IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 40, NO. 5 , MAY 1993 343

Computer Generated Lyapunov Functions for a Class of Nonlinear Systems

Yuzo Ohta, Member, ZEEE, Hiroshi Imanishi, Lei Gong, and Hiromasa Haneda, Fellow, ZEEE

Abstract-In this paper, the problem of constructing Lyapunov functions for a class of nonlinear dynamical systems is considered. The problem of constructing a Lyapunov function is reduced to the construction of a polytope satisfying some conditions. A generalization of the concept of sector condition is proposed so that it makes possible to evaluate a given nonlinear function by using a set of piecewise-linear functions. This improvement reduces the conservatism in the stability analysis of nonlinear systems very much. Two algorithms to construct such polytopes are proposed, and two examples are shown to demonstrate the usefulness of the obtained results.

I. INTRODUCTION

HE LYAPUNOV method is one of the most powerful T methods to analyze stability. Several methods to construct Lyapunov functions have been proposed. Especially, Brayton and Tong [l], [2] proposed a method that constructs a single Lyapunov function which is available for a set of linear systems {j: = A;x I i = 1, 2 , . . . , m}: the problem of con- structing a Lyapunov function is reduced to that of constructing a balanced polytope P such that A;P C pP for any A;, where p E [O, 1) and pP = {px E R" I 1c E P } . They applied the method to analyze the stability of nonlinear systems using the traditional sector condition. Later, Michel, Nam, and Vitta1 [3] improved the result so that they can analyze local stability. Applying the traditional sector condition to a nonlinear system produces a set of linear systems. Polytopes to be constructed need not be balanced,' but constructing nonbalanced polytopes does not bring any merit as long as we treat a set of linear systems.

In this paper, we generalize sector condition so that we can evaluate a given nonlinear function by using a set of piecewise-linear functions. Needless to say, the evaluation of a given nonlinear function using a set of piecewise-linear functions gives the better evaluation of the original nonlinear function than that derived using a set of linear functions. This generalization of sector condition poses a new problem: how to analyze the stability of a set of piecewise-linear systems. We solve this problem using a Lyapunov function given by a gauge function of a polytope. In this case, constructing a nonbalanced polytope makes senses because we treat a set of piecewise linear systems, and reduces the conservatism

Manuscript received February 4, 1992; revised January 19, 1993. This work was supported in part by the Saneyoshi Scholarship Foundation. This paper was recommended by Associate Editor M. Hasler.

The authors are with the Department of Electronics Engineering, Faculty of Engineering, Kobe University, Nada, Kobe 657, Japan.

IEEE Log Number 9208944. ' A polytope P i s said to be balanced if .r E P means --.r E P.

contained in the, $$ability analysis using the traditional sector condition. Moreover, we propose two algorithms to construct polytopes, and we demonstrate the usefulness of our results by using two examples.

Notations: In this paper, R denotes the real number system, and R" is the usual vector space of real n-dimensional vectors x = [xlz2 . . . xnIT. All vectors are to be regarded as column vectors for the purpose of matrix multiplication. For a matrix A , AT denotes the transpose of A. For given integers rn and n such that rrt, 5 n, [ r n . . . n] denotes the set {m, m + l , . . . , n}. The inner product of two vectors x and y in R" is expressed by (x, y) = xlyl + . . . + xnyn. The norm of vector is expressed by I(zJI = (x. x)l/'. For a set P in R", the interior and the boundary of P are denoted by i r t t ( P ) and bd ( P ) , respectively. For IC, y E R", [x,y] denotes a closed segment connecting x and y. Moreover, [x, y), ( x , y] and (x, y) denote segments given by [x, y] - {y}, [x, y] - {x} and (2, y] - {Y}, respectively. For a set V of vectors in R", aff (V), conv (V), and cc ( V ) denote the affine hull, the convex hull and convex conical hull of V, respectively. For a polytope P, ext ( P ) and F ( P ) denote the set of all extreme points of P and the set of all facets of P, respectively. See Appendix A concerning notions and properties relating polytopes. For a set V having a finite number of elements (VI denotes the number of elements of V. Symbol V (universal quantifier) and 3 (existential quantifier) represent "for all" and "there exists," respectively. Moreover, the symbol 31 stands for "there exists a unique element."

11. SYSTEM DESCRIPTION AND STABILITY CRITERION

Let us consider a nonlinear dynamical system given by

s: j: = f ( z ) (2.1)

where x ( t ) E R" is the state vector of S , and f : R" + R" is a continuous function satisfying f (0) = 0. We assume that for each ( to , zo) a solution of (2.1) exists and is unique. We denote the solution of (2.1) passing through zo at the initial time t o by x ( t ) = IC@; t o , xo).

Concerning the stability of (2. I ) , the following result is well known.

2.2. Lemma

Suppose that there exists a set M C R" including an open neighborhood of 0 and a function U : R" + R, satisfying the following conditions.

A) There exist positive constants cy and such that

Vx E R": cyllxII 5 U(.) 5 Pllxll. (2.3)

1057-7122/93$03.00 0 1993 IEEE

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 40, NO. 5, MAY 1993

There exists a positive constant Q such that

vz, Y E R": Iv(z) - v ( Y ) ~ I Q I I ~ - Y I I (2.4)

and There exists a positive constant y such that

vx E O(wmax): ~ ; 2 . 1 ) ( z ) L - Y V ( ~ ) (2.5)

where w,, is a positive constant such that w,,, = max{a > 0 I R(a) M } ,

Q(a) {z E R" I w ( z ) I a} (2.6)

and

Then, the solution z( t ) 0 is exponentially stable. In fact, for any zo E R(wmax), the solution z(t; 0, zo), t 2 0, stays in R(v,,), and satisfies

(2.8)

0 In Lemma 2.2, the condition B) that w ( z ) is Lipschitzian is

a sufficient condition to guarantee the following equality [4]:

V t 2 0: Ilz(t; 0, .'))I1 1. (P/a)llzollexp(-rt>.

Proof: See [4, theorem 4.11, for example.

VP E R": = w ' [ z ( t ; t , P)] (2.9)

where

111. A POLYTOPE AND A LYAPUNOV FUNCTION

In this section, we consider the problem to construct a Lyapunov function w ( z ) by using a polytope P. We assume that P includes an open neighborhood of 0, that is, 0 E int ( P ) . Let us define a function U(.) by

(3.1)

where pP is called a scalar multiple and is defined by pP = {pz I z E P} . Since P is a convex and compact set including an open neighborhood of 0, we have

Vz E R", 31p 2 0, 312 E bd(P) : z = p P , [0, 2 )

Therefore, we have the second formula to calculate w ( z ) :

w ( z ) = inf { p > 0 I z E pP} ,

int ( P ) . (3.2)

w ( z ) = ~ ~ z ~ ~ / ~ ~ i ~ [ 2 0, z = pd, E E bd(P) . (3.3)

Let Fj E F ( P ) be a facet of the polytope P, and let Hj be the corresponding hyperplane given by

Hj = aff [ext (Fj )] = H p ( a j , a j ) = {x E R" I (x, ai) = a j } (3.4)

where uj is a normal vector of the hyperplane Hj and aj is a constant (see Fig. 1).

