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CLC number: TP13

On-line Access: 2014-01-07

Received: 2013-05-09

Revision Accepted: 2013-10-22

Crosschecked: 2013-12-16

Cited: 2

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Citations:  Bibtex RefMan EndNote GB/T7714

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Journal of Zhejiang University SCIENCE C 2014 Vol.15 No.1 P.31-42

http://doi.org/10.1631/jzus.C1300123


Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems


Author(s):  Xiao-li Zhang, An-hui Lin, Jian-ping Zeng

Affiliation(s):  School of Information Science and Technology, Xiamen University, Xiamen 361005, China

Corresponding email(s):   zhxl@xmu.edu.cn

Key Words:  Average dwell time, Impulse, Exponential stability, Robustness


Xiao-li Zhang, An-hui Lin, Jian-ping Zeng. Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems[J]. Journal of Zhejiang University Science C, 2014, 15(1): 31-42.

@article{title="Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems",
author="Xiao-li Zhang, An-hui Lin, Jian-ping Zeng",
journal="Journal of Zhejiang University Science C",
volume="15",
number="1",
pages="31-42",
year="2014",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.C1300123"
}

%0 Journal Article
%T Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems
%A Xiao-li Zhang
%A An-hui Lin
%A Jian-ping Zeng
%J Journal of Zhejiang University SCIENCE C
%V 15
%N 1
%P 31-42
%@ 1869-1951
%D 2014
%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.C1300123

TY - JOUR
T1 - Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems
A1 - Xiao-li Zhang
A1 - An-hui Lin
A1 - Jian-ping Zeng
J0 - Journal of Zhejiang University Science C
VL - 15
IS - 1
SP - 31
EP - 42
%@ 1869-1951
Y1 - 2014
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/jzus.C1300123


Abstract: 
exponential stability and robust exponential stability relating to switched systems consisting of stable and unstable nonlinear subsystems are considered in this study. At each switching time instant, the impulsive increments which are nonlinear functions of the states are extended from switched linear systems to switched nonlinear systems. Using the average dwell time method and piecewise Lyapunov function approach, when the total active time of unstable subsystems compared to the total active time of stable subsystems is less than a certain proportion, the exponential stability of the switched system is guaranteed. The switching law is designed which includes the average dwell time of the switched system. Switched systems with uncertainties are also studied. Sufficient conditions of the exponential stability and robust exponential stability are provided for switched nonlinear systems. Finally, simulations show the effectiveness of the result.

有稳定和不稳定子系统的非线性脉冲切换系统的指数稳定性

研究目的:对一类同时包含稳定和不稳定子系统的切换非线性系统的稳定性及鲁棒稳定性进行研究。旨在针对在切换时刻包含非线性脉冲,且每个子系统都具有非线性级联形式的切换非线性系统,给出其稳定的充分条件,为此类系统的稳定性问题研究提供理论依据。
重要结论:将在切换线性系统切换时刻的非线性脉冲处理方法和手段推广至切换非线性系统中。在应用非线性系统的Lyapunov函数处理非线性脉冲时,适当地利用了矩阵不等式的相关方法。当不稳定子系统和稳定子系统的活跃时间小于一定比例,并且在切换时刻存在满足相应界的非线性脉冲时,切换非线性系统仍能保持其指数稳定性。在不确定性满足相应界的条件下,切换非线性系统也能保持其鲁棒指数稳定性。

关键词:平均驻留时间,脉冲,指数稳定性,鲁棒性

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

[1]de la Sen, M., 2006. Stability of impulsive time-varying systems and compactness of the operators mapping the input space into the state and output spaces. J. Math. Anal. Appl., 321(2):621-650.

[2]de la Sen, M., Luo, N.S., 2003. A note on the stability of linear time-delay systems with impulsive inputs. IEEE Trans. Circ. Syst. I, 50(1):149-152.

[3]Guan, Z.H., Hill, D.J., Shen, X.M., 2005. On hybrid impulsive and switching systems and application to nonlinear control. IEEE Trans. Automat. Control, 50(7):1058-1062.

[4]Hespanha, J.P., Morse, A.S., 1999. Stability of switched systems with average dwell-time. Proc. 38th IEEE Conf. on Decision and Control, p.2655-2660.

[5]Hespanha, J.P., Liberzon, D., Teel, A.R., 2008. Lyapunov conditions for input-to-state stability of impulsive systems. Automatica, 44(11):2735-2744.

