# Autonomous Pointing Control of a Large Satellite Antenna Subject to Parametric Uncertainty
## 3 Autonomous Controller Design
### Adaptive Filter Control Design  
In practice, the flexible satellite and LSA will be subject to weight loss, thermal radiation and temperature variations once in orbit. The modal parameters may well be uncertain, and even time-varying, which would render achieving the actual bending frequency extremely difficult. If the first-order modal parameters presented in Table 1 were to change, for example in the case that the values of \(\Lambda_{1}\), \(\xi_{1}\) and \(B_{rot1}\) changed to 0.082, 0.004 and 93, respectively, and the controller from Equation (13) was still adopted in the control loop to perform pointing maneuvers, then the corresponding Bode magnitude plot is that shown in Figure 3.  
As can be seen, the magnitude presents significant discontinuity at the bending frequencies of 0.11 rad/s and 0.68 rad/s although the controller in Equation (13) is included in the system. This is because a particular SF can only remove particular bending vibration signals. Thus, in order to meet the principle of the same SF frequency as that of the bending vibration, an adaptive algorithm that estimates the actual system bending frequency is required. Based on the estimated values, the controller from Equation (13) is then redesigned. The block diagram of the LSA pointing control system is shown in Figure 4. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(j\) & \(\Lambda_{j}\) & \(B_{rotj}\) & \(\xi_{j}\) & \(L_{y}\mathbf{CE}_{lay}\) \\ \hline
1 & 0.1270 & 103.491 & 0.005 & \(1.01\times 10^{-2}\) \\
2 & 0.1420 & 6.485 & 0.005 & \(5.51\times 10^{-4}\) \\
3 & 0.3401 & 2.667 & 0.005 & \(4.433\times 10^{-4}\) \\
4 & 0.5510 & \(-55.281\) & 0.005 & \(7.48\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The modal parameters of pitch axis. Figure 2: Bode magnitude diagram of satellite and antenna pitch axis. The least squares method (LSM) is proposed to estimate the actual bending frequency. The transfer function from Equation (11) is discretized, and its difference equation can be written as:  
\[A(z^{-1})y(k)=B(z^{-1})u(k-d)+\varepsilon(k) \tag{14}\]  
where \(A(z^{-1}),\ B(z^{-1})\) are discrete unit operator polynomials, \(\varepsilon(k)\) represents the external disturbances and \(d\) denotes the order of delay links. \(A(z^{-1})\) and \(B(z^{-1})\) can be developed as series expansions:  
\[A(z^{-1})=1+a_{1}z^{-1}+a_{2}z^{-2}...+a_{n_{a}}z^{-n_{a}} \tag{15}\]  
\[B(z^{-1})=b_{0}+b_{1}z^{-1}+b_{2}z^{-2}...+b_{n_{b}}z^{-n_{b}} \tag{16}\]  
where \(n_{a}\) and \(n_{b}\) denote the system orders, while \(a_{\rm i}\) and \(b_{\rm i}\) represent the parameters to be estimated. Equation (14) is rewritten as:  
\[ \begin{array}{lll}y(k)&=&-a_{1}y(k-1)-a_{n_{a}}y(k-n_{a})\\ &&+b_{0}u(k-d)+b_{n_{b}}u(k-d-n_{b})+\varepsilon(k)\\ &=&\mathbf{\varphi}^{T}(k)\delta+\varepsilon(k)\end{array}  \tag{17}\]  
Figure 4: Adaptive filter control system. PD: proportional–derivative; LSM: least squares method; LSA: large satellite antenna. Figure 3: Bode magnitude diagram subject to modal parameter uncertainty. where \(\mathbf{\varphi}(k)\) and \(\delta\) are the observation vector of system inputs and outputs and the coefficient matrix, and are respectively given by:  
\[\mathbf{\varphi}(k)=[-y(k-1),...,-y(k-n_{n_{a}}),u(k-d),...,u(k-d-n_{n_{a}})]^{\rm T} \tag{18}\]  
\[\delta=[a_{1},...,a_{n_{a}},b_{0},...,b_{n_{b}}]^{\rm T} \tag{19}\]  
The quadratic performance index is defined as:  
\[ \begin{array}{l}J=\varepsilon^{\rm T}\varepsilon=\left(\mathbf{\varphi}^{\rm T} \delta-y\right)^{\rm T}\left(\mathbf{\varphi}^{\rm T}\delta-y\right)\\ =\delta^{\rm T}\left(\mathbf{\varphi}^{\rm T}\mathbf{\varphi}\right)\delta-2y^{\rm T }\mathbf{\varphi}^{\rm T}\delta+y^{\rm T}y\end{array}  \tag{20}\]  
where \(\delta\) is the estimated value of \(\delta\). We can minimize \(J\) to obtain \(\delta\), namely \(\frac{\partial J}{\partial\delta}=0\).