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 Wchter, Gottfried Sachs, Florent le Bras

The 8th International Conference on Optimization: Techniques and Applications (ICOTA 8), Shanghai, China, Dec. 10-14, 2010

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

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

Introduction: Supersonic Aircraft

Introduction: Hypersonic Passenger Jets

Introduction: Hypersonic Passenger Jets http://www.reactionengines.co.uk/lapcat_anim.htmlProject LAPCATReading Engines, UK

Motivation: Hypersonic Passenger Jets Project LAPCATReading Engines, UK2 box constraints1 control-state constraint1 state constraintquasilinear PDEnon-linear boundary conditionscoupled with ODE

Introduction: The German Snger II ProjectChudej: 1989; Schnepper: 1999 Collaborative Research Center, Munich: 1989 - 2003 air-breathing turbojet - ramjet / scramjet

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

Model: Atmosphereairtemperature [K]air density [kg/m]airpressure [bar]altititude [km]Markus Wchter & Gottfried SachsMunich U of Technology, Germany

Heat conductivityHeat capacityair temperature Model: Atmosphere

Model: Dynamics: Forces

Model: Dynamics: Equations of Motion instantaneous fuel consumptionfor thrustTwo-dimensional flight over a great circle of a rotational Earthinstantaneousfuel consumptionfor active cooling

Model: Dynamics: Boundary ConditionsHouston Rome: 9163 km = 5693 mMunich Houston: 8714 km = 5414 m

Model: The Optimal Control ProblemObjective functionConstraintsstate constr.: dynamic pressurecontrol-state constr.: load factorangle of attack : box constraints : throttle setting

Model: Active Engine Cooling

Model: Active Engine Coolingcontrol-state constraintfuel is reused for thrustinstantaneous fuel consumptionfor coolinginstantaneous fuel consumptionfor thrust

Numerical Results: State Variableswithwithoutactive coolingvelocity [m/s]altitude [10,000 m]mass [100,000 kg]path length [1,000 km][s][s][s][s]Markus Wchter, Kurt ChudejFlorent Le Bras

Numerical Results: Control Variablesangle of attack [deg]throttle setting[s][s]

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

Model: Instationary Heat Constraint: Thermal Protection System

Model: Instationary Heat Constraint: Equationsquasi-linear parabolic initial-boundary value problemwith nonlinear boundary conditions

Model: Instationary Heat Constraint: Boundary Conditions (1)radiationconvectionconvectionradiation

Model: Instationary Heat Constraint: Boundary Conditions (2)in case ofinterior layersmultipointboundary conditionsair temperature after shockto be determined iteratively

Model: Instationary Heat Constraint: State ConstraintODE-PDE state-constrained optimal control problemPDE: quasilinear parabolic with nonlinear bound. conds.

CONTROL: boundary controls indirectly via ODE states and controls

CONSTRAINT: state constraintState-constraint for the temperature:

Model: Instationary Heat Constraint: State ConstraintODE-PDE state-constrained optimal control problemState-constraint for the temperature:

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

Numerical Method: Semi-Discretization in SpaceconvectionradiationconductionFinite Volume Method: locally and globally conservative second order convergent

Numerical Method: Semi-Discretization in Space

Numerical Method: Semi-Discretization in Space1D case

Numerical Method: Method of Linesb.c. towards airb.c. towards interiorlarge scale multiply constrained ODE optimal control problemcoupled with ODE.DIRCOL (O. v. Stryk) with SNOPT (P. Gill)alternatively: NUDOCCCS (C. Bskens) IPOPT (A. Wchter) with AMPL WORHP (Bskens, Gerdts)

Numerical Results: Stagnation Point (1D)

Numerical Results: Stagnation Point: States, Heat Loadsvelocity [m/s]altitude [10,000 m]flight path angle [deg]temperature [K]temperature [K]temperature [K]1st layer2nd layer3rd layerlimittemperature1000 Kon a boundary arc

order concept?[s][s]

Numerical Results: Stagnation Point: Heat Load vs Fuel+ 1 %- 10 %

Numerical Results: Lower Surface (near Engine) (2D)stagnation point (1300C)upper surface (600C)lower surface (700C)engine (1350C)leading edge (1200C)

Numerical Results: in Front of Engine (2D)1st layer2nd layerdue to restrictionof time intervalstate constraintnot activeinfluence of tank

Numerical Results: close to tank (2D)state constraintnot activedue to restrictionof time interval

Tupolew: Absturz der 4. Maschine in Le Bourget 19731975: Frachtflge zwischen Moskau und Almaty, Kasachstan1977: Passagierflge, 82 Rubel = *monatl. Durchschnittsverdienst der UdSSRLAPCAT: Long Term Advanced Propulsion Concepts and Technologies