## 4 Coupled Dynamic and Control for Thermally Induced Vibration and Attitude Motion
### Attitude Controller Design Based on Slide Mode Control Theory  
Sliding mode control is a practical method that can overcome system uncertainty and has robustness to disturbance and unmodeled dynamics. As for spacecraft equipped with  
Figure 12: The displacement and rotation angle response of node 25.  
large antennas, the divergence caused by the uncertainty of thermally induced vibration and the system nonlinearity can be suppressed with sliding mode control. It is also a relatively simple algorithm with high robustness to nonlinearity and is easy to realize in satellite. Examples of applications of sliding mode control on large spacecrafts can be found in reference [31; 32; 33]. By substituting Equation (23) into Equation (22), the dynamic equation of the attitude motion can be expressed as:  
\[\mathbf{J}\dot{\mathbf{\omega}}=-\mathbf{\Omega}\mathbf{J}\mathbf{\omega}+ \mathbf{T}_{c}+\mathbf{D} \tag{24}\]  
where \(\mathbf{T}_{c}\) denotes the external control torque, and:  
\[\mathbf{J}=\Big{(}\mathbf{I}_{b}+\mathbf{I}_{ab}^{b}-\mathbf{H}_{ab}\mathbf{P }_{a}(\mathbf{M}_{a}\mathbf{P}_{a})^{-1}(\mathbf{S}_{b}+\mathbf{S}_{ba}) \mathbf{A}_{ab}\Big{)} \tag{25}\]  
and \(\mathbf{\Omega}=[\mathbf{\omega}]^{X}\), \(\mathbf{D}\) denotes disturbances:  
\[\mathbf{D}=\mathbf{H}_{ab}\mathbf{P}_{a}(\mathbf{M}_{a}\mathbf{P}_{a})^{-1} \mathbf{F}-\mathbf{H}_{ab}\mathbf{P}_{a}(\mathbf{M}_{a}\mathbf{P}_{a})^{-1} \mathbf{K}_{a}\eta_{a} \tag{26}\]  
By designing the external control torque \(\mathbf{T}_{c}\), the system output can follow the ideal attitude angular velocity \(\mathbf{\omega}_{c}\). In addition, based on the attitude dynamics, we can obtain that:  
\[\dot{\mathbf{\theta}}=\mathbf{R}(\varphi,\ \theta,\ \psi)\mathbf{\omega} \tag{27}\]  
where \(\mathbf{\theta}=[\varphi\ \theta\ \psi]^{\mathrm{T}}\), the \(\varphi\), \(\theta\), \(\psi\) are the roll, pitch, and yaw angle, respectively, and  
\[\mathbf{R}(\varphi,\ \theta,\ \psi)=\left[ \begin{array}{ccc}\cos\theta&0&- \cos\varphi\sin\theta\\ 0&1&\sin\varphi\\ \sin\theta&0&\cos\varphi\cos\theta\end{array} \right] \tag{28}\]  
The double-loop sliding mode variable structure control method is adopted, which is shown in Figure 13. The control system is composed of the inner angular velocity loop and the outer angular loop. The switching function can be designed by using the integral sliding mode control. The outer loop is mainly used to design attitude angular velocity command \(\mathbf{\omega}_{c}\). The inner loop gives out the torques commands of the satellites to follow the command attitude angular velocity \(\mathbf{\omega}_{c}\) through the sliding mode control law. Defining \(\theta_{e}=\theta_{c}-\theta\) where \(\theta_{c}\) is the command attitude angle and the \(\theta\) is the real attitude angle, the outer loop sliding model surface is defined as follows:  
\[\mathbf{s}_{w}=\mathbf{\theta}_{e}+\mathbf{K}_{1}\mathbf{\int}_{0}^{t}\mathbf{ \theta}_{e}dt \tag{29}\]  
where \(\mathbf{K}_{1}=\mathrm{diag}[k_{11},\,k_{12},\,k_{13}]>0\) is the gain matrix. \[\dot{\mathbf{s}}_{w}=\dot{\mathbf{\theta}}_{e}+\mathbf{K}_{1}\mathbf{\theta}_ {e}=\dot{\mathbf{\theta}}_{c}-\dot{\mathbf{\theta}}+\mathbf{K}_{1}\mathbf{ \theta}_{e}=\dot{\mathbf{\theta}}_{c}-\mathbf{R}\mathbf{\omega}_{c}+\mathbf{R }\mathbf{\omega}_{e}+\mathbf{K}_{1}\mathbf{\theta}_{e} \tag{30}\]  
The \(\mathbf{\omega}_{c}\) needs to be designed as a differentiable form as:  
\[\mathbf{\omega}_{c}=\mathbf{R}^{-1}\Big{(}\dot{\mathbf{\theta}}_{c}+\mathbf{K }_{1}\mathbf{\theta}_{e}\Big{)}+\mathbf{R}^{-1}\mathbf{\rho}_{1}\mathbf{s}_{w }\ (\mathbf{\rho}_{1}>0) \tag{31}\]  
Figure 13: The structure diagram of the control system. The Lyapunov method is used to determine the stability of the system. A generalized Lyapunov function, which characterizes the motion of the state trajectory to the sliding surface, is defined in terms of the sliding surface.