W-Stability and Local Small Gain Theorem

Copyright © IFAC System Structure and Control, Nantes, France, 1995

W -STABILITY AND LOCAL SMALL GAIN THEOREM H. BOURLEs* , and F. COLLEDANI** *Electricite de France, Direction des Etudes et Rechercbes, 1. Avenue du General de Gaulle.

92141 Clamart. France **Ecole Superieure d·Electricite. Plateau du Moulon. 92192 Gif sur Yvette. France

Resume. Une nouvelle sorte de stabilite entree-sortie est definie sur la base d'un espace de Sobolev. Des versions locales du tbeoreme des petits gains et du tbeoreme de passivite sont etablies dans ce formalisme . Abstract. A new type of input-output stability is defined. based on the use of a Sobolev space. A local version of the Small Gain Theorem is established in tbis framework. as well as a local version of the Passivity Theorem . Key Words. Local small gain; Local passivity: Sobolev space; Input-output stablity; Asymptotic stability

1. INTRODUCTION

is well suited to lead to stability characterizations in the time domain as well as in the frequency domain. This approach was limited by the fact that only global stability results are available in the literature; the very aim of this paper is to establish local results in a (more or less) equivalent framework. Note that the internal and the input-output stability are closely related in the case of reachable and uniformly observable systems (Vidyasagar, 1993; Hill and Moylan, 1982; Vidyasagar and Vannelli, 1982).

As is well known. two different types of stability can be considered for a system (Vidyasagar, 1993). The first one is the "internal stability" (e.g., asymptotic stability); the input of the system is then assumed to be zero, and. roughly speaking, one looks if the system state tends to zero from a non zero initial value; the basic tool to analyze this type of stability is the use of Lyapunov functions . This approach is well suited to obtain global or local stability results: the internal stability is global when (roughly speaking again) the state tends to zero from any initial value, and is local when this occurs for "sufficiently small" initial values of the state.

A new input-output approach, leading to global as well as local stability results has recently been developed: this is the so-called "input-to-state stability" (Sontag, 1989. 1990; Pan et aI., 1993). In this framework, (though it seems promising), characterizations of the stability in the frequency domain cannot be obtained, because the function space which is considered is Loo. Moreover, as Loo is not an inner product space, notions such as passivity cannot be defined in this context.

The second main type of stability is the "input-output stability". In this framework, the initial value of the stale is assumed to be zero, so that the system output is a function of the input! ; and one looks if the output belongs to a specific function space (e.g. a Lebesgue space Lp, l::;p::;oo) when the input belongs to the same space; moreover, one looks if the norm of the output (in this function space) is smaller than some constant multiplied by the norm of the input. The greatest lower bound of the constants satisfying this property is called the gain of the system. The case p=2 is particularly interesting, because the L2-gain of any linear time-invariant (L TI) system is the Hoo-norm of its transfer matrix [where Hoo denotes the well known Hardy space (Francis, 1987)]; moreover, as the Fourier transform is an isomorphism of L2, this space

The function space used in this paper is the set of functions x such that x and its derivative belong to L2· This is the well known Sobolev space W 1.2 (Treves. 1967). This space is interesting for several reasons: on the one hand, it will be shown that all nice properties of L2 are still satisfied by this space; in particular, the gain of any LTI system remains the H",,-norm of its transfer matrix. On the other hand, if a function x belongs to W I. 2 and is smooth enough [more precisely. if it is absolutely continuous; see, e.g., (Alexeev et al., 1982»). then x belongs to Loo, and its norm in this latter space is upper bounded by its norm in W 1.2 if the initial value of x is zero. This is the key property which will enable us to obtain simple

! In what follows. this function is called the input-output operator associated to the system (Bourles and Aloun.

1994 ).

