The rest of this paper is organized as follows. Compared with the existing results, our results are more easily verified via MATLAB. Using the controllability matrix, we present some necessary and sufficient conditions for the controllability, reachability, and stabilizability of finite automata. The main contribution of this paper is to construct a controllability matrix for finite automata based on the algebraic form. In this paper, we investigate the controllability, reachability, and stabilizability of deterministic finite automata by using STP. It should be pointed out that although the concepts of controllability, reachability, and stabilizability of finite automata come from classic control theory, there exist fewer results on the construction of controllability matrix for finite automata. studied the controllability and stabilizability analysis of finite automata based on STP and presented some novel results. Xu and Hong provided a matrix-based algebraic approach for the reachability analysis of finite automata with the help of STP.
Thus, STP provides a convenient way for the construction and analysis of finite automata. The main feature of STP is to convert a finite-valued system into an equivalent algebraic form. Up to now, STP has been successfully applied to many research fields related to finite-valued systems like Boolean networks, multivalued logical networks, game theory, finite automata, and so on. Recently, a new matrix product, namely, the semitensor product (STP) of matrices, has been proposed by Cheng et al. investigated the state feedback stabilization of a deterministic finite automaton and presented some new results. The controllability of a deterministic Rabin automaton was studied in by defining the “controllability subset.” Kobayashi et al. The concepts of controllability, reachability, and stabilizability of finite automata were defined in by resorting to the classic control theory. The study of finite automata has received many scholars’ research interest in the last century due to its wide applications in engineering, computer science, and so on.Īs we all know, controllability and stabilizability analysis of finite automata are fundamental topics, which are important and necessary to the solvability of many related problems. It receives a discrete sequence of inputs from the outside world and changes its state according to the inputs. Finite automaton is a device whose states take values from a finite set. In the research field of theoretical computer science, finite automaton is one of the simplest models of computation. Finally, an illustrative example is given to support the obtained new results. Thirdly, some necessary and sufficient conditions are presented for the controllability, reachability, and stabilizability of finite automata by using the controllability matrix. Secondly, based on the algebraic form, a controllability matrix is constructed for finite automata. Firstly, by expressing the states, inputs, and outputs as vector forms, an algebraic form is obtained for finite automata. This paper investigates the controllability, reachability, and stabilizability of finite automata by using the semitensor product of matrices.