# Article

Full entry | PDF   (0.4 MB)
Keywords:
switched linear systems; finite-time boundedness; multiple Lyapunov-like functions; single Lyapunov-like function; common Lyapunov-like function
Summary:
In this paper, finite-time boundedness and stabilization problems for a class of switched linear systems with time-varying exogenous disturbances are studied. Firstly, the concepts of finite-time stability and finite-time boundedness are extended to switched linear systems. Then, based on matrix inequalities, some sufficient conditions under which the switched linear systems are finite-time bounded and uniformly finite-time bounded are given. Moreover, to solve the finite-time stabilization problem, stabilizing controllers and a class of switching signals are designed. The main results are proven by using the multiple Lyapunov-like functions method, the single Lyapunov-like function method and the common Lyapunov-like function method, respectively. Finally, three examples are employed to verify the efficiency of the proposed methods.
References:
[1] Amato, F., Ambrosino, R., Ariola, M., Cosentino, C.: Finite-time stability of linear time-varying systems with jumps. Automatica 45 (2009), 1354–1358. DOI 10.1016/j.automatica.2008.12.016 | MR 2531617 | Zbl 1162.93375
[2] Amato, F., Ariola, M.: Finite-time control of discrete-time linear system. IEEE Trans. Automat. Control 50 (2005), 724–729. DOI 10.1109/TAC.2005.847042 | MR 2141582
[3] Amato, F., Ariola, M., Dorato, P.: Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica 37 (2001), 1459–1463. DOI 10.1016/S0005-1098(01)00087-5 | Zbl 0983.93060
[4] Amato, F., Merola, A., Cosentino, C.: Finite-time stability of linear time-varying systems: Analysis and controller design. IEEE Trans. Automat. Control 55 (2010), 1003–1008. DOI 10.1109/TAC.2010.2041680 | MR 2654445
[5] Amato, F., Merola, A., Cosentino, C.: Finite-time control of discrete-time linear systems: Analysis and design conditions. Automatica 46 (2010) 919–924. DOI 10.1016/j.automatica.2010.02.008 | MR 2877166
[6] Bhat, S. P., Bernstein, D. S.: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38 (2000), 751–766. DOI 10.1137/S0363012997321358 | MR 1756893 | Zbl 0945.34039
[7] Branicky, M. S.: Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Automat. Control 43 (1998), 475–482. DOI 10.1109/9.664150 | MR 1617575 | Zbl 0904.93036
[8] Cheng, D., Guo, Y.: Advances on switched systems. Control Theory Appl. 22 (2005), 954–960. Zbl 1112.93365
[9] Dorato, P.: Short time stability in linear time-varying systems. In: Proc. IRE Internat. Conv. Rec. Part 4, New York 1961, pp. 83–87.
[10] Engell, S., Kowalewski, S., Schulz, C., Strusberg, O.: Continuous-discrete interactions in chemical processing plants. Proc. IEEE 88 (2000), 1050–1068.
[11] Gayek, J. E.: A survey of techniques for approximating reachable and controllable sets. In: Proc. 30th IEEE Conference on Decision and Control, Brighton 1991, pp.  1724–1729.
[12] Hespanha, J. P., Liberzon, D., Morse, A. S.: Stability of switched systems with average dwell time. In: Proc. 38th Conference on Decision and Control, Phoenix 1999, pp. 2655–2660.
[13] Kamenkov, G.: On stability of motion over a finite interval of time. J. Applied Math. and Mechanics 17 (1953), 529–540. MR 0061237
[14] Liberzon, D.: Switching in Systems and Control. Brikhauser, Boston 2003. MR 1987806 | Zbl 1036.93001
[15] Lin, H., Antsaklis, P. J.: Hybrid state feedback stabilization with $l_2$ performance for discrete-time switched linear systems. Internat. J. Control 81 (2008), 1114–1124. DOI 10.1080/00207170701654354 | MR 2431162 | Zbl 1152.93472
[16] Lin, H., Antsaklis, P. J.: Stability and stabilizability of switched linear systems: A survey of recent results. IEEE Trans. Automat. Control 54 (2009), 308–322. DOI 10.1109/TAC.2008.2012009 | MR 2491959
[17] Li, S., Tian, Y.: Finite-time stability of cascaded time-varying systems. Internat. J. Control 80 (2007), 646-657. DOI 10.1080/00207170601148291 | MR 2304124 | Zbl 1117.93004
[18] Morse, A. S.: Basic problems in stability and design of switched systems. IEEE Control Systems Magazine 19 (1999), 59–70. DOI 10.1109/37.793443
[19] Orlov, Y.: Finite time stability and robust control synthesis of uncertain switched systems. SIAM J. Control Optim. 43 (2005), 1253–1271. DOI 10.1137/S0363012903425593 | MR 2124272 | Zbl 1085.93021
[20] Pepyne, D., Cassandaras, C.: Optimal control of hybrid systems in manufacturing. Proc. IEEE 88 (2000), 1108–1123.
