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On Characterizations of Input-to-State Stability with Respect to Compact Sets
Copyright © IFAC Nonlinear Control Systems Design, Tahoe City, California, USA, 1995
On Characterizations of Input-to-State Stability with Respect to Compact Sets Eduardo D. Sontag*
and Yuan Wang**
*Department of Mathematics , Rutgers Uni1lersity, New Brunswick, NJ 08909, USA "Department of Mathematics, Florida Atlantic Uni1lersity, Boca Raton, FL 99-191 , USA Abstract. Previous characterizations of IS5-stability are shown to generalize without change to the case of stability with respect to sets. Some results on lS5-stabilizability are mentioned as well.
Key Words. Set Stability; input-to-state stability; Lyapunov functions ; control Lyapunov functions ; practical stability
1. Introduction
least for the case of compact invariant sets A , the results in (Sontag and Wang, 1995) indeed can be generalized with little or no change; in particular, ISS with respect to a given A is equivalent to the existence of an "Iss-Lyapunov function" relative to A. (Most of the proofs valid in the equilibrium case extend easily to the set case.) The property of there existing some invariant compact set A so that there is ISS with respect to A will be called the "compact-Iss" property; various remarks about this notion are included, including the equivalence with what is sometimes called "practical stability." Finally, we also sketch some applications of the ISS notion to feedback design and disturbance attenuation.
System stability with respect to input perturbations is one of the central issues to be studied in control. During the past few years , the property called "input to state stability" (ISS) has been proposed (originally in (Sontag, 1989a)) as a foundation for the study of such problems, and has been subsequently employed by many authors in areas ranging from robust control to highly nonlinear small-gain theorems (see for instance (Jiang et al. , 1994; Tsinias, 1989)). In the paper (Sontag and Wang, 1995), the authors established the equivalence among several natural characterizations of the ISS property, stated in terms of dissipation inequalities, robustness margins, and Lyapunov-like functions.
2. Set Input to State Stability
The ISS property was originally stated for stability with respect to a given equilibrium state of interest. On the other hand, in many applications it is of interest to study stability with respect to an invariant set A , where A does not necessarily consist of a single point. Examples of such applications include problems of robust control and the various notions usually encompassed by the term "practical stability." This motivated the study of the "set" version of the ISS property, originally in (Sontag and Lin, 1992), and developed with applications to the study of parameterized families of systems in (Lin et al., 1995). Given the interest in set-Iss , it is an obvious question to ask whether the equivalent characterizations given in (Sontag and Wang, 1995) for the special case A = equilibrium extend to the more general set case. It is the main purpose of this paper to point out that, at
We deal with systems of the following general form:
x=
I(x , u), where I : IRn x IRID -> IRn is a locally Lipschitz map, and we interpret x and u as functions of time i E IR, with values xCi) E IRn and u(i) E IRm , for some positive integers nand m. (Generalizations to systems evolving on manifolds , and/ or restricted control value sets, are also of interest , but in this paper, all spaces are Euclidean.) From now on, one such system is assumed given. A control or input is a measurable locally essentially bounded function u : IR>o -> IRID. We use the notation Ilull to indicate the L~-norm of u , and 1·1for Euclidean norm in IRn and IRm. For each ~ E IRn and each u E L~ , we denote by x(t , ~ , u) the trajectory of the system ~ with initial state x(O) = ~ and the input u. (This solution is a priori only defined on some maximal interval [0, Te, u) , with Te , u ::; +00, but the main definition will include the requirement that Te,u = +00 .) We recall in an appendix the no-
• Supported in part by US Air Force Grant F49620-95-10101 •• Supported in part by NSF Grants DMS-9457826 and DMS-9403924
203
we only need to prove that O(A) is compact, or equivalently, that O(A) is bounded. From the K-asymptotic gain property, we know that there exists some T :::: 0 so that, for each ~ E A, Ix(t,~, O)I A :::; 1 for all t :::: T. By continuity at states in A of solutions of Eo with respect to initial conditions, and compactness of A, there is some constant c so that Ix(t,~, O)I A :::; c for all t E [0, TJ. Thus O(A) is included in a ball of radius max{l, cl. •
tions of positive definite and class K, Krxn and KC comparison functions. Let A be a nonempty subset of ~ E ]Rn, I~IA
]Rn.
For each
= d(~,A) = inf{d(17,~), 17EA}
will denote the usual point-to-set distance from ~ to A. (So for the special case A = {O}, 1~I{o} = I~I is the usual Euclidean norm.) We say that such an A is a O-invariant set for E, or more precisely, for the associated "zero-input" or "undisturbed" system x fo(x) f(x, 0), (Eo)
=
=
Definition 2.4 The system E is compact-1ss if it is 1SS with respect to some compact set A. 0 By Lemma 2.3, this is equivalent to simply asking that system E has K-asymptotic gain with respect to some compact set A. Remark 2.5 The definition of the 1SS property is in terms of a uniform attraction property. Just as for systems with no controls (cf. (Bhatia and Szego, 1970), Theorem 1.5.28, or (Hahn, 1967), Theorem 38.1), compactness of A insures that the following stability-like property is automatically satisfied as well:
if it holds that x(t,~, 0) E A, I:/t:::: 0, I:/~ EA. That is, when the control is u == 0, the solution when starting from A is defined for all t :::: 0 and stays in A. Definition 2.1 The system E has K-asymptotic gain with respect to the nonempty subset A ~ ]Rn
if there is some l' E K (an "asymptotic gain") so that, for all u, solutions exist for all t :::: 0 and (1)
uniformly on compacts for
~.
For each c
The uniformity requirement (which, as will be shown in a forthcoming paper, can be substantially relaxed) means, precisely: solutions exist for all initial states, controls, and times, and for each real numbers r, c > 0, there is some T = T(r,c):::: 0 so that Ix(t,~,u)IA:::; c +I'(IlulD for all u, all I~I :::; r, and all t :::: T. This generalizes the idea of finite (linear) gain, classically used in input/output stability theory. Clearly, if a system has K-asymptotic gain with respect to a set A, and if A' is any subset of ]Rn containing A, then the system also has K-asymptotic gain with respect to A'.
there exists a 0 > 0 so that
I~IA:::; o,lIull:::; 0 ~ Ix(t,~,u)IA:::; c, I:/t:::: O.
Indeed, assume given c > O. Let T = T(I, c/2). Pick any 01 > 0 so that 1'(01) < c /2; then, if I~IA :::; 1 and lIull :::; 01, and t :::: T, Ix(t,~, u)IA :::; c/2 + 1'(lIull) < c . (2) By continuity (at u == 0 and states in A) of solutions with respect to controls and initial conditions, and compactness and O-invariance of A, there is also some 02 > 0 so that, if 117IA :::; 02 and lIull :::; fJ 2 then Ix(t, 17, u)IA :::; c for all t E [0, TJ. Together with (2), this gives the desired property with 0 := min {I, 01, 02}. 0 In a manner entirely analogous to the equilibrium case in (Sontag and Wang, 1995), Lemma 2.7, one can show the following characterization in terms of decay estimates: Lemma 2.6 The system E is 1SS with respect to the compact set A if and only if there are a KCfunction /3 and a K-function , so that
The main concept to be studied is as follows. Definition 2.2 Let A ~ ]Rn be a non empty compact set. The system E is input-to-state stable (1SS) with respect to A if it has K-asymptotic gain with respect to A and A is O-invariant. 0
Note that, in particular, if this property holds then the autonomous system Eo is globally asymptotically stable with respect to A (in the sense in (Lin et al., to appear». The 1SS property can also be definied with respect to non-compact 0invariant sets A (cf. (Lin et al., 1995», but then an additional technical condition must be imposed (see Remark 2.5 below).