49

conditions for nonlinear input-output operators to be "locally stable" (in a sense precise
such that x and its distributional-derivative n

xbelong

to L2. The inner product of two functions x and y in The paper is organized as follows: in Section 2, various signal spaces are defined, and their relations are clarified. In Section 3. "local W-stability" is defined, and then studied in various cases of systems. The notion of "local W-gain" is defined; the local-W-gain of a system G is denoted as 'YWl(G). The relationship between W-stability and asymptotic stability is studied in Section 4. In Section 5, a local version of the Small Gain Theorem is established. According to this theorem, if Systems Gland G2 in Fig.! are locally W-stable, and if YWl(G 1) 'YWl(G2) < I, then the closed loop system in Fig.1 is locally W-stable. +

Ul

~

el

G1

n

..

W 1,2 is defined as W = 2 + 2 and n

the nonn of x in W 1 ,2 is defined by IIxllw = n

n

[w)1I2. Obviously, W1,2 is included in L2 and n

for any function x in Wl ,2 one has IIxll2 ~ IIxllw. Acn denotes the set of functions x : R+~ Rn which are absolutely continuous. 2.2. Definitions and useful properties By Lebesgue's theorem, if x belongs to AC n, then x is almost everywhere differentiable (in the usual sense),

Yl

and its derivative

x is

integrable on any bounded

interval [a,b) included in R +; moreover, one has Y2

G2

e2

u2

b

J~('t) d't = x(b) -

x(a). The following facts are well

known (Alexeev et al., 1982) : AC n is a vector space; if x and y belong to AC n, then the function t ~ x(t) T y(t) belongs to AC 1. Let be a function : n ~ RID, where n denotes some open subset of Rn; if is Lipschitz-continuous and if x belongs to AC n and takes its values in n, then the function t ~ (x(t» belongs to ACID.

Fig. 1. Standard closed loop system In Section 6, as a consequence of this theorem, a local version of the Passivity Theorem is obtained. Due to the lack of place, most of results are stated without proof. The complete proofs will be found in (Bourles, 1995).

Definition 1. wn is the space of functions x belonging n n to W 1,2 n AC n and such that x(O)=O; We is the space of functions x belonging to AC n and such that x(O)=O

2. SIGNAL SPACES AND THEIR RELATIONS 2.1. Notation

.

n

S in Rn or C n is denoted as

and for any finite T>O, xT and (xh belong to L2. For

111;11. L2 denotes the usual Lebesgue-space of functions x: R+ ~ Rn, which are Lebesgue-measurable and square-integrable; 2 denotes the inner product of

any functions x and y belonging to We, we set

The norm of any vector

n

n

2,T and IIxllw,T = [W,T]I/2. Obviously, ILllw is a nonn on Wn,

n

two functions x and y in L2 , and we set IIxll2 = [2)l/2 . For any function x: R+ ~ Rn, xT denotes the truncation of x to the interval [O,T), i.e. the function such that XT(t) = x(t) if t ~ T, and XT(t) = 0 if t > T; L2ne denotes the "extended L~-space", i.e. the space of functions x such that for any T in R+, xT

whereas

is a family of semi-nonns on

W~. (These spaces can be replaced by their completions, and then become a Hilbert space and a Frechet space, respectively). n

Proposition 1. Let x EWe ; then for any finite T>O the following inequality holds:

n

belongs to L2. For any functions x and y belonging to n

L2e, we set 2.T = 2 and IIxll2,T =

IIxlloo,T ~ IlxllW,T

n

IIXTII2. In the same manner, Loo denotes the space of functions x : R + ~ Rn, which are Lebesguemeasurable and essentially bounded; the norm in this space is defined as IIxll oo = ess.sup {lIx(t)lI; t E R +} .

(I)

n

Wn is included in Loo; for any x E Wn, one has 11 x1100 ~ IIxllw and x(t) tends to 0 as t tends to 00 .

