Averaging Results for Homogeneous Differential Equations

Copyright 0 IFAC System Structure and Control, Nantes, France, 1998

AVERAGING RESULTS FOR HOMOGENEOUS DIFFERENTIAL EQUATIONS

J oan Peuteman, Dirk Aeyels

SYSTeMS Universiteit Gent Technologiepark- Zwijnaarde, 9 9052 GENT (Zwijnaarde) BELGIUM e-mail : [email protected] e-mail : [email protected]

Abstract: Within the Liapunov framework , a sufficient condition for uniform asymptotic stability of ordinary differential equations is proposed. Unlike with classical Liapunov theory, the time derivative of the Liapunov function, taken along solutions of the system, may have positive and negative values. It is shown that the proposed condition is useful for the study of uniform asymptotic stability of homogeneous systems with order T > O. In particular, it is established that asymptotic stability of the averaged homogeneous system implies local uniform asymptotic stability of the original time-varying homogeneous system. This shows that averaging techniques may play a prominent role in the study of homogeneous -not necessarily fast timevarying- systems. Serniglobal stability results may be obtained by 'speeding up' the system by means of a change of time-scale. Copyright © 1998IFAC Resume: Dans le contexte de Liapunov, nous proposons une condition suffisante de stabilite uniformement asymptotique pour une equation differentielle ordinaire. La derivee de la function de Liapunov, dans la direction des solutions du systeme, peut prendre des valeurs positives et negatives. Cette condition va servir a. etudier la stabilite uniformement asymptotique des systemes homogenes avec un degre r > O. En particulier, nous montrons que la stabilite asymptotique du systeme homogene moyennise implique la stabilite uniformement asymptotique dans le voisinage du point d'equilibre. Ce result at montre que les techniques de moyennisation permettent d' etudier les systemes homogenes a. l'aide d'un champ de vecteurs dont la dependence du temps n'est pas necessairement acceleree. Des proprietes de stabilite semi-globale sont obtenues si on accelere l'axe reel du temps du champ de vecteurs. Keywords: Nonlinear systems, time-varying systems, asymptotic stability, averaging control.

1. INTRODUCTION

V(x, t) whose derivative along the solutions of the system is negative definite. When this derivative is negative semi-definite, stability rather than asymptotic stability follows . When the derivative of the Liapunov function is negative semi-definite and the differential equation is autonomous, the

The classical Liapunov approach to uniform asymptotic stability of the null solution of a dynamical system x(t) = !(x, t) requires the existence of a positive definite, decrescent Liapunov function 65

Barbashin-Krasovskii theorem or LaSalle's invariance principle may be helpful in proving asymptotic stability; extension to the periodic case is possible. For nonperiodic systems, Narendra and Annaswamy (1987) show that with V(x, t) ::; 0 uniform asymptotic stability can be proven if there exists aT> 0 such that '
when a increases. In Section 5, it is shown that we obtain semiglobal uniform asymptotic stability i.e. any bounded region of attraction can be guaranteed by choosing a sufficiently large.

negative semi-definiteness condition on the timederivative of the Liapunov function can be dispensed with: the origin of a dynamical system is

t k·

2. SUFFICIENT CONDITION FOR UNIFORM ASYMPTOTIC STABILITY In this section, sufficient conditions for uniform asymptotic stability of a dynamical system are proposed. In classical Liapunov theory, the time derivative of the Liapunov function along the solutions of the system is required to be negative definite. In the present case, the derivative of the Liapunov function may have positive and negative values. The theorerp only requires the existence of a Liapunov function and a sequence of times such that the value of this Liapunov function decreases when evaluated along the solutions at

t;

