The viability of switched nonlinear systems with piecewise smooth Lyapunov functions

Jianfeng Lv; Yan Gao; Na Zhao

doi:10.3934/jimo.2020048

Article Contents

2021, Volume 17, Issue 4: 1825-1843. Doi: 10.3934/jimo.2020048

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The viability of switched nonlinear systems with piecewise smooth Lyapunov functions

1.
School of Management, University of Shanghai for Science and Technology Shanghai, 200093, China
2.
School of Science, Inner Mongolia University of Science and Technology Baotou, 014010, China

^* Corresponding author: Yan Gao
^* Corresponding author: Yan Gao

Received: April 30, 2019

Revised: October 31, 2019

Early access: March 2020

Published: July 2021

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Abstract

In this paper, we focus on the viability and attraction for switched nonlinear systems with nonsmooth Lyapunov functions. We determine the viable set and region of attraction for switched systems in which Lyapunov functions are piecewise smooth. The switching law is constructed by using the directional derivatives of a piecewise smooth Lyapunov function along the trajectories of the subsystems. Sufficient conditions are derived to guarantee the viability and attraction of switched nonlinear systems on the level set of a piecewise smooth Lyapunov function. We further extend the method to switched systems involving possible sliding motions. The approach in the paper provides a unified framework for studying viability and attraction with a systematic consideration of sliding motions. Finally, considering two certain classes of piecewise smooth functions, the related conditions of the viability and attraction for the level set are developed.

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Mathematics Subject Classification: Primary: 93C10, 93C60; Secondary: 93C05.

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Figure 1. Trajectories of switched systems excluding sliding motions

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Figure 2. Trajectories of switched systems including sliding motions

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Figure 3. The phase portraits of subsystems 1 and 2

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Figure 4. State responses under the switching law

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