Kybernetika 54 no. 4, 699-717, 2018

Robust sampled-data observer design for Lipschitz nonlinear systems

Yu Yu and Yanjun ShenDOI: 10.14736/kyb-2018-4-0699


In this paper, a robust sampled-data observer is proposed for Lipschitz nonlinear systems. Under the minimum-phase condition, it is shown that there always exists a sampling period such that the estimation errors converge to zero for whatever large Lipschitz constant. The optimal sampling period can also be achieved by solving an optimal problem based on linear matrix inequalities (LMIs). The design methods are extended to Lipschitz nonlinear systems with large external disturbances as well. In such a case, the estimation errors converge to a small region of the origin. The size of the region can be small enough by selecting a proper parameter. Compared with the existing results, the design parameters can be easily obtained by solving LMIs.


nonlinear systems, sampled-data observer, Lipschitz, sampling period, LMIs


93C57, 93B51


  1. J. Ahrens, X. Tan and H. Khalil: Multirate sampled-data output feedback control with application to smart material actuated systems. IEEE Trans. Automat. Control 54 (2009), 2518-2529.   DOI:10.1109/tac.2009.2031204
  2. D. Boutat: Extended nonlinear observer normal forms for a class of nonlinear dynamical systems. Int. J. Robust Nonlinear Control 25 (2015), 461-474.   DOI:10.1002/rnc.3102
  3. S. Boyd, L. Ghaoui and E. Feron et al.: Linear Matrix Inequalities in System and Control Theory. Society for Industrial and Applied Mathematics, ch. 1.2, Philadelphia 1994.   DOI:10.1137/1.9781611970777
  4. M. Chen and C. Chen: Robust nonlinear observer for Lipschitz nonlinear systems subject to disturbances. IEEE Trans. Automat. Control 52 (2007), 2365-2369.   DOI:10.1109/tac.2007.910724
  5. T. Dezuo and A. Trofino: LMI conditions for designing rational nonlinear observers. In: 2014 American Control Conference. 47 (2014), 5343-5348.   DOI:10.1109/acc.2014.6858805
  6. T. Dinh, V. Andrieu, M. Nadri and et al.: Continuous-discrete time observer design for Lipschitz systems with sampled measurements. IEEE Trans. Automat. Control 60 (2015), 787-792.   DOI:10.1109/tac.2014.2329211
  7. Y. Dong, J. Liu and S. Mei: Observer design for a class of nonlinear discrete-time systems with time-delay. Kybernetika 49 (2013), 341-358.   CrossRef
  8. J. Doyle and G. Stein: Robustness with observers. IEEE Trans. Automatic Control 24 (1979), 607-611.   DOI:10.1109/tac.1979.1102095
  9. M. Ekramian, F. Sheikholeslam and S. Hosseinnia et al.: Adaptive state observer for Lipschitz nonlinear systems. Systems Control Lett. 62 (2013), 319-323.   DOI:10.1016/j.sysconle.2013.01.002
  10. M. Gupta, N. Tomar and S. Bhaumik: Observer Design for Descriptor Systems with Lipschitz Nonlinearities: An LMI Approach. Nonlinear Dynamics Systems Theory 14 (2014), 291-301.   CrossRef
  11. W. Kang, A. Krener and M. Xiao et al.: A survey of observers for nonlinear dynamical systems. In: Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications, Vol. II, Springer Berlin Heidelberg 2013, pp. 1-25.   DOI:10.1007/978-3-642-35088-7\_1
  12. H. Khalil: Nonlinear System. Upper Saddle River, Prentice Hall, ch. 14.5, NJ 2000.   CrossRef
  13. H. Kwakernaak and R. Sivan: Linear Optimal Control Systems. Wiley, ch. 3, Theorem 3.14, New York 1972.   CrossRef
  14. F. Lewis: Applied Optimal Control and Estimation. Englewood Cliffs, Prentice-Hall, ch. 3, Theorem 2, NJ 1992.   CrossRef
  15. R. Marino and P. Tomei: Nonlinear control design. Automatica 33 (2009), 1769-1770.   DOI:10.1016/s0005-1098(97)82237-6
