一种基于边界特征线的二阶非线性离散跟踪微分器

A Discrete Second-Order Nonlinear Tracking-Differentiator Based on Boundary Characteristic Curves

利用状态反推方法确定最速离散二阶系统的线性区域的边界特征线、控制特征线以及开关曲线,确定两步可达的区域,若点位于两步可达区外,则做平行辅助直线与上述三条曲线相交于3个不同的特征点,并根据点的位置判断是否按线性比例来确定控制量大小,从而替换非线性边界变换,并依此构造最速分段线性形式的跟踪微分器(TD),不包含任何根号运算,使控制综合函数的形式极大简化.由于线性区域内的3个特征点完全落在特征线上,因而本文的算法与非线性边界变换算法一致.数值仿真的结果说明本文的算法具有无颤振、无超调、快速跟踪输入信号的特点,能得到较好的微分信号,效果与非线性边界变换法一致.最后用扫频算法验证了本算法与非线性边界变换算法幅频、相频特性完全一致.本跟踪微分器算法简单,计算量小,具有较强非线性特征,易于工程实现.

The boundary characteristic curves, control characteristic curves and switching curves of linear region with second-order discrete time optimal control system are presented by state backstep method. The two-step reachable region is also acquired. If the point is not located in the two step reachable region, a parallel auxil- iary line is drawn, which intersects above three curves at three different points. The control variable is ac- quired according to linear proportion of the three characteristic points about the above three curves, which re- place nonlinear boundary transformation, then the time optimal segmental linearized tracking differentiator is constructed, and the control synthetic function is greatly simplified. This algorithm is consistent with nonlin- ear boundary transform algorithm if the three characteristic points in linear region fall completely in the char- acteristic curves. Numerical simulation results show that this discrete form of tracking-differentiator can quick- ly track an input signal without overshooting and chattering, and can produce an excellent differential signal. The sweep-frequency algorithm is employed to compare the above two kinds of tracking differentiator in the field of amplitude-phase frequency characteristic. This tracking-differentiator has the benefits of concision and nonlinearity, and also requires less calculation. It is convenient for engineering applications.