Abstract and Applied Analysis

Further Result on Finite-Time Stabilization of Stochastic Nonholonomic Systems

Abstract

This paper further investigates the problem of finite-time state feedback stabilization for a class of stochastic nonholonomic systems in chained form. Compared with the existing literature, the stochastic nonholonomic systems under investigation have more uncertainties, such as the ${x}_{0}$-subsystem contains stochastic disturbance. This renders the existing finite-time control methods highly difficult to the control problem of the systems or even inapplicable. In this paper, by extending adding a power integrator design method to a stochastic system and by skillfully constructing ${C}^{2}$ Lyapunov function, a novel switching control strategy is proposed, which renders that the states of closed-loop system are almost surely regulated to zero in a finite time. A simulation example is provided to demonstrate the effectiveness of the theoretical results.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 439482, 8 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511945

Digital Object Identifier
doi:10.1155/2013/439482

Mathematical Reviews number (MathSciNet)
MR3070188

Zentralblatt MATH identifier
1291.93315

Citation

Gao, Fangzheng; Yuan, Fushun; Zhang, Jian; Wu, Yuqiang. Further Result on Finite-Time Stabilization of Stochastic Nonholonomic Systems. Abstr. Appl. Anal. 2013 (2013), Article ID 439482, 8 pages. doi:10.1155/2013/439482. https://projecteuclid.org/euclid.aaa/1393511945

References

• 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.
• A. Astolfi, “Discontinuous control of nonholonomic systems,” Systems & Control Letters, vol. 27, no. 1, pp. 37–45, 1996.
• 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.
• 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.
• 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.
• 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.
• I. Kolmanovsky and N. H. McClamroch, “Hybrid feedback lawsfor a class of cascade nonlinear control systems,” IEEE Transactions on Automatic Control, vol. 41, no. 9, pp. 1271–1282, 1996.
• Z. P. Jiang, “Robust exponential regulation of nonholonomic systems with uncertainties,” Automatica, vol. 36, no. 2, pp. 189–209, 2000.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• W. Chen and L. C. Jiao, “Finite-time stability theorem of stochastic nonlinear systems,” Automatica, vol. 46, no. 12, pp. 2105–2108, 2010.
• 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.
• 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.
• H. J. Liu and X. W. Mu, “A converse lyapunov theorem forstochastic finite-time stability,” in Proceedings of the 30th Chinese Control Conference, pp. 1419–1423, 2011.
• 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.
• 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.
• 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.
• 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.