Since, for any p > 0, H p ( a j , a j ) and H p ( p a j , p a j ) give the same hyperplane, without loss of generality, we can assume that aj = 1 (in the following, we say that ai is normalized if aj = 1).

Fig. 1 . A polytope P and one of its facet FJ- and the corresponding normalized normal vector a3 and the hyperplane H 3 .

Since P includes an open neighborhood of 0, R" is decom- posed into the union of a finite number of cones, that is, R" = cc(F1) U . . . U cc(qp(p)l) and int [cc(F;)] n int [cc(Fj)] = q5 if i # j.

Given an arbitrary 5 E cc(Fj). Since Fj is a part of bd(P), there exist p > 0 and P E Fj such that z = pf. Therefore, by (3.3), (3.4), and aj = 1, we have the third formula to calculate U ( z) :

VX E C C ( F ~ ) : W ( X ) = p = p ( P , U') = (2, U') (3.5)

where uj is the normalized normal vector of Fj. Now we consider the condition A) of Lemma (2.2). Since P

is compact and includes an open neighborhood of 0, we have

3a, b E bd(P), V i E bd(P): 0 < llbll 5 IlPll 1. Ilall.

Therefore, from (3.3), we have

vz E R": 1 1 ~ 1 1 / 1 1 ~ 1 1 = Qll4l 1. 4.) = 1 1 ~ 1 1 / 1 1 ~ 1 1 I l l ~ l l / l l ~ l l = PIIxlI,

where a l/llbll, and P E bd(P) is the vector such that z = p i for some p > 0. Thus, we have the following.

l/llall, P

3.6. Lemma

Let P be a polytope such that 0 E int (P). Then the function w(z) defined by (3.1) satisfies the condition A) of Lemma (2.2).

Proof: Omitted. 0 Relating with the condition B) of Lemma (2.2), we have

the following.

3.7. Lemma

Let P be a polytope such that 0 E int (P). Then the function w ( z ) defined by (3.1) satisfies the condition B) of Lemma (2.2), that is,

vz, y E R": lv(x) - v ( Y ) I L QIIZ - YII (3.8)

where Q = max { llaj 1 1 I uj is the normalized normal vector of Fj E F ( P ) } .

Proofi See Appendix B . 0 A formula to calculate W ( ~ , ~ ) ( X ) is given by the following.

OHTA er al.: COMPUTER GENERATED LYAPUNOC FUNCTIONS 345

3.9. Lemma

Let P be a polytope and U(.) is defined by (3.1). Given an arbitrary x E R", we have

3Fj E F ( P ) , 3Ao > 0, VA E [0, A,]: X: z + At(.) E C C ( F ~ ) . (3.10)

Moreover, we have

where u j is the normalized normal vector of Fj. Prooj? See Appendix B. 0

Note that by the definition (3.1) of w(z), we have n(1) = { x E R" I w(x) 5 l} = P. From Lemmas (2.2), (3.6), (3.7), and (3.9), the following result is automatic.

3.12. Theorem: Assume that there exist a polytope P and a constant y > 0 satisfying

A) 0 E int(P); and B) V f j E F ( P ) , V X E C C ( F ~ ) n P : ( f ( ~ ) , aj) I:

- y (x) . (3.13) Then, for any xo E P, the solution x ( t ; 0, xo) of (2.1)

stays in P and satisfies (2.8). Prooj? Omitted. 0

The condition B) of Theorem 3.12 requires to examine that the inequality holds for all x E P, and it is almost impossible. We want to derive a set of conditions that guarantees the condition B) of Theorem 3.12 but only requires us to examine the condition just for a finite number of points. Let n: be an arbitrary point in P. Then, there exist an Fj E F ( P ) , a unique i E FJ C bd(P) and a unique constant p 2 0 such that n: = pi. Note that we have 7 4 ~ ) = p and 742) = 1, because Fj C bd(P). Moreover, i is represented by i = X1xJ1 + . . . + X L ( ~ ) ~ J L ( J ) , where 2 3 % ' ~ are extreme points of Fj, L ( j ) = lext(Fj)I, and X,'s are nonnegative constants such that Xl+...+XL(j) = 1. If f(x) is linear in c c ( F j ) n P , then we get

3.18. Theorem: Assume that there exists a polytope P and

A) 0 E int (P); B) for each FJ E F ( P ) , f(x) is linear in cc(FJ) n P; and C) VFJ E F ( P ) , Vx E ext(F,): (f(x), d) 5 -y =

Then, for any zo E P , the solution x ( t ; 0, z') of (2.1)

a positive constant y satisfying

-7vix).

stays in P and satisfies (2.8). Prooj? Omitted. 0

Note that, in the condition C) of Theorem 3.18, if 5 E ext (F,) nex t ( F z ) , then it is required that both ( f ( z ) , u J ) 5 -y and (f(x). a') 5 -y hold.

The condition C) of Theorem 3.18 is not the suitable representation to examine whether it is satisfied or not. We say that f is inward to P at x if x + Af(x) E P for any sufficiently small A > 0. Suppose that the condition C) is satisfied, then f is inward to P at each extreme point x of P . Now a question arises: If f is inward to P at z, then does the condition C) hold? The answer is yes, and we have the following.

3.19. Lemma

by (3.1). Give an arbitrary x E R", we have

3Fj E F ( P ) , 3Ao > 0, VA E [0, A,]:

Let P be a polytope such that 0 E P and U(.) be defined

5, LC + Af(z) E cc(Fj). (3.20)

Moreover, the following conditions are equivalent to each other:

A) 3y > 0: ~ ; ~ , ~ ) ( x ) L -yu(x); B) 3y > 0: ( f ( ~ ) , aj) I -yv(z), where u j is the

C) 3y > 0, VA E [O, A,]: x + A[f(x) + 7x1 E .v(z)P;

D) 37 > 0, V,Fi[F; E F ( P ) : 5 E C C ( F ~ ) ] : ( f ( ~ ) , ai) 5

normalized normal vector of Fj;

and

-yw(x) , where ai is the normalized normal vector of Fa.

Proofi See Appendix B. 0 f(x) = f[p(X1xJ1 + . . . + XL(J)xJL(J) )] From Theorem 3.18 and Lemma 3.19, we immediately have

3.21. Theorem: Assume that there exists a subset M C R",

1) P g M ;

= v ( z ) [ X ~ ~ ( I C " ) +. . . + X ~ ( ~ ) f ( x ~ ~ ( 3 ) ) ] . (3.14) the following.

a polytope P, and positive constants A and y satisfying Therefore, we have the following.

