Tracking aircraft trajectories in the presence of wind disturbances

Nikolai Botkin; Varvara Turova; Barzin Hosseini; Johannes Diepolder; Florian Holzapfel

doi:10.3934/mcrf.2021010

Article Contents

2021, Volume 11, Issue 3: 499-520. Doi: 10.3934/mcrf.2021010 open access

This issue Previous Article Optimal control of elliptic variational inequalities with bounded and unbounded operators Next Article Optimal control of a non-smooth quasilinear elliptic equation

Tracking aircraft trajectories in the presence of wind disturbances

1.
Technische Universität München, Department of Mathematics, Boltzmannstr. 3, 85748 Garching near Munich, Germany
2.
Technische Universität München, Institute of Flight System Dynamics, Boltzmannstr. 15, 85748 Garching near Munich, Germany

^* Corresponding author: V. Turova
^* Corresponding author: V. Turova

Received: March 2019

Revised: July 2020

Early access: March 2021

Published: September 2021

The work has been supported by the DFG grant TU427/2-2 and HO4190/8-2

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Abstract

A method of path following, utilized in the theory of position differential games as a tool for establishing theoretical results, is adopted in this paper for tracking aircraft trajectories under windshear conditions. It is interesting to note that reference trajectories, obtained as solutions of optimal control problems with zero wind, can very often be tracked in the presence of rather severe wind disturbances. This is shown in the present paper for rather realistic and highly nonlinear models of aircraft dynamics.

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Mathematics Subject Classification: Primary: 49N70, 49N90; Secondary: 70Q05, 34D99.

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Figure 1. Tracking of the landing trajectory in the case of a Dryden disturbance model with the characteristic value of $30\, \mathrm{m} / \mathrm{s}$. The black line presents the aircraft motion, and the grey line shows the reference trajectory

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Figure 2. The angle $ \gamma_K\,\hbox[\text{deg}] $ in the case of a Dryden disturbance model with the characteristic value of 30 $\mathrm{m} / \mathrm{s}$. In the upper plot the black line corresponds to the aircraft motion, and the grey line stands for the reference trajectory. The solid line in the lower plot shows the absolute tracking error using a semi-logarithmic scale

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Figure 3. The angle $ \chi_K\,\hbox[\text{deg}] $ in the case of a Dryden disturbance model with the characteristic value of 30 $\mathrm{m} / \mathrm{s}$. In the upper plot the black line corresponds to the aircraft motion, and the grey line stands for the reference trajectory. The solid line in the lower plot shows the absolute tracking error using a semi-logarithmic scale

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Figure 4. The velocity $ V_K\,[\mathrm{m} / \mathrm{s}] $ in the case of a Dryden disturbance model with the characteristic value of 30 $\mathrm{m} / \mathrm{s}$. In the upper plot the black line corresponds to the aircraft motion, and the grey line stands for the reference trajectory. The solid line in the lower plot shows the absolute tracking error using a semi-logarithmic scale

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Figure 5. The wind components $ W_x\,[\mathrm{m} / \mathrm{s}] $, $ W_y\,[\mathrm{m} / \mathrm{s}] $, and $ W_z\,[\mathrm{m} / \mathrm{s}] $ in the case of a Dryden disturbance model with the characteristic value of $ 30\, \mathrm{m} / \mathrm{s} $

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Figure 6. Tracking of the landing trajectory in the case of a Dryden disturbance model with the characteristic value of 45 $\mathrm{m} / \mathrm{s}$. The black line presents the aircraft motion, and the grey line shows the reference trajectory

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Figure 7. The angle $ \gamma_K\,\hbox[\text{deg}] $ in the case of a Dryden disturbance model with the characteristic value of 45 $\mathrm{m} / \mathrm{s}$. In the upper plot the black line corresponds to the aircraft motion, and the grey line stands for the reference trajectory. The solid line in the lower plot shows the absolute tracking error using a semi-logarithmic scale

Download: Full-size image PowerPoint slide

Figure 8. The angle $ \chi_K\,\hbox[\text{deg}] $ in the case of a Dryden disturbance model with the characteristic value of 45 $\mathrm{m} / \mathrm{s}$. In the upper plot the black line corresponds to the aircraft motion, and the grey line stands for the reference trajectory. The solid line in the lower plot shows the absolute tracking error using a semi-logarithmic scale

Download: Full-size image PowerPoint slide

Figure 9. The velocity $ V_K\,[\mathrm{m} / \mathrm{s}] $ in the case of a Dryden disturbance model with the characteristic value of 45 $\mathrm{m} / \mathrm{s}$. In the upper plot the black line corresponds to the aircraft motion, and the grey line stands for the reference trajectory. The solid line in the lower plot shows the absolute tracking error using a semi-logarithmic scale

Download: Full-size image PowerPoint slide

Figure 10. The wind components $ W_x\,[\mathrm{m} / \mathrm{s}] $, $ W_y\,[\mathrm{m} / \mathrm{s}] $, and $ W_z\,[\mathrm{m} / \mathrm{s}] $ in the case of a Dryden disturbance model with the characteristic value of $ 45\, \mathrm{m} / \mathrm{s} $

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Figure 11. The kinematic velocity $ V_K\,[\mathrm{m} / \mathrm{s}] $ for the case of repulsive disturbance, see (24). The straight line at $ V_K = 150\,\mathrm{m} / \mathrm{s} $ corresponds to the reference

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Figure 12. The angle $ \gamma_K\,\hbox[\text{deg}] $ for the case of repulsive disturbance, see (24). The straight line at $ \gamma_K = 0\,\text{deg} $ corresponds to the reference

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Figure 13. The angle $ \chi_K\,\hbox[\text{deg}] $ for the case of repulsive disturbance, see (24). The straight line at $ \chi_K = 0\,\text{deg} $ corresponds to the reference

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Figure 14. The position component $ y_N\,[{\rm{m}}] $ for the case of repulsive disturbance, see (24). The straight line at $ y_N = 0\,{\rm{m}} $ corresponds to the reference

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Figure 15. The altitude $ h = -z_N\,[{\rm{m}}] $ for the case of repulsive disturbance, see (24). The straight line at $ h = 5000\,{\rm{m}} $ corresponds to the reference

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