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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 439482, 8 pages
http://dx.doi.org/10.1155/2013/439482
Research Article

Further Result on Finite-Time Stabilization of Stochastic Nonholonomic Systems

1School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China
2Department of Mathematics, Zhengzhou University, Zhengzhou 450001, China
3Institute of Automation, Qufu Normal University, Qufu 273165, China

Received 25 February 2013; Revised 6 May 2013; Accepted 29 May 2013

Academic Editor: Ahmed El-Sayed

Copyright © 2013 Fangzheng Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. W. Brockett, “Asymptotic stability andfeed back stabilization,” in Differential Geometric Control Theory, R. W. Brockett, R. S. Millman, and H. J. Sussmann, Eds., pp. 2961–2963, 1983.
  2. A. Astolfi, “Discontinuous control of nonholonomic systems,” Systems & Control Letters, vol. 27, no. 1, pp. 37–45, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. W. L. Xu and W. Huo, “Variable structure exponential stabilization of chained systems based on the extended nonholonomic integrator,” Systems & Control Letters, vol. 41, no. 4, pp. 225–235, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. M. Murray and S. S. Sastry, “Nonholonomic motion planning: steering using sinusoids,” IEEE Transactions on Automatic Control, vol. 38, no. 5, pp. 700–716, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Z. P. Jiang, “Iterative design of time-varying stabilizers for multi-input systems in chained form,” Systems & Control Letters, vol. 28, no. 5, pp. 255–262, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Y. P. Tian and S. Li, “Exponential stabilization of nonholonomic dynamic systems by smooth time-varying control,” Automatica, vol. 38, no. 8, pp. 1139–1146, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. I. Kolmanovsky and N. H. McClamroch, “Hybrid feedback laws for a class of cascade nonlinear control systems,” IEEE Transactions on Automatic Control, vol. 41, no. 9, pp. 1271–1282, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Z. P. Jiang, “Robust exponential regulation of nonholonomic systems with uncertainties,” Automatica, vol. 36, no. 2, pp. 189–209, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Z. Xi, G. Feng, Z. P. Jiang, and D. Cheng, “A switching algorithm for global exponential stabilization of uncertain chained systems,” IEEE Transactions on Automatic Control, vol. 48, no. 10, pp. 1793–1798, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  10. S. S. Ge, Z. Wang, and T. H. Lee, “Adaptive stabilization of uncertain nonholonomic systems by state and output feedback,” Automatica, vol. 39, no. 8, pp. 1451–1460, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Y. G. Liu and J. F. Zhang, “Output-feedback adaptive stabilization control design for non-holonomic systems with strong non-linear drifts,” International Journal of Control, vol. 78, no. 7, pp. 474–490, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  12. Z. Xi, G. Feng, Z. P. Jiang, and D. Cheng, “Output feedback exponential stabilization of uncertain chained systems,” Journal of the Franklin Institute, vol. 344, no. 1, pp. 36–57, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. X. Zheng and Y. Wu, “Adaptive output feedback stabilization for nonholonomic systems with strong nonlinear drifts,” Nonlinear Analysis, vol. 70, no. 2, pp. 904–920, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. F. Gao, F. Yuan, and H. Yao, “Robust adaptive control for nonholonomic systems with nonlinear parameterization,” Nonlinear Analysis, vol. 11, no. 4, pp. 3242–3250, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Z. Y. Liang and C. L. Wang, “Robust stabilization of nonholonomic chained form systems with uncertainties,” Acta Automatica Sinica, vol. 37, no. 2, pp. 129–142, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J. Wang, H. Gao, and H. Li, “Adaptive robust control of nonholonomic systems with stochastic disturbances,” Science in China F, vol. 49, no. 2, pp. 189–207, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Y. L. Liu and Y. Q. Wu, “Output feedback control for stochastic nonholonomic systems with growth rate restriction,” Asian Journal of Control, vol. 13, no. 1, pp. 177–185, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Y. Zhao, J. Yu, and Y. Wu, “State-feedback stabilization for a class of more general high order stochastic nonholonomic systems,” International Journal of Adaptive Control and Signal Processing, vol. 25, no. 8, pp. 687–706, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. P. Bhat and D. S. Bernstein, “Continuous finite-time stabilization of the translational and rotational double integrators,” IEEE Transactions on Automatic Control, vol. 43, no. 5, pp. 678–682, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Y. Hong, J. Wang, and Z. Xi, “Stabilization of uncertain chained form systems within finite settling time,” IEEE Transactions on Automatic Control, vol. 50, no. 9, pp. 1379–1384, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  21. J. Wang, G. Zhang, and H. Li, “Adaptive control of uncertain nonholonomic systems in finite time,” Kybernetika, vol. 45, no. 5, pp. 809–824, 2009. View at Zentralblatt MATH · View at MathSciNet
  22. J. Yin, S. Khoo, Z. Man, and X. Yu, “Finite-time stability and instability of stochastic nonlinear systems,” Automatica, vol. 47, no. 12, pp. 2671–2677, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. F. Gao and F. Yuan, “Finite-time stabilization of stochastic nonholonomic systems and its application to mobile robot,” Abstract and Applied Analysis, Article ID 361269, 18 pages, 2012. View at Zentralblatt MATH · View at MathSciNet
  24. X. Huang, W. Lin, and B. Yang, “Global finite-time stabilization of a class of uncertain nonlinear systems,” Automatica, vol. 41, no. 5, pp. 881–888, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. J. Li, C. Qian, and S. Ding, “Global finite-time stabilisation by output feedback for a class of uncertain nonlinear systems,” International Journal of Control, vol. 83, no. 11, pp. 2241–2252, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. C. Qian and W. Lin, “A continuous feedback approach to global strong stabilization of nonlinear systems,” IEEE Transactions on Automatic Control, vol. 46, no. 7, pp. 1061–1079, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. W. Chen and L. C. Jiao, “Finite-time stability theorem of stochastic nonlinear systems,” Automatica, vol. 46, no. 12, pp. 2105–2108, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  28. W. Chen and L. C. Jiao, “Authors' reply to comments on “Finite-time stability theorem of stochastic nonlinear systems” [Automatica 46 (2010) 2105–2108],” Automatica, vol. 47, no. 7, pp. 1544–1545, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. R. Situ, Thoery of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analysis Techniques with Applications to Engineering, Springer, New York, NY, USA, 2005. View at MathSciNet
  30. H. J. Liu and X. W. Mu, “A converse lyapunov theorem for stochastic finite-time stability,” in Proceedings of the 30th Chinese Control Conference, pp. 1419–1423, 2011.
  31. J. Polendo and C. Qian, “A generalized homogeneous domination approach for global stabilization of inherently nonlinear systems via output feedback,” International Journal of Robust and Nonlinear Control, vol. 17, no. 7, pp. 605–629, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. B. Yang and W. Lin, “Nonsmooth output feedback design with a dynamic gain for uncertain systems with strong nonlinearity,” in Proceedings of the 46th IEEE Conference on Decision and Control (CDC '07), pp. 3495–3500, New Orieans, La, USA, December 2007. View at Publisher · View at Google Scholar · View at Scopus
  33. W. Li, X. J. Xie, and S. Zhang, “Output-feedback stabilization of stochastic high-order nonlinear systems under weaker conditions,” SIAM Journal on Control and Optimization, vol. 49, no. 3, pp. 1262–1282, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. C. Qian and W. Lin, “Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization,” Systems & Control Letters, vol. 42, no. 3, pp. 185–200, 2001. View at Publisher · View at Google Scholar · View at MathSciNet