3.15. Lemma

Let FJ be a facet of a polytope P and let uJ be the normalized normal vector of FJ. Assume that f(x) is linear in cc(F,) n P. Then the following two conditions are equivalent each other.

1) 3yJ > 0, VxJt E ext(F,): ( f ( + ) , u J ) 5 -yJ : and (3.16)

-734x). (3.17)

From Theorem 3.12 and Lemma 3.14, we immediately have

2) 3yJ > 0, VL E cc (FJ) n P : ( f ( ~ ) , U J ) 5

Prooj? Omitted. 0

the following.

2) 0 E int(P); 3) for each Fj E F ( P ) , f(x) is linear in cc(Fj) n P ; and 4) Vx E ext ( P ) : x + A[f(x) + 7x1 E P. Then, for any xo E P, the solution x ( t ; 0, xo) of (2.1)

stays in P and satisfies (2.8). Pruufi Omitted. 0

Under the condition C), the condition D) means that the mapping h(x) = x + A[f(x) + 7x1 maps the polytope P into itself. In other words, to construct a Lyapunov function, we need to find a polytope P which satisfies the conditions A), B), and C), and has the property that h ( P ) C P. This suggests the following iteration scheme: Give an appropriate given polytope P O ; for /c = 0, I , . . . , calculate pk+l = P k ' M k ,

346 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 40, NO. 5, MAY 1993

where Mk = lext(Pk)l, and

P',~+' = conv [ext (P"') U { h ( x k i i ) } ] , xkli E ext ( p k ) } for i = 1, 2 , . . . , Mk.

If Pk+l = Pk for some k , then stop. The polytope P = P k thus obtained satisfies the condition D). In Section V, we will discuss the procedure to construct such a polytope P . The problem to calculate Pi+l = conv [ext (Pi) U {xi}], i = 1, 2,. . . , is called a dynamic convex hull problem [6].

Iv . UNCERTAIN NONLINEAR SYSTEMS

In the previous section, we derived some results which are applicable to piecewise-linear systems. In this section, we apply those results to analyze the stability of the uncertain nonlinear system (2. I), where the nonlinear function f need not be piecewise linear. From Theorem 3.12, and Lemmas 3.15 and 3.19, we have the following.

Theorem 4.1: Assume that there exist a subset M C R", a set of cones {Kk I k E [ l . . . K ] } , a set of functions { ~ Q ( x ) 1 q E [l . . .SI}, and a polytope P satisfying the following conditions.

A) R" = K1 U . . . U KK and Vj , k [ j # k ] : int ( K j ) n int(Kk) = (p, (4.2)

B) VX E M , f ( x ) E c0nv[{fq(x) I 4 E [ I . . . Ql}], (4.3) C) for each q E [ l . . . Q ] and Kk, k E [ l . - . K ] , f*(x) is

D ) P G M E) 0 E int(P);

G) 3A > 0, 3y > 0, VX E ext(P) , Vq E [ l . . - Q ] : x + A[fq(x) + y z ] E P. (4.4)

Then, for any z0 E P, the solution x( t ; 0, xo) of (2.1)

Proof: See Appendix B. U

linear in Kk n M ,

F) V F j E 8'(P), 3Kk E K : cc(Fj) G Kk; and

stays in P and satisfies (2.8).

At this point we need some remarks.

4.1. Remark

The conditions A), B), and C) of Theorem 4.1 are satisfied if we evaluate appropriately the characteristic of the given function f ( x ) . An example of a nonlinear function f(z) , a subset M 5 R", a set of cones {Kk I k E [l . . . K]} , a set of functions {fQ(x) I q E [l satisfying the conditions of A), B), and C) of Theorem 4.1 is the following. Let us consider the case when f(z) is given by

f (x) 1 AX + blgl((cl, x)) + ... + bLgL((CL, x)) (4.6)

where A is an n x n-matrix, @ and cj are n-dimensional column vectors, and gj's are functions from R into itself. The nonlinear function f of (4.6) appears when we consider a feedback system

i = AX + Bu, u = g(y), y = CX (4.7)

where B = [b1b2-..bL], C = [c1c2-..cLIT, and g(y) = [gl(yl)gz(y2) . . . g ~ ( y ~ ) ] ~ . Suppose that gi is any function satisfying

(4.8a) Vyi E M: = [0, U:]: a+y? 5 gi(y;)yi L pi + 2 yi

Fig. 2. An example of function 9;.

and

Vyi E MC = [-~i, 01: aY;?/: 5 gi(yi)yi 5 pz7p? (4.8b)

where U:, cay, a:, p:, at:, and pt: are given constants satisfying U? > 0, c2: > 0, a: 5 ,$, and a; 5 ,B,' (see Fig. 2). For each i, let us define two functions by

and

(4.9b)

For each q E [l . . . 2L], there exists a unique set of integers {qi E (0, 1) 1 i E [ l . . . L ] } such that q = l+q1+2q2+ . . .+ 2L-1q~, and define the function fq(x) by

f"(.) = Ax + blgl,ql((C1l 4) + .. . + bLSL,QL((CL, 4). (4.10)

(4.11)

Then, we have

Vx E M : f ( z ) E conv[{fq(x) I q E [ l . . .Q]}]

where Q = 2L and M G R" is a subset defined by

M = {X E R" I (ci, X) E ( M y U&!:), i E [ l - . . L ] } (4.12)

Therefore, the condition B) of Theorem 4.1 is met. Next we show that the conditions A) and C) hold. Let

L

K k = n { x E R" 1 S:(Ci, X) 2 0) (4.13)

where s t E (-1, 1) is used to determine to which side of the hyperplane Hp(ci, 0) = {x E R" 1 (ci, x) = 0) belongs to

i=l

OHTA e l al.: COMPUTER GENERATED LYAPUNOC FUNCTIONS 347

the cone Kk. The whole space R" is decomposed into, say, K-number of such cones, that is,

R" = K1 U ' . ' U K K and

V j , k [ j # k ] : int ( K j ) fi int (Kk) = 4. (4.14)

Therefore, the condition A) of Theorem 4.1 is met. It is easy to see that for each q and Kk, fq(x) is linear in Kk n M , and hence, the condition C) of Theorem 4.1 is met. In the following, we say that f satisfies the generalized sector condition if f satisfies (4.3) for a set of cones {Kk I k E [ l . . . K ] } and a set of piecewise-linear functions {fq(s) I q E [l . . . Q]} satisfying the conditions A) and C) of Theorem 4.1. 0

4.15. Remark

Suppose that f(x) is given by (4.6). Note that a+(@, respectively) and a i ( p i , respectively), in (4.8) need not be the same constant, in other words, we allow that constants of the sector condition may differ according to cones. On the other hand, if we apply the traditional sector condition, then a: (@, respectively) and a; (p,:, respectively) in (4.8) must be the same constant, and we have

and

4.22. Remark

Theorem 4.1 can treat a nonlinear system x = f(s) whose characteristic f(x) is evaluated by the generalized sector condition (4.3) using a set of piecewise-linear functions {f4(x)}. This is one of the main contribution of this paper. In the papers by Brayton and Tong [l], [2], the functions fq(x)'s are assumed to be linear functions (see [ l , theorem 101 and [2, theorem 5.11). In [2, theorem 5.11 W is assumed to be a convex, bounded neighborhood of the origin, and need not be polytope; however,'he constructing algorithm proposed in [ 11 and [2] is confined to thk construction of a polytope. In this respect, it is natural to consider the case when W is a polytope. If we take [2, theorem 4.1 and corollary] into account, then [2, theorem 5.11 can be stated as follows.

