Optimal control in aerospace elat 1 Emmanuel Tr 1 Sorbonne Universit - PowerPoint PPT Presentation

Optimal control in aerospace elat 1 Emmanuel Tr´ 1 Sorbonne Universit´ e (Paris 6), Labo. J.-L. Lions Mathematical Models and Methods in Earth and Space Sciences Rome Tor Vergata, March 2019

The orbit transfer problem with low thrust Controlled Kepler equation q = − q µ | q | 3 + F ¨ m R 3 : position, F : thrust, m mass: q ∈ I m = − β | F | ˙ Maximal thrust constraint Orbit transfer 3 ) 1 / 2 � F max ≃ 0 . 1 N | F | = ( u 2 1 + u 2 2 + u 2 from an initial orbit to a given final orbit Controllability properties studied in B. Bonnard, J.-B. Caillau, E. Tr´ elat, Geometric optimal control of elliptic Keplerian orbits , Discrete Contin. Dyn. Syst. Ser. B 5 , 4 (2005), 929–956. B. Bonnard, L. Faubourg, E. Tr´ elat, M´ ecanique c´ eleste et contrˆ ole de syst` emes spatiaux , Math. & Appl. 51 , Springer Verlag (2006), XIV, 276 pages.

The orbit transfer problem with low thrust Controlled Kepler equation q = − q µ | q | 3 + F ¨ m R 3 : position, F : thrust, m mass: q ∈ I m = − β | F | ˙ Maximal thrust constraint Orbit transfer 3 ) 1 / 2 � F max ≃ 0 . 1 N | F | = ( u 2 1 + u 2 2 + u 2 from an initial orbit to a given final orbit Controllability properties studied in B. Bonnard, J.-B. Caillau, E. Tr´ elat, Geometric optimal control of elliptic Keplerian orbits , Discrete Contin. Dyn. Syst. Ser. B 5 , 4 (2005), 929–956. B. Bonnard, L. Faubourg, E. Tr´ elat, M´ ecanique c´ eleste et contrˆ ole de syst` emes spatiaux , Math. & Appl. 51 , Springer Verlag (2006), XIV, 276 pages.

Modelling in terms of an optimal control problem � q ( t ) � State: x ( t ) = ˙ q ( t ) Control: u ( t ) = F ( t ) Optimal control problem x ( t ) = f ( x ( t ) , u ( t )) , ˙ x ( t ) ∈ M , u ( t ) ∈ Ω x ( 0 ) = x 0 , x ( T ) = x 1 � T f 0 ( x ( t ) , u ( t )) dt min C ( T , u ) , where C ( T , u ) = 0

Optimal control problem ˙ x ( t ) = f ( x ( t ) , u ( t )) , x ( 0 ) = x 0 ∈ M , u ( t ) ∈ Ω � T f 0 ( x ( t ) , u ( t )) dt x ( T ) = x 1 , min C ( T , u ) with C ( T , u ) = 0 Definition End-point mapping E x 0 , T : L ∞ ([ 0 , T ] , Ω) − → M u �− → x ( T ; x 0 , u )

Optimal control problem ˙ x ( t ) = f ( x ( t ) , u ( t )) , x ( 0 ) = x 0 ∈ M , u ( t ) ∈ Ω � T f 0 ( x ( t ) , u ( t )) dt x ( T ) = x 1 , min C ( T , u ) with C ( T , u ) = 0 Definition End-point mapping E x 0 , T : L ∞ ([ 0 , T ] , Ω) − → M u �− → x ( T ; x 0 , u )

Optimal control problem ˙ x ( t ) = f ( x ( t ) , u ( t )) , x ( 0 ) = x 0 ∈ M , u ( t ) ∈ Ω � T f 0 ( x ( t ) , u ( t )) dt x ( T ) = x 1 , min C ( T , u ) with C ( T , u ) = 0 Definition End-point mapping E x 0 , T : L ∞ ([ 0 , T ] , Ω) − → M u �− → x ( T ; x 0 , u ) − → Optimization problem min C ( T , u ) E x 0 , T ( u )= x 1

Optimal control problem ˙ x ( t ) = f ( x ( t ) , u ( t )) , x ( 0 ) = x 0 ∈ M , u ( t ) ∈ Ω � T f 0 ( x ( t ) , u ( t )) dt x ( T ) = x 1 , min C ( T , u ) with C ( T , u ) = 0 Definition End-point mapping E x 0 , T : L ∞ ([ 0 , T ] , Ω) − → M u �− → x ( T ; x 0 , u ) Definition A control u (or the trajectory x u ( · ) ) is singular if dE x 0 , T ( u ) is not surjective.

Lagrange multipliers (or KKT in general) A control u (or the trajectory x u ( · ) ) is singular if dE x 0 , T ( u ) is not surjective. Optimization problem min C ( T , u ) E x 0 , T ( u )= x 1 R m ) Lagrange multipliers (if Ω = I ∃ ( ψ, ψ 0 ) ∈ ( T ∗ ψ. dE x 0 , T ( u ) = − ψ 0 dC T ( u ) x ( T ) M × I R ) \ { ( 0 , 0 ) } | In terms of the Lagrangian L T ( u , ψ, ψ 0 ) = ψ. E x 0 , T ( u ) + ψ 0 C T ( u ) : ∂ L T ∂ u ( u , ψ, ψ 0 ) = 0 ψ 0 � = 0 ( → ψ 0 = − 1). - Normal multiplier: - Abnormal multiplier: ψ 0 = 0 R m ). ( ⇔ u singular, if Ω = I

Pontryagin Maximum Principle Optimal control problem x ( t ) = f ( x ( t ) , u ( t )) , x ( 0 ) = x 0 ∈ M , ˙ u ( t ) ∈ Ω � T f 0 ( x ( t ) , u ( t )) dt x ( T ) = x 1 , min C ( T , u ) , where C ( T , u ) = 0 Pontryagin 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 , where H ( x , p , p 0 , u ) = � p , f ( x , u ) � + p 0 f 0 ( x , u ) . An extremal is said normal whenever p 0 � = 0, and abnormal whenever p 0 = 0.

