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

In this paper the input-to-state stability (iss) property is studied for discrete-time nonlinear systems. We show that many iss results for continuoustime 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 discretetime 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 continuoustime 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.