Theorem of abstraction for equivalent controllers in hybrid systems

In the analysis and synthesis of controllers for hybrid systems, it is often required to verify whether two controllers for a given hybrid plant are equivalent. This is not easy to do because one needs to verify that the two controlled hybrid systems generate the same sequence of discrete events and the same trajectory for continuous variables for the given initial conditions. In this paper, we prove that under certain conditions, the verification of equivalence can be done at the abstract level of discrete event systems. More precisely, two controllers are equivalent if and only if the languages (that is, the sets of all possible sequences of discrete transitions) generated by two controlled systems are the same. This theorem of abstraction implies that many control problems of hybrid systems can be studied at the abstract level of discrete event systems. Therefore, it can be 0020-0255/$ see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2004.07.012 q This research is supported in part by NSF under grants ITR-0082784 and INT-0213651 and by NASA under grant NAG2-1043. * Address: Department of Electrical and Computer Engineering, Wayne State University, 5050 Anthony Wayne Drive, Detroit, MI 48202, USA. Tel.: +1 313 577 3428; fax: +1 313 577 1101. E-mail address: [email protected] 182 F. Lin / Information Sciences 173 (2005) 181–195 used in the control synthesis for decentralized controllers and partial observation controllers because techniques developed for discrete event systems can now be applied to hybrid systems. 2004 Elsevier Inc. All rights reserved.