Observability analysis of combined finite automata based upon semi-tensor product of matrices approach

Abstract

As a fundamental subject, the state estimation of deterministic finite automata has received considerable attention. Especially, it is increasingly necessary to study various problems based on more complex systems. In this paper, the observability of three kinds of combining automata, structured in parallel, serial and feedback manners, are investigated based on an algebraic state space approach. Compared with the formal language method, the matrix approach has great advantages in problem description and solution. Because of inconsistent frameworks of these combined automata, we optimize structure matrices by pseudo-commutation of semi-tensor product and power-reducing matrix. In addition, we construct corresponding incidence matrices by labelling to avoid superfluous null elements in the logical matrix occupying storage space. It follows that the observability analysis could be carried out under two polynomial matrices, established from the above algebraic form. Meanwhile, two algorithms, judging whether a combined automaton is initial state observable or current state observable, are presented. Finally, there are two representative examples to actualize our approach.