Kybernetika 49 no. 2, 319-340, 2013

Global finite-time observers for a class of nonlinear systems

Yunyan Li, Yanjun Shen and Xiaohua Xia


Global finite-time observers are designed for a class of nonlinear systems with bounded varying rational powers imposed on the increments of the nonlinearities whose solutions exist and are unique for all positive time. The global finite-time observers designed in this paper are with two homogeneous terms. The global finite-time convergence of the observation error system is achieved by combining global asymptotic stability and local finite-time stability.


homogeneity, nonlinear system, global finite-time observer




  1. D. Bestle and M. Zeitz: Cannonical form observer design for non-linear time-variable systems. Internat. J. Control 38 (1983), 419-431.   CrossRef
  2. S. P. Bhat and D. S. Bernstein: Finite-time stability of continous autonomous systems. SIAM J. Control Optim. 38 (2000), 751-766.   CrossRef
  3. S. P. Bhat and D. S. Bernstein: Geometric homogeneity with applications to finite-time stability. Math. Control Sign. Systems 17 (2005), 101-127.   CrossRef
  4. M. S. Chen and C. C. Chen: Robust nonlinear observer for Lipschitz nonlinear systems subject to disturbances. IEEE Trans. Automat. Control 52 (2007), 2365-2369.   CrossRef
  5. R. Engel and G. Kreisselmeier: A continuous-time observer which converges in finite time. IEEE Trans. Automat. Control 47 (2002), 1202-1204.   CrossRef
  6. J. P. Gauthier, H. Hammouri and S. Othman: A simple observer for nonlinear systems applications to bioreactors. IEEE Trans. Automat. Control 37 (1992), 875-880.   CrossRef
  7. H. Hammouri, B. Targui and F. Armanet: High gain observer based on a triangular structure. Internat. J. Robust Nonlinear Control 12 (2002), 497-518.   CrossRef
  8. Y. Hong, Y. Xu and J. Huang: Finite-time control for manipulators. Systems Control Lett. 46 (2002), 243-253.   CrossRef
  9. Ü. Kotta: Application of inverse system for linearization and decoupling. Systems Control Lett. 8 (1987), 453-457.   CrossRef
  10. A. J. Krener and A. Isidori: Linearization by output injection and nonlinear observers. Syst.ems Control Lett. 3 (1983), 47-52.   CrossRef
  11. P. Krishnamurthy, F. Khorrami and R. S. Chandra: Global high-gain-based observer and backstepping controller for generalized output-feedback canonical form. IEEE Trans. Automat. Control 48 (2003), 2277-2284.   CrossRef
  12. J. Levine and R. Marino: Nonlinear systems immersion, observers and finite dimensional filters. Systems Control Lett. 7 (1986), 133-142.   CrossRef
  13. J. Li, C. Qian and M. T. Frye: A dual-observer design for global output feedback stabilization of nonlinear systems with low-order and high-order nonlinearities. Internat. J. Robust Nonlinear Control 19 (2009), 1697-1720.   CrossRef
  14. T. Ménard, E. Moulay and W. Perruquetti: A global high-gain finite-time observer. IEEE Trans. Automat. Control 55 (2010), 1500-1506.   CrossRef
  15. E. Moulay and W. Perruquetti: Finite time stability and stabilization of a class of continuous systems. J. Math. Anal. Appl. 323 (2006), 1430-1443.   CrossRef
  16. E. Moulay and W. Perruquetti: Finite-time stability conditions for non-autonomous continuous systems. Internat. J. Control 81 (2008), 797-803.   CrossRef
  17. W. Perruquetti, T. Floquet and E. Moulay: Finite-time observers: application to secure communication. IEEE Trans. Automat. Control 53 (2008), 356-360.   CrossRef
  18. A. M. Pertew, H. J. Marquez and Q. Zhao: $H_\infty$ observer design for Lipschitz nonlinear systems. IEEE Trans. Automat. Control 51 (2006), 1211-1216.   CrossRef
  19. L. Praly: Asymptotic stabilization via output feedback for lower triangular systems with output dependent incremental rate. IEEE Trans. Automat. Control 48 (2003), 1103-1108.   CrossRef
  20. S. Raghavan and J. K. Hedrick: Observer design for a class of nonlinear systems. Internat. J. Control {\mi 59} (1994), 515-528.   CrossRef
  21. R. Rajamani: Observers for Lipschitz nonlinear systems. IEEE Trans. Automat. Control 43 (1998), 397-401.   CrossRef
  22. L. Rosier: Homogeneous Lyapunov function for homogeneous continuous vector field. Systems Control Lett. 19 (1992), 467-473.   CrossRef
  23. Y. Shen and Y. Huang: Uniformly observable and globally Lipschitzian nonlinear systems admit global finite-time observers. IEEE Trans. Automat. Control 54 (2009), 2621-2625.   CrossRef
  24. Y. Shen and X. Xia: Semi-global finite-time observers for nonlinear systems. Automatica 44 (2008), 3152-3156.   CrossRef
  25. Y. Shen and X. Xia: Semi-global finite-time observers for a class of non-Lipschitz systems. In: Nolcos, Bologna 2010, pp. 421-426.   CrossRef
  26. Y. Shen and X. Xia: Global asymptotical stability and global finite-time stability for nonlinear homogeneous systems. In: 18th IFAC World Congress, Milan 2011, pp. 4644-4647.   CrossRef
  27. F. E. Thau: Observing the state of nonlinear dynamic systems. Internat. J. Control 17 (1973), 471-479.   CrossRef
  28. S. T. Venkataraman and S. Gulati: Terminal slider control of nonlinear systems. In: Proc. IEEE International Conference of Advanced Robotics, Pisa 1990, pp. 2513-2514.   CrossRef
  29. X. Xia and W. Gao: Nonlinear observer design by observer error linearization. SIAM J. Control Optim. 27 (1989), 199-216.   CrossRef
  30. M. Zeitz: The extended Luenberger observer for nonlinear systems. Systems Control Lett. 9 (1987), 149-156.   CrossRef