Ix(t, ~,u)IA :::; /3(I~IA' t) + ,(lIull) (3) holds for each t :::: 0, u EL:, and ~ E ]Rn. 0 Remark 2.7 In the paper (Jiang et al., 1994), a system E is said to be "input-to-state practically stable" (1SpS) if there exist a KC-function /3, a K-function , and a constant c :::: 0 such that
For any A ~ ]Rn, we consider the zero-input orbit from A: O(A) := {17 : 17 = x(t,~, 0), t :::: 0, ~ EA} and let O(A) be the closure of O(A). Lemma 2.3 Let A
> 0,
Ix(t,~, u)1 :::; /3(I~I, t) + 1'(lIull) + c (4) holds for each input u E and each ~ E ]Rn.
L:
~ ]Rn
be a nonempty compact set. If the system E has K-asymptotic gain with respect to A, then it is 1SS with respect to O(A).
This concept is equivalent to the system being compact-1ss. Indeed, the concept implies the K-
asymptotic gain property with respect to the ball of radius c. Conversely, if the system is 1SS with respect to A, and 0 E A (which may be assunmed without loss of generality, by first enlarging
Proof: Since O(A) includes A, the system has Kasymptotic gain with respect to O(A), and the latter set is O-invariant (since O(A) is). Thus
204
set for the system~. Then the system is ISS with respect to A iff it admits an Iss-Lyapunov function with respect to A.
the set and then considering O(A», then it is also ISpS, because Ix(t,e,u)1 ~ Ix(t,e,u)I A +c ~ ,6(leI A , t) + 'Y(llull) + c ~ ,6(lel, t) + 'Y(llull) + c with c:= sup{lel ,e EA}. 0
The proof follows essentially the same lines as the proof of the analogous Theorem 1 (for equilibria) in (Sontag and Wang, 1995). We omit the details for reasons of space, but the main point is that care must be taken to establish all estimates in terms of 1.1 04 rather than Euclidean norm.
2.1. ISS-Lyapunov Functions An Iss-Lyapunov function with respect to the compact subset A ~ ]Rn for system ~ is a smooth function V : ]Rn -+ ]R>o which satisfies the following conditions: (a) V is proper and positive definite with respect to the set A, that is, there exist 0:1, 0:2 E K,oo such that for all e E ]Rn, O:l(leI A) ~ V(O ~ 0:2(leI A), and (b) there exist functions 0:3 E K,oo and such that
(j
One may modify V in an essentially arbitrary way inside the set A, so as to make V positive definite with respect to subsets of A:
(5) E K,
Lemma 2.10 Assume that ~ is ISS with respect to a given compact set A, and let A be any nonempty compact subset of A. Then there is a smooth function V : ]Rn -+ ]R~o which satisfies the following conditions: (a) V is proper and positive definite with respect to the set A, that is, there exist a1, a2 E K,oo such that for all e E ]Rn,
V'V(e)f(e, v) ~ -0:3(leI A ) + (j(lvl) (6) for all e E ]Rn and for all v E ]RID. Note that such a V is automatically also a Lyapunov function for the zero-input system ~o with respect to A. Also, observe that, since here A is compact, the existence of a function 0:2 as stated is in fact a consequence of the continuity of V. Remark 2.8 There are two other, equivalent, ways of defining Iss-Lyapunov function. The first is as follows. One asks that V satisfy (5) and that, for some K,oo-functions 0:4 and X there hold the implication
al(lelA') ~ V(e) ~ a2(lelA')' (9) and (b) there exist a function E K,oo and a non decreasing continuous function such that
a3
V'V(e)f(e, v) ~ -a3(lelA') + a(lvi)
(10)
for all e E ]Rn and for all v E ]RID. Proof: Let V be an Iss-Lyapunov funtion for ~ satisfying (6). Consider the set Al = {e: lel A ~ I}. Since this is disjoint from A, there is a smooth function I{) : ]Rn -+ [0,1] so that I{)(e) = 0 if e E A and 1 if e E Al. Similarly, there is some smooth, nonnegative function A defined on ]Rn which vanishes exactly on A. Now we define V(e) := A(e)(1 - 1, also
lel A ~ x(lvl) ==>
V'V(e)f(e, v) ~ -0:4(leI A) (7) (for each state e E ]Rn and control value v E ]RID). A second variant is to drop the requirement that 04 be of class K,oo; that is, one asks only that there is a X so that lel A ~ x(lvl) ==> V'V(e)f(e,v) < 0 (8) for all e :f. 0 and all v E ]RID. These two variants are equivalent: if (8) holds, then there exists a positive definite function 04 such that (7) holds (define for instance 04( r) as the supremum of the values of V'V(e)f(e, v) on the compact subset consisting of all lel A = r and v so that x(lvl) ~ r); then, for some properly chosen q, q(V) satisfies (7) with a new 04 E K,oo. Extending the proof of Remark 2.4 in (Sontag and Wang, 1995), it can be shown that all variants of the definition are equivalent. 0 Remark 2.9 If A is not compact, then properties (6) and (7) may no longer be equivalent. As an example, consider the following system: Xl = -Xl, X2 = -x2 + x1q(U -lx21), where q : ]R -+ ]R is a smooth function satisfying q( r) = 0 for all r ~ 0 and q( r) > 0 for all r > O. Let A be the set {(el, 6) : 6 O}. Consider the function V(e) = Clearly V satisfies (5) (as I~IA = 161), and I~IA ~ Ivl ==> V'V(e)f(e,v) ~ -e~·Butitis easy to see that V fails to satisfy property (6). 0 The main result is as follows. Theorem 1 Let the compact set A be a O-invariant
eV2.
a
V'V(e) f(e, v) ~ -03(leI A) + 'Y(lvi), (where 0:3 and l' are the comparison functions associated to V) for all v E ]RID and all lel A > l. Since both A and A are compact, there exists some So ~ 0 such that lelA' ~ lel A + So; thus V'V(e) f(e, v) ~ -a3(lelA') + 'Y(lvi), (11) whenever lelA' ~ 1 + So, where a3 is any K,oo function which satisfies a3 (r) ~ 0:3 (r - so) for all r 2: So + 1. The proof is completed by
a
taking any nondecreasing continuous function which majorizes both 1'( r) and the maximum of V'V(e) f(e, v) over alllelA' ~ 1 + So, Ivl ~ r. • This construction can be aEPlied in particular when A contains the origin A = {O} to conclude that if the system ~ is compact-Iss, then ~ admits a semi-ISs-Lyapunov function, that is, a smooth function V satisfying ol(lel) ~ V(e) ~ 02(IW for suitable 01,02 E K,oo and so that
=
(12)
205
closed-loop system a ICC-function f3 such that
for some aE /Coo and nondecreasing continuous function (J'. (Note that, in general, (J' cannot be picked of class /C ; there is no need for such a function to vanish at O. However, one may always write (J' ~ c + (j , for some class-/C function er and some positive constant c.)
