On global exponential stability of discrete-time switching systems with dwell-time ranges: Novel induced LMIs for linear systems with delays

Abstract

In this paper, we provide necessary and sufficient Lyapunov conditions for discrete-time switching systems to be globally exponentially stable, when the switching signal obeys to a switches digraph and is subject to dwell-time constraints. In order to best exploit the information on switching-dwelling constraints, conditions are given by means of multiple Lyapunov functions. The number of involved Lyapunov functions is equal to the number of switching modes. To avoid a pileup of Lyapunov functions, we do not introduce dummy vertices that account for dwell-time ranges. For example, in the linear case, such a pileup corresponds to a pileup of decision matrices related to some linear matrix inequalities. A link between global exponential stability and exponential input-to-state stability is provided. The following result is proved: if, in the case of zero input, the discrete-time switching system is globally exponentially stable, and the functions describing the dynamics of the subsystems, with input, are suitably globally Lipschitz, then the switching system is exponentially input-to-state stable. Finally, exploiting the well known relationship between discrete-time systems with delays and discrete-time switching systems, the provided results are shown for the former systems, in the linear case. In particular, linear matrix inequalities, by which the global exponential stability of linear discrete-time systems with constrained time delays can be possibly established, are provided. The utility of these linear matrix inequalities is shown with a numerical example taken from the literature.