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Model Numerical methods System analysis Numerical results Open problems Optimal control of the atmospheric re-entry of a space shuttle Emmanuel Trlat 1 1 MAPMO Universit dOrlans EDF-CEA-INRIA School, 2007 E. Trlat Atmospheric
Model Numerical methods System analysis Numerical results Open problems
Emmanuel Trélat1
1MAPMO
Université d’Orléans EDF-CEA-INRIA School, 2007
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 Aerodynamic configuration. Range 30 km < altitude < 120 km.
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
Forces Gravity force: −m g Aerodynamic force:
1
drag: 1
2ρSCDv2 2
lift:
1 2ρSCLv2(cos µ
k) Control Bank angle µ (the shuttle is a glider). Optimization criterion Total thermal flux Φ = Z tf
t0
Cq√ρv3dt. ρ(r) = ρ0e−r: atmospheric density.
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
˙ r = v sin γ ˙ v = −g sin γ − ρ SCD 2m v2 + Ω2r cos L(sin γ cos L − cos γ sin L cos χ) ˙ γ = cos γ “ − g v + v r ” + ρ SCL 2m vcos µ + 2Ω cos L sin χ + Ω2 r cos L v (cos γ cos L + sin γ sin L cos χ) ˙ L = v r cos γ cos χ ˙ l = − v r cos γ sin χ cos L ˙ χ = ρ SCL 2m v cos γ sin µ + v r cos γ tan L sin χ + 2Ω(sin L − tan γ cos L cos χ) + Ω2 r v sin L cos L sin χ cos γ State x = (r, v, γ, L, l, χ) r radius, v relative speed, γ flight angle, L latitude, l longitude, χ azimuth. Control µ (bank angle) System of the form ˙ x = X(x) + u1Y1(x) + u2Y2(x), u2
1 + u2 2 = 1
(with u1 = cos µ, u2 = sin µ)
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
State constraints Thermal flux: Φ = Cq√ρv3 Φmax Normal acceleration: γn = γn0CDρv2 γmax
n
Dynamic pressure: P = 1
2ρv2 Pmax initial conditions final conditions altitude 119.82 km 15 km speed 7404.95 m/s 445 m/s flight angle
free azimuth free free latitude 0 deg 10.99 deg longitude free or fixed to 116.59 deg 166.73 deg
Harpold and Graves method (1979) Saturate the constraints on the state along the flight. → not optimal for the optimization criterion: Φ = Z tf
t0
Cq√ρv3 dt
normal acceleration dynamic pressure thermal flux
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
State: x(t) Control: u(t) Optimal control problem ˙ x(t) = f(x(t), u(t)), x(t) ∈ I Rn, u(t) ∈ Ω ⊂ I Rm, x(0) = x0, x(T) = x1, c(x(t)) 0, min C(T, u), where C(T, u) = Z T f 0(x(t), u(t))dt
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
Control system ˙ x(t) = f(x(t), u(t)) (1) Cost C(T, u) = Z T f 0(x(t), u(t))dt Optimal control problem Determine the trajectories x(·) solutions of (1), satisfying x(0) ∈ M0, x(T) ∈ M1, and c(x(t)) 0, and minimizing the cost C(T, u).
Numerical methods
1
direct methods
2
indirect methods
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
Both state and control are discretized ⇒ finite dimensional nonlinear optimization problem min
g(Z)=0 h(Z)0
F(Z), where Z = (x1, . . . , xN, u1, . . . , un). → Numerical solving: Gradient methods, penalization, SQP (sequential quadratic programming), etc.
∂S ∂t + H1(x, ∂S ∂x ) = 0, where H1(x, p) = maxu∈Up, f(x, u) − f 0(x, u). → Numerical solving: explicite methods, level set methods, ...
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
Based on the Maximum Principle (Pontryagin, Maurer, Jacobson, etc): H(x, p, p0, u) = p, f(x, u) + p0f 0(x, u) + ηc(x). Maximum Principle Every minimizing trajectory x(·) is the projection of an extremal (x(·), p(·), p0, u(·)) solution of ˙ x = ∂H ∂p , ˙ p = − ∂H ∂x , H(x, p, p0, u) = max
v∈Ω H(x, p, p0, v).
