Input-to-state stability for discrete-time nonlinear systems

Abstract

In this paper the input-to-state stability (ISS) property is studied for discrete-time nonlinear systems. We show that many ISS results for continuons-time nonlinear systems in earlier papers (Sontag, 1989; Sontag, 1990; Sontag and Wang, 1996; Jiang et al ., 1994; Coron et al ., 1995) can be extended to the discrete-time case. More precisely, we provide a Lyapunov-like sufficient condition for ISS, and we show the equivalence between the ISS property and various other properties. Utilizing the notion of ISS, we present a small gain theorem for nonlinear discrete time systems. ISS stabilizability is discussed and connections with the continuous-time case are made. As in the continuous time case, where the notion ISS found wide applications, we expect that this notion will provide a useful tool in areas related to stability for nonlinear discrete time systems as well.