| Title:
|
Continuous feedback stabilization for a class of affine stochastic nonlinear systems (English) |
| Author:
|
Oumoun, Mohamed |
| Author:
|
Maniar, Lahcen |
| Author:
|
Atlas, Abdelghafour |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
56 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
500-515 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We investigate the state feedback stabilization, in the sense of weak solution, of nonlinear stochastic systems when the drift is quadratic in the control and the diffusion term is affine in the control. Based on the generalised stochastic Lyapunov theorem, we derive the necessary conditions and the sufficient conditions, respectively, for the global asymptotic stabilization in probability by a continuous feedback explicitly computed. The interest of this work is that the existing control methods are inapplicable to a lot of systems contained in the class of stochastic systems considered in this paper. (English) |
| Keyword:
|
continuous state feedback |
| Keyword:
|
control stochastic nonlinear systems |
| Keyword:
|
global asymptotic stability in probability |
| MSC:
|
60H10 |
| MSC:
|
93C10 |
| MSC:
|
93D05 |
| MSC:
|
93D15 |
| MSC:
|
93E15 |
| idZBL:
|
Zbl 07250734 |
| idMR:
|
MR4131740 |
| DOI:
|
10.14736/kyb-2020-3-0500 |
| . |
| Date available:
|
2020-09-02T09:22:54Z |
| Last updated:
|
2021-02-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148311 |
| . |
| Reference:
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[1] Abedi, F., Leong, W. J., Chaharborj, S. S.: On the asymptotical and practical stability of stochastic control systems..Math. Problems Engrg. (2013), 1-10. MR 3035628, 10.1155/2013/560647 |
| Reference:
|
[2] Artstein, Z.: Stabilization with relaxed control..Nonlinear Anal. Theory Methods Appl. 7 (1983), 1163-1173. MR 0721403, 10.1016/0362-546x(83)90049-4 |
| Reference:
|
[3] Chabour, R., Oumoun, M.: On a universal formula for the stabilization of control stochastic nonlinear systems..Stochast. Anal. Appl. 17 (1999), 359-368. MR 1686995, 10.1080/07362999908809606 |
| Reference:
|
[4] Daumail, L., Florchinger, P.: A constructive extension of Artsteins's theorem to the stochastic context..Stochast. Dynamics 2 (2002), 251-263. MR 1912143, 10.1142/s0219493702000418 |
| Reference:
|
[5] Deng, H., Krstic, M., Williams, R. J.: Stabilization of stochastic nonlinear systems driven by noise of unknown covariance..IEEE Trans. Automat. Control 46 (2001), 1237-1253. MR 1847327, 10.1109/9.940927 |
| Reference:
|
[6] Florchinger, P.: A universal formula for the stabilization of control stochastic differential equations..Stochast. Anal. Appl. 11 (1993), 155-162. MR 1214577, 10.1080/07362999308809308 |
| Reference:
|
[7] Florchinger, P.: A universal design of Freeman's formula for the stabilization of stochastic systems..Stochast. Anal. Appl. 34 (2016), 137-146. MR 3437083, 10.1080/07362994.2015.1108203 |
| Reference:
|
[8] Florchinger, P.: Stabilization of nonlinear stochastic systems without unforced dynamics via time-varying feedback..Kybernetika 54 (2018), 321-335. MR 3807718, 10.14736/kyb-2018-2-0321 |
| Reference:
|
[9] Fontbona, J., Raminez, H., Riquelme, V., Silva, F.: Stochastic modeling and control of bioreactors..IFACPapersOnLine 50 (2017), 12611-12616. 10.1016/j.ifacol.2017.08.2203 |
| Reference:
|
[10] Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes..Amsterdam, North-Holland 1981. MR 0637061, 10.1002/bimj.4710280425 |
| Reference:
|
[11] Gao, F., Wu, Y., Yu, X.: Global state feedback stabilization of stochastic high-order nonlinear systems with high-order and low-order nonlinearities..Int. J. Systems Sci. 47 (2016), 16, 3846-3856. MR 3512588, 10.1080/00207721.2015.1129678 |
| Reference:
|
[12] Khalil, H. K.: Nonlinear Systems..Upper Saddle River, Prentice-Hall, NJ 2002. Zbl 1194.93083 |
| Reference:
|
[13] Khasminskii, R. Z.: Stochastic Stability of Differential Equations..Sijthoff and Noordhoff International Publishers 1980. Zbl 1241.60002, MR 0600653, 10.1007/978-3-642-23280-0 |
