Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple Lyapunov functions

SIAM Journal on Control and Optimization

Volumn 45, Issue 1, 2006, Pages 174-206

  Chatterjee, Debasish ^a   Liberzon, Daniel ^a  

^a United States of America   (United States)

Author keywords

Comparison principle; Multiple Lyapunov functions; Stability analysis; Stochastic stability; Switched systems; Two measures

Indexed keywords

COMPARISON PRINCIPLE; DETERMINISTIC SWITCHED SYSTEMS; MULTIPLE LYAPUNOV FUNCTIONS; STABILITY ANALYSIS; STOCHASTIC STABILITY; STOCHASTIC SWITCHED SYSTEMS;

LYAPUNOV METHODS; PERTURBATION TECHNIQUES; SWITCHING FUNCTIONS; SWITCHING SYSTEMS; SYSTEM STABILITY;

STOCHASTIC CONTROL SYSTEMS;

EID: 33847030396     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/040619429     Document Type: Article

References (46)

1. L. ARNOLD, Stochastic Differential Equations: Theory and Applications, Krieger, Melbourne, FL, 1992.
2. S. BATTILOTTI AND A. D. SANTIS, Dwell time controllers for stochastic systems with switching Markov chain, Automatica, 41 (2005), pp. 923-934.
3. S. P. BHAT AND D. S. BERNSTEIN, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., 38 (2000), pp. 751-766.
4. P. BOLZERN, P. COLANERI, AND G. D. NICOLAO, On almost sure stability of discrete-time Markov jump linear systems, in Proceedings of the 43rd Conference on Decision and Control, 2004, pp. 3204-3208.
5. S. BOYD, L. E. GHAOUI, E. FERON, AND V. BALAKRISHNAN, Linear Matrix Inequalities in System and Control Theory, SIAM Stud. Appl. Math. 15, SIAM, Philadelphia, 1994.
6. M. S. BRANICKY, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automat. Control, 43 (1998), pp. 475-482.
7. R. W. BROCKETT, Lecture Notes on Stochastic Control, Harvard University, Cambridge, MA, 1995.
8. D. CHATTERJEE AND D. LIBERZON, On stability of stochastic switched systems, in Proceedings of 43rd Conference on Decision and Control, vol. 4, 2004, pp. 4125-4127.
9. D. CHATTERJEE, Stability Analysis of Deterministic and Stochastic Switched Systems via a Comparison Principle and Multiple Lyapunov Functions, Master's thesis, University of Illinois, Urbana, IL, 2004.
10. F. H. CLARKE, Optimization and Nonsmooth Analysis, Classics in Appl. Math. 5, 2nd ed., SIAM, Philadelphia, 1990.
11. P. FLORCHINGER, Lyapunov-like techniques for stochastic stability, SIAM J. Control Optim., 33 (1995), pp. 1151-1169.
12. M. K. GHOSH, A. ARAPOSTATHIS, AND S. I. MARCUS, Ergodic control of switching diffusions, SIAM J. Control Optim., 35 (1997), pp. 1952-1988.
13. W. HAHN, Stability of Motion, Springer-Verlag, Berlin, 1967.
14. R. Z. HAŚMINSKII, Stochastic Stability of Differential Equations, Sijthoff & Noordhoff, Groningen, The Netherlands, 1980.
15. J. P. HESPANHA, D. LIBERZON, AND A. R. TEEL, On input-to-state stability of impulsive systems, in Proceedings of the 44th Conference on Decision and Control, 2005, pp. 3992-3997.
16. 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, vol. 3, 1999, pp. 2655-2660.
17. J. P. HESPANHA, A model for stochastic hybrid systems with application to communication networks, Nonlinear Analysis Special Issue on Hybrid Systems, 62 (2005), pp. 1353-1383.
18. J. P. HESPANHA, Uniform stability of switched linear systems: Extensions of LaSalle's invariance principle, IEEE Trans. Automat. Control, 49 (2004), pp. 470-482.
19. L. HOU, A. N. MICHEL, AND H. YE, Stability analysis of switched systems, in Proceedings of the 35th IEEE Conference on Decision and Control, vol. 2, 1996, pp. 1208-1212.
20. J. HU, J. LYGEROS, AND S. SASTRY, Towards a theory of stochastic hybrid systems, Lecture Notes in Comput. Sci. 1790, Springer-Verlag, 2000, pp. 160-173.