Let S be a set of matrices with the property that for all s E R", there exists a matrix H ( z ) such that f(z) = H ( z ) z and let A = { I + AS}. If there exists a polytope W and positive constant S < 1 and A such that for each M E E(A), M W C SW, then s = 0 is globally exponentially stable for the system i = f(s), where E(A) denotes the set of extreme matrices of A.

If we assume K = 1 in Theorem 4.1, which means that K1 = R" and that all functions are linear functions, then we have the above result for 6 = 1 - Ay. In Theorem 4.1, we do not assume fq(x) to be linear. In this respect, Theorem 4.1 is a generalization of the above result which is the restatement of [2, theorem 5.11. If fq(x) = Aqx, then the set S in the above is given by

a; = min {a;, a:}, pi = max {,&-, @}. (4.17) S = conv [ {A , I q E [l . . . Q]}] - = {Ala, + x z A z + . . ' + x,A, I

For each i, let us define two constants by A 1 + A2 + . ' . + xg = 1,

2 0, 4 E [l...Ql} (4.23) Si , o = a; and j;, = pi. (4.18)

and the relation (4.21) is rewritten as

conv[{fq(x) I q E [l...Q]}] For each q E [l . . . a"], there exists a unique set of integers {q; E (0, l} I i E [ l . . .L ]} such_that q = l + q l + 2 q z + . . . + 2L-1q~, and define the matrix Aq by C conv [{A,x I q E [l . . . Q]}] = {Hs 1 H E S}. (4.24)

Then, we have

Vx E M : f(z) E conv [{&IC 1 q E [l . . . Q]}] (4.20)

where Q = 2L. Note that conv [{A,. I q E [l . . .SI}] is the best estimate of the region containing f(x) if we apply the traditional sector condition. Moreover, we note that

conv[{f4(x) 1 4 E [l . . .Q]}] C conv[{A,z 14 E [l . . .Q]}] (4.21)

where fq(x) is defined by (4.10). The relation (4.21) means that the generalized sector condition gives a better evaluation of the nonlinear characteristic f(x) than the traditional sector condition. This is the reason why Theorem 4.1 reduces the

0 conservatism of stability conditions very much.

The relation (4.24) shows the reason why Theorem (4.1) reduces the conservatism of the stability condition of the above result (the restatement of [2, theorem 5.11) very much. In fact, as we will demonstrate in Example 6.1, there exists a nonlinear system such that there exists a set of piecewise-linear functions { fq (s ) } satisfying all the conditions of Theorem 4.1 but there does not exist a set of linear functions {fq(x)} satisfying the conditions. 0

4.25. Remark

To construct a polytope satisfying the conditions D), E), and G) of Theorem 4.1 is rather easy. The condition F) requires that any facets Fj of P must be contained some cone Kk. This is the point that makes it difficult to construct a polytope P satisfying the conditions D), E), F), and G) of Theorem 4.1. If fq (x) ' s are linear, then K = 1 in Theorem 4.1 and the

0 condition F) is not required.

I - - -

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v . CONSTRUCTION OF POLYTOPE

In the following, we assume that the characteristic of the given function f ( z ) is evaluated by the generalized sec- tor condition 4.3 using a set of piecewise-linear functions {f,(z)}. The condition F) requires that all the extreme points of P k P n Kk except 0 are those of P . In this respect, we choose the initial polytope P so that P satisfies this property: Let F(K,) = {B,,j 1 j E [ l . . . ( F ( K , ) ( ] } be the set of all facets of K,, where JF(K,)J denotes the number of facets of K,. Let W,,j = {wq>jt E B,,j 1 i E be a set of vectors such that B,, j = cc(W,, j ) . We assume an initial polytope P = c o n v [ { w Q ~ j ~ E W,,j I ~ E [ l . . . I F ( K , ) ( ] , q E [ l . . . Q]} ] is given. If 0 # int (P) , then we add an appropriate set V of vectors to ext ( P ) so that 0 E int (conv [ext (P) u V ] ) . Because of the condition G) of Theorem 4.1, to construct a polytope satisfying the conditions of Theorem 4.1 is much more difficult than to construct a polytope satisfying the conditions of Theorem 3.21.

In this section, we propose two schemes to construct a polytope P satisfying the conditions of Theorem 4.1.

The first one is the following. 1) Suppose we are given appropriate positive constants

A, 7, and T. Construct a polytope P satisfying the conditions D), E), and F) of Theorem 4.1 and the condition

Vx E ext(P): x + A[f(x) + ;U4 E P (5.1)

where

and 2) check the condition G) of Theorem 4.1. The first step will be executed using Procedure const-A,

which will be shown later.

5.3. Remark

The above scheme is closely related to the algorithm pro- posed in [3] (consider the case when the number of subsystem is 1). However, they are different-in the following two points: The first point is that we allow f not to be a linear function, which makes difficult the construction of a polytope P in step 1). If we choose f(x) E conv[{f,(x) 1 q E [ l . . . Q ] } ] as a linear function, then the construction of a polytope P will be much easier. But, as we showed in (4.21), the generalized sector condition gives the better evaluation of the nonlinear characteristic f(x) than the traditional sector condition, and this fact plays a significant role in the step 2). The second point is that we examine whether the stability margin ;U dominates the effect of nonlinearity of f (that is, the difference of f and f ) by checking directly the condition G) of Theorem 4.1. It gives a less conservative result than by examining that using the inequality ;Y - p ~ / a > 0, where Q > 0 and a > 0 are constants satisfying (2.4) and (2.3), respectively, and p 2 0 is a constant satisfying Ilf(x) - f"(x)II 5 pJlz(( for all z E R". Moreover, when a polytope P is given, checking the condition G) of Theorem 4.1 is not so difficult. On the other hand, to determine a minimum constant p satisfying

\lf(x) - J ( Z ) I I 5 p ~ ~ x l l for all x E R" requires more computational efforts. 0

x E R". (5.4)

We say that J is inward to K k at z if (z, h(x; j, A, 9>] n Kk # 4 for sufficiently small A > 0. Note that if z E int (Kk) then f is inward to Kk at x. For convenience, we define a function H ( x ; f , A, ?, Kk) by

Let us define a function h(x; fl, A, 9 ) by

h(x; f , A, 7) = 2 + A{f(x) + Tx},

H ( x ; f , A7 ?, Kk) - - { d K k ) ,

h(x; f"7 A7 ?), if [z, h(x; 17 A, ' U ) ] c Kk (5.5) if L.7 h(x; f , A, ? ) I Kk

where Kk is the cone to which j is inward at z, and y(Kk) is a unique point in [x, h(z; f , A, ; U ) ] n bd(Kk).

