Fixed-Order H-infinity Optimization of Time-Delay Systems

  • Suat Gumussoy
  • Wim Michiels
Conference paper


H-infinity controllers are frequently used in control theory due to their robust performance and stabilization. Classical H-infinity controller synthesis methods for finite dimensional LTI MIMO plants result in high-order controllers for highorder plants whereas low-order controllers are desired in practice. We design fixedorder H-infinity controllers for a class of time-delay systems based on a non-smooth, non-convex optimization method and a recently developed numerical method for H-infinity norm computations.


Linear Matrix Inequality Nonsmooth Optimization Nonconvex Optimization Nonlinear Eigenvalue Problem Exogenous Disturbance 
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.



This article presents results of the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Ministers Office for Science, Technology and Culture, the Optimization in Engineering Centre OPTEC of the K.U.Leuven, and the project STRT1-09/33 of the K.U.Leuven Research Foundation.


  1. 1.
    A. Blomqvist, A. Lindquist and R. Nagamune (2003) Matrix-valued Nevanlinna-Pick interpolation with complexity constraint: An optimization approach.IEEE Transactions on Automatic Control, 48:2172–2190.CrossRefMathSciNetGoogle Scholar
  2. 2.
    S. Boyd and V. Balakrishnan (1990) A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its \(\mathcal{L}_\infty\)-norm. Systems & Control Letters, 15:1–7.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    D. Breda, S. Maset and R. Vermiglio (2006) Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions. Applied Numerical Mathematics, 56:318–331.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    N.A. Bruinsma and M. Steinbuch (1990) A fast algorithm to compute the \(\mathcal{H}_\infty\)-norm of a transfer function matrix. Systems & Control Letters, 14:287–293.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    J.V. Burke, D. Henrion, A.S. Lewis and M.L. Overton (2006). Stabilization via nonsmooth, nonconvex optimization. IEEE Transactions on Automatic Control, 51:1760-1769.CrossRefMathSciNetGoogle Scholar
  6. 6.
    J.V. Burke, A.S. Lewis and M.L. Overton, (2003) A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM Journal on Optimization, 15:751–779.CrossRefMathSciNetGoogle Scholar
  7. 7.
    R. Byers (1988) A bisection method for measuring the distance of a stable matrix to the unstable matrices. SIAM Journal on Scientific and Statistical Computing, 9:875–881.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J.C. Doyle, K. Glover, P.P. Khargonekar and B.A. Francis (1989) State-Space solutions to standard \(\mathcal{H}^2\) and \(\mathcal{H}^\infty\) control problems. IEEE Transactions on Automatic Control 46:1968–1972.MathSciNetGoogle Scholar
  9. 9.
    P. Gahinet and P. Apkarian (1994) An Linear Matrix Inequality Approach to \(\mathcal{H}_\infty\) Control. International Journal of Robust and Nonlinear Control 4:421–448.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    S. Gumussoy and M.L. Overton, (2008) Fixed-Order H-Infinity Controller Design via HIFOO, a Specialized Nonsmooth Optimization Package. Proceedings of the American Control Conference 2750-2754.Google Scholar
  11. 11.
    R. Hryniv and P. Lancaster (1999) On the perturbation of analytic matrix functions. Integral Equations and Operator Theory, 34:325–338.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    A.S. Lewis and M.L.Overton, (2009) Nonsmooth optimization via BFGS. Submitted to SIAM Journal on Optimization.Google Scholar
  13. 13.
    W. Michiels and S. Gumussoy, (2009) Computation of H-infinity Norms for Time-Delay Systems. Accepted to SIAM Journal on Matrix Analysis and Applications. See also Technical Report TW551, Department of Computer Science, K.U.Leuven, 2009.Google Scholar
  14. 14.
    M. Millstone, (2006) HIFOO 1.5: Structured control of linear systems with a non-trivial feedthrough. Master’s Thesis, New York University.Google Scholar
  15. 15.
    J. Vanbiervliet, K. Verheyden, W. Michiels and S. Vandewalle (2008) A nonsmooth optimization approach for the stabilization of time-delay systems. ESAIM Control, Optimisation and Calcalus of Variations 14:478–493.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    K. Zhou, J.C. Doyle and K. Glover (1995) Robust and optimal control. Prentice Hall.Google Scholar

Copyright information

© Springer -Verlag Berlin Heidelberg 2010

Authors and Affiliations

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

Personalised recommendations