The "extended L~-space", L~e, is the space of functions x such that for any T in R +, XT belongs to n

(1I.IIw,T)T~O

Let K 1 denote the causal LTI operator with transfer matrix K 1(s) = (1 +s)-I In, where In denotes the identity matrix of dimension n The following result gives a useful characterization of signals in wn:

n

Loo, and Ilxlloo,T denotes the quantity IlxTlloo. W 1.2 denotes the Sobolev space of functions x : R + ~ Rn

50

n

n

Proposition 2. K 1 is one-to-one from L2e onto We

y;2

and K I-I is the operator y ~ y + for any finite T>O, lIyllW,T S; lIuIl2.T; y = KI u belongs to wn if and only if (ifO u e

L~, and then lIyllw = lIull2.

Corollary 1. Assume that G: L;e ~ W~ is such that K I-I G is L2-stable, and let y = G u; then, y belongs

3. LOCAL W-STABILITY

The notion of W -gain can be defined in an usual n

manner (Desoer and Vidyasagar, 1975): let G : We

~

n

Definition 2. Let G : We

~, and K = {bO: IIG ullw,T::S; k lIullW,T, 'v'u E W~, 'v'T>O}. If K is non empty, G is said to be

n

sup UEWD_{O}

IIGullw U

n

Remark 2. This definition of the "local gain" is close, but not identical to the definition of the "small-signalgain" in (Vidyasagar, 1993). This last definition corresponds to the case where the inequality lIullW.T < E above is replaced by lIull oo ,T < E. By Proposition 1, any small-signal-W-stable system is loW-so Obviously, if G is linear, then G is l-W-s iff G is W-s, and Ywl(G) = yw(G).

W

ID

Let G: W e ~ We; by Proposition 2, one has the following result., where Y2 denotes the L2-gain:

Let us consider the case of a nonlinear memory less operator G, defined as

Proposition ':l. G is W-stable iff KI-I G KI is L2stable, and yw(G) = Y2(KI-I G KI).

(G u)(t) = (t., u(t» Remark 1 Assume that G is a L TI operator with transfer matrix G; as G and KI commute, one has yw(G) = n(G) = IIGlloo (note that with respect to transfer matrices, 11.1100 denotes the norm in Hoo, whereas with respect to time domain signals, this symbol denotes the norm in Loo). n

For any operator G: L2 e Proposition 2 sup uEL~-{O}

IIG ullw lIull2

~

=

Y2

'v'~0

(4)

where : lR + x n ~ lR ID (n denoting some open neighborhood of 0 in lRn) is a Cl function satisfying n

(t,O) = 0, 'v't, and where u is any function in We , taking its values in n. Let at and a u respectively denote the partial derivative with respect to t and u. The following notion will be useful :

ID

We, one has by

(K -I G) 1

Definition ':l. Let u* E n; we say that is differentiable with respect to u at point u* uniformly with respect to t, if there exists a function 11 : n ~ R+ such that 11 (u*+h) tends to 0 as h tends 0 and that

(3)

'v'~0

1I<1>(t,u*+h) - (t.,u*) - au(t,u*) hll = 11(u*+h) IIhll.

(this quantity being finite iff KI-I G is L2-stable). In particular, for 't>0, let K't be the LTI operator with transfer matrix K't(s) = (1 +'ts)-I In. Obviously, K I-I

(5)

The only difference, compared with the usual notion of differentiability with respect to u, is that .., is independent of t.

n

K't is L2-stable, so that for any function u E L2, K't

U

n

E

ID

We, and Kl = {bO : 3

>0, 'v'u EWe, 'v'T>O, lIullW.T < E => IIG ullw.T::S; k lIullw.T}. If Kl is non empty, then we say that G is locally-W-stable (1- W os), and Y'Vn(G) = inf(Kl) is called the local-W-gain (l-W-g) of G. If Kl is empty, we set YWl(G) = 00. 3

(2)

11 11

~

£

W-stable (W-s), and yw(G) = inf(K) is called the Wgain of G (note that, according to the terminology used in the reference above, this is the defmition of the W-gain with zero bias); if K is empty, we set yw(G)=oo. All operators considered in this paper are ~sumed to be causal, hence one has (Willems, 1971) yw(G) =

