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**Supervisory control of a class of discrete event processes.**
*(English)*
Zbl 0618.93033

Qualitative and structural aspects of control for a class of discrete event processes (DEP) are discussed. The processes are assumed to be discrete, asynchronous and in some sense nondeterministic. Classical automata and formal languages theory is used to derive abstract models of DEP and of controlled DEP (CDEP). These models are generators of some formal languages. Control of the CDEP consists in sequential switching of control patterns (some binary assignments to the specified subset of the input alphabet). This is done by a finite automaton, which together with some state feedback mapping constitutes a control structure called supervisor. Conditions for the completeness of the supervisor w.r.t. process model are given.

Several definitions of languages associated with CDEP which is controlled by the supervisor are introduced in order to define the controllability property of the language, generated by the closed-loop structure. A proper supervisor is then defined and a theorem saying that for a nonempty, marked and controllable language there exists proper supervisor which, in closed-loop structure with the model, generates that language. The languages corresponding to the minimal acceptable behavior of the closed-loop system and to the legal behavior of that system are introduced. Then some synthesis problems are addresed, namely supervisory control problems (SCP) and supervisory marking problems (SMP). It is possible to order solutions to SMP and SCP in a lattice-theoretic sense and to define minimally restrictive solutions satisfying behavioral constraints.

The first main result is a theorem on the solvability of SMP and SCP in a minimally restrictive manner. Then projection of supervisors is defined. The next chapter gives an abstract characterization of efficient constructions of the supervisor for a given nonempty, controllable, marked and closed language. A second result is a theorem stating roughly that ”every efficiently constructed supervisor is a quotient of the desired closed-loop behavior”. Two examples illustrating the notions introduced in the paper sum up the presentation.

Several definitions of languages associated with CDEP which is controlled by the supervisor are introduced in order to define the controllability property of the language, generated by the closed-loop structure. A proper supervisor is then defined and a theorem saying that for a nonempty, marked and controllable language there exists proper supervisor which, in closed-loop structure with the model, generates that language. The languages corresponding to the minimal acceptable behavior of the closed-loop system and to the legal behavior of that system are introduced. Then some synthesis problems are addresed, namely supervisory control problems (SCP) and supervisory marking problems (SMP). It is possible to order solutions to SMP and SCP in a lattice-theoretic sense and to define minimally restrictive solutions satisfying behavioral constraints.

The first main result is a theorem on the solvability of SMP and SCP in a minimally restrictive manner. Then projection of supervisors is defined. The next chapter gives an abstract characterization of efficient constructions of the supervisor for a given nonempty, controllable, marked and closed language. A second result is a theorem stating roughly that ”every efficiently constructed supervisor is a quotient of the desired closed-loop behavior”. Two examples illustrating the notions introduced in the paper sum up the presentation.

Reviewer: A.Kasiński

### MSC:

93C65 | Discrete event control/observation systems |

68Q45 | Formal languages and automata |

93C10 | Nonlinear systems in control theory |

93B05 | Controllability |

93B20 | Minimal systems representations |