Formulae relating controllability, observability, and co-observability. (English) Zbl 0911.93020

In the context of supervisory control of discrete-event systems, the notion of controllability can be used to establish necessary and sufficient conditions for the existence of a supervisor that achieves a desired controlled behavior of a given discrete-event plant under complete observation of events. If the events are not completely observed by the supervisor, but rather their observations are filtered by an observation mask, then an additional condition of observability is needed to guarantee its existence. In the more general case of decentralized supervision, more than one supervisor control the plant; the uncontrollable event set for the \(i\)-th supervisor is \(\Sigma_{ui}\), and its observation is filtered through the mask \(M_i\). In this setting, the existence condition for co-observability has been established for the case when the controlled behaviour is given as a prefix-closed language; it can also be generalized to the non-prefix-closed case.
In the paper under review, the authors first obtain a relationship between controllability, observability and co-observability. Given a plant \(G\) with event set \(\Sigma\), uncontrollable event sets \(\Sigma_{ui}\), and observation masks \(M_i\) for the \(i\)-th supervisor, they prove that the infimal, prefix-closed, \((L(G),\bigcap_i \Sigma_{ui})\)-controllable and \((L(G),\Sigma_{ui},M_i)\)-co-observable superlanguage of the specification language equals the intersection (taken over all \(i\)’s) of infimal prefix-closed \((L(G),\Sigma_{ui})\)-controllable and \((L(G),M_i)\)-observable superlanguages. This result can be used to obtain the individual supervisors for decentralized control, where the \(i\)-th supervisor generates the infimal prefix-closed, \((L(G),\Sigma_{ui})\)-controllable and \((L(G),M_i)\)-observable superlanguage of the given specification.


93B07 Observability
93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
93B05 Controllability
93A14 Decentralized systems
93A13 Hierarchical systems
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