Tracking aircraft trajectories in the presence of wind disturbances

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

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

Tracking aircraft trajectories in the presence of wind disturbances

Nikolai Botkin ^1, , Varvara Turova ^1,, , Barzin Hosseini ^2, , Johannes Diepolder ^2, and Florian Holzapfel ^2,

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.

Keywords: Differential games, trajectory tracking, flight dynamics, windshear conditions, landing phase, cruise flight.

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

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

show all references

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

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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} $

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download as 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

Download as 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

Download as 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} $

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

[1]	Marcio A. Jorge Silva, Vando Narciso, André Vicente. On a beam model related to flight structures with nonlocal energy damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3281-3298. doi: 10.3934/dcdsb.2018320
[2]	Gafurjan Ibragimov, Askar Rakhmanov, Idham Arif Alias, Mai Zurwatul Ahlam Mohd Jaffar. The soft landing problem for an infinite system of second order differential equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 89-94. doi: 10.3934/naco.2017005
[3]	Simone Göttlich, Camill Harter. A weakly coupled model of differential equations for thief tracking. Networks & Heterogeneous Media, 2016, 11 (3) : 447-469. doi: 10.3934/nhm.2016004
[4]	Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451
[5]	E. Fossas, J. M. Olm. Galerkin method and approximate tracking in a non-minimum phase bilinear system. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 53-76. doi: 10.3934/dcdsb.2007.7.53
[6]	John A. Morgan. Interception in differential pursuit/evasion games. Journal of Dynamics & Games, 2016, 3 (4) : 335-354. doi: 10.3934/jdg.2016018
[7]	Sylvain Sorin, Cheng Wan. Finite composite games: Equilibria and dynamics. Journal of Dynamics & Games, 2016, 3 (1) : 101-120. doi: 10.3934/jdg.2016005
[8]	Karol Mikula, Róbert Špir, Nadine Peyriéras. Numerical algorithm for tracking cell dynamics in 4D biomedical images. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 953-967. doi: 10.3934/dcdss.2015.8.953
[9]	Marianne Akian, Stéphane Gaubert, Antoine Hochart. Ergodicity conditions for zero-sum games. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3901-3931. doi: 10.3934/dcds.2015.35.3901
[10]	Ciprian G. Gal, T. Tachim Medjo. Approximation of the trajectory attractor for a 3D model of incompressible two-phase-flows. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2229-2252. doi: 10.3934/cpaa.2014.13.2229
[11]	Jingzhen Liu, Ka-Fai Cedric Yiu. Optimal stochastic differential games with VaR constraints. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1889-1907. doi: 10.3934/dcdsb.2013.18.1889
[12]	Alain Bensoussan, Jens Frehse, Christine Grün. Stochastic differential games with a varying number of players. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1719-1736. doi: 10.3934/cpaa.2014.13.1719
[13]	Ellina Grigorieva, Evgenii Khailov. Hierarchical differential games between manufacturer and retailer. Conference Publications, 2009, 2009 (Special) : 300-314. doi: 10.3934/proc.2009.2009.300
[14]	Leon Petrosyan, David Yeung. Shapley value for differential network games: Theory and application. Journal of Dynamics & Games, 2021, 8 (2) : 151-166. doi: 10.3934/jdg.2020021
[15]	Saul Mendoza-Palacios, Onésimo Hernández-Lerma. Stability of the replicator dynamics for games in metric spaces. Journal of Dynamics & Games, 2017, 4 (4) : 319-333. doi: 10.3934/jdg.2017017
[16]	Miguel A. Dumett, Roberto Cominetti. On the stability of an adaptive learning dynamics in traffic games. Journal of Dynamics & Games, 2018, 5 (4) : 265-282. doi: 10.3934/jdg.2018017
[17]	Martin Burger, Marco Di Francesco, Peter A. Markowich, Marie-Therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1311-1333. doi: 10.3934/dcdsb.2014.19.1311
[18]	Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020180
[19]	Kuang Huang, Xuan Di, Qiang Du, Xi Chen. A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4869-4903. doi: 10.3934/dcdsb.2020131
[20]	Pierluigi Colli, Antonio Segatti. Uniform attractors for a phase transition model coupling momentum balance and phase dynamics. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 909-932. doi: 10.3934/dcds.2008.22.909

2020 Impact Factor: 1.284

Tools

Metrics

Other articles
by authors

on AIMS
Nikolai Botkin Varvara Turova Barzin Hosseini Johannes Diepolder Florian Holzapfel
on Google Scholar
Nikolai Botkin Varvara Turova Barzin Hosseini Johannes Diepolder Florian Holzapfel

2020 Impact Factor: 1.284

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.284

Tools

Figures and Tables

2020 Impact Factor: 1.284

Tools

Metrics

Other articles
by authors

on AIMS
Nikolai Botkin Varvara Turova Barzin Hosseini Johannes Diepolder Florian Holzapfel
on Google Scholar
Nikolai Botkin Varvara Turova Barzin Hosseini Johannes Diepolder Florian Holzapfel

[Back to Top]