ifac control of distributed parameter systems, july 20-24, 2009, toulouse inverse method for...
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IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Inverse method for pyrolysable and ablative materials with
optimal control formulation
S.Alestra1, J.Collinet2, and F.Dubois3
[email protected] [email protected]
2EADS ASTRIUM ST
Les Mureaux, France
1EADS Innovation Works
Toulouse, FRANCE
3Conservatoire National des Arts et Métiers
Paris, France
19/12/2006 p2
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Atmospheric re-entry missions Reentry Aerothermodynamics Team at EADS ASTRIUM-ST
Multiphysics : aerodynamics, aerothermodynamics, plasma
design and sizing of the Thermal Protection System (TPS) of the aerospace vehicles
the identification of heat fluxes is of great industrial interest
ARD
Industrial problem
Huygens probe(on Titan)
19/12/2006 p3
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Internal energy balance
Pyrolysis:
Arrhenius equation
Surface energy balance
Surface Recession
x
hm
tdTAhF
x
T
xt
TC gg
T
T
gp
0
1
T
Bn
v
c
v
eAt
1
ConductionAblation
PyrolysisRadiationConvection
)(
)()()(
2
144
0
x
ThhHm
hhHmTThh
wrvc
wrcgrwwr
hydrchemmechc sss
ms
)(0 t ),( txT
x
m
tg
Decompositionand mass balance
General equations of direct probleminput data: heat flux output data: temperature
19/12/2006 p4
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Its c
onte
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Evaluate the heat fluxes from temperature measurements
on thermal protection with ablation and pyrolysis
Inverse Heat problem is hard !! : see theory, diffusion aspect, ..
« Monopyro » 1D numerical tool, EADS ASTRIUM ST
The inverse method
(t), t in [O,T]p(t)= (t) ?
T in [0,T]
Pyrolysis gas
Material
Pyro
lysed
Pyro
lysisZ
on
e
Virg
inM
aterial
Material Pyrolysable Ablative
Radiation Flux
Convection Flux
Heat Flux ofPyrolysis gas
Ablation heat flux
Radaition Flux
Blocking Flux
Radiation Flux
Convection Flux
Fib
res
Resin
Coke
Pyro
lytic
Carb
on
p(t)= (t) ?
19/12/2006 p5
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Its c
onte
nt s
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not b
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Direct Problem
vector of temperature T and ablation s,
functions of time t and position x.
=> system of coupled nonlinear time domain evolution differential equations:
The other variables described above are hidden in the formulation of FSystem is rewritten in reduced variables
etsx )(1
etsxtt
xsTxT
pWFdt
dW
f ,,,0
0)0,()0,(
,
0
)(
),(
ts
txTW
1,0 ,t
s
e
T1 T2 T3
(t)
X (t)
p(t)= (t) Heat Flux
19/12/2006 p6
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Its c
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Direct Discrete scheme
Heat Flux N~=2000K number of one-dimensional grid points (~100), N number of time (N~=2000) iterations in the numerical scheme
The equation is written at time (n+1) :
Linearization at time n forward time discrete linearized Euler scheme, with initial condition vanishing: stability
Nnw
pwft
ww nnn
00
,
0
11
Nnw
wwpwdfpwft
ww nnnnnn
00
,,
0
11
nnK
nnn sTTTw ,,,, 21
19/12/2006 p7
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Cost Function
time domain unknown heat flux convection coefficient
Quadratic error or cost function j(p)
Measured temperatureComputed temperatureTo minimize this quantity, by optimization algorithm we need the derivatives of J(p), with respect to p.
