## 2 Radar Position Servo System Modeling
### Proposed Improved LADRC Design
\begin{bmatrix}\rho_{11}&\rho_{12}&\rho_{13}\\ \rho_{21}&\rho_{22}&\rho_{23}\\ \rho_{31}&\rho_{32}&\rho_{33}\end{bmatrix} \), then  
\[ \begin{array}{c}\big{|}\big{(}M^{-1}e^{w_{0}Mt}N\big{)}_{i}\big{|}=|v_{1i} \rho_{13}+v_{2i}\rho_{23}+v_{3i}\rho_{33}|\\ \leq\left\{\begin{array}{cc}\frac{1}{w_{0}{}^{3}},&i=1\\ \frac{4}{w_{0}{}^{3}},&i=2,3\end{array}\right.\end{array}  \tag{49}\]  
For simultaneous Equations (46), (47) and (49), Equation (46) can be written as  
\[|G_{i}(t)|\leq\frac{3E}{w_{0}{}^{3}}+\frac{4E}{w_{0}{}^{6}} \tag{50}\]  
For all \(t\geq T\), \(i=1,2,3\), order \(\Im_{\max}(0)=|\Im_{1}(0)|+|\Im_{2}(0)|+\Im_{3}(0)\),  
\[\Big{|}\Big{[}e^{w_{0}Mt}\Im(0)\Big{]}_{i}\Big{|}\leq\frac{\Im_{\max}(0)}{w_{ 0}{}^{3}} \tag{51}\]  
\[|\Im_{i}(t)|\leq\frac{\Im_{\max}(0)}{w_{0}{}^{3}}+\frac{3E}{w_{0}{}^{3}}+\frac {4E}{w_{0}{}^{6}} \tag{52}\]  
Order \(\chi_{\max}(0)=|\chi_{1}(0)|+|\chi_{2}(0)|+|\chi_{3}(0)|\), in addition, according to \(\Im_{i}(t)=\chi_{i}(t)/w_{0}{}^{i-1}\), and according to Formulas (50)-(52)  
\[|\chi_{i}(t)|\leq\frac{\Im_{\max}(0)}{w_{0}{}^{3}}+\frac{3E}{w_{0}{}^{4-i}}+ \frac{4E}{w_{0}{}^{7-i}}=\alpha_{i} \tag{53}\]  
Equation (53) shows that if the differential of the total disturbance is bounded, the prediction error of LESO is bounded, and when the \(w_{0}\) increases, \(|\chi_{i}(t)|\to 0\). Then, we analyze the stability of the whole control system. Consider the following:  
\[\dot{V}(t)=JV(t)+p(t) \tag{54}\]  
Among them, \(V(t)=[V_{1}(t),V_{2}(t),\cdots,V_{n}(t)]^{T}\in R^{n}\), \(p(t)=[p_{1}(t),p_{2}(t),\cdots,p_{n}(t)]^{T}\)\(\in R^{n}\), \(J\) is \(n\times n\) matrix. **Lemma 2**. : _If \(J\) is a Hurwitz matrix and \(\lim_{t\rightarrow\infty}\lVert p(t)\rVert=0\), then \(\lim_{t\rightarrow\infty}\lVert V(t)\rVert=0\)._  
**Theory 2**. : _If \(q\) is bounded, there is a constant \(w_{0}>0\), \(k_{p}>0\), \(k_{d}>0\) to make the closed-loop system stable._  
**Definition 1**. : \(v_{a}=\widetilde{v}_{1}\)_, \(\dot{v}_{a}=\widetilde{v}_{2}\), \(\tilde{v}_{a}=\widetilde{v}_{3}\), \(\delta_{1}=\widetilde{v}_{1}-x_{1}\), \(\delta_{2}=\widetilde{v}_{2}-x_{2}\), \(\delta_{3}=\widetilde{v}_{3}-x_{3}\), because \(\chi_{i}(t)=x_{i}(t)-\zeta_{i}(t),i=1,\ 2,\ 3\), Equation (12) can be written as_  
\[u=\left[k_{p}(\delta_{1}+\chi_{1})+k_{d}(\delta_{2}+\chi_{2})-(x_{3}-\chi_{3} )+\widetilde{v}_{3}\right]/d \tag{55}\]  
Because  
\[\dot{\delta}_{1}=\dot{\widetilde{v}}_{1}-\dot{x}_{1}=\widetilde{v}_{2}-x_{2} =\delta_{2} \tag{56}\]  
\[\dot{\delta}_{2}=\dot{\widetilde{v}}_{2}-\dot{x}_{2}=\widetilde{v}_{2}-(x_{3 }+b_{0}u)=-k_{p}\delta_{1}-k_{d}\delta_{2}-k_{p}\chi_{1}-k_{d}\chi_{2}-\chi_ {3} \tag{57}\]  
Then  
\[\dot{\delta}(t)=\widetilde{\Gamma}_{1}\delta(t)+\widetilde{\Gamma}_{2}\chi(t) \tag{58}\]  
Among them, \(\widetilde{\Gamma}_{1}= \begin{bmatrix}0&1\\ -k_{p}&-k_{d}\end{bmatrix} \), \(\widetilde{\Gamma}_{2}= \begin{bmatrix}0&0&0\\ -k_{p}&-k_{d}&-1\end{bmatrix} \). According to Theory 1, \(|\chi_{i}(t)|\leq\frac{\chi_{\max}(0)}{w_{0}^{3}}+\frac{3E}{w_{0}^{-1}}+\frac {4E}{w_{0}^{7}-1},\ i=1,2,3\), and when \(w_{0}\) increases \(|\chi_{i}(t)|\to 0\), available \(\lim_{t\rightarrow\infty}\left\lVert\widetilde{\Gamma}_{2}\chi(t)\right\rVert=0\), the size selection of \(k_{p}\) and \(k_{d}\) makes \(\widetilde{\Gamma}_{1}\) a Hurwitz matrix. Because the DRNN optimizes \(k_{p}\) and \(k_{d}\), it must make \(k_{p}>0\), \(k_{d}>0\), in conclusion, according to Theory 1 and Lemma 1, the existence constant \(w_{0}>0,\ k_{d}>0,\ \text{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{\rm{ \rm{\rm{\rm{ \rm{ \rm{        }}}}}}}}}}}}}}}\)\(\phi_{i}(t)=0,\ i=1,\ 2\) and the closed loop system is stable. (3) BPNN-based LADRC Design  
**Remark 3**.