A Comparison of Control Problems for Timed and Hybrid Systems

Abstract

In the literature, we find several formulations of the control problem for timed and hybrid systems.We argue that formulations where a controller can cause an action at any point in dense (rational or real) time are problematic, by presenting an example where the controller must act faster and faster, yet causes no Zeno effects (say, the control actions are at times 0, ¹ ₂, 1, 1¹ ₄, 2, 2¹ ₈, 3, 3¹ ₁₆, ... ). Such a controller is, of course, not implementable in software. Such controllers are avoided by formulations where the controller can cause actions only at discrete (integer) points in time. While the resulting control problem is wellunderstood if the time unit, or “sampling rate” of the controller, is fixed a priori, we define a novel, stronger formulation: the discrete-time control problem with unknown sampling rate asks if a sampling controller exists for some sampling rate. We prove that this problem is undecidable even in the special case of timed automata.

Partially supported by the FNRS, Belgium, under grant 1.5.096.01.

Partially supported by the DARPA SEC grant F33615-C-98-3614, the AFOSR MURI grant F49620-00-1-0327, the NSF Theory grant CCR-9988172, and the MARCO GSRC grant 98-DT-660.

Partially supported by a “Crédit aux chercheurs” from the Belgian National Fund for Scientific Research.