×

\(H_\infty\) finite-time control for switched nonlinear discrete-time systems with norm-bounded disturbance. (English) Zbl 1214.93043

Summary: Finite-time stability concerns the boundedness of system during a fixed finite-time interval. For switched systems, finite-time stability property can be affected significantly by switching behavior; however, it was neglected by most previous research. In this paper, the problems of finite-time stability analysis and stabilization for switched nonlinear discrete-time systems are addressed. First, sufficient conditions are given to ensure a class of switched nonlinear discrete-time system subjected to norm bounded disturbance finite-time bounded under arbitrary switching, and then the results are extended to \(H_\infty\) finite-time boundedness of switched nonlinear discrete-time systems. Finally based on the results on finite-time boundedness, a state feedback controller is designed to \(H_\infty\) finite-time stabilize a switched nonlinear discrete-time system. A numerical design example is given to illustrate the proposed results within this paper.

MSC:

93B36 \(H^\infty\)-control
93C10 Nonlinear systems in control theory
93C55 Discrete-time control/observation systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Weiss, L.; Infante, E., Finite time stability under perturbing forces and on product spaces, IEEE Transactions on Automatic Control, 12, 54-59 (1967) · Zbl 0168.33903
[2] Michel, A. N.; Wu, S. H., Stability of discrete systems over a finite interval of time, International Journal of Control, 9, 679-693 (1969) · Zbl 0174.40404
[3] D’Angelo, H., Linear Time-Varying Systems: Analysis and Synthesis (1970), Allyn and Bacon: Allyn and Bacon Boston, MA · Zbl 0202.08502
[4] Amato, F.; Ariola, M.; Cosentino, C., Finite-time stabilization via dynamic output feedback, Automatica, 41, 337-342 (2006) · Zbl 1099.93042
[5] Amato, F.; Ariola, M., Finite-time control of discrete-time linear systems, IEEE Transactions on Automatic Control, 50, 724-729 (2005) · Zbl 1365.93182
[6] Meng, Q.; Shen, Y., Finite-time \(H_∞\) control for linear continuous system with norm bounded disturbance, Communications in Nonlinear Science and Numerical Simulations, 14, 1043-1049 (2009) · Zbl 1221.93066
[7] Liberzon, D., Switching in Systems and Control (2003), Birkhauser: Birkhauser Boston, MA · Zbl 1036.93001
[8] Sun, Z.; Ge, S. S., Switched Linear Systems—Control and Design (2005), Springer-Verlag: Springer-Verlag London, UK · Zbl 1074.93025
[9] R.A. Decarlo, M.S. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilization of hybrid systems, in: Proceedings of the IEEE, vol. 88, 2000, pp. 1069-1082.; R.A. Decarlo, M.S. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilization of hybrid systems, in: Proceedings of the IEEE, vol. 88, 2000, pp. 1069-1082.
[10] A. Balluchi, M.D. Benedetto, C. Pinello, C. Rossi, A. Sangiovanni-Vincentelli, Cut-off in engine control: a hybrid system approach, in: Proceedings of the 36th IEEE Conference on Decision and Control, 1997, pp. 4720-4725.; A. Balluchi, M.D. Benedetto, C. Pinello, C. Rossi, A. Sangiovanni-Vincentelli, Cut-off in engine control: a hybrid system approach, in: Proceedings of the 36th IEEE Conference on Decision and Control, 1997, pp. 4720-4725.
[11] B.E. Bishop, M.W. Spong, Control of redundant manipulators using logic-based switching, in: Proceedings of the 36th IEEE Conference on Decision and Control, 1998, pp. 16-18.; B.E. Bishop, M.W. Spong, Control of redundant manipulators using logic-based switching, in: Proceedings of the 36th IEEE Conference on Decision and Control, 1998, pp. 16-18.
[12] Zhang, W.; Branicky, M. S.; Phillips, S. M., Stability of networked control systems, IEEE Control Systems Magazine, 21, 84-99 (2001)