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
Based on the Maximum Principle (Pontryagin, Maurer, Jacobson, etc): H(x, p, p0, u) = p, f(x, u) + p0f 0(x, u) + ηc(x). Maximum Principle Every minimizing trajectory x(·) is the projection of an extremal (x(·), p(·), p0, u(·)) solution of ˙ x = ∂H ∂p , ˙ p = − ∂H ∂x , H(x, p, p0, u) = max
v∈Ω H(x, p, p0, v).
u(t) = u(x(t), p(t)) “ locally, e.g. under the strict Legendre assumption: ∂2H ∂u2 (x, p, u) negative definite ”
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
Based on the Maximum Principle (Pontryagin, Maurer, Jacobson, etc): H(x, p, p0, u) = p, f(x, u) + p0f 0(x, u) + ηc(x). Maximum Principle Every minimizing trajectory x(·) is the projection of an extremal (x(·), p(·), p0, u(·)) solution of ˙ x = ∂H ∂p , ˙ p = − ∂H ∂x , H(x, p, p0, u) = max
v∈Ω H(x, p, p0, v).
u(t) = u(x(t), p(t)) “ locally, e.g. under the strict Legendre assumption: ∂2H ∂u2 (x, p, u) negative definite ”
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
The extremals (x, p) are solutions of ˙ x = ∂H ∂p , x(0) = x0, x(T) = x1, ˙ p = − ∂H ∂x , p(0) = p0, where the optimal control maximizes the Hamiltonian. − → Shooting method: determine p0 s.t. x(T) = x1.
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.
in optimal control, ESAIM Control Optim. Calc. Var. (2007).
http://www.n7.fr/apo/cotcot.
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
Optimal control problem
Discretization Euler, Runge-Kutta, etc.
Dualization Kuhn-Tucker, then Newton method direct methods Dualization Maximum principle
Discretization Euler, Runge-Kutta, etc, then Newton (shooting method) indirect methods
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
Optimal control problem
Discretization Euler, Runge-Kutta, etc.
Dualization Kuhn-Tucker, then Newton method direct methods
Dualization Maximum principle
Discretization Euler, Runge-Kutta, etc, then Newton (shooting method) indirect methods No commutation in general. Commutation for Runge-Kutta methods with positive coefficients (cf Hager, 2000).
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
Direct methods indirect methods simple implementation, without a priori knowledge a priori knowledge of the 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 state constraints difficult to take into account state constraints (globally) optimal controls in closed loop (locally) optimal controls 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) ...
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
In aeronautics, we use more frequently shooting methods, more precise but more difficult to make converge. Problem How to make the shooting method converge? Different tools:
1
a priori implementation of a direct method
2
continuation methods
3
geometric study (→ atmospheric re-entry)
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
˙ r = v sin γ ˙ v = −g sin γ − ρ SCD 2m v2+Ω2r cos L(sin γ cos L − cos γ sin L cos χ) ˙ γ = cos γ “ − g v + v r ” + ρ SCL 2m vcos µ+2Ω cos L sin χ +Ω2 r cos L v (cos γ cos L + sin γ sin L cos χ) ˙ L = v r cos γ cos χ ˙ l = − v r cos γ sin χ cos L ˙ χ = ρ SCL 2m v cos γ sin µ + v r cos γ tan L sin χ +2Ω(sin L − tan γ cos L cos χ) + Ω2 r v sin L cos L sin χ cos γ
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
System ˙ x = X(x) + u1Y1(x) + u2Y2(x), u2
1 + u2 2 = 1
If the rotation Ω of the planet is neglected: First subsystem (u1 = cos µ) ˙ r = v sin γ ˙ v = −g sin γ − ρ SCD 2m v2 ˙ γ = cos γ “ − g v + v r ” + ρ SCL 2m vu1 Second subsystem (u2 = sin µ) ˙ L = v r cos γ cos χ ˙ l = v r cos γ sin χ cos L ˙ χ = v r cos γ tan L sin χ + ρ SCL 2m v cos γ u2
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
System ˙ x = X(x) + u1Y1(x) + u2Y2(x), u2
1 + u2 2 = 1
If the rotation Ω of the planet is neglected: First subsystem (u1 = cos µ) ˙ r = v sin γ ˙ v = −g sin γ − ρ SCD 2m v2 ˙ γ = cos γ “ − g v + v r ” + ρ SCL 2m vu1 − → ˙ q = X(q) + u1Y1(q), |u1| 1 where q = @ r v γ 1 A ∈ I R3.
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
System in I R3: ˙ q = X(q) + uY(q), |u| 1 Theorem
(Bressan, Krener-Schättler, Kupka, Sussmann, ...)