  16. D. Nešić and A. Teel: A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models. IEEE Trans. Automat. Control 49 (2004), 1103-1122.   DOI:10.1109/tac.2004.831175
  17. N. Oucief, M. Tadjine and S. Labiod: Adaptive observer-based fault estimation for a class of Lipschitz nonlinear systems. Archives Control Sci. 26 (2016), 245-259.   DOI:10.1515/acsc-2016-0014
  18. J. Pan, M. Meng and J. Feng: A note on observers design for one-sided Lipschitz nonlinear systems. In: Control Conference IEEE (2015), pp. 1003-1007.   DOI:10.1109/chicc.2015.7259771
  19. C. Perez and M. Mera: Robust observer-based control of switched nonlinear systems with quantized and sample output. Kybernetika 54 (2015), 59-80.   DOI:10.14736/kyb-2015-1-0059
  20. B. Rehák: Sum-of-squares based observer design for polynomial systems with a known fixed time delay. Kybernetika 51 (2015), 856-873.   DOI:10.14736/kyb-2015-5-0856
  21. A. Saberi, P. Sannuti and B. Chen: $H_2$ Optimal Control. Englewood Cliffs, Prentice-Hall, ch. 4, Theorem 4.1.2, NJ 1995.   CrossRef
  22. Y. Shen, D. Zhang and X. Xia: Continuous output feedback stabilization for nonlinear systems based on sampled and delayed output measurements. Internat. J. Robust and Nonlinear Control 26 (2016), 3075-3087.   DOI:10.1002/rnc.3491
  23. Y. Shen, D. Zhang and X. Xia: Continuous observer design for a class of multi-output nonlinear systems with multi-rate sampled and delayed output measurements. Automatica 75 (2017), 127-132.   DOI:10.1016/j.automatica.2016.09.028
  24. G. Stein and M. Athans: The LQG/LTR procedure for multivariable feedback control design. IEEE Trans. Automat. Control 32 (1987), 105-114.   DOI:10.1109/tac.1987.1104550
  25. A. Tahir, A. Magri, T. Ahmed-Ali and et al.: Sampled-data nonlinear observer design for sensorless synchronous PMSM. IFAC-Papers OnLine 48 (2015), 327-332.   DOI:10.1016/j.ifacol.2015.09.206
  26. F. Thau: Observing the state if nonlinear dynamic systems. Int. J. Control 17 (1973), 471-479.   DOI:10.1080/00207177308932395
  27. Y. Wang, X. Liu, J. Xiao and Y. Shen: Output formation-containment of interacted heterogeneous linear systems by distributed hybrid active control. Automatica 93 (2018), 26-32.   DOI:10.1016/j.automatica.2018.03.020
  28. L. Yu: Robust Control: Linear Matrix Inequality Approach. Tsinghua University Press 2002.   CrossRef
  29. A. Zemouche and M. Boutayeb: On LMI conditions to design observers for Lipschitz nonlinear systems. Automatica 49 (2013), 585-591.   DOI:10.1016/j.automatica.2012.11.029
  30. D. Zhang and Y. J. Shen: Continuous sampled-data observer design for nonlinear systems with time delay larger or samller than the sampling period. IEEE Trans. Automat. Control 62 (2017), 5822-5829.   DOI:10.1109/tac.2016.2638043
  31. D. Zhang, Y. Shen and X. Xia: Globally uniformly ultimately bounded observer design for a class of nonlinear systems with sampled and delayed measurements. Kybernetika 52 (2016), 441-460.   DOI:10.14736/kyb-2016-3-0441
  32. W. Zhang, H. Su and S. Su et al.: Nonlinear $H_\infty$ observer design for one-sided Lipschitz systems. Neurocomputing (2014), 505-511.   CrossRef
  33. W. Zhang, H. Su and F. Zhu et al.: A note on observers for discrete-time Lipschitz nonlinear systems. IEEE Trans. Circuits Systems II Express Briefs 29 (2012), 123-127.   DOI:10.1109/tcsii.2011.2174671
  34. Y. Zhou, Y. Soh and J. Shen: High-gain observer with higher order sliding mode for state and unknown disturbance estimations. Int. J. Robust abd Nonlinear Control 24 (2016), 2136-2151.   DOI:10.1002/rnc.2982