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Improved LMI conditions for H ∞ output feedback stabilization of linear discrete-time systems

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Abstract

The problem of H ∞ output feedback control for discrete-time systems is investigated in this paper. The main contribution is to provide a uniform framework for generating sufficient linear matrix inequality conditions. Those conditions are classified into two parallel parts based on the way of slack variable selection. Moreover, it is shown that the existing result is a special case of the new conditions by taking few of the additional free matrix parameters to be zero. This directly leads to more flexibility and less conservativeness in the H ∞ output feedback control design. Numerical examples are carried out for illustration.

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References

  1. V. L. Syrmos, C. T. Abdallah, P. Dorato, and K. Grigoriadis, “Static output feedback-a survey,” Automatica, vol. 33, no. 2, pp. 125–137, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994.

    MATH  Google Scholar 

  3. P. Gahinet and P. Apkarian, “A linear matrix inequality approach to H ∞ control,” Int. J. Robust Nonlinear Control, vol. 4, no. 4, pp. 421–448, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  4. C. Scherer, P. Gahinet, and M. Chilali, “Multiobjective output-feedback control via LMI optimization,” IEEE Trans. Autmat. Contr., vol. 42, no. 7, pp. 896–911, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox, The Mathworks, Natick, MA, 1995.

    Google Scholar 

  6. M. Mattei, “Sufficient conditions for the synthesis of H ∞ fixed-order controllers,” International J. Robust Nonlinear Control, vol. 10, no. 15, pp. 1237–1248, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  7. D. W. C. Ho and G. P. Lu, “Robust stabilization for a class of discrete-time non-linear systems via output feedback: the unified LMI approach,” International J. Control, vol. 76, pp. 105–115, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. C. Lo and M. L. Lin, “Robust H ∞ nonlinear control via fuzzy static output feedback,” IEEE Trans. on Circuits Systems, vol. 50, no. 11, pp. 1494–1502, 2003.

    Article  MathSciNet  Google Scholar 

  9. C. A. R. Crusius and A. Trofino, “Sufficient LMI conditions for output feedback control problems,” IEEE Trans. Autmat. Contr., vol. 44, no. 5, pp. 1053–1057, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  10. C. E. da Souza and A. Trofino, “An LMI approach to stabilization of linear discrete-time periodic systems,” International J. Control, vol. 73, no. 8, pp. 696–703, August 2000.

    Article  MATH  Google Scholar 

  11. E. Prempain and I. Postlethwaite, “Static output feedback stabilization with H ∞ performance for a class of plants,” System & Control Letters, vol. 43, no. 3, pp. 159–1665, July 2001.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. I. Bara and M. Boutayeb, “Static output feedback stabilization with H ∞ performance for linear discrete-time systems,” IEEE Trans. Automat. Contr., vol. 50, no. 2, pp. 250–254, Feb 2005.

    Article  MathSciNet  Google Scholar 

  13. K. H. Lee, J. H. Lee, and W. H. Kwon, “Sufficient LMI conditions for H ∞ output feedback stabilization of linear discrete-time systems,” IEEE Trans. Autmat. Contr., vol. 51, no. 4, pp. 675–680, April 2006.

    Article  MathSciNet  Google Scholar 

  14. J. Dong and G.-H. Yang, “static output feedback control synthesis for linear systems with timeinvariant parametric uncertainties,” IEEE Trans. Autmat. Contr., vol. 52, no. 10, pp. 1930–1936, Oct 2007.

    Article  MathSciNet  Google Scholar 

  15. M. C. De Oliveira, J. C. Geromel, and J. Bernussou, “Extended H 2 and H ∞ norm characterizations and controller parametrizations for discrete-time systems,” International Journal of Control, vol. 75, no. 9, pp. 666–679, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  16. X. Du and G.-H. Yang, “LMI conditions for H ∞ static output feedback control of discrete-time systems,” Proc. of the 47th Conf. Decision and Control, pp. 5450–5455, 2008.

