef=lai(ai-(fi(ai)/'i))fi(ai>to D v ( 2 . 8 . 5 3 ) ( ~we ) > obtain

where wT = (zIT,. .., z1T, fl (al),. ..,A(a,)), where

is an n x n matrix, where S s.. =

= [sJ

is an n x I matrix, where each sG,

+A:CidiiaiPi + &&ai

.

z =J

- PibiUi,

.

i#j is an n,-vector, and R r.. =

=

[rJ is an I x 1matrix, and where

- ai pidiiC:bi - (Ui/ki),

i=j

-3(aiBidijcj7bj+aiPjdjic:bi),

i

+j.

It now follows that system (2.8.53) is absolutely stable if the symmetric ( n + / ) x ( n + / ) matrix (2.8.58) is negative definite. I t is interesting to note that this result is somewhat similar to one obtained by Lefschetz [I], using quite different methods.

2.8

APPLICATIONS

87

Under suitable conditions, frequency domain techniques can be used to construct Lyapunov functions of the form (2.8.56) for subsystem (q)(see Bose [l], Bose and Michel [l, 21). If the pairs ( A i , b i )and (A?,ci) are completely controllable and completely observable, respectively, (see Kalman, Ho, and Narendra [l]) then the Popov criterion (see Popov [ l ] ) along with additional reasonable conditions guarantee the existence of a Lyapunov function for (Yi) of the Lurk type (by the Kalman-Yacubovich Lemma) whose derivative is negative definite (see Kalman [l], Yacubovich [l]). In fact, an algorithm based on the proof of the Kalman-Yacubovich Lemma can be implemented on the computer to construct the Lyapunov function (2.8.56). However, this approach is not particularly satisfactory, for it is not explicit. We will make extensive use of graphical methods (including Popov-type conditions) in Chapter VI, where we establish results which are much easier to use and which are applicable to a larger class of problems. 2.8.59. Example. (Multiple Feedback Systems: Indirect Control Case.) Consider systems described by the set of equations i i =

A i z i - bifi(oi)

ki = diTy =

c d..y .

j= 1

B

(2.8.60)

J’

i = 1, ...,I, where y i > 0 is a constant and all other symbols are defined similarly as those in Eq. (2.8.53). In addition, assume that lim,o,+mJ;A(q)dq = a. This system may be viewed as a nonlinear interconnection of isolated subsystems (Sq.), 2. = A 1. z1. - b1. f I. ( o1. ) ki = dii[~;zi-Yif,(oi)],

(.4”i>

i = 1, ... , I . For each (q) we choose a Lurt type Lyapunov function

v i ( z i ci) , = z?Pi zi

+ Pi l i f i ( q ) dq

(2.8.61)

where Pi = PT is a positive definite matrix and pi > 0 is a constant. For system (2.8.60) we choose a Lyapunov function of the form v(x,a) =

I

CliUi(Zi,fJi) i= 1

(2.8.62)

where oT= (ol,..., ol), xT= ( z l T ,. .., z L T )and , uT = ( a l , ..., a,)> 0. This function is clearly positive definite and radially unbounded. Along solutions of

88

11

SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

Eq. (2.8.60) we have

where w is defined as in Example 2.8.55, of (2.8.58), 3 = [S,] is given by s.. -v =

and

w

=

[TJ

c" is of the same form as submatrix C

[

+ c i d i i a i / j i- aiPibi,

i

12 c1. dJ .I . J./j. J'

i # .i,

=j

is specified by F . . = + ( g .I / j .I d .y .Y j + a J. / j j. dj.i .Y i ).

Noting that (xT,oT)= 0 is the only invariant subset of the set E = {(xT, )'a E R"" : h ( , , 8 , , 0 ) ( x , o) = 0}, it follows from Theorem 2.2.30 that the origin (x', )'a = 0 of system (2.8.60) is asymptotically stable in the large for all admissible nonlinearities if the matrix

is negative definite. This result is somewhat similar to one obtained by Lefschetz [l], using a different approach. It is also interesting to note that Pai and Narayan [ I ] have used a set of equations similar to Eq. (2.8.60) and a Lyapunov function of the form (2.8.62) to analyze multi-machine power systems.

2.8.63. Example. (Stabilization.) Consider systems described by equations of the form

ij = A,zi

+ C 1

C,zj,

i

j=l,i+j

=

I , ..., I,

(2.8.64)

where z , E Rnt,A i is an n, x n , matrix, and C, is an n , x n j matrix. Suppose this system, called an uncompensated system, is unstable or that its degree of stability is not acceptable. Associated with (2.8.64), consider the compensated system ii =

A;zi

+

I j = 1, i # j

C,zj

+ Biui,

i = I,

..., I,

where ui E Rm,and B; is an n i x mi matrix. Let Biui be of the form I

Biu; = Bi

1 E,zj

j = 1

I

4

C F,zj,

j= 1

(2.8.65)

2.9

NOTES AND REFERENCES

89

where E, is an m i x n j matrix. System (2.8.65) may be viewed as a linear interconnection of 1 isolated subsystems (q.),

i i = AiZi + BiEiiZi = AiZi

+ FiiZi,

(%)

i = 1, ..., 1. If (Ai,Bi) is controllable, we can choose the local feedback matrix Eii in such a fashion that (9J has a desired degree of exponential stability. To determine this property, choose Ui(Zi) = Z?PiZi,

Pi

=

P?

where Pi is a positive definite matrix which needs to be determined for some choice of a positive definite matrix Ci, via the Lyapunov equation - ci = (Ai+Fii)TPi

+ Pi(Ai+Fii).

