## 2 Radar Position Servo System Modeling
### Proposed Improved LADRC Design
In [38], the sensitivity is given as  
\[O_{\lambda}(k)=w_{\lambda}{}^{D}\cdot\Psi_{\lambda}(k-1)+\sum_{\varepsilon}w_ {\epsilon\lambda}{}^{I}\cdot I_{c}(k) \tag{21}\]7  
And the gradients of (32) with respect to \(k_{p}(k)\) and \(k_{d}(k)\) can be obtained as  
\[O_{\lambda}(k)=w_{\lambda}{}^{D}\cdot\Psi_{\lambda}(k-1)+\sum_{\varepsilon}w_ {\epsilon\lambda}{}^{I}\cdot I_{c}(k) \tag{21}\]8  
\[O_{\lambda}(k)=w_{\lambda}{}^{D}\cdot\Psi_{\lambda}(k-1)+\sum_{\varepsilon}w_ {\epsilon\lambda}{}^{I}\cdot I_{c}(k) \tag{21}\]9  
Substitute (37) and (38) in (34) and (35), respectively:  
\[G_{DR}(k)=\sum_{\lambda}w_{\lambda}{}^{O}\cdot\Psi_{\lambda}(k) \tag{22}\]0  
\[G_{DR}(k)=\sum_{\lambda}w_{\lambda}{}^{O}\cdot\Psi_{\lambda}(k) \tag{22}\]1  
Therefore, the gradient descent algorithm can be utilized to optimize the parameters and the tuning role is given as  
\[G_{DR}(k)=\sum_{\lambda}w_{\lambda}{}^{O}\cdot\Psi_{\lambda}(k) \tag{22}\]2  
where \(k_{\ell}(k)\) is \(k_{p}(k)\) or \(k_{d}(k)\), \(\mu_{\ell}\) is the learning rate with respect to \(k_{p}(k)\) or \(k_{d}(k)\). According to (41), it can be considered that \(k_{p}(k)\) and \(k_{d}(k)\) can change with the change of system error, so as to achieve the best control effect. The control target is to promise that the desired trajectory can be tracked with a small domain of the origin in presence of dead zone and friction disturbances. The scheme map of the compound controller DRNN-LADRC is given in \[G_{DR}(k)=\sum_{\lambda}w_{\lambda}{}^{O}\cdot\Psi_{\lambda}(k) \tag{22}\]3  
(2) System stability analysis  
Since the discrete form can be converted to the continuous form, the convergence of LESO and the stability analysis of the whole system have been derived in the continuous form in this section. Firstly, the convergence of LESO is analyzed. Order, \(\chi_{i}(t)=x_{i}(t)-\zeta_{i}(t),\ i=1,\ 2,\ 3\)  
\(\Im_{i}(t)=\chi_{i}(t)/w_{0}^{\ i-1},\ i=1,\ 2,\ 3\), combining (4) and (5), the error equation of LESO estimation can be obtained:  
\[\left\{ \begin{array}{l}\dot{\Im}_{1}(t)=-\frac{\eta_{1}}{w_{0} }\Im_{1}+w_{0}\Im_{2}\\ \dot{\Im}_{2}(t)=-\frac{\eta_{2}}{w_{0}}\Im_{1}+w_{0}\Im_{3}\\ \dot{\Im}_{3}(t)=-\frac{\eta_{3}}{w_{0}}\Im_{1}+\frac{q}{w_{0}^{2}}\end{array} \right. \tag{42}\]  
Because \(\eta_{1}=3w_{0}\), \(\eta_{2}=3w_{0}{}^{2}\), \(\eta_{3}=w_{0}{}^{3}\), then (42) can be expressed as  
\[\dot{\Im}(t)=w_{0}M\sigma+N\frac{q}{w_{0}{}^{2}} \tag{43}\]  
Among them, \(M= \begin{bmatrix}-3&1&0\\ -3&0&1\\ -3&0&0\end{bmatrix} \),\(N= \begin{bmatrix}0\\ 0\\ 1\end{bmatrix} \). **Theory 1**. : _Assuming \(q\) is bounded, then there is a positive number \(\alpha_{i}>0\), limited time \(T>0\), making \(|\chi_{i}(t)|\leq\alpha_{i},\ i=1,2,3,\forall t\geq T>0\), \(|\chi_{i}(t)|\leq\alpha_{i},\ i=1,\ 2,\ 3,\ \forall t\geq T>0\)._  
Figure 3: Block diagram of whole system, where \(e_{i}\) represents \(e_{1}\) or \(e_{2}\). Solve Equation (43) to get  
\[\Im(t)=e^{w_{0}Mt}\Im(0)+\int\limits_{0}^{t}e^{w_{0}M(t-\tau)}N\frac{q}{w_{0}{}^ {2}}d\tau \tag{44}\]  
Order:  
\[G(t)=\int\limits_{0}^{t}e^{w_{0}M(t-\tau)}N\frac{q}{w_{0}{}^{2}}d\tau \tag{45}\]  
Because \(q\) bounded, \(q\leq E\), \(E>0\), then  
\[|G_{i}(t)|\leq\frac{E}{w_{0}{}^{3}}\Big{[}\Big{|}\Big{(}M^{-1}N\Big{)}_{i} \Big{|}+\Big{|}\Big{(}M^{-1}e^{w_{0}At}N\Big{)}_{i}\Big{|}\Big{]} \tag{46}\]  
And because \(M^{-1}= \begin{bmatrix}0&0&-1\\ 1&0&-3\\ 0&1&-3\end{bmatrix} \), then  
\[\Big{|}\Big{(}M^{-1}N\Big{)}_{i}\Big{|}=\left\{ \begin{array}{cc}1,&i=1\\ 3,&i=2,3\end{array} \right. \tag{47}\]  
Because \(A\) is a Hurwitz matrix, there is a finite time \(T>0\), so that all \(t\geq T\), \(i,j=1,2,3\) have:  
\[\left\{ \begin{array}{c}\Big{|}\big{(}e^{w_{0}Mt}\big{)}_{ij}\Big{|}\leq\frac {1}{w_{0}{}^{3}}\\ \Big{|}\big{(}e^{w_{0}Mt}N\big{)}_{i}\Big{|}\leq\frac{1}{w_{0}{}^{3}}\end{array} \right. \tag{48}\]  
Set up \(M^{-1}= \begin{bmatrix}v_{11}&v_{12}&v_{13}\\ v_{21}&v_{22}&v_{23}\\ v_{31}&v_{32}&v_{33}\end{bmatrix} \), \(N^{-1}=