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Delay Systems pp 243-256 | Cite as

Eigenvalue Based Algorithms and Software for the Design of Fixed-Order Stabilizing Controllers for Interconnected Systems with Time-Delays

  • Wim Michiels
  • Suat Gumussoy
Part of the Advances in Delays and Dynamics book series (ADVSDD, volume 1)

Abstract

An eigenvalue based framework is developed for the stability analysis and stabilization of coupled systems with time-delays, which are naturally described by delay differential algebraic equations. The spectral properties of these equations are analyzed and their stability properties are studied, taking into account the effect of small delay perturbations. Subsequently, numerical methods for stability assessment and for designing stabilizing controllers with a prescribed structure or order, based on a direct optimization approach, are briefly addressed. The effectiveness of the approach is illustrated with a software demo. The paper concludes by pointing out the similarities with the computation and optimization of \(\mathcal{H}_{\infty}\) norms.

Keywords

Exponential Stability Characteristic Root Static Controller Strong Stability Null Solution 
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.

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References

  1. 1.
    Avellar, C.E., Hale, J.K.: On the zeros of exponential polynomials. Mathematical Analysis and Applications 73, 434–452 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press (2004)Google Scholar
  3. 3.
    Fridman, E., Shaked, U.: H  ∞ -control of linear state-delay descriptor systems: an LMI approach. Linear Algebra and its Applications 351-352, 271–302 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gumussoy, S., Michiels, W.: Fixed-order H-infinity control for interconnected systems using delay differential algebraic equations. SIAM Journal on Control and Optimization 49(5), 2212–2238 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hale, J.K., Verduyn Lunel, S.M.: Strong stabilization of neutral functional differential equations. IMA Journal of Mathematical Control and Information 19, 5–23 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Michiels, W., Vyhlidal, T.: An eigenvalue based approach to the robust stabilization of linear time-delay systems of neutral type. Automatica 41(6), 991–998 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Michiels, W., Gumussoy, S.: Eigenvalue based algorithms and software for the design of fixed-order stabilizing controllers for interconnected systems with time-delays. In: 10th IFAC Workshop on Time Delay Systems, June 22-24. IFAC-PapersOnLine, pp. 144–149. Northeastern University, USA (2012), doi:10.3182/20120622-3-US-4021.00015Google Scholar
  8. 8.
    Michiels, W., Engelborghs, K., Roose, D., Dochain, D.: Sensitivity to infinitesimal delays in neutral equations. SIAM Journal on Control and Optimization 40(4), 1134–1158 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Michiels, W.: Spectrum based stability analysis and stabilization of systems described by delay differential algebraic equations. IET Control Theory and Applications 5(16), 1829–1842 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Michiels, W., Vyhlídal, T., Zítek, P., Nijmeijer, H., Henrion, D.: Strong stability of neutral equations with an arbitrary delay dependency structure. SIAM Journal on Control and Optimization 48(2), 763–786 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Overton, M.: HANSO: a hybrid algorithm for nonsmooth optimization (2009), http://cs.nyu.edu/overton/software/hanso/
  12. 12.
    Vanbiervliet, J., Vandereycken, B., Michiels, W., Vandewalle, S.: A nonsmooth optimization approach for the stabilization of time-delay systems. ESAIM Control, Optimisation and Calculus of Variations 14(3), 478–493 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Vyhlidal, T., Michiels, W., McGahan, P.: Synthesis of strongly stable state-derivative controllers for a time delay system using constrained non-smooth optimization. IMA Journal of Mathematical Control and Information 27(4), 437–455 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceKU LeuvenHeverleeBelgium
  2. 2.MathWorksNatickUSA

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