A Small Model Theorem for Bisimilarity Control Under Partial Observation

This paper extends our prior result on decidability of bisimulation equivalence control from the setting of complete observations to that of partial observations. Besides being control compatible, the supervisor must now also be observation compatible. We show that the "small model theorem" remains valid by showing that a control and observation compatible supervisor exists if and only if it exists over a certain finite state space, namely the power set of the Cartesian product of the system and the specification state spaces. Note to Practitioners-Non-determinism in discrete-event systems arises due to abstraction and/or unmodeled dynamics. This paper addresses the issue of control of non-deterministic systems subject to non-deterministic specifications, under a partial observation of events. Non-deterministic plant and specification are useful when designing a system at a higher level of abstraction so that lower level details of the system and its specification are omitted to obtain higher level models that are non-deterministic. The control goal is to ensure that the controlled system has an equivalent behavior as the specification system, where the notion of equivalence used is that of bisimilarity. Bisimilarity requires the existence of an equivalence relation between the states of the two systems so that transitions on common events beginning from a pair of equivalent states end up in a pair of equivalent successor states. Supervisors are also allowed to be nondeterministic, where the nondeterminism in control is implemented by selecting control actions nondeterministically from among a set of precomputed choices. The main contribution of this paper is to show that a supervisor exists if and only if one exists where the size of its state-space upper bounded and so it suffices to search over this state space. We illustrate our results through a manufacturing example