Robust guaranteed cost control for uncertain linear time-delay systems

  • Huaizhong Li
  • Silviu-Iulian Niculescu
  • Luc Dugard
  • Jean-Michel Dion
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 228)


This chapter is concerned with robust guaranteed cost control for uncertain linear time-delay systems with quadratically constrained uncertainty using a linear matrix inequality (LMI) approach. We only consider the case of using memoryless static state feedback in this chapter. Two specific problems are considered in this chapter, namely the robust guaranteed cost control problem for linear systems with single state delay and the one for systems with mixed state and input delays. We show that feasibility of some LMIs guarantees the solvability of the corresponding robust guaranteed cost control problem.


Linear Matrix Inequality Null Space Uncertain Variable Input Delay Quadratic Cost Function 
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  1. 1.
    S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15. Philadelphia: SIAM Studies in Appl. Math., 1994.Google Scholar
  2. 2.
    J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “State-space solutions to standard \(\mathcal{H}_2\) and \(\mathcal{H}_\infty\) control problems,” IEEE Trans. Auto. Contr., vol. 34, no. 8, pp. 831–847, 1989.Google Scholar
  3. 3.
    E. Feron, V. Balakrishnan and S. Boyd, “A design of stabilizing state feedback for delay systems via convex optimization,” Proc. 31st IEEE Conf. Dec. Contr., Tuscon, Arizona, USA, pp. 147–148, 1992.Google Scholar
  4. 4.
    P. Gahinet and P. Apkarian, “A linear matrix inequality approach to \(\mathcal{H}_\infty\) control,” Int. J. Robust and Nonlinear Contr., vol. 4, pp. 421–428, 1994.Google Scholar
  5. 5.
    P. Gahinet, A. Nemirovskii, A. J. Laub and M. Chilali, LMI Control Toolbox, The MathWorks, Inc., 1995.Google Scholar
  6. 6.
    H. Górecki, S. Fuksa, P. Gabrowski and A. Korytowski, Analysis and Synthesis of Time Delay Systems, John Wiley & Sons, Warszawa, Poland, 1989.Google Scholar
  7. 7.
    V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations. New York: Academic Press, 1986.Google Scholar
  8. 8.
    N. N. Krasovskii, Stability of motion: Application of Lyapunov's Second Method to Defferential Systems and Equations with Delay. Stanford, California: Stanford University Press, 1963.Google Scholar
  9. 9.
    X. Li and C. E. de Souza, “LMI approach to delay-dependent robust stability and stabilization of uncertain linear delay systems,” in Proc. 34th IEEE CDC, New Orleans, Louisiana, 1995.Google Scholar
  10. 10.
    J. E. Marshall, H. Górecki, K. Walton and A. Korytowski, Time-delay systems: Stability and performance criteria with applications. Ellis Horwood, New York, 1992.Google Scholar
  11. 11.
    S.-I. Niculescu, On the stability and stabilization of linear systems with delayed-state, (in French). Ph.D. Thesis, Laboratoire d'Automatique de Grenoble, INPG, February, 1996.Google Scholar
  12. 12.
    S. I. Niculescu, C. E. de Souza, J.-M. Dion and L. Dugard, “Robust stability and stabilization for uncertain linear systems with state delay: Single delay case (I),” Proc. IFAC Workshop on Robust Control Design, Rio de Janeiro, Brazil, pp. 469–474, 1994.Google Scholar
  13. 13.
    S.-I. Niculescu, E. I. Verriest, L. Dugard and J.-M. Dion, “Stability and Robust Stability of Time-Delay Systems: A Guided Tour,” this monography (first chapter), LNCIS, Springer-Verlag, London, 1997.Google Scholar
  14. 14.
    I. R. Petersen and D. C. MacFarlane, “Optimal guaranteed cost control and filtering for uncertain linear systems,” IEEE Trans. Automat. Contr., vol. 39, pp. 1971–1977, 1994.Google Scholar
  15. 15.
    V. RĂsvan, Absolute stability of automatic control systems with delays (in Romanian). Eds. Academiei RSR, Bucharest, Romania, 1975.Google Scholar
  16. 16.
    S. O. Reza Moheimani and I. R. Petersen, “Optimal quadratic guaranteed cost control of a class of uncertain time-delay systems,” in Proc. 34th IEEE CDC, New Orleans, U.S.A., pp. 1513–1518, 1995.Google Scholar
  17. 17.
    A. Thowsen, “Uniform ultimate boundness of the solutions of uncertain dynamic delay systems with state-dependent and memoryless feedback control,” Int. J. Contr., vol. 37, pp. 1153–1143, 1983.Google Scholar
  18. 18.
    L. Xie and C. E. de Souza, “Robust stabilization and disturbance attenuation for uncertain delay system,” Proc. 2nd European Contr. Conf. Groningen, The Netherlands, pp. 667–672, 1993Google Scholar
  19. 19.
    V. A. Yakubovich, “S-procedure in nonlinear control theory,” Vestnik Leningradskogo Universiteta, Ser. Matematika, pp. 62–77, 1971.Google Scholar
  20. 20.
    K. Zhou, J. Doyle and K. Glover, Robust and optimal control, Prentice Hall, New Jersey, 1995.Google Scholar

Copyright information

© Springer-Verlag London Limited 1998

Authors and Affiliations

  • Huaizhong Li
    • 1
  • Silviu-Iulian Niculescu
    • 2
  • Luc Dugard
    • 1
  • Jean-Michel Dion
    • 1
  1. 1.Laboratoire d'Automatique de Grenoble (CNRS-INPG-UJF)ENSIEGSaint-Martin-d'HèresFrance
  2. 2.Laboratoire de Mathématiques AppliquéesEcole Nationale Supérieure de Techniques AvancéesParisFrance

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