# Re-Optimized Design for the ADR-Based Dual-Loop Controller of 100 m Class Aperture Radio Telescope
## 3 Re-Optimized Design of the Dual-Loop Controller
### Velocity Controller Re-Optimization  
From Figure 2, it can be easily seen that the load torque \(\tau_{\rm d}\) is an external disturbance for the motor's speed control system. The load torque \(\tau_{\rm d}\) can be represented as  
\[\tau_{\rm d}=k(K_{\rm t}r_{i}-J_{\rm m}\bar{\theta}_{\rm g}), \tag{15}\]  
where \(\bar{\theta}_{\rm g}\) is the motor angular acceleration. If the disturbance can be compensated, the motor speed control performance can be improved, which is the basis of designing an active disturbance suppression-based controller. Active disturbance rejection control (ADRC) [28; 29] is a valuable engineering application approach in the disturbance/uncertainty estimation and attenuation (DUEA) methods. The method has now been developed as linear ADRC (LADRC) [30] and other derivatives. Since only one integrator exists in the closed-current-loop motor's speed control system, we have used a first-order LADRC. The equation of the linear extended state observer (LESO) of the first-order LADRC algorithm that we employed to observe and compensate for the load torque disturbance \(\tau_{\rm d}\) is  
\[\left[ \begin{array}{c}\dot{z}_{1}\\ \dot{z}_{2}\end{array} \right]=\left[ \begin{array}{cc}0&1\\ 0&0\end{array} \right]\left[ \begin{array}{c}z_{1}\\ z_{2}\end{array} \right]+\left[ \begin{array}{c}b_{0}\\ 0\end{array} \right]u+\left[ \begin{array}{c}\beta_{1}\\ \beta_{2}\end{array} \right](\dot{\theta}_{\rm g}-z_{1}), \tag{16}\]  
where \(b_{0}\) is the estimated forward gain of the motor from the input voltage to the output velocity, \(u\) is the controller output, \(z_{1}\) and \(z_{2}\) are the velocity and disturbance observed by the LESO, \(\left[ \begin{array}{cc}\beta_{1}&\beta_{2}\end{array} \right]^{T}\) is the observation vector of the LESO to be designed. The equation for the first-order LADRC's typical output is shown below. \[u_{c1}=\frac{K_{\rm pv}(r_{\rm v}-z_{1})-z_{2}}{b_{0}}, \tag{17}\]  
where \(K_{\rm pv}\) is the controller gain. Theoretically, the disturbance can be suppressed entirely. In practice, the disturbance \(\tau_{\rm d}\) cannot be directly measured but observed/identified with the help of an observer. It is well known that the estimated disturbance lags relative to the real one. Thus, only compensating for the disturbance has a limited effect, and the impact of the load disturbance is still significant on the motor's velocity. It can be seen from Equation (16) that the error \((\dot{\theta}_{\rm g}-z_{1})\) used to calculate the disturbance is approximately equal to the disturbance velocity. The anti-disturbance performance of the system is further enhanced if it compensates for both the observed disturbance and its velocity. Therefore, it can be considered to add a disturbance-velocity compensation term with a factor of \(K_{\rm av}\) to the conventional first-order LADRC control equation, and then, the new controller equation is  
\[u_{c}=\frac{K_{\rm pv}(r_{\rm v}-z_{1})-z_{2}-K_{\rm av}(\dot{\theta}_{\rm g} -z_{1})}{b_{0}}, \tag{18}\]  
In addition to this, we also utilize a notch filter \(F_{\rm n}\) to filter the fundamental mechanical resonance frequency of the antenna and a first-order low-pass filter \(F_{\rm v}\) to limit the controller's bandwidth. It is valuable to note that applying these two filters mainly attenuates the antenna resonances during the motor control rather than achieving better motor velocity control. The transfer functions of the low-pass filter \(F_{\rm v}\) and notch filter \(F_{\rm n}\) are shown in Equations (18) and (19), respectively. \[F_{\rm v}=\frac{1}{s/\omega_{\rm F}+1}, \tag{19}\]  
\[F_{\rm n}=\frac{s^{2}+2\xi_{1}\omega_{\rm n1}s+\omega_{\rm n1}^{2}}{s^{2}+2\xi_{2 }^{2}\omega_{\rm n1}s+\omega_{\rm n1}^{2}}, \tag{20}\]  
where \(\omega_{\rm F}\) is the filter cutoff frequency, \(\omega_{\rm n1}\) is the resonant frequency to be filtered, and the value \(20\lg(\xi_{1}/\xi_{2})\) is the notch depth.