Value function and optimal trajectories for a control problem with supremum cost function and state constraints Hasnaa Zidani ENSTA ParisTech, University of Paris-Saclay joint work with: A. Assellaou, & O. Bokanowski, & A. Desilles Workshop ”Numerical methods for Hamilton-Jacobi equations in optimal control and related fields” Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 1 / 29
Consider the following state constrained control problem: � � � θ ∈ [0 , t ] Φ( y u � u ∈ U , y u ϑ ( t , y ) := inf max y ( θ )) y ( s ) ∈ K , s ∈ [0 , t ] (1) where y u y denotes the solution of the controlled differential system: � ˙ y ( s ) := f ( y ( s ) , u ( s )) , a.e s ∈ [0 , t ] , y (0) := y , K is a closed set of R d and U is a set of admissible control inputs. Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 2 / 29
Hamilton-Jacobi approach ➤ How can we handle correctly the state constraints ? ➤ What boundary conditions should be considered for the HJB equation? ➤ Trajectory reconstruction and feedback control law. ➤ When Φ ≥ 0, we know that �� t � 1 2 p y ( s )) 2 p ds Φ( y u s ∈ [0 , t ] Φ( y u → max y ( s )) , as p → + ∞ . 0 So, it is possible to approximate the maximum running cost problem by a Bolza problem. Does this approximation work well in practice? Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 3 / 29
Outline 1 Motivation: Abort landing problem in presence of ”Wind Shear” 2 Optimal control problem with maximum-cost and state constraints 3 Optimal trajectories 4 Numerical simulations Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 4 / 29
Outline 1 Motivation: Abort landing problem in presence of ”Wind Shear” 2 Optimal control problem with maximum-cost and state constraints 3 Optimal trajectories 4 Numerical simulations Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 5 / 29
Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 6 / 29
Abort landing problem in presence of windshear (Miele, Wang and Melvin(1987,1988); Bulirsch, Montrone and Pesch (1991..); Botkin-Turova(2012 ...)) Consider the flight motion of an aircraft in a vertical plane: x = V cos γ + w x ˙ ˙ h = V sin γ + w h ˙ V = F T m cos( α + δ ) − F D m − g sin γ − ( ˙ w x cos γ + ˙ w h sin γ ) γ = 1 V ( F T m sin( α + δ ) + F L ˙ m − g cos γ + ( ˙ w x sin γ − ˙ w h cos γ )) where ∂ w x ∂ x ( V cos γ + w x ) + ∂ w x w x ˙ = ∂ h ( V sin γ + w h ) ∂ w h ∂ x ( V cos γ + w x ) + ∂ w h ˙ = ∂ h ( V sin γ + w h ) w h and F T := F T ( V ) is the thrust force F D := F D ( V , α ) and F L := F L ( V , α ) are the drag and lift forces w x := w x ( x ) and w h := w h ( x , h ) are the wind components m , g , and δ are constants. Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 7 / 29
Controlled system Consider the state y ( . ) = ( x ( . ) , h ( . ) , V ( . ) , γ ( . ) , α ( . )). The control variable u is the angular speed of the angle of attack α . Let T be a fixed time horizon and let U be the set of admissible controls � � U := u : (0 , T ) → R , measurable, u ( t ) ∈ U a.e where U is a compact set. The controlled dynamics in this case is: x = V cos γ + w x , ˙ ˙ h = V sin γ + w h , V = F T ˙ m cos( α + δ ) − F D m − g sin γ − ( ˙ w x cos γ + ˙ w h sin γ ) , γ = 1 V ( F T m sin( α + δ ) + F L ˙ m − g cos γ + ( ˙ w x sin γ − ˙ w h cos γ )) , α = u . ˙ Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 8 / 29
Formulation of the optimal control problem Aim: Maximize the minimal altitude over a time interval: θ ∈ [0 , t ] h ( θ ) min while the aircraft stays in a given domain K . Consider the following optimal control problem: � � � � u ∈ U , and y u θ ∈ [0 , t ] Φ( y u ( P ) : ϑ ( t , y ) = inf max y ( θ )) , y ( s ) ∈ K , ∀ s ∈ [0 , t ] where Φ( y u y ( . )) = H r − h ( . ), H r being a reference altitude, and K is a set of state constraints. Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 9 / 29
Formulation of the optimal control problem Aim: Maximize the minimal altitude over a time interval: θ ∈ [0 , t ] h ( θ ) min while the aircraft stays in a given domain K . Consider the following optimal control problem: � � � � u ∈ U , and y u θ ∈ [0 , t ] Φ( y u ( P ) : ϑ ( t , y ) = inf max y ( θ )) , y ( s ) ∈ K , ∀ s ∈ [0 , t ] where Φ( y u y ( . )) = H r − h ( . ), H r being a reference altitude, and K is a set of state constraints. Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 9 / 29
