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Closed curve solutions and limit cycles in a class of second-order switched nonlinear systems. (English) Zbl 1236.93092
Summary: This paper studies the problem of finding the initial states for which the solution of a class of switched systems consisting of unstable second-order nonlinear subsystems is convergent. A method is described and applied to establish the regions in the plane where it is possible to define a switching law such that the solution of a class of switched nonlinear systems converges to the origin. We prove that, under certain conditions, these regions are delimited by closed curve solutions of the switched system. Furthermore, a sufficient condition for the closed curve solution to be a limit cycle is presented. Finally, a numerical example is included in order to illustrate the results.
MSC:
93C30Control systems governed by other functional relations
93C15Control systems governed by ODE
References:
[1]Dayawansa, W. P.; Martin, C. F.: A converse Lyapunov theorem of a class of dynamical systems which undergo switching, IEEE trans. Automat. control 44, 751-760 (1999) · Zbl 0960.93046 · doi:10.1109/9.754812
[2]Liberzon, D.; Morse, A. S.: Basic problems in stability and design of switched systems, IEEE control syst. Mag. 19, 59-70 (1999)
[3]Xie, G.; Wang, L.: Controllability and stabilizability of switched linear systems, Syst. control lett. 48, 135-155 (2003) · Zbl 1134.93403 · doi:10.1016/S0167-6911(02)00288-8
[4]Sun, Z.; Ge, S. S.; Lee, T. H.: Controllability and reachability criteria for switched linear systems, Automatica 38, 775-786 (2002) · Zbl 1031.93041 · doi:10.1016/S0005-1098(01)00267-9
[5]Sun, Z.; Zhen, D.: On reachability and stabilization of switched linear systems, IEEE trans. Automat. control 46, 291-295 (2001) · Zbl 0992.93006 · doi:10.1109/9.905696
[6]Boyd, S.; Ghaoui, L. E.; Feron, E.; Balakrishnan, V.: Linear matrix inequalities in system and control theory, SIAM stud. Appl. math. 15 (1994) · Zbl 0816.93004
[7]Liberzon, D.; Hespanha, J. P.; Morse, A. S.: Stability of switched linear systems: a Lie-algebraic condition, Syst. control lett. 37, 117-122 (1999) · Zbl 0948.93048 · doi:10.1016/S0167-6911(99)00012-2
[8]Mason, P.; Boscain, U.; Chitour, Y.: Common polynomial Lyapunov functions for linear switched systems, SIAM J. Control optim. 45, 226-245 (2006) · Zbl 1132.93038 · doi:10.1137/040613147
[9]Narendra, K. S.; Balakrishnan, J.: A common Lyapunov function for stable LTI systems with commuting A-matrices, IEEE trans. Automat. control 39, 2469-2471 (1994) · Zbl 0825.93668 · doi:10.1109/9.362846
[10]R.N. Shorten, K.S. Narendra, A sufficient condition for the existence of a common Lyapunov function for two second order linear systems, in: Proceedings of the 36th Conference on Decision and Control, San Diego, CA, 1997, pp. 3521–3522.
[11]R.N. Shorten, K.S. Narendra, On the stability and existence of common Lyapunov functions for stable linear switching systems, in: Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, FL, 1998, pp. 3723–3724.
[12]E. Feron, Quadratic stabilizability of switched system via state and output feedback, MIT Technical Report CICS-P-468, 1996.
[13]M.A. Wicks, R.A. DeCarlo, Solution of coupled Lyapunov equations for the stabilization of multimodal linear systems, in: Proceedings of 1997 American Control Conference, pp. 1709–1713.
[14]M.A. Wicks, P. Peleties, R.A. DeCarlo, Construction of piecewise Lyapunov functions for stabilizing switched systems, in: Proceedings of the 33rd Conference on Decision and Control, Lake Buena Vista, FL, 1994, pp. 3492–3497.
[15]Xu, X.; Antsaklis, P.: Stabilization of second-order LTI switched systems, Int. J. Control 73, 1261-1279 (2000) · Zbl 0992.93078 · doi:10.1080/002071700421664
[16]Hu, B.; Xu, X.; Antsaklis, P. J.; Michel, A. N.: Robust stabilizing control laws for a class of second-order switched systems, Syst. control lett. 38, 197-207 (1999) · Zbl 0948.93013 · doi:10.1016/S0167-6911(99)00065-1
[17]Goebel, R.; Sanfelice, R. G.; Teel, A. R.: Invariance principles for switching systems via hybrid systems techniques, Syst. control lett. 57, 980-986 (2008) · Zbl 1148.93028 · doi:10.1016/j.sysconle.2008.06.002
[18]Sanfelice, R. G.; Goebel, R.; Teel, A. R.: Invariance principles for hybrid systems with connections to detectability and asymptotic stability, IEEE trans. Automat. control 52, 2282-2297 (2007)
[19]Sun, Z.; Ge, S. S.: Switched linear systems: control and design, (2005)
[20]Milnor, J.: Morse theory, (1973)
[21]Artstein, Z.: Examples of stabilization with hybrid feedback, Hybrid systems III: Verification and control, 173-185 (1996)
[22]D. Liberzon, Stabilizing a linear system with finite-state hybrid output feedback, in: Proceedings of the 7th IEEE Mediterranean Conference on Control and Automation, 1999, pp. 176–183.
[23]Apostol, T. M.: Mathematical analysis, (1974) · Zbl 0309.26002
[24]Perko, L.: Differential equations and dynamical systems, (1991)