# A Robust Control Algorithm of a Descent Vehicle Angular Motion in the Earth's Atmosphere
## 5 Linearization of Angular Motion Model of the Lander  
We linearize the system of Equations (4) at each computational step of the on-board computer by expanding the right-hand sides into a Taylor series in terms of the coordinates of the vector \(\mathbf{x}_{att}\) near their programmed values  
\[\mathbf{x}_{att,pr}= \begin{bmatrix}\mathbf{\theta}_{pr}^{E+Q_{v}}\\ \mathbf{\omega}_{E_{v}}^{Q_{grn}}\Big{(}\mathbf{x}_{trj},\mathbf{\theta}_{pr}^{E+Q_{v}} \Big{)}\end{bmatrix} ,\]  
which are constant per cycle and written taking into account equalities (1) and (7) at the current values of time t and coordinates of the CM \(\mathbf{x}_{trj}\) from model (3). This linearization is possible because in the considered range of heights (above 40 km), the parameters of the CM motion of the lander change more slowly than the parameters of the angular motion. After linearization, a system of approximate equations is formed  
\[ \begin{cases}\dot{\mathbf{x}}_{att}=\underbrace{f_{att}\big{(}\mathbf{x}_{ trj},\mathbf{x}_{att,0}\mathbf{0}_{3\times 1}\big{)}}_{\xi}+\underbrace{f^{\prime\mathbf{x}_{att}} \big{(}\mathbf{x}_{trj},\mathbf{x}_{att,0}\mathbf{0}_{3\times 1}\big{)}}_{A} \underbrace{\big{(}\mathbf{x}_{att}-\mathbf{x}_{att,pr}\big{)}}_{\Delta\mathbf{x}_{att}}+ \underbrace{\begin{bmatrix}\mathbf{0}_{3\times 3}\\ I_{E_{v}}^{-1}\end{bmatrix}\mathbf{u}}_{B},\\ \mathbf{y}_{att}=\underbrace{\mathbf{g}_{att}\big{(}\mathbf{x}_{trj},\mathbf{x}_{att, pr}\big{)}}_{y_{att,pr}=y_{att}-\Delta y_{att}}+\underbrace{\mathbf{g}^{\prime\mathbf{x}_{att}} \big{(}\mathbf{x}_{trj},\mathbf{x}_{att,pr}\big{)}}_{\Delta\mathbf{x}_{att}}\underbrace{ \big{(}\mathbf{x}_{att}-\mathbf{x}_{att,pr}\big{)}}_{\Delta\mathbf{x}_{att}}.\end{cases}  \tag{8}\]  
In deviations from the programmed values, this system takes the classical form with a perturbation  
\[ \begin{cases}\Delta\dot{\mathbf{x}}_{att}=\mathbf{\zeta}+\mathbf{A}\Delta\mathbf{x}_{att}+\bm {B}\mathbf{u},\\ \Delta\mathbf{y}_{att}=\mathbf{C}\Delta\mathbf{x}_{att},\end{cases}  \tag{9}\]where \(\xi-\) vector of disturbances, \(A\), \(B\) and \(C\) - the matrices of state, control and observation, respectively. According to the theory presented in [29], the control for such a model that ensures the fulfillment of the condition  
\[\mathrm{eig}(\mathbf{A}-\mathbf{B}\mathbf{F}\mathbf{C})=\Lambda^{*} \tag{10}\]  
for the desired spectrum  
\[\Lambda^{*}=\big{\{}\phi_{1},\ \ \ \phi_{2},\ \ \phi_{3},\ \ \phi_{4},\ \ \phi_{5},\ \ \ \phi_{6}\big{\}}\]  
has the form (5). The static addition caused by the presence of a perturbation (zero Taylor term) with the matrix of controlled parameters  
\[\mathbf{N}=\big{[}\mathbf{I}_{3}\ \ \ \mathbf{0}_{3\times 3}\big{]}\]  
(the angular position \(\theta^{E\div Q_{\
u}}\) is regulated) is equal to  
\[\mathbf{u}_{0}=-\big{[}\mathbf{F}\mathbf{C}\ \ \ \mathbf{I}_{3}\big{]} \begin{bmatrix}\mathbf{A}&\mathbf{B} \\ \mathbf{N}&\mathbf{0}_{3\times 3}\end{bmatrix} ^{-1} \begin{bmatrix}\mathbf{I}_{6}\\ \mathbf{0}_{3\times 6}\end{bmatrix} \xi, \tag{11}\]  
here and below, \(\mathbf{I}_{n}\) is the identity matrix of order \(n\). Thus, the control problem is reduced to finding the output controller matrix \(\mathbf{F}\) for a triple of matrices \(\mathbf{A}(t)\), \(\mathbf{B}\), and \(\mathbf{C}(t)\) (matrices \(\mathbf{A}\) and \(\mathbf{C}\), due to linearization (8), change from cycle to cycle), written in block form  
\[\mathbf{A}(t)= \begin{bmatrix}\mathbf{A}_{[1,1]}(t)&\mathbf{A}_{[1,2]}(t)\\ \mathbf{A}_{[2,1]}(t)&\mathbf{A}_{[2,2]}(t)\end{bmatrix} ,\ \ \ \mathbf{B}= \begin{bmatrix}\mathbf{0}_{3\times 3}\\ \mathbf{J}^{-1}\end{bmatrix} ,\ \ \ \mathbf{C}(t)= \begin{bmatrix}\mathbf{c}^{T}(t)&\mathbf{0}_{1 \times 3}\\ \mathbf{0}_{3\times 3}&\mathbf{I}_{3}\end{bmatrix}