Model Numerical methods System analysis Numerical results Open problems Optimal control of the atmospheric re-entry of a space shuttle Emmanuel Trélat 1 1 MAPMO Université d’Orléans EDF-CEA-INRIA School, 2007 E. Trélat Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems Optimal control of a space shuttle during its re-entry phase (CNES project) Motivations aeroassisted orbit transfers development of reusable launchers problems of atmospheric re-entry: Mars Sample Return (CNES) Coron-Praly: stabilization. Bonnard-Faubourg-T: optimal control Role of the atmospheric arc reduction of the kinetic energy by friction with the atmosphere steer the engine from an initial point (position + speed) to a final point take into account some state constraints (thermal flux, normal acceleration, dynamic pressure) Control Range Aerodynamic configuration. 30 km < altitude < 120 km. E. Trélat Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems Model Forces Gravity force: − m � g Aerodynamic force: drag: 1 2 ρ SC D v 2 1 2 ρ SC L v 2 ( cos µ 1 � j + sin µ� 2 lift: k ) Control Bank angle µ (the shuttle is a glider ). Optimization criterion ρ ( r ) = ρ 0 e − r : atmospheric Z t f C q √ ρ v 3 dt . density. Total thermal flux Φ = t 0 E. Trélat Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems System ˙ r = v sin γ v = − g sin γ − ρ SC D 2 m v 2 + Ω 2 r cos L ( sin γ cos L − cos γ sin L cos χ ) State ˙ x = ( r , v , γ, L , l , χ ) − g v + v + ρ SC L “ ” r radius, γ = cos γ ˙ 2 m v cos µ + 2 Ω cos L sin χ r v relative speed, + Ω 2 r cos L ( cos γ cos L + sin γ sin L cos χ ) γ flight angle, v L = v L latitude, ˙ r cos γ cos χ l longitude, l = − v cos γ sin χ ˙ χ azimuth. r cos L χ = ρ SC L cos γ sin µ + v v ˙ r cos γ tan L sin χ 2 m Control µ + 2 Ω( sin L − tan γ cos L cos χ ) + Ω 2 r sin L cos L sin χ (bank angle) v cos γ System of the form x = X ( x ) + u 1 Y 1 ( x ) + u 2 Y 2 ( x ) , u 2 1 + u 2 ˙ 2 = 1 (with u 1 = cos µ, u 2 = sin µ ) E. Trélat Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems State constraints initial conditions final conditions Thermal flux: Φ = C q √ ρ v 3 � Φ max altitude 119.82 km 15 km speed 7404.95 m/s 445 m/s Normal acceleration: flight angle -1.84 deg free γ n = γ n 0 C D ρ v 2 � γ max azimuth free free n latitude 0 deg 10.99 deg 2 ρ v 2 � P max Dynamic pressure: P = 1 longitude free or fixed to 116.59 deg 166.73 deg Harpold and Graves method (1979) normal acceleration Saturate the constraints on the state along dynamic pressure the flight. thermal flux → not optimal for the optimization criterion: Z t f C q √ ρ v 3 dt Φ = t 0 E. Trélat Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems Modelization in the form of an optimal control problem State: x ( t ) Control: u ( t ) Optimal control problem R n , R m , ˙ x ( t ) = f ( x ( t ) , u ( t )) , x ( t ) ∈ I u ( t ) ∈ Ω ⊂ I x ( 0 ) = x 0 , x ( T ) = x 1 , c ( x ( t )) � 0 , Z T f 0 ( x ( t ) , u ( t )) dt min C ( T , u ) , where C ( T , u ) = 0 E. Trélat Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems Cost Control system Z T x ( t ) = f ( x ( t ) , u ( t )) ˙ (1) f 0 ( x ( t ) , u ( t )) dt C ( T , u ) = 0 Optimal control problem Determine the trajectories x ( · ) solutions of (1), satisfying x ( 0 ) ∈ M 0 , x ( T ) ∈ M 1 , and c ( x ( t )) � 0, and minimizing the cost C ( T , u ) . ↓ Numerical methods 1 direct methods 2 indirect methods E. Trélat Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems Direct methods 1. Total discretization Both state and control are discretized ⇒ finite dimensional nonlinear optimization problem F ( Z ) , min g ( Z )= 0 h ( Z ) � 0 where Z = ( x 1 , . . . , x N , u 1 , . . . , u n ) . → Numerical solving: Gradient methods, penalization, SQP (sequential quadratic programming), etc. 2. Hamilton-Jacobi equation ∂ S ∂ t + H 1 ( x , ∂ S ∂ x ) = 0 , where H 1 ( x , p ) = max u ∈ U � p , f ( x , u ) � − f 0 ( x , u ) . → Numerical solving: explicite methods, level set methods, ... E. Trélat Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems Indirect methods Based on the Maximum Principle (Pontryagin, Maurer, Jacobson, etc): H ( x , p , p 0 , u ) = � p , f ( x , u ) � + p 0 f 0 ( x , u ) + η c ( x ) . Maximum Principle Every minimizing trajectory x ( · ) is the projection of an extremal ( x ( · ) , p ( · ) , p 0 , u ( · )) solution of x = ∂ H p = − ∂ H H ( x , p , p 0 , u ) = max v ∈ Ω H ( x , p , p 0 , v ) . ˙ ∂ p , ˙ ∂ x , E. Trélat Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems Indirect methods Based on the Maximum Principle (Pontryagin, Maurer, Jacobson, etc): H ( x , p , p 0 , u ) = � p , f ( x , u ) � + p 0 f 0 ( x , u ) + η c ( x ) . Maximum Principle Every minimizing trajectory x ( · ) is the projection of an extremal ( x ( · ) , p ( · ) , p 0 , u ( · )) solution of x = ∂ H p = − ∂ H H ( x , p , p 0 , u ) = max v ∈ Ω H ( x , p , p 0 , v ) . ˙ ∂ p , ˙ ∂ x , ւ u ( t ) = u ( x ( t ) , p ( t )) “ locally, e.g. under the strict Legendre assumption: ∂ 2 H ” ∂ u 2 ( x , p , u ) negative definite E. Trélat Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems Indirect methods Based on the Maximum Principle (Pontryagin, Maurer, Jacobson, etc): H ( x , p , p 0 , u ) = � p , f ( x , u ) � + p 0 f 0 ( x , u ) + η c ( x ) . Maximum Principle Every minimizing trajectory x ( · ) is the projection of an extremal ( x ( · ) , p ( · ) , p 0 , u ( · )) solution of x = ∂ H p = − ∂ H H ( x , p , p 0 , u ) = max v ∈ Ω H ( x , p , p 0 , v ) . ˙ ∂ p , ˙ ∂ x , տ ւ u ( t ) = u ( x ( t ) , p ( t )) “ locally, e.g. under the strict Legendre assumption: ∂ 2 H ” ∂ u 2 ( x , p , u ) negative definite E. Trélat Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems Indirect methods - Simple shooting method The extremals ( x , p ) are solutions of x = ∂ H ˙ ∂ p , x ( 0 ) = x 0 , x ( T ) = x 1 , p = − ∂ H ˙ ∂ x , p ( 0 ) = p 0 , where the optimal control maximizes the Hamiltonian. − → Shooting method: determine p 0 s.t. x ( T ) = x 1 . - Multiple shooting method E. Trélat Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems Software: COTCOT (Conditions of Order Two and COnjugate Times) http://www.n7.fr/apo/cotcot/ automatic generation in Fortran of the equations of the maximum principle (automatic differentiation, Adifor ); automatic creation of mex files for Matlab ; codes Fortran for numerical integration, shooting methods, and conjugate points computation, interfaced with Matlab . B. Bonnard, J.-B. Caillau, E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control , ESAIM Control Optim. Calc. Var. (2007). B. Bonnard, J.-B. Caillau, E. Trélat, Cotcot: short reference manual , Technical report RT/APO/05/1, http://www.n7.fr/apo/cotcot. E. Trélat Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems Optimal control problem ւց Dualization Discretization Maximum principle Euler, Runge-Kutta, etc. ↓ ↓ Discretization Dualization Euler, Runge-Kutta, etc, Kuhn-Tucker, then Newton (shooting method) then Newton method direct methods indirect methods E. Trélat Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems Optimal control problem ւց Dualization Discretization Maximum principle Euler, Runge-Kutta, etc. ↓ ↓ Discretization Dualization ? ← → Euler, Runge-Kutta, etc, Kuhn-Tucker, then Newton (shooting method) then Newton method direct methods indirect methods No commutation in general. Commutation for Runge-Kutta methods with positive coefficients (cf Hager, 2000). E. Trélat Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems Direct methods indirect methods simple implementation, a priori knowledge of the without a priori knowledge structure of the optimal trajectory not sensitive w.r.t. the initial condition very sensitive w.r.t. the initial condition easy to take into account difficult to take into account state constraints state constraints (globally) optimal controls (locally) optimal controls in closed loop in open loop low to average numerical precision high numerical precision efficient in low dimension efficient in every dimension memory demanding parallel computations local minima problem small domain of convergence hybrid methods continuation methods (homotopic methods) ... E. Trélat Atmospheric re-entry of a space shuttle
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