Application of dynamic programming approach to aircraft take-off in a windshear

Transcription

1 Application of dynamic programming approach to aircraft take-off in a windshear N. D. Botkin, V. L. Turova Technische Universität München, Mathematics Centre Chair of Mathematical Modelling 10th International Conference of Numerical Analysis and Applied Mathematics Kos, Greece, September 19-26, 2012

2 1 Outline Dynamic programming approach Differential game formulation Viscosity solutions of H.-J. equations Numerical method: direct and upwind operators Application to aircraft take-off Dynamic and kinematic equations Microburst models Notes on previous results Three different problem statements Simulation results

3 Dynamics Differential game 2 Cost functional: (1) State constraint: Hamiltonian: Contra-strategies of the second player (N.N.Krasovskii, A.I.Subbotin, Game-Theoretical Control Problems, Springer, 1988). Hamilton-Jacobi-Bellman-Isaacs equation:

4 Viscosity solutions 3 A Lipschitz function is the value function of (1) if and only if (i) for any ; and (ii) for any point and function s.t. attains a local minimum at the following inequality holds (iii) for any point s.t. and function s.t. attains a local maximum at the following inequality holds

5 Numerical method 4 are time and space discretization steps 1. Direct operator defined on continuous functions Introduce grid functions

6 Let be an interpolation operator that maps grid functions into continuous functions and satisfies the condition 5 for any smooth function, where is the restriction of to the grid. y 1-x-y x linear interpolation bilinear interpolation Finite difference scheme y(1-x) x y (1-x)(1-y) x(1-y) (2) Theorem The grid function obtained by (2) converges point-wise to the value function of the differential game (1) as. The convergence rate is. Botkin et al.: SIAM J. on S.C. 33(2), 2011 & Analysis 31, 2011

7 2. Upwind operator 6 where, are the right and left divided differences with spatial steps are the RHS of the control system Finite difference scheme (3) O.A.Malafeyev, M.S.Troeva, A weak solution of Hamilton-Jacobi equation for a differential two-person zero-sum game, in: Proceedings of 8th Int. Symp. on Differential Games and Applications, Maastricht, 1998.

8 Theorem If Let be a bound of the RHS of the control system., then the grid function obtained by the procedure 7 (3) converges pointwise to the value functions of problem (1) as and the convergence rate is. Proof 1. Monotonicity of the operator : 2. Generator property of the operator : for any, and fixed. Botkin et al.: Applied Math. Modeling 35, 2011 & Analysis 31, 2011

9 Application to aircraft take-off in a windshear 8

10 Aircraft dynamics 9 aircraft speed relative to wind aircraft speed relative to ground relative path inclination angle of attack thrust; drag; lift (1) (2)

11 Forces 10 Thrust: Drag: Lift:

12 Wind (microburst) 11

13 Microburst: Model 1 12 Y. H. Chen, S. Pandey, Robust control strategy for take-off performance in a windshear, Optimal Control Applications and Methods 10 (1), 65-79, 1989.

14 Microburst: Model 2 13 h (ft) x (ft) G. Leitmann, S. Pandey, Aircraft control under conditions of windshear, Control and Dynamic Systems 34, part 1, C.T. Leondes (Ed.), Academic Press, New York, 1990.

15 14 A. Miele, T. Wang, W. W. Melvin, Optimal take-off trajectories in the presence of windshear, J. Opt. Theory and Appl. 49 (1), 1-45, Known wind, open-loop control A. Miele, T. Wang, W.W. Melvin, Guidance strategies for near-optimum take-off performance in a windshear, J. Opt. Theory and Appl. 50(1), 1-47, Local information on wind, feedback control Y. H. Chen, S. Pandey, Robust control strategy for take-off performance in a windshear, Optimal Control Applications and Methods 10 (1), 65-79, Unknown wind, feedback robust control G. Leitmann. S. Pandey, Aircraft control for flight in an uncertain environment: takeoff in windshear, J. Opt. Theory and Appl. 70(1), 25-55, Unknown wind, feedback control stabilizing the climb rate G. Leitmann, S. Pandey, E. Ryan, Adaptive control of aircraft in windshear, Int. J. of Robust and Nonlinear Control, 3, , Unknown wind, adaptive feedback control stabilizing the climb rate V. L. Turova, Take-off control in a windshear, Preprint of the Inst. of Math. and Mech., Ekaterinburg, Unknown wind, feedback control based on solving linearized differential games N. D. Botkin, V. M. Kein, V. S. Patsko, V. L. Turova, M. A. Zarkh, Differential games and aircraft control problems in the presence of wind disturbances, Izv. Ross. Akad. Nauk, Tekhn. Kibernet., no. 1, 68-76, 1993 (in Russian), translated as J. of Computer and System Sciences International 32(3), , 1994.

16 P1 Take-off control providing for any 15 Full 4D dynamics Cost functional: State constraint:

17 P2 Take-off with tracking reference trajectory 16 - reference velocity - reference path inclination Substitute and into (2) to arrive at the differential game: Control : Disturbance : :

18 P2 Take-off with tracking reference trajectory 17 - reference velocity - reference path inclination Substitute and into (2) to arrive at the differential game: Control : Disturbance : :

19 P3 Take-off control using the climb rate 18 Differential game Control : Disturbances The structure of the wind velocity field is supposed to be unknown. G. Leitmann, S. Pandey, E. Ryan, Adaptive control of aircraft in windshear, Int. J. of Robust and Nonlinear Control 3, , 1993.

20 19 Example of computed value function (P3) Contour lines

21 Control design 20 interpolation of is stored in an array

22 Simulation of the original nonlinear system 21 P1 (4d, control prob.) P2 (2d, diff. game) Microburst model 1 (k=60) (ft) Altitude (ft/s) Velocity

23 Microburst model 1 (k=60) P1 (4d, control prob.) P2 (2d, diff. game) P1 P2 22 Path inclination Angle of attack (deg) (deg)

24 Microburst model 1 (k=50) 23 Altitude P1 (4d, control prob.) P2 (2d, diff. game) P1 P2 Angle of attack (ft) (deg)

25 Microburst model 2 24 P3 (2d, diff. game, no wind information) Altitude Velocity (ft) (ft/s)

26 25 Wind velocity Control (averaging of bang-bang) (ft/s) (deg)

27 Number of time steps 800 Numerical characteristics 26 Linux SMP-Computer with 8xQuad-Core AMD Opteron processors (2,7 GHz) Shared 64 Gb memory Efficiency of parallelization up to 80% Grid size P1 250 x 200 x 100 x 20 Run time 8 min P2 P x 400 (200 x 40) 400 x 200 (200 x 80) 1,5 min (10 s) 40 s ( 20 s)

28 Conclusion 27 What we can do: Nonlinear dynamics State constraints Fixed/nonfixed termination time, integral objective functionals Dimensions: 2, 3, 4 Future: Higher dimensions using sparse tensor representations of functions* *Tensor train format, or simply TT-format and Quantized-TT (QTT)- format. *Sergey Dolgov, Boris N. Khoromskij, and Ivan V. Oseledets, Fast solution of multi-dimensional parabolic problems in the TT/QTT-format with initial application to the Fokker-Planck equation, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig,Preprint no.: 80, Dimension = 9!