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Stability of switched nonlinear systems via extensions of LaSalle’s invariance principle

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Abstract

This paper studies the extension of LaSalle’s invariance principle for switched nonlinear systems. Unlikemost existing results in which each switching mode in the system needs to be asymptotically stable, this paper allows the switching modes to be only stable. Under certain ergodicity assumptions of the switching signals, two extensions of LaSalle’s invariance principle for global asymptotic stability of switched nonlinear systems are obtained using the method of common joint Lyapunov function.

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References

  1. Liberzon D, Morse A S. Basic problems in stability and design of switched systems. IEEE Control Syst Mag, 1999, 19(5): 59–70

    Article  Google Scholar 

  2. Agrachev A A, Liberzon D. Lie-algebraic stability criteria for switched systems. SIAM J Control Optim, 2001, 40(1): 253–269

    Article  MATH  MathSciNet  Google Scholar 

  3. Zhao J, Dimirovski G. Quadratic stability of a class of switched nonlinear systems. IEEE Trans Autom Control, 2004, 49(4): 574–578

    Article  MathSciNet  Google Scholar 

  4. Xue J U, Zheng N N, Zhong X P. Sequential stratified sampling belief propagation for multiple targets tracking. Sci China Ser F-Inf Sci, 2006, 49(1): 48–62

    Article  MATH  MathSciNet  Google Scholar 

  5. Sira-Ranirez H. Nonlinear P-I controller design for switch mode DC-to-DC power converters. IEEE Trans Circuits Syst, 1991, 38(4): 410–417

    Article  Google Scholar 

  6. Jadbabaie A, Lin J, Morse A S. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans Autom Control, 2003, 48(6): 998–1001

    Article  MathSciNet  Google Scholar 

  7. Cheng D, Wang J, Hu X. Stabilization of Switched Linear Systems via LaSalle’s Invariance Principle. In: Proc. 6th IEEE Inter. Conf. Contr. Auto., 2007

  8. Moreau L. Stability of multiagent systems with time-dependent communication links. IEEE Trans Autom Control, 2005, 50(2): 169–182

    Article  MathSciNet  Google Scholar 

  9. Dayawansa W P, Martin C F. A converse Lyapunov theorem for a class of dynamic systems which undergo switching. IEEE Trans Autom Control, 1999, 44(4): 751–760

    Article  MATH  MathSciNet  Google Scholar 

  10. Mancilla-Aguilar J L, Garcia R A. A converse Lyapunov theorem for nonlinear switched systems. Syst Control Lett, 2000, 41: 67–71

    Article  MATH  MathSciNet  Google Scholar 

  11. Cheng D, Guo L, Huang J. On quatratic Lyapunov function. IEEE Trans Autom Control, 2003, 48(5): 885–890

    Article  MathSciNet  Google Scholar 

  12. Shorten R N, Narendra K S, Mason O. A result on common quadratic Lyapunov functions. IEEE Trans Autom Control, 2003, 48(1): 618–621

    Article  MathSciNet  Google Scholar 

  13. Liberzon D, Hespanha J P, Morse A S. Stability of switched systems: a Lie-algebraic condition. Syst Control Lett, 1999, 37(3): 117–122

    Article  MATH  MathSciNet  Google Scholar 

  14. Branicky M. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans Autom Control, 1998, 43(4): 475–482

    Article  MATH  MathSciNet  Google Scholar 

  15. Hespanha J P. Uniform stability of switched linear sysems: Extensions of LaSalle’s Invariance Principle. IEEE Trans Autom Control, 2004, 49(4): 470–482

    Article  MathSciNet  Google Scholar 

  16. Bacciotti A, Mazzi L. An invariace principle for nonlinear switched systems. Syst Control Lett, 2005, 54: 1109–1119

    Article  MATH  MathSciNet  Google Scholar 

  17. Hespanha J P, Liberzon D, Angeli E, et al. Nonlinear observability notion and stability of switched systems. IEEE Trans Autom Control, 2005, 50(2): 154–168

    Article  MathSciNet  Google Scholar 

  18. Mancilla-Aguilar J L, Garcia R A. An extension of LaSalle’s invariance principle for switched systems. Syst Control Lett, 2006, 55: 376–384

    Article  MATH  MathSciNet  Google Scholar 

  19. Khalil Hassan K. Nonlinear Systems. Upper Saddle River, NJ: Prentice Hall, 2002

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    JinHuan Wang & DaiZhan Cheng

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  1. JinHuan Wang
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  2. DaiZhan Cheng
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Correspondence to JinHuan Wang.

Additional information

Supported partly by the National Natural Science Foundation of China (Grant Nos. 60221301, 60674022 and 60736022)

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Wang, J., Cheng, D. Stability of switched nonlinear systems via extensions of LaSalle’s invariance principle. Sci. China Ser. F-Inf. Sci. 52, 84–90 (2009). https://doi.org/10.1007/s11432-009-0006-z

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  • DOI: https://doi.org/10.1007/s11432-009-0006-z

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