Equation (5.1) is equivalent to the following condition:

vx E ext (PI: ~ ( z ; f , A, ;U> E P. (5.6)

A procedure to construct a polytope satisfying the conditions E) and F) of Theorem 4.1 and (5.6) is the following.

Procedure const-A(var P : POLYTOPE; A, ?: POSI- TIVE; f": FUNCTION); varP: POLYTOPE; x, w: POINT; begin

1. P : = P ; 2. for w E ext(P) do begin

2.1. x:= H(w; f", A, ;U); Find Kk to which J is

2.2. while x # P do begin inward at w;

2.2.l.while (z E int(Kk) and x $! P ) do z:=

2.2.2.if x E b d ( K k ) then P : = conv[ext(P) U

2.2.3.if (there exists z E ext(P) such that z E int (P)) then stop; {Procedure const-A. fails to construct a polytope P}

fuz; f", A, ?);

{ 4 l ;

2.2.4.if x # P then begin

2.2.4.1.2: = H ( x ; f", A, ;Ub 2.2.4.2.Find Kk to which f is inward at x

2.2.4.end;

2.2. end

2. end

4. for each w E ext ( P ) do begin

4.1. z:= H ( w ; J, A, 5); 4.2. done: = false; 4.3. while (not done) do begin

3. P : = P ;

4.3.l.if z # P then begin

4.3.1.13: = conv [ext (P ) U { x } ] ; 4.3.1.2.if (there exists z E ext (P ) such that z E

int (P)) then stop; {Procedure const-A fails to construct a polytope P}

OHTA et al.: COMPUTER GENERATED LYAPUNOC FUNCTIONS 349

4.3.1.3.2: = H(x; J, A, 5); 4.3.1 .end 4.3.2.else done: = true

4.3. end

4. end end

At this point we need some remarks.

5.7. Remark

In Procedure const-A, several Pascal reserved words (var, for, while, do, begin, end, if . . . then. . . else) are used with almost the same meaning as in the standard Pascal. The for statement of the form for x E I do S implies to evaluate S once for each value of x E I . POLYTOPE, POSITIVE, FUNCTION, and POINT denote, respectively, abstract data types representing a polytope P such that 0 E int(P), a

0 positive number, a function and a vector in R".

5.8. Remark

There are three while-loops 2.2, 2.2.1, and 4.3, which may not terminate. If the while-loops 2.2 and 2.2.1 terminate, then the while-loop 4.3 terminates. L_et us consider a sequence {x_"} generated by xktl = H(xk; f , A, 7) where xo E ext(P) . If {xk} converges to 0, then both while-loops 2.2 and 2.2.1 terminate. Needless to say, in advance, we cannot know whether {xk} converges to 0 or not. Moreover, we cannot know how large the integer k' such that xk' E P is. In those respects, Procedure const-A is not an algorithm in the exact sense. However, if x = f (x ) + Tx is exponentially stable, we have zk' E P for a moderate value of k' in many practical applications. From that point of view, we believe that it is worthwhile to be able to know that the system under consideration is exponentially stable by just executing the simulation with a finite number of different initial conditions. Furthermore, to construct a polytope P is useful not only in examining stability but also in estimating a stability region.0

5.9. Remark

In [2], asymptotic stability (in the discrete time sense) of a set of matrix A = { I + AS}2 is considered and it is shown that A is asymptotically stable if and only if the constructive algorithm presented in [2] terminates "stable" in a finite number of steps (See [2, theorems 3.1 and 4.11). Suppose that the whole space R" is discomposed into K number of cones {Kk- 1 k E [ l . . . K ] } as was assumed in Theorem (4.1) and that f(x) is given by J(x) = Akx for x E Kk. Let A = { I + A(& + TI)} . If positive numbers A and y are chose? appropriately and if A is asymptotically stable, then i = f(x) + Tx is exponentially stable and Procedure const-A terminates in finite steps. We also note that Procedure const-A may termiante in finite steps even if A is not asymptotically stable. 0

'See Remark 4.22.

5.10. Remark

In Procedure const-A, the procedure to decide whether x E P or not, and the procedure to compute conv [ext ( P ) U {x}] are required. In [l] and [2], it is proposed to use the linear programming to decide whether x E P or not:

minimize 0 subject to x = Xlz' + . . . + Xmxm,

xi 2 0, A 1 +...+A, = 1

where {d, . . . , P} = ext (P) . This method gives the answer that the new point x is included in P or not. Suppose that x P. Then the new polytope is given by P' = conv [ext ( P ) U {x}]. To save the future computational time, we want to throw out those point y E ext ( P ) which are not extreme points of P'. To do so, we need to solve another problem that y E ext ( P ) or not, and this may consume very much time. If we adopt the beneath-beyond method [6], we

It is not difficult to modify the procedure so that the obtained polytope satisfies the condition D) of Theorem 4.1 in addition to the conditions E) and F) of Theorem 4.1.

Needless to say, the above scheme is just a heuristic scheme, and it might exist a polytope P' satisfying the conditions of Theorem 4.1 even if the polytope obtained in step 1) does not satisfy condition G) of Theorem 4.1.

Procedure const-B(var P: POLYTOPE; A, y: POSI-

var P : POLYTOPE;

1. P:= P; 2. for q E [l do const-A(P, A, y, f q ) ;

3.

can circumvent this problem. 0

The second scheme is the following.

TIVE; f ' , f 2 , . . . , fQ: FUNCTION);

while P # P do begin

3.1. P : = P; 3.2. for q E [l . . . Q] do const-A(P, A, y, f q )

3. end

end.

At this point we need some remarks.

5.11. Remark

Procedure const-B is almost the same as the algorithm proposed in[l] and [2]. The difference is that we allow f 4 not to be linear. 0

5.12. Remark

Suppose that f"x) = Aq, k X for x E Kk and let A = { I + A(A,, k + T I ) } . If A is asymptotically stable (in the discrete time sense) then we can show that Procedure const-B terminates in finite steps. We note that Procedure const-B may terminate in finite steps even if A is not asymptotically stable (See Remark 5.9). Moreover, as we said in Remark 5.8, if we choose positive numbers A and y appropriately, then exponential stability of i = fq(x) implies that Procedure const-A(P, A, y, f q )

~

350 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 40, NO. 5, MAY 1993

/ / ‘A

Fig. 3 . The constructed polytope P (Example (6.1)).

terminates. However, it does not guarantee that Pro- cedure const-B(P, A, y, f l , f’, . . . , f & ) terminates. Con- versely, Procedure const-A(P, A, y, f) terminates if Proce- dure constB(P, A, y, f l , f ’ , . . . , f Q ) terminates. We also note that Procedure const-B(P, A, y, f l , f 2 , . . . , f Q ) may succeed in constructing a polytope P satisfying the conditions E), F), and G) of Theorem 4.1 when the polytope P’ constructed by using Procedure const-A(P, A, y, f”) does not

0 satisfy condition G) of Theorem 4.1.