[21] Pettersson, S.: Synthesis of switched linear systems. In: Proc. 42nd Conference on Decision and Control, Maui 2003, pp. 5283–5288.
[22] Sun, Z.: Stabilizing switching design for switched linear systems: A state-feedback path-wise switching approach. Automatica 45 (2009), 1708–1714. DOI 10.1016/j.automatica.2009.03.001 | MR 2879485 | Zbl 1184.93077
[23] Sun, Z., Ge, S. S.: Analysis and synthesis of switched linear control systems. Automatica 41 (2005), 181–195. DOI 10.1016/j.automatica.2004.09.015 | MR 2157653 | Zbl 1074.93025
[24] Sun, X., Zhao, J., Hill, D. J.: Stability and $L_2$ gain analysis for switched delay systems: A delay-dependent method. Automatica 42 (2006), 1769–1774. DOI 10.1016/j.automatica.2006.05.007 | MR 2249722 | Zbl 1114.93086
[25] Tanner, H. G., Jadbabaie, A., Pappas, G. J.: Flocking in fixed and switching networks. IEEE Trans. Automat. Control 52 (2007), 863–868. DOI 10.1109/TAC.2007.895948 | MR 2324246
[26] Safonov, M. G., Goh, K. G., Ly, J.: Control system synthesis via bilinear matrix inequalities. In: Proc. American Control Conference, Baltimore 1994, pp. 45–49.
[27] Antwerp, J. G. Van, Braat, R. D.: A tutorial on linear and bilinear matrix inequalities. J. Process Control 10 (2000), 363–385. DOI 10.1016/S0959-1524(99)00056-6
[28] Varaiya, P.: Smart cars on smart roads: Problems of control. IEEE Trans. Automat. Control 38 (1993), 195–207. DOI 10.1109/9.250509 | MR 1206801
[29] Wang, J., Zhang, G., Li, H.: Adaptive control of uncertain nonholonomic systems in finite time. Kybernetika 45 (2009), 809–824. MR 2599114 | Zbl 1190.93086
[30] Weiss, L., Infante, E. F.: Finite time stability under perturbing forces and on product spaces. IEEE Trans. Automat. Control 12 (1967), 54–59. DOI 10.1109/TAC.1967.1098483 | MR 0209589 | Zbl 0168.33903
[31] Xu, X., Zhai, G.: Practical stability and stabilization of hybrid and switched systems. IEEE Trans. Automat. Control 50 (2005), 1897–1903. DOI 10.1109/TAC.2005.858680 | MR 2182748
[32] Yang, H., Cocquempot, V., Jiang, B.: On stabilization of switched nonlinear systems with unstable modes. Systems Control Lett. 58 (2009), 703–708. DOI 10.1016/j.sysconle.2009.06.007 | MR 2584005 | Zbl 1181.93074
[33] Zhao, J., Hill, D. J.: On stability, $L_2$-gain and $H_{\infty }$ control for switched systems. Automatica 44 (2008), 1220–1232. DOI 10.1016/j.automatica.2007.10.011 | MR 2531787
[34] Zhao, S., Sun, J., Liu, L.: Finite-time stability of linear time-varying singular systems with impulsive effects. Internat. J. Control 81 (2008), 1824–1829. DOI 10.1080/00207170801898893 | MR 2462577 | Zbl 1148.93345
[35] Zhu, L., Shen, Y., Li, C.: Finite-time control of discrete-time systems with time-varying exogenous disturbance. Communications in Nonlinear Science and Numerical Simulation 14 (2009), 361–370. DOI 10.1016/j.cnsns.2007.09.013 | MR 2458814 | Zbl 1221.93240

Partner of