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Optimal Control of Coupled Systems of ODEs and PDEs with Applications to Hypersonic Flight

Optimal Control of Coupled Systems of ODEs and PDEs with Applications to Hypersonic Flight Part 1 Hans Josef Pesch University of Bayreuth, Germany Part 1: Kurt Chudej, Markus Wächter, Gottfried Sachs, Florent le Bras The 8th International Conference on Optimization:

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Optimal Control of Coupled Systems of ODEs and PDEs with Applications to Hypersonic Flight

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  1. Optimal Control of Coupled Systems of ODEs and PDEs with Applications to Hypersonic Flight Part 1 Hans Josef Pesch University of Bayreuth, Germany Part 1: Kurt Chudej, Markus Wächter, Gottfried Sachs, Florent le Bras The 8th International Conference on Optimization: Techniques and Applications (ICOTA 8), Shanghai, China, Dec. 10-14, 2010

  2. Outline • Introduction/Motivation • The hypersonic trajectory optimization problem • The instationary heat constraint • Numerical results • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion

  3. Outline • Introduction/Motivation • The hypersonic trajectory optimization problem • The instationary heat constraint • Numerical results • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion

  4. Concorde first flight: March 2, 1969; first regular flight: 1976; 20 planes Mach 2.23 (2400 km/h, 18 km altitude) Paris – New York 3-3.5 h Crash after take-off: July 25, 2000, Paris, Charles de Gaule Tupolew Tu 144 (Charger, Konkordski) first flight: Dec 31, 1968; first regular flight: 1975; 16 planes Mach 2.35 (2500 km/h, 18 km altitude) Introduction: Supersonic Aircraft

  5. From Times Online February 5, 2008 Hypersonic jet could offer day trips Down Under Day trips to Australia came a step closer to reality today when a British firm unveiled plans for a hypersonic passenger jet which could fly from Europe to Sydney in less than five hours. Reaction Engines of Oxfordshire says that its A2 plane could be in service within 25 years, carrying 300 passengers at a top speed of almost 4,000 mph, five times the speed of sound and twice the speed of Concorde. The jet may be more environmentally friendly than other plains because it uses liquid hydrogen for power rather than fossil fuel. Introduction: Hypersonic Passenger Jets

  6. Introduction: Hypersonic Passenger Jets http://www.reactionengines.co.uk/lapcat_anim.html Project LAPCAT Reading Engines, UK

  7. PDE ODE Motivation: Hypersonic Passenger Jets quasilinear PDE non-linear boundary conditions coupled with ODE 2 box constraints 1 control-state constraint 1 state constraint Project LAPCAT Reading Engines, UK

  8. Introduction: The German Sänger II Project Collaborative Research Center, Munich: 1989 - 2003 air-breathing turbojet -ramjet / scramjet Chudej: 1989; Schnepper: 1999

  9. Outline • Introduction/Motivation • The hypersonic trajectory optimization problem • The instationary heat constraint • Numerical results • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion

  10. Model: Atmosphere air temperature [K] air pressure [bar] air density [kg/m³] altititude [km] Markus Wächter & Gottfried Sachs Munich U of Technology, Germany

  11. Model: Atmosphere Heat conductivity Heat capacity air temperature

  12. lift thrust drag due to hypersonic flight weight Model: Dynamics: Forces

  13. homotopy parameter Model: Dynamics: Equations of Motion Two-dimensional flight over a great circle of a rotational Earth instantaneous fuel consumption for thrust instantaneous fuel consumption for active cooling

  14. Model: Dynamics: Boundary Conditions Houston – Rome: 9163 km = 5693 m Munich – Houston: 8714 km = 5414 m

  15. Model: The Optimal Control Problem Objective function Constraints angle of attack : box constraints : throttle setting control-state constr.: load factor state constr.: dynamic pressure

  16. stagnation point (1300°C) engine (1350°C) lower surface (700°C) upper surface (600°C) leading edge (1200°C) Model: Active Engine Cooling

  17. H2 tank H2 pump cold air compressor 4 6 cold air cooler air 7 8 2 5 3 1 turbo/ramjet engine Model: Active Engine Cooling instantaneous fuel consumption for cooling instantaneous fuel consumption for thrust control-state constraint fuel is reused for thrust

  18. Numerical Results: State Variables velocity [m/s] altitude [10,000 m] without with active cooling [s] [s] mass [100,000 kg] path length [1,000 km] [s] [s] Markus Wächter, Kurt Chudej Florent Le Bras

  19. Numerical Results: Control Variables angle of attack [deg] throttle setting [s] [s]

  20. Outline • Introduction/Motivation • The hypersonic trajectory optimization problem • The instationary heat constraint • Numerical results • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion • Appendix: more applications

  21. convection radiation outer surface hot air flow multiwall packages (metalic sandwich structure) conduction SiO2 fleece (variable) multiwall package damping layer fuselage wall convection radiation inner surface Model: Instationary Heat Constraint: Thermal Protection System

  22. quasilinear PDE affine linear functions nonlinear boundary conditons Model: Instationary Heat Constraint: Equations quasi-linear parabolic initial-boundary value problem with nonlinear boundary conditions

  23. Model: Instationary Heat Constraint: Boundary Conditions (1) radiation convection radiation convection

  24. Model: Instationary Heat Constraint: State Constraint State-constraint for the temperature: ODE-PDE state-constrained optimal control problem PDE: quasilinear parabolic with nonlinear bound. conds. CONTROL: boundary controls indirectly via ODE states and controls CONSTRAINT: state constraint

  25. Outline • Introduction/Motivation • The hypersonic trajectory optimization problem • The instationary heat constraint • Numerical results • The hypersonic rocket car problems • Theoretical results • New necessary conditions • Numerical results • Conclusion

  26. Numerical Method: Semi-Discretization in Space radiation convection conduction Finite Volume Method: locally and globally conservative second order convergent

  27. 18° K tank convection 300° K conduction radiation conservation law in each volume second order convergent FVM Numerical Method: Semi-Discretization in Space

  28. Numerical Method: Semi-Discretization in Space 1D case

  29. Numerical Method: Method of Lines b.c. towards air b.c. towards interior coupled with ODE. large scale multiply constrained ODE optimal control problem DIRCOL (O. v. Stryk) with SNOPT (P. Gill) alternatively: NUDOCCCS (C. Büskens) IPOPT (A. Wächter) with AMPL WORHP (Büskens, Gerdts)

  30. stagnation point (1300°C) engine (1350°C) lower surface (700°C) upper surface (600°C) leading edge (1200°C) Numerical Results: Stagnation Point (1D)

  31. Numerical Results: Stagnation Point: States, Heat Loads altitude [10,000 m] flight path angle [deg] velocity [m/s] [s] temperature [K] temperature [K] temperature [K] limit temperature 1000 K on a boundary arc order concept? 2nd layer 3rd layer 1st layer [s]

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