Automating the addition of fault tolerance with discrete controller synthesis
Formal Methods in System Design
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This paper deals with a new type of optimal control for discrete event systems. Our control problem extends the theory of [R. Sengupta and S. Lafortune, SIAM J. Control Optim., 36 (1998), pp. 488--541] that is characterized by the presence of uncontrollable events, the notion of occurrence and control costs for events, and a worst-case objective function. A significant difference with [R. Sengupta and S. Lafortune, SIAM J. Control Optim., 36 (1998), pp. 488--541] is that our aim is to make the system evolve through a set of multiple goals, one by one, with no order necessarily prespecified, whereas the previous theory only deals with a single goal. Our solution approach is divided into two steps. In the first step, we use the optimal control theory in [R. Sengupta and S. Lafortune, SIAM J. Control Optim., 36 (1998), pp. 488--541] to synthesize individual controllers for each goal. In the second step, we develop the solution of another optimal control problem, namely, how to modify if necessary and piece together, or schedule, all of the controllers built in the first step in order to visit each of the goals with the least total cost. We solve this problem by defining the notion of a scheduler and then by mapping the problem of finding an optimal scheduler to an instance of the well-known traveling salesman problem (TSP) [E. L. Lawler, J. K. Lenstra, A. H. G. Rinooy Kan, and D. B. Shmoys, The Traveling Salesman Problem, John Wiley, 1985]. We finally suggest various strategies to reduce the complexity of the TSP resolution while still preserving global optimality.