# Autonomous Pointing Control of a Large Satellite Antenna Subject to Parametric Uncertainty
## 3 Autonomous Controller Design
### Active Disturbance Rejection Filter  
Thermal deformation errors, installation errors and deployment errors can all negatively affect the accuracy and stability of LSA pointing. The installation error and the deployment error can generally be assumed to be constant, and thus can be compensated through precise calibration and adding an integrator in a closed-loop system. During in-orbit operations, solar radiation pressure may periodically lead to antenna thermal deformation, which brings about periodic errors in the LSA pointing. After successful stabilization of the satellite and flexible LSA, to decrease the periodic pointing error, the active disturbance rejection technique is introduced into the feedback loop. The solar radiation-induced periodic error, \(\theta_{they}\), is firstly modeled as:  
\[\theta_{they}=\theta_{1}\sin(p_{1}t+\phi_{1}) \tag{24}\]where \(\theta_{1}\) and \(\phi_{1}\) are the unknown amplitude and phase angle, while \(p_{1}\) denotes the known frequency. In general, the error \(\theta_{they}\) can be described by a Laplace transformation:  
\[\theta_{they}(s)=\frac{N_{they}(s)}{D_{they}(s)} \tag{25}\]  
where \(N_{they}(s)\) is arbitrary, while the roots of \(D_{they}(s)\) correspond to the frequencies at which periodic excitation takes place. The active disturbance rejection filter in the control loop provides a solution for effective cancellation of the poles of \(\theta_{they}(s)\), which is on the basis of the internal model principle. As shown in Figure 5, the closed-loop transfer function is:  
\[y(s)=\frac{D_{SF}(s)D_{\text{they}}(s)}{D_{they}(s)D_{SF}(s)D(s)+N_{SF}(s)N_{PD }(s)}\frac{N_{\text{they}}(s)}{D_{\text{they}}(s)} \tag{26}\]  
The presence of \(1/D_{\text{they}}(s)\) in the control loop results in the effective cancellation of the poles of \(\theta_{they}(s)\), provided that no root of \(D_{\text{they}}(s)\) is a zero of the system transfer function. Then, an active disturbance rejection filter can be designed that has proper transfer function and uses \(1/D_{\text{they}}(s)\). Besides, a proper numerator is chosen to go with \(1/D_{\text{they}}(s)\), which is of the same order as \(D_{\text{they}}(s)\), and as such that there is a zero for each pole. The active disturbance rejection filter can employ a zero-pole combination, and is given by:  
\[\frac{s^{2}/{z_{1}}^{2}+2\xi_{z}s/z_{1}+1}{s^{2}/{p_{1}}^{2}+1} \tag{27}\]  
where \(z_{1}\) and \(p_{1}\) are a pair of zero poles, and \(\xi_{z}\) is a gain. The modified controller is thus given by:  
\[T_{z}(s)=-\big{(}K_{d}s+K_{p}\big{)}\cdot\frac{s^{2}/{\omega_{z}}^{2}+2\xi_{z} s/{\omega_{z}}+1}{s^{2}/{\omega_{p}}^{2}+2\xi_{p}s/{\omega_{p}}+1}\cdot\frac{s^{2}/ {z_{1}}^{2}+2\xi_{z}s/z_{1}+1}{s^{2}/{p_{1}}^{2}+1}\cdot\psi(s) \tag{28}\]  
Equation (28) can then be rewritten as:  
\[T_{z}(s)=-\big{(}K_{d}s+K_{p}\big{)}\cdot\frac{s^{2}/{\omega_{z}}^{2}+D_{1z} s+1}{s^{2}/{\omega_{p}}^{2}+D_{2z}s+1}\cdot\frac{T_{z1}s^{2}+D_{z1}s+1}{s^{2}/ {p_{1}}^{2}+1}\cdot\psi(s) \tag{29}\]  
where \(K_{d}\), \(K_{p}\), \(D_{1z}\), \(D_{2z}\), \(T_{z1}\) and \(D_{z1}\) are the design parameters. For the solar radiation induced error \(p_{1}=0.01\) rad/s [16], hence, the modified autonomous pointing controller is governed by the following equations:  
\[ \begin{array}{c}T_{z}(s)=-(520s+10)\cdot\frac{1\times 10^{6}s^{2}+200s+1}{1 \times 10^{6}s^{2}+1}\cdot F(s)\psi(s)\\ F(s)=\frac{25.5s^{2}+0.1s+1}{25.5s^{2}+18.2s+1}\cdot\frac{1.83s^{2}+0.01s+1}{ 1.83s^{2}+2s+1}\end{array}  \tag{30}\]  
Figure 5: Active disturbance rejection control system. SF: structural filter. **Remark 3**: _The separation between the zero and the pole affects the setting time of the closed-loop system. Generally, the larger the separation is, the shorter the settling time will be. This is a consequence of the position of the closed-loop eigenvalue corresponding to the zero-pole. As the separation is increased, the eigenvalue is pushed farther to the left, speeding up the response time of the rejection filter.