METTI V Thermal Measurements and Inverse Technique Benjamin REMY, - - PowerPoint PPT Presentation
METTI V Thermal Measurements and Inverse Technique Benjamin REMY, Stphane ANDRE & Denis MAILLET Benjamin.remy@ensem.inpl-nancy.fr L.E.M.T.A, U.M.R.- C.N.R.S. 7563 / E.N.S.E.M 02, avenue de la Fort de Haye, B.P 160 54 504
METTI V – Thermal Measurements and Inverse Technique
Benjamin REMY, Stéphane ANDRE & Denis MAILLET
Benjamin.remy@ensem.inpl-nancy.fr
ROSCOFF(France) – June 13-18 2011 L.E.M.T.A, U.M.R.- C.N.R.S. 7563 / E.N.S.E.M 02, avenue de la Forêt de Haye, B.P 160
54 504 Vandoeuvre-Lès-Nancy Cedex - FRANCE
1
I.
Definition and Vocabulary
II.
Useful Tools to investigate NLPE Problems
Contrast Method)
technique
estimated parameters (Case of the classical “Flash” method)
2
3
Measuring a physical quantity
j
β requires a specific experiment allowing for this quantity to "express
itself as much as possible" (notion of sensitivity). This experiment is requires a system onto which inputs u(t ) are applied (stimuli) and whose
pure dynamical experiment. A model M is required to mathematically express the dependence of the system's response with respect to quantity
j
β and to other additional parameters
( )
β k j ≠
:
η( , , ) = y t β u
4
( )
k
β k j ≠
:
mo
η( , , ) = y t β u
Many candidates may exist for function η -depending on the degree of complexity reached for modelling the physical process- which may exhibit different mathematical structure –depending for example on the type of method used to solve the model equations. Once this model is established, the physical quantities in vector β acquire the status of model parameters. This model (called knowledge model if it is derived from physical laws and/or conservation principles) is initially established in a direct formulation. Knowing inputs u(t ) and the value taken by parameter β , the output(s) can be predicted.
The linear or non linear character of the model has to be determined: A Linear model with respect to its Inputs (LI structure) is such as:
1 1 2 2 1 1 2 2 mo mo mo
y ( t,β,α u α u ) α y ( t,β,u ) α y ( t,β,u ) + = +
(1) A Linear model with respect to its parameters (LP structure) is such as:
1 1 2 2 1 1 2 2 mo mo mo
y ( t,α β α β ,u ) α y ( t,β ,u ) α y ( t,β ,u ) + = +
(2) The inverse problem consists in making the direct problem work backwards with the objective of getting (extracting) β from
mo
y ( t,β,u ) for given inputs and observations y . This is an identification
5
mo
process. The difficulty stems here from two points: (i) Measurements y are subjected to random perturbations (intrinsic noise ε ) which in turn will generate perturbed estimated values ˆ
β of β , even if the model is perfect: this constitutes an
estimation problem. (ii) the mathematical model may not correspond exactly to the reality of the experiment. Measuring the value of β in such a condition leads to a biased estimation
true
ˆ Bias E( β ) β = −
: this corresponds to an identification problem (which model η to use ?) associated to an estimation problem (how to estimate β for a given model?).
The estimation/identification process basically tends to make the model match the data (or the contrary). This is made by using some mathematical "machinery" aiming at reducing some gap (distance or norm)
mo
( ) ( , , ) = − r β y y t β u
(3) One of the obvious goal of NLPE studies is then to be able to assess the performed estimation through the production of numerical values for the variances
ˆ V( ) β obtained on the estimators (set of
estimated values parameter. This allows to give the order of magnitude of confidence bounds for the
6
estimated values parameter. This allows to give the order of magnitude of confidence bounds for the estimate). NLPE problems require the use of Non Linear statistics for studying such properties of the estimates. Because of the two above-mentioned drawbacks of MBM, the estimated or measured value of a parameter
j
β will be considered as "good" if it is not biased and if its variance is minimum.
Quantifying the bias and variance is also helpful to determine which one of two rival experiments is the most appropriate for measuring the searched parameter (Optimal design). In case of multiple parameters (vector β ) and NLPE problems, it is also helpful to determine which components of vector β are correctly estimated in a given experiment.
