On convergence of tracking differentiator and application to frequency estimation of sinusoidal signals

The tracking differentiator was first proposed by Jingqing Han in 1989 and the proof of convergence was presented the first time in [J.Q. Han and W. Wang, Journal of Systems Science and Mathematical Science, 14(2)(1994), 177–183]. Unfortunately, the proof there is incorrect. This problem has been open for over two decades. In this paper, we give a rigorous proof under some additional conditions. An application to online estimate the unknown frequencies for the finite sum of the sinusoidal signals is presented. The numerical simulations illustrate the effectiveness of the estimation from both linear and nonlinear tracking differentiators.

[1]  Dong Sun,et al.  A simple nonlinear velocity estimator for high-performance motion control , 2005, IEEE Transactions on Industrial Electronics.

[2]  Takanori Emaru,et al.  Research on estimating smoothed value and differential value by using sliding mode system , 2003, IEEE Trans. Robotics Autom..

[3]  J. Han,et al.  NONLINEAR TRACKING-DIFFERENTIATOR , 1994 .

[4]  X. Xia,et al.  Global frequency estimation using adaptive identifiers , 2002, IEEE Trans. Autom. Control..

[5]  Jingqing Han,et al.  From PID to Active Disturbance Rejection Control , 2009, IEEE Trans. Ind. Electron..

[6]  P. Olver.  Nonlinear Systems , 2013 .

[7]  Salim Ibrir,et al.  Linear time-derivative trackers , 2004, Autom..

[8]  Yuxin Su,et al.  Disturbance-rejection high-precision motion control of a Stewart platform , 2004, IEEE Transactions on Control Systems Technology.

[9]  Bao-Zhu Guo,et al.  Linear tracking-differentiator and application to online estimation of the frequency of a sinusoidal signal with random noise perturbation , 2002, Int. J. Syst. Sci..

[10]  Zhiqiang Gao,et al.  A DSP-based active disturbance rejection control design for a 1-kW H-bridge DC-DC power converter , 2005, IEEE Trans. Ind. Electron..

[11]  H. Khalil,et al.  Discrete-time implementation of high-gain observers for numerical differentiation , 1999 .

[12]  A. Levant.  Robust exact differentiation via sliding mode technique , 1998 .

[13]  Geng Yang,et al.  Finite-Time-Convergent Differentiator Based on Singular Perturbation Technique , 2007, IEEE Transactions on Automatic Control.

[14]  Shankar P. Bhattacharyya,et al.  New results on the synthesis of PID controllers , 2002, IEEE Trans. Autom. Control..

[15]  Wenchao Xue,et al.  What kinds of system can be used as tracking-differentiator , 2010, Proceedings of the 29th Chinese Control Conference.