# A Robust Control Algorithm of a Descent Vehicle Angular Motion in the Earth's Atmosphere
## 6 Robust Output Regulator
 \tag{39}\]where \(s_{x}=-\phi_{x1}-\phi_{x2}\), \(s_{y}=-\phi_{y1}-\phi_{y2}\), \(m_{x}=\phi_{x1}\phi_{x2}\) - positive constants and \(m_{y}=\phi_{y1}\phi_{y2}=a_{5,2}>0\), provides a spectrum  
\[\text{eig}\Big{(}A_{RY}-B_{RY}\,F_{RY}|_{\phi_{y1}\phi_{y2}=a_{5,2}}C_{RY} \Big{)}=\big{\{}\phi_{x1},\ \ \phi_{x2},\ \ \phi_{y1},\ \ \phi_{y2}\big{\}},\]  
which, according to the [PERSON] criterion [29], corresponds to a stable system. Matrix (39) does not contain cross coefficients between the \"Roll\" and \"Yaw\" channels. Moreover, it is robust because does not depend on the variable parameters \(a_{4,2}\) and \(a_{5,2}\) of the state matrix \(\mathbf{A}_{RY}\). Next, we will consider the autonomous problem of modal output control in the \"Pitch\" channel, described by a completely controllable and completely observable (by state) triple of matrices (16) and spectrum (17). This is a problem with one control input, which means that the state regulator matrix is uniquely found using the Ackerman formula [30]:  
\[\mathbf{K}_{P}= \begin{bmatrix}0&1\end{bmatrix}\begin{bmatrix}B_{P}&A_{P}B_{P} \end{bmatrix} ^{-1}(A_{P}-\phi_{5}I_{2})(A_{P}-\phi_{6}I_{2})=J_{z} \begin{bmatrix} m_{z}-a_{6,3}&s_{z}\end{bmatrix} ,\]  
where \(s_{z}=-\phi_{z1}-\phi_{z2}\), \(m_{z}=\phi_{z1}\phi_{z2}\), \(\phi_{z1}=\phi_{5}\), \(\phi_{z2}=\phi_{6}\). Output control is possible if the equation as below is fulfilled. \[\underbrace{\mathbf{K}_{P} \begin{bmatrix}1\\ 0\end{bmatrix} }_{\mathbf{C}_{P}^{\perp R}}=0\ \ \Leftrightarrow\ \ m_{z}=a_{6,3}.\]  
The output regulator matrix will be  
\[\mathbf{F}_{P}|_{m_{z}=a_{6,3}}=\mathbf{K}_{P}\,\underbrace{\mathbf{C}_{P}^{T}\Big{(}C_{P }C_{P}^{T}\Big{)}^{-1}}_{\mathbf{C}_{P}^{T}}=J_{z}s_{z}. \tag{40}\]  
If \(s_{z}\) is a positive constant, then, since \(m_{z}=\phi_{z1}\phi_{z2}=a_{6,3}>0\), matrix (40) will provide the spectrum  
\[\text{eig}\Big{(}A_{P}-\mathbf{B}_{P}\,\mathbf{F}_{P}|_{\phi_{z1}\phi_{z2}=a_{6,3}}C_{ P}\Big{)}=\big{\{}\phi_{z1},\ \ \phi_{z2}\big{\}},\]  
which, according to the [PERSON] criterion [29], corresponds to a stable system. Moreover, matrix (40) is robust, since does not depend on the variable parameter \(a_{6,3}\) of the state matrix \(A_{P}\). Combining the results (39) and (40), we write down the robust matrix of the output regulator, which does not contain cross coefficients between the \"Roll\", \"Yaw\" and \"Pitch\" channels:  
\[\mathbf{F}|_{ \begin{subarray}{c}m_{y}=a_{6,2}\\ m_{z}=a_{6,3}\end{subarray} }= \begin{bmatrix}m_{x}&s_{x}&0&0\\ 0&0&s_{y}&0\\ 0&0&0&s_{z}\end{bmatrix} . \tag{41}\]  
Thus, a robust output regulator (Figure 2) has been synthesized, in which there are no cross-connections between control channels, and the remaining coefficients do not depend on the variable parameters of the state matrix. The resulting regulator with matrix (41) is universal in the sense that it is determined only by the inertial characteristics of the lander and the desired poles, which allows its use on vehicles with any aerodynamic characteristics, including any aerodynamic quality [23; 25]. It is shown that a robust regulator can be synthesized on the basis of a modified van der Wood approach using parameterization and methodically different calculations of matrices with the desired spectra at the zero and first decomposition levels (at the zero level, the output regulator is calculated, and due to the first level, its desired properties are provided). Two separate parametrized modal control problems have been solved: in the \"Roll-Yaw\" and \"Pitch\" channels. In the first task, the goal was set to zero cross-connections between the \"Roll\" and \"Yaw\" channels. To achieve it, complex sets and systems of equations and inequalities were compiled, and 5 cumbersome solutions were obtained. The complexity is caused by the presence of decomposition, and the robustness of the output regulator for a specific linear model is obtained as a concomitant factor in zeroing cross-connections.