7
In the case of a single output signal y with m sampling points for the explanatory variable t and for a model involving n parameters, the sensitivity matrix is (
)
m n ×
defined as
k
nom mo i i j j , β pour k j
y (t ; ) S β
≠
∂ = ∂
t
β
As the problem is NL, the sensitivity matrix has only a local meaning. It is calculated for a given nominal parameter vector
nom
β
. If the model has a LP structure, this means that the sensitivity matrix is independent from β . It can be
8
If the model has a LP structure, this means that the sensitivity matrix is independent from β . It can be expressed as (Lecture 2)
1 n mo j j j
y (t, ) S (t )β
=
=∑ β
The sensitivity coefficient
j
S (t ) to the
th
j parameter
j
β corresponds to the
th
j column of matrix S .
The primary way of getting information about the identifiability of the different parameters is to analyse sensitivity the coefficients through graphical observations. This is possible only when considering reduced sensitivity coefficients
* j
S because the parameters of a model do not have in
general the same units.
k k
nom nom mo mo j j j j j j , β pour k j , β pour k j
( ; ) ( ; ) * β β β (lnβ )
≠ ≠
∂ ∂ = = = ∂ ∂
t t
y t β y t β S S
TOOL Nr1: A superimposed plot of reduced sensitivity coefficients
* j( )
S t gives a first idea
about the more influent parameters of a problem (largest magnitude) and about possible correlations (sensitivity coefficients following the same evolution). Example: Measurement of thermophysical properties of coatings through Flash method using thermal contrast principle. Case
2 n =
e1 e
2
9
1 2 1 2 1
e a K e a =
and
1 1 1 2 2 2 2
λ ρ c K λ ρ c =
Experiment A ϕ (2) a λ ρ C
p
T
A
experiment B ϕ (2 T
B
(1)
Reduced sensitivity coefficients for
1
0 1 K . = and
2
1 36 K . =
INVERSE ANALYSIS :
The Observable:
i
The model :
005 . =
i
ε
σ
The experimental noise corrupt the data:
Id cov
2 2
) ( ) var( ) ( σ ε σ ε ε = = =
i i i
E
=
n 1 i 2 i i
allows to get an estimate of β β β β via minimization
10
Sensitivities to parameters
) ( ˆ ˆ
) ( ) ( ) ( ) ( ) ( ) ( k k 1 k k k 1 k
t t
β T Y X X X β β − + =
− +
The minimization process indicates the basic tools for inverse analysis
= Matrix analysis
INVERSE ANALYSIS : Variance-covariance matrix
1 2
−
T
= ∂ ∂ = ∇ =
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
p n p p n
t T t T t T t T t T t T
t t
β β β β β β ) , ( ) , ( ) , ( ) , ( ) , ( ) , (
2 1 1 1 2 1 1
) , ( ) , (
β β β β β β T β
β β T β T X L M M M L
11
Variance-covariance matrix
1 2
−
T
≈ L L ) ˆ var( ) ˆ , ˆ cov( ) ˆ , ˆ cov( ) ˆ var( ) ˆ (
j j i j i i
β β β β β β β cov
≈ L L 1 1
ij ij
ρ ρ ) ˆ (β cor
2 ˆ 2 ˆ
) ˆ , ˆ cov(
j i ij
j i β β σ
σ β β ρ =
≈ O M M L ) var( ) , cov( ) (
j j i
β β β β cov
≈ O M M L 1
ij
ρ ) (β cor
≈ O M M L L
j j
i i
β β
β β
ˆ / ˆ var
ˆ / ˆ var
) ˆ (β Vcor
relative error
Correlation coefficients
12
TOOL Nr2: Matrix
ˆ ( )
cor
V β gives a quantitative point of view about the identifiability of the
the estimated parameters (due to the sole stochastic character of the noise, supposed unbiased). The off-diagonal terms (correlation coefficients) are generally of poor interest because of their too global character. Values very close to 1
± may explain very large variances (errors) on the
parameters through a correlation effect.
13
0.11 0.12
S=0.07
0 07 J . =
0.11 0.12
S=0.07
0 07 J . =
Parameter Local Minima
The thermal characterization of a semi-transparent material implies at least three basic parameters: the thermal diffusive characteristic time
2 d
t e a =
, the dimensionless optical thickness
τ and the
dimensionless Planck number N and so
[ ]
T d
t ,τ ,N = β
.
