Convergent discrete-time nonlinear systems: The case of PWA systems

In this paper we extend the notion of convergence, as defined for continuous-time dynamical systems, to the realm of discrete-time systems. A system is said to be convergent if it exhibits a unique, globally asymptotically stable solution that is defined and bounded on the entire time axis. The convergence property is highly instrumental in solving output regulation, tracking, synchronization and observer design problems. First, we provide a general sufficient condition for the convergence of nonlinear discrete-time systems. Next, we propose constructive sufficient conditions for convergence of discrete-time piecewise afflne (PWA) systems. These conditions are given in the form of matrix inequalities. The proposed results are illustrated by an example in which a tracking control problem for a discrete-time PWA system is tackled.