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Aerospace Dynamics and Optimal Control workshopParis, 23 May 2008 - 1

Application of Optimal Control to Space Transportation System Design

Aerospace Dynamics and Optimal Control WorkshopParis, May 23rd 2008

Nicolas BérendLong-Term Design and System Integration Department


Summary

• Introduction about space transportations applications and optimal control.

• Presentation of Onera’s FLOP/OLGA solver and applications.

• Interior point method.

• Other methods and applications.

• Needs in optimal control.


Applications

Space Transportation Systems


Main types of trajectory optimization problems

• Ascent.• Reentry.• Orbital Transfer.

Load factor

Equilibrium glide boundary

Heat Flux

Entry point

Drag deceleration

Velocity

Dynamic pressure

[Drag, Velocity] profile for a reentry phase

Dynamic pressure : 1/2.ρ.Vr2

Fairing Heat Flux : Cq.ρ1/2.Vr3.15

Heat flux peak : 1/2.ρ.Vr3

Verticality barrier

Re-entry

Visibility

Fairing Jettisoning

AOA ~ 0°

Altitude

Velocity

[Altitude, Velocity] profile for an ascent phase

Low-thrust orbital transfer


Special needs for RLV studies (1/2):Branching trajectories

• In branching cases:• 2 simultaneous controlled phases

follows a single one, starting from the same position (e.g. staging).

• the simultaneous phases have their own constraints and/or performance index.

Branching cases are complex to study.

May lead to multi-objective optimization problems.

The problem of initialization is more

acute.

Using existing solvers may not be the best solution.

Range

Altitude Orbiter trajectory

Booster return trajectory

Joint flight

Staging point


Special needs for RLV studies (2/2):abort trajectories

• RLV: specifications of high probability for vehicle recovery (10-4).

study of trajectories in case of a non-catastrophic failure (e.g. 1 engine out of n).

trajectories may be different from one case to another (e.g. RTLS, TAL, ATO).

complex problem: choice of the abort strategy as a function of the failure time.

many cases to study.

R a n g e

A l t i t u d e N o m i n a l t r a j .

R T L S

T A L

A T O

E n g i n e f a i l u r e

Need for a very robust tool, fast convergence.

Example of abort strategies for differentfailure times


General statement of a trajectory optimization problem

• Trajectory dynamics:

Different state vector possible (orbital parameters, position+velocity vectors, etc.)

• Performance index. Examples: max(payload), max(range), ...

• Local constraints: Examples: altitude, velocity, thermal flux (fairing separation).

• Path constraints: Examples: mechanical and thermal loads.

x¿

t = f x t , u t , a

J = φ x t f , a ,t f ∫0

tfL x t , u t , a ⋅dt

ψ i x t i , a , t i = 0 or ≤ 0

γ i x t , u t , a = 0 or ≤ 0


Generalized Projected GradientFLOP/OLGA: Future Launcher Optimization Program

(C. Aumasson, Onera)• Main features:

• Adapted to both expendable and reusable LV.• Multi-phases trajectories: ascent and/or reentry.• 3 d.o.f. dynamics (controls= attitude angles).• Solves for both controls (u(t)) and design parameters (a).• Library of:

• Vehicles characteristics (aerodynamics, propulsion, etc.)• Performance indexes (payload mass, thermal flux, etc.).• Constraints (thermal flux, transverse load factor, etc).

• Optimization method• OLGA optimization core: generalized projected gradient (direct method).• Main features:

• Management of multi-phases dynamics.• two-level optimization problem using a fixed “global step”.• Explicit expression of “current” Lagrange multipliers instead of the solving NLP

problem.


OLGA optimization coreFormulation of the global optimization problem

• Find the optimal controls u*(t) and parameters a*(t) such that:

• The following performance index is minimized:

• The following constraints are satisfied:

• for i = 1 to q with:

• Possible constraints:• Final, intermittent constraints (ϕi)• Final/integral constraints (Li)• Path constraints, formulated as integral constraints (Li)

J = φ0 x t f , a , t f ∫0

tfL0 x t , u t , a ⋅dt

ψ i ≤ 0

ψ i = φi x t f , a , t f ∫0

tfL i x t , u t , a ⋅dt


OLGA optimization corePrinciple of the method

• At each iteration: calculation of controls variations δu(t) and parameters variations δa for given objectives of performance index variations δJ and constraints variations δΨ

• First order sensibility equations are used in the process:

calculation of the Hamiltonians and adjoint vectors.

