Coalgebra and coinduction in discrete-event control with partial observations

Coalgebra and coinduction provide new results and insights for the supervisory control of discrete-event systems (DES) with partial observations. The paper is based on the formalism developed for supervisory control of DES in the full observation case, i.e. the notion of bisimulation, its generalizations (partial bisimulation and control relation), and the finality of the automaton of partial languages. The concept of nondeterministic weak transitions introduced in this paper yields a definition of deterministic weak transitions. These are shown to be useful in the study of partially observed DES. They give rise to the relational characterizations of normality and observability. These characterizations lead to new algorithms for supremal normal and supremal normal and controllable sublanguages that are compared to the ones known in the literature. Coinduction is used to define an operation on languages called supervised product, which represents the language of the closed-loop system, where the first language acts as a supervisor and the second as an open-loop system. This technique can be used to define many important languages, e.g. supremal controllable sublanguages, infimal controllable or/and observable superlanguages. A variation of supervised product corresponding to the permissive control policy with full controllability is given. It is shown to be equal to the infimal observable superlanguage. We have obtained as a byproduct coinductive definitions of these important languages. We show that antipermissive control policy cannot be captured by coinduction. However, we present an algorithm based on the antipermissive control policy for the computation of an observable sublanguage that contains the supremal normal sublanguage. Using a similar method monolithic algorithms for computation of supremal normal and supremal normal and controllable sublanguages are developed. Finally, the lattice theoretic continuity of the supervised product (i.e. the distributivity of the supervised product with respect to partial language unions) is studied