A matrix-based approach to verifying stability and synthesizing optimal stabilizing controllers for finite-state automata

摘要

In this paper, we develop a matrix-based methodology to investigate the problems of stability and stabilizability for a deterministic finite automaton (DFA) in the framework of the semi-tensor product (STP) of matrices. First, we discuss the equilibrium point stability (resp., set stability) of a DFA, i.e., verifying whether or not all state trajectories starting from a subset of states converge to a specified equilibrium point (resp., subset of states). The necessary and sufficient conditions for verifying both stabilities are given, respectively. Second, equilibrium point stabilizability (resp., set stabilizability) of a DFA is investigated as verifying the issue of whether or not a DFA can be globally or locally stabilized to a specified equilibrium point (resp., subset of states) by a permissible state-feedback controller. Based on the pre-reachability set and invariant-subset defined in this paper, the matrix-based criteria for verifying equilibrium point stabilizability and set stabilizability are derived, respectively. Furthermore, for each type of stabilizability, all permissible state-feedback controllers for the case of minimal length state trajectories, called optimal state-feedback controllers, are characterized by using the proposed polynomial algorithms. Finally, two examples are presented to illustrate the effectiveness of the theoretical results. (c) 2018 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.