Optimal space launcher trajectories in case of a singular arc eric - PowerPoint PPT Presentation

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Optimal space launcher trajectories in case of a singular arc eric Bonnans 1 , Pierre Martinon 1 and Emmanuel Tr´ elat 2 Fr´ ed´ CEA-EDF-INRIA School Optimal Control : Algorithms and Applications May 30 - June 1st 2007 1 INRIA FUTURS, team COMMANDS, and CMAP, Ecole Polytechnique 2 Universit´ e d’Orl´ eans, MAPMO 1 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Problem statement - Shooting method 1 Minimum principle - Singular arcs 2 Continuation approach - Simplicial homotopy 3 Numerical Experiments 4 Ongoing work: A realistic launcher model 5 2 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Problem statement - Shooting method Problem statement - Shooting method 1 Minimum principle - Singular arcs 2 Continuation approach - Simplicial homotopy 3 Numerical Experiments 4 Ongoing work: A realistic launcher model 5 3 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Problem statement Context: generalized Goddard problem We consider optimal 3D trajectories for a launch vehicle. Objective: payload maximization ie reach a prescribed position with a maximal final mass. Due to aerodynamic forces, possible presence of singular arcs , which means here that optimal trajectories may include arcs with a non maximal thrust. 4 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work General formulation and resolution methods Optimal control problem � t f  Min g 0 ( t f , x ( t f )) + t 0 l ( t , x , u ) dt Objective   x = f ( t , x , u ) ˙ Dynamics    ( P ) u ∈ U Admissible controls  ψ 0 ( t 0 , x ( t 0 )) = 0 Initial conditions     ψ 1 ( t f , x ( t f )) = 0 Terminal conditions • Direct methods - state-control discretization → nonlinear optimization problem (NLP) • Indirect methods - optimality necessary conditions (Pontryagin’s Minimum Principle) → non linear system (shooting methods) 5 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Pontryagin’s Minimum Principle We introduce the costate p , and define the Hamiltonian: H ( t , x , p , u ) = l + < p | f > Under the assumptions: - ∃ ( x , u ) admissible for ( P ), with x AC and u measurable - f , l are C 0 wrt u , and C 1 wrt t , x - ψ 0 , ψ 1 are C 1 wrt x Then any optimal pair (¯ x , ¯ u ) satisfies: � ˙ x = ∂ H ¯ ∂ p ( t , ¯ x , ¯ p , ¯ u ) - ∃ ¯ p � = 0, AC, such that p = − ∂ H ˙ ¯ ∂ x ( t , ¯ x , ¯ p , ¯ u ) - ¯ u minimizes H a.e. over [ t 0 , t f ]. - ¯ x , ¯ p satisfy the “transversality conditions” at t 0 and t f 6 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Boundary Value Problem (BVP) By noting y = ( x , p ), we obtain the following problem � ˙ y ( t ) = ϕ ( t , y ( t )) ( BVP ) Boundary conditions at t 0 and t f Initial value problem (IVP) By noting z the part of ( x ( t 0 ) , p ( t 0 )) not set by the boundary conditions, we define � ˙ y ( t ) = ϕ ( t , y ( t )) ( IVP ) y ( t 0 ) = y 0 ( z ) Shooting function We note ˜ y ( t , z ) the solution of ( IVP ) and define the shooting function S : z �→ value of boundary conditions for ˜ y ( t f , z ) Then solving ( BVP ) is equivalent to finding a zero of S . 7 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Shooting method overview Optimal control problem ( P ) ↓ PMP, Boundary Value Problem ( BVP ) Hamiltonian system, H-minimal control ↓ Initial value problem ( IVP ) Shooting function S ↓ Shooting method: solve S ( z ) = 0 (typically with a Newton method) • Fast and accurate method • Can be extremely sensitive to the initial point 8 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Minimum principle - Singular arcs Problem statement - Shooting method 1 Minimum principle - Singular arcs 2 Continuation approach - Simplicial homotopy 3 Numerical Experiments 4 Ongoing work: A realistic launcher model 5 9 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Problem modelization State (Position - Speed - Mass)  r ˙ = v   − D ( r , v ) � v � − g ( r ) + C u v ˙ = v m m  m ˙ = − b � u �  with ( r ( t ) , v ( t ) , m ( t )) ∈ R 3 × R 3 × R , D ( r , v ) > 0 the drag component, g ( r ) the gravity, C > 0 the maximal thrust and b > 0. Control u ( t ) ∈ R 3 is the normalized control corresponding to the thrust, submitted to the constraint � u ( t ) � ≤ 1. 