# A Robust Control Algorithm of a Descent Vehicle Angular Motion in the Earth's Atmosphere
## 6 Robust Output Regulator  
The simplified linear model described by the triple matrices (13) can be split into two components:  
* autonomous model in the Roll-Yaw channel \[\mathbf{A}_{RY}= \begin{bmatrix}\mathbf{0}_{2\times 2}&\mathbf{I}_{2}\\ \mathbf{A}_{RY[2,1]}&\mathbf{0}_{2\times 2}\end{bmatrix} ,\quad\mathbf{B}_{RY}= \begin{bmatrix} \mathbf{0}_{2\times 2}\\ \mathbf{J}_{RY}^{-1}\end{bmatrix} ,\quad\mathbf{C}_{RY}= \begin{bmatrix}\mathbf{e}_{R}^{T}& \mathbf{0}_{1\times 2}\\ \mathbf{0}_{2\times 2}&\mathbf{I}_{2}\end{bmatrix} \] (14) where \[\mathbf{A}_{RY[2,1]}= \begin{bmatrix}0&-a_{4,2}\\ 0&-a_{5,2}\end{bmatrix} ,\quad \begin{array}{c}a_{4,2}>0,\\ a_{5,2}>0,\end{array} \quad\mathbf{I}_{RY}= \begin{bmatrix}J_{x_{2}}&-J_{x_{2}y_{2}} \\ -J_{x_{2}y_{2}}&y_{y_{2}}\end{bmatrix} ,\quad\mathbf{e}_{R}= \begin{bmatrix}1\\ 0\end{bmatrix} ,\] with the desired spectrum eigenvectors \[\Lambda_{RY}=\big{\{}\phi_{1},\quad\phi_{2},\quad\phi_{3},\quad\phi_{4}\big{\}};\] (15)
* autonomous model in the Pitch channel \[\mathbf{A}_{P}= \begin{bmatrix}0&1\\ -a_{6,3}&0\end{bmatrix} ,\quad\mathbf{B}_{P}= \begin{bmatrix}0\\ \int_{-2}^{-1}\end{bmatrix} ,\quad\mathbf{C}_{P}= \begin{bmatrix}0&1\end{bmatrix} ,\] (16) where \(a_{63}>0\), with the desired spectrum eigenvectors \[\Lambda_{P}=\big{\{}\phi_{5},\quad\phi_{6}\big{\}}.\] (17)  
In records (14) and (16), the axial moments of inertia \(J_{x_{\varphi}}\), \(J_{y_{\varphi}}\), \(J_{z_{\varphi}}\) and the centrifugal moment of inertia \(J_{z_{\varphi}y_{\varphi}}\) in the SC \(E_{\varphi}\) were used. Let us consider an autonomous problem of modal output control in the Roll-Yaw channel, described by a completely controllable and completely observable (by state) triple of matrices (14) and spectrum (15). We obtain a parameterized set of its solutions based on a modification of the direct van der Wood approach [21; 30]. Figure 1: Onboard model of the spacecraft angular motion. At the zero level of decomposition  
\[\mathbf{A}_{RY0}=\mathbf{A}_{RY},\ \ \ \mathbf{B}_{RY0}=\mathbf{B}_{RY}\]  
the left annihilator and its pseudoinverse matrix [31] are respectively equal to  
\[\mathbf{B}_{RY0}^{\perp L}= \begin{bmatrix}\mathbf{I}_{2}&\mathbf{0}_{2\times 2}\end{bmatrix} ,\ \ \ \mathbf{B}_{RY0}^{\perp L+}=\mathbf{B}_{RY0}^{\perp LT}\Big{(}\mathbf{B}_{RY0}^{\perp L} \mathbf{B}_{RY0}^{\perp LT}\Big{)}^{-1}=\mathbf{B}_{RY0}^{\perp LT}.\]  
The first level of decomposition  
\[\mathbf{A}_{RY1}=\mathbf{B}_{RY0}^{\perp L}\mathbf{A}_{RY0}\mathbf{B}_{RY0}^{\perp L+}=\mathbf{0} _{2\times 2},\ \ \ \mathbf{B}_{RY1}=\mathbf{B}_{RY0}^{\perp L}\mathbf{A}_{RY0}\mathbf{B}_{RY0}=\mathbf{J}_{RY}^{-1}\]  
is finite due to the invertibility of matrix \(\mathbf{B}_{RY1}\). The controller matrix at the first decomposition level is  
\[\mathbf{K}_{RY1}=\mathbf{B}_{RY1}^{-1}\mathbf{A}_{RY1}-\mathbf{\Phi}_{RY1}\mathbf{B}_{RY1}^{-1}= -\mathbf{\Phi}_{RY1}\mathbf{J}_{RY}=-\mathbf{J}_{RY}\mathbf{\bar{\Phi}}_{RY1},\]  
where \(\mathbf{\Phi}_{RY1}\) and \(\mathbf{\Phi}_{RY1}=\mathbf{J}_{RY}\mathbf{\Phi}_{RY1}\mathbf{J}_{RY}^{-1}\) are mutually similar matrices with the spectrum  
\[\mathrm{eig}\,\mathbf{\bar{\Phi}}_{RY1}=\mathrm{eig}\,\mathbf{\Phi}_{RY1}=\{\phi_{1}, \ \ \phi_{2}\}. \tag{18}\]  
The pseudoinverse matrix and the auxiliary matrix at the zero decomposition level are, respectively, equal  
\[ \begin{array}{l}\mathbf{B}_{RY0}^{+}=\big{(}\mathbf{B}_{RY0}^{T}\mathbf{B}_{RY0}\big{)} ^{-1}\mathbf{B}_{RY0}^{T}=\big{[}\mathbf{0}_{2\times 2}\ \ \mathbf{J}_{RY}\big{]},\\ \mathbf{B}_{RY0}^{-}=\mathbf{B}_{RY0}^{+}+\mathbf{K}_{RY1}\mathbf{B}_{RY0}^{\perp L}=\mathbf{J}_{ RY}\underbrace{\big{[}-\mathbf{\bar{\Phi}}_{1}\ \ \mathbf{I}_{2}\big{]}} _{\mathbf{B}_{RY0}^{-}},\end{array} \]  
and the controller matrix at this level is  
\[