Finite-time boundedness and H<sub>∞</sub> finite-time boundedness of switched linear systems with time-varying delay and exogenous disturbances are addressed. Based on average dwell time (ADT) and free-weight matrix technologies, sufficient conditions which can ensure finite-time boundedness and H<sub>∞</sub> finite-time boundedness are given. And then in virtue of the results on finite-time boundedness, the state memory feedback controller is designed to H<sub>∞</sub> finite-time stabilize a time-delay switched system. These conditions are given in terms of LMIs and are delay-dependent. An example is given to illustrate the efficiency of the proposed method.
J. Liu, X. Z. Liu and W. C. Xie, “Delay-Dependent Robust Control for Uncertain Switched Systems with Time-Delay,” Nonlinear Analysis: Hybrid Systems, Vol. 2, No. 1, 2008, pp. 81-95. doi:10.1016/j.nahs.2007.04.001
X. M. Sun, G. M. Dimirovski, J. Zhao and W. Wang, “Exponential Stability for Switched Delay Systems Based on Average Dwell Time Technique and Lyapunov Function Method,” Proceedings of American Control Conference, Minneapolis, 14-16 June 2006, pp. 1539-1543.
L. Hetel, J. Daafouz and C. Iung, “Stability Analysis for Discrete Time Switched Systems with Temporary Uncertain Switching Signal,” Proceedings of 46th IEEE Conference on Decision and Control, New Orleans, 12-14 December 2007, pp. 5623-5628.
X. M. Sun, J. Zhao and D. J. Hill, “Stability and L2-Gain Analysis for Switched Delay Systems: A Delay-Dependent Method,” Automatica, Vol. 42, No. 10, 2006, pp.1769-1774. doi:10.1016/j.automatica.2006.05.007
Y. G. Sun, L. Wang and G. Xie, “Exponential Stability of Switched Systems with Interval Time-Varying Delay,” IET Control Theory & Applications, Vol. 8, No. 3, 2009, pp. 1033-1040.
L. V. Hien, Q. P. Ha and V. N. Phat, “Stability and Stabilization of Switched Linear Dynamic Systems with Time Delay and Uncertainties,” Applied Mathematics and Computation, Vol. 210, No. 1, 2009, pp. 223-231. doi:10.1016/j.amc.2008.12.082
L. Vu and K. A. Morgansen, “Stability of Time-Delay Feedback Switched Linear Systems,” IEEE Transactions on Automatic Control, Vol. 10, No. 55, 2010, pp.2358-2390.
X. M. Sun, W. Wang, G. P. Liu and J. Zhao, “Stability Analysis for Linear Switched Systems with Time-Varying Delay,” IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, Vol. 38, No. 2, 2008, pp. 528-533. doi:10.1109/TSMCB.2007.912078
I. Malloci, J. Daafouz and C. Iung, “Stability and Stabilization of Two Time Scale Switched Systems in Discrete Time,” IEEE Transactions on Automatic Control, Vol. 55, No. 6, 2010, pp. 1434-1438. doi:10.1109/TAC.2010.2044277
G. Feng, “Stability Analysis of Piecewise Discrete-Time Linear Systems,” IEEE Transactions on Automatic Control, Vol. 47, No. 7, 2002, pp. 1108-1112. doi:10.1109/TAC.2002.800666
D. Liberzon and A.S. Morse, “Basic Problems in Stability and Design of Switched Systems,” IEEE Control Systems Magazine, Vol. 19, No. 5, 1999, pp. 59-70. doi:10.1109/37.793443
Z. D. Sun and S. S. Ge, “Analysis and Synthesis of Switched Linear Control Systems,” Automatica, Vol. 41, No. 2, 2005, pp. 181-195. doi:10.1016/j.automatica.2004.09.015
H. Lin and P. J. Antsaklis, “Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results,” IEEE Transactions on Automatic Control, Vol. 54, No. 2, 2009, pp. 308-322. doi:10.1109/TAC.2008.2012009
M.S. Branicky, “Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid Systems,” IEEE Transactions on Automatic Control, Vol. 43, No. 4, 1998, pp. 475-482. doi:10.1109/9.664150
K. S. Narendra and J. A. Balakrishnan, “Common Lyapunov Function for Stable LTI Systems with Commuting A-Matrices,” IEEE Transactions on Automatic Control, Vol. 39, No. 12, 1994, pp. 2469-2471. doi:10.1109/9.362846
J. H. Wang, D. Z. Cheng and X. M. Hu, “An Extension of LaSalle’s Invariance Principle for a Class of Switched Linear Systems,” Systems & Control Letters, Vol. 58, No. 10-11, 2009, pp. 754-758. doi:10.1016/j.sysconle.2009.08.008
H. Ye, A. N. Michel and L. Hou, “Stability Theory for Hybrid Dynamic Systems,” IEEE Transactions on Automatic Control, Vol. 43, No. 4, 1998, pp. 461-474. doi:10.1016/j.sysconle.2009.08.008
L. Zhang and P. Shi, “Stability, L2 Gain and Asynchronous Control of Discrete-Time Switched Systems with Average Dwell Time,” IEEE Transactions on Automatic Control, Vol. 54, No. 9, 2009, pp. 2193-2200.
