Bisimulation, the Supervisory Control Problem and Strong Model Matching for Finite State Machines

Abstract

A fundamental relationship between the controllability of a language with respect to another language and a set of uncontrollable events in the Supervisory Control Theory initiated by (Ramadge and Wonham, 1989) and bisimulation of automata models is derived. The theoretical results relating bisimulation to controllability support an efficient solution to the Basic Supervisory Control Problem. Using (Fernandez, 1990) generalization of the partition refinement algorithm of (Paige and Tarjan, 1987), it is possible to find a partition which represents the supremal controllable sublanguage of an automaton with respect to the language of another automaton and a set of events in a worst-case running time of O( m log( n )), where m is the number of transitions and n is the number of states. Utilizing the bisimulation property of language controllability and derived relationships between automata languages and input/output finite-state machine behaviors, a precise relationship is formally derived between Supervisory Control Theory and the system-theoretic problem posed by (DiBenedetto et al., 1994) called Strong Input/Output FSM Model Matching. Specifically, it is proven that in deterministic settings instances of each problem can be mapped to the other framework and solved.