## 5 The SDRE Integral Type Servo Controller
### Design of SDW Matrix
#### 5.3.3 Design of SDW Matrix-Cooperative Control of RW and DRC Antenna  
From results of simulation I, we find that it is necessary to switch the characteristics of SDRE servo controller from regulator to servo according to whether the reference trajectory is time-variant or not in order to shorten the time of settling time. More specifically, when the reference trajectory is time-variant, we are designing a large SDW matrix for an integral state error vector. When the reference trajectory is fixed, and the steady state error is small, we design a large SDW matrix for a state error vector as shown in Table 4. In order to switch the SDW matrix depending on reference trajectory, we augment the state vector and SDLR as follows:  
\[\hat{\mathbf{x}}= \begin{bmatrix}\mathbf{q}_{e}^{T}&\mathbf{\omega}_{e}^{T}&\mathbf{ \phi}_{e}^{T}&\mathbf{\Omega}_{e}^{T}&\mathbf{\Omega}_{e}^{T}&\int\mathbf{q}_{e}^{T}\mathrm{ d}t&\int\mathbf{\omega}_{e}^{T}\mathrm{d}t&\int\mathbf{\phi}_{e}^{T}\mathrm{d}t&\mathbf{ \omega}_{r}^{T}&\mathbf{\phi}_{r}^{T}\end{bmatrix} ^{T},\ \ \mathbf{u}= \begin{bmatrix}\mathbf{\Omega}^{T}&\mathbf{\tau}_{\phi}^{T} \end{bmatrix} ^{T} \tag{42}\] \[\hat{\mathbf{x}}=\hat{\mathbf{A}}_{c}(\hat{\mathbf{x}})\hat{\mathbf{x}}+\hat{\mathbf{ B}}_{c}(\hat{\mathbf{x}})u\] \[\hat{\mathbf{A}}_{c}(\hat{\mathbf{x}})= \begin{bmatrix}\tilde{\mathbf{A}}_{c} (\hat{\mathbf{x}})&\mathbf{0}_{25\times 3}&\mathbf{0}_{25\times 2}\\ \dot{\mathbf{\omega}}_{r}/q_{e0}\,\mathbf{0}_{3\times 24}&\kappa\mathbf{E}_{3\times 3}&\mathbf{0 }_{3\times 2}\\ \dot{\mathbf{\phi}}_{r}/q_{e0}\,\mathbf{0}_{3\times 24}&\mathbf{0}_{2\times 3}&\kappa\mathbf{E}_{ 2\times 2}\end{bmatrix} \mathbf{B}_{c}(\hat{\mathbf{x}})= \begin{bmatrix}\tilde{\mathbf{B}}_{c}( \hat{\mathbf{x}})\\ \mathbf{0}_{3\times 5}\\ \mathbf{0}_{2\times 5}\end{bmatrix} \]  
To switch the SDW matrix, we define the sigmoid function as follows:  
\[K(x)=1-1/(1+\exp(-1000(|x|-0.01))) \tag{43}\]  
We show the sigmoid function in Figure 10. Next, we define the gain function as follows:  
\[K_{Q(q_{e},\mathbf{\omega}_{r})}=K_{(q_{e})}K_{(\mathbf{\omega}_{r})},\ K_{Q(q_{e},\bm {\omega}_{r})}^{int}=1-K_{Q(q_{e},\mathbf{\omega}_{r})} \tag{44}\]  
\(K_{Q}\) becomes 1 when the steady state error is small and reference trajectory is fixed. \(K_{Q}^{int}\) becomes 1 when the steady error is large or reference trajectory is time-variant. \begin{table}
\begin{tabular}{c c} \hline \hline reference trajectory is time-variant & \(\mathbf{Q}_{x_{e}}^{int}\) is large \\ \hline reference trajectory is fixed & \(\mathbf{Q}_{x_{e}}\) is large \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Design guidelines. Figure 10: Sigmoid function of proposed method \(K(x)\). \begin{table}
\begin{tabular}{c c c} \hline \hline  & **SDRE** & **SDRE Servo** \\ \hline settling time & 52.82 s & 57.04 s \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Settling time of simulation II. On the other hand, from the results of simulation II, we can confirm that we should make the SDW matrix for state error vector \(\mathbf{x}_{e}\) when the steady error is large in order to reduce the overshoot that occurs between 38 s and 48 s. Therefore, we design the SDW matrix with the design guidelines as shown in Table 5. In Table 5 we make the SDW matrix for integral of state error vector large when the reference attitude is fixed and state error is large because we consider the case where unknown disturbance occurs when satellite is steady-state. With the design guidance Table 5, we design the SDW matrix of state as follows:  
\[