I

n

ID.

to Loo If u belongs to L2, and lIylloo ::s; Y2(K 1- G) lIull2·

wn. Hence, wn (rep. We) is the space of functions n

n

Proposition 4. Assume that is locally Lipschitzcontinuous at point u*=O, is differentiable with respect to u at point 0 uniformly with respect to t, and satisfies the following condition:

in L2 (resp. L2e) filtered by first order low-pass filters. Note that by Proposition 3, one has min('t, 't- I ) n(K't- 1 G K't)::S; yw(G)::S; max('t, 't-I) n(K't- 1 G K't). The following result is a consequence of (3) and Proposition 1, and can be of practical interest:

n

As YEW e. the derivative Y is defined almost everywhere in the usual sense. Note that K (1 is causal. 2

3

Note that 'YWI(G) = inf

sup

£>0 O
causal: see (Willems, 1971).

51

IIG ullw Ilullw ' because G is

°

sup IId t(t,u)1I

°

lim.sup £?::o u~ lIull

=

~<

Then, if is a locally exponentially stable eqUilibrium point of (8) (with u = 0), the operator G associated to Lis I-W-s.

(6)

00

This result is closely related to (Vidyasagar, 1993, Sec. 6.3, Theorem 15); the main difference (apart from the fact that W-stability, instead of Lp-stability is obtained) is Condition (10). For instance, in the case of a linear output equation y ;::: C(t) x, this condition

Then, G is 1-W -s and "YWI(G) ~ ~ + sup cr (du(t,O» . £?::O

(7)

For example, in the case n=l, m=l, assume that (t,u) = cos rot (e U - 1), ro;;::O; one then has "YWI(G) ~ ro+l.

reduces to sup C1 (C(t» <

Now, let us consider the case of an operator G associated to a nonlinear time-invariant system L

We now assume that L is time-invariant, Le. f(t,x,u) ;::: f(x,u), g(t,x,u) = g(x,u), where f and g are Cl in a neighborhood of z* ;::: and satisfy f(O,O) = 0, g(O.O) ;::: 0 . Let us denote as 4>(t, to, xo , u) the state x(t) satisfying the initial condition x(to) = xo.

4 .2. Time-invariant case

°

x

described by state-space equations = f(x,u) , y = g(x,u) (see footnote 1). where f and g are C I in a neihborhood of (0,0) and satisfy f(O,O) = 0, g(O,O) = 0. Set dxf(O,O) = A, duf(O,O) = B, dxg(O,O) = C, dug(O,O) = D, and let G(s) be the transfer matrix of the linear approximation LI of L around (0,0) . The theorem below is very useful:

Definition 4. L is locally W -reachable if there exists a function ~ of class K and a neighborhood U of in Rn such that for any x in U, there exists a finite time 't and a control u E wm such that 4>('t, 0, 0, u) = x, u(t) = for t;;:: 't and lIullw ~ ~(lIxll) . If U ;::: R n, L is said to be globally W-reachable.

°

°

Theorem 1. Assume that (C, A) is detectable, (A, B) is stabilizable and G E Hoo. Then, L is I-W-s and "YWI(L) = IIG11oo .4

W -reach ability is a natural generalization of controllability in the case of linear systems possibily fed back by a time-invariant memoryless nonlinearity (see Proposition 6 and Remark 3):

°

Sketch of the Droof. The equilibrium is exponentially stable for the unforced system Lt. and therefore for the unforced system L. Now, L can be considered as a perturbed version of L , with a perturbation whose local W -gain is equal to (applying Proposition 4) . The theorem is then a consequence of Remark I.

