Shape Optimization of Transfer Functions

  • Jiawang Nie
  • James W. Demmel


We show how to optimize the shape of the transfer function of a linear time invariant (LTI) single-input-single-output (SISO) system. Since any transfer function is rational, this can be formulated as an optimization problem for the coefficients of polynomials. After characterizing the cone of polynomials which are nonnegative on intervals, we formulate this problem using semidefinite programming (SDP), which can be solved efficiently. This work extends prior results for discrete LTI SISO systems to continuous LTI SISO systems.

Key words

Linear system transfer function shape optimization nonnegative polynomials convex cone semidefinite programming 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AV02]
    B. Alkire and L. Vandenberghe, Convex optimization problems involving finite autocorrelation sequences. Mathematical Programming Series A 93 (2002), 331–359.MathSciNetCrossRefGoogle Scholar
  2. [CD91]
    Frank M. Callier, Charles A. Desoer, Linear System Theory, Springer-Verlag, New York, 1991.Google Scholar
  3. [Fab02]
    L. Faybusovich, On Nesterov’s approach to semi-infinite programming, Acta Applicandae Mathematicae 74 (2002), 195–215.MATHMathSciNetCrossRefGoogle Scholar
  4. [GHN00]
    Y. Genin, Y. Hachez, Yu. Nesterov, P. Van Dooren, “Convex Optimization over Positive Polynomials and filter design”, Proceedings UKACC Int. Conf. Control 2000, page SS41, 2000.Google Scholar
  5. [GHY03]
    Y. Genin, Y. Hachez, Yu. Nesterov, P. Van Dooren, Optimization problems over positive pseudopolynomial matrices, SIAM Journal on Matrix Analysis and Applications 25 (2003), 57–79.MathSciNetCrossRefGoogle Scholar
  6. [KS95]
    T. Kailath and A.H. Sayed, “Displacement Structure: theory and applications”, SIAM Rev. 37 (1995), 297–386.MathSciNetCrossRefGoogle Scholar
  7. [Luk18]
    Lukacs, “Verscharfung der ersten Mittelwersatzes der Integralrechnung für rationale Polynome”, Math. Zeitschrift, 2, 229–305, 1918.MathSciNetGoogle Scholar
  8. [Mar48]
    A.A. Markov, “Lecture notes on functions with the least deviation from zero”, 1906. Reprinted in Markov A.A. Selected Papers (ed. N. Achiezer), GosTechIzdat, 244–291, 1948, Moscow(in Russian).Google Scholar
  9. [NN94]
    Yu. Nesterov and A. Nemirovsky, “interior-point polynomial algorithms in convex programming”, SIAM Studies in Applied Mathematics, vol. 13, Society of Industrial and Applied Mathematics(SIAM), Philadelphia, PA, 1994.Google Scholar
  10. [Nes00]
    Yu. Nesterov, “Squared functional systems and optimization problems”, High Performance Optimization (H. Frenk et al., eds), Kluwer Academic Publishers, 2000, pp.405–440.Google Scholar
  11. [Pol97]
    E. Polak, “Optimization: Algorithms and Consistent Approximations”. Applied Mathematical Sciences, Vol. 124, Springer, New York, 1997.Google Scholar
  12. [PS76]
    G. Pólya and G. Szegö, Problems and Theorems in Analysis II, Springer-Verlag, New York, 1976Google Scholar
  13. [PR00]
    V. Powers and B. Reznick, “Polynomials That are Positive on an Interval”, Transactions of the American Mathematical Society, vol. 352, No. 10, pp. 4677–4692, 2000.MathSciNetCrossRefGoogle Scholar
  14. [WBV97]
    S.-P. Wu, S. Boyd, and L. Vandenberghe, “FIR filter design via spectral factorization and convex optimization”, Applied and Computational Control, Signals and Circuits, B. Datta, ed., Birkhäuser, 1997, ch.2, pp.51–81.Google Scholar
  15. [Par01]
    P.A. Parrilo. Semidefinite Programming relaxations for semialgebraic problems. Math. Prog., No. 2, Ser. B, 293–320, 96 (2003).MATHMathSciNetCrossRefGoogle Scholar
  16. [Stu99]
    J.F. Sturm, “SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones”, Optimization Methods and Software, 11&12 (1999) 625–653.MATHMathSciNetGoogle Scholar
  17. [VB96]
    L. Vandenberghe and S. Boyd, “Semidefinite Programming”, SIAM Review, 38(1):49–95, 1996.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Jiawang Nie
    • 1
  • James W. Demmel
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations