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Further results on exponential stability of switched systems

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Abstract

Addressed in this paper is the stability issue of continuous-time switched linear systems consisting of both stable and unstable subsystems under switching signals satisfying the constraint of mode-dependent average dwell time. A multiple discontinuous homogeneous polynomial Lyapunov function (MDHPLF) method is proposed, which results in, compared with existing works, less conservative sufficient conditions guaranteeing exponential stability of systems under consideration. It is proven that the conservatism of stability condition reduces as the degree of MDHPLF increases. In addition, all main results can be applied to scenarios where all subsystems are stable or unstable.

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Notes

  1. In some papers, \({\varvec{x}}_m\) is denoted by \({\varvec{x}}^{\left\{ m\right\} }\). However, the notation \({\varvec{x}}^{\left\{ m\right\} }\) will make some expressions in the sequel complicated and is thus not adopted here.

  2. If \({\varvec{y}}\in {\mathbb {R}}^{\vartheta (n,dm)}\) and \({\varvec{y}}\ne {\varvec{0}}\) then \(K_1{\varvec{y}}\ne {\varvec{0}}\) since \(K_1\) is full-column rank. Let \({\varvec{z}}=K_1{\varvec{y}}\ne {\varvec{0}}\), then one obtains that \({\varvec{z}}^\textrm{T}\left( P^\ell _i\right) ^{\otimes d}{\varvec{z}}>0\), i.e., \({\varvec{y}}^\textrm{T}K_1^\textrm{T}\left( P^\ell _i\right) ^{\otimes d}K_1{\varvec{y}}>0\) provided that \({\varvec{y}}\ne {\varvec{0}}\). Therefore, \(K_1^\textrm{T}\left( P^\ell _i\right) ^{\otimes d}K_1>0\).

  3. For example, let \(d=2,c=1\), and the polynomial is \(\eta _iP^{\ell -1}_i\otimes L\left( {\varvec{v}}^{\ell ,\ell -1}_i\right) +L\left( {\varvec{v}}^{\ell ,\ell -1}_i\right) \otimes \eta _iP^{\ell -1}_i\).

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Acknowledgements

This work was partially supported by National Nature Science Foundation (62073270), State Ethnic Affairs Commission Innovation Research Team, 5. Fundamental Research Funds for the Central Universities (2021HQZZ02), and Innovative Research Team of the Education Department of Sichuan Province (15TD0050).

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Authors and Affiliations

  1. Key Laboratory of Electronic Information of State Ethnic Affairs Commission, College of Electrical Engineering, Southwest Minzu University, Chengdu, 610225, Sichuan, China

    Yaping Li

  2. Laboratory of Complex Systems and Intelligent Control, College of Electrical Engineering, Southwest Minzu University, Chengdu, 610225, Sichuan, China

    Xingwen Liu, Shiquan Shao, Jun Yang & Jun Wang

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  1. Yaping Li
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Correspondence to Xingwen Liu.

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Communicated by Nadhir Messai.

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Li, Y., Liu, X., Shao, S. et al. Further results on exponential stability of switched systems. Comp. Appl. Math. 42, 47 (2023). https://doi.org/10.1007/s40314-022-02181-x

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