°

4.1. Time-varying case

°

Proposition 5. Consider the system L, described by the state-space equations

x

(8) (9)

Recall that the nonlinear time-invariant system 1: is said to be locally uniformly observable iff there exist a function 0. of class K and a neighborhood V of in R n such that V'XE V, IIg(4)('' 0, x, 0), 0)112;;:: o.(lIxll): if V;::: R n, then L is said to be globally uniformly observable (Vidyasagar, 1993). For LTI systems, uniform observability is equivalent to the standard notion of observability; moreover, the closed loop system Lf above is globally uniformly observable if (C, A) is observable. The theorem below can be considered as the reciprocal of Proposition 5 (with additional minimality assumptions on the state-space realization) . Its proof is close to the one of (Vidyasagar, 1993, Sec . 6 .3, Theorem 39), with appropriate modifications.

x(t) E R n, u(t) E Rm, yet) E RP, where fis Cl , fand g are locally Lipschitz-continuous in a neighborhood of z*;:::O, where z ;::: [x T uT]T, and f(t,O,O) ;::: 0 and g(t,O,O) ;::: 0, Vt. Assume that g is differentiable with respect to z at point 0, uniformly with respect to t, and that sup lIa~(t,z)1I .

t?!O

hm .sup ="::-II-zl-I--<

z

~

00

PTQoosition 6. Assume that L is linear, i.e. f(x ,u) ;::: A x + B u , g(x,u) ;::: C x + D u. Then, (A, B) is controllable iffL is globally W-reachable. Remark 3 . Consider the feedback system in Fig .l , where Gl is the system L of Proposition 6, and where G 2 is a time-invariant memory less nonlinearity defined by G2(U)(t) ;::: (u(t», where is differentiable in an open neighborhood U of in Rm and satisfies <1>(0) ;::: O. Assume that U2;:::0 and consider the system Lf with input u I. Then, Lf is W -reachable (locally is U is stricly included in Rm, and globally if U=Rm) if (A, B) is controllable.

4. RELATIONSHIP BETWEEN W -STABILITY AND ASYMPTOTIC STABILITY

= f(t,x,u) y;::: g(t,x,u)

00.

t?!O

°

( 10)

0

For the sake of simplicity. the system :E and the associated operator are denoted by the same symbol.

4

52

m n operator G l : W e ~ We satisfying Ywl(GZ) < 1, iff Yw(G 1) ~ l.

Theorem 2. Assume that the non linear time-invariant system L is locally (resp. globally) W -reachable, locally (resp. globally) unifonnly observable and I-W-s (resp. W os). Then, 0 is a locally (resp. globally) asymptotically stable equilibrium for the unforced system.

6. LOCAL PASSIVITY As wn is an inner product space. the notions of Wpassivity and strict W-passivity can obviously be defined (Willems, 1971; Desoer and Vidyasagar, 1975). Local versions of these notions will now be defined.

5. LOCAL SMALL GAIN THEOREM Let us consider the standard closed loop system in Fig. n

m

m

n

n

Let G : W e

I, where G 1 : We ~ W e and G 2 : W e ~ We are

I-W-s: set u = [UI T U2T]T, Y = [YI T Y2T]T and e = [e I T e2 T] T. All signals are assumed to be zero at initial time O. Assume that this closed loop system is well posed (Willems, 1971). i.e. there exist two n+m

~

n

We be a causal operator.

Definition 4. G is locally W -passive if there exists n

£>0 such that W.T ~ 0 whenever u EWe and T>O are such that lIullw.T < £ 5; G is locally strictly W-passive if there exists 11>0 such that G -11 I is locally W-passive.

n+m

operators HI and H2 : W e ~ We such that e = HI u and y = H2 u. We will say that this closed loop system is I-W-s if HI and H2 are loW-so

n

n

Theorem 3 IfYWI(Gl) YWl(G2) < 1, then the closed loop system in Fig. 1 is I-W-s.