VI. EXAMPLES To demonstrate the usefulness of Theorem 4.1 and Proce-

dure const-A we show two examples. First of all, we note that if we treat a set of linear systems then Theorem 4.1 is same as [2, theorem 5.11. The first example is a case where the results in [1]-[3] cannot show stability but Theorem 4.1 and Procedure const-A succeeds in constructing a Lyapunov function.

6.1. Example

A, b l , b’, c1, and c’ are given by

-0.9 0.0 A = [ 0.0 -0.9

Let us consider a system given by (4.6) where L

) b1 = [! 1.0 0.0 -0.9 0

b’= [A], c1= [H]. c’= [%] and gl(yl) and g’(y2) satisfy (4.9) with cy: = 0.485, p,’ = 0.515, ay = 0.98, pc = 1.02, cy: = 0.78, ,@ = 0.82, a; = 0.48, and ,&’ = 0.52. Choosing A = 0.5, = 0.05 and W = {el, e’, e3 , -e1, -e’, - e 3 } , where ei denotes the ith coordinate vector, Procedure const-A(P, A, y, f) constructs a polytope P with 13 extreme points, 33 edges, and 22 facets, and P is shown in Fig. 3. Moreover, it can be shown that condition G) of Theorem 4.1 is satisfied, and hence, P generates a Lyapunov function, which is defined by (3.1), for the system considered here. In the previous papers by Brayton and Tong [ 11, [2], it is assumed that fq(z)’s are linear functions (See theorems 10 of [l] and 5.1 of [2]). Let S be

I “ I ’

Re

, . ,,,’ . Popov Plot

Fig. 4. Checking the Popov criterion.

a set of matrices with the property that for all z E R”, there exists H such that f ( z ) = H z . It is easy to see that matrix A4 = A + ,8;b1(c1)T + ,@b2(c2)T is one of the extreme matrices of S (see Remark 4.15 and (4.23)). The eigenvalue of A4 are 0.042189, and -1.3711 f 0.81596j, and hence, it is unstable. Therefore, the result in [ l ] and [2] cannot show the stability of the system for this example. Moreover, we note that any member of S must be stable if the result in [3] can guarantee stability. But, as we saw, A4 E S is unstable. Therefore, the results in [3] never succeed to guarantee the stability of the system considered here, and hence, Theorem 4.1 and Procedure const-A are more useful than the previous results [1]-[3] concerning at least this example. Here, we assumed implicitly that M = R3. Since fQ(z) ’s are linear in each cone, for any p > 0, p P satisfies all the conditions of Theorem 4.1 if P does. Therefore, the system considered here is exponentially stable in the large. If M is not R3, then we need to find the largest constant p > 0 such that p P M . 0

The second example compares our result with the Popov criterion [7] and the circle criterion [7]-[9].

6.3. Example

A, bl, and c1 are given by Let us consider a system given by (4.6) where L = 1, and

-1.0 0.0 -0.385 A = [ 3.0 -2.0

c1 = [ 81 (6.4)

and gl(y1) satisfies (4.9) with CY: = 0, ,@ = 0.3, a; = 6.595, and ,8; = 6.635. If we represent nonlinear function gl(y1) by the usual sector condition, then it belongs to the sector [0, 6.6351. As we show in Figs. 4 and 5, neither the Popov condition nor the circle condition is satisfied. On

0.01, b1 = [-A], 0.0 0.5 -0.5

OHTA et al.: COMPUTER GENERATED LYAPUNOC FUNCTIONS 35 1

Fig. 5. Checking the circle criterion.

Fig. 6. The constructed polytope P (Example (6.4)).

the other hand, choosing A = 0.5, .?. = 0.05, and W = {e ' , e', e3, -e', -e2, - e 3 } , Procedure const-A(P, A, ?, f) constructs a polytope P with 13 extreme points, 33 edges, and 22 facets, and P is shown in Fig. 6 . Moreover, it can be shown that the condition G ) of Theorem (4.1) is satisfied. Therefore, Theorem 4.1 and Procedure const-A are more useful than both the Popov criterion [7] and the circle criterion [7]-[9]

0 concerning at least this example.

VII. CONCLUSION In this paper, we considered the problem to construct

a Lyapunov function for a class of nonlinear systems. We derived some stability results which gave the foundation for the construction of a Lyapunov function defined by a polytope. Theorem 4.1 is one of the main results of this paper, and is a generalization and an improvement of [2, theorem 5.11. We proposed two schemes to construct a polytope. Moreover, we showed two examples to demonstrate the usefulness of our results: The first example is a nonlinear system that is characterized by an unstable matrix in a region (cone), and hence, the methods in [1]-[3] cannot show stability. The

second example is a nonlinear system that satisfies the sector condition in each region (cone), and satisfies neither the Popov condition nor the circle condition. For these systems, we can construct polytopes that satisfy the conditions of Theorem 4.1, and hence these system are exponentially stable in the large (A4 = R3). Unfortunately, the proposed algorithm has no guarantee to stop. In this respect, these are not algorithms in the exact sense; however, if we choose positive numbers y and A appropriately, Prqedure const-A stops whenever the system considered is exponentially stable. In this point of view, we believe that it is worthwhile to be able to know that the system under consideration is exponentially stable by just executing simulations with a finite number of different initial conditions. Furthermore, to construct a polytope P is useful not only in examining stability but also in estimating a stability region.

VIII. APPENDIX A For the convenience of readers, we summarize several

notions relating polytopes.

A. 1. Definition

I ) Segment: When x and y are distinct points in R", then the set [IC, y] = { z = Ax + (1 - X)y E R" I 0 5 X I. 1) is called the closed segment between x and y. Moreover, [x, y), (x, y] and (IC, y) denote segments given

2) Convex set: A subset P of R" is said to be convex if and only if the closed segment [x, y] is contained in P for a n y x ~ P a n d g ~ P .

3) Supporting halfspace and supporting hyperplane: Let P be a nonempty closed convex set in R". A supporting half space of P is a closed half space S- = Hs-(a , a ) = {IC E R" I (IC, a ) 5 a } such that P C: S- and H n P # 4, where H = H p = {x E R" I (x, a ) = a} , a is a normal vector of the hyperplane H , and a is a number. And the hyperplane H is called supporting hyperplane. If P is not contained in H , H is called proper supporting hyperplane.

