Formulae relating controllability, observability, and co-observability

We present formulae relating controllability, observability, and co-observability arising in the context of supervisory control of discrete event systems. Given a discrete event plant G with event set Σ, uncontrollable event set Σui, and observation mask Mi of the ith supervisor, we first show that the infimal prefix-closed, (L(G), ⋂ i Σui)controllable, and (L(G),Σui,Mi)–co-observable superlanguage equals the intersection (taken over all i’s) of the infimal prefix-closed, (L(G),Σui)-controllable, and (L(G),Mi)-observable superlanguage. Next we show that the infimal prefix-closed, and (Σ∗,M)-observable superlanguage computation preserves (Σ∗,Σu)-controllability. These results can be used to compute individual supervisors in the decentralized control setting.