14
0.12 0.14 0.16 0.18 0.2 0.22 0.05 0.06 0.07 0.08 0.09 0.1 Planck Number Optical Thickness
S=0.07
Planck number N Optical thickness Optical thickness
0 07
OLS
J . =
0.12 0.14 0.16 0.18 0.2 0.22 0.05 0.06 0.07 0.08 0.09 0.1 Planck Number Optical Thickness
S=0.07
Planck number N Optical thickness Optical thickness
0 07
OLS
J . =
Level sets for
( )
OLS
J β
in the (
)
τ ,N parameter space
vector components (found using either deterministic
N°1 N°2 N°3 N°4
a (107 m²/s)
5.2 4.9 5.85 4.8
N
0.6 0.74 0.16 0.82
τ
0.38 0.5 0.076 0.56 Rr = ( )
1 N
Pl
+ τ τ
2.18 2.22 2.26 2.28
Example of local minima found ˆ
β
TOOL Nr3: In given conditions of noise and for a given model, it may be interesting to look at the level-set representation of the optimisation criterium in appropriate cut-planes (for given pair of parameters if n>3), and compare it with the minimum achievable criterium given by
2
J mσ =
.
We focus here on the reduced sensitivity matrix. This (m, n) matrix is composed of n column vectors, the
15
We focus here on the reduced sensitivity matrix. This (m, n) matrix is composed of n column vectors, the reduced sensitivity coefficients
* j
S
[ ]
j k pour , j nom j * j * n * * *
k
ith
≠
∂ ∂ = =
β
β β
t
β t η S S S S S ) ; ( w
2 1
L
These n column vectors
* j
S are in fact just the components of a set of n vectors
* j
S r
in a m-dimension vector
* 1
S r
,
* 2
S r
,…,
* n
S r
} is linearly independent only if: n j j
j n j * j j
≤ ≤ = ⇒ =
=
1 as such any for
1
α α S
Reduced sensitivity vectors
a - independent sensitivities (r = n = 2) b - dependent sensitivities c- nearly dependent sensitivities
16
Sensitivity (A.U.) Sensitivity (A.U.)
1 2 * * β βS f( S ) = −
2 3 * * β βS f ( S ) − = −
1 3 * * β βS f (S ) =
Sensitivity (A.U.) Sensitivity (A.U.)
1 2 * * β βS f( S ) = −
2 3 * * β βS f ( S ) − = −
1 3 * * β βS f (S ) = 3 *
S
1 1 2 2 * *
ˆ ˆ c c + S S
3 *
S
1 1 2 2 * *
ˆ ˆ c c + S S
Sensitivities plotted by pairs Evidence of Linear Combination between all three parameters
TOOL Nr4: The SVD of the normalized sensitivity matrix around nominal values of the parameter vector β can be advantageously calculated to get valuable information.
One way to analyse the results of the estimation process is to calculate the residuals (equation 10) at
curve
ˆ ( , ) r t β is equal to a null function:
( )
( ) ( )
( ) ( )
( )
1 1 T T T T i mo i ˆ
ˆ y y t ,
− −
− = − = − = − E r = E E S E S S S S S S S S E β β β ε ε β β β ε ε β β β ε ε β β β ε ε Since
( ) =
E ε 0 ,
( ) =
E r 0 which means that if the model used for describing the experiment is
adapted, the residuals curve is “unsigned” (unbiased theoretical model). On the contrary, "signed"
17
adapted, the residuals curve is “unsigned” (unbiased theoretical model). On the contrary, "signed" residuals can be considered as the manifestation of some biased estimation. The bias can originates from different sources and mainly: (i) the a priori decision that some parameters of the model are known and therefore fixed at some given value (maybe measured by another experiment). As authentic parameters of the PEP, they can alter the estimates of the remaining unknown parameters. (ii) Experimental imperfections which makes the model idealized with respect to the reality of the phenomena. The existence of a bias means that there exists a systematic and generally unknown inconsistency between the model and the experimental data.
An artificial bias is introduced under the form of a linear drift superimposed to the output simulated
situation before the experiment starts. A noise respecting is also added to the simulation of the measurements so that we have:
η( , ) b( ) = + + y t β t ε
Time Interval 70 s 150 s 300 s a (m²/s) 3.76.10-6 3.22.10-6 2.21.10-6 λ λ λ λ (W/m.°C) 0.031 0.064 0.084
18
λ λ λ λ (W/m.°C) 0.031 0.064 0.084
Signed character of "post-estimation" residuals in the presence
Influence of the existence of some bias on the parameter estimates for a badly conditioned problem
TOOL Nr5: The "post-estimation"residuals have to be analysed carefully to check the instance of a bias
sensor to check whether this bias may introduce too large confidence intervals of the estimates (with respect to the pure stochastic estimation of the variances of parameter estimates in the absence of any bias). Relative invariance of the estimates with respect to the identification intervals may suggest that the bias is acceptable. In the opposite case, strategies must begin either to change the nature of the estimation problems (reduce initial goals) or to use residuals to give a fait quantitative evaluation of confidence bounds of the estimates.
