Skip to main content
SpringerLink
Log in

A bisection method for computing the H∞ norm of a transfer matrix and related problems

  • Published:
Mathematics of Control, Signals and Systems Aims and scope Submit manuscript

Abstract

We establish a correspondence between the singular values of a transfer matrix evaluated along the imaginary axis and the imaginary eigenvalues of a related Hamiltonian matrix. We give a simple linear algebraic proof, and also a more intuitive explanation based on a certain indefinite quadratic optimal control problem. This result yields a simple bisection algorithm to compute the H∞ norm of a transfer matrix. The bisection method is far more efficient than algorithms which involve a search over frequencies, and the usual problems associated with such methods (such as determining how fine the search should be) do not arise. The method is readily extended to compute other quantities of system-theoretic interest, for instance, the minimum dissipation of a transfer matrix. A variation of the method can be used to solve the H∞ Armijo line-search problem with no more computation than is required to compute a single H∞ norm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. D. O. Anderson, An algebraic solution to the spectral factorization problem,IEEE Trans. Automat. Control,12 (1967), 410–414.

    Article  Google Scholar 

  2. S. Barnett and D. D. Siljak, Routh’s algorithm: a centennial survey,SIAM Rev.,19 (1977), 472–489.

    Article  MathSciNet  Google Scholar 

  3. S. Boyd and J. Doyle, Comparison of peak and RMS gains for discrete-time systems,Systems Control Lett.,9 (1987), 1–6.

    Article  MathSciNet  Google Scholar 

  4. R. Byers, A HamiltonianQR algorithm,SIAM J. Sci. Statist. Comput.,7 (1986), 212–229.

    Article  MathSciNet  Google Scholar 

  5. R. Byers, A Bisection Algorithm for Measuring the Distance of a Stable Matrix to the Unstable Matrices, Technical Report, North Carolina State University at Raleigh, 1987; to appear inSIAM J. Sci. Statist. Comput.

  6. C. A. Desoer and M. Vidyasagar,Feedback Systems: Input-Output Properties, Academic Press, New York, 1975.

    MATH  Google Scholar 

  7. D. F. Enns, Model reduction with balanced realizations: an error bound and a frequency weighted generalization,Proceedings of the 23rd IEEE Conference on Decision and Control, Las Vegas, NV, 1984, pp. 127–132.

  8. B. A. Francis,A Course in H ∞ Control Theory, Lecture Notes in Control and Information Sciences, Vol. 88, Springer-Verlag, New York, 1987.

    Book  MATH  Google Scholar 

  9. K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and theirL ∞-error bounds,Internat. J. Control,39 (1984), 1115–1193.

    MathSciNet  Google Scholar 

  10. G. H. Golub, S. Nash, and C. Van Loan, A Hessenberg-Schur method for the matrix problemAX+XB=C, IEEE Trans. Automat. Control,24 (1979), 909–913.

    Article  MathSciNet  Google Scholar 

  11. G. H. Golub and C. F. Van Loan,Matrix Computations, Johns Hopkins University Press, Baltimore, MD, 1983.

    MATH  Google Scholar 

  12. P. Henrici,Applied and Computational Complex Analysis, Vol. 1, Wiley-Interscience, New York, 1978.

    Google Scholar 

  13. T. Kailath,Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980.

    MATH  Google Scholar 

  14. A. J. Laub, Efficient calculation of frequency response matrices from state space models,ACM Trans. Math. Software,12 (1986), 26–33.

    Article  Google Scholar 

  15. D. G. Luenberger,Linear and Nonlinear Programming, 2nd edn., Addison-Wesley, Reading, MA, 1984.

    MATH  Google Scholar 

  16. C. F. Van Loan, A symplectic method for approximating all eigenvalues of a Hamiltonian matrix,Linear Algebra Appl.,61 (1984), 233–251.

    Article  MathSciNet  Google Scholar 

  17. C. F. Van Loan, How near is a stable matrix to an unstable one?Contemp. Math.,47 (1985), 465–478.

    Google Scholar 

  18. M. Vidyasagar,Control System Synthesis: A Factorization Approach, MIT Press, Cambridge, MA, 1985.

    MATH  Google Scholar 

  19. J. C. Willems, Least squares stationary optimal control and the algebraic Riccati equation,IEEE Trans. Automat. Control,16 (1971), 621–634.

    Article  MathSciNet  Google Scholar 

  20. G. Zames and B. A. Francis, Feedback, minimax sensitivity, and optimal robustness,IEEE Trans. Automat. Control,28 (1983), 585–601.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Information Systems Laboratory, Electrical Engineering Department, Stanford University, 94305, Stanford, California, USA

    S. Boyd & V. Balakrishnan

  2. Aerospace Engineering Department, University of Michigan, Ann Arbor, Michigan, USA

    P. Kabamba

Authors
  1. S. Boyd
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Balakrishnan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. Kabamba
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported in part by NSF under Grant ECS-85-52465, ONR under Grant N00014-86-K-0112, an IBM faculty development award, and Bell Communications Research.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Boyd, S., Balakrishnan, V. & Kabamba, P. A bisection method for computing the H∞ norm of a transfer matrix and related problems. Math. Control Signal Systems 2, 207–219 (1989). https://doi.org/10.1007/BF02551385

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02551385

Key words

Search

Navigation