Kybernetika 54 no. 2, 321-335, 2018

Stabilization of nonlinear stochastic systems without unforced dynamics via time--varying feedback

Patrick FlorchingerDOI: 10.14736/kyb-2018-2-0321

Abstract:

In this paper we give sufficient conditions under which a nonlinear stochastic differential system without unforced dynamics is globally asymptotically stabilizable in probability via time-varying smooth feedback laws. The technique developed to design explicitly the time-varying stabilizers is based on the stochastic Lyapunov technique combined with the strategy used to construct bounded smooth stabilizing feedback laws for passive nonlinear stochastic differential systems. The interest of this work is that the class of stochastic systems considered in this paper contains a lot of systems which cannot be stabilized via time-invariant feedback laws.

Keywords:

stochastic differential systems, Smooth time-varying feedback law, Global asymptotic stability in probability

Classification:

60H10, 93C10, 93D05, 93D15, 93E15

References:

  1. F. Abedi, M. A.Hassan and N. Arifin: Control Lyapunov function for feedback stabilization of affine in the control stochastic time-varying systems. Int. J. Math. Anal. 5 (2011), 175-188.   CrossRef
  2. F. Abedi, W. J.Leong and S. S. Chaharborj: On the aymptotic and practical stability of stochastic control systems. Math. Prob. Engrg. 2013 (2013), Article ID 560647, 1-10.   DOI:10.1155/2013/560647
  3. F. Abedi, W. J. Leong and M. Abedi: Lyapunov characterization of the stability of stochastic control systems. Math. Prob. Engrg. 2015 (2015), Article ID 584935, 1-7.   DOI:10.1155/2015/584935
  4. R. Brockett: Asymptotic stability and feedback stabilization. In: Differential Geometric Control Theory (R. Brockett, R. Millman, and H. Sussmann, eds.), Birkhäuser, Basel, Boston 1983, pp. 181-191.   CrossRef
  5. G. Campion, B. d'Andréa-Novel and G. Bastin: Controllability and state feedback stabilization of nonholonomic mechanical systems. In: International Workshop in Adaptive and Nonlinear Control: Issues in Robotics, Springer-Verlag 1990.   DOI:10.1007/bfb0039268
  6. J. M. Coron: Global asymptotic stabilization for controllable systems without drift. Math. Control Signal Systems 5 (1992), 295-312.   DOI:10.1007/bf01211563
  7. J. M. Coron and J. B. Pomet: A remark on the design of time-varying stabilization feedback laws for controllable systems without drift. In: Proc. IFAC NOLCOS, Bordeaux 1992, pp. 413-417.   CrossRef
  8. P. Florchinger: Lyapunov-like techniques for stochastic stability. SIAM J. Control Optim. 33 (1995), 4, 1151-1169.   CrossRef
  9. P. Florchinger: A stochastic Jurdjevic-Quinn theorem. SIAM J. Control Optim. 41 (2002), 83-88.   CrossRef
  10. P. Florchinger: Global asymptotic stabilization in probability of nonlinear stochastic systems via passivity. Int. J. Control 89 (2016), 1406-1415.   DOI:10.1080/00207179.2015.1132009
  11. P. Florchinger: Time-varying stabiliziers for stochastic systems with no unforced dynamics. Asian J. Control xx (2018), xxx-xxx.   DOI:10.1002/asjc.1703
  12. V. Jurdjevic and J. P. Quinn: Controllability and stability. J. Differential Equations 28 (1978), 381-389.   DOI:10.1016/0022-0396(78)90135-3
  13. R. Z. Khasminskii: Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Alphen aan den Rijn 1980.   CrossRef
  14. H. J. Kushner: Stochastic stability. In: Stability of Stochastic Dynamical Systems (R. Curtain, ed.), Lecture Notes in Mathematics 294, Springer Verlag, Berlin, Heidelberg, New York 1972, pp. 97-124.   DOI:10.1007/bfb0064937
  15. W.Lin: Time-varying feedback control of nonaffine nonlinear systems without drift. Systems Control Lett. 29 (1996), 101-110.   DOI:10.1016/s0167-6911(96)00050-3
  16. J. B. Pomet: Explicit design of time-varying stabilizing control law for a class of controllable systems without drift. Systems Control Lett. 18 (1992), 147-158.   DOI:10.1016/0167-6911(92)90019-o
  17. C. Samson: Time-varying stabilization of a nonholonomic car-like mobile robot. Rapport de recherche 1515, INRIA Sophia-Antipolis 1991.   CrossRef
  18. R. Sepulchre, G. Campion and V. Wertz: Some remarks about periodic feedback stabilization. In: Proc. IFAC NOLCOS, Bordeaux 1992, pp. 418-423.   DOI:10.1177/027836499301200104
  19. E. Sontag: Feedback stabilization of nonlinear systems. In: Robust Control of Linear Systems and Nonlinear Control (M. K. Kaashoek et al., eds.), Birkhäuser, Boston 1990, pp. 61-81.   DOI:10.1007/978-1-4612-4484-4_4
  20. E. D.Sontag and H. J.Sussmann: Remarks on continuous feedback. In: Proc. 19th IEEE Conference on Decision and Control, Albuquerque 1980, pp. 916-921.   DOI:10.1109/cdc.1980.271934