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.
Similar content being viewed by others
References
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. [doi:10.1016/j.jmaa.2005.08.038]
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. [doi:10.1109/TCSI.2002.807514]
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. [doi:10.1109/TAC.2005.851462]
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. [doi:10.1109/CDC.1999.831330]
Hespanha, J.P., Liberzon, D., Teel, A.R., 2008. Lyapunov conditions for input-to-state stability of impulsive systems. Automatica, 44(11):2735–2744. [doi:10.1016/j.automatica.2008.03.021]
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. [doi:10.1109/.2001.980200]
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. [doi:10.1109/TCSI.2005.856666]
Lennartson, B., Tittus, M., Egardt, B., et al., 1996. Hybrid systems in process control. IEEE Control Syst. Mag., 16(5):45–56. [doi:10.1109/37.537208]
Liberzon, D., 2003. Switching in Systems and Control. Birkhäuser, Boston. [doi:10.1007/978-1-4612-0017-8]
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. [doi: 10.1109/TAC.2008.2007476]
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. [doi:10.1016/j.automatica. 2011.01.061]
Marchenko, V.M., Zaczkiewicz, Z., 2009. Representation of solutions of control hybrid differential-difference impulse systems. Differ. Equat., 45(12):1811–1822. [doi:10.1134/S0012266109120118]
Petersen, I.R., Hollot, C.V., 1986. A Riccati equation approach to the stabilization of uncertain linear systems. Automatica, 22(4):397–411. [doi:10.1016/0005-1098(86)90045-2]
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. [doi:10.1109/ICII.2001.983729]
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. [doi:10.1109/ISIC.2007.4450886]
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. [doi:10.1109/TSMCB.2007.912078]
Varaiya, P., 1993. Smart cars on smart roads: problems of control. IEEE Trans. Automat. Control, 38(2):195–207. [doi:10.1109/9.250509]
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. [doi:10.3182/20080706-5-KR-1001.01292]
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. [doi: 10.1016/S0947-3580(98)70108-6]
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. [doi:10. 1109/TAC.2010.2060173]
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.
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. [doi: 10.1109/TSMCB.2008.925747]
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. [doi:10.1016/j.sysconle.2006.05.002]
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. [doi:10.1016/S0016-0032(01)00030-8]
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. [doi:10.1080/00207720116692]
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. [doi:10.1080/00207170500274966]
Zhu, W., 2010. Stability analysis of switched impulsive systems with time delays. Nonl. Anal. Hybr. Syst., 4(3):608–617. [doi:10.1016/j.nahs.2010.03.009]
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. [doi:10.1016/j.nahs.2008.09.018]
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Natural Science Foundation of China (Nos. 61074004 and 61374037)
Rights and permissions
About this article
Cite this article
Zhang, Xl., Lin, Ah. & Zeng, Jp. Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems. J. Zhejiang Univ. - Sci. C 15, 31–42 (2014). https://doi.org/10.1631/jzus.C1300123
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1631/jzus.C1300123