Exponential stability of switched stochastic delay systems with non-linear uncertainties

International Journal of Systems Science

Volumn 40, Issue 6, 2009, Pages 637-648

  Liu, Jun ^a   Liu, Xinzhi ^a   Xie, Wei Chau ^a  

^a UNIVERSITY OF WATERLOO   (Canada)

Author keywords

Average dwell time; Exponential stability; Linear matrix inequality; Non linear uncertainties; Stochastic systems; Switched systems; Time delay system

Indexed keywords

AVERAGE DWELL-TIME; EXPONENTIAL STABILITY; NON-LINEAR UNCERTAINTIES; SWITCHED SYSTEMS; TIME-DELAY SYSTEM;

DELAY CONTROL SYSTEMS; FUZZY CONTROL; LINEAR CONTROL SYSTEMS; MULTIVARIABLE CONTROL SYSTEMS; ROBUSTNESS (CONTROL SYSTEMS); STABILITY CRITERIA; STOCHASTIC SYSTEMS; SWITCHING SYSTEMS; SYSTEM STABILITY; TIME DELAY; UNCERTAIN SYSTEMS;

LINEAR MATRIX INEQUALITIES;

EID: 67651227092     PISSN: 00207721     EISSN: 14645319     Source Type: Journal    
DOI: 10.1080/00207720902755770     Document Type: Article

References (55)

