Sampling–based verification of Lyapunov’s inequality for piecewise continuous nonlinear systems

R.V. Bobiti, M. Lazar

Research output: Contribution to journalArticleAcademic

Abstract

This paper considers a sampling-based approach to stability verification for piecewise continuous nonlinear systems via Lyapunov functions. Depending on the system dynamics, the candidate Lyapunov function and the set of initial states of interest, one generally needs to handle large, possibly non-convex or non-feasible optimization problems. To avoid such problems, we propose a constructive and systematically applicable sampling-based method to Lyapunov's inequality verification. This approach proposes verification of the decrease condition for a candidate Lyapunov function on a finite sampling of a bounded set of initial conditions and then it extends the validity of the Lyapunov function to an infinite set of initial conditions by automatically exploiting continuity properties. This result is based on multi-resolution sampling, to perform efficient state- space exploration. Using hyper-rectangles as basic sampling blocks, to account for different constraint scales on different states, further reduces the amount of samples to be verified. Moreover, the verification is decentralized in the sampling points, which makes the method scalable. The proposed methodology is illustrated through examples.
LanguageEnglish
Pages1-14
Number of pages14
JournalarXiv
Issue number1609.00302
StatePublished - 2016

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Nonlinear systems
Lyapunov functions
Sampling
Dynamical systems

Cite this

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abstract = "This paper considers a sampling-based approach to stability verification for piecewise continuous nonlinear systems via Lyapunov functions. Depending on the system dynamics, the candidate Lyapunov function and the set of initial states of interest, one generally needs to handle large, possibly non-convex or non-feasible optimization problems. To avoid such problems, we propose a constructive and systematically applicable sampling-based method to Lyapunov's inequality verification. This approach proposes verification of the decrease condition for a candidate Lyapunov function on a finite sampling of a bounded set of initial conditions and then it extends the validity of the Lyapunov function to an infinite set of initial conditions by automatically exploiting continuity properties. This result is based on multi-resolution sampling, to perform efficient state- space exploration. Using hyper-rectangles as basic sampling blocks, to account for different constraint scales on different states, further reduces the amount of samples to be verified. Moreover, the verification is decentralized in the sampling points, which makes the method scalable. The proposed methodology is illustrated through examples.",
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Sampling–based verification of Lyapunov’s inequality for piecewise continuous nonlinear systems. / Bobiti, R.V.; Lazar, M.

In: arXiv, No. 1609.00302, 2016, p. 1-14.

Research output: Contribution to journalArticleAcademic

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