# Autonomous Pointing Control of a Large Satellite Antenna Subject to Parametric Uncertainty
## 3 Autonomous Controller Design
### PD Plus Structural Filter Design  
It is assumed that the satellite has only one flexible appendage, the LSA. We firstly propose a pointing controller, and the transfer function of the satellite pitch axis can be then given by:  
\[\frac{\theta_{y}(s)}{T_{y}(s)}=\frac{1}{I_{y}\mathrm{s}^{2}\left(1-\sum\limits _{j}\frac{k_{j}\mathrm{s}^{2}}{\mathrm{s}^{2}+2\mathrm{c}_{j}\Lambda_{j} \mathrm{s}+\Lambda_{j}}^{2}\right)} \tag{9}\]  
Figure 1: A satellite with flexible appendages. where \(\theta_{y}\) and \(T_{y}\) are the satellite pitch angle and pitch-axis control torque respectively, \(I_{y}\) is the pitch inertia, \(s\) is the Laplace transform variable, \(k_{j}\) is the \(j\)th modal gain of LSA, \(\varsigma_{j}\) and \(\Lambda_{j}\) are the modal damping ratio and the modal frequency of LSA pitch axis. The LSA pointing can be transformed as:  
\[\frac{\theta_{ay}}{\theta_{y}}=1-s^{2}{\sum\limits_{j}{\frac{{L_{y}\phi_{lay} B_{rotj}}}{{s^{2}+2\varsigma_{j}\Lambda_{j}s+{\Lambda_{j}}^{2}}}}} \tag{10}\]  
where \(\theta_{ay}\) is the LSA pitch pointing angle. According to Equations (9) and (10), the transfer function of LSA pitch pointing in the satellite body coordinate system is given by:  
\[\frac{\theta_{ay}}{T_{y}}=\frac{1-s^{2}{\sum\limits_{j}{\frac{{L_{y}\phi_{lay} B_{rotj}}}{{s^{2}+2\varsigma_{j}\Lambda_{j}s+{\Lambda_{j}}^{2}}}}}}{{I_{y}{s^{2} \left({1-\sum\limits_{j}{\frac{{k_{j}{s^{2}}}}{{s^{2}+2\varsigma_{j}\Lambda_{ j}s+{\Lambda_{j}}^{2}}}}}\right)}}} \tag{11}\]  
where the subscript y represents pitch axis variables of satellite and LSA. For the system presented in Equation (11), the conventional PD feedback can stabilize the antenna pointing control system, but cannot effectively suppress structural vibrations of the antenna. Figure 2 shows the open-loop Bode diagram of the LSA pointing control system (dashed line). As can be seen, the magnitude is amplified and exhibits a large discontinuity at the bending frequencies of 0.198 rad/s and 0.745 rad/s corresponding to unstable poles for the control system. This arises from the vibrations of the antenna structure, which can seriously affect pointing precision and wreck system stability. The structural filter (SF) can provide a possible solution to suppress the vibration, and improve system performance. The SF can be realized from a second-order filter, represented as:  
\[\frac{{s^{2}/{\omega_{z}}^{2}+2\varsigma_{z}s/\omega_{z}+1}}{{s^{2}/{\omega_{p }}^{2}+2\varsigma_{p}s/\omega_{p}+1}} \tag{12}\]  
where \(\omega_{z}\) and \(\omega_{p}\) are the frequencies of SF zeros and poles, and \(\varsigma_{z}\) and \(\varsigma_{p}\) are the damping ratios. For different choices of \(\omega_{z}\), \(\omega_{p}\), \(\varsigma_{z}\) and \(\varsigma_{p}\), different filters can be achieved. The principle and design process of the SF have been clearly investigated by the authors of [16]. The notch filter is a kind of gain-stable filter, and can be used to suppress structure vibration for LSA pointing control. Let the frequencies of SF zeros equal to the frequencies of unstable poles, then the vibration caused by the unstable poles could be suppressed in the closed-loop system. The inertia matrix of the pitch-axis is \(I_{y}=18,050\,\mathrm{kg}\cdot\mathrm{m}^{2}\), and other representative parameters are given in Table 1[16]. As shown in Figure 2, the magnitude jumps at the frequencies of 0.198 rad/s and 0.745 rad/s. Then controller is therefore given by:  
\[T_{y}(s)=-(520s+10)\cdot\frac{25.5s^{2}+0.1s+1}{25.5s^{2}+18.2s+1}\cdot\frac{1. 83s^{2}+0.01s+1}{1.83s^{2}+2s+1}\cdot\psi(s) \tag{13}\]  
where \(\psi(s)\) denotes the pitch pointing angle error, the subscript j represents the modal order, and the first four-order modal parameters are chosen. The magnitude plot of open loop system with controller from Equation (13) is shown in Figure 2 (solid line). Obviously, the magnitude at the frequencies of 0.198 rad/s and 0.745 rad/s is well reduced through introducing a notch filter into feedback loop.