# American Institute of Mathematical Sciences

September  2021, 11(3): 499-520. doi: 10.3934/mcrf.2021010

## 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

Received  March 2019 Revised  July 2020 Published  September 2021 Early access  March 2021

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

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.

Citation: Nikolai Botkin, Varvara Turova, Barzin Hosseini, Johannes Diepolder, Florian Holzapfel. Tracking aircraft trajectories in the presence of wind disturbances. Mathematical Control & Related Fields, 2021, 11 (3) : 499-520. doi: 10.3934/mcrf.2021010
##### References:
 [1] J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Advances in Design and Control, 19. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898718577.  Google Scholar [2] N. Botkin, J. Diepolder, V. Turova, M. Bittner and F. Holzapfel, Viability approach to aircraft control in windshear conditions, in Advances in Dynamic and Mean Field Games (eds. J. Apaloo and B. Viscolani), Birkhäuser, (2017), 325–343. doi: 10.1007/978-3-319-70619-1.  Google Scholar [3] N. Botkin, V. Turova, J. Diepolder, M. Bittner and F. Holzapfel, Aircraft control during cruise flight in windshear conditions: Viability approach, Dynamic Games and Applications, 7 (2017), 594-608.  doi: 10.1007/s13235-017-0215-9.  Google Scholar [4] H. Bouadi and F. Mora-Camino, Aircraft trajectory tracking by nonlinear spatial inversion, in AIAA Guidance, Navigation, and Control Conference, Minneapolis, Minnesota, August (2012), 13–16. doi: 10.2514/6.2012-4613.  Google Scholar [5] R. Brockhaus, W. Alles and R. Luckner, Flugregelung, Springer, 2011. Google Scholar [6] G. Brüning, X. Hafer and G. Sachs, Flugleistungen, 2$^{nd}$ edition, Springer, 1986. doi: 10.1007/978-3-662-07259-2.  Google Scholar [7] C. R. Chalk, T. P. Neal, T. M. Harris, F. E. Pritchard and R. J. Woodcock, Background Information and User Guide for Mil-F-8785B (ASG), 'Military Specification-Flying Qualities of Piloted Airplanes', Cornell Aeronautical Lab Inc Buffalo Ny, 1969. Google Scholar [8] J. Diepolder, P. Piprek, N. Botkin, V. Turova and F. Holzapfel, A robust aircraft control approach in the presence of wind using viability theory, in 2017 Australian and New Zealand Control Conference (ANZCC), IEEE, (2017), 155–160. doi: 10.1109/ANZCC.2017.8298503.  Google Scholar [9] A. Farooq and D. J. N. Limebeer, Path following of optimal trajectories using preview control, in Proceedings of the 44th IEEE Conference on Decision and Control, Seville, Spain, (2005), 2787–2792. doi: 10.1109/CDC.2005.1582585.  Google Scholar [10] F. Fisch, Development of a Framework for the Solution of High-Fidelity Trajectory Optimization Problems and Bilevel Optimal Control Problems, Ph.D. thesis, Chair of Flight System Dynamics, Technical University of Munich, 2011. Google Scholar [11] N. N. Krasovski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over l} }}$ and A. I. Subbotin, Game-Theoretical Control Problems, Springer-Verlag, New York, 1988.  Google Scholar [12] Y. S. Osipov and A. V. Kryazhimskiy, Inverse Problems for Ordinary Differential Equations: Dynamical Solutions, Gordon and Breach Science Publishers, Basel, 1995.  Google Scholar [13] A. V. Kryazhimskiy and V. I. Maksimov, Resource-saving tracking problem with infinite time horizon, Differential Equations, 47 (2011), 1004-1013.  doi: 10.1134/S001226611107010X.  Google Scholar [14] A. V. Kryazhimskii and V. I. Maksimov, On combination of the processes of reconstruction and guaranteeing control, Automation and Remote Control, 74 (2013), 1235-1248.  doi: 10.1134/S0005117913080018.  Google Scholar [15] G. Leitmann, S. Pandey and E. Ryan, Adaptive control of aircraft in windshear, Int. Journal of Robust and Nonlinear Control, 3 (1993), 133-153.  doi: 10.1002/rnc.4590030206.  Google Scholar [16] I. Lugo-Cárdenas, S. Salazar and R. Lozano, Lyapunov based 3D path following kinematic controller for a fixed wing UAV, IFAC-PapersOnLine, 50 (2017), 15946-15951.  doi: 10.1016/j.ifacol.2017.08.1747.  Google Scholar [17] V. I. Maksimov, The tracking of the trajectory of a dynamical system, J. Appl. Math. Mech., 75 (2011), 667-674.  doi: 10.1016/j.jappmathmech.2012.01.007.  Google Scholar [18] V. I. Maksimov, Differential guidance game with incomplete information on the state coordinates and unknown initial state, Differential Equations, 51 (2015), 1656-1665.  doi: 10.1134/S0012266115120137.  Google Scholar [19] A. Miele, T. Wang and W. W. Melvin, Optimization and gamma/theta guidance of flight trajectories in a windshear, in ICAS, Congress, 15th, London, England, September 7-12, 1986, Proceedings Volume 2,878–899. Google Scholar [20] J. -J. E. Slotine and W. Li, Applied Nonlinear Control, Taipei : Prentice Education Taiwan Ltd., 2005. Google Scholar [21] A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