19
experiment A ϕ (2) a λ ρ C
p
T
A
experiment B ϕ (2) e
1
e
2
(1) T
B
0.2 0.4 0.6 0.8 1 1.2 K1 = 0.1 - K2 = 1.36 Substrate Bi-layer Contrast
Measurement of the thermophysical properties of deposits by the bilayer or "thermal contrast" technique
( ) ( ) ( ) ( ) ( )
∫
∞
− = = exp , , , dt pt t z T t z T p z
L
θ =
in in
i i i i i i i i
D C B A φ θ φ θ
i i i i i i i i i i i i i i
a pe a p = λ C a pe a p λ = B , a pe = = D A
2 2 2
sinh et sinh 1 cosh λ λ
( ) ( ) ( ) ( ) ( )
* * * * * ~ *
dt t p z,t T = z,t T = z,p ~ −
∫
∞
exp
L
θ
Laplace Transform Reduced Laplace Transform
2 2 2 *
= e t a t
2 2 2 *
= a e p p
*
p s =
0.2 0.4 0.6 0.8 1
t*=a2t/e2
2
Flash Experiment on the substrate:
D C B A =
in
= ϕ =
2 2 2 2 2 2 2 int 2 2
φ θ φ θ ϕ ϕ
2 2 2 2 2 2 2 2
sinh
2
a pe a p = C =
λ θ
( )
s s = T T ~ = ~
s
*
sinh 1
2 2 2
∞
L
θ
Flash Experiment
the bi-layer material:
= ϕ =
/
2 1 2 1 2 1 2 1
2 1
/
/ eq eq eq eq in / in /
D C B A = φ θ φ θ + +
1 2 2 1 2 1 2 1 2 2 1 1
B A B A C B A A = B A B A = B A
eq eq
With:
eq eq
D A ≠ + + + +
1 2 2 1 1 2 2 1 1 2 2 1 2 1 2 1 2 2 2 2 1 1 1 1
C B A A C A C A B A B A C B A A = D C B A D C B A = D C
eq eq eq eq
With:
C +A C A = C =
/ eq s /
/
1 2 2 1 2 1 2 1
2 1
ϕ ϕ θ + ϕ
1 1 2 2 2 2 2 2 1 1 1 1 2 1 2 / 1
cosh sinh cosh sinh = a pe a pe a p a pe a pe a p
2 2 2 2 /
λ λ θ
2 2 2 1 1 1 2 / 1
e c e c T ρ ρ + ϕ =
1/2 ∞
and
( ) ( )
+ + s a a e e s s s a a e e c c s e c e c a e =
* /
1 2 2 1 1 2 2 1 2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 2 1
cosh sinh cosh sinh 1 ρ λ ρ λ ρ ρ θ
eq eq
D A ≠
1 2 2 1 1
a a e e K =
2 2 2 1 1 1 2
c c K ρ λ ρ λ =
Let introduce now the parameters: ratio of the root of characteristic times ratio of the thermal effusivities
( ) ( ) ( ) ( )
cosh sinh cosh sinh 1 1 ~
2 2 1 * 2 / 1
+ + = s K s s s K K K K s
1 1
θ
Contrast curve:
( ) ( )
~ ~ ~ ~ ~
* *
L L
*
(depends on the thicknesses of the materials) (intrinsic to the nature of the two layers)
Contrast curve:
( ) ( )
T ~ = T T ~ ~ ~ = ~
* * * / * * /
∆ − = − ∆
L L
2 2 1 2 2 1 *
θ θ θ
( ) ( ) ( ) ( ) ( )
− + + = ∆ s s K s s s K K K K s
1 1
sinh 1 cosh sinh cosh sinh 1 1 ~
2 2 1 *
θ
2 2 2 1 1 1 2 1 3
e c e c K K K ρ ρ = =
1 2 2 1 2 1 4
λ λ e e K K K = =
thermal capacities ratio thermal resistances ratio In all cases, the corresponding substrate properties must to be known
Case 1 : Case 2 :
Conductive coating / Insulating substrate Insulating film / Conductive substrate