Assume that (I + G )-1 : We ~ We is well defined, and let H = (G - 1)(1 + G)-I. It is well k'llown (Desoer and Vidyasagar, 1975, p. 216) that Yw(H) ~ 1 if G is W -passive, and Yw(H) < 1 if G is W -s and W -strictly passive. These properties still hold if local gains and local passivity are considered (instead of "global" ones). Therefore, by a standard scheme equivalence and by Theorem 3, one obtains the following result:

Remark 4. This theorem is still true if Lp-stability

Theorem 4. Let us consider the standard closed loop

(l~p
system in Fig.I, where Gl and G2 : We ~ We are such that this closed loop system is well posed. If Gl is locally W-passive and if G2 is I-W-s and locally strictly W -passive, then this closed loop system is 1W-s.

The following theorem is a local version of the well known Small Gain Theorem (Zames, 1966a). Its proof is very close to the one given in (Bouries, 1994) for discrete-time systems.

n

is considered instead of W-stability. However, W-stability is more interesting because Proposition 4 and Theorem 1 do not hold in Lp framework. Consider for instance the case where one of the operators in the loop is memory less and only locally bounded (in the sense where the associated function cl> satisfies the assumptions of Proposition 4); a signal x in L2 can take very large values, even if IIxli2 is very small. and then nothing can be said about the signal t ~ cI>(t,x(t)); hence. local L2-stability cannot be established (in an input-output approach). Small-signal-L2-stability can be proved in this case by first proving internal stability (via Lyapunov theory) and then using (Vidyasagar, 1993, Sec. 9.3, Theorem 39) . But to prove internal stability can be difficult when one of the systems in the loop is infinite dimentional (see all examples below).

n

n

Proposition 7 Let G : We ~ We be a LTI operator; then, G is locally (strictly) W-passive iff G is (strictly) W-passive, and this condition is satisfied if G is (strictly) L2-passive. Therefore, W -passivity or strict W -passivity of a LTI operator can be analyzed in the frequency domain. Now, let us consider the case of a nonlinear memory less operator G, defined by (4), where cl> is like in Proposition 4. Let !l(.) denote the usual matrix measure (Desoer and Vidyasagar, 1975), i.e. for any matrix A E C nxn , Il(A) = Amax (A + A *)12.

Example 1. Assume that Gl is the L 11 operator with e- ts transfer function k 1 + Ts (1)0, T>O) and that G2 is

Proposition 8. G is locally strictly W -passive if i) inf {- 11[- aucl>(t.O)]} = 8 > 0 and ii) ~ < 28, where ~

an operator of the fonn (4), where cI>(t,u) = cos cot (e U -1), co~O. By Theorem 3 and Proposition 4, the

~o

is defined by (6).

resulting closed loop system is I-W-s if Ikl < _1_. 1 + ro

Example 2. Let GIbe the LTI operator with transfer

Theorem 3 gives a sufficient condition for I-Wstability; conversely, using Remark 1. a necessary condition can also be obtained: n

n

function

b . lal < 1. b>O. T>O. and let 1 + a e- ts + Ts

5 In other words. G is locally W -passive iff inf

m

Corollary 2. Let G 1 : We ~ W e be a L TI operator. The closed-loop system in Fig. 1 is I-W -stable for any

w

53

~

O.

inf

G2 be the operator G defined by (4), with (t,u)