• Validity of sensibility equations is ensured by fixing a « global variation step » N:

Γ = global sensibility matrix.

δJ = [∂ K0

∂ a∫0

tf ∂ H 0

∂ a⋅dt ]⋅δa∫0

tf [ ∂H 0

∂ u⋅δu ]⋅dt δΨ i = [ ∂K i∂ a

∫0

tf ∂H i

∂ a⋅dt ]⋅δa∫0

tf [∂ H i

∂ u⋅δu]⋅dt

N 2 = [δJ δψ T ]⋅Γ−1⋅[δJδψ ]


OLGA optimization coreOrganization of the optimization process (1/2)

• At each iteration:

Subsidiary optimization problem

Find the objectives for constraint variations δΨ*i such that:

• Norm of the new constraints values is minimized:

• Variations stay within the « normalized constraint step » NΨ≤N :• (to ensure the validity of the calculus of variations)

Solution:

η = Lagrange multiplierΓΨ= constraints sensibility matrix.

min ∥ψδψ ¿∥S ∥δψ¿∥=N ψ

δψ ¿ =−Γ ψ⋅Γ ψη⋅S−1 −1⋅ψ


OLGA optimization coreOrganization of the optimization process (2/2)

• Local optimization problemFind control variations δu*(t) and parameters variations δa such that:

• Variations on performance index is minimized:(if <0: the performance index is reduced)

• Variations on constraints are equals to objectives:for i =1 to q

• Solution:

min δJ

δu¿ t =W−1⋅[ ∂H 0T

∂ u. ..

∂ HjT

∂ u ]⋅µ δa¿ = B−1⋅[ ∂K 0

∂ a∫0

tf ∂H 0

∂ a⋅dt

T

.. . ∂ Kq∂a∫0

tf ∂ H q

∂a⋅dt

T ]⋅µµ = Γ−1⋅[δJ ¿

δψ¿ ] = [ µ0 µ1 . . . µq ]T ν i=µ iµ0

δψ i = δψi¿

= “current” Lagrange multipliers (i = 1 to q)


« Classical » Projected Gradient Method

Normalized step domain (N)

Domain for zero constraint variations

Projected gradient

Gradient of the performance

index


Generalized Projected Gradient Method

Gradient of the performance

index

Modified projected gradient

Domain for desired constraint variations

Domain for satisfied

constraints

Normalized step domain (N)


FLOP/OLGA – Generalized Projected GradientAdvantages and Drawbacks

• Advantages:• Starts with unsatisfied constraints easier initialization process

(no adjoint/Lagrange multiplier initialization).• Full support of multi-phases problems.• General formulation of performance index and constraint function easier adaptation to new problems.

• Drawbacks:• Slow convergence (1st order method).• Yields local optimal solution (like others gradient-based methods).• Convergence sometimes difficult because of the formulation of

path constraints as final-integral constraints.


Example of trajectory optimization using FLOP/OLGA:Ascent trajectory for a classical (expendable) launcher

Ariane 5 trajectory


Example of trajectory optimization using FLOP/OLGA:SOH RLV ascent and reentry trajectory

• Performance index:• max(payload mass)

• Path constraints: - thermal flux - dynamic pressure - load factor• • Local constraints: - orbital parameters at

injection - final altitude and

velocity


Example of trajectory optimization using FLOP/OLGA:booster return trajectories for a TSTO concept

Booster return trajectories for variants of a TSTO concept with different staging Mach number

• Performance index:min(range to landing site)• Path constraints: - thermal flux - dynamic pressure - load factor• Local constraints: - final altitude and

velocity.


Interior point optimization method

• J. Laurent-Varin PhD Thesis (2005): « RLV Trajectory Optimization using an Interior Point Method » (Cnes/Onera/Inria, PhD supervisor F. Bonnans).

• Interior point formulation of the optimal control problem:• Augmented performance index with log-barrier penalization:

Original problem Unconstrained problem withwith path constraints penalized performance index

• Solving of the unconstrained problem with decreasing penalization factor ε.• Other features of the PhD work:

• Runge-Kutta discretization with analysis of the discretization error and mesh refinement.• Dedicated linear algebra solvers (QR factorization for band matrices).