Ix(t ,e,d)1 ~ f3(t , e), Vt 2: 0 for all E ]Rn and all dEMo . We next apply to (15) the converse Lyapunov Theorem in (Lin et al., to appear) for systems with no controls but with time-varying parametric uncertainties. We conclude (using the value u = k( x)) that there exists also a smooth, proper and positive definite function V such that
e
On the other hand, if ~ admits a semi-IssLyapunov function satisfying (12) , then it follows that lel 2: x(lvl) =:} Y'V(e)/(e , v) ~ -0:3(lel)/2 where x(r) 0: 31 0 2(J'(r) . Observe that X is a nondecreasing function. Using exactly the same arguments as used on page 441 of (Sontag, 1989a) , one can show that there exist some /cC-function f3 and some nondecreasing function 'Y such that
=
inf .. {a(e ) + Bo(e ) u + cp(e)B 1 (e) d} < 0 for all Idl ~ 1 and :I 0, where we are using the notations a(e)=Y'V(e)/(e), Bo(e)=Y'V(e)G(e) , and B 1 (e)=Y'V(e)D(O. Consequently, for any e:l 0, inf .. {a(e) + Bo(Ou+ cp(e)IB 1 (e)l} < O. (16) For any given real-valued function V on IR.n , let Bo be the set {e: Bo(e) = 0, :I O}. Thus, for the above V , (17) a(e) < 0 if E Bo
e
Ix(t , e, u)1 ~ f3( lel , t) + 'Y( lui) for each input u and each initial state It is easy to see from this that the system is compact-Iss . So we proved the following conclusion:
e.
e
Corollary 2.11 The system ~ is compact-Iss iff it admits a semi-Iss-Lyapunov function. •
e
and
3. ISS-Control Lyapunov Functions
lim lel- oo
In this section, we provide a preliminary discussion of some questions related to input to state stabilizability under feedback. (For simplicity, we consider only the case of stabilization with respect to the equilibrium A = {O} , but the theory could be developed in more generality.) Now inputs in ~ will be assumed to be partitioned into two components, one of which corresponds to true controls u : lIho --4 IR.m , and the other to "disturbances" w : IR>o --4 IR.P acting on the system. Furthermore, ;.re will assume that the right-hand side of ~ is affine in u and w . That is to say, we consider systems on IR.n of the general form:
= I(x)
eEBo
(We make the convention that -a(e)/IB 1 (OI = those points where B 1 (e) = O. The meaning of the limit along Bo is the obvious one, namely that values are large as long as le I is large and E Bo. If Bo happens to be bounded, this condition is vacuous.) Motivated by these considerations, we say that a smooth, proper and positive definite function V is an Iss-control-Lyapunov-function (Iss-clf) for (13) if (17)-(18) hold. An Iss-clf V is said to satisfy the small control property (scp) iffor any c > 0, there is a 6 > 0 such that for each 0 < lel < 6, there is some Ivl < c such that a(e) + Bo(e)v < O. (To simplify, we also say that the corresponding pair (a(e) , Bo(e)) "satisfies scp." ) These definitions constitute one possible generalization of the corresponding notion for systems with no "w", for which only equation (17) is relevant; see (Sontag, 1989b; Isidori, 1995). Theorem 2 If there is an Iss-clf V for system (13 ), then there exists a feedback law k : ]Rn --4 ]Rm which is smooth on ]Rn \ {O} and is such that the closed-loop system (14) is ISS. If V satisfies the small control property, then k can be chosen to be continuous at O.
+00 at
e
+ G(x)u + D(x)w ,
(13) where 1 and the columns of the matrices G and D are smooth vector fields . We also assume that 1(0) = o. :i;
Our main objective is to study conditions for the existence of a smooth feedback law u = k( x) such that the closed-loop system :i; = I(x) + G(x)k(x) + D(x )w (14) is ISS , with w seen as the external input. As when studying (non-Iss) stabilizability under feedback , we search for conditions expressed in terms of "control Lyapunov functions ."
Observe that if there exists a feedback law as above, then, by the main result in (Sontag and Wang, 1995), there exists a positive definite, proper function cp such that the system :i;
= I(x)
+ G(x)k(x) + cp(x)D(x)d
(18)
This result is not really new: (Freeman and Kokotovic, to appear) give a very similar theorem which applies to systems not necessarily affine in controls, but we believe that the method of proof given here is very natural and constructive.
(15)
We will say that a feedback k is almost smooth if it is smooth on ]Rn \ {O} and continuous at O.
is uniformly globally asymptotically stable (UGAS ) with respect to all dEMo (= the set of all measurable functions from IR.>0 to the unit ball in ]Rm ) . That is, there exists for solutions of this
Remark 3.1 The converse of the above Theorem is also true , that is, if system (13) is Iss-stabilizable
206
IR n \ {O }. (Note here that ko may fail to be smooth on IRn \ {O} because the function e t-+ IB 1(e) I may fail to be smooth.) To obtain a feedback that is smooth on 1R.n \ {O}, it is sufficient to approximate ko on 1R.n \ {O} by a smooth k so that Equation (19) still holds. It then follows that
by an almost smooth feedback k(·), then there exists an Iss-clf with scp. However, in order to prove the stronger assertion, we would need to make some modifications to the proof of Theorem 1 in (Lin et al. , to appear) , but we don't have the space to do so in this conference paper. We may sketch the main steps, however. Basically, and using the notations in that paper, one needs to show that, when the O-invariant set A used there is compact, the functions g and U constructed for obtaining V are locally Lipschitz outside A without needing to assume that f is locally Lipschitz on A (which, when k is almost smooth but not smooth, cannot be guaranteed). To show the Lipschitz continuity of g, one only needs to notice that the trajectories startipg outside A never enter A in negative time; hence, the original proof is still valid. To prove the Lipschitz condition on U , one first notices that the stability of the system implies that trajectories starting from a compact set always stay in a compact set in positive time. Using this, one can show that x(t,~, de , ~) always stays in a compact set disjoint from A, for any t E [0 , te,~l . This then allows one to apply the Lipschitz condition for f outside A, together with the Gronwall inequality, to conclude that U is lo0 cally Lipschitz outside A.
+ Bo(Ok(e) + Bl(e)v < 0 ~ :I 0 and all Ivl ~ '!/J(IW.
a(~)
(20) for all Therefore, V is an Iss-Lyapunov function for the closed-loop system, and consequently, the closed-loop system is ISS. To show that if in addition V satisfies the scp then the feedback law can be chosen to be continuous everywhere, observe that one may always choose .1. I IB,({),P(ell - 0 If' in such a manner that liml {_O a(e) . For instance, one can pick up any smooth Koofunction;j so that ;j(r) ~ min{1/J(r)(-a(O) : e E Bo , I~I = r} in a neighborhood of 0 where -a(~) ~ 1, and let ;j(r) ~ '!/J(r) everywhere. Then replace '!/J by;j. With this new choice of '!/J and the resulting
Sketch of Proof of Theorem 2. Assume that (13 ) admits an Iss-clf V satisfying (17)-(18). If BD is unbounded, we let
. {
aCe)
'!/Jl(r) = mf -IB1(e)l : Bo(~) = 0, I~I = r
(21)
}
The above arguments also show that if there exists a smooth, positive definite and proper function
(values may be infinite). If BD is bounded, we let '!/Jl (r) be defined in this way for those r for which there is some e E BD with I~I 2 r, but let '!/Jl (r) = 00 otherwise. In either case, it holds that inf {1/Jl(r) , r E [a , b]} > 0 for each two numbers 0 < a < b, so there is a smooth Koo-function '!/J(r ) such that '!/J( r) ~ min {r 2, '!/Jl (r)} for all r 2 o. Let
=::}
a(~)
+
inf u {a(~) + Bo(e)u + IB1(e)1
O.
