Stability Analysis of Linear Switched Systems with Mixed Delays
This paper addresses the stability of the switched systems with discrete and distributed time delays. By applying Lyapunov functional and function method, we show that, if the norm of system matrices Bi is small enough, the asymptotic stability is always achieved. Finally, a example is provided to verify technically feasibility and operability of the developed results.
[1] A. Morse, "Control using logic-based switching," Lecture Notes in
Control and Information Science, vol. 222, London: Springer, 1997.
[2] A. Van Der Schaft and H. Schumacher, "An introduction to hybrid
dynamical systems," Lecture Notes in Control and Information Science,
vol. 251, London: Springer, 2000.
[3] M. S. Branicky, "Multiple Lyapunov functions and other analysis tools for
switched and hybrid systems," IEEE Transaction Automatomatic Control,
vol. 43, pp. 475-482, 1998.
[4] W. P. Dayawansa, and C. F. Martin, "A converse Lyapunov theorem for a
class of dynamical systems which undergo switching," IEEE Transaction
Automatomatic Control, vol. 44, pp. 751-760, 1999.
[5] J. Daafouz, P. Riedinger, and C. Lung, "Stability analysis and control
synthesis for switched systems: a switched Lyapunov function approach,"
IEEE Transaction Automatomatic Control, vol. 47, pp. 1883-1887, 2002.
[6] E. Feron, "Quadratic stabilizability of switched systems via state and
output feedback," Center for Intelligent Control Systems, Massachusetts
Institute of Technology, Cambridge, MA, Technical Report CICS, pp. 468,
1996.
[7] Z. G. Li, C. Y. Wen, and Y. C. Soh, "Stabilization of a class of switched
systems via designing switching laws," IEEE Transaction Automatomatic
Control, vol. 46, pp. 665-670, 2001.
[8] Z. G. Li, C. Y. Wen, and Y. C. Soh, "Observer-based stabilization of
switching linear systems," Automatica, vol. 39, pp. 517-524, 2003.
[9] D. Liberzon, J. P. Hespanha, and A. S. Morse, "Stability of switched
systems: a Lie-algebraic condition," Systems & Control Letters, vol. 37,
pp. 117-122, 1999.
[10] D. Liberzon and A. S. Morse, "Basic problems in stability and design of
switched systems," IEEE Control Systems Magazine, vol. 19, pp. 59-70,
1999.
[11] A. S. Morse, Control Using Logic-Based Switching. London: Springer,
1997.
[12] Y. G. Sun, L. Wang, and G. Xie, "Delay-dependent robust stability
and stabilization for discrete-time switched systems with modedependent
time-varying delays," Applied Mathematics and Computation, vol. 180,
pp. 428-435, 2006.
[13] M. A. Wicks, P. Peleties, and R. A. DeCarlo, "Construction of piecewise
Lyapunov functions for stabilizing switched systems," in Proceedings of
IEEE Conference on Decision and Control, Lake Buena Vista, December
1994, pp. 3492-3497.
[14] M. A. Wicks, P. Peleties, and R. A. DeCarlo, "Switched controller
synthesis for the quadratic stabilization of a pair of unstable linear
systems," European Journal of Control, vol. 4, pp. 140-147, 1998.
[15] Y. Wang, L. Xie, and C. E. De Souza, "Robust control of a class of
uncertain nonlinear systems," Systems & Control Letters, vol. 19, pp.
139-149, 1972.
[16] D. Xie, L. Wang, F. Hao, and G. Xie, "LMI approach to gain analysis
and control synthesis of uncertain switched systems," IEE Proceedings-
Control Theory and Applications, vol. 151, pp. 21-28, 2004.
[17] G. Xie and L. Wang, "Controllability and stabilizability of switched
linear systems," Systems & Control Letters, vol. 48, pp, 135-155, 2003.
[18] G. Xie, D. Zheng, and L. Wang, "Controllability of switched linear
systems," IEEE Transaction Automatomatic Control, vol. 47, pp. 1401-
1405, 2002.
[19] G. Zhai, B. Hu, K. Yasuda, and A. N. Michei, "Disturbance attenuation
properties of time- controlled switched systems," Journal of Franklin
Institute, vol. 338, pp. 765-779, 2001.
[20] P. Peleties, M. Wicks, and R. DeCarlo, "Switched controller synthesis
for quadratic stabilization of a pair of unstable linear systems," European
Journal of Control, val. 4, pp. 140-147, 1998.
[21] M. S. Branicky, "Stability of hybrid systems: State of the art," in:
Proceeding of the 36th Conference on Decision & Control, San Diego,
CA, USA, 1997.
