Reliable decentralized supervisory control for marked language specifications

We consider a discrete event system controlled by a decentralized supervisor consisting of n local supervisors. In our previous work, we have studied reliable decentralized supervisory control for closed language specifications. In this paper, we extend this work to the specifications given by marked languages. A decentralized supervisor is said to be k-reliable (1 less than or equal to k less than or equal to n) if it exactly achieves a specification language without blocking under possible failures of any less than or equal to n - k local supervisors. We present necessary and sufficient conditions for the existence of a k-reliable decentralized supervisor. Then we define a weaker version of k-reliability, called weak k-reliability, by relaxing the nonblocking requirement. We obtain necessary and sufficient conditions for the existence of a weakly k-reliable decentralized supervisor. Moreover, we propose an iterative scheme for computing a sublanguage of a specification for which the existence conditions of a weakly k-reliable decentralized supervisor are satisfied.