Approximately Bisimilar Symbolic Models for Incrementally Stable Switched Systems
Switched systems constitute an important modeling para- digm faithfully describing many engineering systems in which software interacts with the physical world. Despite considerable progress on stability and stabilization of switched systems, the constant
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Laboratoire Jean Kuntzmann, Universit´e Joseph Fourier B.P. 53, 38041 Grenoble, France [email protected] Department of Electrical Engineering, University of California at Los Angeles Los Angeles, CA 90095-1594 {pola,tabuada}@ee.ucla.edu Department of Electrical and Information Engineering, University of L’Aquila Poggio di Roio, 67040 L’Aquila, Italy [email protected]
Abstract. Switched systems constitute an important modeling paradigm faithfully describing many engineering systems in which software interacts with the physical world. Despite considerable progress on stability and stabilization of switched systems, the constant evolution of technology demands that we make similar progress with respect to different, and perhaps more complex, objectives. This paper describes one particular approach to address these different objectives based on the construction of approximately equivalent (bisimilar) symbolic models for a switched system. The main contribution of this paper consists in showing that under standard assumptions ensuring incremental stability of a switched system (i.e. existence of common or multiple Lyapunov functions), it is possible to construct a symbolic model that is approximately bisimilar to the original switched system with a precision that can be chosen a priori. To support the computational merits of the proposed approach we present a realistic example of a boost dc-dc converter and show how to synthesize a switched controller that regulates the output voltage at a desired level.
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Introduction
Switched systems constitute an important modeling paradigm faithfully describing many engineering systems in which software interacts with the physical world. Although this fact already amply justifies its study, switched systems are also quite intriguing from a theoretical point of view. It is well known that by judiciously switching between stable subsystems one can render the overall system unstable. This motivated several researchers over the years to understand which classes of switching strategies or switching signals preserve stability (see
This work was partially supported by the ANR SETIN project VAL-AMS and by the NSF CAREER award 0717188.
M. Egerstedt and B. Mishra (Eds.): HSCC 2008, LNCS 4981, pp. 201–214, 2008. c Springer-Verlag Berlin Heidelberg 2008
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A. Girard, G. Pola, and P. Tabuada
e.g. [1]). Despite considerable progress on stability and stabilization of switched systems, the constant evolution of technology demands that we make similar progress with respect to different, and perhaps more complex, objectives. These comprise the synthesis of control strategies guiding the switched systems through predetermined operating points while avoiding certain regions in the state space, enforcing limit cycles and oscillatory behavior, reconfiguration upon the occurrence of faults, etc. This paper describes one particular approach to address these different objectives based on the construction of symbolic models in which sets of states in the switched system are represented
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