Abstract
This paper presents a new improved nonlinear tracking differentiator (INTD) with hyperbolic tangent function in the state space system. The stability and convergence of the INTD are thoroughly investigated and proved. Through the error analysis, the proposed INTD can extract differentiation of any piecewise smooth nonlinear signal to reach a high accuracy. the INTD has the required filtering features and can cope with th nonlinearities caused by the niose. Through simulations, the INTD is implemented as signal derivative generator for the closed loop feedback control system with a nolinear PID controller for the nonlinear Mass Spring Damper system and showed that it could achieve the signal tracking and differentiation faster with a minimum mean square error.
Highlights
Differentiation of signals in real time is an old and wellknown problem
The peaking phenomenon is presented through time domain analysis, while frequency domain analysis proves that the proposed nonlinear tracking differentiator attenuates signals with a certain frequency band
Jing Han has made some investigations on traditional structures and essential properties of nonlinear tracking differentiator
Summary
Differentiation of signals in real time is an old and wellknown problem. An ideal differentiator would have to differentiate measurement noise with possibly large derivatives along with the signal [1]. In [11], two particular high-gain tracking differentiators were proposed This differentiator was based on the Taylor expansion, the time lagging phenomenon of the traditional high-gain differentiator is reduced effectively. The peaking phenomenon is presented through time domain analysis, while frequency domain analysis proves that the proposed nonlinear tracking differentiator attenuates signals with a certain frequency band. The paper is organized as follows: in Section II an improved nonlinear tracking differentiator is proposed, and the main convergence results are presented. Lemma 1: (Convergence of the INTD system): the improved tracking differentiator described by (1) with its design parameters is globally asymptotically stable. If z2(0) = 0, z1(t) is a decreasing function for t > 0 until it reaches the tracking phase where βz1−(1−α)v ≪ 1. The system has the attenuation effect for ω ≫ ωn. □
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