$\omega-$nonblocking supervisory control of discrete-event systems with infinite behavior

TL;DR

This paper introduces a method for synthesizing supervisors for discrete-event systems with infinite behavior that guarantees nonblocking, deadlock-free, and livelock-free operation by ensuring all infinite cycles include marker states.

Contribution

It proposes the concept of markability and an algorithm to synthesize an $oldsymbol{ extomega}$-nonblocking supervisor ensuring infinite behavior visits marker states.

Findings

Algorithm computes supremal controllable and markable sublanguages.

Ensures supervisors are deadlock-free and livelock-free.

Validated with a robot example.

Abstract

In the supervisory control framework of discrete-event systems (DES) with infinite behavior initiated by Thistle and Wonham, a supervisor satisfying the minimal acceptable specification and the maximal legal specification is synthesized. However, this supervisor may incur livelocks as it cannot ensure that the infinite behavior under supervision will always visit some marker states. To tackle this problem, we propose the definition of markability by requiring that all infinite cycles include at least one marker state. Then we formulate the problem of $\omega-$nonblocking supervisory control of DES with infinite behavior to synthesize an $\omega-$nonblocking (i.e. nonblocking, deadlock-free and livelock-free) supervisor. An algorithm is proposed to achieve $\omega-$nonblockingness by computing the supremal $*-$controllable, $*-$closed, $\omega-$controllable and markable sublanguage. We utilize the example of a robot as a running example.