New Characterizations of Input-to-State Stability - Department of

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 41, NO. 9, SEPTEMBER 1996 1283

New Characterizations of Input-to-State Stability Eduardo D. Sontag, Fellow, IEEE, and Yuan Wang

Abstruct- We present new characterizations of the input-to- state stability property. As a consequence of these results, we show the equivalence between the ISS property and several (apparent) variations proposed in the literature.

I. INTRODUCTION

HIS paper studies stability questions for systems of the general form

E: i = f ( z , U ) (1)

with states z ( t ) evolving in Euclidean space EL” and controls U ( . ) taking values u(t) E W & Et” for some positive integers w and m (in all the main results, W = R”). The questions to be addressed all concern the study of the size of each solution z(t)-its asymptotic behavior as well as maximum value-as a function of the initial condition z(O) and the magnitude of the control U ( . ) .

One of the most important issues in the study of control systems is that of understanding the dependence of state trajectories on the magnitude of inputs. This is especially relevant when the inputs in question represent disturbances acting on a system. For linear systems, this leads to the consideration of gains and the operator-theoretic approach, including the formulation of H““ control. For not-necessarily linear systems, there is no complete agreement as yet regarding what are the most useful formulations of system stability with respect to input perturbations. One candidate for such a formulation is the property called “input-to-state stability” (ISS), introduced in [12]. Various authors, e.g., [4]-[6], [lo], and [17], have subsequently employed this property in studies ranging from robust control and highly nonlinear small-gain theorems to the design of observers and the study of parameterization issues; for expositions see [14] and most especially [7] and [XI . The ISS property is defined in terms of a decay estimate of solutions and is known (cf. [15]) to be equivalent to the validity of a dissipation inequality

T

holding along all possible trajectories (this is reviewed below) for an appropriate “energy storage” function V and comparison

Manuscript received July 21, 1995; revised April 29, 1996. Recommended by Associate Editor, A. M. Bloch. This research was supported in part by the US Air Force under Grant F49620-95-1-0101 and also in part by the NSF under Grants DMS-9457826 and DMS-9403924.

E. D. Sontag is with the Department of Mathematics, SYCON-Rutgers Center for Systems and Control, Rutgers University, New Brunswick, NJ 08903 USA (e-mail: [email protected]).

Y. Wang is with Department of Mathematics, Florida Atlantic University, Boca Raton, FL 33431 USA.

Publisher Item Identifier S 001 8-9286(96)06777-3.

functions o, Q. (A dual notion of “output-to-state stability” (OSS) can also be introduced and leads to the study of nonlinear detectability; see [ 161.)

In some cases, notably in [2], [6], and [18], authors have suggested apparent variations of the ISS property which are more natural when solving particular control problems. The main objective of this paper is to point out that such variations are in fact theoretically equivalent to the original ISS definition. (This does not in any way diminish the interest of these other authors’ contributions; on the contrary, the alternative characterizations are of great interest, especially since the actual estimates obtained may be more useful in one form than another. For instance, the “small-gain theorems” given in [2] and [6] depend, in their applicability, on having the ISS property expressed in a particular form. This paper merely states that from a theoretical point of view, the properties are equivalent. For an analogy, the notion of “convergence” in R” is independent of the particular norm used-e.g., all Lp norms are equivalent-but many problems are more naturally expressed in one norm than another.)

One of the main conclusions of this paper is that the ISS property is equivalent to the conjunction of the following two properties: i) asymptotic stability of the equilibrium z = 0 of the unforced system (that is, of the system defined by (1) with U = 0) and ii) every trajectory of (1) asymptotically approaches a ball around the origin whose radius is a function of the supremum norm of the control being applied. We prove this characterization along with many others. Since it is not harder to do so, the results are proved in slightly more generality, for notions relative to an arbitrary compact attractor rather than the equilibrium z = 0.

A. Basic Definitions and Notations

Euclidean norm in R” or Rm is denoted simply as 1 . 1 . More generally, we will study notions relative to nonempty subsets A of R”; for such a set A, 1tl.A = d(E, A) = inf { d ( q , E ) , 11 E A} denotes the point-to-set distance from

E R” to A. (So for the special case A = {0}, IEl{o> = It\.) We also let, for each E > 0 and each set A

Most of the results to be given are new even for A = {0}, so the reader may wish to assume this and interpret 1El.A simply as the norm of E. (We prefer to deal with arbitrary A because of potential applications to systems with parameters as well as the “practical stability” results given in Section VI.)

00 18-9286/96$05.00 0 1996 IEEE

1284 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 41, NO. 9, SEPTEMBER 1996

The map f : R" x IR" -+ IR" in (1) is assumed to be locally Lipschitz continuous. By a control or input we mean a measurable and locally essentially bounded function U : Z + R", where Z is a subinterval of IR which contains the origin, so that u( t ) E W for almost all t . Given a system with control-value set U, we often consider the same system but with controls restricted to take values in some subset 0 C W; we use M O for the set of all such controls.

Given any control U defined on an interval Z and any E IR", there is a unique maximal solution of the initial value

problem x = f(., U ) , ~ ( 0 ) = E. This solution is defined on some maximal open subinterval of Z, and it is denoted by x(., t , U ) . (For convenience, we allow negative times t in the expression z ( t , E , U ) , even though the interest is in behavior for t 2 0.) A forward complete system is one such that for each U defined on Z = IR20, and each E , the solution z ( t , <, U,) is defined on the entire interval IR>o. The L",nonn (possibly infinite) of a control U is denoted by IIull,. That is, llull, is the smallest number c such that lu(t)l 5 c for almost all t E Z. Whenever the domain Z of a control U is not specified, it will be understood that Z = R ~ O .

A function F : S -+ IR defined on a subset S of IR" containing zero is positive dejinite if E(.) > 0 for all x E S , I(: # 0, and F ( 0 ) = 0. It is proper if the preimage F-'(-D, U ) is bounded for each D > 0. A function y: R20 - I R > o is of class N (or an "N function") if it is continuous andnondecreasing; it is of class No (os an "No function") if in addition it satisfies y(0) = 0. A function y: R2o -+ IR>o - is of class IC (or a "IC function") if it is continuous, positive definite, and strictly increasing and is of class I C , if it is also unbounded (equivalently, it is proper, or y(s) 4 +m as s i +m). Finally, recall that p: x IR20 + R>o is said to be a function of class ICC if for each fixed t 2 Or@(., t ) is of class K: and for each fixed s 2 0, P ( s , t ) decreases to zero as t i 00. (The notations IC, ICm, and ICG are fairly standard; the notations N and No are introduced here for convenience.)

B. A Catalog of Properties We catalog several properties of control systems which will

be compared in this paper. Much of the terminology-except for "ISS" and the names for properties of unforced systems-is not standard and should be considered tentative.

A zero-invariant set A for a system C as in (I) is a subset A C IR" with the property that z ( t , E , 0) E A for all t 2 0 and all < E A, where 0 denotes the control which is identically equal to zero on R ~ o .

From now on, all definitions are with respect to a given forward-complete system C as in (1) and a given compact zero-invariant set A for this system. The main definitions follow.

