## 2 Radar Position Servo System Modeling
### Proposed Adaptive Tracking Differentiator Design  
**Remark 1**.: _As the input signal change rate increases, the output signal will have an obvious tracking lag in the traditional TD. The elimination of the lag is achieved by increasing the speed factor, but it will reduce the filtering effect of the TD and reproduce the noise of the input signal in the output differential signal. At the same time, a spike error of the output differential signal will exist. The ATD, which can adaptively adjust the speed factor and filter factor according to the change rate of the input signal, is designed in this paper, to manage the contradiction between filtering performance and tracking speed. Moreover, the LADRC introducing the proposed ATD will consume less energy than the LADRC [34], eliminating the peak value of the control signal and reducing the impact of the control effect on the final controlling element._  
The traditional TD [34] is described as follows:  
\[\left\{\begin{matrix}v_{1}(k+1)=v_{1}(k)+h\cdot v_{2}(k)\\ v_{2}(k+1)=v_{2}(k)+h\cdot fst(v_{1}(k)-v_{d}(k),v_{2}(k),\kappa,h_{0})\end{matrix}\right. \tag{13}\]  
where \(v_{d}(k)\) is the desired trajectory, \(h\) is the sampling step, \(\kappa\) is the speed factor, \(h_{0}\) is the filter factor, \(v_{1}(k)\) and \(v_{2}(k)\) track \(v_{d}(k)\) and \(\dot{v}_{d}(k)\), respectively, and \(fst(\cdot)\) is a nonlinear composite function, which can be expressed as  
\[fst(z_{1},z_{2},\kappa,h)=\left\{\begin{matrix}-\kappa\cdot\left(\frac{\sigma }{\tau}\right),|a|\leq l\\ -\kappa\cdot\text{sgn}(a),|a|>l\end{matrix}\right.\]  
\[a=\left\{\begin{matrix}z_{2}+\frac{a_{0}-l}{2}\text{sgn}(f),|f|>l_{0}\\ z_{2}+\frac{\tau}{l},|f|\leq l_{0}\end{matrix}\right.\]  
\[l=\kappa\cdot h_{t}l_{0}=l\cdot h,a_{0}=\sqrt{l^{2}+8\kappa|f|},f=z_{1}+h \cdot z_{2}\]  
In order to manage the above problem of the traditional TD, it is necessary to design a continuous and bounded function as the adaptive control function of the speed factor \(\kappa\), which can increase rapidly with the increase in the differential signal. At the same time, the filter factor \(h_{0}\) is supposed to be taken large when the differential signal is around zero in order to eliminate spike error.  
The Cauchy distribution [35] is considered, and its probability density function \(f(x)\) and cumulative distribution function \(F(x)\) are given as  
\[f(x)=\begin{matrix}1\\ \end{matrix}\left[\pi\cdot\sigma\cdot\left[1+\left(\frac{x-x_{0}}{\sigma} \right)^{2}\right]\right] \tag{14}\]  
\[F(x)=\frac{1}{\pi}\cdot\arctan\frac{x-x_{0}}{\sigma}+\frac{1}{2} \tag{15}\]  
where \(\sigma\) is the scale factor, and \(x_{0}\) is the position factor.  
It is well known that the position of two function curves can be adjusted by tuning \(x_{0}\); the height and width of the curves can be adjusted by tuning \(\sigma\). Additionally, the value of \(f(x)\) is large enough when \(x\approx x_{0}\), \(F(x)\) is a bounded function that increases rapidly with the increase of \(x\). Then, \(F(x)\) and \(f(x)\) are utilized to design the adaptive function of \(\kappa\) and \(h_{0}\) by setting \(x_{0}=0\), respectively, which are described as  
\[\kappa(v_{2})=K\cdot\left(\frac{1}{\pi}\cdot\arctan\frac{|v_{2}|}{\sigma_{1}} \right)+\frac{1}{2}+L \tag{16}\]  
where \(K\) is the variety range of \(\kappa\), and \(L\) is the initial value of \(\kappa\).  
\[h_{0}(v_{2})=\frac{1}{\pi\cdot\sigma_{2}\cdot\left(1+\frac{v_{2} ^{2}}{\sigma_{2}}\right)} \tag{17}\]  
Therefore, (13) can be rewritten as  
\[\left\{\begin{matrix}v_{1}(k+1)=v_{1}(k)+h\cdot v_{2}(k)\\ v_{2}(k+1)=v_{2}(k)+h\cdot fst(v_{1}(k)-v_{d}(k),v_{2}(k),\kappa(v_{2}),h_{0} (v_{2}))\end{matrix}\right. \tag{18}\]  
After introducing (18) into the LADRC, LSEF (12) can be rewritten as  
\[\left\{\begin{matrix}e_{1}=v_{1}-\zeta_{1},e_{2}=v_{2}-\zeta_{2}\\ u_{0}=k_{p}\cdot e_{1}+k_{d}\cdot e_{2}\\ u=u_{0}-\frac{\zeta_{3}}{h_{0}}\end{matrix}\right. \tag{19}\]