Discrete-Time Convergent Nonlinear Systems

The convergence property of discrete-time nonlinear systems is studied in this article. The main result provides a Lyapunov-like characterization of the convergence property based on two distinct Lyapunov-like functions. These two functions are associated with the incremental stability property and the existence of a compact positively invariant set, which together guarantee the existence of a well-defined, bounded, and unique steady-state solution. The links with the conditions available in recent literature are discussed. These generic results are subsequently used to derive constructive conditions for the class of discrete-time Lur'e-type systems. Such systems consist of an interconnection between a linear system and a static nonlinearity that satisfies cone-bounded (incremental) sector conditions. In this framework, the Lyapunov-like functions that characterize convergence are determined by solving a set of linear matrix inequalities. Several classes of Lyapunov-like functions are considered: both Lyapunov-Lur'e-type functions and quadratic functions. A numerical example illustrates the applicability of the results.