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 Sigma, uncontrollable event set Sigma(ui), and observation mask M-i of the ith supervisor, we first show that the infimal prefix-closed, (L(G), boolean AND(i) Sigma(ui))-controllable, and (L(G), Sigma(ui), M-i)-co-observable superlanguage equals the intersection (taken over all i's) of the infimal prefix-closed, (L(G), Sigma(ui))-controllable, and (L(G), M-i)-observable superlanguage. Next, we show that the infimal prefix-closed, and (Sigma*,M)-observable superlanguage computation preserves (Sigma*, Sigma(u))-controllability. These results can be used to compute individual supervisors in the decentralized control setting. (C) 1998 Elsevier Science Ltd. All rights reserved.