We first recall the definition of the (ISS) property

3y E IC, p E ICCst: v t E R"vU( . )v t 2 0

b(t, E , 5 P(IEId, t ) + r ( l / u l l c ) . (ISS)

This was the form of the original definition of (ISS) given in [12]. It is known that a system is (ISS) if and only if it

satisfies a dissipation inequality, that is to say, there exists a smooth V : IR" i I R > o and there are functions aL E IC,, i = 1, 2, 3 , and o E so that

a l ( l t l A ) 5 V(<) 5 aZ(/ 'dA)

ov(E)f(E, U) 5 ~ ( I U I ) - a s ( l t l ~ )

( 2 )

( 3 )

and

for each E IR" and w E R". See [14] and [I51 for proofs and an exposition, respectively. A very useful modification of this characterization due to [ l l ] is the fact that the (ISS) property is also equivalent to the existence of a smooth V satisfying (2) and (3) replaced by an estimate of the type VV(<)f(<, U) 5 -V([) - o l y ( I [ l ~ ) . (This can be understood as: "for some positive definite and proper functions y = V ( x ) and 'U = W ( U ) of states and outputs, respectively, along all trajectories of the system we have = -y + w.") The main purpose of this paper is to establish further equivalences for the (ISS) property.

It will be technically convenient to first introduce a local version of the property (ISS), by requiring only that the estimate hold if the initial state and the controls are small, as follows:

3 P > 0, Y E IC, P E U s t : v I4ln I p, q ~ l l m I p

144 E , U)lA i P(IEIA, t ) + r(llullm) 2 0. ( L I W

Several standard properties of the "unforced" system obtained when U = 0 will appear as technical conditions. We review these now. The 0-global attraction property with respect to A (0-GATT) holds if every trajectory x( . ) of the zero-input system

(CO) : x = f(., 0) (4)

satisfies lim lx(t, I, Q ) / A + 0; if this is merely required of trajectories with initial conditions satisfying Iz(0)l~ < p, for some p > 0, we have the 0-local attraction property with respect to A (0-LATT). The 0-local stability property with respect to A (0-LS) means that for each E > 0 there is a S > 0 so that l t l ~ < 6 implies that jz(t, E . 0 ) l ~ < E for all t 2 0. Finally, the 0-asymptotic stability property with respect to A (0-AS) is the conjunction of (0-LATT) and (0-LS), and the 0- global asymptotic stability property with respect to A (0-GAS) is the conjunction of (0-GATT) and (0-LS). Note that (0-GAS) is equivalent to the conjunction of (0-AS) and (0-GATT). It is useful (see, e.g., [3] , [7], and [ 121) to express these properties in terms of comparison functions

t i ,

3 p E ICCst: v < E R"Vt 2 0

Iz(t, <, o)ln i P(l<lA, t ) (0-GAS) and

3 p > 0, p E ICCst: V)I<IA < pvt 2 0 Iz(t, E , o)lA 5 P(IE/A, t ) (0-AS)

respectively. Next we introduce several new concepts. The limit property

with respect to A holds if every trajectory must at some time

SONTAG AND WANG: INPUT-TO-STATE STABILITY 1285

get to within a distance of A which is a function of the magnitude of the input

Theorem 1: Assume that given any forward-complete system C as in (l), with U = R" and a compact zero-invariant set A for this system, the following properties are equivalent:

3 7 E Nost : E R"VU(*) 4 (ISS);

Observe that if this property holds, then it also holds with some y E IC,. However, the case y 5 0 will be of interest since it corresponds to a notion of attraction for systems in which controls U are viewed as disturbances.

The asymptotic gain property with respect to A holds if every trajectory must ultimately stay not far from A, depending on the magnitude of the input

d) (LIM) and (0-GAS); e) (AG) and (0-GAS); f) (AG) and (LISS); g) (AG) and (LS); h) (LIM) and (LS); i) (LIM) and (GS); j ) (AG) and (GS). This theorem will follow from several technical facts which

are stated in the next section and proved later in the paper. These technical results are of interest in themselves.

C. List of Main Technical Steps

Again, if the property holds, then it also holds with some y t ICm, but the case y G 0 will be of interest later. The uniform asymptotic gain property with respect to A holds if the limsup in (AG) is attained uniformly with respect to initial states in compacts and all U

We assume a given forward-complete system C as in (1) with U = IR" and a compact zero-invariant set A for this system. For ease of reference, we first list several obvious implications:

(UAG) * (AG); ( 5 ) (AG) * (LIM); (6) (ISS) ==+ (0-GAS); (7)

(LISS) =+ (0-AS); (8) (LISS) * (LS). (9)

The boundedness property with respect to A holds if bounded initial states and controls produce uniformly bounded trajectories

Because W M ) implies (O-GATT), and @-GAS) is the Same as @AS) Plus (@GATT), we have

(LIM) and (0-GAS) (LIM) and (0-AS). (10)

301, 02 E N s t : V < E R" V U ( , ) SUP l ~ ( t , E , u)lA It was shown in [15] that t>o

L m={~l(lEIA), a2(11ullco)>. (BND) (ISS) (UAG) and (LS). (11)

It turns out that (LS) is redundant, so (UAG) is in fact equivalent to (Iss) as stated next. (This is sometimes called the "UBIBS" or "uniform bounded-

input bounded-state'' property.) The global stability property with respect to A holds if, in addition, small initial states and

Proposition I.I:

controls produce uniformly small trajectories (UAG) + (LS).

This observation generalizes a result which is well known for systems with no controls (see, e.g., [I , Th. 1.5.281 or [3,

3g1, ff2 E Nost: V t E IR" V U ( ' ) sup I X ( t , E , u)lA t > O -

5 max{al(lEIA), a z ( l l U l l w ) > . (GS) Th. 38.11). It should be noted that the-standing hypothesis that A is compact is essential for this implication; in the general

respect to A is not redundant. From Proposition 1.1 and (7), we know then that

Observe that if this property holds, then it also holds with both Of noncompact sets A, the local property with

fft E IC,. The local stability property with respect to A holds if we merely require a local estimate of this type

(UAG) * (0-GAS).

Lemma 1.2: (BND) and (LS) a (GS). Lemma 1.3: (LIM) and (GS) If this property holds, then it also holds with both ai E IC,,

2 = 1, 2 . Lemma1.4: (LIM) + (BND). (AG) and (GS).


The converse of Lemma 1.5 is of course false, as illustrated by the autonomous system x = 0 (with n = m = 1) which even satisfies (GS) but does not satisfy (LIM). From Lemmas 1.3 and 1.5, we have that

(LIM) and (LS) e (LIM) and (GS).

The most interesting technical result will be Proposition 1.6.

(13)

Proposition 1.6:

(LIM) and (LS) +- (UAG).

We now indicate how the proof of Theorem 1 follows from

(A e C): by Proposition 1.1 and (11). (C j E): by (5) and (12).

* ( E + F ) : by Lemma 1.2. ( F + G): by (9). (G 3 H ) : by (6).

* ( H 3 I ) : by (13). ( I + J ) : by Lemma 1.4. ( J + G): obvious. ( H + C): this is Proposition 1.6. ( E + 0): by (6). ( B 0): by (10). ( D j H ) : by Lemma 1.2 and (9).