Under certain assumptions on the Lie Brackets: the small time accessible set is boundered by the surfaces formed by the arcs q+q− and q−q+. Every interior point is reached with q−q+q− or q+q−q+. Small time accessible set in I R3.
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
Attempt with such a strategy:
500 1000 1500 2 4 6 8 10 12 x 10
4
Altitude (m) 500 1000 1500 2000 4000 6000 8000 Velocity (m/s) 500 1000 1500 !0.5 0.5 Flight angle (rad) 500 1000 1500 0.05 0.1 0.15 0.2 0.25 Latitude (rad) 500 1000 1500 2 2.2 2.4 2.6 2.8 3 Longitude (rad) 500 1000 1500 1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.42 Azimuth (rad)
→ State constraints violated.
200 400 600 800 1000 1200 1400 0.5 1 1.5 2 2.5 3 3.5 Bank angle (rad)
200 400 600 800 1000 1200 1400 1 2 3 x 10
6
Thermal flux (W/m2) 200 400 600 800 1000 1200 1400 50 100 Normal acceleration (m/s2) 200 400 600 800 1000 1200 1400 2 4 6 x 10
4
Dynamic pressure (kPa)
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
System in I R3: ˙ q = X(q) + uY(q), |u| 1, State constraint: c(q) 0. Minimal time problem. Theorem
(Bonnard-Faubourg-T, 2003)
Let q0 ∈ {c = 0}. Assume:
1 At x0, X, Y et [X, Y] form a frame, and [X ± Y, [X, Y]](x0) = aX(x0) + bY(x0) + c[X, Y](x0), with a < 0. 2 The constraint is of order 2 and the boundary control is not saturating.
Then, the boundary arc passing through q0 is locally time minimizing if and only if the arc q− is contained in the nonadmissible domain c 0. In this case, the local minimal time synthesis is of the form q−q+qbq+q−.
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
Problem 1 : free initial longitude. Problem 2 : fixed initial longitude. initial conditions final conditions altitude 119.82 km 15 km speed 7404.95 m/s 445 m/s flight angle
free azimuth free free latitude 0 deg 10.99 deg longitude free or fixed to 116.59 deg 166.73 deg
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
500 1000 2 4 6 8 10 12 x 10
4
Altitude (m) 500 1000 2000 4000 6000 8000 Velocity (m/s) 500 1000 !0.3 !0.25 !0.2 !0.15 !0.1 !0.05 0.05 Flight angle (rad) 500 1000 0.05 0.1 0.15 0.2 0.25 Latitude (rad) 500 1000 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Longitude (rad) 500 1000 0.5 1 1.5 2 Azimuth (rad)
100 200 300 400 500 600 700 800 900 1000 !2 !1 1 2 3 4 Bank angle (rad) seconds
100 200 300 400 500 600 700 800 900 1000 2 4 6 8 x 10
5Thermal flux (W/m2) 100 200 300 400 500 600 700 800 900 1000 10 20 30 40 Normal acceleration (m/s2) 100 200 300 400 500 600 700 800 900 1000 1 2 3 x 10
4Dynamic pressure (kPa) seconds
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
500 1000 1500 2 4 6 8 10 12 x 10
4
Altitude (m) 500 1000 1500 2000 4000 6000 8000 Velocity (m/s) 500 1000 1500 !0.5 !0.4 !0.3 !0.2 !0.1 0.1 Flight angle (rad) 500 1000 1500 0.05 0.1 0.15 0.2 0.25 Latitude (rad) 500 1000 1500 2 2.2 2.4 2.6 2.8 3 Longitude (rad) 500 1000 1500 1.5 Azimuth (rad)
200 400 600 800 1000 1200 1400 !1.5 !1 !0.5 0.5 1 1.5 2 2.5 3 3.5 Bank angle (rad) seconds
200 400 600 800 1000 1200 1400 2 4 6 8 x 10
5Thermal flux (W/m2) 200 400 600 800 1000 1200 1400 10 20 30 Normal acceleration (m/s2) 200 400 600 800 1000 1200 1400 1 2 3 x 10
4Dynamic pressure (kPa) seconds
Atmospheric re-entry of a space shuttle
Model Numerical methods System analysis Numerical results Open problems
general conjugate point theory, involving singular arcs and state constraints HJB method for aeronautics problems (study in the pole OPALE) commutation problem "discretization <–> dualization"
Atmospheric re-entry of a space shuttle