Consider

uniformly asymptotically stable under the condition that for a positive definite de crescent V(x, t) , 3T > 0 and a sequence of times t; such that V(X(tk+1),t k+1 ) - V(x(tk),t k ) ::; - , (llx(tk)11) with ,( .) : R + -+ R which is continuous, strictly increasing and passing through the origin and t k+1 - tk ::; T Vk E Z and tk -+ 00 as k -+ 00 and -+ -00 as k -+ -00. Compared to (Narendra and Annaswamy, 1987), V(x, t) ::; 0 is no longer required. Compared to (Aeyels and Peuteman, accepted), the asymptotic stability condition needs to be satisfied only for a sequence t k, not for all t . Unlike (Aeyels and Peuteman, submitted), this paper is focused towards uniform asymptotic stability, not exponential stability. In Section 2 of this paper, a proposition is stated, giving a sufficient condition which guarantees uniform asymptotic stability of a nonlinear dynamical system. In Section 3 and 4, uniform asymptotic stability of time-varying homogeneous systems ±(t) = f(x, at) (Va E Rt) with order T > 0 is discussed using the proposition of Section 2. It is shown that asymptotic stability of the averaged system guarantees local uniform asymptotic stability of the original time-varying homogeneous system. An important - perhaps surprising feature of our result is that it is valid independent of the scale factor a . This shows that averaging techniques may play a prominent role in the study of homogeneous -not necessarily fast timevarying- systems of order T > O. When the order T = 0 (M 'Closkey and Murray, 1993), this result is only obtained for a sufficiently large. The region of attraction of such a homogeneous system x(t) = I(x, at) (a E Rci) with order T > 0 depends on a. This region of attraction increases

x(t) = f(x , t)

(1)

with 1 : W x R -+ Rn , W open, W C Rn . Let 1(0, t) = 0 '
t;

Proposition 1. Let U C W be an open neighborhood of O. Consider a closed ball BI'(O) c U then ' 0, T finite: 3J.L' > 0 such that (xo, to) E BI" (0) x R implies that x(t) E BI'(O) for t E [to, to + T) . Here x(t) is the solution of (1) evaluated at t with initial condition Xo at to . (the solutions are assumed to exist over the considered time interval) Remark 2. The proof of Proposition 1 shows that J.L' = J.Le- KT is an appropriate choice of J.L' . K is the maximum of the upper bounds of the Lipschitz functions lx (t) corresponding to the finite subcovering of BI'(O) which consists of neighborhoods N(x) . Proposition 3. Consider a function V : U x R -+ R, with U C W an open neighborhood of O. We assume that the following additional conditions are satisfied:

66

• Condition 1: V (x, t) is positive definite and decrescent; i.e. V(O, t) = 0 'r/t and 'r/x E U : a(lIxll) :s: V(x, t) :s: ,8(11xll)· The functions a(·) : R+ -+ R and ,8(.) : R+ -+ R

is homogeneous of order T with dilation <5(s,x) = (STi X1 , ... , sTnxn)T ('r/i E {I, ... , n} : 0 < ri < 00) if for each i E {I, .. . , n}, 'r/x E Rn and 'r/s ~ 0:

are strictly increasing continuous functions passing through the origin. • Condition 2: There exists an increasing sequence of times t'k (k E Z) with t'k -+ 00 as k -+ 00 and t'k -+ -00 as k -+ -00, 3 finite T > 0 : t'k+l - t'k :s: T (Vk E Z) , there exists a /( .) : R+ -+ R which is continuous, strictly increasing and passing through the origin and an open set U ' C U which contains the origin such that 'r/k E Z, 'r/x(t'k) E u' \ {O}:

AV(t'k+l' t'k)

Definition The homogeneous p-norm Pp with dilation <5(s, x) = (STiXl, ... ,sTnXn)T is a continuous map from Rn to R+, X -+ pp(x) such that (5) i=l

:s: -/(llx(t'k)11) (2) = V(x(t'k+l),t'k+l)-

with pERt .

Here, AV(t'k+l,t'k) V(x(t'k)' tk) and X(t'k+l) is the solution of (1) with initial condition x(t'k) at t'k .

Remark 6. Calling the function Pp a "norm" is a misnomer. In general Pp does not satisfy the triangle inequality neither the scale property.

Then the equilibrium point x = 0 of (1) is uniformly asymptotically stable.

