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Stability and stabilization of stochastic systems with multiplicative noise

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  • Control Theory
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Abstract

In this paper, stability and stabilization of linear stochastic time-invariant systems are studied based on spectrum technique. Firstly, the relationship among mean square exponential stability, asymptotical mean square stability, second-order moment exponential stability and the spectral location of the systems is revealed with the help of a spectrum operator L A,C . Then, we focus on almost sure exponential stability and stochastic stabilization. A criterion on almost sure exponential stability based on spectrum technique is obtained. Sufficient conditions for mean square exponentially stability and asymptotic mean square stability are given via linear matrix inequality approach and some numerical examples to illustrate the effectiveness of our results are presented.

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References

  1. L. J. Hu and X. R. Mao, “Almost sure exponential stabilisation of stochastic systems by state-feedback control,” Automatica, vol. 44, no. 2, pp. 465–471, February 2008.

    Article  MathSciNet  Google Scholar 

  2. J. Feng, J. Lam, S. Xu, and Z. Shu, “Stabilisation of stochastic systems with optimal decay rate,” IET Control Theory Appl., vol. 2, no. 1, pp. 1–6, February 2008.

    Article  MathSciNet  Google Scholar 

  3. J. L. Willems and J. C. Willems, “Feedback stabilizability for stochastic system with state and control dependent noise,” Automatica, vol. 12, no. 3, pp. 277–283, May 1976.

    Article  MATH  MathSciNet  Google Scholar 

  4. W. H. Zhang and B. S. Chen, “On stabilizability and exact observability of stochastic systems with their application,” Automatica, vol. 40, no. 1, pp. 87–94, January 2004.

    Article  MATH  MathSciNet  Google Scholar 

  5. M. A. Rami and X. Y. Zhou, “Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic control,” IEEE Trans. on Automatic Control, vol. 45, no. 6, pp. 1131–1142, June 2000.

    Article  MATH  MathSciNet  Google Scholar 

  6. Z. Wang, H. Qiao, and K. J. Burnham, “On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters,” IEEE Trans. on Automatic Control, vol. 47, no. 4, pp. 640–646, April 2002.

    Article  MathSciNet  Google Scholar 

  7. W. H. Zhang and L. H. Xie, “Interval stability and stabilization of linear stochastic systems,” IEEE Trans. on Automatic Control, vol. 54, no. 4, pp. 810–815, April 2009.

    Article  MathSciNet  Google Scholar 

  8. T. Damm, Rational Matrix Equations in Stochastic Control, Springer, Berlin-Heidelberg, 2004.

    MATH  Google Scholar 

  9. M. D. Fragoso, O. L. V. Costa, and C. E. De Souza, “A new approach to linearly perturbed Riccati equations arising in stochastic control,” Applied Mathematics and Optimization, vol. 37, no. 1, pp. 99–126, June 1998.

    Article  MATH  MathSciNet  Google Scholar 

  10. W. H. Zhang, J. E. Feng, B. S. Chen, and Z. L. Cheng, “On spectral assignment and detectability of linear stochastic systems,” Proc. of American Control Conference, pp. 386–387, 2005.

  11. J. E. Feng, S. Y. Xu, and W. H. Zhang, “H ∞ output feedback control with spectrum constraints for uncertain stochastic systems,” Circuits, Systems, and Signal Processing, vol. 26, no. 2, pp. 193–214, April 2007.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. Z. Has’minskii, Stochastic Stability of Differential Equations, Sijtjoff and Nordhoff, Alphen, 1980.

    Google Scholar 

  13. X. R. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, 1997.

    MATH  Google Scholar 

  14. D. Hinrichsen and A. J. Pritchard, “Stochastic H ∞,” SIAM Journal on Control and Optimization, vol. 36, no. 5, pp. 1504–1538, September 1998.

    Article  MATH  MathSciNet  Google Scholar 

  15. J. Xu and L. H. Xie, “Dilated LMI characterization and a new stability criterion for polytopic uncertain systems,” Proc. 6th World Congress on Intelligent Control and Automation, pp. 234–247, 2006.

  16. L. E. Ghaoui and M. A. Rami, “Robust state-feedback stabilization of jump linear systems via LMIs,” Int. J. Robust Nonlinear Control, vol. 6, pp. 1015–1022, 1996.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Key Laboratory for Robot and Intelligent Technology of Shandong Province, the College of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdao, 266510, China

    Huiying Sun, Meng Li & Weihai Zhang

Authors
  1. Huiying Sun
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  2. Meng Li
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  3. Weihai Zhang
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Corresponding author

Correspondence to Huiying Sun.

Additional information

Recommended by Editorial Board member Myotaeg Lim under the direction of Editor Young Il Lee. This work was supported by the National Natural Science Foundation of China (Grant No.60874032) and the Key Project of Natural Science Foundation of Shandong Province (Grant No.ZR2009GZ001) and Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20103718110006). The authors gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.

Huiying Sun received her M.S. degree from Qingdao University of Science and Technology and her Ph.D. degree from the Ocean University of China, in 2004 and 2007, respectively. She is currently an associate professor of Shandong University of Science and Technology. Her main research interests include linear stochastic control and stochastic differential games.

Meng Li received his Bachelor degree majoring in Communication Engineering from Shandong University of Science and Technology, China, in 2008. Since September 2008, he has been pursuing his M.S. degree in the same University. His main research interests include linear stochastic control and stochastic differential games.

Weihai Zhang received his M.S. degree from Hangzhou University, and his Ph.D. degree from Zhejiang University, Hangzhou, China, in 1994 and 1998, respectively. From August 1998 to May 2001, he worked at Shandong Institute of Light Industry as an associate professor. He was a postdoctoral researcher from May 2001 to July 2003 at National Tsing Hua University, Hsinchu, Taiwan. He rejoined Shandong Institute of Light Industry in August 2003 as a professor. From November 2006 to May 2007, he visited Nanyang Technological University as a visiting research scientist. He is currently a professor of Shandong University of Science and Technology. His research interests include linear and nonlinear stochastic optimal control, robust H infinity control and stochastic stability.

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Sun, H., Li, M. & Zhang, W. Stability and stabilization of stochastic systems with multiplicative noise. Int. J. Control Autom. Syst. 9, 211–217 (2011). https://doi.org/10.1007/s12555-011-0202-7

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  • DOI: https://doi.org/10.1007/s12555-011-0202-7

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