Uniform tube based stabilization of switched linear systems with mode-dependent persistent dwell-time

In this note, the stabilization problem for a class of discrete-time switched linear systems with additive disturbances is investigated. The considered switching signals are of mode-dependent persistent dwell-time (MPDT) property and the disturbances are assumed to be amplitude-bounded. By constructing a quasi-time-varying (QTV) Lyapunov function, a QTV stabilizing controller is designed for the nominal system such that the resulting closed-loop system is globally uniformly asymptotically stable. In the presence of bounded additive disturbances, a MPDT robust positive invariant set is determined for the error system between the nominal system and disturbed system. A concept of generalized robust positive invariant (GRPI) set under admissible MPDT switching is further proposed for the error system. It is demonstrated that the disturbed system is also asymptotically stable in the sense of converging to the MPDT GRPI set that can be regarded as the cross section of a uniform tube of the disturbed system. A numerical example is provided to verify the theoretical findings.