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New Results on Deterministic Networked Supervisory Control and Relative Delay Observability

This paper considers supervisory control of discrete event systems with observation delays and control delays, which is termed deterministic networked supervisory control. Delay observability, together with delay controllability, is required for obtaining a networked supervisor so that the language generated by the supervised system is deterministic and is equal to a given specification language. In this article, we simplify this existence condition and prove such a networked supervisor also exists if controllability and delay observability are satisfied. A new language property called relative delay observability, which accommodates observation delays in relative observability, is further defined. Relatively delay observability implies delay observability and enjoys an important property that is closed under union. We then show how to check the relative delay observability in polynomial time complexity. A general algorithm used to calculate the supremal sublanguage satisfying controllability and another user-defined language property in a solution space of interest, is presented. As a special case, we calculate the supremal controllable and relatively delay observable sublanguage. Finally, a practical example is given to illustrate the application of relative delay observability.