[13] I.V. Kolmanovsky, J. Sun, A multi-mode switching-based command tracking in network controlled systems with pointwise-in-time constraints and disturbance inputs, in: Proceedings of the Sixth WCICA, 2006, pp. 199-104.; I.V. Kolmanovsky, J. Sun, A multi-mode switching-based command tracking in network controlled systems with pointwise-in-time constraints and disturbance inputs, in: Proceedings of the Sixth WCICA, 2006, pp. 199-104.
[14] Narendra, K. S.; Driollet, O. A.; Feiler, M.; George, K., Adaptive control using multiple models, switching and tuning, International Journal of Adaptive Control and Signal Processing, 17, 87-102 (2003) · Zbl 1016.93034
[15] Phat, V. N., Switched controller design for stabilization of nonlinear hybrid systems with time-varying delays in state and control, Journal of the Franklin Institute, 347, 195-207 (2010) · Zbl 1298.93290
[16] Sreekumar, C.; Agarwal, V., A hybrid control algorithm for voltage regulation in DC-DC boost converter, IEEE Transactions on Industrial Electronics, 55, 2530-2538 (2008)
[17] Narendra, K. S.; Balakrishnan, J. A., Common Lyapunov function for stable LTI systems with commuting A-matrices, IEEE Transactions on Automatic Control, 39, 2469-2471 (1994) · Zbl 0825.93668
[18] Branicky, M. S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Transactions on Automatic Control, 43, 475-482 (1998) · Zbl 0904.93036
[19] Ye, H.; Michel, A. N.; Hou, L., Stability theory for hybrid dynamic systems, IEEE Transactions on Automatic Control, 43, 461-474 (1998) · Zbl 0905.93024
[20] Zhang, L.; Wang, C.; Chen, L., Stability and stabilization of a class of multimode linear discrete-time systems with polytopic uncertainties, IEEE Transactions on Industrial Electronics, 56, 3684-3692 (2009)
[21] Morse, A. S., Supervisory control of families of linear set-point controllers, Part 1: exact matching, IEEE Transactions on Automatic Control, 41, 1413-1431 (1996) · Zbl 0872.93009
[22] J.P. Hespanha, D. Liberzon, 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.; J.P. Hespanha, D. Liberzon, 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.
[23] 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, in: Proceedings of the American Control Conference, 2000, pp. 200-204.; 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, in: Proceedings of the American Control Conference, 2000, pp. 200-204. · Zbl 1022.93043
[24] Zhang, L.; Shi, P., Stability, \(L_2\) gain and asynchronous control of discrete-time switched systems with average dwell time, IEEE Transactions on Automatic Control, 54, 2193-2200 (2009)
[25] Lin, H.; Antsaklis, P. J., Stability and stabilizability of switched linear systems: a survey of recent results, IEEE Transactions on Automatic Control, 54, 308-322 (2009) · Zbl 1367.93440
[26] Leith, D. J.; Shorten, R. N.; Leithead, W. E.; Mason, O.; Curran, P., Issues in the design of switched linear control systems: a benchmark study, International Journal of Adaptive Control and Signal Processing, 17, 103-118 (2003) · Zbl 1016.93026
[27] Liberzon, D.; Morse, A. S., Basic problems in stability and design of switched systems, IEEE Control Systems Magazine, 19, 59-70 (1999) · Zbl 1384.93064
[28] Mahmoud, M. S.; Nounou, H. N.; Xia, Y., Robust dissipative control for internet-based switching systems, Journal of the Franklin Institute, 347, 154-172 (2010) · Zbl 1298.93138
[29] Zhao, X.; Zeng, Q., New robust delay-dependent stability and \(H_∞\) analysis for uncertain Markovian jump systemswith time-varying delays, Journal of the Franklin Institute, 347, 863-874 (2010) · Zbl 1286.93199
[30] Zhai, G.; Hu, B.; Yasuda, K.; Michel, A. N., Disturbance attenuation properties of time-controlled switched systems, Journal of the Franklin Institute, 338, 765-779 (2001) · Zbl 1022.93017
[31] Wang, Y.; Xie, L.; De Souza, C. E., Robust control of a class of uncertain nonlinear systems, System and Control Letters, 19, 139-149 (1992) · Zbl 0765.93015
[32] Boyd, S.; Ghaoui, L. E.; Feron, E.; Balakrishnan, V., Linear Matrix Inequalities in Systems and Control Theory (1994), SIAM: SIAM Philadelphia, PA
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.