ABSTRACT

Finite-time boundedness and H_{∞} 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_{∞} 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_{∞} 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.

Finite-time boundedness and H

Cite this paper

nullH. Liu and Y. Shen, "*H*_{∞} Finite-Time Control for Switched Linear Systems with Time-Varying Delay," *Intelligent Control and Automation*, Vol. 2 No. 3, 2011, pp. 203-213. doi: 10.4236/ica.2011.23025.

nullH. Liu and Y. Shen, "

References

[1] 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

[2] 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.

[3] 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.

[4] 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

[5] 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.

[6] 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

[7] 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.

[8] 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

[9] 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

[10] 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

[11] 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

[12] 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

[13] 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

[14] 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

[15] 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

[16] 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

[17] 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

[18] 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.

[19] 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.

[20] 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

[21] 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

[22] 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

[23] 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

[24] 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.

[25] 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

[26] 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

[27] 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

[28] 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

[29] 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

[30] P. Dorato, “Short Time Stability in Linear Time-Varying Systems,” Proceedings of IRE International Convention Record, New York, 9 May 1961, pp. 83-87.

[31] 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

[32] 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

[33] 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

[34] 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.

[35] S. Boyd, L. E. Ghaoui and E. Feron, “Linear matrix inequalities in systems and control theory,” SIAM, Philadelphia, 1994.

[36] 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.

[37] 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

[38] M. Wu and Y. He, “Robust Control of Delay Systems-Free Weight Matrix Method,” Science Press, Beijing, 2008.

[1] 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

[2] 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.

[3] 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.

[4] 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

[5] 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.

[6] 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

[7] 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.

[8] 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

[9] 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

[10] 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

[11] 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

[12] 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

[13] 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

[14] 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

[15] 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

[16] 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

[17] 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

[18] 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.

[19] 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.

[20] 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

[21] 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

[22] 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

[23] 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

[24] 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.

[25] 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

[26] 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

[27] 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

[28] 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

[29] 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

[30] P. Dorato, “Short Time Stability in Linear Time-Varying Systems,” Proceedings of IRE International Convention Record, New York, 9 May 1961, pp. 83-87.

[31] 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

[32] 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

[33] 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

[34] 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.

[35] S. Boyd, L. E. Ghaoui and E. Feron, “Linear matrix inequalities in systems and control theory,” SIAM, Philadelphia, 1994.

[36] 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.

[37] 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

[38] M. Wu and Y. He, “Robust Control of Delay Systems-Free Weight Matrix Method,” Science Press, Beijing, 2008.