A Novel Mathematical Characterization for Switched Linear Systems Based on Automata and Its Stabilizability Analysis

Mathematical characterization of switched systems based on automata is one of the most challenging problems in the context of hybrid dynamical systems. This article investigates the mathematical representation and stabilization of a class of slowly and fastly switched linear systems with automata in the hybrid coupling framework. To address such an issue, by using the semitensor product of matrices, a so-called “automation-dependent switched linear system” is established to generalize the traditional switched system, which is characterized mathematically in the new form of a hybrid dynamic system consisting of a continuous-time differential equation and a discrete-time difference equation. Moreover, a novel automation-dependent constrained switching scheme is developed for the stabilization of the proposed model with unstable subsystems. Next, by designing slow mode-dependent average dwell time (S-MDADT), fast mode-dependent average dwell time (F-MDADT), slow cycle-dependent cycle dwell time (S-CDCDT), and fast cycle-dependent cycle dwell time (F-CDCDT) switching laws, a generalized stabilization lemma for automation-dependent switched nonlinear systems with both stable and unstable modes inside and outside the cycle is derived in the terms of digital transformations and multiple Lyapunov functions. Furthermore, based on the stabilizability criteria of the nonlinear case, by setting digital transformations and linear matrix inequalities, a new stabilization theorem for automation-dependent switched linear systems with unstable subsystems is obtained. It should be noted that the proposed scheme provides a more precise estimation of the upper and lower bounds for the dwell time of the system with unstable modes. Finally, a practical converter circuit example and a numerical example are given to show the validity of the designed techniques.