# Re-Optimized Design for the ADR-Based Dual-Loop Controller of 100 m Class Aperture Radio Telescope
## 4 Simulation Results and Analysis  
We apply the set of parameters of a 100 m class radio telescope in [19] to the established multiple-motor-driven 4 DOF model shown in Equation (14) to verify the performance of the re-optimized dual-loop controller, where we set the number of the motors in the driving arm to one. The quantitative feedback theory (QFT)-based robust PID (RPID) controller in [19] is used for comparison.  
Inspired by the bandwidth parameterization tuning approach of LADRC [30], we also tuned LADRC here via the system's frequency, based on the fundamental resonance frequency but not the closed-loop bandwidth frequency. The system's feedforward gain \(b_{0}\) and the fundamental resonant frequency \(\omega_{n1}\) are the first factors to determine. We obtain \(b_{0}\) by an open-loop test, and thus, it includes the information of the antenna load. The 100 m class antenna's fundamental resonance frequency is 4.27 rad/s (0.68 Hz), and consequently, the central angular frequency of the notch filter can be chosen as 4 rad/s; the low-pass filter's cutoff frequency is usually chosen to be two times bigger than \(\omega_{n1}\) and is five times it here (during tuning an interesting phenomenon is found that a cutoff frequency not less than \(2\omega_{n1}\) is sufficient, indicating the robustness of the LADRC). The bandwidth of the first-order LESO is chosen to be five times \(\omega_{n1}\). The prominent difference to the bandwidth  
Figure 4: Block diagram of the antenna’s re-optimized dual-loop control structure of P-DLADRC, where the DLADRC is rewritten as the combination of \(C(s)\) and \(H(s)\).  
parameterization approach is that the choice of the LESO's gains is not according to pole assignment but to the open-loop test. The factor \(K_{\mathrm{PV}}\) is chosen by an empirical formula of \(\omega_{n1}/4\), while \(K_{\mathrm{av}}\) is tuned by trial and error.  
The well-tuned parameters of the velocity-loop DLADRC controller and the filters are shown in Equations (24) and (4), respectively.  
\[b_{0} = 5.6\times 10^{-5},\omega_{n1}=4,\omega_{o}=k\omega_{n1}=20\] \[\beta_{1} = k_{1}\omega_{o}=8\times 20,\beta_{2}=k_{2}\omega_{o}^{2}=2\times 2 0^{2},K_{\mathrm{PV}}=0.8,K_{\mathrm{av}}=200, \tag{24}\]  
\[F_{\mathrm{n}}=\frac{s^{2}+2\times 0.1\times 4s+4^{2}}{s^{2}+2\times 1\times 4s+4^{ 2}},F_{\mathrm{v}}=\frac{1}{s/20+1}, \tag{25}\]  
The robust PID controller in the velocity-loop is as below.  
\[G_{\mathrm{v}}(s)=\frac{1900(s/0.2+1)}{s(s/5+1)}\cdot\frac{s^{2}+0.8s+16}{s^{ 2}+8s+16}, \tag{26}\]  
Wind loads are the most significant environmental disturbances for an antenna's pointing. The velocity \(v_{\mathrm{wind}}\) of the wind loads can be expressed as the sum of the steady-state (average) velocity \(V\) and the transient gust velocity \(\Delta v\), i.e., \(v_{\mathrm{wind}}=(V+\Delta v)\). The wind loads impact the antenna's primary reflector (dish) and the feed arms in terms of disturbance torque, causing a static displacement and dynamic effects, consequently deteriorating the antenna's pointing accuracy. The wind-induced torque \(\tau_{\mathrm{wind}}\) is related to the square of the wind speed \(v_{\mathrm{wind}}^{2}\), air density \(\rho_{\mathrm{air}}\), antenna aperture \(D\), as well as relative angle \(\Delta_{\theta}\). Assume that the 100 m aperture antenna suffers from oncoming wind with an average speed \(V=6\) m/s and turbulivity (it can be used for calculating a gust velocity) of 0.15 from the time of 35 s. The wind-induced torque can be written in the following form:  
\[\tau_{\mathrm{wind}}=\frac{1}{2}\rho_{\mathrm{air}}C_{\mathrm{T}}(V+\Delta v)^ {2}\cdot\frac{\pi D^{2}}{8}\frac{1}{3}D, \tag{27}\]  
where the air density is \(\rho_{\mathrm{air}}=1.225\) kg/m\({}^{3}\), and the aerodynamic thrust coefficient is \(C_{\mathrm{T}}=0.1\).