Observability and Decentralized Control of Fuzzy Discrete Event Systems

TL;DR

This paper extends supervisory control theory to fuzzy discrete event systems, establishing conditions for the existence of decentralized supervisors under partial observation, and discusses related superlanguages and sublanguages.

Contribution

It introduces the concepts of observability, normality, and co-observability for fuzzy languages and proves their role in supervisory control for fuzzy DES with partial observation.

Findings

Existence of a partially observable fuzzy supervisor requires observability and controllability.

Decentralized control is feasible if the fuzzy language is controllable and co-observable.

The paper discusses infimal controllable and observable superlanguages and supremal controllable and normal sublanguages.

Abstract

Fuzzy discrete event systems as a generalization of (crisp) discrete event systems have been introduced in order that it is possible to effectively represent uncertainty, imprecision, and vagueness arising from the dynamic of systems. A fuzzy discrete event system has been modelled by a fuzzy automaton; its behavior is described in terms of the fuzzy language generated by the automaton. In this paper, we are concerned with the supervisory control problem for fuzzy discrete event systems with partial observation. Observability, normality, and co-observability of crisp languages are extended to fuzzy languages. It is shown that the observability, together with controllability, of the desired fuzzy language is a necessary and sufficient condition for the existence of a partially observable fuzzy supervisor. When a decentralized solution is desired, it is proved that there exist local fuzzy supervisors if and only if the fuzzy language to be synthesized is controllable and co-observable. Moreover, the infimal controllable and observable fuzzy superlanguage, and the supremal controllable and normal fuzzy sublanguage are also discussed. Simple examples are provided to illustrate the theoretical development.