So inf v {al(~) + Bo(e) v} < 0 for all ~ :I 0, where al(e) := ace) +
+ Bo(e)ko(O +
Ix(t , ~ , w)1 ~ f3(lel , t) + -y(llwll). (22) where -y(r) = 0:1100:20 '!/J-l(r). We say that a function -y is a feedback assignable gain if there is some k so that an estimate of this form holds (for some f3), under some almost smooth feedback law. When studying the problem of ISS stabilizability for a system of the form (13), a special case is that of "matching uncertainties" when G = D ; then the equations have the form
(19)
where 0:( r, s) is the function defined by
- r±v0'±?"
o:(r, s)= { 0,
.
'
if s ..... O' . r, Ifs=O.
As it was shown in (Sontag, 1989b) , the function 0: is analytic on the set
:i;
S = {(oS, r) E lR,z: r:l 0 or s < O} . This then implies that ko is locally Lipschitz on
= f(x)
+ G(x)(u + w) .
(23)
In (Sontag, 1989a) , it was shown that if system (23) is stabilizable by a smooth feedback
207
°
4. Appendix
when w = in (23), then the system is also ISSstabilizable by a smooth feedback. In (Praly and Wang, submitted), this result was generalized to obtain the following "gain assignability" result, rederived here in a somewhat simpler manner by means of Iss-Lyapunov functions.
We recall the definitions of the standard classes of comparison functions. A function -y : IR>o -+ R>o is positive definite if -y( s) > 0 for all s > 0, -and -y(0) = 0; the function -y is a K,-function if it is continuous, positive definite, and strictly increasing; and -y is a K,oo-function if it is a K,function and -y(s) -+ 00 as s -+ 00. Finally, a function 13: IR>o x IR>o -+ IR>o is a K,C-function if for each fixed- t :2: 0-the function 13(', t) is a K,function, and for each fixed s :2: 0,13(8, t) decreases to zero as t -+ 00.
Proposition 3.2 If system (23) is stabilizable by an almost smooth feedback k when w = 0, then every -y E K,oo is an assignable gain. Proof: By standard converse Lyapunov theorems, the hypotheses imply the existence of a controlLyapunov function V satisfying (21) and infu{a(~)
+ B(Ou} < 0,
'V~
f.
0,
5. REFERENCES
(24)
Bhatia, Nam Parshad and George Philip Szego (1970). Stability Theory of Dynamical Systems. SpringerVerlag. New York.
where a(~) = VV(~)f(o, B(O = VV(~)G(~), and in addition, V satisfies the scp. Let 1/J be a K,oo-function such that 1/J(r) :2: 2-y-l 0 all 0 a2(r) . Let t.p be a almost smooth function so that t.p(~) :2: 1/J(I~)1 /2 for all ~ E IRn. It follows from (24) that infu{al(~) +B(~)u} < 0, 'V~ f. 0, where al(~) = a(~) + IB(~)I t.p(~) . Arguing as earlier, one knows that there exists a k that is smooth on IRn \ {O} such that
Freeman, Randy and Petar Kokotovic (to appear). Inverse optimality in robust stabilization. SIAM J. Control and Optimization, to appear. Hahn, Wolfgang (1967). Stability of Motion. SpringerVerlag. Berlin. Isidori, Alberto (1995). Nonlinear Control Systems: An Introduction. third ed. Springer-Verlag. New York.
(25)
Jiang, Zhong-Ping, Andrew Teel and Laurent Praly (1994). Small-gain theorem for ISS systems and applications. Mathematics of Control, Signals, and Systems, 7, pp. 95-120.
for all ~ f. 0. Note again that if the pair (a, B) satisfies the scp, the pair (ab B) also satisfies the scp, so the resulting k is continuous everywhere. From (25), it follows that for each ~ f. 0, a(~)
+ B(~)k(~) + B(~)w <
°
Lin, Yuandan, Eduardo D. Sontag and Yuan Wang (1995) . Input to state stabilizability for parameterized families of systems. Int. Journal of Robust and Nonlinear Control, 5, pp. 187-205.
provided Iwl ~ 1/J(1~1)/2. From here one obtains the desired conclusion. •
Lin, Yuandan, Eduardo D. Sontag and Yuan Wang (to appear). A smooth converse Lyapunov theorem for robust stability. SIAM Journal on Control and Optimization. (Preliminary version in "Recent results on Lyapunov-theoretic techniques for nonlinear stability", in Proc. Amer. Automatic Control Conference, Baltimore, June 1994, pp. 1771-1775.).
Finally, we relate our results to those in (Isidori, 1995), Section 9.5, where the problem of disturbance attenuation with stability is discussed in the context of finiteness of L2 gains (we assume for this comparison that the output function is y = x). It was shown there that if system (13) admits a positive definite proper smooth function V such that for each ~ f. 0, Bo(~)
Praly, Laurent and Yuan Wang (submitted). Stabilization in spite of matched unmodelled dynamics and an equivalent definition of input-to-state stability.
= 0 =>
a(O
1
+ 4-y2 IB1(~)12 + 1~12 < 0,
Sontag, Eduardo D. (1989a). Smooth stabilization implies coprime factorization. IEEE Transactions on Automatic Control AC-34, 435-443.
(26)
°
where -y > is some constant, then there exists an almost smooth feedback k such that the closedloop system is ISS with an Iss-Lyapunov function satisfying 2 VV(f(~) + g(~)k(~) + p(e)w) ~ -y21w1 Observe that if (26) holds, then V is also an ISSLyapunov function as defined above. This follows immediately from the following observation:
Sontag, Eduardo D. (1989b). A 'universal' construction of Artstein's theorem on nonlinear stabilization. Systems fj Control Letters 13, 117-123.
e.
Sontag, Eduardo D. and Yuan Wang (1995). On characterizations of the input-to-state stability property. Systems fj Control Letters 24, pp 351-359. Sontag, Eduardo D. and Yuandan Lin (1992). Stabilization with respect to noncompact sets: Lyapunov characterizations and effect of bounded inputs. In Nonlinear Control Systems Design 1992, IFAC Symposia Series, 1999, Number 7, M. Fliess Ed., Pergamon Press, Oxford, 1993,
Lemma 3.3 Let a, b, c : IRn -+ IR be continuous functions. Assume that c is positive definite and proper. If a(e) + (b(e))2 + c(~) < for all ~ f. 0, then liIllJel--+oo = 00 .
Ib(m
°
Tsinias, John (1989). Sufficient Lyapunovlike conditions for stabilization. Mathematics of Control, Signals, and Systems 2, 343-357.
