Journal of Optimization Theory and Applications

, Volume 142, Issue 3, pp 603–618 | Cite as

H Control and Exponential Stability of Nonlinear Nonautonomous Systems with Time-Varying Delay

  • V. N. PhatEmail author
  • Q. P. Ha


This paper addresses the design of H state feedback controllers for a class of nonlinear time-varying delay systems. The interesting features here are that the system in consideration is nonautonomous with fast-varying delays, the delay is also involved in the observation output, and the controllers to be designed satisfy some exponential stability constraints on the closed-loop poles. By using the proposed Lyapunov functional approach, neither a controllability assumption nor a bound restriction on nonlinear perturbations is required to obtain new sufficient conditions for the H control. The conditions are derived in terms of a solution to the standard Riccati differential equations, which allows for simultaneous computation of the two bounds that characterize the stability rate of the solution.


H control Exponential stability Nonlinear perturbation Time-varying delay Riccati equations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chukwu, E.N.: Stability and Time-Optimal Control of Hereditary Systems. Academic Press, San Diego (1992) zbMATHGoogle Scholar
  2. 2.
    Udwadia, F.E., von Bremen, H., Phohomsiri, P.: Time-delayed control design for active control of structures: principles and applications. Struct. Control Health Monit. 14, 27–61 (2007) CrossRefGoogle Scholar
  3. 3.
    Udwadia, F.E., Hosseini, M., Chen, Y.: Robust control of uncertain systems with time varying delays in control input. In: Proceedings of the American Control Conference, USA, pp. 3840–3845 (1997) Google Scholar
  4. 4.
    Zhong, Q.-C.: Robust Control of Time-delay Systems. Springer, New York (2006) zbMATHGoogle Scholar
  5. 5.
    Xie, L., Fridman, E., Shaked, U.: Robust H control of distributed delay systems with application to combustion control. IEEE Trans. Automat. Contr. 46(12), 1930–1935 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Niculescu, S.I.: H memoryless control with stability constraint for time-delay systems: an LMI approach. IEEE Trans. Automat. Contr. 43, 739–743 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Shaked, U., Yaesh, I.: H static output-feedback control of linear continuous-time systems with delay. IEEE Trans. Automat. Contr. 43, 1431–1436 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fridman, E., Shaked, U., Xie, L.: Robust H filtering of linear systems with time-varying delay. IEEE Trans. Automat. Contr. 48(1), 159–165 (2003) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Trinh, H., Ha, Q.P.: State and input simultaneous estimation for a class of time-delay systems with uncertainties. IEEE Trans. Circuits Syst. II 54, 527–531 (2007) CrossRefGoogle Scholar
  10. 10.
    Phat, V.N., Niamsup, P.: Stabilization of linear non-autonomous systems with norm bounded controls. J. Optim. Theory Appl. 131, 135–149 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ravi, R., Nagpal, K.M., Khargonekar, P.P.: H control of linear time-varying systems: A state-space approach. SIAM J. Control Optim. 29, 1394–1413 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Zhang, H., Xie, L., Wang, W., Lu, X.: An innovation approach to H fixed-lag smoothing for continuous time-varying systems. IEEE Trans. Automat. Contr. 49(12), 2240–2244 (2004) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Phat, V.N., Vinh, D.Q.: Controllability and H control for linear continuous time-varying uncertain systems. Differ. Equ. Appl. 4, 105–111 (2007) MathSciNetGoogle Scholar
  14. 14.
    Phat, V.N., Vinh, D.Q., Bay, N.S.: L 2-stabilization and H control for linear non-autonomous time-delay systems in Hilbert spaces via Riccati equations. Adv. Nonlinear Var. Inequal. 11, 1–12 (2008) MathSciNetGoogle Scholar
  15. 15.
    Phat, V.N.: Nonlinear H optimal control in Hilbert spaces via Riccati operator equations. Nonlinear Funct. Anal. Appl. 9, 79–92 (2004) zbMATHMathSciNetGoogle Scholar
  16. 16.
    Boyd, S., Il Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in Systems and Control Theory. Studies in Applied Mathematics. SIAM, Philadelphia (1994) Google Scholar
  17. 17.
    Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993) zbMATHGoogle Scholar
  18. 18.
    Trinh, H., Aldeen, M.: On robustness and stabilization of linear systems with delayed nonlinear perturbations. IEEE Trans. Automat. Contr. 42, 1005–1007 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Zevin, A., Pinsky, M.: Exponential stability and solution bounds for systems with bounded nonlinearities. IEEE Trans. Automat. Contr. 48, 1799–1804 (2003) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Parlakci, M.N.A.: Improved robust stability criteria for time delay systems with nonlinear perturbations. In: Proc. of ICCA Conference, Budapest, June 2005, pp. 341–346 Google Scholar
  21. 21.
    Laub, A.J.: Schur techniques for solving Riccati differential equations. IEEE Trans. Automat. Contr. 24, 913–921 (1979) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    William Thomas, R.: Riccati Differential Equations. Academic Press, San Diego (1972) Google Scholar
  23. 23.
    Abou-Kandil, H., Freiling, G., Ionescu, V., Jank, G.: Matrix Riccati Equations in Control and Systems Theory. Birkhäuser, Basel (2003) zbMATHGoogle Scholar
  24. 24.
    Klamka, J.: Controllability of Dynamical Systems. Kluwer Academic, Dordrecht (1991) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institute of MathematicsHanoiVietnam
  2. 2.Faculty of Engineering and Information TechnologyUniversity of Technology SydneySydneyAustralia

Personalised recommendations