## 4 DDPG Optimization of Control Parameters for Active Disturbance Rejection Controller
### Stability Analysis
First, let \(\beta=-w_{0}\frac{\lambda min\big{(}Q_{\eta}\big{)}}{2\lambda max\big{(}P_{ \eta}\big{)}},\alpha=\frac{F_{h}\lambda_{max}\big{(}P_{\eta}\big{)}}{\omega_{o }^{3}\sqrt{\lambda_{min}\big{(}P_{\eta}\big{)}}}\). By applying the inequality, we can obtain the following. Assume \(\dot{W}=\beta W+\alpha\). Thus, there is \(W\leq(W(t_{0})-\frac{\alpha}{B})e^{\beta(t-t_{0})}+\frac{\alpha}{B}\). By organizing, we can obtain the following expression:  
\[ \begin{array}{l}W\leq(\frac{2F_{h}\lambda_{\max}^{2}(P_{q})}{\omega_{0}^{4} \sqrt{\lambda_{\min}\big{(}P_{q}\big{)}\lambda_{\min}\big{(}Q_{q}\big{)}}}-W(t_ {0}))e^{-\omega_{0}\frac{\lambda_{\min}\big{(}Q_{q}\big{)}}{2\lambda_{\min} \big{(}Q_{q}\big{)}}(t-t_{0})}\\ +\frac{2F_{h}\lambda_{\max}^{2}(P_{q})}{\omega_{0}^{4}\sqrt{\lambda_{\min} \big{(}P_{q}\big{)}\lambda_{\min}\big{(}Q_{q}\big{)}}}\end{array}  \tag{41}\]  
From \(W=\sqrt{V(q)}=\sqrt{q^{T}P_{q}q}\), \(\frac{V(q)}{\lambda_{\max}\big{(}P_{q}\big{)}}\leq\|\ q\|^{2}\leq\frac{V(q)}{ \lambda_{\min}\big{(}P_{q}\big{)}}\), we can obtain Equation (42):  
\[\|\ q\|\leq\frac{\sqrt{V}}{\sqrt{\lambda_{min}\big{(}P_{q}\big{)}}}=\frac{W}{ \sqrt{\lambda_{min}\big{(}P_{q}\big{)}}} \tag{42}\]  
When \(t\rightarrow\infty\), the Expression (43) can be obtained:  
\[ \begin{array}{l}\|\ q\ \|\leq\frac{\sqrt{V(q)}}{\sqrt{\lambda_{\min} \big{(}P_{q}\big{)}}}\\ \leq\frac{2F_{h}\lambda_{\max}^{2}\big{(}P_{q}\big{)}}{\omega_{0}^{4}\lambda_ {\min}\big{(}P_{q}\big{)}\lambda_{\min}\big{(}Q_{q}\big{)}}\\ =\frac{\frac{\alpha}{B}}{\omega_{0}^{4}}\end{array}  \tag{43}\]  
In Formula (43), \(k=\frac{2F_{h}\lambda_{\max}^{2}\big{(}P_{q}\big{)}}{\lambda_{\min}\big{(}P_ {q}\big{)}\lambda_{\min}\big{(}Q_{q}\big{)}}\) is a normal constant, and because \(P_{q}\), \(Q_{q}\) are independent of \(\omega_{0}\), the Equation (43) can demonstrate that \(\underset{\omega_{0}\rightarrow\omega_{0}\rightarrow\infty}{lim}||q||=0\), which, together with \(q_{i}=\frac{\varepsilon_{i}}{\omega_{0}^{2}}(i=1,2,3)\), is produced. Therefore, \(\underset{\omega_{0}\rightarrow\omega_{0}\rightarrow\infty}{lim}||e||\leq||q|| \omega_{0}^{3}\leq\frac{k}{\omega_{0}}=0\), thus completing the proof. Define \(H_{e}\) as \(H_{e}=\{e|||||e||\leq E\}\), where \(E\) is a positive constant. By adjusting \(\omega_{0}\) to ensure \(\frac{k}{\omega_{0}}\leq E\), the estimated error of LESO will remain within \(H_{e}\), Theorem 1 has been proven. \(\square\)  
Theorem 2: _According to the error fechack control law given by Equation (19), we can ensure the closed-loop stability of the control system. According to Theorem 1, the convergence of LESO can be guaranteed by carefully selecting \(\omega_{0}\),b, and the estimation error of LESO will be constrained within \(H_{e}\). Translate Equation (19) into the Equation (14) to obtain:_  
\[\left\{ \begin{array}{l}\dot{e}_{\varphi_{1}}=e_{\psi_{2}}\\ \dot{e}_{\varphi_{2}}=-k_{p}e_{\psi_{1}}-k_{d}e_{\psi_{2}}+k_{p}e_{1}+k_{d}e_{2 }+e_{3}\end{array} \right. \tag{44}\]  
Referencing Equation (44), \(e_{\psi 1}=\psi_{r}-\psi_{1}\) and \(e_{\psi 2}=\dot{\psi}_{r}-\psi_{2}\) are defined as tracking errors. The above equation can be rewritten as:  
\[ \begin{array}{l}\dot{e}_{\psi}=Ce_{\psi}+Ge\\ e_{\psi}=\begin{bmatrix}e_{\psi 1},e_{\psi 2}\end{bmatrix}^{T},C=\begin{bmatrix}0&1\\ -k_{p}&-k_{d}\end{bmatrix}\\ G=\begin{bmatrix}k_{p}&k_{d}&1\end{bmatrix}\end{array}  \tag{45}\]  
Since the controller output is positive, ensure that \(k_{p},k_{d}\geq 0\). Then, the characteristic roots of \(C\) can be expressed as:  
\[\lambda_{1,2}=-\frac{k_{d}}{2}\pm\sqrt{-k_{p}+\frac{k_{d}^{2}}{4}} \tag{46}\]Therefore, the designed linear active disturbance rejection controller is stable, and Theorem 2 has been proven. The tracking error of the observer is also bounded.