A very particular consequence of the main theorem is worth focusing upon: A J , i.e., (ISS) is equivalent to having both the global stability property with respect to A and the asymptotic gain property with respect to A. Consider the property

all these technical facts.

3y E No st: V t E R” Vu( . ) r 1

(the limsup is understood in the “essential” sense of holding up to a set of measure zero; note also that since y is continuous and nondecreasing, the right-hand term equals limtitooy[/u(t)I]). It is easy to show (see Lemma 11.1) that this is equivalent to (AG). The conjunction of (14) and (GS) is the “asymptotic C, stability property” proposed by Tee1 and discussed in [2] (in that paper, A = (0)); it thus follows that asymptotic C, stability is precisely the same as (ISS).

In [18], Tsinias considered the following property (in that paper, A = (0)):

-

3y E ICst: v t E R” V u ( . )

{Iz(t, E , .)Id 2 y[lu(t)/ lvt 2 0)

which obviously implies (LIM). The author considered the conjunction of (15) and (LS) (more precisely, the author also assumed a local stability property that implies (LS), namely f ( z , U ) = Az+Bu+o(z, U ) , with A Hurwitz); because of the equivalence A J H , this conjunction is also equivalent to (ISS).

The outline of the rest of the paper is as follows. In Section I1 we first prove Proposition 1.1, Lemmas 1.2-1.4, and the equivalence between Property (14) and (AG), all of which

are elementary. Section I11 contains the proof of the basic technical step needed to prove the main result, as well as a proof of Lemma 1.5. After this, Section IV establishes a result showing that uniform global asymptotic stability (UGAS) of systems with disturbances (or equivalently, of an associated differential inclusion) follows from the nonuniform variant of the concept; this would appear to be a rather interesting result in itself, and in any case it is used in Section V to provide the proof of Proposition 1.6. Finally, in Section VI we make some remarks characterizing so-called “practical” ISS stability in terms of ISS stability with respect to compact sets.

11. SOME SIMPLE IMPLICATIONS We start with the proof of Proposition 1.1.

Proof: We will show the following property which is equivalent to (LS):

v & > 0 3 6 > O s t : V l < / A < 6 v l IU l lm 5 6 SUP Iz(t, E , .)la 5 E . (16)

Indeed, assume a given E > 0. Let T = T ( E / ~ , 1). Pick any 61 > 0 so that ~(61) < & / a . Then for all 5 1 and

t2o

llullm 5 61

(17) E

Sup b(t, t, .)Id 5 5 + r ( l l U l l m ) < E . t>T

By continuity (at u E 0 and states in A) of solutions with respect to controls and initial conditions and compactness and zero-invariance of A, there is some S2 = & ( E , T ) > 0 so that

I d 4 5 62

and

5 E .

Together with (17), this gives the desired property with 6 := min(1, 61, S,}.

We now prove Lemma 1.2. Proof: We first note that the 0-global asymptotic stability

property with respect to A implies the existence of a smooth function V such that

al(/dd) <v(t) 5 aZ(ltld) v t IR“ for some al , a2 E IC,, and

ov(<)f(t, 0) 5 -OY(/<lA) Vt E R”

for some a3 E IC, (this is well known; see, for instance, [9] for one such converse Lyapunov theorem). Following exactly the same steps as in the proof of [13, Lemma 3.21, one can show that there exists some function x E IC, such that for all X(lUl) I ltla 5 1

(Here we note that in the proof of [13, Lemma 3.21, the function g ( s ) = 1 for s E [ O , 11.)


Using exactly the same arguments used in [12, p. 4411, one can show that there exist a KL-function P and a IC,-function y so that if Iz(t, E , U ) ~ A 5 1 for all t E [0, T ) for some T > 0, then it holds that

for all t E [0, T ) . Let p = min{K1(l/2), y-'(1/2)}, where K ( T ) = P(T, 0) for T 2 0. Note here that p 5 K1(1/2) 5 l / 2 . We now show that the (LISS) property holds with these ,El, y, and p. Fix any < and U with 5 p and llu\lm 5 p. First note that lz(t, <, U ) [ < 1 for t small enough.

Claim: Is@, E , U ) ~ A 5 1 for all t 2 0. Assume the claim is false. Then with

it holds that 0 < tl < 00. Note that Iz(t, E , U ) ~ A < 1 for all t E [0, t l ) . This then implies that

By continuity Iz(t1, t, U ) ~ A < 1, contradicting the definition of tl . This shows that tl = 00, i.e., Iz(t, E , u ) l ~ 5 1 for all t 2 0. Thus the estimate in (19) holds for all t , as desired.

Next we prove Lemma 1.3: Boundedness property with respect to A and local stability property with respect to A implies global stability property with respect to A (the converse is obvious).

Proof: Assume that (BND) and (LS) hold for a given choice of 6,a1, az, c q , az. Pick a constant c 2 0 and two class-K functions PI and P 2 so that for each i = 1, 2, a,(s) 5 Pz(s) + c for all s 2 0. Pick two class-K functions y1 and 7 2 so that for each i = 1, 2, it holds that

Consider any E and U . Then (GS) holds. Indeed, if both \C IA 5 6 and \lullw 5 6, then this follows from (LS). Assume now that \C IA > 6. Thus (BND) implies that for all t 2 0

The case JIu(I, > 6 is similar. Lemma 1.4 says that the limit property with respect to A

plus the global stability property with respect to A imply the asymptotic gain property with respect to A; it is shown as follows.

Proof: Let 01, a2, y E n/, be as in (LIM) and (GS). We claim that (AG) holds with

3 s ) := max((a1 O Y ) ( S ) , 0 2 ( 4 ) .

Pick any I , U , and any E > 0. By (LIM), there is some T 2 0 so that Iz(T, E , U)~A L y(l lul lm) + E . Applying (GS) to the initial state z(T, E , U ) and the control w ( t ) := u(t + T ) , we conclude that

5 max{al[r(llUllm) + El , a2(llullm)}

and taking E + 0 provides the conclusion. Finally, we show the following. Lemma ZZ.1: Property (14) is equivalent to (AG).

Proof: Since y[Kt++, lu(t)I] 5 y(l\uIl,), (14) implies (AG) with the same y. Conversely, assume that (AG) holds; we next show that (14) holds with the same y. Pick any E E IR", control U , and E > 0. Let T := E+,+m Iu(t)I. Let h > 0 be such that y(r + h) - y ( ~ ) < E. Pick T > 0 so that lu(t)l 5 T + h for almost all t 2 T , and consider the functions x ( t ) := z ( t + T ) and v ( t ) := u(t + T ) defined on R>o. Note that 'U is a control with l l w l l m 5 T + h and that - ~ ( t ) = z( t , <, w), where C = the asymptotic gain property initial state < and control w

-

lim Iz(t, <, U ) ( d t-+m

Letting E -+ 0 gives (14).

z(T, E , U ) . By the definition of with respect to A, applied with

-

= lim lz(t, <, V ) l A t-+m

L r ( l lv l lm)

L r(' + h) < Y ( T ) + E .

111. UNIFORM REACHABILITY TIME

Let (1) be a forward-complete system. For each subset 0 of the input-value space W, each T 2 0, and each subset C & R", we denote

RT,(C) : = { ~ ( t , E , U ) 10 5 t 5 T , U E M u , E E C } and

R o ( C ) := { ~ ( t , E , U ) I t 2 0, U E M O , E E C } = U RT,(C).

