## 2 Radar Position Servo System Modeling
### Proposed Improved LADRC Design  
**Remark 2**. : _In most cases, \(k_{p}\) and \(k_{d}\) as the gains of the PD controller are acquired by an experiential or trial-and-error approach [36]. Moreover, \(k_{p}\) and \(k_{d}\) may be simplified by placing both poles for the closed loop equivalent system at the same critically damped location so that people need only consider adjusting the controller bandwidth [35]. In this paper, the DRNN is introduced to estimate the sensitivity of the plant with respect to its input on-line and tune the \(k_{p}\) and \(k_{d}\) in real time, making the LADRC have self-learning ability and adapt well to the changes of the radar servo system._  
(1) DRNN-based LADRC Design  
The DRNN is better suited for estimating the system model than the static feed-forward networks, and the DRNN has the ability to manage time-varying input or output through its natural temporal operation since the signals within the hidden layer are the feedback signals from the output layer and have different time delays [28]. Additionally, the structure of the DRNN is shown in Figure 2, where the dashed part denotes the diagonal weights that transmit the unit delay output of the recurrent neuron as an input to the same recurrent neuron. The DRNN has a three-layer structure, namely the input layer, the hidden layer and the output layer. The input weight vector is denoted by \(Q^{I}{}_{DR}=\left[w_{11}{}^{I},w_{12}{}^{I},\cdot\cdot\cdot,w_{\kappa}{}^{I} \right]^{T}\), where \(\varepsilon\) denotes the number of input neurons in the input layer, and \(\lambda\) denotes the number of recurrent neurons in the hidden layer. Additionally, the diagonal weight vector and output weight vector are expressed by \(Q^{D}{}_{DR}=\left[w_{1}{}^{D},w_{2}{}^{D},\cdot\cdot\cdot,w_{\lambda}{}^{D} \right]^{T}\), and \(Q^{O}{}_{DR}=\left[w_{1}{}^{O},w_{2}{}^{O},\cdot\cdot\cdot,w_{\lambda}{}^{O} \right]^{T}\). The external input vector is denoted by \(I(k)=\left[I_{1}(k),I_{2}(k),\cdot\cdot\cdot,I_{c}(k)\right]^{T}\). The output of any \(\lambda-th\) recurrent neuron is described by \(\Psi_{\lambda}(k)\), which is evaluated as  
\[\Psi_{\lambda}(k)=f_{act}(N_{\lambda}(k)) \tag{20}\]  
where \(f_{act}(\cdot)\) is an activation function, which is expressed by \(f_{act}(x)=\left.e^{x}-e^{-x}\right/e^{x}+e^{-x}\); \(O_{\lambda}(k)\) is the input of any \(\lambda-th\) recurrent neuron, which is evaluated as  
\[O_{\lambda}(k)=w_{\lambda}{}^{D}\cdot\Psi_{\lambda}(k-1)+\sum_{\varepsilon}w_ {\epsilon\lambda}{}^{I}\cdot I_{c}(k) \tag{21}\]There is only one neuron in the output layer, and its activation function is linear so there is no restriction on its value. Thus, the output of the DRNN can be obtained as  
\[G_{DR}(k)=\sum_{\lambda}w_{\lambda}{}^{O}\cdot\Psi_{\lambda}(k) \tag{22}\]  
Then, an error function for a training cycle can be defined as  
\[E_{D}=\frac{1}{2}[y(k)-y_{D}(k)]^{2}=\frac{1}{2}e_{d}{}^{2}(k) \tag{23}\]  
where \(y(k)\) is the output of the plant, \(y_{D}(k)=Y_{D}(k)\). The gradient of error function (23) with respect to an arbitrary weight vector simply becomes  
\[\frac{\partial E_{D}}{\partial Q}=-e_{d}(k)\cdot\frac{\partial y_{D}(k)}{ \partial Q}=-e_{d}(k)\cdot\frac{\partial G_{DR}(k)}{\partial Q} \tag{24}\]  
where \(Q\) represents \(Q^{O}{}_{DR}\), \(Q^{D}{}_{DR}\) or \(Q^{I}{}_{DR}\). **Lemma 1**[37].