10 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Objective: payload maximization We maximize the final mass m ( t f ), which is equivalent to minimizing the cost � t f � u ( t ) � dt 0 Initial and final conditions  ( r ( t 0 ) , v ( t 0 ) , m ( t 0 )) = ( r 0 , v 0 , m 0 )  r ( t f ) = r f , v f is free , m f is free. Final time t f is free.  11 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Application of the Minimum principle Hamiltonian � � p v , − D ( r , v ) � v � − g ( r ) + C u v H = (1 − bp m ) � u � + � p r , v � + m m Costate dynamics � �  � p v , v � p v , ∂ g = 1 ∂ D p r ˙ ∂ r + m � v � ∂ r    � p v , v � = − p r + 1 ∂ D p v ∂ v + D � v � − D v ˙ m � p v , v � p v � v � � v � 3 m m  � � p v , − D ( r , v ) = 1 � v � + C u v  p m ˙  m m m 12 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Optimal control - Study of singular arcs Optimal control u ∗ minimizes H, and therefore (1 − bp m ) � u � + C m � p v , u � . For a nondegenerate extremal: 1 The set T := { t ∈ [0 , t f ] | p v ( t ) = 0 } has a finite cardinal. 2 There exists a measurable function α on [0 , t f ], with values in [0 , 1], such that u ( t ) = − α ( t ) p v ( t ) a.e. on [0 , t f ] . � p v ( t ) � , 3 Define the switching function ψ ( t ) = 1 − bp m ( t ) − C � p v ( t ) � . m ( t ) Then � if ψ ( t ) < 0 then α ( t ) = 1 , if ψ ( t ) > 0 α ( t ) = 0 . then 13 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Singular arcs A singular arc occurs if ψ vanishes on a subset of positive measure of [0 , t f ]. The optimal control (more precisely α ( t )) is then obtained by derivating ψ = 0 until u appears explicitely. • First derivative: ˙ ψ = 0 → u disappears (as expected) • Second derivative: ¨ ψ = 0 → symbolic calculus yields A ( r , v , m , p r , p v , p m ) α = B ( r , v , m , p r , p v , p m ) In the numerical simulations, we check that over a singular arc - A is non-zero so the relation is not trivial - B A ∈ [0 , 1] so that � u ( t ) � ≤ 1 is satisfied - A < 0 (generalized Legendre-Clebsch optimality condition). 14 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Control structure Knowing the expression of the singular control is not enough. We also need to know the control structure : - at time t , are we on a singular arc or not ? How to determine the number and location of singular arcs ? We regularize the problem, and try to detect the structure as the regularization tends to 0. → Continuation approach 15 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Continuation approach - Simplicial homotopy Problem statement - Shooting method 1 Minimum principle - Singular arcs 2 Continuation approach - Simplicial homotopy 3 Numerical Experiments 4 Ongoing work: A realistic launcher model 5 16 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Continuation approach We consider a family of problems ( P λ ) λ ∈ [0 , 1] , such that - ( P 1 ) is the target problem ( P ) - we known how to solve ( P 0 ). The homotopy H connects ( P 0 ) to ( P 1 ): H : ( λ, z ) → S λ ( z ) Continuation: follow the zero path of H , from λ = 0 to λ = 1. - start from the known zero of H ( · , 0) = S 0 . - if we reach λ = 1 we have a zero of H ( · , 1) = S 1 = S . Note: the existence of such a path is not automatic... 17 / 41

Shooting method Singular arcs Simplicial homotopy Numerical Experiments Ongoing work Main homotopy: regularization We regularize the problem by adding a quadratic term to the criterion: � t f � u ( t ) � + (1 − λ ) � u ( t ) � 2 dt , Min λ ∈ [0 , 1] 0 For λ < 1 the regularized Hamiltonian is strictly convex wrt u . The new control law is then u ∗ = − p v ( t )  if ψ ( t ) < − 2(1 − λ ) � p v ( t ) �   then u ∗ = 0 if ψ ( t ) > 0 else u ∗ = − p v � p v � C / m − (1 − bp m )   � p v � 2(1 − λ ) → no commutations or singular arc anymore 18 / 41