TLO

In [9, Proposition 5.11, the following is shown. Fact Ill. I : Let (1) be a forward-complete system. For each

bounded subset 0 of the input-value space U, each T 2 0, and each bounded subset C C IR", R g ( C ) is bounded. 0

Given a fixed system (1) which is forward-complete, a point E E Rn, a subset S C_ R", and a control U , one may consider the "first crossing time"

r(E, S , U ) := inf {t 2 0 I z(t , E , U ) E S }

with the convention that T ( < , S , U ) = $00 if z ( t , E , U )

for all t 2 0. S


The following result and its corollary are central. They state in essence that for bounded controls, if T ( < . S, U ) is finite for all U , then this quantity is uniformly bounded over U up to small perturbations of < and S and (the corollary) uniformly on compact sets of initial states as well. (Observe that we are not making the assumption that f is convex on control values and that the set of such values is compact and convex which would make the result far simpler, by means of a routine weak- * compactness argument.) The result will be mainly applied in the following special case: 0 is a closed ball in Rm, W = IR”, and for a given compact set A, C (in the Corollary) IS a closed ball of the type B(A, as), p E C, R = B(A, 2s), and K = B[A, $SI. But the general case is not harder to prove, and it is of independent interest.

Lemma 111.2: Let (1) be a forward-complete system. As- sume the following are given:

an open subset 0 of the state-space IR”; a compact subset K c $ 2 ; a bounded subset 0 of the input-value space U; a point p E R“; a neighborhood W of p ;

so that

sup .(p, R, U ) = +oo. (20) uEMo

Then there is some point q E W and some v E M O such that

(21) ~ ( q , K , W) = +W.

Proof: Let pa = p be as in the hypotheses. Thus for each integer k 2 1 we may pick some d k E M O so that z ( t , p o , d k ) $ R for all 0 5 t 5 k . For each j 2 1, we let

Consider first {Oj(t)}j>l as a sequence of functions defined on [0, 11. From Fact 111.1 we know that there exists some compact subset 5’1 of IR” such that z ( t , p o , d j ) E S1 for all 0 5 t 5 1, for all j 2 1. Let M = sup{lf((, X)l: E 5’1, X E 0). Then I(d/dt)Qj(t)i 5 M for all j and almost all 0 5 t 5 1. Thus the sequence { O j ( t ) ) j 2 1 is uniformly bounded and equicontinuous on [0, 11, so by the Arzela-Ascoli theorem, we may pick a subsequence {o~(j)}j?l of {j}j?l

with the property that { Q c l ( , ) ( t ) } j ? 1 converges to a continuous function ~ , l ( t ) uniformly on [0, 11. Now we consider {ODl(,)(t)}j21 as a sequence of functions defined on [l, 21. Using the same argument as above, one proves that there exists a subsequence {oz(j)}j?~ of {OI(J’)}~?I such that {d,,cj,(t)}j21 converges uniformly to a function & z ( t ) for t E [l, 21. Since {oz(j)} is a subsequence of {ol(j)}, it follows that ~ ( 1 ) = ~ 1 ( 1 ) . Repeating the above procedure, one obtains inductively on k 2 1 a subsequence {ok+l(j)}j?l of {ak(j)}jrl such that the sequence {d,b+l(j)(t)}j21 converges uniformly to a continuous function ~ , k + l on [ k , k+1]. Clearly, ~ , k ( k ) = ~ ~ k + ~ ( k ) for all k 2 1. Let IC be the continuous function defined by ~ ( t ) = ~ k ( t ) for t E [k - 1, k ) for each k 2 1. Then on each interval [k - 1, k ] , ~ ( t ) is the uniform limit of {d,,( ,)( t)}.

Since the complement of 62 is closed and the O j ’ s have images there, it is clear that K, remains outside R and hence

Q,(t) = x ( t , p o , d j ) , t 2 0.

outside K . If K, would be a trajectory of the system corresponding to some control U , the result would be proved (with q = P O ) . The difficulty lies, of course, in the fact that there is no reason for IC to be a trajectory. However, IC can be well approximated by trajectories, and the rest of the proof consists of carrying out such an approximation.

Some more notations are needed. For each control d with values in 0, we will denote by Ad the control given by Ad(t) = d ( t + 1) for each t in the domain of d (so, for instance, the domain of Ad is [-1, +a) if the domain of d was IR>O). We will also consider iterates of the A operator, A‘d, corresponding to a shift by k . Since K is compact and R is open, we may pick an T > 0 such that B ( K , T ) C R. We pick an r0 smaller than 7-/2 so that the closed ball of radius To around po is included in the neighborhood W in which q must be found. Finally, let p k = ~ ( k ) for each k 2 1. Observe that both po and p l are in 5’1 by construction.

Next, for each j 2 I, we wish to study the trajectory z[-t, p l , Ad,,,j,] for t E [0, 11. This may be a priori undefined for all such t. However, since S1 is compact, we may pick another compact set ,’?I containing B(S1, T ) in its interior, and we may also pick a function f : R” x IR“ -+ IR” which is equal to f for all (x, U ) E SI x 0 and has compact support; now the system x = f(x, U) is complete, meaning that solutions exist for all t E (-m, CO). We use ?(t, <, U )

to denote solutions of this new system. Observe that for each trajectory 2( t , E , U ) which remains in SI, z ( t , E , U )

is also defined and coincides with 2 ( t , E , U ) . In particu-

each j , since these both equal x[1- t , p o , d,,(,)] for each t E [0, 11. The set of states reached from 5’1, using the modified system in negative times t E [-1, 01, is included in some compact set (because the modified system is complete, and again appealing to Fact 111. I) . Thus, by Gronwall’s estimate, there is some L 2 0 so that for all j 2 1 and all t E [0, 11

1 2 x 9 w, Qul(,)(l), A d 4 = 4-6 b ( j ) ( 1 ) , Ad,,,j)l for

IZ[-t, P I , AdD,(j)l - +t> Q,,,j&), Ad,,(j)lI

(no “N” needed in the second solution, since it is also a solution of the original system). Since Ou,(j)(l) -+ pl, it follows that there exists some jl such that for all j 2 j ,

for all t E [0, 11. Note that this means in particular that Z [ - t , p 1 , Ad,,(,)] E B(S1, 7-/4) 3, for all such t , for all j 2 j1, so “-”can be dropped in (22) for all j 2 jl. Now let 0 < r1 < T O be such that

for all t E [0, I], for all p B ( p l , T I ) . As this implies in particular that 5[-t, p , Ad,,(,,)] E B(S1, r / 2 ) C 51, again tildes can be dropped. Combining (22) and (23), it follows that for each p E B(p1, T I ) , x[-t, p i Ad,,(,,)] is defined for all t E [0, 11 and

P , ~ d o , ( , , ) I - +t, QC1(,,)(1), ~d , , ( , , ) l l < To (24)


for all t E [0, 11. Let wl(t) = d,l(,l)(t). Then (24) implies that for each p E B(pl , r1) it holds that x(-1, PI , Awl) E B ( p o , T O ) , and, since z[-t, Bal(,l), Ad,,(,,)] for all t E [O, 11

for all t E [0, 11. Then for such a choice of rk+l, it holds for each p E B ( p k + l , rk+l), by (27), that

I+t? P? AWk+l) - +t, ~,r+1(3k+,)(k + I), AWk+lll 3rh r < - < - 4 2 4 - 6 P , Awl) e ( K ? ;) v t E 10, I].

