Stability of Discrete-Time Systems Under Restricted Switching via Logic Dynamical Generator and STP-Based Mergence of Hybrid States

In this article, we propose a novel technique for the stability analysis of discrete-time switched systems under constrained switching based on the semitensor product (STP) of matrices and vector-representation of logic. We assume that the admissible switching signals are generated by a logic dynamical system, referred to as the logic dynamical generator (LDG) of the admissible switching. We demonstrate that switching with minimum dwell-time and restricted successors represent two special cases of this type, by constructing their respective LDGs. First, we transfer the LDG into algebraic form, based on the vector-representation of logic. Next, we merge the logic state of the LDG and the continuous state of the original switched system using the STP and derive the dynamics of the merged state; a switched system without constraints on switching. The stability of the switched system under constrained switching is equivalent to the stability of the merged system under arbitrary switching. Based on this, sufficient conditions for the asymptotical stability of nonlinear switched systems with analytic subsystems under constrained switching are obtained. Specifically, in the case of all subsystems being linear, necessary and sufficient conditions for asymptotical stability under constrained switching are proposed.