Extended lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems

Guisheng Zhai; Xuping Xu; Hai Lin; Derong Liu

International Journal of Applied Mathematics and Computer Science (2007)

  • Volume: 17, Issue: 4, page 447-454
  • ISSN: 1641-876X

Abstract

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We analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result.

How to cite

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Zhai, Guisheng, et al. "Extended lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems." International Journal of Applied Mathematics and Computer Science 17.4 (2007): 447-454. <http://eudml.org/doc/207849>.

@article{Zhai2007,
abstract = {We analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result.},
author = {Zhai, Guisheng, Xu, Xuping, Lin, Hai, Liu, Derong},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {exponential stability; switched systems; arbitrary switching dwell time scheme; Lie algebra; common quadratic Lyapunov functions; arbitrary switching; dwell time scheme},
language = {eng},
number = {4},
pages = {447-454},
title = {Extended lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems},
url = {http://eudml.org/doc/207849},
volume = {17},
year = {2007},
}

TY - JOUR
AU - Zhai, Guisheng
AU - Xu, Xuping
AU - Lin, Hai
AU - Liu, Derong
TI - Extended lie algebraic stability analysis for switched systems with continuous-time and discrete-time subsystems
JO - International Journal of Applied Mathematics and Computer Science
PY - 2007
VL - 17
IS - 4
SP - 447
EP - 454
AB - We analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result.
LA - eng
KW - exponential stability; switched systems; arbitrary switching dwell time scheme; Lie algebra; common quadratic Lyapunov functions; arbitrary switching; dwell time scheme
UR - http://eudml.org/doc/207849
ER -

References

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  3. Liberzon D. and Tempo R. (2003): Gradient algorithm for finding common Lyapunov functions. Proceedings of the 42nd IEEE Conference on Decision and Control, Hawaii, USA, pp. 4782-4787. 
  4. Liberzon D. (2003): Switching in Systems and Control. Boston: Birkhäuser. Zbl1036.93001
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  8. Samelson H. (1969): Notes on Lie Algebra. New York: Van Nostrand Reinhold. Zbl0209.06601
  9. Zhai G. (2003): Stability and L2 gain analysis of switched symmetric systems, In: Stability and Control of Dynamical Systems with Applications, (D. Liu and P.J. Antsaklis, Eds.), Boston: Birkhäuser, pp. 131-152. Zbl1044.93060
  10. Zhai G. (2001a): Quadratic stabilizability of discrete-time switched systems via state and output feedback. Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, pp. 2165-2166. 
  11. Zhai G., Hu B., Yasuda K. and Michel A.N. (2002a): Stability and L2 gain analysis of discrete-time switched systems. Transactions of the Institute of Systems, Control and Information Engineers, Vol. 15, No. 3, pp. 117-125. 
  12. Zhai G., Chen X., Ikeda M. and Yasuda K. (2002b): Stability and L2 gain analysis for a class of switched symmetric systems. Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, pp. 4395-4400. 
  13. Zhai G., Lin H., Michel A.N., and Yasuda K. (2004): Stability analysis for switched systems with continuous-time and discrete-time subsystems. Proceedings of the American Control Conference, Boston, MA, pp. 4555-4560. 
  14. Zhai G., Xu X., Lin H., and Michel A.N. (2006): Analysis and design of switched normal systems. Nonlinear Analysis, Vol. 65, No. 12, pp. 2248-2259. Zbl1119.34042
  15. Zhai G., Hu B., Yasuda K. and Michel A.N. (2001b): Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach. International Journal of Systems Science, Vol. 32, No. 8, pp. 1055-1061. Zbl1022.93043

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