In what follows we will prove, by induction, that for each z 2 1, there exist 0 < T, < ~ , - 1 and w, of the form w, = for some j so that for all p E B(p,, T,),

for all t E 10, 11 which implies that for each P E B(pk+l, 'Q+l)

x( - t , p , Aw,) is defined for all t E [0, 11 4 - 6 P? AWk+l) B ( K , ;) \Jt E [O, 11.

Moreover, it follows from (28) and (29) that

and 4-1, P ? AWk+l) E B(Pk, T k )

+I? P , E B(p,-1, Tt-1).

The case i = 1 has already been shown in the above argument. Assume now that the above conclusion is true for

i = 1, 2 , . . + , k . Consider, for each j , the trajectory x[- t , P ~ + I , Ak+1d,k+1(3)]. Again, this may be a priori undefined for all such t . But, again, by modifying the system in a compact set 3 k + l containing a neighborhood of Rk+'(p~) , one can show that there exists some yk+l 2 5 + 1 so that for all j 2 yk+l, z[- t , p k + l , Ak+1d,k+l(3)] is defined for all t E [0, 11, and

for all p E B(pk+l, T ~ + I ) . This completes the induction step. Finally, we define a control w on R2o as follows:

~ ( t ) = wk(t - k + 1)

for each integer k 2 1. Then 'U E M O . For each k , we consider the trajectory z ( - t , p k , A'.) for

t E [O,k]. Inductively, x(- i , p k , A%) E B ( p k - , , r k - , )

for each z 5 k , so this trajectory is indeed well defined. We now let z k = z ( - k , p k , A%). Note that also from ~ ( - 2 , p k , A") E B(pk- , , r k - , ) we can conclude that

if t E [k - I, k )

~ ( - 4 P ~ C , A") 6 B ( K , 5) V t E [O, k] (30) M-t, Pk+l? Ak+ld,,+,(,)I T k

- 4-t , Q,,+,(,)(k + I)? Ak+147k+l(j)ll < 4 (25) which is equivalent to

for all 0 5 t 5 1. We note here that for any j , z[-t, d , k + l ( j ) , Ak+1d,k+1(3k+l)] is defined for all t E [0, k + 11 and

~ ( t , z k , U) B (IT, f> V t E [O, k ] . (31)

As xk E B ( p 0 , T O ) for all k , there exists a subsequence of { z k } that converges to some point q E W . To simplify notation, we

4-1, f l c T k + l ( j ) ( k + 11, Ak+1dc7k+l(,)1 = QOk+,(,)(k).

function of initial states i n B ( p o , r0) and using the control 'U

on the interval [O, NI, we know that there exists some Ll > 0 such that

for all t E [o, 11, for all j 2 j k + l . Let j k + l =

max{?k+l, h + 1 > , and k t wk+l(t) = Akd,k+l(3k+l)(t) for t E [0, I]. Then (25), applied with j = jk+1, says that

for all t E [0, 11. Using the case t = 1 of this equation, as well as (26) applied with j = j k + ~ and t = 0 and the

we conclude that equality ~ [ - l , K T k + l ( , k + l ) ( k + 11, AWk+lI = Q Uk+l(,k+l)(k)'

4-1, p k + l , A w ~ c + I ) E B ( ~ k , ?). (28)

Pick any r k + l so that 0 < rk+l < r k and for every p E

B(pli+l, rlc+l), z(-t, p , Awk+l) is defined for all t E [O, 11, and

Without loss of generality, we assume that ko 2 N . Combin- ing (32) with (31), it follows that x ( t , q , U) e B ( K , r / 4 ) for all t E [0, NI. As N was arbitrary, it follows that

The contrapositive of Lemma 111.2 gives that if from each point we can reach R in finite time, then the set K can be reached in uniform time from each state. This can in tum be

~ ( t , q , w) @ K for all t >_ 0.

T k made stronger to provide, in addition, uniformity on compacts for initial stateS as follows. M-t? P , A W k + l ) - X ( - C Pk+l, AWk+l)I < y (29)

1290 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 41, NO. 9, SEPTEMBER 1996

Corollary 111.3: Let (1) be a forward-complete system. As-

a compact subset C of the state-space R"; an open subset R of the state-space R"; a compact subset K C R; a bounded subset 0 of the input-value space U;

sume the following are given:

so that

V t E RTL V U E M O 3 t 2 0 s.t. ~ ( t , E , U ) E K.

Then

SUP { ~ ( p , Q, U ) 1 p E C, U E M O } < fa.

Proof: Pick an open set Ro and an E > 0 so that K c Ro c B(Ro, E ) C R. By Lemma 111.2, applied with QO in place of R and W = R", we know that for each p E C there is some T = Tp so that for each U t M O there is some t E [O, TI so that z( t , p i U ) E 00. (Otherwise, we would have supuEMo ~ ( p , 00, U ) = +CO, and thus by the Lemma there is some q E W = IR" and some 'U E M O such that ~ ( q , K , w) = +cc so that z ( t , q , w)&tinK for all t 2 0, contradicting the assumption.) By Fact 111.1 and Gronwall's lemma, there is also some L = L, > 0 [Lp is obtained from the Lipschitz constant for f on R z ( C ) ] so that

146 E , U ) - 4 4 P , U)l 5 LIE -PI

for all t E [0, TI and all ( E C. Denote 6, := Lp/&, pick a finite cover of C by sets of the form B ( p , 6,), and let T be the largest of the Tp's in this cover. Now given any E C and any U E M O , choose p so that E B(p, 6,) and t E [O. Tp] so that ~ ( t , p , U ) E 00; then also z ( t , E , U ) E R and t 5 T .

Corollary 111.4: Under the hypotheses of Corollary 111.3, and if in addition R c C, there is some T 2 0 so that

RO(C) = R S ( C )

and in particular R o ( C ) is bounded. Prooj? Let T = s u p { r ( p > s1, U ) Ip E C, U E M O } .

Pick any y E Ro(C); thus y = z(to, E , U ) , for some E E C, U E M O , and t o 2 0. Let tl := max{t 5 t o Iz(t, E , U ) t C}, and p := z(t1, E , U ) . By maximality of tl, x ( t , p, U) 61 C for any t E [0, t o - t l ] , where v( t ) = w ( t + t 1 ) . Thus, from the definition of T , t o - tl < T . So y = to - t l , p , w) E

As a simple application, we now prove Lemma 1.5: If C is forward-complete and satisfies the limit property with respect to A, then it satisfies the boundedness property with respect to A.

Proof: Assume without loss of generality that y E IC, For each s > 0, define

RT, (C) . a

m ( s ) := S U P (144 E , U ) / A I E E R A , as ) , U E MO, t 2 0, c3 = B[O, y-l(s)]}.

