Online computation of supremal relatively observable sublanguage of discrete-event systems

Recently we studied a new observability concept, called relative observability, in supervisory control of discrete-event systems under partial observation. Relative observability is preserved under set union, and thus there exists the supremal relatively observable sublanguage of a given language. An algorithm was proposed for computing the supremal sublanguage, but requires double-exponential complexity in the worst case. In this paper, we study an efficient online algorithm to compute the supremal relatively observable sublanguage of a closed (regular) language. Our algorithm computes a state-output function online, where the states correspond to the observed event sequences and the output defines the set of events to be disabled at each state; under the restriction of the state-output function, the system generates exactly the supremal relatively observable sublanguage. At each step of determining the output, i.e. the set of events to be disabled, the algorithm is of polynomial complexity, making the online control under partial observation computationally feasible.