[22] M. S. Branicky, "Multiple Lyapunov functions and other analysis tools
for switched and hybrid systems," IEEE Transaction Automatic Control,
vol. 43, no. 4, pp. 475-482, 1998.
[23] M. A. Wicks and P. Peleties, "Construction of piecewise Lyapunov
functions for stabilizing switched systems," in: Proceeding of the 33rd
Conference on Decision & Control, Lake Buena, Vista, FL, Dec. 1994,
pp. 3492-3497.
[24] A. V. D. Schaft and H. Schumacher, An Introduction to Hybrid Dynamical
Systems. London Limited: Springer-Verlag, 2000.
[25] J. P. Hespanha and A. S. Morse, "Stability of switched systems with
average dwell-time," in: Proceeding of the 38th Conference on Decision
& Control, Phoenix, Arizona, USA, Dec. 1999, vol. 88, pp. 2660-2655.
[26] D. Liberzon and A. S. Morse, "Basic problems in stability and design
of switchedsystems," IEEE Control Systems Magazine, Oct. 1999.
[27] G. Zhai, B. Hu, K. Yasuda, and A. N. Michael, "Stability analysis of
switched systems with stable and unstable subsystems: An average dwell
time approach," in: Proceeding of the American Control Conference,
Chicago, Illinois, USA, Jun. 2000, pp. 200-204.
[28] X. Li, C. E. Souza, "Criteria for robust stability and stabilization of
uncertain linear system with state delay," Automatica, vol. 33, pp. 1657-
1662, 1997.
[29] H. Kokame, H. Kobayashi, and T. Mori, "Robust performance for
linear delay-differential systems with time-varying uncertainties," IEEE
Transactions on Automatic Control, vol. 43, pp. 223-226, 1998.
[30] J. K. Hale, S. M. Verduyn Lunel, Introduction to Functional Differential
Equations. New York: Springer-Verlag, 1993.
[31] S. j. Kim, S. A. Campbell, and X. Z. Liu, "Delay independent stability
of linear switching systems with time delay," Journal of Mathematical
Analysis and Applications, vol. 339, pp. 785-801, 2008.
[32] J. Hale, S. V. Lunel, Introduction to Functional Differential Equations.
New York: Springer-Verlag, 1993.
[1] A. Morse, "Control using logic-based switching," Lecture Notes in
Control and Information Science, vol. 222, London: Springer, 1997.
[2] A. Van Der Schaft and H. Schumacher, "An introduction to hybrid
dynamical systems," Lecture Notes in Control and Information Science,
vol. 251, London: Springer, 2000.
[3] M. S. Branicky, "Multiple Lyapunov functions and other analysis tools for
switched and hybrid systems," IEEE Transaction Automatomatic Control,
vol. 43, pp. 475-482, 1998.
[4] W. P. Dayawansa, and C. F. Martin, "A converse Lyapunov theorem for a
class of dynamical systems which undergo switching," IEEE Transaction
Automatomatic Control, vol. 44, pp. 751-760, 1999.
[5] J. Daafouz, P. Riedinger, and C. Lung, "Stability analysis and control
synthesis for switched systems: a switched Lyapunov function approach,"
IEEE Transaction Automatomatic Control, vol. 47, pp. 1883-1887, 2002.
[6] E. Feron, "Quadratic stabilizability of switched systems via state and
output feedback," Center for Intelligent Control Systems, Massachusetts
Institute of Technology, Cambridge, MA, Technical Report CICS, pp. 468,
1996.
[7] Z. G. Li, C. Y. Wen, and Y. C. Soh, "Stabilization of a class of switched
systems via designing switching laws," IEEE Transaction Automatomatic
Control, vol. 46, pp. 665-670, 2001.
[8] Z. G. Li, C. Y. Wen, and Y. C. Soh, "Observer-based stabilization of
switching linear systems," Automatica, vol. 39, pp. 517-524, 2003.
[9] D. Liberzon, J. P. Hespanha, and A. S. Morse, "Stability of switched
systems: a Lie-algebraic condition," Systems & Control Letters, vol. 37,
pp. 117-122, 1999.
[10] D. Liberzon and A. S. Morse, "Basic problems in stability and design of
switched systems," IEEE Control Systems Magazine, vol. 19, pp. 59-70,
1999.
[11] A. S. Morse, Control Using Logic-Based Switching. London: Springer,
1997.
[12] Y. G. Sun, L. Wang, and G. Xie, "Delay-dependent robust stability
and stabilization for discrete-time switched systems with modedependent
time-varying delays," Applied Mathematics and Computation, vol. 180,
pp. 428-435, 2006.