We claim that a ~ ( s ) < 00 for all s > 0. To see this, pick an s > 0 and let C = B(d, as) , R = B(A. a s ) , and K = B(A, 3s /2) . Pick any U E M O , 0 = B[O, y-l(s)]. Note that a l ( s ) is the largest possible value of l v l ~ for

v E %?,(C). Property (LIM) implies that for each E E R", there is some t 2 0 so that

144 E , U)IA I ;r(ll&o) 5 9 [ Y 1 ( S ) I

3s 2 - - -

that is, z( t , <, U ) E K . We may then apply Corollary 111.4, to conclude that Ro (C) is bounded so 01 (s) is indeed finite. We also define, for each r > 0, the (finite) number

c2(7-) := m[y(r)I.

Finally, we define ( ~ ~ ( 0 ) := 0 for z = 1, 2. Note that both (TI

and ( ~ 2 are nondecreasing functions. Assume without loss of generality that both (T, E Jf (if this does not hold, one may pick a larger 5, 2 0,). We next show that (BND) holds with these definitions.

E R" and a control U , and let s := lcln and r := ~ ~ u ~ ~ ~ . Let x ( t ) = z(t ,<, U ) for all t 2 0. I f s = T = 0 then E E A, so zero-invariance of A means that the right-hand side in the estimate (BND) vanishes. So assume s > 0 or r > 0. Consider the two following cases.

Case 1: Assume that s 2 y ( ~ ) . Note that necessarily s > 0. We have a trajectory with E E B(A, 2s) and with llull, = r 5 ~ - ' ( s ) , so by definition of a l ( s ) it holds that

Case 2: Assume that s 5 y(r). Note that necessarily T > 0. Define SX := y(r). We have that ~ [ I A = s I 2 < 22 and that llulio0 = r = y-'(i). Thus for all t it holds that I x ( t ) l ~ 5 al(E) = a l [y ( r ) ] = a~()Iu11,) as desired.

Pick

I.(t)lA I Cl(S) = fll(iElA) for all t .

IV. UNIFORM STAEILITY

The notions introduced next are motivated by thinking of inputs not as controls but as "time-varying parameters" or "multiplicative uncertainties." Again we assume a given forward-complete system C as in (1) and a compact zero- invariant set A for this system.

The global asymptotic stability property with respect to A (GAS) holds if the system is uniformly stable with respect to A

V & > 0 3 s > 0 s.t.: V I(lA I s SUP I4t, E , .)I/. I E V u ( . ) (33) t>o

and is attractive with respect to A

Note that this last condition is just as in (AG) with y = 0. Observe also that if (33) holds, then the set A must be invariant in the strong sense that ~ ( t , ( , U ) E A whenever E A for all inputs U (not just if U = 0).

The uniform global asymptotic stability property with respect to d holds if the system is uniformly stable with respect to A and is uniformly attractive with respect to A

V E > 0 V K > 0 3T = T ( e , 6) 2 0 s.t.:

I t lA 5 * SUP Iz(t, E , U)lA I E V U ( ' ) . t t T


This is the same as asking that (UAG) hold with y = 0. The main result in this section is as follows.

Theorem 2: Assume a given forward-complete system C as in (1) and a compact zero-invariant set A for this system. Furthermore, assume that the set of control values U is compact. Then the system satisfies (GAS) if and only if it satisfies (UGAS).

Proofi Given E > 0 and /c, > 0, first find S as in (33). Let 0 = U, K = B(A, 6/2), 0 = B(A, S), and C = B(A, K ) .

By (34), the hypotheses of Corollary 111.3 hold so there is some T 2 0 such that whenever l [ l ~ 5 K , and for each control U ,

there is some t o = to([ , U ) 5 T so that (x( t0, [, U ) ~ A 5 S. From the choice of 6, it follows that \z ( t , (, U ) \ A 5 E for all t 2 t o and hence for all t 2 T.

v. PROOF OF THE MAIN PROPOSITION

We now prove Proposition 1.6. We must show that (LIM) and (LS) together imply (UAG) (the converse is obvious). By (13), we may assume that (GS) holds, so by Lemma 1.4, (AG) is also true. So from now on we assume both (AG) and (GS). The proof is based on first introducing a new system-which appears also in an argument used in [ 151-having a compact input-value set and satisfying (GAS), then using the equivalence (UGAS) = (GAS) for this auxiliary system to conclude that the original system is (UAG). (An intuitive interpretation is that this allows one to restrict attention to input values that are subject to a state-dependent constraint of the type y[ (u( t ) ( ] 5 I z ( t ) l ~ ; inputs not satisfying this constraint do not matter, since for them already z ( t ) is bounded by a function of the input magnitude.)

We start with a general construction. Take any locally Lipschitz function p: R” + R>o which vanishes on the set A. Consider the auxiliary system with the same state-space R”, input-value set U0 equal to the closed unit ball B(0, 1) in U = R”, and equations as follows:

(qJ: 5 = f[., 4 x 1 dl = f d x , 4.

We use ‘‘d(.)” to denote inputs to (E,) to avoid confusion with inputs to the original system, and for each E R” and each d we use z,(t, E , d) to denote the trajectory of (E,) with initial state E and input d. The system (E,) may not be complete even if the original system C is (for example, x = U and ~ ( z ) = z2), but on the domain of the definition of the solution one has that zp(t, E , d ) = z ( t , E , U ) , where u(t) := p[z,(t, E , d)] d ( t ) . Note that for (E,), A is invariant in the strong sense that all trajectories starting in A remain there (since for each the solution of 2 = f ( z , 0), ~ ( 0 ) = E , satisfies cp[z(t)] = 0, by zero-invariance of A and hence also satisfies i ( t ) = f{z(t),cp[x(t)] d ( t ) } for all d).

Let 01, a 2 , and y be so that (GS) and (14) [already shown to be equivalent to (AG)] both hold. Without loss of generality, we assume that these are of class IC, and that y = 0 2 . Pick any smooth IC,-function p such that

and let p([) = p ( l [ I ~ ) . We claim, with this choice of cp, the system (E,) is complete and satisfies the global asymptotic stability property with respect to A.

We first show that for each [ E R” and each input d, and denoting zp(t, E , d) simply as x v ( t )

02{(P[Z~(~)l) 5 $al(lEIA) (35)

for each t in the maximal interval [0, T,,,) of the definition of xv. Once this is established, it will follow that T,,, = fcc (completeness) and that every trajectory is bounded. If [ E A, then (35) holds for all t 2 0, since both sides vanish. So assume that [ E lR” \ A and d is an input. Denote x p ( t , [, d) simply as z,(t). For each t small enough, az{cp[z(t)]) < 2 gal ( l [ lA) Since

”) = ~Z[P(151A)I 5 f l t l A I $ 4 t I A ) .

