Stability of sampled-data composite systems with many nonlinearities

Abstract

A sampleddata composite system given by a set of vector difference equations is dealt wi t h. The system given by X~( T + 1) -XXT) = A i i f i [ x i ( r ) ] is referred to as the ith isolated subsystem. It is shown that the composite system is asymptotically stable in the large if the A satisfy certain conditions and the leading principal minors of the determinant Ibijl, i, j = 1, . . . , n. are all positive. Here, the diagonal element bii is a positive number such that IIxi(r + 1)li -IXi(T)li 5 -hill fi[Xi(r)]I holds with regard to the motion of the ith isolated subsystem, and the nondiagonal element bij, i # j , is the minus of IIAijll, which is defined as the maximum of llAirrjll, for IIxjl = I. Some extensions of this result are also given. Composite relay controlled systems are studied as examples.

Stability and compensations of systems with multiple nonlinearities

“…Similar results have been established for systems described by difference Equations [3], [6] , [14] and also for systems described by functional differential Equa tions [14] . More recently, this work has been extended to stochastic systems [16] and to the analysis of bounded-input bounded-output stability [20].…”

“…Thompson [22] used a scalar Lyapunov function to establish conditions for exponential stability. Araki, Ando and Kondo [3] used a particular Lyapunov function to obtain stability results for a class of systems described by difference equations. Michel and Porter [13] generalized this approach to systems with nonlinear, time-varying interconnections described by ordinary differential equations or difference equations.…”

Stability of large-scale nonlinear systems--Quadratic-order theory of composite-system method using M-matrices

“…Michel [4] made basic study about use of Lyapunov's second method for CS. The simple M-matrix condition as given in the text was obtained by Araki et al [5], [7] and by Siljak [8] almost contemporarily in different contexts, whereas the Lyapunov-type theorem on M -matrices ((vi) of Theorem Al) was proved in [7] and its discrete-time version (Theorem A2) in 1201. Siljak [8] emphasized the aspect that arbitrary time-varyingness within given bounds is allowed for interconnections and introduced the idea of "connective stability."…”