反正切形式跟踪微分器设计及相平面分析

Design and phase plane analysis of an arctangent-based tracking differentiator

跟踪微分器可以从不连续或含有噪声的测量信号中提取连续滤波信号和微分信号.本文从二阶跟踪微分器的加速度函数式入手,提出一种反正切形式的跟踪微分器结构,利用李雅普诺夫第二方法证明其具有全局一致渐进稳定性,利用相平面分析方法研究,说明原点是稳定结点或稳定焦点,并给出跟踪微分器参数调节的定性规律和约束条件.最后在含有噪声的方波输入下进行仿真,结果表明反正切形式跟踪微分器可以兼顾跟踪快速性与过渡过程平稳性,并且具有很好的滤波效果,可以给出连续平滑的跟踪信号和微分信号,而且待调节的参数较少,具有工程实用价值.

Tracking differentiator（TD）can produce continuous tracking signal and its differential signal from discontinuous and noisy inputs.An arctangent-based TD is presented based on acceleration function analysis,and then its global uniformly asymptotical stability is proved using Lyapunov＇s direct method.Moreover,phase plane analysis is conducted to show that the origin（0,0）is a stable nodal point or a stable focus point.By these results,we propose the parameter constraint conditions of the arctangent-based TD.The TD is simulated to track square wave input signals,giving three conclusions：first,the tracking output and its differential signals are smooth and continuous;second,the filter performance of the TD is satisfactory;third,the setting time of the response is short and the overshoot is small.