4 ) Linear (afJine, convex, nonnegative, respectively) combi- nation: A linear combination of vectors v', v2, . . . , um is a vector of the form x = Xlv' + X2uz + . . . + Xmum, where A', Xz,. . . , A, are real numbers. A linear combination is called an affine combination if X1 + + . . f + Am = 1 is satisfied. An affine combination is called a convex combination if A; 2 0 for all i E [l . . . m]. A linear combination is called nonnegative combination if A; 2 0 for all i E [l . . . m].

5) Linear (afJine, convex, convex conical, respectively) hull: Let V be any subset of R". A linear (affine, convex, con- vex conical, respectively) hull of V is denoted by span (V) (aff (V), conv (V), cc (V), respectively). And they are given, respectively, by

by [IC, YI - {Y), [IC, YI - {IC) and (x, Y1 - {PI, respectively.

span(V) = { A ~ z ' + XZIC' I X i E R, xi E V, i = 1, 2} aff (V) = {Alzl + X2x2 I A; E R, xi E V,

i = 1, 2, A' + Xz = 1)

i = 1, 2, X I + A2 = l} c o n v ( ~ ) = { X ~ I C ' + ~ 2 2 I A; E [o, CO), IC; E V,

-

352 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 40, NO. 5, MAY 1993

and

cc ( V ) = {px I x E conv (V) , p >_ o}.

In particular, if V = {xl, x2,. . . , xm} , conv(V) is given by

conv (V) = {A,z' + x2x2 + . . . + A,P I xi E [O, 11, A, + A2 + . . . + A, = 11.

The dimension dim ( A ) of a nonempty affine space A is the dimension of the linear subspace L such that A = x + L = { z = x+y I y E L}. Therefore, dim [aff (V)] = dim [span (V)] if 0 E aff (V), and dim [aff (V)] = dim [span (V)] - 1 if 0 $! aff (V) . Note that dim [conv (V)] = dim [aff (V)].

6) Face, facet and extreme point: A convex subset F of a convex set P is called a face of P if for any two distinct points x, y E P such that (5, y) n F is nonempty, we actually have [E, y] G F . Note that in order to have [x, y] C F it suffices to have x, y E F. The empty set 4 and P itself are faces of P. A face F is called k-face if dim ( F ) = k. A facet of P is a face F such that 0 5 dim (8') = dim ( P ) - 1. A point x in a convex set P is called an extreme point of P if {x} is a face of P. The set of extreme points of P is denoted by ext (P ) .

7) Polytope: The polytope is the convex hull of a nonempty 0 finite set {U', v2, . . . , V m ) .

IX. APPENDIX B

Proof of Lemma 3.7

Given arbitrary x, y E R". Since cc (Fj) 's are convex, there exist nonnegative constants X i ' s and vectors xi's such that 0 = A0 < A 1 < . . . < A, = 1 and that [xi+', xi] belong to a cone, say cc(Fi+,), where s is a positive integer and x2 = (1-Xi)x+X;y, i E [O...s] . Then,from(3.5), itfollows

w(y) - U(.) = w ( 2 S ) - w(xS-1)

+ w(xs-l) - . . . - U(.') + w(d) - w(x0) - - ( x S - z S - 1 , U S ) +. . . + (x' - 2 0 , U ' )

= ( A S - AS-l)(Y - 5 , U S )

+ . . . + (A, - Xo)(y - 2, U')

I ( A s - AS-l) l lY - 211 . )luSIl + . .. + (A1 - A0)l lV - $11 . Ilalll I max {Ila'll I j E [1 . . .SI) . 1/31 - XI[,

where ai is the normalized normal vector of Fj E F ( P ) . Therefore, we have (3.8). 0

Proof of Lemma 3.9

Since P is a polytope such that 0 E int(P), we have

Therefore, for any given x E R", there exists F k E F ( P ) such that x E cc (Fk). Suppose that x E int [cc (Fk)]. Then there exists a A0 > 0 such that (3.10) holds for j = k, since cc(Fj) is convex. Suppose now that x E bd[cc(Fk)] and let {Fi I i E 1(x) C [l...Q]} be the set of facets such that x E bd[cc(Fi)]. Since cc(Fi)'s are convex, there exists at

R" = CC(F~) U CC(F~) U . . . U CC(FQ), Q = IF (P) ) .

least one j E I(x) such that (3.10) holds. By (3.5) and (3.10), we get

'~[x + Af(.)] - U(.) = (X + Af(x), U ' ) - (z, U' )

= A(f(x), U ' ) .

Therefore, using above equality and (2.7), we have (3.1 1). 0 To prove Lemma 3.19, we need the following lemmas.

B.I. Lemma

If

IF3 E F ( P ) , 3Ao > 0, VA E [O, A,]: x, x + Af(x) E cc (F3) (€3.2)

then

VA E [0, Ao], Vy 2 0: 2 + A[f(x) + 7x1 E CC (F3). (B.3)

Pro08 Let ext (F3) = {xJ1, 2 3 2 , . . . , xJL(3)). Then x and x+Af(x) are represented by 2 = p[X1x31 +. . . + X L ( 3 ) x J ~ ( 3 ) ]

and x + Af(s) = C J [ ( ~ X ~ ~ +. . . + ( L L ( ~ ) X ~ L ( J ) ] , where A,'s and (''s are nonnegative constants such that A1 i- . . . + AL(3) = <I + . . . + < L ( ~ ) = 1 and p and CJ are nonnegative constants. We calculate

2 + A[f(x) + 7x1 = C J [ ( ~ X ' ~ + . . . + ( L ( 3 ) d L ( 3 ) ]

+ Ayp[A1x3' + . . . + A L ( J ) ~ 3 L ( ~ ) ]

= (0 + Ay p)[CixJ1 + . . . + &5(3)23L(3)]

where & = (~4 + AypAz)/(g + Ayp) L 0, z E [l . . .L( j ) ] and [I+. . .+&) = 1. Therefore, x+A[f(x)+yx] E cc (F3), and hence we have (B.3). 0

B.4. Lemma

Let P be a polytope and U(.) is defined by (3.1). Given an arbitrary y E R". Let Fj be a facet such that y E cc (Fj). Then,

VFi E F ( P ) : ~ ( y ) = (y, U' ) 2 (y, u Z ) (B.5)

where U' and ui are normalized normal vectors of Fj and Fi, respectively.