The Lemma follows directly from the inequality p + ; :2: 2yi7 for all p, (J' > 0. • Proof:
208
On Characterizations of Input-to-State Stability with Respect to Compact Sets Eduardo D. Sontag*
and Yuan Wang**
*Department of Mathematics , Rutgers Uni1lersity, New Brunswick, NJ 08909, USA "Department of Mathematics, Florida Atlantic Uni1lersity, Boca Raton, FL 99-191 , USA Abstract. Previous characterizations of IS5-stability are shown to generalize without change to the case of stability with respect to sets. Some results on lS5-stabilizability are mentioned as well.
Key Words. Set Stability; input-to-state stability; Lyapunov functions ; control Lyapunov functions ; practical stability
1. Introduction
least for the case of compact invariant sets A , the results in (Sontag and Wang, 1995) indeed can be generalized with little or no change; in particular, ISS with respect to a given A is equivalent to the existence of an "Iss-Lyapunov function" relative to A. (Most of the proofs valid in the equilibrium case extend easily to the set case.) The property of there existing some invariant compact set A so that there is ISS with respect to A will be called the "compact-Iss" property; various remarks about this notion are included, including the equivalence with what is sometimes called "practical stability." Finally, we also sketch some applications of the ISS notion to feedback design and disturbance attenuation.
System stability with respect to input perturbations is one of the central issues to be studied in control. During the past few years , the property called "input to state stability" (ISS) has been proposed (originally in (Sontag, 1989a)) as a foundation for the study of such problems, and has been subsequently employed by many authors in areas ranging from robust control to highly nonlinear small-gain theorems (see for instance (Jiang et al. , 1994; Tsinias, 1989)). In the paper (Sontag and Wang, 1995), the authors established the equivalence among several natural characterizations of the ISS property, stated in terms of dissipation inequalities, robustness margins, and Lyapunov-like functions.
2. Set Input to State Stability
The ISS property was originally stated for stability with respect to a given equilibrium state of interest. On the other hand, in many applications it is of interest to study stability with respect to an invariant set A , where A does not necessarily consist of a single point. Examples of such applications include problems of robust control and the various notions usually encompassed by the term "practical stability." This motivated the study of the "set" version of the ISS property, originally in (Sontag and Lin, 1992), and developed with applications to the study of parameterized families of systems in (Lin et al., 1995). Given the interest in set-Iss , it is an obvious question to ask whether the equivalent characterizations given in (Sontag and Wang, 1995) for the special case A = equilibrium extend to the more general set case. It is the main purpose of this paper to point out that, at
We deal with systems of the following general form:
x=
I(x , u), where I : IRn x IRID -> IRn is a locally Lipschitz map, and we interpret x and u as functions of time i E IR, with values xCi) E IRn and u(i) E IRm , for some positive integers nand m. (Generalizations to systems evolving on manifolds , and/ or restricted control value sets, are also of interest , but in this paper, all spaces are Euclidean.) From now on, one such system is assumed given. A control or input is a measurable locally essentially bounded function u : IR>o -> IRID. We use the notation Ilull to indicate the L~-norm of u , and 1·1for Euclidean norm in IRn and IRm. For each ~ E IRn and each u E L~ , we denote by x(t , ~ , u) the trajectory of the system ~ with initial state x(O) = ~ and the input u. (This solution is a priori only defined on some maximal interval [0, Te, u) , with Te , u ::; +00, but the main definition will include the requirement that Te,u = +00 .) We recall in an appendix the no-
• Supported in part by US Air Force Grant F49620-95-10101 •• Supported in part by NSF Grants DMS-9457826 and DMS-9403924
203
we only need to prove that O(A) is compact, or equivalently, that O(A) is bounded. From the K-asymptotic gain property, we know that there exists some T :::: 0 so that, for each ~ E A, Ix(t,~, O)I A :::; 1 for all t :::: T. By continuity at states in A of solutions of Eo with respect to initial conditions, and compactness of A, there is some constant c so that Ix(t,~, O)I A :::; c for all t E [0, TJ. Thus O(A) is included in a ball of radius max{l, cl. •
tions of positive definite and class K, Krxn and KC comparison functions. Let A be a nonempty subset of ~ E ]Rn, I~IA
]Rn.
For each
= d(~,A) = inf{d(17,~), 17EA}
will denote the usual point-to-set distance from ~ to A. (So for the special case A = {O}, 1~I{o} = I~I is the usual Euclidean norm.) We say that such an A is a O-invariant set for E, or more precisely, for the associated "zero-input" or "undisturbed" system x fo(x) f(x, 0), (Eo)
=
=
Definition 2.4 The system E is compact-1ss if it is 1SS with respect to some compact set A. 0 By Lemma 2.3, this is equivalent to simply asking that system E has K-asymptotic gain with respect to some compact set A. Remark 2.5 The definition of the 1SS property is in terms of a uniform attraction property. Just as for systems with no controls (cf. (Bhatia and Szego, 1970), Theorem 1.5.28, or (Hahn, 1967), Theorem 38.1), compactness of A insures that the following stability-like property is automatically satisfied as well:
if it holds that x(t,~, 0) E A, I:/t:::: 0, I:/~ EA. That is, when the control is u == 0, the solution when starting from A is defined for all t :::: 0 and stays in A. Definition 2.1 The system E has K-asymptotic gain with respect to the nonempty subset A ~ ]Rn
if there is some l' E K (an "asymptotic gain") so that, for all u, solutions exist for all t :::: 0 and (1)
uniformly on compacts for
~.
For each c
The uniformity requirement (which, as will be shown in a forthcoming paper, can be substantially relaxed) means, precisely: solutions exist for all initial states, controls, and times, and for each real numbers r, c > 0, there is some T = T(r,c):::: 0 so that Ix(t,~,u)IA:::; c +I'(IlulD for all u, all I~I :::; r, and all t :::: T. This generalizes the idea of finite (linear) gain, classically used in input/output stability theory. Clearly, if a system has K-asymptotic gain with respect to a set A, and if A' is any subset of ]Rn containing A, then the system also has K-asymptotic gain with respect to A'.
there exists a 0 > 0 so that
I~IA:::; o,lIull:::; 0 ~ Ix(t,~,u)IA:::; c, I:/t:::: O.
Indeed, assume given c > O. Let T = T(I, c/2). Pick any 01 > 0 so that 1'(01) < c /2; then, if I~IA :::; 1 and lIull :::; 01, and t :::: T, Ix(t,~, u)IA :::; c/2 + 1'(lIull) < c . (2) By continuity (at u == 0 and states in A) of solutions with respect to controls and initial conditions, and compactness and O-invariance of A, there is also some 02 > 0 so that, if 117IA :::; 02 and lIull :::; fJ 2 then Ix(t, 17, u)IA :::; c for all t E [0, TJ. Together with (2), this gives the desired property with 0 := min {I, 01, 02}. 0 In a manner entirely analogous to the equilibrium case in (Sontag and Wang, 1995), Lemma 2.7, one can show the following characterization in terms of decay estimates: Lemma 2.6 The system E is 1SS with respect to the compact set A if and only if there are a KCfunction /3 and a K-function , so that
The main concept to be studied is as follows. Definition 2.2 Let A ~ ]Rn be a non empty compact set. The system E is input-to-state stable (1SS) with respect to A if it has K-asymptotic gain with respect to A and A is O-invariant. 0
Note that, in particular, if this property holds then the autonomous system Eo is globally asymptotically stable with respect to A (in the sense in (Lin et al., to appear». The 1SS property can also be definied with respect to non-compact 0invariant sets A (cf. (Lin et al., 1995», but then an additional technical condition must be imposed (see Remark 2.5 below).