This yields the estimates Am(P)(ziI22 ui(zi)I A,(Pi)l~i(2, /Vui(zi)/I 2,I,,,,(Pi)~zi~,and Dui(yi)(zi) I -A,(C,) Izi12. In accordance with Corollary 2.5.21, the compensated system (2.8.65) will be exponentially stable in the large if the principal minors of the test matrix D = [d,] are all positive, where

This is the case if for example

Thus, for given matrices Bi, i = 1, ..., 1, we need to choose feedback matrices EG,i, j = 1, ..., I, and matrices C i ,i = 1, ..., I, in such a fashion that the ratios Am(Ci)/AM(Pi),i = 1 , ..., I are maximized and such that IlC,+F,II i \\CG\l, i, j = 1, ..., I, i # j . Finally, we can invoke Corollary 2.5.23 to obtain an indication of the degree of stability of the compensated system, by computing p = mink Re[A,(D)], k = 1, ...,1. 2.9 Notes and References

For existence and uniqueness results of solutions for ordinary differential equations refer to Coddington and Levinson [1). Standard references on the Lyapunov theory include Hahn [2], Yoshizawa [J], Krasovskii [l], and LaSalle and Lefschetz [l]. Our exposition is heavily influenced by Hahn [2], where an extensive discussion of comparison functions of class K is given (Hahn [2, pp. 95-97]). For a good overview of comparison theorems and the comparison principle, refer to Lakshmikantham and Leela [l, 21, Szarski [1],

90

11 SYSTEMS DESCRIBED BY ORDINARY DIFFERENTIAL EQUATIONS

and Walter [l]. The original results in this area include Miiller [l] and Kamke [l]. For subsequent work, refer to Wazewski [I], Grimmer [l] and LaSalle [I]. Standard references on absolute stability are the monographs by Aizerman and Gantmacher [I] and Lefschetz [l]. Nice sources on M-matrices include the papers by Ostrowski [1] and Fiedler and Ptak [l] and the books by Bellman [3] and Gantmacher [ l , 21. See also the report by Araki [2]. Sections 2.3 and 2.4 are based on references by Porter and Michel [l, 23, Michel and Porter 131, Michel [S, 71, Bose and Michel [ l , 21, and Bose [l]. For related results refer to Thompson [I, 21 and Thompson and Koenig [l]. Section 2.5 is an adaptation and expansion of results reported by Michel and Porter [3], Araki and Kondo [l], Michel (75-7, 91, Araki [l, 41, Rasmussen and Michel [2,4], and Rasmussen [l]. The introduction of vector Lyapunov functions by Bellman [2] gave rise to the qualitative analysis of large scale systems, using the approach advanced herein. Bailey [ l , 21 was the first to apply the comparison principle to vector Lyapunov functions in analyzing nonlinear composite systems with linear interconnecting structure. Subsequent extensions include the work of Matrosov [ 1, 21 whose approach we follow in Section 2.6. Refer also to Piontkovskii and Rutkovskaya [I]. Additional related references on Lyapunov stability of composite systems include Weissenberger [I], Matzer [ 11, Grujii. and Siljak [ 2 ] , Tokumaru, Adachi, and Amemiya [2], and Athans, Sandell, and Varaiya [I]. Section 2.7 is based on results by Michel [2,4, 91. Related work is contained i n Michel [ I , 31, Michel and Heinen [I-41, and Michel and Porter [ I , 21. For results on practical stability and finite time stability, refer to LaSalle and Lefschetz [I], Weiss and lnfante [I], and Michel and Porter [4]. The source of Examples 2.8.1, 2.8.9, and 2.8.22 is Piontkovskii and Rutkovskaya [I]. Our treatment of these examples follows Michel [ S , 71. For a good qualitative treatment of economic systems refer to Quirk and Saposnik [I]. The sources of Examples 2.8.34, 2.8.39, and 2.8.41 are Sandberg [lo], Mitra and So [I], and the related work by Rosenbrock [ I , 21. In our approach to these examples we follow Michel [9]. Multiple feedback systems similar to those of Examples 2.8.52 and 2.8.59 are treated by Lefschetz [l] .The present results, which are not entirely identical to those of Lefschetz [l], are based on references by Bose [I] and Bose and Michel [1,2]. Sources concerned with systems containing many nonlinearities are numerous. For some of these refer to Narendra and Neuman [l], Narendra and Taylor [l], McClamroch and Janculescu [I], and Blight and McClamroch [l].