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##### References:
 [1] J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Advances in Design and Control, 19. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898718577.  Google Scholar [2] N. Botkin, J. Diepolder, V. Turova, M. Bittner and F. Holzapfel, Viability approach to aircraft control in windshear conditions, in Advances in Dynamic and Mean Field Games (eds. J. Apaloo and B. Viscolani), Birkhäuser, (2017), 325–343. doi: 10.1007/978-3-319-70619-1.  Google Scholar [3] N. Botkin, V. Turova, J. Diepolder, M. Bittner and F. Holzapfel, Aircraft control during cruise flight in windshear conditions: Viability approach, Dynamic Games and Applications, 7 (2017), 594-608.  doi: 10.1007/s13235-017-0215-9.  Google Scholar [4] H. Bouadi and F. Mora-Camino, Aircraft trajectory tracking by nonlinear spatial inversion, in AIAA Guidance, Navigation, and Control Conference, Minneapolis, Minnesota, August (2012), 13–16. doi: 10.2514/6.2012-4613.  Google Scholar [5] R. Brockhaus, W. Alles and R. Luckner, Flugregelung, Springer, 2011. Google Scholar [6] G. Brüning, X. Hafer and G. Sachs, Flugleistungen, 2$^{nd}$ edition, Springer, 1986. doi: 10.1007/978-3-662-07259-2.  Google Scholar [7] C. R. Chalk, T. P. Neal, T. M. Harris, F. E. Pritchard and R. J. Woodcock, Background Information and User Guide for Mil-F-8785B (ASG), 'Military Specification-Flying Qualities of Piloted Airplanes', Cornell Aeronautical Lab Inc Buffalo Ny, 1969. Google Scholar [8] J. Diepolder, P. Piprek, N. Botkin, V. Turova and F. Holzapfel, A robust aircraft control approach in the presence of wind using viability theory, in 2017 Australian and New Zealand Control Conference (ANZCC), IEEE, (2017), 155–160. doi: 10.1109/ANZCC.2017.8298503.  Google Scholar [9] A. Farooq and D. J. N. Limebeer, Path following of optimal trajectories using preview control, in Proceedings of the 44th IEEE Conference on Decision and Control, Seville, Spain, (2005), 2787–2792. doi: 10.1109/CDC.2005.1582585.  Google Scholar [10] F. Fisch, Development of a Framework for the Solution of High-Fidelity Trajectory Optimization Problems and Bilevel Optimal Control Problems, Ph.D. thesis, Chair of Flight System Dynamics, Technical University of Munich, 2011. Google Scholar [11] N. N. Krasovski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over l} }}$ and A. I. Subbotin, Game-Theoretical Control Problems, Springer-Verlag, New York, 1988.  Google Scholar [12] Y. S. Osipov and A. V. Kryazhimskiy, Inverse Problems for Ordinary Differential Equations: Dynamical Solutions, Gordon and Breach Science Publishers, Basel, 1995.  Google Scholar [13] A. V. Kryazhimskiy and V. I. Maksimov, Resource-saving tracking problem with infinite time horizon, Differential Equations, 47 (2011), 1004-1013.  doi: 10.1134/S001226611107010X.  Google Scholar [14] A. V. Kryazhimskii and V. I. Maksimov, On combination of the processes of reconstruction and guaranteeing control, Automation and Remote Control, 74 (2013), 1235-1248.  doi: 10.1134/S0005117913080018.  Google Scholar [15] G. Leitmann, S. Pandey and E. Ryan, Adaptive control of aircraft in windshear, Int. Journal of Robust and Nonlinear Control, 3 (1993), 133-153.  doi: 10.1002/rnc.4590030206.  Google Scholar [16] I. Lugo-Cárdenas, S. Salazar and R. Lozano, Lyapunov based 3D path following kinematic controller for a fixed wing UAV, IFAC-PapersOnLine, 50 (2017), 15946-15951.  doi: 10.1016/j.ifacol.2017.08.1747.  Google Scholar [17] V. I. Maksimov, The tracking of the trajectory of a dynamical system, J. Appl. Math. Mech., 75 (2011), 667-674.  doi: 10.1016/j.jappmathmech.2012.01.007.  Google Scholar [18] V. I. Maksimov, Differential guidance game with incomplete information on the state coordinates and unknown initial state, Differential Equations, 51 (2015), 1656-1665.  doi: 10.1134/S0012266115120137.  Google Scholar [19] A. Miele, T. Wang and W. W. Melvin, Optimization and gamma/theta guidance of flight trajectories in a windshear, in ICAS, Congress, 15th, London, England, September 7-12, 1986, Proceedings Volume 2,878–899. Google Scholar [20] J. -J. E. Slotine and W. Li, Applied Nonlinear Control, Taipei : Prentice Education Taiwan Ltd., 2005. Google Scholar [21] A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.  Google Scholar
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
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
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
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
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 $30\, \mathrm{m} / \mathrm{s}$
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
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
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
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
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 $45\, \mathrm{m} / \mathrm{s}$
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
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
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
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
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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