Now let

t l := inf{t E [O, T,,,) I cJ2{p[z,(i)]} 1 : m ( ) s I A ) }

with tl := T,,, if the set is empty. Then (35) holds on [0, t l ) . The restriction to the interval [0, t l ) of z, is the same as the restriction to that interval of z( t , E , U), where u(t) = cp[z,(t)]d(t) for t E [0, tl) and u(t) = 0 for t 2 t 1 . Note that Iu(t>l = Icp[o,(t)]d(t)) 5 p[zp(t)] for

on t E [0, t l ) implies that uz(JIuJJ,) 5 $al(l[I~). Together with (GS), we conclude that

I%(t)lA I m(lEl”4) (36)

t E LO, tl), so ~ 2 ( 1 1 ~ l l c o ) L suPt,[,,tl) cJz{cp[z,(t)I), and (35)

whenever t E [0, t l ) . From this it follows that

a2{‘P[Z~( t )1} 5 a2{P[al(l<lA)l}

5 ;ffl(lElA)

for all t E [ O , t l ) . If tl < T,,,, then by continuity as t -+ t;, we conclude that 02{p[xp(t1)]} 5 $al(lElA) which contradicts the definition of t l . Thus tl = T,,,, and in particular (35) and (36) hold for all t 2 0, and T’,, = - fco. Equation (36) shows uniform stability and also that Emt,, I zv ( t ) I~ < CO for each trajectory. Attraction follows because for any trajectory xp, using the fact that for all C

Property (14) implies

from which it follows that

and so the system (E,) is (GAS) as claimed.


It then follows from Theorem 2 that (E,) is (UGAS). To complete the proof of Proposition 1.6, we need to show that this implies that the original system C is (UAG). There are at least two ways to prove this fact. The first is very short but uses a converse Lyapunov theorem; the second is self-contained but longer. Each proof is of interest, so we provide both.

By the main result in [91, (E,) being (UGAS) implies that there exists a smooth, proper, positive definite function V : IR" i IR with the property that VV(<)f,(<, U ) < 0 for all < # 0 and all vectors v E a(0, 1) in R". This means that V V ( < ) f ( [ , p) < 0 whenever p E R" and p ( l < I ~ ) 2 (1-1.1 which, since p E IC,, implies the uniform asymptotic gain property with respect to A (see, e.g., [12]). Another proof, not using the existence of V , is as follows.

Pick any compact subset C c R" and any E > 0. Since (E,) is (UGAS), there is some T 2 0 so that

SUP SUP I",.(t, I, d)lA L E . S E C , d t2T

Pick any [ E C and any control U . Consider z ( t ) = z ( t , E , U ) .

Let t

0 .- 1nf{t I I I 4 l m 2 P[lz(t)lAl) (to = 00 if the set is empty), where ut denotes the restriction of U to the interval [t, CO). Define

for each t E [O, t o ) and d ( t ) := 0 for t 2 t o . Note that l ld/ lm I 1, i.e., d is an input for the system (E,). Thus

.(t) = f [ z ( t ) , 4t) l = f { x ( t ) , d(t)cp[4t)l} = f, [ 4 t ) , 4t)l

so z ( t ) = x,(t, E . d ) for all 0 5 t < to. From the choice of T , it follows that Ix,(t, 4, d ) l ~ 5 E for all t 2 T . Thus either

to IT (38 )

Iz(t)lA I C for all t E [T, t o ) . (39)

or

If to = 00, then this means that Iz(t)lA 5 E for all t 2 T . Otherwise, there is some sequence t , 4 t$ such that

Il'Ullm 2 /IutnIlm

2 P[l4tn)lal by definition of t o , where we denote 'U := ut. Therefore p [ I z ( t o ) l ~ ] 5 l l w l l m as well, by continuity of Iz(t)ln. So for each t 2 t o , using (GS)

Iz(t) Id 5 max {Ol [ / x ( t O ) la], OZ(1 I u 1 loo)}

I maxi(a1 oP-l)(ll'Ullm), m(Il'Ullm)}

= a(114lm) L Q(IIU11m)

where we defined

a ( s ) := max((a1 o p - ' ) ( s ) , oz(s)}.

We conclude that t 2 T ; in the case of (38), since t 2 to, /z(t)lA 5 a(IIuII..), and in the case of (39) we have that

t 2 t o . In either case, t 2 T implies I x ( t ) / ~ I E when t E [ T , ~ o ) and 1 4 t h 5 a(llull,) when

lz(t>ln 5 m a x i & , Q(llullm)> 5 a(llullm) + E

completing the proof of condition (UAG) (with ''7" function a).

VI. A REMARK ON "PRACTICAL" ISS Assume a given forward-complete system C as in (1) with

U = IR". In [ 6 ] , the system C is said to be "input-to-state practically stable'' (ISpS) if there exist a ICC-function p, a IC-function 7, and a constant c 2 0 such that

Iz(t, E , .)I 5 P(I<I, t ) + r(ll.llm) + c (40)

holds for each control U and each [ E IR". Compared with the definition of the ISS property, the difference is in the possibly nonzero constant e. In this section, we remark that this property can be rephrased in terms of the plain ISS property.

DeJinition VZ. I : The system C has the compact ZSSproperty if there is some compact zero-invariant set A such that C has the uniform asymptotic gain property with respect to A. 0

For any subset A C IR", not necessarily zero-invariant, we may consider the zero-input orbit from A

O ( A ) := (7 : Q = z( t , E , O ) , t 2 0 , < E A}

and let O ( A ) be the closure of O ( A ) . Lemma VZ.2: If property (UAG) holds for a set A, not

necessarily zero-invariant, then 0 (A) is bounded and the system is 1SS with respect to O(A) .

Proofi Since O ( A ) includes A, the system satisfies (UAG) with respect to O ( A ) , and the latter set is 0-invariant (since O(A) is). Thus we only need to prove that O ( A ) is compact, or equivalently, that O ( A ) is bounded. From property (UAG), we know that there exists some T 2 0 so that for each E A, Ix(t, E , O ) I A 5 1 for all t 2 T . By continuity, at states in A of solutions to the system with U E 0 with respect to initial conditions and compactness of A, there is some constant c so that lx(t, <, 0 ) l ~ 5 c for all t E [OjT]. Thus O ( A ) is included in a ball of radius max { 1, e}.

Proposition V1.3: The system C is ISpS if and only if it is compact-ISS.

Proo$ If (40) holds, then the property (UAG) holds with respect to the set A := B(0, e) , and hence by Lemma VI.2, the system is ISS with respect to O(A) . Conversely, assume that C satisfies the ISS property with respect to some compact (zero-invariant) set A. Without loss of generality, we assume that 0 E A (otherwise, we enlarge A; zero-invariance is not required). Then

with c := sup { 1 [ 1 , E A}.

SONTAG AND WANG: INPUT-TO-STATE STABILITY

As a consequence of the equivalence between these concepts, one may apply the general theory of ISS systems to the study of the ISpS property. As a simple illustration, we show next that the latter property can be characterized in terms of Lyapunov functions.

Proposition VZ.4: The system C is compact-ISS if and only if there are a smooth function V: R” + R>o - and functions a; E K m , i = 1, 2, 3 and CT E N so that

and

ov(<)f(E, U) I g(I.1) - a s ( k l ) . (41)

Proofi The sufficiency part is routine: If V is as stated, then ov(E)f(E, U) 5 - ~ 3 ( 1 E I ) / 2 whenever IEI 2 x ( I v I ) , where X ( T ) = CY;’ O 2a(r). Observe that x is a nondecreasing function. Using exactly the same arguments as used in [ 12, p. 4411, one can then show that there exist a ICL-function and some nondecreasing function y such that

144 E , .)I I P(III, 4 + r(llullC0)

for each input U and each initial state E , and it follows from here that the system is indeed compact-ISS.

