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LMI-Based Optimization

  • Mohammad Fathi
  • Hassan Bevrani
Chapter

Abstract

In this chapter, following an introduction on the fundamentals of linear matrix inequalities (LMIs), the application of LMIs to solve convex optimization problems, using numerical examples, is explained. Then, the robust optimization problems are formulated and solved via the LMI-based H and mixed H2H optimization techniques. In order to solve non-convex optimization problems an iterative LMI is addressed. Finally, the effectiveness of given LMI-based optimization techniques in control synthesis is emphasized and several illustrative examples are given.

Keywords

Linear matrix inequalities (LMI) Robust optimization Optimal H control Multi-objective optimization Performance index Non-convex optimization 

References

  1. 1.
    S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory (Philadelphia, SIAM Books, 1994)CrossRefGoogle Scholar
  2. 2.
    H. Bevrani, Robust Power System Frequency Control (Basel, Springer, 2014)zbMATHGoogle Scholar
  3. 3.
    T. Iwasaki, R.E. Skelton, J.C. Geromel, Linear quadratic suboptimal control with static output feedback. Syst. Control Lett. 23, 421–430 (1994)MathSciNetCrossRefGoogle Scholar
  4. 4.
    F. Leibfritz, Static Output Feedback Design Problems, PhD Dissertation, Trier University, 1998Google Scholar
  5. 5.
    P. Gahinet, P. Apkarian, A linear matrix inequality approach to H control. Int. J. Robust Nonlinear Control. 4, 421–448 (1994)MathSciNetCrossRefGoogle Scholar
  6. 6.
    J.C. Geromel, C.C. Souza, R.E. Skeltox, Static output feedback controllers: stability and convexity. IEEE Trans. Autom. Control. 42, 988–992 (1997)CrossRefGoogle Scholar
  7. 7.
    T. Iwasaki, R.E. Skelton, All controllers for the general H control problem: LMI existence conditions and state space formulas. Automatica. 30, 1307–1317 (1994)MathSciNetCrossRefGoogle Scholar
  8. 8.
    R.E. Skelton, J. Stoustrup, T. Iwasaki, The H control problem using static output feedback. Int. J. Robust Nonlinear Control. 4, 449–455 (1994)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Y.Y. Cao, Y.X. Sun, W.J. Mao, Static output feedback stabilization: an ILMI approach. Automatica 34, 1641–1645 (1998)CrossRefGoogle Scholar
  10. 10.
    Y.S. Hung, A.G.J. MacFarlane, Multivariable feedback: a quasi-classical approach, in: Lecture Notes in Control and Information Sciences (Springer, New York, 1982)CrossRefGoogle Scholar
  11. 11.
    P.P. Khargonekar, M.A. Rotea, Mixed H2/H control: a convex optimization approach. IEEE Trans. Autom. Control. 39, 824–837 (1991)CrossRefGoogle Scholar
  12. 12.
    C.W. Scherer, Multiobjective H2/H control. IEEE Trans. Autom. Control. 40, 1054–1062 (1995)CrossRefGoogle Scholar
  13. 13.
    C.W. Scherer, P. Gahinet, M. Chilali, Multiobjective output-feedback control via LMI optimization. IEEE Trans. Autom. Control. 42, 896–911 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali. LMI Control Toolbox (The MathWorks, Natick, 1995)Google Scholar
  15. 15.
    F. Zheng, Q.G. Wang, H.T. Lee, On the design of multivariable PID controllers via LMI approach. Automatica 38, 517–526 (2002)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Mohammad Fathi
    • 1
  • Hassan Bevrani
    • 1
  1. 1.University of KurdistanKurdistanIran

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