# A Robust Control Algorithm of a Descent Vehicle Angular Motion in the Earth's Atmosphere
## 6 Robust Output Regulator
\begin{array}{l}\mathbf{K}_{RY}=\mathbf{K}_{RY0}=\mathbf{B}_{RY0}^{-}\mathbf{A}_{RY0}-\mathbf{ \Phi}_{RY0}\mathbf{B}_{RY0}^{-}=\\ =\mathbf{J}_{RY}\big{[}\mathbf{A}_{RY[2,1]}+\mathbf{\bar{\Phi}}_{RY0}\mathbf{\bar{\Phi}}_{RY1 }\ \ \ \ \ -\mathbf{\bar{\Phi}}_{RY0}-\mathbf{\bar{\Phi}}_{RY1}\big{]},\end{array}  \tag{19}\]  
where \(\mathbf{\bar{\Phi}}_{RY0}\) and \(\mathbf{\Phi}_{RY0}=\mathbf{J}\mathbf{\bar{\Phi}}_{RY0}\mathbf{I}^{-1}\) are mutually similar matrices with the spectrum  
\[\mathrm{eig}\,\mathbf{\bar{\Phi}}_{RY0}=\mathrm{eig}\,\mathbf{\Phi}_{RY0}=\big{\{} \phi_{3},\ \ \phi_{4}\big{\}}. \tag{20}\]  
Matrix (19) characterizes the modal state controller for the pair of matrices \((\mathbf{A}_{RY},\,\mathbf{B}_{RY})\) from the recorder (14) and the spectrum (15). To calculate the modal output controller, we write the right annihilator of the matrix \(\mathbf{C}_{RY0}=\mathbf{C}_{RY}\)  
\[\mathbf{C}_{RY0}^{\perp R}=- \begin{bmatrix}0&1&0&0\end{bmatrix} ^{T}\]  
and in the equation:  
\[\mathbf{\Phi}_{RY0}\underbrace{\mathbf{B}_{RY0}^{-}\mathbf{C}_{RY0}^{\perp R}}_{\mathbf{ \bar{R}}_{RY0}}=\underbrace{\mathbf{B}_{RY0}^{-}\mathbf{A}_{RY0}\mathbf{C}_{RY0}^{\perp R }}_{\mathbf{\bar{R}}_{RY0}}, \tag{21}\]  
calculate the matrix coefficients  
\[\mathbf{\bar{G}}_{RY0}=\mathbf{\bar{\Phi}}_{RY1} \begin{bmatrix}0\\ 1\end{bmatrix} ,\ \ \ \mathbf{\bar{H}}_{RY0}= \begin{bmatrix}a_{4,2}\\ a_{5,2}\end{bmatrix} . \tag{22}\]  
From equations (22) it can be seen that the right side of equation (21) is unchanged, and the matrix coefficient for the calculated matrix \(\mathbf{\bar{\Phi}}_{RY0}\) on the left side can be changeddepending on the value of the matrix \(\mathbf{\tilde{\Phi}}_{RY1}\). Let us write the value of this coefficient in a general parameterized form \((\mu_{1},\ \mu_{2}\in\mathbb{C})\):  
\[\mathbf{\tilde{G}}_{RY0}^{*}= \begin{bmatrix}\mu_{1}\\ \mu_{2}\end{bmatrix} . \tag{23}\]  
Let us find the matrix \(\Phi_{1}\) with the desired spectrum (18) at the first level of decomposition, at which the equality \(G_{1}=G_{2}\) is fulfilled. To do this, let us consider the expression for the matrix G1 from the record (22) as the equation  
\[\mathbf{\tilde{\Phi}}_{RY1}\underbrace{ \begin{bmatrix}0\\ 1\end{bmatrix} }_{\mathbf{\tilde{G}}_{RY1}}=\underbrace{ \begin{bmatrix}\mu_{1} \\ \mu_{2}\end{bmatrix} }_{\mathbf{\tilde{H}}_{RY1}}, \tag{24}\]  
solvable with respect to the matrix \(\mathbf{\tilde{\Phi}}_{RY1}\). To provide the spectrum (5.5) to the matrix \(\mathbf{\tilde{\Phi}}_{RY1}\), we form a pair of state and observation matrices \(\mathbf{\tilde{\Phi}}_{RY1}\). \[ \begin{split}\mathbf{A}_{\mathbf{\tilde{\Phi}}_{RY1}}=\mathbf{ \tilde{H}}_{RY1}\underbrace{\begin{bmatrix}\mathbf{\tilde{G}}_{RY1}^{T} \mathbf{\tilde{G}}_{RY1}\end{bmatrix}^{-1}\mathbf{\tilde{G}}_{RY1}^{T}\\ \mathbf{\tilde{G}}_{RY1}^{+}\end{bmatrix}}_{\mathbf{\tilde{G}}_{RY1}^{+}}= \begin{bmatrix}0&\mu_{1}\\ 0&\mu_{2}\end{bmatrix},\ \ \ \mathbf{C}_{\mathbf{\tilde{\Phi}}_{RY1}}=\mathbf{\tilde{G}}_{RY1}^{ \perp L}=\begin{bmatrix}1&0\end{bmatrix}.\end{split}  \tag{25}\]  
For this pair, the observability matrix and its determinant are  
\[\mathbf{N}_{\mathbf{\tilde{\Phi}}_{RY1}}= \begin{bmatrix}\mathbf{C}_{\mathbf{\tilde{ \Phi}}_{RY1}}\\ \mathbf{C}_{\mathbf{\tilde{\Phi}}_{RY1}}\mathbf{A}_{\mathbf{\tilde{\Phi}}_{RY1}} \end{bmatrix} = \begin{bmatrix}1&0\\ 0&\mu_{1}\end{bmatrix} ,\ \ \ \det\mathbf{N}_{\mathbf{\tilde{\Phi}}_{RY1}}=\mu_{1},\]  
i.e., full observability takes place if  
\[\mu_{1}\
eq 0. \tag{26}\]  
Taking into account the Ackerman formula [30], we calculate the state observer matrix for a pair of matrices (25) and spectrum (18):  
\[\mathbf{L}_{\mathbf{\tilde{\Phi}}_{RY1}}=\Big{(}\mathbf{A}_{\mathbf{\tilde{\Phi}}_{RY 1}}-\mathbf{\phi}_{1}\mathbf{I}_{2}\Big{)}\Big{(}\mathbf{A}_{\mathbf{\tilde{\Phi}}_{RY1}} -\mathbf{\phi}_{2}\mathbf{I}_{2}\Big{)}\mathbf{N}_{\mathbf{\tilde{\Phi}}_{RY1}}^{-1} \begin{bmatrix}0\\ 1\end{bmatrix} = \begin{bmatrix}\tilde{\mu}_{2}\\ \tilde{\mu}_{1}\end{bmatrix}