=(0

Feedback for FrequencylPower Control of Power Plants, Proc. 32 nd CDC, December 13-17, San Antonio (Texas), 3740-3741. Desoer, C. A., and Vidyasagar, M. (1975), Feedback Systems: Input-Output Properties. Academic Press, New-York . Francis, B. A. (1987). A Course in Hoc Control Theory. Springer Verlag, Berlin . Hill, D. J., and Moylan, P. J: (1982). Connections Between Finite-Gain and Asymptotic Stability, IEEE Trans. on Autom. Control,25, 931-936. Pan, D.-J., Han, Z.-Z., and Zhang, Z.-J . (1993). Bounded-input-bounded-output stabilization of nonlinear systems using state detectors, Syst. Contr. Letters, 21, 189-198. Safonov, M. G., and Athans, M. (1981). A Multiloop Generalization of the Circle Criterion for Stability Margin Analysis, IEEE Trans. on Autom. Control, 26, 415-422. Sontag, E. D. (1989). Smooth Stabilization Implies Coprime Factorization, IEEE Trans. on Autom. Control, 34, 435-443. Sontag, E. D . (1990). Further Facts About Input-ToState Stabilization, IEEE Trans. on Autom. Control, 35, 473-476. Treves, F. (1967). Topological Vector Spaces, Distributions and Kernels. Academic Press, London. Vidyasagar, M. (1993). Nonlinear Systems Analysis (second ed.). Prentice-Hall, Englewood Cliffs, N. J. Vidyasagar, M ., and Vannelli, A. (1982). New Relationships Between Input-Output and Lyapunov Stability, IEEE Trans. on Autom. Control, 27, 481-483. Willems, J. C. (1971). The Analysis of Feedback Systems. The MIT Press, Cambridge (MA) . Zames, G. (l966a). On the Input-Output Stability of Time-Varying Nonlinear Feedback Systems. Part I: Conditions Derived Using Concepts of Loop Gain, Conicity, and Positivity", IEEE Trans. on Autom. Control, 11, 228-238. Zames, G. (1966b). On the Input-Output Stability of Time-Varying Nonlinear Feedback Systems . Part II: Conditions Involving Circles in the Frequency Plane and Sector Nonlinearities, IEEE Trans. on Autom. Control, 11, 465-476.

+ 1 + cos rot) sin u; Gl is W-passive and G2 is I-W-s and locally strictly W-passive if 0 ~ ro < 20 (by Proposition 8); hence, by Theorem 4, the closed loop system in Fig. 1 is I-W-s if the latter condition is satisfied. Remark ". Other standard loop transformations can be made, in order to obtain a local version of the Circle Criterion (Zames, 1966b; Safonov and Athans, 1981). 7. CONCLUSION For several reasons, W -stability and local-W -stability are of practical interesl Roughly speaking, a system is 1-W -s if, when excited by a signal with small energy and filtered by a first order low-pass filter K" its output, "filtered" by the inverse filter, is of small energy (Section 2). Therefore, this type of stability is of signifiance in cases where disturbances, reference signals, etc., acting on the system can be considered as filtered signals (all signals are probably of this kind in practice). Moreover, W -stability and asymptotic stability are closely related in case of a minimal realization (Section 4). The local-W-gain of a timeinvariant non linear system is nothing but the Hoonorm of the transfer matrix of its linear approximation (Theorem 1), so this notion is clear. W-stability is well suited to formulate a useful Local Small Gain Theorem and a Local Passivity Theorem (Remark 4). An interesting possible application of the local stability results stated in this paper is the robustness study of feedback systems perturbed by nonlinearities which are only locally bounded (Colledani et al., 1993). Usual input-output theorems indeed cannot be applied in this case, and to prove internal stability can be difficult (especially in case of infinite dimentional systems). Local input-output stability results can also be obtained for discrete-time systems (Bourles, 1994). The relationship between W-stability and asymptotic stability in the case of infinite-dimensional systems will be studied elsewhere. 7. REFERENCES Alexeev, V. ,Tikhomirov, V., and Fomine, S. (1982). Commande optimale. Mir, Moscow. Bourles, H. (1994). A Local Small Gain Theorem for Discrete-Time Systems, Proc . 33 rd CDC, December 14-16, Lake Buena Vista (Florida), 2137-2138. Bourles, H., and Aloun, F. (1994). In: La Robustesse, Analyse et Synthese de Commandes Robustes, (A. Oustaloup, Ed.), Chap. 3, pp. 163-235. Hermes, Paris. Bourles, H., and Colledani, F. (1995). W-Stability and Local Input-output Stability Results, IEEE Trans. on Autom. Control (to appear). Colledani, F. , Bourles, H., and Vanhersecke, M . P. (1993). Robust Controller with Local Linearizing

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