About this Journal Submit a Manuscript Table of Contents
Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 808035, 21 pages
http://dx.doi.org/10.1155/2012/808035
Research Article

Nonsmooth Adaptive Control Design for a Large Class of Uncertain High-Order Stochastic Nonlinear Systems

School of Control Science and Engineering, Shandong University, Jinan 250061, China

Received 27 September 2011; Accepted 3 November 2011

Academic Editor: Xue-Jun Xie

Copyright © 2012 Jian Zhang and Yungang Liu. 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. H. J. Kushner, Stochastic Stability and Control, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1967. View at Zentralblatt MATH
  2. R. Z. Has’minskii, Stochastic Stability of Differential Equations, vol. 7 of Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1980.
  3. M. Krstić and H. Deng, Stabilization of Nonlinear Uncertain Systems, Communications and Control Engineering Series, Springer, New York, NY, USA, 1998. View at Zentralblatt MATH
  4. P. Florchinger, “Lyapunov-like techniques for stochastic stability,” SIAM Journal on Control and Optimization, vol. 33, no. 4, pp. 1151–1169, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. Z. Pan and T. Başar, “Backstepping controller design for nonlinear stochastic systems under a risk-sensitive cost criterion,” SIAM Journal on Control and Optimization, vol. 37, no. 3, pp. 957–995, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. H. Deng and M. Krstić, “Output-feedback stochastic nonlinear stabilization,” IEEE Transactions on Automatic Control, vol. 44, no. 2, pp. 328–333, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. H. Deng, M. Krstić, and R. J. Williams, “Stabilization of stochastic nonlinear systems driven by noise of unknown covariance,” IEEE Transactions on Automatic Control, vol. 46, no. 8, pp. 1237–1253, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. S.-J. Liu, J.-F. Zhang, and Z.-P. Jiang, “Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems,” Automatica, vol. 43, no. 2, pp. 238–251, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. S.-J. Liu, Z.-P. Jiang, and J.-F. Zhang, “Global output-feedback stabilization for a class of stochastic non-minimum-phase nonlinear systems,” Automatica, vol. 44, no. 8, pp. 1944–1957, 2008. View at Publisher · View at Google Scholar
  10. Y. Liu, Z. Pan, and S. Shi, “Output feedback control design for strict-feedback stochastic nonlinear systems under a risk-sensitive cost,” IEEE Transactions on Automatic Control, vol. 48, no. 3, pp. 509–513, 2003. View at Publisher · View at Google Scholar
  11. Y. Liu and J.-F. Zhang, “Reduced-order observer-based control design for nonlinear stochastic systems,” Systems & Control Letters, vol. 52, no. 2, pp. 123–135, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. Y. Liu and J.-F. Zhang, “Practical output-feedback risk-sensitive control for stochastic nonlinear systems with stable zero-dynamics,” SIAM Journal on Control and Optimization, vol. 45, no. 3, pp. 885–926, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. Z. Pan, Y. Liu, and S. Shi, “Output feedback stabilization for stochastic nonlinear systems in observer canonical form with stable zero-dynamics,” Science in China (Series F), vol. 44, no. 4, pp. 292–308, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. W. Lin and C. Qian, “Adding one power integrator: a tool for global stabilization of high-order lower-triangular systems,” Systems & Control Letters, vol. 39, no. 5, pp. 339–351, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. M. Krsti'c, I. Kanellakopoulos, and P. V. Kokotovi'c, Nonlinear and Adaptive Control Design, John Wiley & Sons, New York, NY, USA, 1995.
  16. W. Lin and C. Qian, “Adaptive control of nonlinearly parameterized systems: the smooth feedback case,” IEEE Transactions on Automatic Control, vol. 47, no. 8, pp. 1249–1266, 2002. View at Publisher · View at Google Scholar
  17. Z. Sun and Y. Liu, “Adaptive state-feedback stabilization for a class of high-order nonlinear uncertain systems,” Automatica, vol. 43, no. 10, pp. 1772–1783, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. W. Lin and C. Qian, “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
