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Robust Control pp 202–209Cite as

The mixed H 2 and H ∞ control problem

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Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 183))

Abstract

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.

supported by the Netherlands Organization for Scientific Research (NWO).

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References

  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.

    Article  MATH  MathSciNet  Google Scholar 

  2. —, “Robust stability and performance analysis for state-space systems via quadratic Lyapunov bounds”, SIAM J. Matrix. Anal. & Appl., 11 (1990), pp. 239–271.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. B.A. Francis, A course inH∞ control theory, vol. 88 of Lecture notes in control and information sciences, Springer-Verlag, 1987.

    Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. I.R. Petersen, “A stabilization algorithm for a class of uncertain linear systems”, Syst. & Contr. Letters, 8 (1987), pp. 351–357.

    Article  MATH  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  11. A.A. Stoorvogel, “The robust H2 control problem: a worst case design”, Submitted for publication.

    Google Scholar 

  12. —, “The singular H ∞ control problem with dynamic measurement feedback”, SIAM J. Contr. & Opt., 29 (1991), pp. 160–184.

    Article  MATH  MathSciNet  Google Scholar 

  13. H.L. Trentelman, Almost invariant subspaces and high gain feedback, vol. 29 of CWI Tracts, Amsterdam, 1986.

    Google Scholar 

  14. J.C. Willems, “Least squares stationary optimal control and the algebraic Riccati equation”, IEEE Trans. Aut. Contr., 16 (1971), pp. 621–634.

    Article  MathSciNet  Google Scholar 

  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. K. Zhou and P.P. Khargonekar, “An algebraic Riccati equation approach to H∞ optimization”, Syst. & Contr. Letters, 11 (1988), pp. 85–91.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Author notes

  1. A. A. Stoorvogel

    Present address: Dept. of Mathematics, Eindhoven University of Technology, the Netherlands

Authors and Affiliations

  1. Dept. of Electrical Engineering, University of Michigan, 48109-2122, Ann Arbor, MI, U.S.A.

    A. A. Stoorvogel

  2. Mathematics Institute, University of Groningen, P.O. Box 800, 9700 AV, Groningen, The Netherlands

    H. L. Trentelman

Authors
  1. A. A. Stoorvogel
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  2. H. L. Trentelman
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Editor information

L. D. Davisson A. G. J. MacFarlane H. Kwakernaak J. L. Massey Ya Z. Tsypkin A. J. Viterbi Shigeyuki Hosoe

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© 1992 Springer-Verlag

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Stoorvogel, A.A., Trentelman, H.L. (1992). The mixed H 2 and H ∞ control problem. In: Davisson, L.D., et al. Robust Control. Lecture Notes in Control and Information Sciences, vol 183. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0114667

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  • DOI: https://doi.org/10.1007/BFb0114667

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55961-0

  • Online ISBN: 978-3-540-47320-6

  • eBook Packages: Springer Book Archive

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