Thickness (µ µ µ µm) a (m2/s) λ λ λ λ (W/m.°K) ρ ρ ρ ρCp (J/m3.°K) Case 1 : Aluminium coating on a Cobalt/Nickel substrate Film (1) 220 9,46.10-5 230 2,43.106 Substrate (2) 1 100 2,36.10-5 84,5 3,57.106 C . W/m h ° =
2
10 1 10 . 3 , 1
4 2 2
<< = =
−
λ he Bi Case 2 : Insulating film on a Alumina substrate Film (1) 247 6,84.10-7 2,23 3,26.106 Substrate (2) 640 7,47.10-6 23 3,08.106 C . W/m h ° =
2
10 1 10 . 8 , 2
4 2 2
<< = =
−
λ he Bi
0.2 0.4 0.6 0.8 1
0.2 0.4 0.6 0.8 1 1.2 t*=a2t/e2
2
K1 = 0.1 - K2 = 1.36 Substrate Bi-layer Contrast 1 2 3 4 5
0.2 0.4 0.6 0.8 1 1.2 t*=a2t/e2
2
K1 = 1.28 - K2 = 0.32 Substrate Bi-layer Contrast
0.5 1 K1 = 1.28 - K2 = 0.32 Constrast Curve Sensitivity - K1 Sensitivity - K2
0.05 0.1 0.15 0.2 K1 = 0.1 - K2 = 1.36 Constrast Curve Sensitivity - K1 Sensitivity - K2
Sensitivity Study Through 2 Examples ( )
( )
1 2 −
= X X K Cov
t c
σ
Case 1 : Conductive coating / Insulating substrate Case 2 : Insulating coating / Conductive substrate
1 2 3 4 5
t*=a2t/e2
2
0.2 0.4 0.6 0.8 1
t*=a2t/e2
2
Variance-Covariance
28.0302 -35.9846
Variance-Covariance
0.1067 3.1409 3.1409 99.1677
Correlation
1.0000 -0.9952
Correlation
1.0000 0.9655 0.9655 1.0000
Case 1 Case 2
Covariance & Correlation Matrices
1
K
is closed to unity
( ) ( ) ( ) ( )
s K s K s s K
1 1 1
cosh sinh ~ cosh sinh −
( ) ( ) ( )
− = ∆ s s K s s
1 *
sinh 1 cosh sinh 1 1 ~ θ
Contrast Curves
1 =
N
σ 1000 = Npt
0.05 0.1 0.15 K3 = 0.136 - K4 = 0.073529 Constrast Curve Sensitivity - K3 Sensitivity - K4
Case 1 : Conductive coating / Insulating substrate Case 2 : Insulating coating / Conductive substrate
Introduction of a new set of parameters : (
)
4 3, K
K
0.5 1 K3 = 0.4096 - K4 = 4 Constrast Curve Sensitivity - K3 Sensitivity - K4 0.2 0.4 0.6 0.8 1
t*=a2t/e2
2
1 2 3 4 5
t*=a2t/e2
2
Variance-Covariance
2.6921 -18.5189
Correlation
1.0000 -0.9346
Case 1
( ) ( ) ( ) ( ) ( )
− + + = ∆ s s K s s s K K K K s
1 1
sinh 1 cosh sinh cosh sinh 1 1 ~
2 2 1 * s
θ
( ) ( )
− − 1 ~ cosh ~ sinh
1 1 1
s K s K s K
( ) ( ) ( )
− + + = ∆ s s s s. K K s sinh 1 sinh cosh 1 1 ~
3 3 * s
θ
Variance-Covariance
103.5845 -97.1801
Correlation
1.0000 -0.9999
Case 2
1
K
is closed to zero Sensitivity Curves
Bi-layer material : P.V.C deposit / Steel substrate
Example
Thickness (mm) a (m2/s) λ λ λ λ (W/m.°K) ρ ρ ρ ρCp (J/m3.°K) Case: P.V.C deposit on a Steel substrate Film (1) 1 1,21.10-7 0.19 1,57.106 Substrate (2) 5 8,33.10-6 30 3,60.106 Substrate (2) 5 8,33.10 30 3,60.10 Nominal values 66 . 1
1 =
K 052 .