1. Arnold, L. (1974), Stochastic Differential Equations: Theory and Applications, London, New York, Sydney: John Wiley and Sons.
2. Arnold, L. (1984), 'A Formula Connecting Sample and Moment Stability of Linear Stochastic Systems', SIAM Journal on Applied Mathematics, 44, 793-802.
3. Arnold, L., Oeljeklaus, E., and Pardoux, E. (1984), Almost Sure and Moment Stability for Linear Itô Equations, Lecture Notes in Mathematics (Vol. 1186), Berlin: Springer-Verlag, 129-159.
4. Boyd, S., Ghaoui, L.E., Feron, E., and Balakrishnan, V. (1994), Linear Matrix Inequalities in System and Control Theory, Philadephia: SIAM.
5. Cao, Y.Y., Sun, Y.X., and Cheng, C. (1998), 'Delay-dependent Robust Stabilisation of Uncertain Systems with Multiple State Delays', IEEE Transactions on Automatic Control, 43, 1608-1612.
6. Chen, W.-H., Guan, Z.H., and Lu, X. (2005), 'Delay-dependent Exponential Stability of Uncertain Stochastic Systems with Multiple Delays: An LMI Approach', Systems and Control Letters, 54, 547-555.
7. Chen, Y., Xue, A., Zhou, S., and Lu, R. (2008), 'Delay-dependent Robust Control for Uncertain Stochastic Time-delay Systems', Circuits, Systems, and Signal Processing, 47, 447-460.
8. Chen, Y., and Xue, A. (2008), 'Improved Stability Criterion for Uncertain Stochastic Delay Systems with Nonlinear Uncertainties', Electronics Letters, 44, 458-459.
9. Cheng, D. (2004), 'Stabilization of Planar Switched Systems', Systems and Control Letters, 51, 79-88.
10. Daafouz, J., Riedinger, P., and Iung, C. (2002), 'Stability Analysis and Control Synthesis for Switched Systems: a Switched Lyapunov Function Approach', IEEE Transactions on Automatic Control, 47, 1883-1887.
11. Fridman, E. (2001), 'New Lyapunov-Krasovskii Functionals for Stability of Linear Retarded and Neutral Type Systems', Systems and Control Letters, 43, 309-319.
12. Fridman, E. (2002), 'An Improved Stabilisation Method for Linear Time-delay Systems', IEEE Transactions on Automatic Control, 47, 1931-1937.
13. Fridman, E. (2003), 'Delay-dependent Stability and H∞ Control: Constant and Time-varying Delays', International Journal of Control, 76, 48-60.
14. Gahinet, P., Nemirovski, A., Laub, A., and Chilali, M. (1995), LMI Control Toolbox User's Guide, Natick, MA: The Mathworks, Inc.
15. Gu, K. (2000), 'An Integral Inequality in the Stability Problem of Time-delay Systems', in Proceeding of the IEEE Conference on Decision Control, Sydney, Australia, 2805-2810.
16. Hale, J., and Lunel, S.M.V. (1993), Introduction to Functional Differential Equations, New York: Springer-Verlag.
17. Han, Q.L., Gu, K., and Yu, X. (2003), 'An Improved Estimate of Robust Stability Bound of Time-delay Systems with Norm-bounded Uncertainty', IEEE Transactions on Automat. Control, 48, 1629-1634.
18. Has'minskii, R.Z. (1980). Stochastic Stability of Differential Equations, (translated from the Russian by D. Louvish), Monographs and Text books on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Alphen, The Netherlands: Sijthoff and Noordhoff.
19. He, Y., Wu, M., She, J.H., and Liu, G.P. (2004), 'Delay-dependent Robust Stability Criteria for Uncertain Neutral Systems with Mixed Delays', System and Control Letters, 51, 57-65.
20. Hespanha, J.P., and Morse, A.S. (1999), 'Stability of Switched Systems with Average Dwell Time', in Proceedings of the 38th IEEE Conference on Decision and Control, 3, 2655-2660.
21. Hespanha, J., Liberzon, D., Angeli, D., and Sontag, E.D. (2005), 'Nonlinear Norm-observability Notions and Stability of Switched Systems', IEEE Transactions on Automatic Control, 50, 154-168.
22. Hu, B., Xu, X., Antsaklis, P.J., and Michel, A.N. (1999), 'Robust Stabilising Control Laws for a Class of Second-order Switched Systems', Systems and Control Letters, 38, 197-207.
23. Kwon, O.M., and Park, J.H. (2004), 'On Improved Delay-dependent Robust Control for Uncertain Time-delay Systems', IEEE Transactions on Automatic Control, 49, 1991-1995.
24. Kolmanovskii, V.B., and Nosov, V.B. (1986), Stability of Functional Differential Equations, London: Academic Press.
25. Kolmanovskii, V.B., and Richard, J.-P. (1999), 'Stability of Some Linear Systems with Delays', IEEE Transactions on Automatic Control, 44, 984-989.
26. Kushner, H.J. (1967), Stochastic Stability and Control, New York/London: Academic Press.
27. Li, X., and De Souza, C.E. (1997a), 'Criteria for Robust Stability and Stabilisation of Uncertain Linear System with State Delay', Automatica, 33, 1657-1662.
