Journal of Intelligent & Robotic Systems

, Volume 94, Issue 2, pp 405–421 | Cite as

Analysis and Design of a Time-Varying Extended State Observer for a Class of Nonlinear Systems with Unknown Dynamics Using Spectral Lyapunov Function

  • Mehran Attar
  • Vahid Johari MajdEmail author
  • Navid Dini


In this study, a novel strategy based on the integration of differential algebraic spectral theory (DAST) and spectral Lyapunov function is presented to analyze and design a time-varying extended state observer (TESO) for a class of nonlinear systems with unknown dynamics. The simultaneous estimation of the lumped disturbance and state vectors are achieved by using a TESO based on the time-varying parallel differential (PD) eigenvalues of the observer. The observer bandwidth design is based on the combination of DAST and spectral Lyapunov function. By using this method, a systematic approach is derived to obtain the observer parameters, which improves boundedness of the observer estimation error in terms of transient and persistent performance. A comparison between TESO and previous similar methods is provided in the simulation part upon the TMUBOT quadruped robot dynamic model which indicates a distinguished answer in the estimation error of the TESO. Moreover, by applying the proposed algorithm to the TMUBOT robot, the superiority of the algorithm in practical schemes will be illustrated.


Active disturbance rejection control Extended state observer Differential algebraic spectral theory Spectral Lyapunov function 


