Decentralized Discrete-Event Control Problems

Supervisory control themy is the study of discrete-event processes, such as computer systems and manufacturing systems, which require control to induce desirable behavior. Informally, a dimte-event system is a process (or set of processes) that s u n s out in some initial state and is transformed from state to state by the occurrence of discrete events. Such a system can be thought of as a set of sequences of events, with each sequence describing a series of actions that occur within the system. Formally, the processes requiring control are modeled by automata and their desired behavior by formal languages. Problems associated with centralized (as opposed to distributed) discrete-event systems have been more extensively explored [1]-[5], and an application within semiconductor manufacturing [6]. [7] provides a compelling argument for considering this class of problems as useful in future engineering practise. More recently, decentralized control has been investigated and possible applications include flexible manufacturing systems [8] and communication systems [9]-[ 111. At this point, examples of decentralized discrete-event systems control have primarily served a pedagogical and mathematical purpose and have been highly simplified versions of physically realistic applications. We believe, however, that sufficient intuitive motivation has been given to justify M e r exploring the question "Given a discrete-event system, do decentralized conmllers exist that satisfy a certain objective?' Moreover, if discrete-event control theory is to play a role in real-world applications, we must address the issue of computer implementation. In particular, we need to develop algor i t h m s for implementing supervisory control solutions. To facilitate software development, it is important to understand the computational complexity of our control problems. In this note, we explore the computational complexity of a class of decentralized discrete-event problems. We generalize the results of Tsitsiklis [12]. These results indicate that testing for solvability within a restricted class of control problems can be done in polynomial time. Even when a solution is proven to exist, however, there is no polynomial-time algorithm for producing it. Moreover, if we move to the more general class of problems, we cannot even test for solvability in polynomial time, let alone produce supervim solutions. The good news is that some interesting problems are captured by the restricted decentralized discrete-event control formulation. In particular, communication protocol verification is one such example. It is not, u priori, obvious that Tsitsiklis' positive complexity results hold for the decentralized case. Just because the centralized version of a problem can be solved efficiently does nat imply that its decentralized counterpart is also solvable efficiently. For example, some NP-complete multiprocessor scheduling problems become trivially solvable in polynomial time when restricted to a single processor [13].