28. Li, X., and De Souza, C.E. (1997b), 'Delay-dependent Robust Stability and Stabilisation of Uncertain Linear Delay Systems: A Linear Matrix Approach', IEEE Transactions on Automatic Control, 40, 1144-1148.
29. Li, X., and De Souza, C.E. (1999), 'Delay-dependent Robust H∞ Control of Uncertain Linear State-delayed Systems', Automatica, 35, 1313-1321.
30. Li, Z.G., Wen, C.Y., and Soh, Y. (2001), 'Stabilization of a Class of Switched Systems via Designing Switching Laws', IEEE Transactions on Automatic Control, 46, 665-670.
31. Liberzon, D. (2003), Switching in Systems and Control, Boston, MA: Birkhäuser.
32. Liu, J., Liu, X., and Xie, W.-C. (2008), 'Delay-dependent Robust Control for Uncertain Switched Systems with Time-delay', Nonlinear Analysis Hybrid Systems, 2, 81-95.
33. Mancilla-Aguilar, J.L., and Garcila, R.A. (2000), 'A Converse Lyapunov Theorem for Nonlinear Switched Systems', Systems and Control Letters, 41, 67-71.
34. Matveev, A., and Savkin, A. (2000), Qualitative Theory of Hybrid Dynamical Systems, Boston, MA: Birkhäuser.
35. Mao, X. (1996), 'Robustness of Exponential Stability of Stochastic Differential Delay Equations', IEEE Transactions on Automatic Control, 41, 442-447.
36. Mao, X. (1997), Stochastic Differential Equations and their Applications, Chichester: Horwood Publications.
37. Mao, X., Koroleva, N., and Rodkina, A. (1998), 'Robust Stability of Uncertain Stochastic Differential Delay Equations', Systems and Control Letters, 35, 325-336.
38. Millerioux, G., and Daafouz, J. (2004), 'Input Independent Chaos Synchronization of Switched Systems', IEEE Transactions on Automatic Control, 49, 1182-1187.
39. Moon, Y.S., Park, P., Kwon, W.H., and Lee, Y.S. (2001), 'Delay-dependent Robust Stabilisation of Uncertain State-delayed Systems', International Journal of Control, 74, 1447-1455.
40. Niculescu, S.-I. (2001), 'On Delay Dependent Stability under Model Trasformaion of Some Neutral Linear Systems', International Journal of Control, 74, 609-617.
41. Orlov, Y. (2003), 'Extended Invariance Principle for Nonautonomous Switched Systems', IEEE Transactions on Automatic Control, 48, 1448-1452.
42. Orlov, Y. (2004), 'Finite Time Stability and Robust Control Synthesis of Uncertain Switched Systems', SIAM Journal of Control Optimization, 43, 1253-1271.
43. Park, P. (1999), 'A Delay-dependent Stability Criterion for Systems with Uncertain Time-invariant Delays', IEEE Transactions on Automatic Control, 44, 876-877.
44. Park, P., Moon, Y.S., and Kwon, W.H. (1998), 'A Delay-dependent Robust Stability Criterion for Uncertain Time-delay Systems', in Proceedings of the American Control Conference, Philadephia, USA, 1963-1964.
45. van der Schaft, A., and Schumacher, H. (2000), An Introduction to Hybrid Dynamical Systems, Lecture Notes in Control and Information Sciences 251, London: Springer-Verlag.
46. 2-gain Analysis for Switched Delay Systems: A Delay-dependent Method', Automatica, 42, 1769-1774.
47. Sun, X.-M., Dimirovski, G.M., Zhao, J., and Wang, W. (2006), 'Exponential Stability for Switched Delay Systems Based on Average Dwell Time Technique and Lyapunov Function Method', in Proceedings of the 2006 American Control Conference, Minnesota, USA, 1539-1543.
48. Verriest, E.I., and Florchinger, P. (1995), 'Stability of Stochastic Systems with Uncertain Time Delays', Systems and Control Letters, 24, 41-47.
49. Xu, H., Liu, X., and Teo, K.L. (2005), 'Robust H∞ Stabilisation with Definite Attenuance of an Uncertain Impulsive Switched System', ANZIAM Journal, 46, 471-484.
50. Xu, S., and Chen, T. (2002), 'Robust H∞ Control for Uncertain Stochastic Systems with State Delay', IEEE Transactions on Automatic Control, 47, 2089-2094.
51. Xu, S., Lam, J., Mao, X., and Zou, Y. (2005), 'A New LMI Condition for Delay-dependent Robust Stability of Stochastic Time-delay Systems', Asian Journal of Control, 7, 419-423.
52. Xu, S., Shi, P., Chu, Y., and Zou, Y. (2006), 'Robust Stochastic Stabilisation and H∞ Control of Uncertain Neutral Stochastic Time-delay Systems', Journal of Mathematical Analysis and Applications, 314, 1-16.
53. Yan, P., and Özbay, H. (2008), 'Stability of Switched Time Delay Systems', SIAM Journal of Control Optimization, 47, 936-949.
54. Yue, D., and Won, S. (2001), 'Delay-dependent Robust Stability of Stochastic Systems with Time Delay and Nonlinear Uncertainties', Electronic Letters, 37, 992-993.
55. Yue, D., and Han, Q.-L. (2005), 'Delay-dependent Exponential Stability of Stochastic Systems with Time-varying Delay, Nonlinearity, and Markovian Switching', IEEE Transactions on Automatic Control, 50, 217-222.