## 4 Case Study
### Case Study 2: Regular Controller Performances  
In this part, the tracking control of the radar system (8) via the proposed compound controller DRNN-LADRC verifies the control performance of the composite controller,  
Figure 8: Tracking outputs of the ATD. (**a**) Tracking the given signal, (**b**) Tracking the derivative signal.  
Figure 7: Tracking outputs of TD while \(\kappa=15\), \(h_{0}=0.2\). (**a**) Tracking the given signal, (**b**) Tracking the derivative signal.  
Figure 9: The adaptive adjustment curves of the speed factor \(\kappa(k)\) and filter factor \(h_{0}(k)\).  
and uses the classic LADRC controller for comparison. The tracking target of the radar system is \(\
u_{d}(k)=0.1\pi(\sin(2t)-\cos(t))\). The parameters of the radar system (8) are given as follows: \(R_{a}=1.25\)\(\Omega\), \(J=0.03\) kg \(\cdot\) m\({}^{2}\), \(K_{E}=3.11\)\(v/\)rad \(\cdot\) s\({}^{-1}\), \(K_{T}=3.2\) N \(\cdot\) m/A, \(K_{P}=3.5\), \(F=0.74\), \(m=8.5\) kg, \(r=0.3\) m, \(J_{L}=2.37\) kg \(\cdot\) m\({}^{2}\), \(\gamma(t)=\sin(t)\), \(d=0.5\), \(F_{c}=0.01\) N \(\cdot\) m, \(F_{s}=0.03\) N \(\cdot\) m, and \(B=0.02\), \(\tau=3\). The parameters of the DRNN-LADRC are expressed as: \(h=0.001\), \(w_{0}=180\), \(k_{p}(0)=35\), \(k_{d}(0)=35\), \(b_{0}=1.8\), \(K=300\), \(L=15\), \(\sigma_{1}=3.5\), \(\sigma_{2}=10.25\), \(\mu_{D}=0.4\chi_{D}=0.04\mu_{\ell}=0.51(k)=\left[u(k-1)\quad y(k-1)\quad 1 \right]^{T}\), \(\mu_{D}=0.4\), \(\chi_{D}=0.04\), and \(\mu_{\uparrow}=0.5\) the number of neurons in the input layer, hidden layer, and output layer are \(3,7\) and \(1\), respectively. The initial value of the weight vectors with respect to the input layer, hidden layer, and output layer, are random vectors in a certain range, and the initial weight is usually set from \(-0.5\) to \(0.5\).  
Tracking outputs of the proposed DRNN-LADRC and traditional LADRC under the same conditions are demonstrated in Figure 10. Additionally, the comparison results of tracking errors between the DRNN-LADRC and LADRC are shown in Figure 11. As seen, the tracking accuracy of the DRNN-LADRC is higher with smaller tracking errors, the disturbance effects caused by the dead zone and friction nonlinearities are alleviated and the tracking performance of the radar system has been improved. The responses of the proposed ATD depicted in Figure 12a show that \(v_{1}\) and \(v_{2}\) can track the initial and derivative signal well, respectively, without phase lags. Additionally, the adaptive adjustment curves of \(\kappa(k)\) and \(h_{0}(k)\) are shown in Figure 12b. Figure 13 shows the adaptive adjustment curves of \(k_{p}(k)\) and \(k_{d}(k)\) in LSEF, which can adjust constantly with the change of system error. The Jacobian information of the plant, namely the sensitivity of the plant with respect to its input, is estimated by the DRNN on-line as shown in Figure 14, which can be introduced into the process of the parameter adjustment in LSEF. The internal disturbances of the radar position servo system are shown in Figure 15, and the curves of friction nonlinearity and dead zone are shown in Figure 15a,b, respectively.  
Figure 11: Comparison result of tracking errors between the DRNN-LADRC and traditional LADRC.  
Figure 12: Responses of the ATD. (**a**) Tracking signals of \(v_{1}\) and \(v_{2}\), (**b**) Adaptive adjustment curves of \(\kappa(k)\) and \(h_{0}(k)\).  
Figure 13: The adaptive adjustment curves of \(k_{p}(k)\) and \(k_{d}(k)\) in DRNN-LADRC.