[13] M. A. Wicks, P. Peleties, and R. A. DeCarlo, "Construction of piecewise
Lyapunov functions for stabilizing switched systems," in Proceedings of
IEEE Conference on Decision and Control, Lake Buena Vista, December
1994, pp. 3492-3497.
[14] M. A. Wicks, P. Peleties, and R. A. DeCarlo, "Switched controller
synthesis for the quadratic stabilization of a pair of unstable linear
systems," European Journal of Control, vol. 4, pp. 140-147, 1998.
[15] Y. Wang, L. Xie, and C. E. De Souza, "Robust control of a class of
uncertain nonlinear systems," Systems & Control Letters, vol. 19, pp.
139-149, 1972.
[16] D. Xie, L. Wang, F. Hao, and G. Xie, "LMI approach to gain analysis
and control synthesis of uncertain switched systems," IEE Proceedings-
Control Theory and Applications, vol. 151, pp. 21-28, 2004.
[17] G. Xie and L. Wang, "Controllability and stabilizability of switched
linear systems," Systems & Control Letters, vol. 48, pp, 135-155, 2003.
[18] G. Xie, D. Zheng, and L. Wang, "Controllability of switched linear
systems," IEEE Transaction Automatomatic Control, vol. 47, pp. 1401-
1405, 2002.
[19] G. Zhai, B. Hu, K. Yasuda, and A. N. Michei, "Disturbance attenuation
properties of time- controlled switched systems," Journal of Franklin
Institute, vol. 338, pp. 765-779, 2001.
[20] P. Peleties, M. Wicks, and R. DeCarlo, "Switched controller synthesis
for quadratic stabilization of a pair of unstable linear systems," European
Journal of Control, val. 4, pp. 140-147, 1998.
[21] M. S. Branicky, "Stability of hybrid systems: State of the art," in:
Proceeding of the 36th Conference on Decision & Control, San Diego,
CA, USA, 1997.
[22] M. S. Branicky, "Multiple Lyapunov functions and other analysis tools
for switched and hybrid systems," IEEE Transaction Automatic Control,
vol. 43, no. 4, pp. 475-482, 1998.
[23] M. A. Wicks and P. Peleties, "Construction of piecewise Lyapunov
functions for stabilizing switched systems," in: Proceeding of the 33rd
Conference on Decision & Control, Lake Buena, Vista, FL, Dec. 1994,
pp. 3492-3497.
[24] A. V. D. Schaft and H. Schumacher, An Introduction to Hybrid Dynamical
Systems. London Limited: Springer-Verlag, 2000.
[25] J. P. Hespanha and A. S. Morse, "Stability of switched systems with
average dwell-time," in: Proceeding of the 38th Conference on Decision
& Control, Phoenix, Arizona, USA, Dec. 1999, vol. 88, pp. 2660-2655.
[26] D. Liberzon and A. S. Morse, "Basic problems in stability and design
of switchedsystems," IEEE Control Systems Magazine, Oct. 1999.
[27] G. Zhai, B. Hu, K. Yasuda, and A. N. Michael, "Stability analysis of
switched systems with stable and unstable subsystems: An average dwell
time approach," in: Proceeding of the American Control Conference,
Chicago, Illinois, USA, Jun. 2000, pp. 200-204.
[28] X. Li, C. E. Souza, "Criteria for robust stability and stabilization of
uncertain linear system with state delay," Automatica, vol. 33, pp. 1657-
1662, 1997.
[29] H. Kokame, H. Kobayashi, and T. Mori, "Robust performance for
linear delay-differential systems with time-varying uncertainties," IEEE
Transactions on Automatic Control, vol. 43, pp. 223-226, 1998.
[30] J. K. Hale, S. M. Verduyn Lunel, Introduction to Functional Differential
Equations. New York: Springer-Verlag, 1993.
[31] S. j. Kim, S. A. Campbell, and X. Z. Liu, "Delay independent stability
of linear switching systems with time delay," Journal of Mathematical
Analysis and Applications, vol. 339, pp. 785-801, 2008.
[32] J. Hale, S. V. Lunel, Introduction to Functional Differential Equations.
New York: Springer-Verlag, 1993.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:49489", author = "Xiuyong Ding and Lan Shu", title = "Stability Analysis of Linear Switched Systems with Mixed Delays", abstract = "This paper addresses the stability of the switched systems with discrete and distributed time delays. By applying Lyapunov functional and function method, we show that, if the norm of system matrices Bi is small enough, the asymptotic stability is always achieved. Finally, a example is provided to verify technically feasibility and operability of the developed results.
", keywords = "Switched system, stability, Lyapunov function, Lyapunov functional, delays.", volume = "4", number = "8", pages = "1061-7", }
