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Finite-time stability analysis of impulsive switched discrete-time linear systems: the average Dwell time approach. (English) Zbl 1267.93099
Summary: This paper investigates the finite-time stability problem for a class of discrete-time switched linear systems with impulse effects. Based on the average dwell time approach, a sufficient condition is established which ensures that the state trajectory of the system remains in a bounded region of the state space over a pre-specified finite time interval. Different from the traditional condition for asymptotic stability of switched systems, it is shown that the total activation time of unstable subsystems can be greater than that of stable subsystems. Moreover, the finite-time stability degree can also be greater than one. Two examples are given to illustrate the merit of the proposed method.

##### MSC:
 93C55 Discrete-time control/observation systems
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##### References:
 [1] F. Amato, M. Ariola, P. Dorato, Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica 37(9), 1459–1463 (2001) · Zbl 0983.93060 · doi:10.1016/S0005-1098(01)00087-5 [2] F. Amato, M. Ariola, Finite-time control of discrete-time linear systems. IEEE Trans. Autom. Control 50(5), 724–729 (2005) · Zbl 1365.93182 · doi:10.1109/TAC.2005.847042 [3] F. Amato, R. Ambrosino, C. Cosentino, G. De Tommasi, Finite-time stabilization of impulsive dynamical linear systems. Nonlinear Anal. Hybrid Syst. 5(1), 89–101 (2011) · Zbl 1371.93159 · doi:10.1016/j.nahs.2010.10.001 [4] F. Amato, R. Ambrosino, M. Ariola, G. De Tommasi, Robust finite-time stability of impulsive dynamical linear systems subject to norm-bounded uncertainties. Int. J. Robust Nonlinear Control 21(10), 1080–1082 (2011) · Zbl 1225.93087 · doi:10.1002/rnc.1620 [5] R. Ambrosino, F. Calabrese, C. Cosentino, G. De Tommasi, Sufficient conditions for finite-time stability of impulsive dynamical systems. IEEE Trans. Autom. Control 54(4), 861–865 (2009) · Zbl 1367.93425 · doi:10.1109/TAC.2008.2010965 [6] S.P. Bhat, D.S. Bernstein, Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38(3), 751–766 (2000) · Zbl 0945.34039 · doi:10.1137/S0363012997321358 [7] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control 43(4), 475–482 (1998) · Zbl 0904.93036 · doi:10.1109/9.664150 [8] M.S. Branicky, Stability of switched and hybrid systems, in Proceedings of the 33rd IEEE Conference on Design and Control (1999), pp. 3498–3503 [9] D. Cheng, Controllability of switched bilinear systems. IEEE Trans. Autom. Control 40(4), 511–515 (2001) · Zbl 1365.93039 [10] J. Daafouz, P. Riedinger, C. Iung, Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach. IEEE Trans. Autom. Control 47(11), 1883–1887 (2002) · Zbl 1364.93559 · doi:10.1109/TAC.2002.804474 [11] R.A. DeCarlo, M. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems. Proc. IEEE 88(7), 1069–1082 (2000) · doi:10.1109/5.871309 [12] H. Du, X. Lin, S. Li, Finite-time boundedness and stabilization of switched linear systems. Kybernetika 46(5), 870–889 (2010) · Zbl 1205.93076 [13] J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time, in Proceedings of the 38th Conference on Decision and Control (1999), pp. 2655–2660 [14] M. Johansson, A. Rantzer, Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans. Autom. Control 43(4), 555–559 (1998) · Zbl 0905.93039 · doi:10.1109/9.664157 [15] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulse Differential Equations (World Scientific, Singapore, 1989) [16] S.H. Lee, J.T. Lim, Stability analysis of switched systems with impulse effects, in Proceedings of the 1999 IEEE International Symposium on Intelligent (1999), pp. 79–83 [17] D. Liberzon, Switching in Systems and Control (Brikhauser, Boston, 2003) · Zbl 1036.93001 [18] H. Lin, P.J. Anstaklis, Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans. Autom. Control 54(2), 2351–2355 (2009) [19] X. Lin, H. Du, S. Li, Finite-time boundedness and L 2-gain analysis for switched delay systems with norm-bounded disturbance. Appl. Math. Comput. 217(12), 5982–5993 (2011) · Zbl 1218.34082 · doi:10.1016/j.amc.2010.12.032 [20] L. Liu, J.T. Sun, Finite-time stabilization of linear systems via impulsive control. Int. J. Control 81(6), 905–909 (2008) · Zbl 1149.93338 · doi:10.1080/00207170701519060 [21] L. Lv, Z. Lin, H. Fang, $$$\backslash$$mathcal{L}_{2}$ gain analysis for a class of switched systems. Automatica 45(4), 965–972 (2009) · Zbl 1162.93355 · doi:10.1016/j.automatica.2008.10.021 [22] S. Pettersson, Analysis of switched linear systems, in Proceedings of the 42nd IEEE Conference on Decision and Control (2003), pp. 5283–5288 [23] S. Pettersson, Controller design of switched linear systems, in Proceedings of the 2004 American Control Conference (2004), pp. 3869–3874 [24] S. Pettersson, Synthesis of switched linear systems handling sliding motions, in Proceedings of the 2005 International Symposium on Intelligent Control (2005), pp. 18–23 [25] Z. Sun, S.S. Ge, Switched Linear Systems: Control and Design (Springer, New York, 2005) · Zbl 1075.93001 [26] J. Xu, J.T. Sun, Finite-time stability of linear time-varying singular impulsive systems. IET Control Theory Appl. 4(10), 2239–2244 (2010) · doi:10.1049/iet-cta.2010.0242 [27] X. Xu, G. Zhai, Practical stability and stabilization of hybrid and switched systems. IEEE Trans. Autom. Control 50(11), 1897–1903 (2005) · Zbl 1365.93359 · doi:10.1109/TAC.2005.858680 [28] H. Ye, A.N. Michael, L. Hou, Stability theory for hybrid dynamical systems. IEEE Trans. Autom. Control 43(4), 461–474 (1998) · Zbl 0905.93024 · doi:10.1109/9.664149 [29] G.S. Zhai, B. Hu, K. Yasuda, A.N. Michel, Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach. Int. J. Syst. Sci. 32(8), 1055–1061 (2001) · Zbl 1022.93043 [30] G. Zong, S. Xu, Y. Wu, Robust H stabilization for uncertain switched impulsive control systems with state delay: an LMI approach. Nonlinear Anal. Hybrid Syst. 2(4), 1287–1300 (2008) · Zbl 1163.93386 · doi:10.1016/j.nahs.2008.09.018 [31] G. Zong, L. Hou, J. Li, A descriptor system approach to l 2 filtering for uncertain discrete-time switched system with mode-dependent time-varying delays. Int. J. Innov. Comput. Inf. Control 7(5A), 2213–2224 (2011) [32] G. Zong, L. Hou, J. Li, Robust l 2 guaranteed cost filtering for uncertain discrete-time switched system with mode-dependent time-varying delays. Circuits Syst. Signal Process. 30(1), 17–33 (2011) · Zbl 1205.93152 · doi:10.1007/s00034-010-9204-6
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