Design of Fixed-Order Stabilizing and \(\mathcal{H}_2\) - \(\mathcal{H}_\infty\) Optimal Controllers: An Eigenvalue Optimization Approach

  • Wim Michiels
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 423)


An overview is presented of control design methods for linear time-delay systems, which are grounded in numerical linear algebra techniques such as large-scale eigenvalue computations, solving Lyapunov equations and eigenvalue optimization. The methods are particularly suitable for the design of controllers with a prescribed structure or order. The analysis problems concern the computation of stability determining characteristic roots and the computation of \(\mathcal{H}_2\) and \(\mathcal{H}_{\infty}\) type cost functions. The corresponding synthesis problems are solved by a direct optimization of stability, robustness and performance measures as a function of the controller parameters.


Delay Differential Equation Characteristic Root Synthesis Problem Nonlinear Eigenvalue Problem Wolfe Line Search 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Boyd, S., Balakrishnan, V.: A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L  ∞ -norm. Systems & Control Letters 15, 1–7 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Breda, D., Maset, S., Vermiglio, R.: Pseudospectral differencing methods for characteristic roots of delay differential equations. SIAM Journal on Scientific Computing 27(2), 482–495 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Breda, D., Maset, S., Vermiglio, R.: TRACE-DDE: a Tool for Robust Analysis and Characteristic Equations for Delay Differential Equations. In: Loiseau, J.J., Michiels, W., Niculescu, S.-I., Sipahi, R. (eds.) Topics in Time Delay Systems. LNCIS, vol. 388, pp. 145–155. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Bruinsma, N.A., Steinbuch, M.: A fast algorithm to compute the \(\mathcal{H}_{\infty}\)-norm of a transfer function matrix. Systems and Control Letters 14, 287–293 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Burke, J.V., Henrion, D., Lewis, A.S., Overton, M.L.: HIFOO - a matlab package for fixed-order controller design and H-infinity optimization. In: Proceedings of the 5th IFAC Symposium on Robust Control Design, Toulouse, France (2006)Google Scholar
  6. 6.
    Burke, J.V., Lewis, A.S., Overton, M.L.: A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM Journal on Optimization 15(3), 751–779 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Curtain, R.F., Zwart, H.: An introduction to infinite-dimensional linear systems theory. Texts in Applied Mathematics, vol. 21. Springer, Heidelberg (1995)zbMATHCrossRefGoogle Scholar
  8. 8.
    Engelborghs, K., Luzyanina, T., Roose, D.: Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Trans. Math. Softw. 28(1), 1–24 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Gu, K., Kharitonov, V.L., Chen, J.: Stability of time-delay systems. Birkhauser (2003)Google Scholar
  10. 10.
    Gumussoy, S., Michiels, W.: Fixed-order strong H-infinity control of interconnected systems with time-delays. TW Report 579, Department of Computer Science, Katholieke Universiteit Leuven, Belgium (October 2010)Google Scholar
  11. 11.
    Gumussoy, S., Michiels, W.: A predictor corrector type algorithm for the pseudospectral abscissa computation of time-delay systems. Automatica 46(4), 657–664 (2010)zbMATHCrossRefGoogle Scholar
  12. 12.
    Gumussoy, S., Overton, M.L.: Fixed-order H-infinity controller design via HIFOO, a specialized nonsmooth optimization package. In: Proceedings of the American Control Conference, Seattle, USA, pp. 2750–2754 (2008)Google Scholar
  13. 13.
    Jarlebring, E., Meerbergen, K., Michiels, W.: A Krylov method for the delay eigenvalue problem. SIAM Journal on Scientific Computing 32(6), 3278–3300 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Krstic, M.: Delay compensation for nonlinear, adaptive, and PDE systems. Birkhauser (2009)Google Scholar
  15. 15.
    Jarlebring, E., Vanbiervliet, J., Michiels, W.: Characterizing and computing the \(\mathcal{H}_2\) norm of time-delay systems by solving the delay Lyapunov equation. Technical Report 553, K.U.Leuven, Leuven (2009), To appear in IEEE Transactions on Automatic ControlGoogle Scholar
  16. 16.
    Kharitonov, V., Plischke, E.: Lyapunov matrices for time-delay systems. Syst. Control Lett. 55(9), 697–706 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Lewis, A., Overton, M.L.: Nonsmooth optimization via BFGS (2009),
  18. 18.
    Mahmoud, M.S.: Robust control and filtering for time-delay systems. Control Engineering, vol. 5. Dekker, New York (2000)zbMATHGoogle Scholar
  19. 19.
    Michiels, W., Gumussoy, S.: Characterization and computation of h-infinity norms of time-delay systems. SIAM Journal on Matrix Analysis and Applications 31(4), 2093–2115 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Michiels, W., Niculescu, S.-I.: Stability and stabilization of time-delay systems. An eigenvalue based approach. An Eigenvalue Based Approach. SIAM (2007)Google Scholar
  21. 21.
    Michiels, W., Vyhlídal, T., Zítek, P.: Control design for time-delay systems based on quasi-direct pole placement. Journal of Process Control 20(3), 337–343 (2010)CrossRefGoogle Scholar
  22. 22.
    Niculescu, S.-I.: Delay effects on stability. A robust control approach. LNCIS, vol. 269. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  23. 23.
    Overton, M.: HANSO: a hybrid algorithm for nonsmooth optimization (2009),
  24. 24.
    Saad, Y.: Numerical methods for large eigenvalue problems. Manchester University Press (1992)Google Scholar
  25. 25.
    Sipahi, R., Niculescu, S.I., Abdallah, C.T., Michiels, W., Gu, K.: Stability and stabilization of systems with time delay. Control Systems Magazine 31(38-65), 3278–3300 (2011)MathSciNetGoogle Scholar
  26. 26.
    Vanbiervliet, J., Michiels, W., Jarlebring, E.: Using spectral discretisation for the optimal \(\mathcal{H}_2\) design of time-delay systems. International Journal of Control (in press, 2011)Google Scholar
  27. 27.
    Vanbiervliet, J., Michiels, W., Vandewalle, S.: Smooth stabilization and optimal h2 design. In: Proceedings of the IFAC Workshop on Control Applications of Optimization, Jyväskylä, Finland (2009)Google Scholar
  28. 28.
    Vanbiervliet, J., Vandereycken, B., Michiels, W., Vandewalle, S.: A nonsmooth optimization approach for the stabilization of time-delay systems. ESAIM Control, Optimisation and Calcalus of Variations 14(3), 478–493 (2008)zbMATHCrossRefGoogle Scholar
  29. 29.
    Vanbiervliet, J., Vandereycken, B., Michiels, W., Vandewalle, S., Diehl, M.: The smoothed spectral abscissa for robust stability optimization. SIAM Journal on Optimization 20(1), 156–171 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Vyhlídal, T., Zítek, P.: Mapping based algorithm for large-scale computation of quasi-polynomial zeros. IEEE Transactions on Automatic Control 54(1), 171–177 (2009)CrossRefGoogle Scholar
  31. 31.
    Vyhlídal, T., Zítek, P., Paulů, K.: Design, modelling and control of the experimental heat transfer set-up. In: Loiseau, J.J., et al. (eds.) Topics in Time Delay Systems. Analysis, Algorithms, and Control. LNCIS, vol. 308, pp. 303–314. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  32. 32.
    Zhou, K., Doyle, J.C., Glover, K.: Robust and optimal control. Prentice-Hall (1995)Google Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Computer ScienceK.U. LeuvenHeverleeBelgium

Personalised recommendations