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Research Papers

# Observability and Distinguishability Properties and Stability of Arbitrarily Fast Switched Systems

[+] Author and Article Information
Najah F. Jasim

Department of Mechanical Engineering,
University of Karbala,
Karbala 56001, Iraq
e-mail: naje76@yahoo.com

It can be shown from the proof of Claim 1 in Ref. [23], p. 220 that properness and positive definiteness of V(x) imply this condition. Here, we just like to highlight this condition explicitly.

The harmonic series is a divergent sequence of numbers in the form ${1,12,13,14,...,1n,...}$ whose terms approach zero and whose sum grows without bound as n gets large.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received November 17, 2009; final manuscript received April 21, 2013; published online June 27, 2013. Assoc. Editor: Swaroop Darbha.

J. Dyn. Sys., Meas., Control 135(5), 051011 (Jun 27, 2013) (5 pages) Paper No: DS-09-1321; doi: 10.1115/1.4024363 History: Received November 17, 2009; Revised April 21, 2013

## Abstract

This paper addresses sufficient conditions for asymptotic stability of classes of nonlinear switched systems with external disturbances and arbitrarily fast switching signals. It is shown that asymptotic stability of such systems can be guaranteed if each subsystem satisfies certain variants of observability or 0-distinguishability properties. In view of this result, further extensions of LaSalle stability theorem to nonlinear switched systems with arbitrary switching can be obtained based on these properties. Moreover, the main theorems of this paper provide useful tools for achieving asymptotic stability of dynamic systems undergoing Zeno switching.

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## References

Lin, H., and Antsaklis, P. J., 2009, “Stability and Stabilizability of Switched Linear System: A Survey of Recent Results,” IEEE Trans. Autom. Control, 54(2), pp. 308–322.
Shorten, R., Wirth, F., Mason, O., Wulff, K., and King, C., 2007, “Stability Criteria for Switched and Hybrid Systems,” SIAM Rev., 49(4), pp. 545–592.
Lee, T. C., and Jiang, Z. P., 2009, “Uniform Asymptotic Stability of Nonlinear Switched Systems With an Application to Mobile Robots,” IEEE Trans. Autom. Control, 53(6), pp. 1235–1252.
Hespanha, J. P., and Morse, A. S., 1999, “Stability of Switched Systems With Average Dwell-Time,” Proceedings of the 38th IEEE Conf. on Decision and Control, pp. 2655–2660.
Hespanha, J. P., 2004, “Uniform Stability of Switched Linear Systems: Extensions of LaSalle's Invariance Principle,” IEEE Trans. Autom. Control, 49(3), pp. 470–482.
Dayawansa, W. P., and Martin, C. F., 1999, “A Converse Lyapunov Theorem for a Class of Dynamical Systems Which Undergo Switching,” IEEE Trans. Autom. Control, 44(4), pp. 751–760.
Liberzon, D., and Morse, A. S., 1999, “Basic Problems in Stability and Design of Switched Systems,” IEEE Control Syst. Mag., 19(5), pp. 59–70.
Hetel, L., Daafouz, J., and Iung, C., 2006, “Stabilization of Arbitrary Switched Linear Systems With Unknown Time-Varying Delays,” IEEE Trans. Autom. Control, 51(10), pp. 1668–1674.
Gurvits, L., Shorten, R., and Mason, O., 2007, “On the Stability of Switched Positive Linear Systems,” IEEE Trans. Autom. Control, 52(6), pp. 1099–1103.
Johansson, K. H., LygerosJ., Sastry, S., and Egerstedt, M., 1999, “Simulation of Zeno Hybrid Automata,” IEEE Conference on Decision and Control, Phoenix, AZ.
Zhang, J., Johansson, K. H., Lygeros, J., and Sastry, S., 2001, “Zeno Hybrid Systems,” Int. J. Robust Nonlinear Control, 11, pp.435–451.
Ames, A. D., Abate, A., and Sastry, S., 2007, “Sufficient Conditions for the Existence of Zeno Behavior in Nonlinear Hybrid Systems via Constant Approximations,” Proceedings of IEEE Conference Decision and Control, pp. 4033–4038.
Ames, A. D., Tabuada, P., and Sastry, S., 2006, “On the stability of Zeno Equilibria,” Hybrid Systems: Computation and Control (Lecture Notes in Computer Science), Springer-Verlag, NewYork, pp. 34–48.
Ames, A. D., Zheng, H., Gregg, R. D., and Sastry, S., 2006, “Is There Life After Zeno? Taking Executions Past The Breaking (Zeno) Point,” Proceedings of American Control Conference, pp. 2652–2657.
Camlibel, M. K., and Schumacher, J. M., 2001, “On the Zeno Behavior of Linear Complementarity Systems,” Proceedings of IEEE Conference Decision Control, pp. 346–351.
Goebel, R., and Teel, A. R., 2008, “Lyapunov Characterization of Zeno Behavior in Hybrid Systems,” Proceedings of 47th IEEE Conference Decision Control, pp. 2752–2757.
Goebel, R., and Teel, A. R., 2008, “Zeno Behavior in Homogeneous Hybrid Systems,” Proc. 47th IEEE Conference Decision Control, pp. 2758–2763.
Or, Y., and Teel, A. R., 2011, “Zeno Stability of the Set-Valued Bouncing Ball,” IEEE Trans. Autom. Control, 56(2), pp. 447–452.
Mancilla, J. L., and Garcia, R. A., 2009, “Input-to-Output Stability Properties of Switched Perturbed Nonlinear Control Systems,” Lat. Am. Appl. Res., 39, pp. 239–244.
Mancilla-Aguilar, J. L., Garcia, R., Sontag, E., and Wang, Y., 2004, “Representation of Switched Systems by Perturbed Control Systems,” 43rd IEEE Conference on Decision and Control, pp. 3259–3264.
Zheng, G., Yu, L., Boutat, D., and Barbot, J.-P., 2009, “Algebraic Observer for a Class of Switched Systems With Zeno Phenomenon,” Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, pp. 3876–3881.
Yu, L., Barbot, J.-P., Boutat, D., and Benmerzouk, D., 2011, “Observability Forms for Switched Systems With Zeno Phenomenon or High Switching Frequency,” IEEE Trans. Autom. Control, 56(2), pp. 436–441.
Sontag, E. D., 1998, Mathematical Control Theory, Deterministic Finite Dimensional Systems, 2nd ed., Springer-Verlag, New York.
Hespanha, J. P., Liberzon, D., Angeli, D., and Sontag, E., 2005, “Nonlinear Norm-Observability Notions and Stability of Switched Systems,” IEEE Trans. Autom. Control, 50(2), pp. 154–168.
Mancilla-Aguilar, J. L., Garcia, R., Sontag, E., and Wang, Y., 2005, “On the Representation of Switched Systems With Inputs by Perturbed Control Systems,” Nonlinear Anal., 60, pp. 1111–1150.
Cichon, M., Kubiaczyk, I., and Sikorska, A., 2004, “The Henstock-Kurweil-Pettis Integral and Existence Theorems for the Cauchy Problem,” Czeckoslovak Math. J., 54, pp. 279–289.
Branicky, M. S., 1998, “Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid Systems,” IEEE Trans. Autom. Control, 43(4), pp. 475–482.
Khalil, H. K., 1996, Nonlinear Systems, Prentice-Hall, Upper Saddle River, NJ.

## Figures

Fig. 1

Spring-damper-mass system

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