L. Wu, T. Qi and Z. Feng, “Average Dwell Time Approach to L2 –L∞ Control of Switched Delay Systems via Dynamic Output Feedback,” IET Control Theory & Applications, Vol. 10, No. 3, 2009, pp. 1425-1436.
A. S. Morse, “Supervisory Control of Families of Linear Set-Point Controllers, Part 1: Exact Matching,” IEEE Transactions on Automatic Control, Vol. 41, No. 10, 1996, pp. 1413-1431. doi:10.1109/9.539424
Q. K. Li, J. Zhao and G. M. Dimirovski, “Tracking Control for Switched Time-Varying Delays Systems with Stabilizable and Unstabilizable Subsystems,” Nonlinear Analysis: Hybrid Systems, Vol. 3, No. 2, 2009, pp. 133-142. doi:10.1016/j.nahs.2008.11.004
J. Liu, X. Z. Liu and W. C. Xie, “Delay-Dependent Robust Control for Uncertain Switched Systems with Time-Delay,” Nonlinear Analysis: Hybrid Systems, Vol. 2, No.1, 2008, pp. 81-95. doi:10.1016/j.nahs.2007.04.001
X. M. Zhang, M. Wu and J. H. She, “Delay-Dependent Stabilization of Linear Systems with Time-Varying State and Input Delays,” Automatica, Vol. 41, No. 8, 2005, pp. 1405-141. doi:10.1016/j.automatica.2005.03.009
Y. P. Zhang, S. R. Kang and D. Loguinov, “Delay-Independent Stability and Performance of Distributed Congestion Control,” IEEE/ACM Translations on Networking, Vol. 15, No. 5, 2007, pp. 838-851.
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,” Proceedings of 22nd IEEE International Symposium on Intelligent Control, Singapore, 12 October 2007, pp. 208-213. doi:10.1109/ISIC.2007.4450886
S. P. Ma, C. H. Zhang and Z. Wu, “Delay-Dependent Stability and H∞ Control for Uncertain Discrete Switched Singular Systems with Time-Delay,” Applied Mathematics and Computation, Vol. 206, No. 1, 2008, pp. 413-424. doi:10.1016/j.amc.2008.09.020
X. Z. Lin, H. B. Du and S. H. Li, “Finite-Time Boundedness and L2-Gain Analysis for Switched Delay Systems with Norm-Bounded Disturbance,” Applied Mathematics and Computation, Vol. 217, No. 12, 2011, pp. 5982-5993. doi:10.1016/j.amc.2010.12.032
L. Weiss and E. F. Infante, “Finite Time Stability under Perturbing Forces and on Product Spaces,” IEEE Transactions on Automatic Control, Vol. 12, No. 1, 1967, pp. 54-59. doi:10.1109/TAC.1967.1098483
A. N. Michel and S. H. Wu, “Stability of Discrete Systems over a Finite Interval of Time,” International Journal of Control, Vol. 9, No. 6, 1969, pp. 679-693. doi:10.1080/00207176908905789
P. Dorato, “Short Time Stability in Linear Time-Varying Systems,” Proceedings of IRE International Convention Record, New York, 9 May 1961, pp. 83-87.
F. Amato and M. Ariola, “Finite-Time Control of Discrete-Time Linear Systems,” IEEE Transactions on Automatic Control, Vol. 50, No.5, 2005, pp. 724-729. doi:10.1109/TAC.2005.847042
G. Garcia, S. Tarbouriech and J. Bernussou, “Finite-Time Stabilization of Linear Time-Varying Continuous Systems,” IEEE Transactions on Automatic Control, Vol. 54, No. 2, 2009, pp. 364-369. doi:10.1109/TAC.2008.2008325
W. M. Xiang and J. Xiao, “H∞ Finite-Time Control for Switched Nonlinear Discrete-Time Systems with Norm-Bounded Disturbance,” Journal of the Franklin Institute, Vol. 348, No. 2, 2011, pp. 331-352. doi:10.1016/j.jfranklin.2010.12.001
H. B. Du, X. Z. Lin and S. H. Li, “Finite-Time Stability and Stabilization of Switched Linear Systems,” Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, 15-18 December 2009, pp. 1938-1943.
S. Boyd, L. E. Ghaoui and E. Feron, “Linear matrix inequalities in systems and control theory,” SIAM, Philadelphia, 1994.
J. P. Hespanha and A. S. Morse, “Stability of Switched Systems with Average Dwell-Time,” Proceedings of IEEE Conference on Decision and Control, Phoenix, December 1999, pp. 2655-2660.
K. Gu, “A Further Refinement of Discretized Lyapunov Functional Method for the Stability of Time-Delay Systems,” International Journal of Control, Vol. 74, No. 10, 2001, pp. 967-976. doi:10.1080/00207170110047190
M. Wu and Y. He, “Robust Control of Delay Systems-Free Weight Matrix Method,” Science Press, Beijing, 2008.