## 3 Design of Composite Control Method
### Optimized Design of Disturbance Observer
#### 3.2.3 Filter Design  
In the HODOB system, the choice of the weighting function in the optimal filter \(Q\)(s) is crucial. First, considering the model uncertainties, for the telescope pointing system, the unmodeled part includes the system delay and parameter variations in the dynamic model (rotational inertia and rotational stiffness). The curve in Figure 13 reflects the frequency and amplitude variations in model perturbations caused by uncertainties in the rotational inertia and rotational stiffness. Considering the possible parameter variations in Figure 13, the upper limit function \(W_{T}\)(s) for uncertainties satisfying the frequency characteristic condition \(\mid\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
overall order \(n=3\) and the relative order \(q=1\) of the filter \(Q(s)\), the weighted function \(W_{TD}(s)\) that satisfies Equation (26) can be chosen as:  
\[W_{TD}(s)=\frac{s^{2}+2700s+8.1\times 10^{4}}{2\times 10^{6}}. \tag{33}\]  
The selection of \(W_{SD}(s)\) considers the overall order of \(Q(s)\) as \(n=n_{w}+k-1\), where \(n_{w}\) is the order of \(W_{SD}(s)\) and \(k\) is the relative order of the controlled object. In the suboptimal solution, when maximizing \(\gamma\) to approach the optimal solution, the order is reduced by 1 [27]. Therefore, the weighting function \(W_{SD}(s))\) for low-frequency disturbance rejection performance can be chosen as:  
\[W_{SD}(s)=\frac{0.5(s+1000)^{2}}{10(s+0.001)^{2}}. \tag{34}\]  
According to Equations (33) and (34), the optimal filter \(Q\) (\(s\)) can be obtained by using a robust control toolbox in Matlab (version R2021b) software  
\[Q(s)=\frac{1.9845\times 10^{6}s+4.316\times 10^{8}}{s^{3}+3129s^{2}+1.9845\times 1 0^{6}s+4.316\times 10^{8}}. \tag{35}\]  
Figure 14: Frequency response of E(\(\omega\)), \(W_{TD}^{-1}(s)\).  
Figure 13: Robust stability analysis of the closed-loop system. (**a**) Perturbation model \(\Delta(s)\) for the moment of inertia (\(J_{t1}=13\)–\(J_{t5}=17\)), with rotational stiffness held constant (\(K=130\)). (**b**) Perturbation model \(\Delta(s)\) for rotational stiffness (\(K_{1}=120\)–\(K_{5}=140\)), with moment of inertia held constant (\(J_{t}=15\)).  
The above formula satisfies the order condition \(n=3\) and the relative order condition \(k=2\). For the general quadratic coefficient filter design [28], the form that satisfies the same conditions is  
\[Q_{b}(s)=\frac{3\sigma s+1}{\left(\sigma s\right)^{3}+3(\sigma s)^{2}+3(\sigma s )+1} \tag{36}\]  
We chose \(\sigma=0.005\) and compared the low-frequency characteristics of this filter with the optimal filter from Equation (35) while ensuring consistency in the high-frequency characteristics.  
Figure 15 presents a comparison of the low-frequency performance between \(1-Q(s)\) and \(1-Q_{b}(s)\). In the frequency range below 900 rad/s, the magnitude of \(1-Q(s)\) is smaller than that of \(1-Q_{b}(s)\). Specifically, in the frequency range below 200 rad/s, the amplitude difference is approximately 20 dB. This implies that under the same order condition, the disturbance suppression capability of the filter \(Q(s)\) optimized by the \(H_{\infty}\) norm is 10 times higher than that of the traditional filter \(Q_{b}(s)\).