Advertisement

Computational complexity in robust controller synthesis

  • Shinji Hara
  • Yuji Yamada
Part A Learning And Computational Issues
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 241)

Abstract

This paper is concerned with the computational complexity analysis in robust control problems. We first formulate a fairly general class of nonconvex optimization problem named “Matrix Product Eigenvalue Problem (MPEP)” and explain the connection to robust control problems. We next summarize the worst case computational complexity results and investigate the computational cost for the actual case. Finally, we make a comparison with an element-wise bounding for the BMI optimization problem and a matrix-based bounding for the MPEP.

Keywords

Linear Matrix Inequality Relaxation Problem Static Output Feedback Nonconvex Optimization Problem Computational Complexity Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S.P. Boyd et al., Linear Matrix Inequalities in System and Control Theory, SIAM, 1994.Google Scholar
  2. [2]
    H. Fujioka and K. Hoshijima, “Bounds for the BMI Eigenvalue Problem,” Trans. SICE, 33(7):616–621, 1997.Google Scholar
  3. [3]
    Hisaya Fujioka and Kohji Yamashita, “Computational Aspects of Constantly Scaled Sampled-Data H Control Synthesis,” Proc. IEEE CDC, 440–445, 1996.Google Scholar
  4. [4]
    P. Gahinet and P. Apkarian, “An LMI-based Parameterization of all H Controllers with Applications,” Proc. IEEE CDC, 656–661, 1993.Google Scholar
  5. [5]
    L. El Ghaoui, F. Oustry, and M. AitRami, “A Cone Complementarity Linearization Algorithm for Static Output-Feedback and Related Problems,” IEEE Trans. Automat. Contr., AC-42(8):1171–1176, 1997.CrossRefGoogle Scholar
  6. [6]
    L. El Ghaoui, V. Balakrishnan, E. Feron and S.P. Boyd, “On maximizing a robustness measure for structured nonlinear perturbations,” Proc. ACC, 2923–2924, 1992.Google Scholar
  7. [7]
    K.G. Goh, M.G. Safonov, and G.P. Papavassilopoulos, “A Global Optimization Approach for the BMI Problem,” Proc. IEEE CDC, 2009–2014, 1994.Google Scholar
  8. [8]
    T. Iwasaki and R.E. Skelton, “All Controllers for the General H Control Problem: LMI Existence Conditions and State Space Formulas,” Automatica, 30(8):1307–1318, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    T. Iwasaki and R.E. Skelton, “The XY-centering algorithm for the dual LMI problem: A new approach to fixed order control design,” Int. J. Control, 62(6): 1257–1272, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    H. Konno and T. Kuno, “Linear multiplicative programming,” Mathematical Programming, 56: 51–64, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    T. Kuno and H. Konno, “A Parametric Successive Underestimation Method for Convex Multiplicative Programming Problems,” Journal of Global Optimization, 1: 267–286, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    M. Mesbahi and G.P. Papavassilopoulos, “On the Rank Minimization Problem Over a Positive Semidefinite Linear Matrix Inequality,” IEEE Trans. Automat. Contr., AC-42(2):239–243, 1997.CrossRefMathSciNetGoogle Scholar
  13. [13]
    G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd (Eds.), Optimization: Handbooks in Operations Research and Management Science, 1, Esevier, 1989.Google Scholar
  14. [14]
    A. Packard, “Gain scheduling via linear fractional transformations,” Syst. Contr. Lett., 22:79–92, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    A. Packard, K. Zhou, P. Pandey and G. Becker, “A Collection of robust control problem leading to LMIs,” Proc. IEEE CDC, 1245–1250, 1991.Google Scholar
  16. [16]
    M.G. Safonov, K.G. Goh, and J.H. Ly, “Control System Synthesis via Bilinear Matrix Inequalities,” Proc. ACC, 45/49, 1994.Google Scholar
  17. [17]
    Y. Takano, T. Watanabe and K. Yasuda, “Branch and Bound Technique for Global Solution of BMI,” Trans. SICE, 33(7):701–708, 1997 (in Japanese).Google Scholar
  18. [18]
    O. Toker and H. Özbay, “On the NP-Hardness of Solving Bilinear Matrix Inequalities and Simultaneous Stabilization with Static Output Feedback,” Proc. ACC, 2525–2526, 1995.Google Scholar
  19. [19]
    H.D. Tuan and S. Hosoe, “DC optimization approach to robust controls: The optimal scaling value problem,” Proc. ACC, 1997.Google Scholar
  20. [20]
    L. Vandenberghe and S. P. Boyd, “Semidefinite Programming,” SIAM Review, 38(1):49–95, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Y. Yamada, “Global Optimization for Robust Control Synthesis based on the Matrix Product Eigenvalue Problem,” Ph. D. dissertation, Tokyo Inst. of Tech., 1998.Google Scholar
  22. [22]
    Y. Yamada and S. Hara, “A Global Algorithm for Scaled Spectral Norm Optimization,” Proc. The 25th SICE Symp. on Control Theory, 177/182, 1996.Google Scholar
  23. [23]
    Y. Yamada and S. Hara, “Global Optimization for H Control with Block-diagonal Constant Scaling,” Proc. IEEE CDC, 1325–1330, 1996.Google Scholar
  24. [24]
    Y. Yamada and S. Hara, “The Matrix Product Eigenvalue Problem — Global optimization for the spectral radius of a matrix product under convex constraints —,” Proc. IEEE CDC, 4926–4931, 1997.Google Scholar
  25. [25]
    Y. Yamada and S. Hara, “Global Optimization for H Control with Constant Diagonal Scaling,” IEEE Trans. AC, AC-43(2): 191–203, 1998.MathSciNetGoogle Scholar
  26. [26]
    Y. Yamada, S. Hara, and H. Fujioka, “Global Optimization for Constantly Scaled H Control Problems,” Proc. ACC, 427–430, 1995.Google Scholar
  27. [27]
    Y. Yamada, S. Hara, and H. Fujioka, “ε-Feasibility for H Control Problem with Constant Diagonal Scaling,” Trans. SICE, 33(3): 155–162, 1997.Google Scholar

Copyright information

© Springer-Verlag London Limited 1999

Authors and Affiliations

  • Shinji Hara
  • Yuji Yamada
    • 1
  1. 1.Department of Computational Intelligence and Systems ScienceTokyo Institute of TechnologyYokohamaJapan

Personalised recommendations