Control of Nondeterministic Discrete-Event Systems for Bisimulation Equivalence

Abstract

Abstract-Most prior work on supervisory control of discrete event systems is for achieving deterministic specifications, expressed as formal languages. In this paper we study supervisory control for achieving nondeterministic specifications. Such specifications are useful when designing a system at a higher level of abstraction so that lower level details of system and its specification are omitted to obtain higher level models that may be nondeterministic. Nondeterministic specifications are also meaningful when the system to be controlled has a nondeterministic model due to the lack of information (caused for example by partial observation or unmodeled dynamics). Language equivalence is not an adequate notion of behavioral equivalence for nondeterministic systems, and instead we use the finest known notion of equivalence, namely the bisimulation equivalence. Choice of bisimulation equivalence is also supported by the fact that bisimulation equivalence specification is equivalent to a specification in the temporal logic of -calculus that subsumes the complete branching-time logic CTL*. Given nondeterministic models of system and its specification, we study the design of a supervisor (possibly nondeterministic) such that the controlled system is bisimilar to the specification. We obtain a small model theorem showing that a supervisor exists if and only if it exists over a certain finite state space, namely the power set of Cartesian product of system and specification state spaces. Also, the notion of state-controllability is introduced as part of a necessary and sufficient condition for the existence of a supervisor. In the special case of deterministic systems, we provide an existence condition that can be verified polynomially in both system and specification states, when the existence condition holds.

The role of supervisory controller synthesis in automatic control software development

“…Subsequent attempts, like [23][24][25], recognized the importance of (bi)simulation-based approaches. A process-theoretic approach to supervisory control theory was proposed in [26], employing the behavioral preorder partial bisimulation.…”

Bisimilarity Control of Nondeterministic Discrete Event Systems under Event and State Observations

“…Due to the model abstraction or the unmodeled dynamics, nondeterminism of transitions can arise in the plant and specification models [2], [3]. Various supervisory control problems for nondeterministic systems were studied in [2]- [10].…”

“…Various supervisory control problems for nondeterministic systems were studied in [2]- [10]. In particular, a general case that both the plant and the specification are nondeterministic, and a supervisor is allowed to be nondeterministic was considered in [2], [9].…”

“…A problem of synthesizing a nondeterministic supervisor enforcing bisimulation equivalence between the supervised plant and the specification was studied in [2], [9]. In [2], a supervisor observes each event occurrence of the plant, and the supervised plant is modeled by the synchronous composition of the plant and the supervisor.…”

“…In [2], a supervisor observes each event occurrence of the plant, and the supervised plant is modeled by the synchronous composition of the plant and the supervisor. small model theorem was provided to show that a supervisor enforcing bisimulation equivalence between the supervised plant and the specification exists if and only if such a supervisor whose state space is the power set of the Cartesian product of the plant and the specification state spaces exists.…”