# A Robust Control Algorithm of a Descent Vehicle Angular Motion in the Earth's Atmosphere
## 7 Selecting the Desired Poles  
Let the control signals (5) be recalculated into the duration of switching on of the relay reaction engines of the system of executive organs according to the standard on-board logic. Let us consider the choice of the desired poles used in the formation of the robust regulator output matrix (41). From the given constraints written in its subscript,  
\[ \begin{array}{l}m_{y}=\phi_{y1}\phi_{y2}=a_{5,2}(t)=\tilde{q}(t)\tilde{a}_{5,2}>0,\\ m_{z}=\phi_{z1}\phi_{z2}=a_{6,3}(t)=\tilde{q}(t)\tilde{a}_{6,3}>0\end{array} \]  
it can be seen that in the \"Roll\" channel the constant poles f1 and f2 can be assigned arbitrarily if the stability conditions are met  
\[s_{x}=-(\phi_{x1}+\phi_{x2})>0,\ \ \ m_{x}=\phi_{x1}\phi_{x2}>0,\]  
and in the channels \"Yaw\" and \"Pitch\" the variable poles have the form  
\[ \begin{array}{ll}\phi_{y1}(t)=-\frac{s_{y}}{2}-\sqrt{\frac{s_{y}^{2}}{4}- \tilde{q}(t)a_{5,2}},&\phi_{y2}(t)=-\frac{s_{y}}{2}+\sqrt{\frac{s_{y}^{2}}{4}- \tilde{q}(t)\tilde{a}_{5,2}};\\ \phi_{z1}(t)=-\frac{s_{z}}{2}-\sqrt{\frac{s_{z}^{2}}{4}-\tilde{q}(t)\tilde{a}_ {6,3}},&\phi_{z2}(t)=-\frac{s_{z}}{2}+\sqrt{\frac{s_{z}^{2}}{4}-\tilde{q}(t) \tilde{a}_{6,3}}.\end{array}  \tag{42}\]  
It is known from model (13) that the coefficients \(\tilde{a}_{5,2}\), \(\tilde{a}_{6,3}\) and \(\tilde{q}(t)\) are positive. Therefore, regardless of the specific values of the constants \(s_{y}=-\left(\phi_{y1}+\phi_{y2}\right)>0\) and \(s_{z}=-\left(\phi_{z1}+\phi_{z2}\right)>0\),the poles \(\phi_{y1}\), \(\phi_{y2}\) and \(\phi_{z1}\), \(\phi_{z2}\) will be located in the left complex half-plane, ensuring, together with the poles, the stability of the closed-loop system (13) and (41). With an appropriate choice of the values of \(s_{y}\) and \(s_{z}\), at the beginning of the flight section under consideration at a low velocity head q, there is a time interval where  
\[\tilde{q}(t)\leq\frac{1}{4}\max\left(\frac{s_{y}^{2}}{\tilde{a}_{5,2}},\frac{ s_{z}^{2}}{\tilde{a}_{6,3}}\right),\]  
Figure 2: Robust control formation scheme. that is, one or both pairs of poles \(\phi_{y1}\), \(\phi_{y2}\) and \(\phi_{z1}\), \(\phi_{z2}\) consist of real numbers. This allows the process of bringing the orientation in the corresponding control channel to be aperiodic (with less overshoot and fuel consumption than during the oscillatory process). Further, with a descend of the lander and an increase in the velocity head \(\mathsf{q}\), a moment of time begins, starting from which  
\[\tilde{q}(t)>\frac{1}{4}\max\left(\frac{s_{y}^{2}}{\overline{a}_{5,2}},\frac{ s_{z}^{2}}{\overline{a}_{6,3}}\right),\]  
and the parameters \(s_{y}\) and \(s_{z}\), defining constant real parts in pairs of complex conjugate poles (42) with variable imaginary parts, provide a fixed stability margin in the Yaw and Pitch channels. Specific positive values of the constants \(s_{x}\), \(s_{y}\), \(s_{z}\) and \(m_{x}\) in formula (41) are selected based on the results of mathematical modeling for a specific object and initial conditions (IC). The selection criterion is the accuracy of maintaining the orientation of the vehicle at a given restriction on the total fuel consumption \(Q_{\max}\). Accuracy (\(\delta\gamma\), \(\delta\beta\), \(\delta\alpha\) in angles and \(\delta w_{x_{x}}\), \(\delta w_{y_{x}}\), \(\delta w_{z_{y}}\) in angular velocities) is understood as the maximum modulus deviations of the state parameters from their programmed values over the time interval \([t_{0}+T_{trans};\ t_{0}+T]\), where \(T_{trans}\) is the duration of the PP of alignment, \(T\) is the total simulation time. As a working variant of the IC of angular motion, one of the most fuel-intensive options is used in terms of initial deviations and signs of angles and angular velocities. Further, for this option, a certain sample of stochastic descent processes (with scatter of the parameters of the Earth's atmosphere, aerodynamics of the lander and measurement errors) is modeled without roll-overs with control according to the existing algorithm.