Ix(t, ~,u)IA :::; /3(I~IA' t) + ,(lIull) (3) holds for each t :::: 0, u EL:, and ~ E ]Rn. 0 Remark 2.7 In the paper (Jiang et al., 1994), a system E is said to be "input-to-state practically stable" (1SpS) if there exist a KC-function /3, a K-function , and a constant c :::: 0 such that
For any A ~ ]Rn, we consider the zero-input orbit from A: O(A) := {17 : 17 = x(t,~, 0), t :::: 0, ~ EA} and let O(A) be the closure of O(A). Lemma 2.3 Let A
> 0,
Ix(t,~, u)1 :::; /3(I~I, t) + 1'(lIull) + c (4) holds for each input u E and each ~ E ]Rn.
L:
~ ]Rn
be a nonempty compact set. If the system E has K-asymptotic gain with respect to A, then it is 1SS with respect to O(A).
This concept is equivalent to the system being compact-1ss. Indeed, the concept implies the K-
asymptotic gain property with respect to the ball of radius c. Conversely, if the system is 1SS with respect to A, and 0 E A (which may be assunmed without loss of generality, by first enlarging
Proof: Since O(A) includes A, the system has Kasymptotic gain with respect to O(A), and the latter set is O-invariant (since O(A) is). Thus
204
set for the system~. Then the system is ISS with respect to A iff it admits an Iss-Lyapunov function with respect to A.
the set and then considering O(A», then it is also ISpS, because Ix(t,e,u)1 ~ Ix(t,e,u)I A +c ~ ,6(leI A , t) + 'Y(llull) + c ~ ,6(lel, t) + 'Y(llull) + c with c:= sup{lel ,e EA}. 0
The proof follows essentially the same lines as the proof of the analogous Theorem 1 (for equilibria) in (Sontag and Wang, 1995). We omit the details for reasons of space, but the main point is that care must be taken to establish all estimates in terms of 1.1 04 rather than Euclidean norm.
2.1. ISS-Lyapunov Functions An Iss-Lyapunov function with respect to the compact subset A ~ ]Rn for system ~ is a smooth function V : ]Rn -+ ]R>o which satisfies the following conditions: (a) V is proper and positive definite with respect to the set A, that is, there exist 0:1, 0:2 E K,oo such that for all e E ]Rn, O:l(leI A) ~ V(O ~ 0:2(leI A), and (b) there exist functions 0:3 E K,oo and such that
(j
One may modify V in an essentially arbitrary way inside the set A, so as to make V positive definite with respect to subsets of A:
(5) E K,
Lemma 2.10 Assume that ~ is ISS with respect to a given compact set A, and let A be any nonempty compact subset of A. Then there is a smooth function V : ]Rn -+ ]R~o which satisfies the following conditions: (a) V is proper and positive definite with respect to the set A, that is, there exist a1, a2 E K,oo such that for all e E ]Rn,
V'V(e)f(e, v) ~ -0:3(leI A ) + (j(lvl) (6) for all e E ]Rn and for all v E ]RID. Note that such a V is automatically also a Lyapunov function for the zero-input system ~o with respect to A. Also, observe that, since here A is compact, the existence of a function 0:2 as stated is in fact a consequence of the continuity of V. Remark 2.8 There are two other, equivalent, ways of defining Iss-Lyapunov function. The first is as follows. One asks that V satisfy (5) and that, for some K,oo-functions 0:4 and X there hold the implication
al(lelA') ~ V(e) ~ a2(lelA')' (9) and (b) there exist a function E K,oo and a non decreasing continuous function such that
a3
V'V(e)f(e, v) ~ -a3(lelA') + a(lvi)
(10)
for all e E ]Rn and for all v E ]RID. Proof: Let V be an Iss-Lyapunov funtion for ~ satisfying (6). Consider the set Al = {e: lel A ~ I}. Since this is disjoint from A, there is a smooth function I{) : ]Rn -+ [0,1] so that I{)(e) = 0 if e E A and 1 if e E Al. Similarly, there is some smooth, nonnegative function A defined on ]Rn which vanishes exactly on A. Now we define V(e) := A(e)(1 -
lel A ~ x(lvl) ==>
V'V(e)f(e, v) ~ -0:4(leI A) (7) (for each state e E ]Rn and control value v E ]RID). A second variant is to drop the requirement that 04 be of class K,oo; that is, one asks only that there is a X so that lel A ~ x(lvl) ==> V'V(e)f(e,v) < 0 (8) for all e :f. 0 and all v E ]RID. These two variants are equivalent: if (8) holds, then there exists a positive definite function 04 such that (7) holds (define for instance 04( r) as the supremum of the values of V'V(e)f(e, v) on the compact subset consisting of all lel A = r and v so that x(lvl) ~ r); then, for some properly chosen q, q(V) satisfies (7) with a new 04 E K,oo. Extending the proof of Remark 2.4 in (Sontag and Wang, 1995), it can be shown that all variants of the definition are equivalent. 0 Remark 2.9 If A is not compact, then properties (6) and (7) may no longer be equivalent. As an example, consider the following system: Xl = -Xl, X2 = -x2 + x1q(U -lx21), where q : ]R -+ ]R is a smooth function satisfying q( r) = 0 for all r ~ 0 and q( r) > 0 for all r > O. Let A be the set {(el, 6) : 6 O}. Consider the function V(e) = Clearly V satisfies (5) (as I~IA = 161), and I~IA ~ Ivl ==> V'V(e)f(e,v) ~ -e~·Butitis easy to see that V fails to satisfy property (6). 0 The main result is as follows. Theorem 1 Let the compact set A be a O-invariant
eV2.
a
V'V(e) f(e, v) ~ -03(leI A) + 'Y(lvi), (where 0:3 and l' are the comparison functions associated to V) for all v E ]RID and all lel A > l. Since both A and A are compact, there exists some So ~ 0 such that lelA' ~ lel A + So; thus V'V(e) f(e, v) ~ -a3(lelA') + 'Y(lvi), (11) whenever lelA' ~ 1 + So, where a3 is any K,oo function which satisfies a3 (r) ~ 0:3 (r - so) for all r 2: So + 1. The proof is completed by
a
taking any nondecreasing continuous function which majorizes both 1'( r) and the maximum of V'V(e) f(e, v) over alllelA' ~ 1 + So, Ivl ~ r. • This construction can be aEPlied in particular when A contains the origin A = {O} to conclude that if the system ~ is compact-Iss, then ~ admits a semi-ISs-Lyapunov function, that is, a smooth function V satisfying ol(lel) ~ V(e) ~ 02(IW for suitable 01,02 E K,oo and so that
=
(12)
205
closed-loop system a ICC-function f3 such that
for some aE /Coo and nondecreasing continuous function (J'. (Note that, in general, (J' cannot be picked of class /C ; there is no need for such a function to vanish at O. However, one may always write (J' ~ c + (j , for some class-/C function er and some positive constant c.)