  19. W. Lin and C. Qian, “Adaptive control of nonlinearly parameterized systems: a nonsmooth feedback framework,” IEEE Transactions on Automatic Control, vol. 47, no. 5, pp. 757–774, 2002. View at Publisher · View at Google Scholar
  20. W. Lin and R. Pongvuthithum, “Nonsmooth adaptive stabilization of cascade systems with nonlinear parameterization via partial-state feedback,” IEEE Transactions on Automatic Control, vol. 48, no. 10, pp. 1809–1816, 2003. View at Publisher · View at Google Scholar
  21. Z. Sun and Y. Liu, “Adaptive practical output tracking control for high-order nonlinear uncertain systems,” Acta Automatica Sinica, vol. 34, no. 8, pp. 984–988, 2008. View at Publisher · View at Google Scholar
  22. Z. Sun and Y. Liu, “Adaptive stabilisation for a large class of high-order uncertain non-linear systems,” International Journal of Control, vol. 82, no. 7, pp. 1275–1287, 2009. View at Publisher · View at Google Scholar
  23. J. Zhang and Y. Liu, “A new approach to adaptive control design without overparametrization for a class of uncertain nonlinear systems,” Science China Information Sciences, vol. 54, no. 7, pp. 1419–1429, 2011. View at Publisher · View at Google Scholar
  24. C. Qian and W. Lin, “Recursive observer design, homogeneous approximation, and nonsmooth output feedback stabilization of nonlinear systems,” IEEE Transactions on Automatic Control, vol. 51, no. 9, pp. 1457–1471, 2006. View at Publisher · View at Google Scholar
  25. X. Xie and J. Tian, “State-feedback stabilization for high-order stochastic nonlinear systems with stochastic inverse dynamics,” International Journal of Robust and Nonlinear Control, vol. 17, no. 14, pp. 1343–1362, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. J. Tian and X. Xie, “Adaptive state-feedback stabilization for more general high-order stochastic nonlinear systems,” Acta Automatica Sinica, vol. 34, no. 9, pp. 1188–1191, 2008. View at Publisher · View at Google Scholar
  27. X. Xie and J. Tian, “Adaptive state-feedback stabilization of high-order stochastic systems with nonlinear parameterization,” Automatica, vol. 45, no. 1, pp. 126–133, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. W. Li and X. Xie, “Inverse optimal stabilization for stochastic nonlinear systems whose linearizations are not stabilizable,” Automatica, vol. 45, no. 2, pp. 498–503, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. X. Xie and N. Duan, “Output tracking of high-order stochastic nonlinear systems with application to benchmark mechanical system,” IEEE Transactions on Automatic Control, vol. 55, no. 5, pp. 1197–1202, 2010. View at Publisher · View at Google Scholar
  30. W. Li, X. 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
  31. X. Xie, N. Duan, and X. Yu, “State-feedback control of high-order stochastic nonlinear systems with SiISS inverse dynamics,” IEEE Transactions on Automatic Control, vol. 56, no. 8, pp. 1921–1926, 2011. View at Publisher · View at Google Scholar
  32. F. C. Klebaner, Introduction to Stochastic Calculus with Applications, Imperial College Press, London, UK, 2nd edition, 2005.
  33. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, The Netherlands, 2nd edition, 1981.
  34. J. Kurzweil, “On the inversion of lyapunov’s theorem on the stability of motion,” American Mathematical Society Translations 2, vol. 24, pp. 19–77, 1956.
  35. X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, London, UK, 2006.
  36. X. Yu and X. Xie, “Output feedback regulation of stochastic nonlinear systems with stochastic iISS inverse dynamics,” IEEE Transactions on Automatic Contro, vol. 55, no. 2, pp. 304–320, 2010. View at Publisher · View at Google Scholar
  37. X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Series in Mathematics & Applications, Horwood, Chichester, UK, 1997.
  38. E. Sontag and A. Teel, “Changing supply functions in input/state stable systems,” IEEE Transactions on Automatic Control, vol. 40, no. 8, pp. 1476–1478, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH