## 4 The SDRE Method
### Design SDRE Controller of the Satellite Mounted with DRC Antenna and RW  
We need to have satellite and antenna attitude track reference attitude \(\mathbf{r}_{c}\) which is obtained by adding the optimized FF control input to an ideal model. We define reference attitude \(\mathbf{r}_{c}\) as follows:  
\[\mathbf{r}_{c}=\left[\mathbf{q}_{r}^{T}(t)\ \ \mathbf{\omega}_{r}^{T}(t)\ \ \mathbf{\phi}_{r}^{T}(t)\ \ \mathbf{\dot{\phi}}_{r}^{T}(t)\ \ \mathbf{\Omega}_{r}^{T}(t)\right]^{T} \tag{27}\]  
where \(\mathbf{q}_{r},\mathbf{\omega}_{r},\mathbf{\dot{\phi}}_{r},\mathbf{\Omega}_{r}\) represent reference of quaternion, reference of satellite angular velocity, reference of DRC antenna joint angle, reference of DRC antenna joint angular velocity and reference of RW wheel speed. In order to apply the SDRE controller, we need to rewrite the kinematics and dynamics of satellite mounted with antenna into an error dynamics representation because SDRE method is regulator controller. We define state vector \(\mathbf{x}\), input vector \(\mathbf{u}\) and error dynamics as follow:  
\[\mathbf{x}=\left[\mathbf{q}_{e}^{T}\ \ \mathbf{\omega}_{e}^{T}\ \ \mathbf{\phi}_{e}^{T}\ \ \mathbf{\dot{\phi}}_{e}^{T}\ \ \mathbf{\Omega}_{e}\right]^{T}=\left[\mathbf{q}_{e}^{T}\ \ \mathbf{\omega}_{s}^{T}-\mathbf{\omega}_{r}^{T}\ \ \mathbf{\phi}^{T}-\mathbf{\phi}_{r}^{T}\ \ \mathbf{\dot{\phi}}^{T}-\mathbf{\dot{\phi}}_{r}^{T}\ \ \mathbf{\Omega}^{T}-\mathbf{\Omega}_{r}^{T}\right]^{T},\ \mathbf{u}=\left[\mathbf{\dot{\Omega}}^{T}\ \ \mathbf{\tau}_{\phi}^{T}\right]^{T}\] \[\mathbf{\dot{x}}=\mathbf{A}_{c}(\mathbf{x})\mathbf{x}+\mathbf{B}_{c}(\mathbf{x})\mathbf{u}\] \[\mathbf{A}_{c}(\mathbf{x})=\left[\begin{array}{cccc}\kappa\mathbf{E}_{4 \times 4}&\mathbf{Z}(\mathbf{q}_{e})&\mathbf{0}_{4\times 2}&\mathbf{0}_{4\times 3}&\\ \dfrac{\mathbf{W}_{R}}{q_{e0}}&\mathbf{0}_{3\times 3}&-\mathbf{I}_{c}^{-1}(\mathbf{I}_{c}+\mathbf{ \omega}_{s}\times\mathbf{I}_{c})&\mathbf{0}_{3\times 2}&\mathbf{W}&-\mathbf{I}_{c}^{-1}\mathbf{ \omega}_{s}\times\mathbf{I}_{RW}\\ \mathbf{0}_{2\times 4}&\mathbf{0}_{2\times 3}&\kappa\mathbf{E}_{2\times 2}&\mathbf{E}_{2 \times 2}&\mathbf{0}_{2\times 3}\\ \dfrac{-\mathbf{\dot{\phi}}_{r}}{q_{e0}}&\mathbf{0}_{2\times 3}&\mathbf{0}_{2\times 3}&\mathbf{0}_{ 2\times 2}&\kappa\mathbf{E}_{2\times 2}&\mathbf{0}_{2\times 3}\\ \dfrac{-\mathbf{\dot{\Omega}}_{r}}{q_{e0}}&\mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 3}&\mathbf{0 }_{3\times 2}&\mathbf{0}_{3\times 2}&\kappa\mathbf{E}_{3\times 3}\\ \end{array}\right]\] \[\mathbf{B}_{c}(\mathbf{x})=\left[\begin{array}{cccc}\mathbf{0}_{4\times 3}&\mathbf{0}_{4 \times 2}\\ -\mathbf{I}_{c}^{-1}\mathbf{I}_{RW}&-\mathbf{I}_{c}^{-1}\mathbf{I}_{ac}\mathbf{e}_{\phi_{1}}/P_{1} -\mathbf{I}_{c}^{-1}\mathbf{I}_{ac}\mathbf{e}_{\phi_{2}}/P_{2}\\ \mathbf{0}_{2\times 3}&\mathbf{0}_{2\times 2}\\ \mathbf{0}_{2\times 3}&\mathbf{\mathrm{diag}}(1/P_{1},1/P_{2})\\ \mathbf{E}_{3\times 3}&\mathbf{0}_{3\times 2}\\ \end{array}\right] \tag{28}\] \[\mathbf{W}=\left[-\mathbf{I}_{c}^{-1}(\mathbf{\dot{I}}_{ac}+\mathbf{\omega}_{s} \times\mathbf{I}_{ac})\mathbf{e}_{\phi_{1}}\ \ -\mathbf{I}_{c}^{-1}(\mathbf{\dot{I}}_{ac}\mathbf{e}_{\phi_{2}}+\mathbf{I}_{ac}\mathbf{\dot{e}}_{ \phi_{2}}+\mathbf{\omega}_{s}\times\mathbf{I}_{ac}\mathbf{e}_{\phi_{2}})\right]\] \[\mathbf{W}_{\mathbf{R}}=-\mathbf{\dot{\omega}}_{r}-\mathbf{I}_{c}^{-1}\left(\mathbf{ \dot{I}}_{c}+\mathbf{\omega}_{s}\times\mathbf{I}_{c}\right)\mathbf{\omega}_{r}\] \[\mathbf{Z}(\mathbf{q}_{e})=\dfrac{1}{2}\begin{bmatrix}-q_{e1}&-q_{e2}&-q_{e 3}\\ q_{e0}&-q_{e3}&q_{e2}\\ q_{e3}&q_{e0}&-q_{e1}\\ -q_{e2}&q_{e1}&q_{e0}\end{bmatrix}\]  
where \(\mathbf{E}\) is the unit matrix, and \(\kappa\) is a small negative error that needs to be chosen not to affect the system, and \(\mathbf{q}_{e}\) represents quaternion error vector. In order to written the nonlinear equations of satellite mounted with DRC antenna kinematics and dynamics in state space form, we substituted Equation (12) into Equation (8).