A contraction theory based nonlinear tracking differentiator with variable damping ratio

Nonlinear tracking differentiator (NTD) is a powerful tool for achieving differential signals. Although numerous studies considering NTD have been carried out, the convergence analysis of NTD in these works is accomplished by the Lyapunov method, which is quite complex and needs unrealistic assumptions. Contraction theory, a newly proposed system analysis tool, is particularly effective in analyzing nonlinear non-autonomous systems. Motivated by these considerations, this paper proposed an NTD with variable damping (VD-NTD) based on contraction theory. Firstly, the convergence of NTD is performed with contraction theory, which is much simpler than the traditional Lyapunov-based method. Secondly, the optimization of NTD is done based on its characteristics. Also, the convergence of the proposed VD-NTD is verified elegantly by contraction theory. Finally, simulations on sinusoidal signals, filtered square waves, and complex signals with sudden changes have been carried out, the proposed method not only acquires better performance but is more robust to unpredictable disturbances.