A necessary condition for local asymptotic stability of discrete-time nonlinear systems with exogenous disturbance

K e y w o r d s D i s c r e t e t i m e nonlinear systems, Control systems, Asymptotic stability, Robust control. 1. I N T R O D U C T I O N Local a sympto t i c s tab i l i za t ion of non l inea r control sys tems is one of the cen t ra l p roblems in control sys tems l i te ra ture . I t is one of the p r im a ry design cr i ter ia in physical , engineer ing, and indus t r i a l problems. The p rob lem of s tab i l izab i l i ty for non l inea r control sys tems can be viewed as a m o d e r n vers ion of the classical s tab i l i ty p rob lem for equi l ibr ia of o r d i n a r y differential equa t ions . Some i m p o r t a n t works on s t ab izab i l i ty of non l inea r works are the works of Jurd jev ic and Q u i n n [3], Brocket t [4], Aeyels [5], Byrnes and Isidori [6-9], S u s s m a n n [10], and Sontag [11]. In a recent work [12], Byrnes and S u n d a r a p a n d i a n derived a necessary cond i t i on for local a sympto t i c s tab i l i ty of con t inuous t ime non l inea r sys tems wi th c o n s t a n t real pa rame t r i c uncerta in ty , which is an e n h a n c e m e n t of Brocke t t ' s necessary cond i t ion [4] for non l inea r a u t o n o m o u s systems. In [1], we der ived the d iscre tet ime vers ion of the necessary cond i t ion ob t a ined by 0898-1221/06/$ see front matter (~ 2006 Elsevier Ltd. All rights reserved. doi: 10.1016/j.camwa.2005.01.031 Typeset by .A~IS-TEX 662 V. SUNDARAPANDIAN Byrnes and Sundarapandian. This paper is a discrete-time analog of our recent work [13] on continuous-time nonlinear systems with time-varying exogenous disturbance. In this paper, we derive a necessary condition for local asymptotic stability of equilibria of continuous-time nonlinear systems with time-varying disturbance. As a corollary of our general result, we deduce our recent necessary condition [1] for discrete-time nonlinear systems with constant real-parametric uncertainty and the necessary condition derived by Lin and Byrnes [2] for local asymptotic stability of equilibria of nonlinear discrete-time autonomous systems. In this paper, we consider a nonlinear system described by xk+l = f ( x k , ~ k ) ~ f , k ( X k ) , (1) where x E R n is the state and # E R m is the parametric uncertainty or disturbance for nonlinear system (2). The disturbance vector # is assumed to satisfy the exosystem dynamics given by