Overview of Controllability Definitions in Supervisory Control Theory

TL;DR

This paper reviews various definitions of controllability in supervisory control theory, analyzes their relationships in deterministic and nondeterministic automata, and clarifies their implications for practical control of supervised plants.

Contribution

It systematically compares different controllability notions in supervisory control theory and clarifies their relationships and practical implications.

Findings

Controllability notions by Flordal and Malik, and Kushi and Takai are equivalent in nondeterministic settings.

These notions imply traditional language controllability.

State controllability by Zhou et al. implies language controllability.

Abstract

In the field of supervisory control theory, the literature often proposes different definitions for the same concept, making it difficult to understand how these definitions are related. This is definitely so for the fundamental notion of controllability of a supervisor w.r.t. a plant. This paper lists definitions of controllability found in the literature and studies their relationships in settings of both deterministic and nondeterministic automata. In the general context, where both the supervisor and the plant are allowed to be nondeterministic, the notions of controllability as described by Flordal and Malik, and uncontrollable event admissibility by Kushi and Takai are equivalent. These are also the only notions that imply the traditional notion of (language) controllability. From a practical perspective, one is often more interested in controllability of a supervised plant w.r.t. a plant. In this context, in addition to the previous two controllability notions, state controllability by Zhou et al. implies language controllability.