# A Robust Control Algorithm of a Descent Vehicle Angular Motion in the Earth's Atmosphere
## 2 Statement of the Research Problem  
The orientation of the capsule-type lander and the associated velocity coordinate system (CS) \(E_{\text{\tiny{p}}}=ox_{\text{\tiny{p}}}y_{\text{\tiny{p}}}z_{\text{\tiny{p}}}\). (rotated relative to the associated geometric CS \(E=oxyz\) around the axis \(z\) by the calculated \"balancing\" angle of attack \(\alpha_{bal}^{*}\)) relative to the reference velocity CS \(Q_{\text{\tiny{p}}}=ox_{\text{\tiny{p}}}Y_{\text{\tiny{p}}}Z_{\text{\tiny{p}}}\) during landing from the near-earth orbit is considered [9; 14; 15].  
For the study, a section of the trajectory is selected, on which the values of the atmosphere density \(\rho\) and the linear velocity \(v\) of the lander contribute to the most effective control of the lander motion. Such a section approximately corresponds to heights \(h\leq 80\) and Mach numbers [22]\(M\geq 6\), and the average time of movement along it is \(T=280\). In this section, the aerodynamic coefficients of the SA are practically independent of the Mach number and altitude, and the balancing position, characterized by the balancing angles of attack \(\alpha_{bal}\) and side slip \(\beta_{bal}\)[23], can be considered constant.  
The task for the study consists in high-precision stabilization of the lander in the programmed balancing position (with tracking the programmed roll angle \(\gamma_{pr}\)), i.e., in maintaining the state vector of angular motion.  
\[\mathbf{x}_{att}=\begin{bmatrix}\mathbf{\theta}^{E+Q_{\text{\tiny{p}}}}\\ \mathbf{\omega}_{E_{\text{\tiny{p}}}}^{E_{\text{\tiny{p}}}}\end{bmatrix}\quad( \mathbf{\theta}^{E+Q_{\text{\tiny{p}}}}=\begin{bmatrix}\gamma&\beta&\alpha\end{bmatrix} ^{T},\quad\mathbf{\omega}_{E_{\text{\tiny{p}}}}^{E_{\text{\tiny{p}}}}=\begin{bmatrix} \omega_{x_{\text{\tiny{p}}}}^{E_{\text{\tiny{p}}}}&\omega_{y_{\text{\tiny{p}} }}^{E_{\text{\tiny{p}}}}&\omega_{z_{\text{\tiny{p}}}}^{E_{\text{\tiny{p}}}} \end{bmatrix}^{T}),\]  
where \(\gamma\) - speed roll angle, \(\beta\) - angles of side slip, \(\alpha\) - angles of attack (GOST 20058-80), near its programmed value  
\[\mathbf{x}_{att,pr}=\begin{bmatrix}\mathbf{\theta}_{pr}^{E+Q_{\text{\tiny{p}}}}\\ \mathbf{\omega}_{E_{\text{\tiny{p}}},\mu}^{E_{\text{\tiny{p}}}}\end{bmatrix}\quad( \mathbf{\theta}_{pr}^{E+Q_{\text{\tiny{p}}}}\!=\begin{bmatrix}\gamma_{pr}&\beta_{ bal}&\alpha_{bal}\end{bmatrix}^{T},\quad\mathbf{\omega}_{E_{\text{\tiny{p}}},\mu}^{E_{ \text{\tiny{p}}}}=\begin{bmatrix}\omega_{x_{\text{\tiny{p}}}}^{Q_{\text{\tiny{p }}}}&\omega_{y_{\text{\tiny{p}}}}^{Q_{\text{\tiny{p}}}}&\omega_{z_{\text{\tiny{p}} }}^{Q_{\text{\tiny{p}}}}\end{bmatrix}^{T}) \tag{1}\]  
with the help of reaction engines of the system of executive organs of landing of constant thrust with variable pulse durations according to information from the gyroscopic angle measurement system (GAMS) (inertial roll angle) and angular velocity sensor (vector of angular velocity \(\mathbf{\omega}_{E_{\text{\tiny{p}}}}^{E_{\text{\tiny{p}}}}\)) about the observation vector  
\[\mathbf{y}_{att}=\begin{bmatrix}\hat{\gamma}\\ \mathbf{\omega}_{E_{\text{\tiny{p}}}}^{E_{\text{\tiny{p}}}}\end{bmatrix}. \tag{2}\]  
Hereinafter, the subscript with the name of the CS or its axes denotes the mappings of vectors (matrix columns) and matrices (tensors of inertia, kinematic equations) to the corresponding bases or projections of vectors on the indicated axes. The superscript indicates the basis, the absolute movement of which characterizes the given vector. Ifrelative movement is considered, the symbol \"\(\div\)\" is added in the superscript followed by the designation of the basis relative to which the movement occurs.  
It is required, using analytical methods of modal control, to increase, as far as possible, the accuracy of stabilization of the lander without increasing fuel consumption in comparison with the standard algorithm.