Algorithmic Analysis of Nonlinear Hybrid Systems

Thomas A. Henzinger, Pei-Hsin Ho, and Howard Wong-Toi

Hybrid systems are digital real-time systems that are embedded in analog environments. Model-checking tools are available for the automatic analysis of linear hybrid automata, whose environment variables are subject to piecewise-constant polyhedral differential inclusions. In most embedded systems, however, the environment variables have differential inclusions that vary with the values of the variables. Such inclusions are prohibited in the linear hybrid automaton model. We present two methods for translating nonlinear hybrid systems into linear hybrid automata. Properties of the nonlinear systems can then be inferred from the automatic analysis of the translated linear hybrid automata.

The first method, called clock translation, replaces constraints on nonlinear variables by constraints on clock variables. The clock translation is efficient but has limited applicability. The second method, called linear phase-portrait approximation, conservatively overapproximates the phase portrait of a hybrid automaton using piecewise-constant polyhedral differential inclusions. Both methods are sound for safety properties; that is, if we establish a safety property of the translated linear system, we may conclude that the original nonlinear system satisfies the property. When applicable, the clock translation is also complete for safety properties; that is, the original system and the translated system satisfy the same safety properties. The phase-portrait approximation method is not complete for safety properties, but it is asymptotically complete; intuitively, for every safety property, and for every relaxed nonlinear system arbitrarily close to the original, if the relaxed system satisfies the safety property, then there is a linear phase-portrait approximation that also satisfies the property.

We illustrate both methods by using HyTech--a symbolic model checker for linear hybrid automata--to automatically check properties of a nonlinear temperature controller and of a predator-prey ecology.

IEEE Transactions on Automatic Control 43:540-554, 1998. Preliminary reports on this work appeared in the T.A. Henzinger and P.-H. Ho, "Algorithmic analysis of nonlinear hybrid systems," Proceedings of the Seventh International Conference on Computer-Aided Verification (CAV), Lecture Notes in Computer Science 939, Springer, 1995, pp. 225-238, and in T.A. Henzinger and H. Wong-Toi, "Linear phase-portrait approximations for nonlinear hybrid systems," in Hybrid Systems III, Lecture Notes in Computer Science 1066, Springer, 1996, pp. 377-388.

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