2 =
K 086 .
3 =
K 92 . 31
4 =
K
2 5
. 10 . 5 m K/W Rc
−
=
( )
3 ,
2 2 *
= = λ e R R
c c
Case 2: Insulating coating / Conductive substrate
2 4 6 8 10 0.2 0.4 0.6 0.8 1 1.2 1.4 K1 = 1.6594 - K2 = 0.052549 Substrate Bi-layer Contrast 2 4 6 8 10 0.2 0.4 0.6 0.8 1 1.2 1.4 K1 = 1.6594 - K2 = 0.052549 - Rc
* = 0.3
Substrate Bi-layer Contrast
Bi-layer material : P.V.C deposit / Steel substrate
Example
2 4 6 8 10
0.5 1 K1 = 1.6594 - K2 = 0.052549 t*=a2t/e2
2
Constrast Curve Sensitivity - K1 Sensitivity - K2 2 4 6 8 10 t*=a2t/e2
2
Contrast Curves Sensitivity Curves
2 4 6 8 10 t*=a2t/e2
2
2 4 6 8 10
0.5 1 t*=a2t/e2
2
K1 = 1.6594 - K2 = 0.052549 - Rc
* = 0.3
Constrast Curve Sensitivity - K1 Sensitivity - K2 Sensitivity - Rc
*
0.2 0.4 0.6 0.8 1 Temperature T,°C K1 = 1.66 - K2 = 0.05 Simulated Thermogram Theoretical Thermogram Residuals x 4
(Perfect Contact)
0.2 0.4 0.6 0.8 1 Temperature T,°C K1 = 1.6582 - K2 = 0.047585 Simulated Thermogram Theoretical Thermogram Residuals x 4
Nominal Values :K1 = 1.66 – K2 = 0.05
Example
2 4 6 8 10
Time t*=at2/e2
2
2 4 6 8 10
Time t*=at2/e2
2
As predicted by theory, without noise we exactly find the nominal values used for the simulation Estimation with noise
1
K
As predicted by theory, the more sensitive parameter in this case is the parameter
Variance-Covariance x
N 2
σ
0.1670 10.9443 10.9443 745.4980
Correlation
1.0000 0.9809 0.9809 1.0000
( )
% 41 . cov 0067 . 6582 . 1
1 * 1 1
1
= = ± = K K K
N K
σ σ
( )
% 27 cov 013 . 047585 .
2 * 2 2
2= = ± = K K K
N K
σ σ
Estimation without noise
Optimization of the experiment
Can the parameter estimation be improved by a change of parameters Note on the change of parameters
( )
b a K
K ,
Let introduce now a new couple of parameters:
( ) ( )
b a K
K t f K K t f , , , ,
2 1
= = ∆θ
( )
, K K F K =
∂ ∂ F F
a a
The new parameters introduced are function of the old ones:
( ) ( )
2 1 2 1
, , K K F K K K F K
b b a a
= =
So it is for the Sensivity and Covariance matrices:
1 12. −
= J X X ab
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
t b b a b a a
J K Var K K Cov K K Cov K Var J K Var K K Cov K K Cov K Var . , , . , ,
2 2 1 2 1 1
=
Sensitivity Covariance
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 1 2 1 1 2 2 2 2 1 1 1 2 1 2 1 2 2 2 1 2 1 2 1 2 1 2 2 2 1 2 1
, Cov Var Var , Cov , Cov 2 Var Var Var , Cov 2 Var Var Var K K b a b a K b a K b a K K K K b b K b K b K K K a a K a K a K
b a b a
+ + + = + + = + + =
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 1 2 1 1 2 1 2 1 2 2 2 1 2 1 1
, Cov Var , Cov , Cov 2 Var Var Var Var Var K K b K b K K K K b b K b K b K K K
b a b a
+ = + + = =
( ) ( )
2 1 1 1
, = K K F K K K F K
b b a a
= =
The standard-deviation of a given parameter does not depend on the choice of the second parameter
= ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =
2 1 2 1 2 1 2 1
b b a a K F K F K F K F J
b b a a
( )
4 3, K
K
Let consider now an estimation with:
Variance-Covariance
767.6456 -745.4210
Correlation
1.0000 -1.0000
( )
% 27 cov 022 . 078902 .