Pro08 By the definition of P and normalized normal vectors d ' s , P is given by P = {x E R" I (2, ui) 5 1, i = 1, 2 , . . . , (F(P)I}. Therefore, for any F;, w(y)P is included in 5';- = {x E R" I (x, ui) 5 v(y)}. On the other hand, by ( 3 3 , we have v(y) = (y, U' ) , since y E cc(Fj). Therefore, we have (B.5). O

Proof of Lemma 3.19: As we saw in the proof of Lemma 3.9, there exists an Fj E F ( P ) and a A0 > 0 such that (3.20) holds. From (3.20) and Lemma (B.l), it follows

VA E [0, A,], Vy 2 0:

x, z + Af(x), II: + A[f(x) + 7x1 E CC (Fj ) . (€3.6)

By (B.6) and ( 3 3 , we have

U(.) = (5 , U ' ) (B.7)

OHTA et al.: COMPUTER GENERATED LYAPUNOC FUNCTIONS

and

VA E [0, A,]: U(Z + A[f(x) + y ~ ] ) = (x + A [ ~ ( z ) + 7x1, U’) (B.8)

where UJ is the normalized normal vector of Fj.

conditions A)-D). In the following, we will show the equivalence of the

1) A)eB). Trivial by Lemma 3.9. 2) B)eC). Observe that

(f(x), a j ) 5 -YV(X) * (f(.), U’) 5 -(v, a j ) by (B.7) * VA E [O, A,]: (Z + A[f(x) + 7x1, U’) 5 (z, U’),

VA E [0, A,]: U ( X + A[f(x) + 7x1) 5 U ( $ )

by (B.7) and (B.8) e VA i [0, A,]: z + A[f(x) + 7x1 E U(.) by (3.1).

Therefore, conditions B) and C) are equivalent. 3) C)+D). Suppose that z E int[cc (F j ) ] , then there is no

Fi E F ( P ) such that x E cc(Fi) and F; # Fj. Therefore, D) coincides with B) and there is nothing to prove. Suppose now that x E bd[cc(Fj)], and let F; be any facet of P such that z E cc (F;) . Then

W(Z) = (x, U%). (B.9)

Observe that

(. + A[f(.) + 7x1, 4 5 (z + A[f(x) + 7x1, uj) by Lemma (B.4) = w(x + A[f(x) + 7x1) by (€3.8) 5 4.) by, C) and (3.1) = (x, U%) by (B.9).

Therefore, we have (f(z) + yx, U;) 5 0, and hence, by (B.9),

(f(x), 2) 5 -y(z, 2) = -yw(Z).

Therefore, we have D). 4) D) + B). Trivial.

Proof of Theorem 4.1: It suffices to show that condition B) of Theorem 3.12 holds. Since x E ext (P) C bd(P) , w(x) = 1. By Lemma 3.19, the condition G) of Theorem 4.1 is equivalent to

This completes the proof. 0

353

REFERENCES

G’) Vx E ext(P), VFj[Fj E F ( P ) : x E Fj], Vq E [ l . . .Q]: (fq(z), uj) 5 -yv(z). (B.lO)

Since fq(x) is linear in cc(F;) by conditions C) and F), (B.lO) and Lemma 3.15 imply

Vq E [ l . . . Q ] , VFj E F ( P ) , VX E CC(F~) n P: (f*(x), uj) 5 -yw(z). (B.ll)

From the condition B) and (B.ll), we have

VFj E F ( P ) , VX E C C ( F ~ ) n P: (f(z), U’) 5 -YV(Z)

which is condition B) of Theorem 3.12. This completes the proof. 0

R. K. Brayton and C. H. Tong, “Stability of dynamical systems: A constructive approach,” IEEE Trans. Circuits Syst., vol. CAS-26, pp. 224-234, 1979. -, “Constructive stability and asymptotic stability of dynamical systems,” IEEE Trans. Circuits Syst., vol. CAS-27, pp. 1121-1 130, 1980. A. N. Michel, B. H. Nam, and V. Vittal, “Computer generalized Lyapunov functions for interconnected systems: Improved results with applications to power systems,” IEEE Trans. Circuits Syst., vol. CAS-3 1, pp. 189-198, 1984. T. Yoshizawa, Stability Theory by Liapunov Second Method. Tokyo: The mathematical Society of Japan, 1966. S. Boyd and Q. Yang, “Structured and simultaneous Lyapunov functions for system stability problems,” Int. J. Control, vol. 49, pp. 2215-2240, 1989. H. Edelsbrunner, Algorithms in Combinatorial Geometry. Berlin: Springer-Verlag, 1987. C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties. New York: Academic, 1975. I. W. Sandberg, “A frequency domain condition for stability of feedback systems containing a single time-varying nonlinear element,” Bell Syst. Tech. J. , vol. 43, pp. 1601-1608, F’t. 11, 1964. G. Zames, “On the input-output stability of systems with nonlinear time- varying feedback systems, Pt. I and U,” IEEE Trans. Automatic Contr.,

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vol. AC-11, pp. 228-238 and pp. 465477, 1966.

Yuzo Ohta (S’72-M’78) received the B.E. and M.E. degrees in electrical engineering from Kobe Univer- sity, Hyogo, Japan, in 1972 and 1974, respectively, and the D.E. degree in electronic engineering from Osaka University, Osaka, Japan, 1977.

From 1977 to 1987, he was at Fukui University, Fukui, Japan. Since 1987, he has been with the Department of Electronics Engineering, at the Fac- ulty of Engineering, Kobe University, Kobe, Japan, where he is currently an Associate Professor. From 1981 to 1982, he was a Visiting Research Associate

in the Department of Electrical Engineering and Computer Science, University of Santa Clara, CA. His research interests include stability theory, robust control, computer aided design, parallel processing, and robot vision.

Hiroshi Imanishi received the B.E. in electrical engineering and the M.E. degree in electronics engi- neering from Kobe University, Kobe, Japan, in 1990 and 1992, respectively.

Since 1992 he has been with Matsushita Electric Industrial Co. Ltd. His research interests include computer generation of Lyapunov functions and its applications.

Lei Gong received the B.E. degree in electric and electronic engineering from Jiao Tong University, Shanghai, China, in 1985, and the M.E. degree in electronics engineering from Kobe University, Hyogo, Japan, in 1991.

Since 1991 he has been with Daifuku Co. Ltd., Osaka, Japan. Currently he works for Contec Co. Ltd. on leave from Daifuku Co. Ltd. His research interests include computer geometry and its appli- cations.

354 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-I: FTJNDAMENTAL THEORY AND APPLICATIONS, VOL. 40, NO. 5, MAY 1993

Hiromasa Haneda (M’72-SM’91-F’93) received the B.S. and M.S. degrees in electrical engineering from Kyoto University, Japan in 1963 and 1965, respectively, and the Ph.D. degree in electrical engi- neering and computer sciences from the University of California, Berkeley, in 1972.

In 1965 he joined Mitsubishi Electric Corporation working in systems and control area. In 1972 he joined the faculty of The Department of Electrical Engineering, Kobe University, and now is a Profes- sor in The Department of Electronics Engineering at

the same university. From 1981 to 1982he was a visiting associate professor at the University of California, Berkeley. His main interests are in the computer- aided design, digital control and digital circuits.

Dr. Haneda is a recipient of the best paper award from The Institute of Electrical Engineers, Japan, 1985.