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Lyapunov Functions, Stability and Input-to-State Stability Subtleties for Discrete-Time Discontinuous Systems

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3 Author(s)
Mircea Lazar ; Dept. of Electr. Eng., Eindhoven Univ. of Technol., Eindhoven, Netherlands ; W. P. Maurice H. Heemels ; Andy R. Teel

In this note we consider stability analysis of discrete-time discontinuous systems using Lyapunov functions. We demonstrate via simple examples that the classical second method of Lyapunov is precarious for discrete-time discontinuous dynamics. Also, we indicate that a particular type of Lyapunov condition, slightly stronger than the classical one, is required to establish stability of discrete-time discontinuous systems. Furthermore, we examine the robustness of the stability property when it was attained via a discontinuous Lyapunov function, which is often the case for discrete-time hybrid systems. In contrast to existing results based on smooth Lyapunov functions, we develop several input-to-state stability tests that explicitly employ an available discontinuous Lyapunov function.

Published in:

IEEE Transactions on Automatic Control  (Volume:54 ,  Issue: 10 )