Kybernetika 51 no. 1, 99-111, 2015

A separation principle for the stabilization of a class of time delay nonlinear systems

Amel BenabdallahDOI: 10.14736/kyb-2015-1-0099


In this paper, we establish a separation principle for a class of time-delay nonlinear systems satisfying some relaxed triangular-type condition. Under delay independent conditions, we propose a nonlinear time-delay observer to estimate the system states, a state feedback controller and we prove that the observer-based controller stabilizes the system.


delay system, output feedback stabilization, nonlinear observer, separation principle


93C10, 93D15, 93D20


  1. A. N. Atassi and H. K. Khalil: A separation principle for the stabilization of a class of nonlinear systems. IEEE Trans. Automat. Control 44 (1999), 1672-1687.   DOI:10.1109/9.788534
  2. A. N. Atassi and H. K. Khalil: Separation results for the stabilization of nonlinear systems using different high-gain observer designs. Systems Control Lett. 39 (2000), 183-191.   DOI:10.1016/s0167-6911(99)00085-7
  3. S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan: Linear matrix inequalities in systems and control theory. In: SIAM Stud. Appl. Math. 15 (1994).   DOI:10.1137/1.9781611970777
  4. H. L. Choi and J. T. Lim: Global exponential stabilization of a class of nonlinear systems by output feedback. IEEE Trans. Automat. Control 50 (2005), 2, 255-257.   DOI:10.1109/tac.2004.841886
  5. A. Germani, C. Manes and P. Pepe: Local asymptotic stability for nonlinear state feedback delay systems. Kybernetika 36 (2000), 31-42.   CrossRef
  6. A. Germani, C. Manes and P. Pepe: An asymptotic state observer for a class of nonlinear delay systems. Kybernetika 37 (2001), 459-478.   CrossRef
  7. A. Germani, C. Manes and P. Pepe: Input-output linearization with delay cancellation for nonlinear delay systems: the problem of the internal stability. Int. J. Robust Nonlinear Control 13 (2003), 909-937.   CrossRef
  8. A. Germani, C. Manes and P. Pepe: Separation theorems for a class of retarded nonlinear systems. In: IFAC-Papers OnLine, Workshop on Time-Delay Systems, Praha 2010.   DOI:10.3182/20100607-3-cz-4010.00006
  9. A. Germani, C. Manes and P. Pepe: Observer-based stabilizing control for a class of nonlinear retarded systems. Lect. Notes Control Inform. Sci. 423 (2012), 331-342.   DOI:10.1007/978-3-642-25221-1_25
  10. J. K. Hale and S. M. Lunel: Introduction to Functional Differential Equations. Applied Mathematical Sciences. Springer-Verlag, New York 1991.   DOI:10.1007/978-1-4612-4342-7
  11. S. Ibrir: Observer-based control of a class of time-delay nonlinear systems having triangular structure. Automatica 47 (2011), 388-394.   DOI:10.1016/j.automatica.2010.10.052
  12. M. Jankovic: Recursive predictor design for state and output feedback controllers for linear time delay systems. Automatica 46 (2010), 510-517.   DOI:10.1016/j.automatica.2010.01.021
  13. H. K. Khalil: Nonlinear Systems. Prentice-Hall, Upper Saddle River, NJ 2001.   DOI:10.1002/rnc.1054
  14. O. M. Kwon, J. H. Park, S. M. Lee and S. C. Won: LMI optimization approach to observer-based controller design of uncertain time-delay systems via delayed feedback. J. Optim. Theory Appl. 128 (2006), 103-117.   DOI:10.1007/s10957-005-7560-3
  15. X. Li and C. de Souza: Output feedback stabilization of linear time-delay systems. Stability and control of time-delay systems. Lect. Notes Control Inform. Sci. (1998), 241-258.   CrossRef
  16. L. A. Marquez, C. Moog and M. Velasco Villa: Observability and observers for nonlinear systems with time delay. Kybernetika 38 (2002), 445-456.   CrossRef
  17. P. Pepe and I. Karafyllis: Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form. Int. J. Control 86 (2013), 232-243.   DOI:10.1080/00207179.2012.723137
  18. C. Qian and W.Lin: Output feedback control of a class of nonlinear systems: A non-separation principle paradigm. IEEE Trans. Automat. Control 47 (2002), 1710-1715.   DOI:10.1109/tac.2002.803542
  19. Y. J. Sun: Global stabilization of uncertain systems with time-varying delays via dynamic observer-based output feedback. Linear Algebra Appl. 353 (2002), 91-105.   DOI:10.1016/s0024-3795(02)00292-6
  20. M. V. Thuan, V. N. Phat and H. Trinh: Observer-based controller design of time-delay systems with an interval time-varying delay. Int. J. Appl. Math. Comput. Sci. 22 (2012), 4, 921-927.   DOI:10.2478/v10006-012-0068-8
  21. J. Tsinias: A theorem on global stabilization of nonlinear systems by linear feedback. Systems Control Lett. 17 (1991), 357-362.   DOI:10.1016/0167-6911(91)90135-2
  22. Z. Wang, D. P. Goodall and K. J. Burnham: On designing observers for time delay systems with nonlinear disturbances. Int. J. Control 75 (2002), 803-811.   DOI:10.1080/00207170210126245
  23. X. Zhang, Z. Cheng and and X. P. Wang: Output feedback stabilization of nonlinear systems with delayed output. In: Proc. American Control Conference, Portland 2005, pp. 4486-4490.   DOI:10.1109/acc.2005.1470769
  24. L. Zhou, X. Xiao and G. Lu: Observers for a class of nonlinear systems with time delay. Asian J. Control 11 (2009), 6, 688-693.   DOI:10.1002/asjc.150