Numerical Methods for Robust Control

  • P. Hr. Petkov
  • A. S. Yonchev
  • N. D. Christov
  • M. M. Konstantinov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4818)


A brief survey on the numerical properties of the methods for \({\cal H}_\infty\) design and μ-analysis and synthesis of linear control systems is given. A new approach to the sensitivity analysis of LMI – based \({\cal H}_\infty\) design is presented that allows to obtain linear perturbation bounds on the computed controller matrices. Some results from a detailed numerical comparison of the properties of the different available \({\cal H}_\infty\) optimization methods are then presented. We also discuss the sensitivity of the structured singular value (μ) that plays an important role in the robust stability analysis and design of linear control systems.


Linear Matrix Inequality Riccati Equation Numerical Property Linear Control System Controller Matrice 
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.
    Balas, G., et al.: Robust Control Toolbox User’s Guide. The MathWorks, Inc., Natick, MA (2006)Google Scholar
  2. 2.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. In: MPS-SIAM Series on Optimization, SIAM, Philadelphia (2001)Google Scholar
  3. 3.
    Christov, N.D., Konstantinov, M.M., Petkov, P.Hr.: New perturbation bounds for the continuous-time H  ∞  - optimization problem. In: Li, Z., Vulkov, L.G., Waśniewski, J. (eds.) NAA 2004. LNCS, vol. 3401, pp. 232–239. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Gahinet, P.: Explicit controller formulas for LMI-based H  ∞  synthesis. Automatica 32, 1007–1014 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gahinet, P., Apkarian, P.: A linear matrix inequality approach to H  ∞  control. Int. J. Robust Non. Contr. 4, 421–448 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Green, M., Limebeer, D.J.N.: Linear Robust Control. Prentice Hall, Englewood Cliffs, NJ (1995)zbMATHGoogle Scholar
  7. 7.
    Higham, N., et al.: Sensitivity of computational control problems. IEEE Control Syst. Magazine 24, 28–43 (2004)CrossRefGoogle Scholar
  8. 8.
    Iglesias, P., Glover, K.: State-space approach to discrete-time H  ∞ -control. International J. of Control 54, 1031–1074 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Nesterov, Y., Nemirovski, A.: Interior–Point Polynomial Algorithms in Convex Programming. SIAM, Philladelphia (1994)CrossRefGoogle Scholar
  10. 10.
    Petkov, P.Hr., Gu, D.-W., Konstantinov, M.M.: Fortran 77 Routines for H  ∞  and H 2 Design of Discrete-Time Linear Control Systems. NICONET Report 1999-5, available electronically at
  11. 11.
    Yonchev, A., Konstantinov, M., Petkov, P.: Linear Matrix Inequalities in Control Theory. Demetra, Sofia (in Bulgarian) (2005) ISBN 954-9526-32-1Google Scholar
  12. 12.
    Young, P.M., Newlin, M.P., Doyle, J.C.: Computing bounds for the mixed μ problem. Int. J. of Robust and Nonlinear Control 5, 573–590 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zhou, K., Doyle, J.C., Glover, K.: Robust and Optimal Control, p. 07458. Prentice-Hall, Upper Saddle River, New Jersey (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • P. Hr. Petkov
    • 1
  • A. S. Yonchev
    • 1
  • N. D. Christov
    • 2
  • M. M. Konstantinov
    • 3
  1. 1.Technical University of SofiaSofiaBulgaria
  2. 2.Université des Sciences et Technologies de LilleVilleneuve d’AscqFrance
  3. 3.Civil Engineering and GeodesyUniversity of ArchitectureSofiaBulgaria

Personalised recommendations