Necessary and sufficient conditions for the absolute stability of discrete type lurie control system

Abstract

In this paper, it is discussed that the obsolute stability for zero solution of the discrete type Lurie control system

$$\left. {\begin{array}{*{20}c} {x(n + 1) = Ax(n) + bf[\sigma (n)]} \\ {\sigma (n) = c^T x(n)} \\ \end{array} } \right\}$$

((1))

in which the nonlinear function f(δ) satisfying conditions as follows

$$\begin{array}{*{20}c} {f(0) = 0,\sigma f(\sigma ) > 0} & {(\sigma \ne 0)} \\ \end{array}$$

((2))

or

$$\begin{array}{*{20}c} {f(0) = 0,0 \leqslant k_1 \leqslant f(\sigma )/\sigma \leqslant k_2< + \infty } & {(\sigma \ne 0)} \\ \end{array}$$

((3))

It gives the necessary and sufficient conditions for the absolute stability for system (1) under conditions (2). We also obtain the sufficient criteria for absolute stability of the simplified system of (1) under conditions (3).