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  1. 1.
    Chen, W.-H., Yang, J., Guo, L., Li, S.: Disturbance observer-based control and related methods—an overview. IEEE Trans. Ind. Electron. 63, 2 (2016)CrossRefGoogle Scholar
  2. 2.
    Sariyildiz, E., Ohnishi, K.: Stability and robustness of disturbance observer-based motion control systems. IEEE Trans. Ind. Electron. 62(1), 414–422 (2015)CrossRefGoogle Scholar
  3. 3.
    Besancon, G.: Nonlinear Observers and Applications. Springer, New York (2007)CrossRefzbMATHGoogle Scholar
  4. 4.
    Nozaki, T., Mizoguchi, T., Ohnishi, K.: Decoupling strategy for position and force control based on modal space disturbance observer. IEEE Trans. Ind. Electron. 61(2), 1022–1032 (2014)CrossRefGoogle Scholar
  5. 5.
    Huang, Y., Xue, W.: Active disturbance rejection control: Methodology and theoretical analysis. ISA Trans. 53(4), 963–976 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Han, J.: A class of extended state observers for uncertain systems. Control Decis. 10(1), 85–88 (1995)Google Scholar
  7. 7.
    Han, J.: From PID to active disturbance rejection control. IEEE Trans. Ind. Electron. 56(3), 900–906 (2009)CrossRefGoogle Scholar
  8. 8.
    Leonard, F., Martini, A., Abba, G.: Robust nonlinear controls of model-scale helicopters under lateral and vertical wind gusts. IEEE Trans. Control Syst. Technol. 20(1), 154–163 (2012)CrossRefGoogle Scholar
  9. 9.
    Gao, Z: From linear to nonlinear control means: A practical progression. ISA Trans. 41(2), 177–189 (2002)CrossRefGoogle Scholar
  10. 10.
    Yao, J., Jiao, Z., Ma, D.: Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with back stepping. IEEE Trans. Ind. Electron. 61(11), 6285–6293 (2014)CrossRefGoogle Scholar
  11. 11.
    Attar, M., Majd, V.J., Dini, N., Edrisi, F.: Estimation of decenteralized unknown dynamics for a 2DOF manipulator using a time varying extended state observer. In: 4th International Conference on Robotics and Mechatronics (ICROM) (2016)Google Scholar
  12. 12.
    Yoo, D., Yau, S., Gao, Z.: Optimal fast tracking observer bandwidth of the linear extended state observer. Int. J. Control 80(1), 102–111 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zheng, Q., Gao, L.Q., Gao, Z.: On stability analysis of active disturbance rejection control for nonlinear time-varying plants with unknown dynamics. In: Proc. 46th Conf. Decision Control, pp. 3501–3506. New Orleans (2007)Google Scholar
  14. 14.
    Madoński, R., Herman, P.: Survey on methods of increasing the efficiency of extended state disturbance observers. ISA Trans. 6, 18–27 (2015)CrossRefGoogle Scholar
  15. 15.
    Gao, Z., Huang, Y., Han, J.: An alternative paradigm for control systems design. In: Proceedings of the 40th IEEE Conference on Decision and Control (2001)Google Scholar
  16. 16.
    Huang, Y., Xue, W.: Active disturbance rejection control: Methodology and theorical analysis. ISA Trans. 53, 963–976 (2014)CrossRefGoogle Scholar
  17. 17.
    Przybyla, M., Kordasz, M., Madoński, R., Herman, P., Sauer, P.: Active disturbance rejection control of a 2DOF manipulator with significant modeling uncertainty. Bull. Polish Acad. Sci. 60, 3 (2012)Google Scholar
  18. 18.
    Khalil, H.K.: Nonlinear Systems, 3rd edn., pp. 610–625. Englewood Cliffs, Prentice-Hall (2002)Google Scholar
  19. 19.
    Guo, B., Zhao, Z.: On the convergence of an extended state observer for nonlinear systems with uncertainty. Syst. Control Lett. 60(6), 420–430 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Han, J., Zhang, R.: Error analysis of the second order ESO. J. Syst. Sci. Math. Sci. 19(4), 465–471 (1999)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Guo, B., Zhao, Z.: On convergence of nonlinear active disturbance rejection for SISO systems. In: Control and Decision Conference the 24th. Taiyuan (2012)Google Scholar
  22. 22.
    Guo, B., Zhao, Z.: On convergence of nonlinear extended state observer for multi-input multi-output systems with uncertainty. IET Control Theory Appl. 6(15), 2375–2386 (2012)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Zhao, Z.L., Guo, B.Z.: On active disturbance rejection control for nonlinear systems using time-varying gain. Eur. J. Control, 62–70 (2015)Google Scholar
  24. 24.
    Zhu, J.: A unified spectral theory for linear time-varying systems— progress and challenges. In: Proc. 34th Conf. Decision Control, pp. 2540–2546. New Orleans (1995)Google Scholar
  25. 25.
    Liu, Y., Zhu, J.: Regular perturbation analysis for trajectory linearization control. In: Proc. Amer. Control Conf., New York, pp. 3053–3058 (2007)Google Scholar
  26. 26.
    Jim Zhu, J., Liu, Y., Hang, R.: A spectral Lyapunov function for exponentially stable LTV systems. Amer. Control Conf., Hyatt Regency Riverfront (2009)Google Scholar
  27. 27.
    Silverman, L.M.: Transformation of time-variable systems to canonical (phase-variable) form. IEEE Trans. Autom. Control 11(2), 300–303 (1966)CrossRefGoogle Scholar
  28. 28.
    Farid, Y., Majd, V.J., Ehsani-Seresht, A.: Fractional-order active fault-tolerant force-position controller design for the legged robots using saturated actuator with unknown bias and gain degradation. Mech. Syst. Signal Process. 104, 465–486 (2018)CrossRefGoogle Scholar
  29. 29.
    Graham, A.: A note on a transformation between two canonical forms in state-space in terms of the eigenvalues of the system matrix. IEEE Trans. Autom. Control 13(4), 448 (1968)CrossRefGoogle Scholar
  30. 30.
    Wolovich, W.A.: On the stabilization of controllable systems. IEEE Trans. Autom. Control 13(5), 569–572 (1968)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Edrisi, F., Majd, V.J., Attar, M., Dini, N.: Modifying the attitude of quadruped robot body against disturbances via data fusion. In: 2016 4th International Conference on Robotics and Mechatronics(ICROM), pp. 55–60. IEEE (2016)Google Scholar
  32. 32.
    Dini, N., Majd, V., Edrisi, F., Attar, M.: Estimation of external forces acting on the legs of a quadruped robot using two nonlinear disturbance observers. In: Proceedings of the 4th International Conference on Robotics and Mechatronics (ICROM), pp. 72–77, Tehran (2016)Google Scholar
  33. 33.
    Li, Z., Xiao, S., Sam, S., Su, H.: Constrained multi-legged robot system modeling and fuzzy control with uncertain kinematics and dynamics incorporating foot force optimization. IEEE Trans. Man Cybern. Syst. (2015)Google Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Intelligent Control Systems Laboratory, School of Electrical and Computer EngineeringTarbiat Modares UniversityTehranIran

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