Pontryagin Maximum Principle H ( x , p , p 0 , u ) = � p , f ( x , u ) � + p 0 f 0 ( x , u ) Pontryagin 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 , ( p ( T ) , p 0 ) = ( ψ, ψ 0 ) up to (multiplicative) scaling . An extremal is said normal whenever p 0 � = 0, and abnormal whenever p 0 = 0. Singular trajectories coincide with projections of abnormal extremals s.t. ∂ H ∂ u = 0.

Pontryagin Maximum Principle H ( x , p , p 0 , u ) = � p , f ( x , u ) � + p 0 f 0 ( x , u ) Pontryagin 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

Pontryagin Maximum Principle H ( x , p , p 0 , u ) = � p , f ( x , u ) � + p 0 f 0 ( x , u ) Pontryagin 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

Shooting method: Extremals z = ( x , p ) are solutions of Exponential mapping x = ∂ H ˙ ∂ p ( x , p ) , x ( 0 ) = x 0 , ( x ( T ) = x 1 ) exp x 0 ( t , p 0 ) = x ( t , x 0 , p 0 ) p = − ∂ H ˙ ∂ x ( x , p ) , p ( 0 ) = p 0 (extremal flow) where the optimal control maximizes the Hamiltonian. − → Shooting method: determine p 0 s.t. exp x 0 ( t , p 0 ) = x 1

Shooting method: Extremals z = ( x , p ) are solutions of Exponential mapping x = ∂ H ˙ ∂ p ( x , p ) , x ( 0 ) = x 0 , ( x ( T ) = x 1 ) exp x 0 ( t , p 0 ) = x ( t , x 0 , p 0 ) p = − ∂ H ˙ ∂ x ( x , p ) , p ( 0 ) = p 0 (extremal flow) where the optimal control maximizes the Hamiltonian. − → Shooting method: determine p 0 s.t. exp x 0 ( t , p 0 ) = x 1 Remark - PMP = first-order necessary condition for optimality. - Necessary / sufficient (local) second-order conditions: conjugate points . → test if exp x 0 ( t , · ) is an immersion at p 0 . (fold singularity)

There exist other numerical approaches to solve optimal control problems: direct methods: discretize the whole problem ⇒ finite-dimensional nonlinear optimization problem with constraints Hamilton-Jacobi methods. The shooting method is called an indirect method. In aerospace applications, shooting methods are privileged in general because of their numerical accuracy. BUT: difficult to make converge... (Newton method) To improve performance and facilitate applicability, PMP may be combined with: (1) continuation or homotopy methods (2) geometric control (3) dynamical systems theory E. Tr´ elat, Optimal control and applications to aerospace: some results and challenges , JOTA 2012.

Minimal time orbit transfer Maximum Principle ⇒ the extremals ( x , p ) are solutions of x = ∂ H p = − ∂ H ˙ ˙ ∂ p , x ( 0 ) = x 0 , x ( T ) = x 1 , ∂ x , p ( 0 ) = p 0 , with an optimal control saturating the constraint: � u ( t ) � = F max . − → Shooting method: determine p 0 s.t. x ( T ; x 0 , p 0 ) = x 1 combined with a homotopy on F max �→ p 0 ( F max ) Heuristic on t f : t f ( F max ) · F max ≃ cste . (the optimal trajectories are ”straight lines”, Bonnard-Caillau 2009) (Caillau, Gergaud, Haberkorn, Martinon, Noailles, ...)

Minimal time orbit transfer P 0 = 11625 km, | e 0 | = 0 . 75, i 0 = 7 o , P f = 42165 km F max = 6 Newton x 10 ! 4 1 2 0 q 3 −2 arcsh det( ! x) 30 0 40 20 10 20 0 0 −10 ! 1 0 100 200 300 400 500 −20 −20 t −30 −40 ! 3 −40 x 10 6 q 2 q 1 5 5 4 " n ! 1 3 20 2 0 1 q 2 q 3 0 0 0 100 200 300 400 500 t −20 −40 −5 −60 −40 −20 0 20 40 −50 0 50 q 1 q 2 Minimal time: 141.6 hours ( ≃ 6 days). First conjugate time: 522.07 hours.

Continuation method Main tool used : continuation (homotopy) method → continuity of the optimal solution with respect to a parameter λ Theoretical framework (sensitivity analysis): F ( p 0 ( λ ) , λ ) = exp x 0 ,λ ( T , p 0 ( λ )) − x 1 = 0 Local feasibility is ensured: Global feasibility is ensured: in the absence of conjugate points in the absence of abnormal minimizers ↓ ↓ Numerical test of Jacobi fields True for generic systems having more than 3 controls (Chitour Jean Tr´ (Bonnard Caillau Tr´ elat, COCV 2007) elat, JDG 2006)