  17. M. C. de Oliveira, J. Bernussou, and J. C. Geromel, “A new discrete-time robust stability condition,” System & Control Letters, vol. 37, no. 4, pp. 261–265, 1999.

    Article  MATH  Google Scholar 

  18. D. Peaucelle, D. Arzelier, O. Bachelier, and J. Bernussou, “A new robust D-stability condition for real convex polytopic uncertainty,” Systems & Control Letters, vol. 40, no. 1, pp. 21–30, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  19. Z. Duan, J. Zhang, C. Zhang, and E. Mosca, “Robust H 2 and H ∞ filtering for uncertain linear systems,” Automatica, vol. 42, no. 11, pp. 1919–1926, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  20. C. W. J. Hol and C. W. Scherer, “A sum of squares approach to fixed-order H ∞ synthesis,” in Positive Polynomials in Control, D. Henrion and A. Garulli, Eds., Springer-verlag, Berlin, Germany, 2005.

    Google Scholar 

  21. P. Apkarian and D. Noll, “Nonsmooth H ∞ synthesis,” IEEE Trans. Autmat. Contr., vol. 51, no. 1, pp. 71–86, 2006.

    Article  MathSciNet  Google Scholar 

  22. F. Leibfritz, “An LMI-based algorithm for designing suboptimal static H 2/H ∞ output feedback controllers,” SIAM J. Control Optimization, vol. 39, no. 6, pp. 1711–1735, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  23. U. Shaked, “An LPD approach to robust H 2 and H ∞ static output-feedback design,” IEEE Trans. Autmat. Contr., vol. 48, no. 5, pp. 866–872, 2003.

    Article  MathSciNet  Google Scholar 

  24. V. Suplin and U. Shaked, “Robust H ∞ outputfeedback control of linear discrete-time systems,” System & Control Letters, vol. 54, no. 8, pp. 799–808, 2005.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Information Science and Engineering, Northeastern University, Shenyang, 110004, China

    Xin Du & Guang-Hong Yang

  2. Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang, 110004, China

    Guang-Hong Yang

Authors
  1. Xin Du
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  2. Guang-Hong Yang
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Corresponding author

Correspondence to Guang-Hong Yang.

Additional information

Recommended by Editorial Board member Myotaeg Lim under the direction of Editor Jae Weon Choi. This work was supported in part by the Funds for Creative Research Groups of China (No. 60821063), National 973 Program of China (Grant No. 2009CB320604), the Funds of National Science of China (Grant No. 60974043), and the 111 Project (B08015). He is also with the Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang 110004, China

Xin Du was born in 1983. He received the B.E. degree from the University of Science and Technology Beijing, China, in 2004. He now is a Ph.D. candidate at Northeastern University. His research interest covers model reduction, controller reduction, and robust control.

Guang-Hong Yang received the B.S. and M.S. degrees in Northeast University of Technology, China, in 1983 and 1986, respectively, and the Ph.D. degree in Control Engineering from Northeastern University, China (formerly, Northeast University of Technology), in 1994. He was a lecturer/associate professor with Northeastern University from 1986 to 1995. He joined the Nanyang Technological University in 1996 as a postdoctoral fellow. From 2001 to 2005, he was a research scientist/senior research scientist with the National University of Singapore. He is currently a professor at the College of Information Science and Engineering, Northeastern University. His current research interests include fault-tolerant control, fault detection and isolation, non-fragile control systems design, and robust control. Dr. Yang is an Associate Editor for the International Journal of Control, Automation, and Systems (IJCAS), and an Associate Editor of the Conference Editorial Board of the IEEE Control Systems Society.

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Du, X., Yang, GH. Improved LMI conditions for H ∞ output feedback stabilization of linear discrete-time systems. Int. J. Control Autom. Syst. 8, 163–168 (2010). https://doi.org/10.1007/s12555-010-0121-z

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  • DOI: https://doi.org/10.1007/s12555-010-0121-z

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