# A Robust Control Algorithm of a Descent Vehicle Angular Motion in the Earth's Atmosphere
## 6 Robust Output Regulator
,\]  
where  
\[\tilde{\mu}_{1}=\tfrac{(\mu_{2}-\phi_{1})(\mu_{2}-\phi_{2})}{\mu_{1}},\ \ \ \tilde{\mu}_{2}=\mu_{2}-\phi_{1}-\phi_{2}.\]  
The matrix with the desired spectrum at the first level of decomposition generally has a form that is neither diagonal nor triangular:  
\[\mathbf{\tilde{\Phi}}_{RY1}=\mathbf{A}_{\mathbf{\tilde{\Phi}}_{RY1}}-\mathbf{L}_{ \mathbf{\tilde{\Phi}}_{RY1}}\mathbf{C}_{\mathbf{\tilde{\Phi}}_{RY1}}=\begin{bmatrix -\tilde{\mu}_{2}&\mu_{1}\\ -\tilde{\mu}_{1}&\mu_{2}\end{bmatrix}}. \tag{27}\]  
Next, from equation (21), we find the matrix \(\mathbf{\tilde{\Phi}}_{RY0}\) with the desired spectrum (20) at the zero level of decomposition. Let the matrix \(\mathbf{\tilde{\Phi}}_{RY1}\) of the form (27) be assigned at the first level of decomposition. Then the matrix coefficients (22) are  
\[\mathbf{\tilde{G}}_{RY0}=\mathbf{\tilde{G}}_{RY0}^{*}= \begin{bmatrix}\mu_{1} \\ \mu_{2}\end{bmatrix} ,\ \ \ \mathbf{\tilde{H}}_{RY0}= \begin{bmatrix}a_{42}\\ a_{52}\end{bmatrix} .\]Hence, when satisfying inequality (26), equation (21) is solvable with respect to the matrix \(\mathbf{\tilde{\Phi}}_{RY0}\). To provide the spectrum (20) to the matrix \(\mathbf{\tilde{\Phi}}_{RY0}\), we form a pair of state and observation matrices  
\[ \begin{split} A_{\mathbf{\tilde{\Phi}}_{RY0}}=\mathbf{\tilde{H}}_ {RY}\underbrace{\left(\mathbf{\tilde{G}}_{RY0}^{T}\mathbf{\tilde{G}}_{RY0} \right)^{-1}}_{\mathbf{\tilde{G}}_{RY0}^{T}}=\frac{1}{\mu}\begin{bmatrix}a_{4,2}\mu_{1}&a_{4,2}\mu_{2}\\ a_{5,2}\mu_{1}&a_{5,2}\mu_{2}\end{bmatrix},\\ C_{\mathbf{\tilde{\Phi}}_{RY0}}=\mathbf{\tilde{G}}_{RY0}^{\text{LL}}=[-\mu_{2} &\mu_{1}],\end{split}  \tag{28}\]  
where \(\mu=\mu_{1}^{2}+\mu_{2}^{2}\). For this pair, the observability matrix and its determinant are  
\[ \begin{split}\mathbf{N}_{\mathbf{\tilde{\Phi}}_{RY0}}=\begin{bmatrix} C_{\mathbf{\tilde{\Phi}}_{RY0}}\\ C_{\mathbf{\tilde{\Phi}}_{RY0}}A_{\mathbf{\tilde{\Phi}}_{RY0}}\end{bmatrix}= -\begin{bmatrix}\mu_{2}&-\mu_{1}\\ \frac{\mu_{1}}{\mu}\delta&\frac{\mu_{2}}{\mu}\delta\end{bmatrix},\quad\det(N_ {\mathbf{\tilde{\Phi}}_{RY0}})=\delta,\end{split} \]  
where \(\delta=a_{42}\mu_{2}-a_{52}\mu_{1}\), i.e., full observability takes place if  
\[\mu_{2}\
eq\frac{a_{5,2}}{a_{4,2}}\mu_{1}. \tag{29}\]  
By Ackerman formula [30], we calculate the state observer matrix for a pair of matrices (28) and spectrum (20):  
\[ \begin{split}\mathbf{L}_{\mathbf{\tilde{\Phi}}_{RY0}}=\left(A_{ \mathbf{\tilde{\Phi}}_{RY0}}-\phi_{3}\mathbf{I}_{2}\right)\left(A_{\mathbf{ \tilde{\Phi}}_{RY0}}-\phi_{4}\mathbf{I}_{2}\right)\mathbf{N}_{\mathbf{\Phi}_ {RY0}}^{-1}\begin{bmatrix}0\\ 1\end{bmatrix}=-\frac{1}{\mu}\begin{bmatrix}\kappa_{11}\mu_{1}+\kappa_{12}\mu_ {2}\\ \kappa_{21}\mu_{1}+\kappa_{22}\mu_{2}\end{bmatrix},\end{split} \]  
where  
\[ \begin{split}\kappa_{11}&=\delta^{-1}(a_{4,2}-\phi_{3} \mu_{1})(a_{4,2}-\phi_{4}\mu_{1}),\\ \kappa_{12}&=\delta^{-1}(a_{4,2}(a_{5,2}-\phi_{3} \mu_{2})-\phi_{4}\mu_{2}(a_{4,2}-\phi_{3}\mu_{1})),\\ \kappa_{21}&=\delta^{-1}(a_{5,2}(a_{4,2}-\phi_{3} \mu_{1})-\phi_{4}\mu_{1}(a_{5,2}-\phi_{3}\mu_{2})),\\ \kappa_{22}&=\delta^{-1}(a_{5,2}-\phi_{3} \mu_{2})(a_{5,2}-\phi_{4}\mu_{2}).\end{split} \]  
The matrix with the desired spectrum at the zero decomposition level will take the form  
\[\mathbf{\tilde{\Phi}}_{RY0}=A_{\mathbf{\tilde{\Phi}}_{RY0}}-L_{\mathbf{\tilde {\Phi}}_{RY0}}C_{\mathbf{\tilde{\Phi}}_{RY0}}= \begin{bmatrix}-\kappa_{12}&\kappa_{11}\\ -\kappa_{22}&\kappa_{21}\end{bmatrix} . \tag{30}\]  
Next, we substitute matrices \(\mathbf{\tilde{\Phi}}_{RY0}\) (30) and \(\mathbf{\tilde{\Phi}}_{RY1}\) (27) into the calculation formula of the state regulator matrix (19) and calculate the output regulator matrix  
\[