## 3 Design of Composite Control Method
### \(H_{\infty}\) Mixed Sensitivity Controller Design  
In general, system models often involve uncertainties, making the robust stability of the control system crucial. \(H_{\infty}\) is a frequency-domain optimization method used for designing robust controllers to ensure stability and meet certain performance requirements in the presence of uncertainties and disturbances. In this paper, we have designed an \(H_{\infty}\) mixed-sensitivity controller to address disturbance rejection and robust stability issues. The \(H_{\infty}\) mixed-sensitivity control diagram is illustrated in Figure 9.  
In Figure 9, \(K(s)\) is the designed feedback controller, \(P_{n}(s)\) is the nominal model of the plant, \(r\) is the reference input model, \(n\) represents the sensor noise, and \(d_{s}\), \(d_{f}\), \(d_{g}\), and \(d_{p}\) respectively denote the spacecraft reaction torque, nonlinear friction, gravity gradient torque, and piezoelectric actuator noise. The sensitivity function \(S_{0}(s)\) and complementary sensitivity function \(T_{0}(s)\) of the system are defined as follows:  
\[S_{0}(s)=\frac{1}{1+P_{n}(s)K(s)},T_{0}(s)=\frac{P_{n}(s)K(s)}{1+P_{n}(s)K(s)}. \tag{12}\]  
The sensitivity function \(S_{0}(s)\) characterizes the transmission influence of the external input \(r\) on the control error \(e\) and the transmission influence of the measurement noise \(n\) on the measurement output \(y\). The smaller the singular values of \(S_{0}(s)\), the stronger the system's tracking ability and the better the robust performance in suppressing input disturbances. To ensure small singular values of \(S_{0}(s)\), it is required to minimize the \(\infty\)-norm of the sensitivity function, that is,  
\[\left\|W_{S0}(j\omega)\cdot S_{0}(j\omega)\right\|_{\infty}<1,\ \forall\omega, \tag{13}\]  
where \(W_{S0}(s)\) is a weighted function of \(S_{0}(s)\), aiming to improve the tracking accuracy and attenuate low-frequency disturbance. The complementary sensitivity function \(T_{0}(s)\) characterizes the robust stability of the system in response to model multiplicative uncertainty. The smaller the singular values of \(T_{0}(s)\), the better the robust performance. To ensure the stability of the telescope pointing  
Figure 8: Composite control block scheme for telescope pointing system. Figure 9: The structure of the \(H_{\infty}\) mixed-sensitivity controller. system under multiplicative perturbation \(\Delta(s)\), according to the small gain theorem [24], the robust stability condition of the closed-loop system is given by:  
\[\overline{\sigma}(\Delta(j\omega)T_{0}(j\omega))<1,\ \forall\omega, \tag{14}\]  
where \(\overline{\sigma}\) represents the maximum singular value. Assuming that the upper bound function of the magnitude frequency response for \(\overline{\sigma}(\Delta(j\omega))<W_{T0}(j\omega)\) is \(W_{T0}(s)\), the sufficient condition for robust stability is  
\[\left\|W_{T0}(s)T_{0}(s)\right\|_{\infty}<1, \tag{15}\]  
where \(W_{T0}(s)\) represents the weighting function of \(T_{0}(s)\), which represents the robustness against system uncertainty and measurement noise. Due to the relationship \(S_{0}(s)+T_{0}(s)=1\) between \(S_{0}(s)\) and \(T_{0}(s)\), it is not possible for both of them to be small simultaneously. Therefore, there is a trade-off between the robustness and the performance of the system. In practical applications, a mixed sensitivity performance index is utilized for representation [25]. \[\min_{K(s)}\left\| \begin{matrix}W_{S0}(s)S_{0}(s)\\ W_{T0}(s)T_{0}(s)\end{matrix} \right\|_{\infty}<\gamma_{0}, \tag{16}\]  
where \(0<\gamma_{0}<1\) represents the performance level of \(H_{\infty}\). Based on the above analysis, the weighted function \(W_{S0}(s)\) should possess low-pass characteristics to attenuate low-frequency disturbances. Meanwhile, the weighted function \(W_{T0}(s)\) should possess high-pass characteristics to handle multiplicative disturbances and sensor noise. The TianQin space gravitational wave detection frequency range is from 0.1 mHz to 1 Hz.