| Title:
|
Stabilization of nonlinear stochastic systems without unforced dynamics via time-varying feedback (English) |
| Author:
|
Florchinger, Patrick |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
52 |
| Issue:
|
6 |
| Year:
|
2016 |
| Pages:
|
988-1002 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
stochastic differential systems |
| Keyword:
|
smooth time–varying feedback law |
| Keyword:
|
global asymptotic stability in probability |
| MSC:
|
60H10 |
| MSC:
|
93C10 |
| MSC:
|
93D05 |
| MSC:
|
93D15 |
| MSC:
|
93E15 |
| idZBL:
|
Zbl 06707384 |
| idMR:
|
MR3607858 |
| DOI:
|
10.14736/kyb-2016-6-0988 |
| . |
| Date available:
|
2017-02-13T11:49:44Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146001 |
| . |
| Reference:
|
[1] Abedi, F., Hassan, M. A., Arifin, N.: Control Lyapunov function for feedback stabilization of affine in the control stochastic time-varying systems..Int. J. Math. Anal. 5 (2011), 175-188. Zbl 1238.93079 |
| Reference:
|
[2] Abedi, F., Leong, W. J., Chaharborj, S. S.: On the aymptotic and practical stability of stochastic control systems..Math. Problems in Engineering 2013 (2013), 560647. 10.1155/2013/560647 |
| Reference:
|
[3] Abedi, F., Leong, W. J., Abedi, M.: Lyapunov characterization of the stability of stochastic control systems..Math. Problems in Engineering 2015 (2015), 584935. MR 3319255, 10.1155/2015/584935 |
| Reference:
|
[4] Brockett, R.: 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. Zbl 0528.93051, MR 0708502 |
| Reference:
|
[5] Campion, G., d'Andréa-Novel, B., Bastin, G.: Controllability and state feedback stabilization of nonholonomic mechanical systems..In: International Workshop in Adaptive and Nonlinear Control: Issues in Robotics, Springer-Verlag 1990. 10.1007/bfb0039268 |
| Reference:
|
[6] Coron, J. M.: Global asymptotic stabilization for controllable systems without drift..Math. Control Signal Systems 5 (1992), 295-312. Zbl 0760.93067, MR 1164379, 10.1007/bf01211563 |
| Reference:
|
[7] Coron, J. M., Pomet, J. B.: 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. |
| Reference:
|
[8] Florchinger, P.: Lyapunov-like techniques for stochastic stability..SIAM J. Control Optim. 33 (1995), 4, 1151-1169. Zbl 0845.93085, MR 1339059, 10.1137/s0363012993252309 |
| Reference:
|
[9] Florchinger, P.: A stochastic Jurdjevic-Quinn theorem..SIAM J. Control Optim. 41 (2002), 83-88. Zbl 1014.60062, MR 1920157, 10.1137/s0363012900370788 |
| Reference:
|
[10] Florchinger, P.: Global asymptotic stabilization in probability of nonlinear stochastic systems via passivity..Int. J. Control 89 (2016), 1406-1415. MR 3494626, 10.1080/00207179.2015.1132009 |
| Reference:
|
[11] Florchinger, P.: Time-varying stabiliziers for stochastic systems with no unforced dynamics..Preprint 2016. |
| Reference:
|
[12] Jurdjevic, V., Quinn, J. P.: Controllability and stability..J. Differential Equations 28 (1978), 381-389. Zbl 0417.93012, MR 0494275, 10.1016/0022-0396(78)90135-3 |
| Reference:
|
[13] Khasminskii, R. Z.: Stochastic Stability of Differential Equations..Sijthoff and Noordhoff, Alphen aan den Rijn 1980. Zbl 1241.60002 |
| Reference:
|
[14] Kushner, H. J.: 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. Zbl 0275.93055, MR 0406657, 10.1007/bfb0064937 |
| Reference:
|
[15] Lin, W.: Time-varying feedback control of nonaffine nonlinear systems without drift..Systems Control Lett. 29 (1996), 101-110. Zbl 0866.93082, MR 1420407, 10.1016/s0167-6911(96)00050-3 |
| Reference:
|
[16] Pomet, J. B.: Explicit design of time-varying stabilizing control law for a class of controllable systems without drift..Systems Control Lett. 18 (1992), 147-158. MR 1149359, 10.1016/0167-6911(92)90019-o |
| Reference:
|
[17] Samson, C.: Time-varying Stabilization of a Nonholonomic Car-like Mobile Robot..Repport de recherche 1515, INRIA Sophia-Antipolis 1991. |
| Reference:
|
[18] Sepulchre, R., Campion, G., Wertz, V.: Some remarks about periodic feedback stabilization..In: Proc. IFAC NOLCOS, Bordeaux 1992, pp. 418-423. |
| Reference:
|
[19] Sontag, E.: 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. Zbl 0735.93063, MR 1115377, 10.1007/978-1-4612-4484-4_4 |
| Reference:
|
[20] D.Sontag, E., Sussmann, H. J.: Remarks on continuous feedback..In: Proc. 19th IEEE Conference on Decision and Control, Albuquerque 1980, pp. 916-921. 10.1109/cdc.1980.271934 |
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