Ix(t ,e,d)1 ~ f3(t , e), Vt 2: 0 for all E ]Rn and all dEMo . We next apply to (15) the converse Lyapunov Theorem in (Lin et al., to appear) for systems with no controls but with time-varying parametric uncertainties. We conclude (using the value u = k( x)) that there exists also a smooth, proper and positive definite function V such that
e
On the other hand, if ~ admits a semi-IssLyapunov function satisfying (12) , then it follows that lel 2: x(lvl) =:} Y'V(e)/(e , v) ~ -0:3(lel)/2 where x(r) 0: 31 0 2(J'(r) . Observe that X is a nondecreasing function. Using exactly the same arguments as used on page 441 of (Sontag, 1989a) , one can show that there exist some /cC-function f3 and some nondecreasing function 'Y such that
=
inf .. {a(e ) + Bo(e ) u + cp(e)B 1 (e) d} < 0 for all Idl ~ 1 and :I 0, where we are using the notations a(e)=Y'V(e)/(e), Bo(e)=Y'V(e)G(e) , and B 1 (e)=Y'V(e)D(O. Consequently, for any e:l 0, inf .. {a(e) + Bo(Ou+ cp(e)IB 1 (e)l} < O. (16) For any given real-valued function V on IR.n , let Bo be the set {e: Bo(e) = 0, :I O}. Thus, for the above V , (17) a(e) < 0 if E Bo
e
Ix(t , e, u)1 ~ f3( lel , t) + 'Y( lui) for each input u and each initial state It is easy to see from this that the system is compact-Iss . So we proved the following conclusion:
e.
e
Corollary 2.11 The system ~ is compact-Iss iff it admits a semi-Iss-Lyapunov function. •
e
and
3. ISS-Control Lyapunov Functions
lim lel- oo
In this section, we provide a preliminary discussion of some questions related to input to state stabilizability under feedback. (For simplicity, we consider only the case of stabilization with respect to the equilibrium A = {O} , but the theory could be developed in more generality.) Now inputs in ~ will be assumed to be partitioned into two components, one of which corresponds to true controls u : lIho --4 IR.m , and the other to "disturbances" w : IR>o --4 IR.P acting on the system. Furthermore, ;.re will assume that the right-hand side of ~ is affine in u and w . That is to say, we consider systems on IR.n of the general form:
= I(x)
eEBo
(We make the convention that -a(e)/IB 1 (OI = those points where B 1 (e) = O. The meaning of the limit along Bo is the obvious one, namely that values are large as long as le I is large and E Bo. If Bo happens to be bounded, this condition is vacuous.) Motivated by these considerations, we say that a smooth, proper and positive definite function V is an Iss-control-Lyapunov-function (Iss-clf) for (13) if (17)-(18) hold. An Iss-clf V is said to satisfy the small control property (scp) iffor any c > 0, there is a 6 > 0 such that for each 0 < lel < 6, there is some Ivl < c such that a(e) + Bo(e)v < O. (To simplify, we also say that the corresponding pair (a(e) , Bo(e)) "satisfies scp." ) These definitions constitute one possible generalization of the corresponding notion for systems with no "w", for which only equation (17) is relevant; see (Sontag, 1989b; Isidori, 1995). Theorem 2 If there is an Iss-clf V for system (13 ), then there exists a feedback law k : ]Rn --4 ]Rm which is smooth on ]Rn \ {O} and is such that the closed-loop system (14) is ISS. If V satisfies the small control property, then k can be chosen to be continuous at O.
+00 at
e
+ G(x)u + D(x)w ,
(13) where 1 and the columns of the matrices G and D are smooth vector fields . We also assume that 1(0) = o. :i;
Our main objective is to study conditions for the existence of a smooth feedback law u = k( x) such that the closed-loop system :i; = I(x) + G(x)k(x) + D(x )w (14) is ISS , with w seen as the external input. As when studying (non-Iss) stabilizability under feedback , we search for conditions expressed in terms of "control Lyapunov functions ."
Observe that if there exists a feedback law as above, then, by the main result in (Sontag and Wang, 1995), there exists a positive definite, proper function cp such that the system :i;
= I(x)
+ G(x)k(x) + cp(x)D(x)d
(18)
This result is not really new: (Freeman and Kokotovic, to appear) give a very similar theorem which applies to systems not necessarily affine in controls, but we believe that the method of proof given here is very natural and constructive.
(15)
We will say that a feedback k is almost smooth if it is smooth on ]Rn \ {O} and continuous at O.
is uniformly globally asymptotically stable (UGAS ) with respect to all dEMo (= the set of all measurable functions from IR.>0 to the unit ball in ]Rm ) . That is, there exists for solutions of this
Remark 3.1 The converse of the above Theorem is also true , that is, if system (13) is Iss-stabilizable
206
IR n \ {O }. (Note here that ko may fail to be smooth on IRn \ {O} because the function e t-+ IB 1(e) I may fail to be smooth.) To obtain a feedback that is smooth on 1R.n \ {O}, it is sufficient to approximate ko on 1R.n \ {O} by a smooth k so that Equation (19) still holds. It then follows that
by an almost smooth feedback k(·), then there exists an Iss-clf with scp. However, in order to prove the stronger assertion, we would need to make some modifications to the proof of Theorem 1 in (Lin et al. , to appear) , but we don't have the space to do so in this conference paper. We may sketch the main steps, however. Basically, and using the notations in that paper, one needs to show that, when the O-invariant set A used there is compact, the functions g and U constructed for obtaining V are locally Lipschitz outside A without needing to assume that f is locally Lipschitz on A (which, when k is almost smooth but not smooth, cannot be guaranteed). To show the Lipschitz continuity of g, one only needs to notice that the trajectories startipg outside A never enter A in negative time; hence, the original proof is still valid. To prove the Lipschitz condition on U , one first notices that the stability of the system implies that trajectories starting from a compact set always stay in a compact set in positive time. Using this, one can show that x(t,~, de , ~) always stays in a compact set disjoint from A, for any t E [0 , te,~l . This then allows one to apply the Lipschitz condition for f outside A, together with the Gronwall inequality, to conclude that U is lo0 cally Lipschitz outside A.
+ Bo(Ok(e) + Bl(e)v < 0 ~ :I 0 and all Ivl ~ '!/J(IW.
a(~)
(20) for all Therefore, V is an Iss-Lyapunov function for the closed-loop system, and consequently, the closed-loop system is ISS. To show that if in addition V satisfies the scp then the feedback law can be chosen to be continuous everywhere, observe that one may always choose .1. I IB,({),P(ell - 0 If' in such a manner that liml {_O a(e) . For instance, one can pick up any smooth Koofunction;j so that ;j(r) ~ min{1/J(r)(-a(O) : e E Bo , I~I = r} in a neighborhood of 0 where -a(~) ~ 1, and let ;j(r) ~ '!/J(r) everywhere. Then replace '!/J by;j. With this new choice of '!/J and the resulting
Sketch of Proof of Theorem 2. Assume that (13 ) admits an Iss-clf V satisfying (17)-(18). If BD is unbounded, we let
. {
aCe)
'!/Jl(r) = mf -IB1(e)l : Bo(~) = 0, I~I = r
(21)
}
The above arguments also show that if there exists a smooth, positive definite and proper function
(values may be infinite). If BD is bounded, we let '!/Jl (r) be defined in this way for those r for which there is some e E BD with I~I 2 r, but let '!/Jl (r) = 00 otherwise. In either case, it holds that inf {1/Jl(r) , r E [a , b]} > 0 for each two numbers 0 < a < b, so there is a smooth Koo-function '!/J(r ) such that '!/J( r) ~ min {r 2, '!/Jl (r)} for all r 2 o. Let
=::}
a(~)
+
inf u {a(~) + Bo(e)u + IB1(e)1
O.