| Reference:
|
[14] Klebaner, F. C.: Introduction to Stochastic Calculus with Applications..Imperial College Press, London 2005. MR 2160228, 10.1142/p386 |
| Reference:
|
[15] Kushner, H. J.: Stochastic Stability and Control..Academic Press, New York 1967. MR 0216894, 10.1002/zamm.19680480428 |
| Reference:
|
[16] Lan, Q., Li, S.: Global output-feedback stabilization for a class of stochastic nonlinear systems via sampled-data control..Int. J. Robust Nonlinear Control 27 (2017), 3643-3658. MR 3733629, 10.1002/rnc.3758 |
| Reference:
|
[17] Lan, Q., Niu, H., Liu, Y., Xu, H.: Global output-feedback finite-time stabilization for a class of stochastic nonlinear cascaded systems..Kybernetika 53 (2017), 780-802. MR 3750103, 10.14736/kyb-2017-5-0780 |
| Reference:
|
[18] Lewis, A. L.: Option Valuation Under Stochastic Volatility II..Finance Press, Newport Beach 2009. MR 3526206, 10.1111/rssa.12262 |
| Reference:
|
[19] Li, F., Liu, Y.: Global stability and stabilization of more general stochastic nonlinear systems..J. Math. Anal. Appl. 413 (2014), 841-855. MR 3159808, 10.1016/j.jmaa.2013.12.021 |
| Reference:
|
[20] Lin, Y., Sontag, E. D.: A universal formula for stabilization with bounded controls..Systems Control Lett. 16 (1991), 393-397. MR 1112756, 10.1016/0167-6911(91)90111-q |
| Reference:
|
[21] Maniar, L., Oumoun, M., Vivalda, J. C.: On the stabilization of quadratic nonlinear systems..Europ. J Control 35 (2017), 28-33. MR 3648351, 10.1016/j.ejcon.2017.03.001 |
| Reference:
|
[22] Mao, X. R.: Stochastic Differential Equations and Their Applications..Horwood Publishing, Chichester 1997. Zbl 0892.60057, MR 1475218 |
| Reference:
|
[23] Mao, X., Truman, A., Yuan, C.: Euler-Maruyama approximations in mean-reverting stochastic volatility model under regime-switching..J. Appl. Math. Stochast. Anal. (2006), 1-20. MR 2237177, 10.1155/jamsa/2006/80967 |
| Reference:
|
[24] Ondreját, M., Seidler, J.: A note on weak solutions to stochastic differential equations..Kybernetika 54 (2018), 888-907. MR 3893126, 10.14736/kyb-2018-5-0888 |
| Reference:
|
[25] Sontag, E. D.: A universal construction of Artstein's theorem on nonlinear stabilization..Systems Control Lett. 13 (1989), 117-123. MR 1014237, 10.1016/0167-6911(89)90028-5 |
| Reference:
|
[26] Yang, H., Kloeden, P. E., Wu, F.: Weak solution of stochastic differential equations with fractional diffusion coefficient..Stochast. Anal. Appl. 36 (2018), 4, 613-621. MR 3854532, 10.1080/07362994.2018.1434005 |
| Reference:
|
[27] Zha, W., Zhai, J., Fei, S.: Global adaptive control for a class of uncertain stochastic nonlinear systems with unknown output gain..Int. J. Control Automat. Systems 15 (2017), 3, 1125-1133. MR 3418397, 10.1007/s12555-016-0023-9 |
| Reference:
|
[28] Zhang, B. L., Han, Q. L., Zhang, X. M.: Recent advances in vibration control of offshore platforms..Nonlinear Dynamics 89 (2017), 755-771. 10.1007/s11071-017-3503-4 |
| Reference:
|
[29] Zhang, B. L., Han, Q. L., Zhang, X. M.: Event-triggered $H_\infty$ reliable control for offshore structures in network environments..J. Sound Vibration 368 (2016), 1-21. 10.1016/j.jsv.2016.01.008 |
| Reference:
|
[30] Zhang, B. L., Han, Q. L., Zhang, X. M., Yu, X.: Sliding mode control with mixed current and delayed states for offshore steel jacket platforms..IEEE Trans. Control Systems Technol. 22 (2014), 1769-1783. 10.1109/tcst.2013.2293401 |
| Reference:
|
[31] Zhang, J., Liu, Y.: Continuous output-feedback stabilization for a class of stochastic high-order nonlinear systems..J. Control Theory Appl. 11 (2013), 343-350. MR 3083980, 10.1007/s11768-013-2166-z |
| Reference:
|
[32] Zhang, X., Xie, X.: Global state feedback stabilization of nonlinear systems with high-order and low-order nonlinearities..Int. J. Control 87 (2014), 642-652. MR 3172535, 10.1080/00207179.2013.852252 |
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