Abstract
This paper investigates the problem of the nonfragile observer design for discrete-time switched nonlinear systems with time delay. Based on the average dwell-time approach and linear matrix inequality (LMI) techniques, an exponential stability criterion for the discrete-time switched delay system with Lipschitz nonlinearity is derived. Based on several technical lemmas, the discrete-time observer design can be transferred to the problem of solving a set of LMIs. Furthermore, in cases when the gain of the state observer varies, a kind of nonfragile observer is proposed, and the solution to the observer gain is also obtained by solving a set of LMIs. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.
Similar content being viewed by others
References
A. Alessandri, P. Coletta, Switching observers for continuous-time and discrete-time linear systems, in Proceedings of the American Control Conference, Arlington, Virginia, 2516–2521 (2001)
S.P. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (SIAM, Philadelphia, 1994)
C.T. Chen, Linear System Theory and Design (Holt, Rinehart and Winston, New York, 1984)
D. Cheng, L. Guo, Y. Lin, Y. Wang, Stabilization of switched linear systems. IEEE Trans. Autom. Control 50(5), 661–666 (2005)
J.P. Hespanha, Uniform stability of switched linear systems: extension of LaSalle’s invariance principle. IEEE Trans. Autom. Control 49(4), 470–482 (2004)
J.P. Hespanha, D. Liberzon, D. Angeli, E.D. Sontag, Nonlinear norm-observability notions and stability of switched systems. IEEE Trans. Autom. Control 50(2), 154–168 (2005)
A.L. Juloski, W.P.M.H. Heemels, S. Weiland, Observer design for a class of piecewise affine systems, in Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, Nevada, USA, 2606–2611 (2002)
Q.K. Li, J. Zhao, G.M. Dimirovski, Tracking control for switched time-varying delays systems with stabilizable and unstabilizable subsystems. Nonlinear Anal., Hybrid Syst. 3(2), 133–142 (2009)
H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans. Autom. Control 54(2), 308–322 (2009)
A.E. Pearson, Y.A. Fiagbedzi, An observer for time lag systems. IEEE Trans. Autom. Control 34(8), 775–777 (1989)
S. Pettersson, Observer design for switched systems using multiple quadratic Lyapunov functions, in Proceedings of the 2005 IEEE International Symposium on Intelligent Control, pp. 262–267 (2005)
Y. Song, J. Fan, M. Fei, T. Yang Robust, H ∞ control of discrete switched system with time delay. Appl. Math. Comput. 205(1), 159–169 (2008)
Z. Sun, A robust stabilizing law for switched linear systems. Int. J. Control 77(4), 389–398 (2004)
Z. Sun, S.S. Ge, Analysis and synthesis of switched linear control systems. Automatica 41(2), 181–195 (2005)
V. Sundarapandian, General observers for discrete-time nonlinear systems. Math. Comput. Model. 39(1), 87–96 (2004)
R. Wang, J. Zhao, Guaranteed cost control for a class of uncertain switched delay systems: an average dwell-time method. Cybern. Syst. 38(1), 105–122 (2007)
Z. Xiang, R. Wang, Non-fragile observer design for nonlinear switched systems with time delay. Int. J. Intell. Comput. Cybern. 2(1), 175–189 (2009)
Z.R. Xiang, W.M. Xiang, Observer design for a class of switched nonlinear systems. Control Intell. Syst. 36(4), 318–322 (2008)
Z.R. Xiang, W.M. Xiang, Stability analysis of switched systems under dynamical dwell time control approach. Int. J. Syst. Sci. 40(4), 347–355 (2009)
D. Xie, L. Wang, F. Hao, Robust stability analysis and control synthesis for discrete-time uncertain switched systems, in Proceedings of Conference Decision and Control, pp. 4812–4817 (2003)
G.S. Zhai, B. Hu, K. Yasuda, A.N. Michel, Disturbance attenuation properties of time-controlled switched systems. J. Franklin Inst. 338(7), 765–779 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (No. 60974027).
Rights and permissions
About this article
Cite this article
Xiang, Z., Wang, R. & Jiang, B. Nonfragile Observer for Discrete-Time Switched Nonlinear Systems with Time Delay. Circuits Syst Signal Process 30, 73–87 (2011). https://doi.org/10.1007/s00034-010-9210-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-010-9210-8