The problem of exponential stability for a class of switched nonlinear systems with discrete and distributed time-varying delays is studied. The constraint on the derivative of the time-varying delay is not required which allows the time delay to be a fast time-varying function. We study the stability properties of switched nonlinear systems consisting of both stable and unstable subsystems. Average dwell-time approached and improved piecewise Lyapunov functional combined with Leibniz-Newton are formulated. New delay-dependent sufficient conditions for the exponential stabilization of the switched systems are first established in terms of LMIs. A numerical example is also given to illustrate the effectiveness of the proposed method. 1. Introduction The switched systems are an important class of hybrid systems. They are described by a family of continuous or discrete-time subsystems and a rule that orchestrates the switching between the subsystems. Recently, switched systems have attracted much attention due to the widespread application in control, chemical engineering processing [1], communication networks, traffic control [2, 3], and control of manufacturing systems [4–6]. A switched nonlinear system with time delay is called switched nonlinear delay system, where delay may be contained in the system state, control input, or switching signals. In [7–9], some stability properties of switched linear delay systems composed of both stable and unstable subsystems have been studied by using an average dwell-time approach and piecewise Lyapunov functions. It is shown that when the average dwell time is sufficiently large and the total activation time of the unstable subsystems is relatively small compared with that of the Hurwitz stable subsystems, global exponential stability is guaranteed. The concept of dwell time was extended to average dwell time by Hespanha and Morse [10] with switching among stable subsystems. Furthermore, [11] generalized the results to the case where stable and unstable subsystems coexist. The stability analysis of nonlinear time-delay systems has received increasing attention. Time-delay systems are frequently encountered in various areas such as chemical engineering systems, biological modeling, and economics. The stability analysis for nonlinear time-delay systems has been investigated extensively. Various approaches to such problems have been proposed, see [12–14] and the references therein. It is well known that the existences of time delay in a system may cause instability and oscillations system. Thus, the stability analysis
S. Engell, S. Kowalewski, C. Schulz, and O. Stursberg, “Continuous-discrete interactions in chemical processing plants,” Proceedings of the IEEE, vol. 88, no. 7, pp. 1050–1068, 2000.
C. Livadas, J. Lygeros, and N. A. Lynch, “High-level modeling and analysis of the traffic alert and collision avoidance system (TCAS),” Proceedings of the IEEE, vol. 88, no. 7, pp. 926–948, 2000.
M. Song and T. J. Tarn, “Integration of task scheduling, action planning, and control in robotic manufacturing systems,” Proceedings of the IEEE, vol. 88, no. 7, pp. 1097–1107, 2000.
D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems,” IEEE Control Systems Magazine, vol. 19, no. 5, pp. 59–70, 1999.
X. M. Sun, D. Wang, W. Wang, and G. H. Yang, “Stability analysis and L2-gain of switched delay systems with stable and unstable subsystems,” in Proceedings of the IEEE 22nd International Symposium on Intelligent Control (ISIC '07), pp. 208–213, October 2007.
G. Zhai, B. Hu, K. Yasuda, and A. N. Michel, “Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach,” International Journal of Systems Science, vol. 32, no. 8, pp. 1055–1061, 2001.
J. P. Hespanha and A. S. Morse, “Stability of switched systems with average dwell-time,” in Proceedings of the 38th IEEE Conference on Decision and Control (CDC '99), pp. 2655–2660, December 1999.
G. Zhai, B. Hu, K. Yasuda, and A. N. Michel, “Piecewise lyapunov functions for switched systems with average dwell time,” Asian Journal of Control, vol. 2, no. 3, pp. 192–197, 2000.
J. H. Park and O. Kwon, “Novel stability criterion of time delay systems with nonlinear uncertainties,” Applied Mathematics Letters, vol. 18, no. 6, pp. 683–688, 2005.
J. H. Park and H. Y. Jung, “On the exponential stability of a class of nonlinear systems including delayed perturbations,” Journal of Computational and Applied Mathematics, vol. 159, no. 2, pp. 467–471, 2003.
Q. L. Han, “Robust stability for a class of linear systems with time-varying delay and nonlinear perturbations,” Computers and Mathematics with Applications, vol. 47, no. 8-9, pp. 1201–1209, 2004.
H. R. Karimi, M. Zapateiro, and N. Luo, “New delay-dependent stability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbations,” Mathematical Problems in Engineering, vol. 2009, Article ID 759248, 22 pages, 2009.
Y. Chen, A. Xue, R. Lu, and S. Zhou, “On robustly exponential stability of uncertain neutral systems with time-varying delays and nonlinear perturbations,” Nonlinear Analysis: Theory, Methods and Applications, vol. 68, no. 8, pp. 2464–2470, 2008.
V. B. Kolmanovskii, S. I. Niculescu, and J. P. Richard, “On the Liapunov-Krasovskii functionals for stability analysis of linear delay systems,” International Journal of Control, vol. 72, no. 4, pp. 374–384, 1999.
V. N. Phat and A. V. Savkin, “Robust state estimation for a class of uncertain time-delay systems,” Systems and Control Letters, vol. 47, no. 3, pp. 237–245, 2002.
V. N. Phat and P. T. Nam, “Exponential stability criteria of linear non-autonomous systems with multiple delays,” Electronic Journal of Differential Equations, vol. 58, pp. 1–9, 2005.
T. Botmart and P. Niamsup, “Robust exponential stability and stabilizability of linear parameter dependent systems with delays,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2551–2566, 2010.
I. Amri, D. Soudani, and M. Benrejeb, “Exponential stability and stabilization of linear systems with time varying delays,” in Proceedings of the 6th International Multi-Conference on Systems, Signals and Devices (SSD '09), March 2009.
L. V. Hien and V. N. Phat, “Exponential stability and stabilization of a class of uncertain linear time-delay systems,” Journal of the Franklin Institute, vol. 346, no. 6, pp. 611–625, 2009.
Q. L. Han, “A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays,” Automatica, vol. 40, no. 10, pp. 1791–1796, 2004.
T. Li, Q. Luo, C. Sun, and B. Zhang, “Exponential stability of recurrent neural networks with time-varying discrete and distributed delays,” Nonlinear Analysis: Real World Applications, vol. 10, no. 4, pp. 2581–2589, 2009.
F. Gao, S. Zhong, and X. Gao, “Delay-dependent stability of a type of linear switching systems with discrete and distributed time delays,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 24–39, 2008.