J =∫0

tfL x t , u t ⋅dt

a ≤ g x t , u t ≤ b

J ε =∫0

tfLε x t , u t ⋅dt

Lε x , u = L x , u −ε⋅∑i=1

ng

[ log g i x , u −ai log b i−g i x , u ]


Example of RLV trajectory optimization with the interior point algorithm

Joint booster +orbiter ascent

Orbiter ascent

1

4

2 Booster return


Trajectory optimization for an Aeroassisted Orbital Transfer Vehicle (1/2)

• Studies in the frame of Onera’s CENTOR project (2004-2007)

• Aeroassistance = use of atmospheric forces in the upper atmosphere to perform an orbit transfer with no propellant consumption.

• Problem similar to atmospheric reentry but:

• Different objective function.• Different flight domain.

Orbit before aeroassistance

Orbit after aeroassistance

α

Drag

Lift

Velocity

µ


Trajectory optimization for an Aeroassisted Orbital Transfer Vehicle (2/2)

• For precise optimization of a given concept and mission analysis:

« Heavy » optimization tool (FLOP/OLGA).

• For performance calculation in the frame of multidisciplinary design optimization:

Simplified optimization tool OCTAVES:

• Direct approach.• Simplified control law defined by

a few parameters ( sub-optimal method).

• SQP solver.

AOTV trajectory optimization with OCTAVES

AOTV trajectory optimization with FLOP

Concept Cryo - Mission 2 - avec et sans borne sur gîte

1 113

1 465

0

20

40

60

80

100

120

140

160

0 100 200 300 400 500 600 700 800 900

Temps (s)

Alti

tude

(km

)

0

200

400

600

800

1000

1200

1400

1600

Flux

ther

miq

ue (k

W/m

²)

alt itude alt itude 2 flux therm flux therm 2

Rn = 1.622m


Optimization of an airlaunch system(Cnes/Onera DEDALUS UAV concept)

• Need for a rapid and robust trajectory optimization method (context: joint UAV+launcher vehicule design)

Simplified optimization tool OSTRAL:

• Direct approach.• Simplified control law defined by a

few parameters ( sub-optimal method).

• SQP solver.


Needs in optimal control applied to launchers(cf. 2003 Onera preparatory study to the creation of Cnes’

OPALE research group)

• Initialisation methods.• Global ascent+reentry optimization and branching

trajectories ( RLV).• Global optimization method (search of the « best »

optimum).• Stochastic optimization ( safety requirements for manned

or unmanned systems).• OC in the context of Multidisciplinary Design Optimization

(MDO)


2008 update: needs in optimal control applied to space transportation systems

• Currently drop of interest for RLV studies less need for global ascent+reentry optimization and branching trajectories (#2).

• Main need: OC in the context of Multidisciplinary Design Optimization (#5).

• Easy integration to a MDO process.• No (or reduced) user intervention for initialization.• Robustness.

• Other needs still existing but with a lower priority:• Initialisation method (#1, linked to #5).• Initialisation method (#3, linked to #5).• Stochastic optimization (#4)


Example of simplified MDO process for a multi-stage launcher

Staging optimization

M ? / Max(Φ(M)) ΨV(M,α,Vc)=0

Convergence ? |Vf-Vtarget|ε

Trajectory optimisation

A ? / Max(Vf(M,A)) Ψhf(M,A)=0 Ψγf(M,A)=0

Volume & geometry design

- Loss coefficients α - Velocity gap : (Vf-Vtarget)

Masses M

Masses M

Surface Sref

Fin

Yes

No

Update : - Loss coefficients α - Caract velocity :

Vc = f(Vf-Vtarget)

- α - (Vf-Vtarget)

- Loss coefficients α - Caracteristic velocity VC

V0 , h0 , γ0

Mu

geometry

ML


Conclusion

• Their exist today different OC methods and tools, for different contexts and applications (reentry, ascent, ...).

• No OC problem in space transportation applications seems out of reach with existing tools

but ...

• No optimization method is perfect in every aspect:• Easiness of initialization.• Rapidity and radius of convergence.• Quality of the optimum found.

• Today, the main challenge is to find easily and rapidly an optimal trajectory for a new launcher ( initialization and convergence domain issues).

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