Proposition 7. Consider the homogeneous system = !(x , t) with order T > 0 and with dilation <5(s, x) = (STi Xl, ... , STn Xn )T . ! is locally Lipschitz i.e. 'r/x, 3 neighborhoodN(x) such that the restricx(t)

Remark 4. The proof of Proposition 3 shows that when the closed ball with center 0 and radius €, B«O) C U' that Bo«)(O) with <5(10) = e- KT {3-1(a(€e- KT » belongs to the region of attraction of (1) where K is the maximum of the upper bounds of the Lipschitz functions lz(t) corresponding to the finite sub covering of B«O) which consists of neighborhoods N(x) .

tion !IN(z) is Lipschitz with Lipschitz function lz(t) and lz(t) is bounded 'r/t E R. If there exists an increasing sequence of times t'k (k E Z) with t'k -+ 00 as k -+ 00 and t'k -+ -00 as k -+ -00, 3 finite T > 0 : t'k+l - t'k :s: T ('r/k E Z) and 3 KI > 0 such that 'r/k and 'r/x with PT(X) = 1 and r > max{rl, ... ,rn }

Remark 5. IT Condition 2 of Proposition 3 is replaced by the condition that there exists an increasing sequence of times t'k (k E Z) with t'k -+ 00 as k -+ 00 and t'k -+ -00 as k -+ -00, 3 finite T > 0 : t'k+l - t'k :s: T ('r/k E Z), there exists an open set U ' C U which contains the origin such that 'r/k E Z, Vx(t'k) E {O}:

(6)

u' \

V(X(t'k+l)' tk+l) - V(x(t'k), t'k) then the equilibrium point x formly stable.

= 0 of

:s: 0

where

(3)

• V(x) is a positive definite continuous homogeneous Liapunov function i.e. 'r/x ERn,

(1) is uni-

'r/s

3. HOMOGENEOUS SYSTEMS

~

0

for some I > O. • ~~ is locally Lipschitz i.e. 'r/x, 3 neighborhood Nv(x) such that &~~%) INv(z) is Lipschitz with Lipschitz function I;'.

Proposition 3 introduces a sufficient condition for uniform asymptotic stability of a dynamical system. Because of Condition 2, it may in general be a difficult task to verify uniform asymptotic stability by means of this proposition. This section can be seen as an elaboration of the previous one. When considering homogeneous systems, we show that Condition 2 of Proposition 3 may be replaced by a condition independent of the flow.

then there exists a P with PT(X(t'k» < p:

Definition

Here, AV(t'k+l,t'k) = V(x(t'k+l» - V(x(t'k» and X(t'k+l) is the solution of x(t) = !(x, t) with initial condition x( t'k) at t'k .

Thesystemx(t)

> 0 such that 'r/x(t'k) =I 0

(8)

= !(x,t) with x = (Xl,X2, ... ,xn )T 67

These conditions also imply that Vo: > 0 : X = f (x, o:t) is locally uniformly asymptotically stable. No global stability property is obtained. The region of attraction of such a system x = f(x,o:t) depends on 0: and this bounded region of attraction can be made arbitrary large by taking 0: sufficiently large.

4. UNIFORM ASYMPTOTIC STABILITY OF TIME-VARYING HOMOGENEOUS SYSTEMS Having Proposition 7 available, it is now possible to establish that asymptotic stability of the averaged system of a time-varying homogeneous system implies local uniform asymptotic stability of the original time-varying homogeneous system. Because of the homogeneity and the order condition 'T > 0, this result is valid when the original system is not fast time-varying.

Theorem 10. The homogeneous system x = f(x,o:t) with order 'T > 0 where x = f(x, t) satisfies all the conditions of Theorem 8 is semiglobally uniformly asymptotically stable i.e .

Theorem 8. Consider the homogeneous system x(t) = f(x , t) with order 'T > 0 and with dilation 6(s, x) = (SrlXl ' ... srnxn)T . f is locally Lipschitz i.e. Vx , 3 neighborhoodN(x) such that the restriction fl.N"(,,) is Lipschitz with Lipschitz function [,,(t) . [,,(t) is bounded Vt ER.

• the system is uniformly stable • VR > 0: 30: > 0 such that VXo E BR(O) , Wo t;: R : x(t) converges uniformly to 0 as t tends. to 00 i.e. Vf. > 0, 3T(f.) > 0: IIxo!l < R implies that !lx(t)" < f., Vto , Vt> to + T(f.).