So inf v {al(~) + Bo(e) v} < 0 for all ~ :I 0, where al(e) := ace) +
+ Bo(e)ko(O +
Ix(t , ~ , w)1 ~ f3(lel , t) + -y(llwll). (22) where -y(r) = 0:1100:20 '!/J-l(r). We say that a function -y is a feedback assignable gain if there is some k so that an estimate of this form holds (for some f3), under some almost smooth feedback law. When studying the problem of ISS stabilizability for a system of the form (13), a special case is that of "matching uncertainties" when G = D ; then the equations have the form
(19)
where 0:( r, s) is the function defined by
- r±v0'±?"
o:(r, s)= { 0,
.
'
if s ..... O' . r, Ifs=O.
As it was shown in (Sontag, 1989b) , the function 0: is analytic on the set
:i;
S = {(oS, r) E lR,z: r:l 0 or s < O} . This then implies that ko is locally Lipschitz on
= f(x)
+ G(x)(u + w) .
(23)
In (Sontag, 1989a) , it was shown that if system (23) is stabilizable by a smooth feedback
207
°
4. Appendix
when w = in (23), then the system is also ISSstabilizable by a smooth feedback. In (Praly and Wang, submitted), this result was generalized to obtain the following "gain assignability" result, rederived here in a somewhat simpler manner by means of Iss-Lyapunov functions.
We recall the definitions of the standard classes of comparison functions. A function -y : IR>o -+ R>o is positive definite if -y( s) > 0 for all s > 0, -and -y(0) = 0; the function -y is a K,-function if it is continuous, positive definite, and strictly increasing; and -y is a K,oo-function if it is a K,function and -y(s) -+ 00 as s -+ 00. Finally, a function 13: IR>o x IR>o -+ IR>o is a K,C-function if for each fixed- t :2: 0-the function 13(', t) is a K,function, and for each fixed s :2: 0,13(8, t) decreases to zero as t -+ 00.
Proposition 3.2 If system (23) is stabilizable by an almost smooth feedback k when w = 0, then every -y E K,oo is an assignable gain. Proof: By standard converse Lyapunov theorems, the hypotheses imply the existence of a controlLyapunov function V satisfying (21) and infu{a(~)
+ B(Ou} < 0,
'V~
f.
0,
5. REFERENCES
(24)
Bhatia, Nam Parshad and George Philip Szego (1970). Stability Theory of Dynamical Systems. SpringerVerlag. New York.
where a(~) = VV(~)f(o, B(O = VV(~)G(~), and in addition, V satisfies the scp. Let 1/J be a K,oo-function such that 1/J(r) :2: 2-y-l 0 all 0 a2(r) . Let t.p be a almost smooth function so that t.p(~) :2: 1/J(I~)1 /2 for all ~ E IRn. It follows from (24) that infu{al(~) +B(~)u} < 0, 'V~ f. 0, where al(~) = a(~) + IB(~)I t.p(~) . Arguing as earlier, one knows that there exists a k that is smooth on IRn \ {O} such that
Freeman, Randy and Petar Kokotovic (to appear). Inverse optimality in robust stabilization. SIAM J. Control and Optimization, to appear. Hahn, Wolfgang (1967). Stability of Motion. SpringerVerlag. Berlin. Isidori, Alberto (1995). Nonlinear Control Systems: An Introduction. third ed. Springer-Verlag. New York.
(25)
Jiang, Zhong-Ping, Andrew Teel and Laurent Praly (1994). Small-gain theorem for ISS systems and applications. Mathematics of Control, Signals, and Systems, 7, pp. 95-120.
for all ~ f. 0. Note again that if the pair (a, B) satisfies the scp, the pair (ab B) also satisfies the scp, so the resulting k is continuous everywhere. From (25), it follows that for each ~ f. 0, a(~)
+ B(~)k(~) + B(~)w <
°
Lin, Yuandan, Eduardo D. Sontag and Yuan Wang (1995) . Input to state stabilizability for parameterized families of systems. Int. Journal of Robust and Nonlinear Control, 5, pp. 187-205.
provided Iwl ~ 1/J(1~1)/2. From here one obtains the desired conclusion. •
Lin, Yuandan, Eduardo D. Sontag and Yuan Wang (to appear). A smooth converse Lyapunov theorem for robust stability. SIAM Journal on Control and Optimization. (Preliminary version in "Recent results on Lyapunov-theoretic techniques for nonlinear stability", in Proc. Amer. Automatic Control Conference, Baltimore, June 1994, pp. 1771-1775.).
Finally, we relate our results to those in (Isidori, 1995), Section 9.5, where the problem of disturbance attenuation with stability is discussed in the context of finiteness of L2 gains (we assume for this comparison that the output function is y = x). It was shown there that if system (13) admits a positive definite proper smooth function V such that for each ~ f. 0, Bo(~)
Praly, Laurent and Yuan Wang (submitted). Stabilization in spite of matched unmodelled dynamics and an equivalent definition of input-to-state stability.
= 0 =>
a(O
1
+ 4-y2 IB1(~)12 + 1~12 < 0,
Sontag, Eduardo D. (1989a). Smooth stabilization implies coprime factorization. IEEE Transactions on Automatic Control AC-34, 435-443.
(26)
°
where -y > is some constant, then there exists an almost smooth feedback k such that the closedloop system is ISS with an Iss-Lyapunov function satisfying 2 VV(f(~) + g(~)k(~) + p(e)w) ~ -y21w1 Observe that if (26) holds, then V is also an ISSLyapunov function as defined above. This follows immediately from the following observation:
Sontag, Eduardo D. (1989b). A 'universal' construction of Artstein's theorem on nonlinear stabilization. Systems fj Control Letters 13, 117-123.
e.
Sontag, Eduardo D. and Yuan Wang (1995). On characterizations of the input-to-state stability property. Systems fj Control Letters 24, pp 351-359. Sontag, Eduardo D. and Yuandan Lin (1992). Stabilization with respect to noncompact sets: Lyapunov characterizations and effect of bounded inputs. In Nonlinear Control Systems Design 1992, IFAC Symposia Series, 1999, Number 7, M. Fliess Ed., Pergamon Press, Oxford, 1993,
Lemma 3.3 Let a, b, c : IRn -+ IR be continuous functions. Assume that c is positive definite and proper. If a(e) + (b(e))2 + c(~) < for all ~ f. 0, then liIllJel--+oo = 00 .
Ib(m
°
Tsinias, John (1989). Sufficient Lyapunovlike conditions for stabilization. Mathematics of Control, Signals, and Systems 2, 343-357.
The Lemma follows directly from the inequality p + ; :2: 2yi7 for all p, (J' > 0. • Proof:
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