21. Y. JI AND H. J. CHIZECK, Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control, IEEE Trans. Automat. Control, 35 (1990), pp. 777-788.
22. H. K. KHALIL, Nonlinear Systems, 3rd ed., Prentice-Hall, Englewood Cliffs, NJ, 2002.
23. M. KRSTIĆ AND H. DENG, Stabilization of Nonlinear Uncertain Systems, Springer-Verlag, Berlin, 1998.
24. P. R. KUMAR AND P. VARAIYA, Stochastic Systems: Estimation, Identification, and Adaptive Control, Prentice-Hall, Englewood Cliffs, 1985.
25. H. J. KUSHNER, Stochastic Stability and Control, Academic Press, New York, 1967.
26. G. S. LADDE AND V. LAKSHMIKANTHAM, Random Differential Inequalities, Academic Press, New York, 1980.
27. V. LAKSHMIKANTHAM, S. LEELA, AND A. A. MARTYNYUK, Practical Stability of Nonlinear Systems, World Scientific, Singapore, 1990.
28. V. LAKSHMIKANTHAM AND S. LEELA, Differential and Integral Inequalities: Theory and Application, Vol. 1, Academic Press, New York, 1969.
29. V. LAKSHMIKANTHAM AND X. LIU, Stability Analysis in Terms of Two Measures, World Scientific, Singapore, 1995.
30. D. LIBERZON, Switching in Systems and Control, Birkhäuser, Boston, 2003.
31. Y. LIN, E. D. SONTAG, AND Y. WANG, A smooth converse Lyapunov theorem for robust stability, SIAM J. Control Optim., 34 (1996), pp. 124-160.
32. V. M. MATROSOV, Vector Lyapunov functions method in the analysis of dynamical properties of nonlinear differential equations, Trends in Theory and Practice of Nonlinear Differential Equations, Lecture Notes in Pure and Appl. Math. 90, Marcel Dekker, New York, 1984, pp. 357-374.
33. Y. N. MERENKOV, Stability-like properties of stochastic differential equations, Differential Equations, 39 (2003), pp. 1703-1712.
34. S. P. MEYN AND R. L. TWEEDIE, Markov Chains and Stochastic Stability, Springer-Verlag, Berlin, 1993.
35. A. A. MOVCHAN, Stability of processes with respect to two metrics, Prikl. Mat. Mekh., 24 (1960), pp. 988-1001.
36. B. K. ØKSENDAL, Stochastic Differential Equations, 5th ed., Springer-Verlag, Berlin, 1998.
37. P. PELETIES AND R. A. DECARLO, Asymptotic stability of m-switched systems using Lyapunovlike functions, in Proceedings of the American Control Conference, 1991, pp. 1679-1684.
38. S. PETTERSSON AND B. LENNARTSON, Stability and robustness for hybrid systems, in Proceedings of the 35th IEEE Conference on Decision and Control, vol. 2, 1996, pp. 1202-1207.
39. M. PRANDINI, J. P. HESPANHA, AND M. CAMPI, Hysteresis-based switching control of stochastic linear systems, in Proceedings of the 2003 European Control Conference, 2003.
40. S. SATHANANTHAN AND L. H. KEEL, Optimal practical stabilization and controllability of systems with Markovian jumps, Nonlinear Anal., 54 (2003), pp. 1011-1027.
41. E. D. SONTAG, Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd ed., Springer-Verlag, Berlin, 1998.
42. J. SUN, Y. ZHANG, AND Q. WU, Less conservative conditions for asymptotic stability of implusive control systems, IEEE Trans. Automat. Control, 48 (2003), pp. 829-831.
43. A. D. TEEL AND L. PRALY, A smooth Lyapunov function from a class-KL, estimate involving two positive semidefinite functions, ESAIM. Control Optim. Calculus Variations, 5 (2000), pp. 313-367.
44. A. VAN DER SCHAFT AND H. SCHUMACHER, An Introduction to Hybrid Dynamical Systems, Springer-Verlag, Berlin, 1999.
45. E. WONG AND B. HAJEK, Stochastic Processes in Engineering Systems, 2nd ed., Springer-Verlag, Berlin, 1985.
46. M. ZAKAI, A Lyapunov criterion for the existence of stationary probability distributions for systems perturbed by noise, SIAM J. Control, 7 (1969), pp. 390-397.