If

Here x(t) is the solution of the homogeneous system x = f(x , o:t) with initial condition Xo at to·

• Condition 1: The averaged system x(t) f(x) is asymptotically stable where +T

f(x)

:= }~oo

1 / 2T

(9)

f(x , t)dt

6. CONCLUSIONS

-T

• Condition 2: There exists a continuous function M : [0, +oo[~ [0, +oo[ with lim M('T) 'T

=0

Proposition 3 gives a sufficient condition for asymptotic stability of a differential equation. This result is related to the result of NarendraAnnaswamy (1987), but the negative semi-definiteness on Vex, t) is no longer needed. This result is useful for the investigation of uniform asymptotic stability of homogeneous sytems with order 'T > O. More precisely, averaging becomes a useful tool to investigate uniform asymptotic stability i.e. asymptotic stability of the averaged system implies local uniform asymptotic stability of the original time-varying system. It is important that this result is not restricted to fast time-varying systems. The region of attraction of x = f(x,o:t) increases with increasing 0:. The uniform asymptotic stability is semiglobal since by taking 0: large enough, every bounded region of attraction can be guaranteed. Comparing. these results with the results of M'Closkey and Murray (1993), it becomes clear that

(10)

T~OO

! t2

11

(f(x, t) - f(x»dtll

~ M(tz -

td(l1)

tl

when Pr(x)

= 1 with r > max{rl ' ... , rn} .

then x(t) = f(x, t) is locally uniformly asymptotically stable.

Remark 9. The proof of Theorem 8 relies upon the result of Rosier (1992). The asymptotic stability of x = f(x) implies the existence of a homogeneous Liapunov function Vex) whose derivative along the solutions of the system x = f(x) is negative definite. Although, in general, the derivative of Vex) along the solutions of x = f(x, t) may attain positive and negative values, it satisfies the conditions of Proposition 7 and therefore, it also satisfies (8) which implies by Proposition 3 the local asymptotic stability of x = f(x, t).

• we are dealing with homogeneous systems of order 'T > 0 while M'Closkey and Murray (1993) are dealing with homogeneous systems of order 'T = 0. • because of the order 'T > 0, asymptotic stability of the averaged system implies asymptotic stability, not exponential stability, of the original time-varying system. M'Closkey and Murray (1993) consider homogeneous systems of order 'T = 0 and therefore they may conclude exponential stability, with repect to

5. SEMIGLOBAL UNIFORM ASYMPTOTIC STABILITY The conditions of Theorem 8 imply local uniform asymptotic stability of the homogeneous timevarying system x f(x, t) with order 'T > O.

=

68

the homogeneous norm , of the original timevarying system . • we obtain local stability results for the homogeneous system i; = f(x, at) with order T > 0 for every a > O. By setting (1 = at and t: = 1. , i; = f(x,at) is equivalent with i; = t:f(x,(1). M'Closkey and Murray (1993) deal with homogenous systems i; = t:f(x, (1, t:) of order T = 0 and therefore, the stability results are only valid for t: sufficiently small. Q

7. ACKNOWLEDGEMENTS This paper presents research results of the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian state, Prime Minister's Office for Science, Technology and Culture. The scientific responsability rests with its authors.

References D. Aeyels (1995) . Asymptotic stability of nonautonomous systems by Liapunov's direct method. Systems f9 Control Letters, 25 , pp. 273-280. D. Aeyels and R. Sepulchre (1994) . On the convergence of a time-variant linear differential equation arising in identification. Kybernetika, 30 , pp. 715-723. D. Aeyels and J. Peuteman (accepted) . A new asymptotic stability criterion for nonlinear time-variant differential equations. IEEE Trans. Automat. Control. D. Aeyels and J. Peuteman (submitted). On exponential stability of non-linear time-variant differential equations. R.T. M'Closkey and R.M . Murray (December 1993) . Nonholonomic Systems and Exponential Convergence: Some Analysis Tools. Proceedings of the 32nd Conference on decision and Control, San Antonio, Texas, pp. 943-948. K.S . Narendra and A.M. Annaswamy (1987) . Persistent excitation in adaptive systems. Int. Journal of Control, 45, pp. 127-160. L. Rosier (1992). Homogeneous Lyapunov function for homogeneous continuous vector field . Systems fj Control Letters, 19, pp. 467-473.

69