To prove the converse, namely the existence of such a function V, we first appeal to the converse Lyapunov result in [9]. Assuming that the uniform asymptotic gain property with respect to A holds, there is some smooth V: R” -+ R ~ O and there are a, E IC,, i = 1, 2, 3 and CT E IC so that both (2) and (3) hold. As earlier, we may assume without loss of generality that A contains the origin. Thus the case A = (0) of the following claim then provides the desired conclusion.

Claim: Let A be any nonempty compact subset of A. Then there is a smooth function V : R” 4 R>o which satisfies the following conditions.

V is proper and positive definite with respect to the set A, that is, there exist 61, 6 2 E K m such that for all < E R”

&(lEld) 5 V(E) I &(I<IA). (42)

There exist a function 63 E IC, and a nondecreasing continuous function 5 such that

(43) vv(E)f(E, U) I ~ ( I u I ) - 6 3 s ( l E I ~ )

for all < E R” and for all w E R”. Indeed, consider the set A1 = { E : l E l ~ 2 l}. Since this is

disjoint from A, there is a smooth function ‘p: R” -+ [0, 11 so that

0, i f < E A , 1 if < E A l .

Similarly, there is some smooth, nonnegative function X defined on IR” which vanishes exactly on a. Now we define

V(E) := X ( I ) [ l - 4 E ) I + V(OP(E).

By construction, V is smooth and is proper and positive definite with respect to A; that is, there are comparison

1293

(where a3 and y are the comparison functions associat5d to V ) for all ‘U E R” and all ~ [ I A > 1. Since both A and A are compact, there exists some SO 2 0 such that l c l ~ 5 ~ [ I A + S O ;

thus

whenever 1 5 1 ~ 2 1 + SO, where (YY is any K m function which satisfies & ( T ) C Y ~ ( T - S O ) for all T 2 SO + 1. The proof of the claim, and hence the proposition, is completed by taking any nondecreasing continuous function 6 which majorizes both y(r) and the maximum of VV( [ ) f (< , U) over

Note that this result, as opposed to the Lyapunov characterization of the ISS property with respect to A = {0}, does not require CT E IC. However, one may always write CT 5 CS 6 for some class- K function 5 and some positive constant e.

all l E l ~ 5 1 + SO, Iw( 5 T .

REFERENCES

N. P. Bhatia and G. P. Szego, Stability Theory of Dynamical Systems. New York: Springer-Verlag, 1970. J.-M. Coron, L. Praly, and A. Ted, “Feedback stabilization of nonlinear systems: Sufficient conditions and Lyapunov and input-output techniques,” in Trends in Control, A. Isidori, Ed. Springer-Verlag, 1995. W. Hahn, Stability of Motion. X. M. Hu, “On state observers for nonlinear systems,” Syst. Contr. Lett., vol. 17, pp. 645-473, 1991. A. Isidori, “Global almost disturbance decoupling with stability for nonminimum-phase single-input single-output nonlinear systems,” Syst. Contr. Lett., to appear. Z.-P. Jiang, A. Teel, and L. Praly, “Small-gain theorem for ISS systems and applications,” Mathematics Contr., Sig. Syst., vol. 7, pp. 95-120, 1994. H. K. Khalil, Nonlinear Systems, 2nd ed. New York: Macmillan, 1996. M. KrstiC, I. Kanellakopoulos, and P. KokotoviC, Nonlinear and Adap- tive Control Design. New York: Wiley, 1995. Y. Lin, E. D. Sontag, and Y. Wang, “A smooth converse Lyapunov theorem for robust stability,” SIAMJ. Contr. Opt., vol. 34, pp. 124-160, 1996. W. M. Lu, “A state-space approach to parameterization of stabilizing controllers for nonlinear systems,” IEEE Trans. Automat. Contr., vol. 40, pp. 1576-1588, 1995. L. Praly and Y. Wang, “Stabilization in spite of matched unmodeled dynamics and an equivalent definition of input-to-state stability,” Math. Contr., Signals, Syst., vol. 9, pp. 1-33, 1996. E. D. Sontag, “Smooth stabilization implies coprime factorization,” IEEE Trans. Automat. Contr., vol. 34, pp. 435443, 1989. -, “Further facts about input to state stabilization,” IEEE Trans. Automat. Contr., vol. 35, pp. 473476, 1990. __, “On the input-to-state stability property,” European J. Contr., vol. 1, pp. 24-36, 1995. E. D. Sontag and Y. Wang, “On characterizations of the input-to-state stability property,” Syst. Contr. Lett., vol. 24, pp. 351-359, 1995; also “Notions equivalent to input-to-state stability,” in Proc. IEEE Con$ Decision Contr. __ , “Output-to-state stability and detectability of nonlinear systems,” submitted; also “Detectability of nonlinear systems,” in Proc. Con$ Information Science Systems. J. Tsinias, “Sontag’s ‘input to state stability condition’ and global stabilization using state detection,” Syst. Contr. Lett., vol. 20, pp. 219-226, 1993. __, “Versions of Sontag’s input to state stability condition and output feedback global stabilization,” J. Math. Syst., Estimation Contr., vol. 6, pp. 113-116, 1996.

Berlin: Springer-Verlag. 1967.

Orlando, FL: IEEE, 1994, pp. 3438-3443.

Princeton, NJ: 1996, pp. 1031-1036.

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Eduardo D. Sontag (SM’87-F’93) received the Licenciado degree from the University of Buenos Aires in 1972 and the Ph.D. degree, under Rudolf E. Kalman at the Center for Mathematical Systems Theory, from the University, of Florida, Gainesville, in 1976, both in mathematics.

Since 1977 he has been with the Department of Mathematics at Rutgers University, New Brunswick, NJ, where he is currently a Professor I1 of Mathematics as well as a Member of the Graduate Faculties of the Department of Computer

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 41, NO. 9, SEPTEMBER 1996

Science and of the Department of Electrical and Computer Engineering. He is also the director of the Rutgers Center for Systems and Control. He has authored over two hundred journal and conference papers and book chapters as well as the following: Topics in Art8cial Intelligence (Buenos Aires: Prolam, 1972), Polynomial Response Maps (Berlin: Springer, 1979), and Mathematical Control Theoty: Deterministic Finite Dimensional Systems (New York: Springer-Verlag, 1990). He has been a consultant for Bell Laboratories and Siemens. His current research interests include control theory and foundations of learning and neural networks.

Dr. Sontag is an Associate Editor for IEEE TRANSACTIONS ON AUTOMATIC CONTROL, Neural Networks, Control-Theory and Advanced Technology, SMAI-COCOV, Journal f o r Computer and Systems Sciences, Dynamics and Control, and Neurocomputing. In addition, he is a cofounder and comanaging editor of Mathematics for Control, Signals, and Systems.

Dr. Wang received an Associate Editor of Syst,

Yuan Wang received the Ph.D. degree in mathematics from Rutgers University, New Brunswick, NJ, in 1990.

She has been with the Department of Mathematics at Florida Atlantic University, Boca Raton, since August 1990, where she is currently an Associate Professor. In 1994 she held a short-term visiting position at LAGEP Universite Claude Bernard Lyon I, France. Her research interests include control theory, particularly realization and stabilization of nonlinear systems. NSF Young Investigator Award in 1994. She is an

ems and Control Letters.

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