Kybernetika 49 no. 4, 590-600, 2013

Global finite-time stabilization for a class of stochastic nonlinear systems by dynamic state feedback

Weiqing Ai, Junyong Zhai and Shumin Fei


This paper addresses the problem of global finite-time stabilization by dynamic state feedback for a class of stochastic nonlinear systems. Firstly, we show a dynamic state transformation, under which the original system is transformed into a new system. Then, a state feedback controller with a dynamic gain is designed for the new system. It is shown that global finite-time stabilization in probability for a class of stochastic nonlinear system under linear growth condition can be guaranteed by appropriately choosing design parameters. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed design scheme.


stochastic nonlinear systems, dynamic state transformation, finite-time stabilization


93E12, 62A10


  1. S. P. Bhat and D. S. Bernstein: Finite-time stability of continuous autonomous systems. SIAM J. Control Optim. 38 (2000), 751-766.   CrossRef
  2. W. S. Chen and L. C. Jiao: Authors' reply to ``Comments on `finite-time stability theorem of stochastic nonlinear systems [Automatica 46(2010)2105-2108]'". Automatica 47 (2011), 1544-1545.   CrossRef
  3. H. Deng and M. Krstić: Stochastic nonlinear stabilizationq,- I: A backstepping design. Syst. Control Lett. 32 (1997), 143-150.   CrossRef
  4. A. Isidori: Nonlinear Control Systems. 3rd edition. Springer-Verlag, New York 1995.   CrossRef
  5. S. Y. Khoo, J. L. Yin, Z. H. Man and X. H. Yu: Finite-time stabilization of stochastic nonlinear systems in strict-feedback form. Automatica 49 (2013), 1403-1410.   CrossRef
  6. M. Kristić and H. Deng: Stabilization of uncertain systems. Springer-Verlag, New York 1998.   CrossRef
  7. W. Q. Li, Y. W. Jing and S. Y. Zhang: Adaptive state-feedback stabilization for a large class of high-order stochastic nonlinear systems. Automatica 47 (2011), 819-828.   CrossRef
  8. W. Q. Li, Y. W. Jing, S. Y. Zhang and G. M. Dimirovski: State-feedback stabilization for high-order stochastic nonlinear systems without strict triangular conditions. In: Proc. Amer. Control Conference 2010, pp. 367-372.   CrossRef
  9. R. Marino and P. Tomei: Nonlinear Control Design. Prentice Hall, U.K. 1995.   CrossRef
  10. L. Praly and Z. P. Jiang: Linear output feedback with dynamic high gain for nonlinear systems. Systems Control Lett. 53 (2004), 107-116.   CrossRef
  11. C. J. Qian and W. Lin: A continuous feedback approach to global strong stabilization of nonlinear systems. IEEE Trans. Automat. Control 46 (2001), 1061-1079.   CrossRef
  12. C. J. Qian and W. Lin: Output feedback control of a class of nonlinear systems: a nonseparation principle paradigm. IEEE Trans. Automat. Control 47 (2002), 1710-1715.   CrossRef
  13. S. Seo, H. Shim and J. Seo: Global finite-time stabilization of a nonlinear system using dynamic exponent scaling. In: Proc. 47th IEEE Conference on Decision Control 2008, pp. 3805-3810.   CrossRef
  14. S. Seo, H. Shim and J. Seo: Finite-time stabilizing dynamic control of uncertain multi-input linear systems. IMA J. Math. Control Inform. 28 (2011), 525-537.   CrossRef
  15. J. Tian and X.J. Xie: Adaptive state-feedback stabilization for more general high-order stochastic nonlinear systems. Acta Autom. Sinica 34 (2008), 1188-1192.   CrossRef
  16. X. J. Xie and N. Duan: Output tracking of high-order stochastic nonlinear systems with application to benchmark mechanical system. IEEE Trans. Automat. Control 55 (2010), 1197-1202.   CrossRef
  17. X. J. Xie and J. Tian: State-feedback stabilization for high-order stochastic nonlinear systems with stochastic inverse dynamics. Internat. J. Robust. Nonlinear Control 17 (2007), 1343-1362.   CrossRef
  18. J. L. Yin, S. Y. Khoo, Z. H. Man and X. H. Yu: Finite-time stability and instability of stochastic nonlinear systems. Automatica 47 (2011), 2671-2677.   CrossRef
  19. J. Y. Zhai and S. M. Fei: Global practical tracking control for a class of uncertain non-linear systems. IET Control Theory Appl. 5 (2011), 1343-1351.   CrossRef
  20. J. Y. Zhai and C. J. Qian: Global control of nonlinear systems with uncertain output function using homogeneous domination approach. Internat. J. Robust Nonlinear Control 22 (2012), 1543-1561.   CrossRef
  21. X. Zhang and G. Feng: Global finite-time stabilisation of a class of feedforward non-linear systems. IET Control Theory Appl. 5 (2011), 1450-1457.   CrossRef
  22. X. Zhang, G. Feng and Y. Sun: Finite-time stabilization by state feedback control for a class of time-varying nonlinear systems. Automatica 48 (2012), 499-504.   CrossRef