Nonlinear tracking differentiator based on improved sigmoid function

Inspired by the characteristic of common activation function of neutral network, which is known as sigmoid function, we propose a nonlinear tracking differentiator (STD) with simple form and fewer tuning parameters. Firstly, exponential and scale factors are introduced to improve the sigmoid function, then the acceleration function is constructed by utilizing the improved sigmoid function. Secondly, the global uniformly asymptotical stability of the tracking differentiator (TD) in non-perturbation form is proved by using Lyapunov direct method. Moreover, the concrete form of TD is presented by the principle of system equivalence, and its frequency-domain characteristic is analyzed by utilizing the frequency-sweep test. Finally, simulations are performed and results are compared with those of linear differentiator, high-speed nonlinear tracking differentiator and improved nonlinear tracking differentiator, arctangent-based TD. It concludes that the sigmoid function-based nonlinear tracking differentiator not only guarantees the response with high speed and smoothness but also presents the behavior with no chattering in the whole course and exhibits excellent performance in approximating and filtering the generalized derivative of the signal.