Mutually Exclusive State-Feedback Control of State Machines and Deadlocks of the Resulting System

This paper treats deadlocks and liveness of the closed-loop discrete event system GSCSM|f consisting of a strongly connected state machine (SCSM), a subclass of Petri nets, and a state-feedback which realizes the mutually exclusive control.The GSCSM|f is not always live even if the original SCSM is live, and deadlocks exist in GSCSM|f.A deadlock marking is newly defined and the D-net is proposed to detect these deadlock markings.Then the relation between deadlock freeness and liveness is analysed.Under an assumption, it is shown that the existence of the place set called DPS in D-net is the necessary and sufficient condition for the existence of deadlock markings, and that deadlock freeness is equivalent to the liveness.Furthermore, defining LPS as a special class of DPS, we show that if DPS/LPS exists, the sufficient condition for liveness is that the initial token count ￨M0￨ satisfies the equation 1≦￨M0￨≦ min ￨DPS/LPS￨-1.