Nonconservative Lifted Convex Conditions for Stability of Discrete-Time Switched Systems Under Minimum Dwell-Time Constraint

In this note, a novel conception called virtual clock, which is defined by an artificial timer over a finite cycle, is introduced for stability analysis of discrete-time switched linear systems under minimum dwell-time constraint. Two necessary and sufficient conditions associated with a virtual clock with a sufficient length are proposed to ensure the global uniform asymptotic stability of discrete-time switched linear systems. For the two nonconservative stability criteria, the lifted version maintains the convexity in system matrices. Based on the lifted convex conditions, the extensions to <inline-formula><tex-math notation="LaTeX">$\ell _2$</tex-math></inline-formula>-gain computation and <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula> control problems are presented in the sequel. In particular, a novel virtual-clock-dependent controller is designed, which outperforms the traditional mode-dependent and common gain controllers. Several numerical examples are provided to illustrate our theoretic results.