Robust Control pp 202-209 | Cite as

The mixed H2 and H control problem

  • A. A. Stoorvogel
  • H. L. Trentelman
Conference paper
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 183)


This paper is concerned with robust stability in combination with nominal performance. We pose an H norm bound on one transfer matrix to guarantee robust stability. Under this constraint we minimize an upper bound for the H 2 norm of a second transfer matrix. This transfer matrix is chosen so that its H 2 norm is a good measure for performance. We extend earlier work on this problem. The intention is to reduce this problem to a convex optimization problem.


The H2 control problem the H control problem robust stability auxiliary cost 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D. S. Bernstein and W.M. Haddad, “LQG control with and H performance bound: a Riccati equation approach”, IEEE Trans. Aut. Contr., 34 (1989), pp. 293–305.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    —, “Robust stability and performance analysis for state-space systems via quadratic Lyapunov bounds”, SIAM J. Matrix. Anal. & Appl., 11 (1990), pp. 239–271.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    J. Doyle, K. Glover, P.P. Khargonekar, and B.A. Francis, “State space solutions to standard H2 and H control problems”, IEEE Trans. Aut. Contr., 34 (1989), pp. 831–847.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    J.C. Doyle, K. Zhou, and B. Bodenheimer, “Optimal control with mixed H2 and H performance objectives”, in Proc. ACC, Pittsburgh, PA, 1989, pp. 2065–2070.Google Scholar
  5. [5]
    B.A. Francis, A course inH control theory, vol. 88 of Lecture notes in control and information sciences, Springer-Verlag, 1987.Google Scholar
  6. [6]
    P.P. Khargonekar, I.R. Petersen, and K. Zhou, “Robust stabilization of uncertain linear systems: quadratic stabilizability and H control theory”, IEEE Trans. Aut. Contr., 35 (1990), pp. 356–361.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    P.P. Khargonekar and M.A. Rotea, “Mixed H2/H control: a convex optimization approach”, To appear in IEEE Trans. Aut. Contr., 1990.Google Scholar
  8. [8]
    I.R. Petersen, “A stabilization algorithm for a class of uncertain linear systems”, Syst. & Contr. Letters, 8 (1987), pp. 351–357.zbMATHCrossRefGoogle Scholar
  9. [9]
    I.R. Petersen and C.V. Hollot, “A Riccati equation approach to the stabilization of uncertain linear systems”, Automatica, 22 (1986), pp. 397–411.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    M.A. Rotea and P.P. Khargonekar, “H2 optimal control with an H constraint: the state feedback case”, Automatica, 27 (1991), pp. 307–316.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    A.A. Stoorvogel, “The robust H2 control problem: a worst case design”, Submitted for publication.Google Scholar
  12. [12]
    —, “The singular H control problem with dynamic measurement feedback”, SIAM J. Contr. & Opt., 29 (1991), pp. 160–184.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    H.L. Trentelman, Almost invariant subspaces and high gain feedback, vol. 29 of CWI Tracts, Amsterdam, 1986.Google Scholar
  14. [14]
    J.C. Willems, “Least squares stationary optimal control and the algebraic Riccati equation”, IEEE Trans. Aut. Contr., 16 (1971), pp. 621–634.CrossRefMathSciNetGoogle Scholar
  15. [15]
    K. Zhou, J. Doyle, K. Glover, and B. Bodenheimer, “Mixed H2 and H control”, in Proc. ACC, San Diego, CA, 1990, pp. 2502–2507.Google Scholar
  16. [16]
    K. Zhou and P.P. Khargonekar, “An algebraic Riccati equation approach to H optimization”, Syst. & Contr. Letters, 11 (1988), pp. 85–91.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • A. A. Stoorvogel
    • 1
  • H. L. Trentelman
    • 2
  1. 1.Dept. of Electrical EngineeringUniversity of MichiganAnn ArborU.S.A.
  2. 2.Mathematics InstituteUniversity of GroningenGroningenThe Netherlands

Personalised recommendations