Semi-tensor product approach to controllability and stabilizability of finite automata

Abstract

Using semi-tensor product of matrices, the controllability and stabilizability of nite automata are investigated. By expressing the states, inputs, and outputs in vector forms, the transition and output functions are represented in matrix forms. Based on this algebraic description, a necessary and suf cient condition is proposed for checking whether a state is controllable to another one. By this condition, an algorithm is established to nd all the control sequences of an arbitrary length. Moreover, the stabilizability of nite automata is considered, and a necessary and suf cient condition is presented to examine whether some states can be stabilized. Finally, the study of illustrative examples veri es the correctness of the presented results/algorithms.

Controllability, Reachability, and Stabilizability of Finite Automata: A Controllability Matrix Method

“…Using the controllability matrix, we present some necessary and sufficient conditions for the controllability, reachability, and stabilizability of finite automata. Compared with the existing results [5,26], our results are more easily verified via MATLAB.…”

“…Using the vector form of elements in and , Yan et al [26] construct the transition structure matrix (TSM) of = ( , , , , , 0 ,…”

“…Up to now, STP has been successfully applied to many research fields related to finite-valued systems like Boolean networks [10][11][12][13][14][15][16][17][18][19][20], multivalued logical networks [21][22][23], game theory [24,25], finite automata [5,26], and so on [27][28][29][30][31][32][33][34][35]. The main feature of STP is to convert a finite-valued system into an equivalent algebraic form [22].…”

“…The main feature of STP is to convert a finite-valued system into an equivalent algebraic form [22]. Thus, STP provides a convenient way for the construction and analysis of finite automata [5,26]. Xu and Hong [5] provided a matrixbased algebraic approach for the reachability analysis of finite automata with the help of STP.…”

“…Xu and Hong [5] provided a matrixbased algebraic approach for the reachability analysis of finite automata with the help of STP. Yan et al [26] studied the controllability and stabilizability analysis of finite automata based on STP and presented some novel results. 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.…”