[6]Hiskens, I.A., 2001. Stability of hybrid system limit cycles: application to the compass gait biped robot. Proc. 40th IEEE Conf. on Decision and Control, p.774-779.

[7]Kim, S., Campbell, S.A., Liu, X.Z., 2006. Stability of a class of linear switching systems with time delay. IEEE Trans. Circ. Syst. I, 53(2):384-393.

[8]Lennartson, B., Tittus, M., Egardt, B., et al., 1996. Hybrid systems in process control. IEEE Control Syst. Mag., 16(5):45-56.

[9]Liberzon, D., 2003. Switching in Systems and Control. Birkhäuser, Boston.

[10]Liu, B., Marquez, H.J., 2008. Controllability and observability for a class of controlled switching impulsive systems. IEEE Trans. Automat. Control, 53(10):2360-2366.

[11]Liu, J., Liu, X.Z., Xie, W.C., 2011. Input-to-state stability of impulsive and switching hybrid systems with time-delay. Automatica, 47(5):899-908.

[12]Marchenko, V.M., Zaczkiewicz, Z., 2009. Representation of solutions of control hybrid differential-difference impulse systems. Differ. Equat., 45(12):1811-1822.

[13]Petersen, I.R., Hollot, C.V., 1986. A Riccati equation approach to the stabilization of uncertain linear systems. Automatica, 22(4):397-411.

[14]Qin, S.Y., Song, Y.H., 2001. The theory of hybrid control systems and its application perspective in electric power systems. Proc. Int. Conf. on Info-Tech and Info-Net, p.85-94.

[15]Sun, X.M., Wang, D., Wang, W., et al., 2007. Stability analysis and L2-gain of switched delay systems with stable and unstable subsystems. IEEE 22nd Int. Symp. on Intelligent Control, p.208-213.

[16]Sun, X.M., Wang, W., Liu, G.P., et al., 2008. Stability analysis for linear switched systems with time-varying delay. IEEE Trans. Syst. Man Cybern. B, 38(2):528-533.

[17]Varaiya, P., 1993. Smart cars on smart roads: problems of control. IEEE Trans. Automat. Control, 38(2):195-207.

[18]Wang, M., Dimirovski, G.M., Zhao, J., 2008. Average dwell-time method to stabilization and L2-gain analysis for uncertain switched nonlinear systems. Proc. 17th IFAC World Congress, p.7642-7647.

[19]Wicks, M., Peleties, P., DeCarlo, R., 1998. Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems. Eur. J. Control, 4(2):140-147.

[20]Xu, H.L., Teo, K.L., 2010. Exponential stability with L2-gain condition of nonlinear impulsive switched systems. IEEE Trans. Automat. Control, 55(10):2429-2433.

[21]Xu, H.L., Liu, X.Z., Teo, K.L., 2005. Robust H stabilization with definite attendance of uncertain impulsive switched systems. ANZIAM J., 46(4):471-484.

[22]Xu, H.L., Teo, K.L., Liu, X.Z., 2008. Robust stability analysis of guaranteed cost control for impulsive switched systems. IEEE Trans. Syst. Man Cybern. B, 38(5):1419-1422.

[23]Yao, J., Guan, Z.H., Chen, G.R., et al., 2006. Stability, robust stabilization and H control of singular-impulsive systems via switching control. Syst. Control Lett., 55(11):879-886.

[24]Zhai, G.S., Hu, B., Yasuda, K., et al., 2001a. Disturbance attenuation properties of time-controlled switched systems. J. Franklin Inst., 338(7):765-779.

[25]Zhai, G.S., Hu, B., Yasuda, K., et al., 2001b. Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach. Int. J. Syst. Sci., 32(8):1055-1061.

[26]Zhai, G.S., Lin, H., Kim, Y., et al., 2005. L2-gain analysis for switched systems with continuous-time and discrete-time subsystems. Int. J. Control, 78(15):1198-1205.

[27]Zhu, W., 2010. Stability analysis of switched impulsive systems with time delays. Nonl. Anal. Hybr. Syst., 4(3):608-617.

[28]Zong, G.D., Xu, S.Y., Wu, Y.Q., 2008. Robust H stabilization for uncertain switched impulsive control systems with state delay: an LMI approach. Nonl. Anal. Hybr. Syst., 2(4):1287-1300.

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