The inverse method
N
n
nm
nm
Wiables
N tTpwpwJpJ1
2
var
1 )(),...,()(
Nppp ,...,1
nmnmT
19/12/2006 p8
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Adjoint SystemAdjoint variable : dual multiplyer ofLagrangian L + calculus of variations
Cancel the variations of L with respect to Direct system, forward in time, initial vanishing condition Cancel the variations of L with respect to w Adjoint system, backward in time, final vanishing condition
The inverse method
1
0
11
2/1
1
2
varint
2/12/1
var
11
,,,
,...,,,...,,,...,,,
N
n
nnnnnn
nN
n
nnm
iablesadjo
N
wiables
N
pparameter
N
wwpwdfpwft
wwtT
wwppLwpL
00
2,,
2/1
22/1122/112/12/1
nN
tTwwpwfdpwdft
N
nm
nm
nnnnnntnn
2/1n nw
19/12/2006 p9
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Gradient computation
With this particular choice of , the gradient of the cost function is simply obtained by :
Variations L function of p discrete gradients
apply an iterative inverse procedure minimizing J(p) to estimate the unknown parameter optimal function
The inverse method
p
L
p
JJ
1
0
12/1 ,N
n
nnnnn wwwp
dfw
p
f
p
J
19/12/2006 p10
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Gradient computation
The inverse method
0
,*
Tft
ttpTW
f
t obs
dt
p
pWf
p
L
p
j t
,
Adjoint State
Final Condition
Gradients
00
),(
tW
pWft
WDirect State
Initial condition
time
time
W = (T,s)
df / dp =
complex, non linear
df / dW = complex, non linear, tables
Measurements
19/12/2006 p11
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
OptimizationThe inverse method
2
2
dpjd
j(p*)
p*j (p)
j(p*)
p*j (p)
)( pj
Direct problemT(p)
Optimization p+ p
1) Steepest Descent to explore2) Quasi Newton to finish convergence
(T(p)-)**2
p0,
P optimal
)( pj
dpdj
Gradient
2
2
dpjd
Approximation of
Hessian
Direct + Adjoint system
Can be computed by Automatic Differentiation tool
19/12/2006 p12
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Automatic Differentiation TAPENADE, INRIA Sophia Antipolis, France
The inverse method
Program (Fortran) : sequence of elementary arithmetic operations
Derivatives can be computed automatically
If the code is modified, it is more easy to compute new adjoints and new gradients
Input
• function f
• cost function J(p)
Output
functions f’
gradient dJ/dp AD Tool
19/12/2006 p13
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Automatic Differentiation
),,,,,,,(1 ptwwwwfww knj
kni
nj
ni
ni
ni
ni
kni
knj
k jkn
i
knj
kni
nj
ni
kjn
ini w
J
w
ptwwwwf
),,,,,,,(1
N
n
nm
nm
Wiables
N tTpwpwJJ1
2
var
1 )(),...,(
The inverse method
Direct problem instruction
Cost Function
Adjoint system instructions : differentiation in reverse mode, with push, pop
Gradient computed by reverse mode
1
0
12/1 ,N
n
nnnnn wwwp
dfw
p
f
p
J
time
time
19/12/2006 p14
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Virgin material, low heat flux)Pseudo measurements very well rebuilt (RMS<1K)Automatic Differentiation (AD) sucessful
Some applications
Heat Flux without AD Heat Flux with AD
Gradient
Quasi Newton
Cost Function
19/12/2006 p15
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Carbon/Resin with ablation, pyrolysis & pseudo measurements Results OK with pyrolysis and ablation (without and with AD) Results OK with 2% noise on pseudo measurement Tichonov regularization to stabilize the solution
Some applications
Convection (noise without regularization)
Convection (noise with regularization)
Cost Function
GradientQuasi Newton
19/12/2006 p16
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
ARD1998 on Ariane Flight 503First use of the inverse method for « in-flight »
rebuilding during ARD post-flight analysis (1999)Last improvements of the method OK
Some applications
19/12/2006 p17
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Plasma Torch Facility
Material to be tested
Nozzle
Ablation compensation
Fluxmeters
Measurements:
• Laser (ablation)
• Pyrometer (surface temperature)
TC1
TC2
TC3
TC4
TC5
TC6
TC7
TC8
19/12/2006 p18
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Plasma torch: First results ONLY ONE SENSOR USED
Influence of sensor used Ablation restitution
Influence of sensor used Temperature at thermocouple
19/12/2006 p19
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Missing sensors Temperature at Thermocouple
Missing sensors convection coefficient restitution
Influence of sensor used Heat Flux restitution
Influence of sensor used Temperature at surface
ONLY ONE SENSOR USED
SEVERAL SENSORS USED AT THE SAME TIME
19/12/2006 p20
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Astrium Space Transportation
IFAC Control of Distributed Parameter Systems , July 20-24, 2009, Toulouse
Conclusion & perspectivesConclusion:
Inverse method sucessfully implemented First tests of Automatic Differentiation promising Validation OK for pseudo-measurements (with or without noise) Good results obtained on hard cases
Perspectives: Theoretical aspects (observability, identificability) to be analyzed Improve robustness of the method (initial guess,uncertainties on
noise, regularization) test on industrial re-entry problems Improve automatic differentiation version for hard cases