A graph-theoretic optimal control problem for terminating discrete event processes
- 42 Downloads
Most of the results to date in discrete event supervisory control assume a “zero-or-infinity” structure for the cost of controlling a discrete event system, in the sense that it costs nothing to disable controllable events while uncontrollable events cannot be disabled (i.e., their disablement entails infinite cost). In several applications however, a more refined structure of the control cost becomes necessary in order to quantify the tradeoffs between candidate supervisors. In this paper, we formulate and solve a new optimal control problem for a class of discrete event systems. We assume that the system can be modeled as a finite acylic directed graph, i.e., the system process has a finite set of event trajectories and thus is “terminating.” The optimal control problem explicitly considers the cost of control in the objective function. In general terms, this problem involves a tradeoff between the cost of system evolution, which is quantified in terms of a path cost on the event trajectories generated by the system, and the cost of impacting on the external environment, which is quantified as a dynamic cost on control. We also seek a least restrictive solution. An algorithm based on dynamic programming is developed for the solution of this problem. This algorithm is based on a graph-theoretic formulation of the problem. The use of dynamic programming allows for the efficient construction of an “optimal subgraph” (i.e., optimal supervisor) of the given graph (i.e., discrete event system) with respect to the cost structure imposed. We show that this algorithm is of polynomial complexity in the number of vertices of the graph of the system.
Key Wordsdiscrete event systems optimal control graph theory dynamic programming
Unable to display preview. Download preview PDF.
- Brave, Y., and Heymann, M., 1990. On optimal attraction of discrete-event processes. Center for Intelligent Systems Report 9010, Technion, Haifa, Israel.Google Scholar
- Chen, E., and Lafortune, S., 1991. Dealing with blocking in supervisory control of discrete event systems.IEEE Trans. Automatic Control, 36(6):724–735.Google Scholar
- Kumar, R., and Garg, V., 1991. Optimal control of discrete event dynamical systems using network flow techniques. Preprint, Department of Electrical and Computer Engineering, University of Texas, Austin.Google Scholar
- Lafortune, S., and Chen, E., 1990. The infimal closed controllable superlanguage and its application in supervisory control.IEEE Trans. Automatic Control, 35(4):398–405.Google Scholar
- Passino, K.M., and Antsaklis, P.J., 1989. On the optimal control of discrete event systems.Proc. 28th IEEE Conf. Decision Control: 2713–2718.Google Scholar
- Ramadge, P.J., and Wonham, W.M., 1987. Supervisory control of a class of discrete event processes.SIAM J. Control Optim. 25(1):206–230.Google Scholar
- Ramadge, P.J., and Wonham, W.M., 1989. The control of discrete event systems.Proc. IEEE, 77(1):81–98.Google Scholar
- Sengupta, R., and Lafortune, S., 1991a. Optimal control of a class of discrete event systems.Preprints, IFAC Int. Symp. on Distributed Intelligence Systems: 25–30.Google Scholar
- Sengupta, R., and Lafortune, S., 1991b. An optimal control problem for a class of discrete event systems. Technical Report CGR-57, College of Engineering Control Group Reports, University of Michigan.Google Scholar