Outline 1 Motivation: Abort landing problem in presence of ”Wind Shear” 2 Optimal control problem with maximum-cost and state constraints 3 Optimal trajectories 4 Numerical simulations Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 10 / 29
A general setting ➤ For a given non-empty compact subset U of R k and a finite time T > 0, define the set of admissible control to be, � � u : (0 , T ) → R k , measurable, u ( t ) ∈ U a.e U := . ➤ Consider the following control system: � ˙ y ( s ) := f ( y ( s ) , u ( s )) , a.e s ∈ [0 , T ] , (2) y (0) := y , where u ∈ U and the function f is defined and continuous on R d × U and that it is Lipschitz continuous w.r.t y , ( i ) f : R d × U → R d is continuous, � ( ii ) ∃ L > 0 s.t. ∀ ( y 1 , y 2 ) ∈ R d × R d , ∀ u ∈ U , | f ( y 1 , u ) − f ( y 2 , u ) | ≤ L ( | y 1 − y 2 | ) . ➤ The corresponding set of feasible trajectories: S [0 , T ] ( y ) := { y ∈ W 1 , 1 (0 , T ; R d ) , y satisfies (2) for some u ∈ U} , Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 11 / 29
A general setting ➤ For a given non-empty compact subset U of R k and a finite time T > 0, define the set of admissible control to be, � � u : (0 , T ) → R k , measurable, u ( t ) ∈ U a.e U := . ➤ Consider the following control system: � ˙ y ( s ) := f ( y ( s ) , u ( s )) , a.e s ∈ [0 , T ] , (2) y (0) := y , where u ∈ U and the function f is defined and continuous on R d × U and that it is Lipschitz continuous w.r.t y , ( i ) f : R d × U → R d is continuous, � ( ii ) ∃ L > 0 s.t. ∀ ( y 1 , y 2 ) ∈ R d × R d , ∀ u ∈ U , | f ( y 1 , u ) − f ( y 2 , u ) | ≤ L ( | y 1 − y 2 | ) . ➤ The corresponding set of feasible trajectories: S [0 , T ] ( y ) := { y ∈ W 1 , 1 (0 , T ; R d ) , y satisfies (2) for some u ∈ U} , Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 11 / 29
State constrained control problem with maximum cost ➤ Consider the following state constrained control problem: � � � � u ∈ U , θ ∈ [0 , t ] Φ( y u y u ϑ ( t , y ) := inf max y ( θ )) y ( s ) ∈ K , s ∈ [0 , t ] (3) ➤ the cost function Φ( · ) is assumed to be Lipschitz continuous and K is a closed set of R d . ➤ For every y ∈ R d , the set f ( y , U ) = { f ( y , u ) , u ∈ U } is assumed to be convex. ➤ The function ϑ is lower semicontinuous (lsc) on K , ϑ ≡ + ∞ outside K . Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 12 / 29
State constrained control problem with maximum cost ➤ Consider the following state constrained control problem: � � � � u ∈ U , θ ∈ [0 , t ] Φ( y u y u ϑ ( t , y ) := inf max y ( θ )) y ( s ) ∈ K , s ∈ [0 , t ] (3) ➤ the cost function Φ( · ) is assumed to be Lipschitz continuous and K is a closed set of R d . ➤ For every y ∈ R d , the set f ( y , U ) = { f ( y , u ) , u ∈ U } is assumed to be convex. ➤ The function ϑ is lower semicontinuous (lsc) on K , ϑ ≡ + ∞ outside K . Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 12 / 29
State constrained control problem with maximum cost ➤ Consider the following state constrained control problem: � � � � u ∈ U , θ ∈ [0 , t ] Φ( y u y u ϑ ( t , y ) := inf max y ( θ )) y ( s ) ∈ K , s ∈ [0 , t ] (3) ➤ the cost function Φ( · ) is assumed to be Lipschitz continuous and K is a closed set of R d . ➤ For every y ∈ R d , the set f ( y , U ) = { f ( y , u ) , u ∈ U } is assumed to be convex. ➤ The function ϑ is lower semicontinuous (lsc) on K , ϑ ≡ + ∞ outside K . Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 12 / 29
Some references Maximum cost problems without state constraints ( K = R d ): Barron-Ishii (99) Bolza or Mayer problems with state constraints: Soner (86), Rampazzo-Vinter (89), Frankowska-Vinter (00), Motta (95), Cardaliaguet-Quincampoix-Saint-Pierre (97), Altarovici-Bokanowski-HZ (13), Hermosilla-HZ (15) Maximum cost problems with state constraints: Quincampoix-Serea (02), Bokanowski-Picarelli-HZ (13), Assellaou-Bokanowski-Desilles-HZ (CDC’16), Assellaou-Bokanowski-Desilles-HZ (preprint’16) Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 13 / 29
Recommend
More recommend