3 * 3 3
3= = ± = K K K
N K
σ σ
( )
% 27 cov 3757 . 9 8478 . 34
4 * 4 4
4= = ± = K K K
N K
σ σ
Nominal Values : K3 = 0.086 – K4 = 31.92
2 4 6 8 10
0.2 0.4 0.6 0.8 1 Temperature T,°C K3 = 0.078902 - K4 = 34.8478 Simulated Thermogram Theoretical Thermogram Residuals x 4 2 4 6 8 10
0.5 1 K3 = 0.083 - K4 = 33.2 t*=a2t/e2
2
Constrast Curve Sensitivity - K3 Sensitivity - K4 2 4 6 8 10 Time t*=at2/e2
2
As expected, the two parameters are not well estimated
0476 , 848 , 34 / 0789 , /
4 3 2
= = = K K K 6581 , 1 848 , 34 . 0789 , .
4 3 1
= = = K K K
We exactly find the same values for (
)
2 1, K
K
( )
2 1, K
K
as those obtained with the estimation in
31
( )
( )
t T C P T grad div
p ∂
∂ = + ⋅ ρ λ Transient Heat Transfer Equation :
Cylindrical Coordinate System :
Boundaries Conditions :
t T a P r T r r T ∂ ∂ = + ∂ ∂ + ∂ ∂ 1 1
2 2
λ
0<r<r1 : Hot-Wire r1<r<r2 : Medium
Assumptions :
1 1
1
Φ = ∂ ∂ −
=r r
r T S λ
2 2
2
Φ = ∂ ∂ −
=r r
r T S λ Boundaries Conditions : Initial Condition :
ext
T T =
Theoretical Model : “quadrupole approach”
( ) ( ) ( ) dt
pt t r p r − = ∫
∞
exp , , ~
0 θ
θ Laplace Transform :
(numerical inversion)
Quadrupole Formulation : Representation in a “T ” - Quadrupole Form :
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 1 1 1 1 2 1 2 1 1 1 1 1 2 1 2 1 2 1 2 2 1 2 1 1 1 2 1 2
2 2 1 kr I kr K kr I kr K kr D kr I kr K kr I kr K r r lk C kr I kr K kr I kr K l B kr I kr K kr I kr K kr A ⋅ + ⋅ = ⋅ − ⋅ − = ⋅ − ⋅ = ⋅ + ⋅ = πλ πλ
( ) ( ) ( ) ( )
Φ = Φ p r p r D C B A p r p r , ~ , ~ , ~ , ~
2 2 2 2 1 1 1 1
θ θ C Z et C D Z C A Z 1 1 , 1
3 2 1
= − = − =
Transfer Matrix : M → Waterfall Setting
( ) ( )
1 1 2 1 2 1
3 2 2 1 1 1 1 1
→ = → = ⋅ =
∞ ∞ ∞
Z kr l Z kr K kr kr K l Z πλ πλ Semi-infinite Medium : (r2>>1) Asymptotic Model : p→ 0 et r1<<1
Bessel’s Functions Approximation :
( ) ( ) ( ) ( ) ( )
= Γ ⋅ − − −
→ → 1 1 1 1 1 1
1 2 1 2 ~ lim ln ~ lim
1 1
kr kr kr K kr kr K
kr kr
( )
1 ~
ln ~ ~ Φ − = Φ = kr Z θ
3
→ =
∞
C Z
Quadrupole Formulation
( )
1 1 1 1 1
~ 2 ln ~ ~ Φ − = Φ =
∞
l kr Z πλ θ
Response to a Step Stimulation :
p
1 1
~ Φ = Φ
( ) ( )
ste
C t l t r + ⋅ Φ = ln 4 ,
1 1 1
λ π θ
( )
1 2 1 1
ln 4 t t l ⋅ Φ = ∆ λ π θ
Hot-Wire effects :
( )
p G V d r r dr rl V
m r r m 1 2 1 1
1 1
2 2 1 ⋅ = Φ ⋅ = ⋅ =
θ π θ θ
New definition of inner parameters : Asymptotic expansion : p→ 0 et r1<<1
Contact Resistance :
m
θ ~
m
Φ ~
1
~ θ
1
~ Φ
C R
c
R
∞ 1
Z
Hot-Wire Contact Resistance Semi-infinite Medium
Capacity p C l r 1 Z Resistance l 8 1 Z Z
p 2 1 3 2 1
→ ⋅ ρ ⋅ π = → πλ = =
m
θ ~
m
Φ ~
1
~ θ
1
~ Φ
C Z =
3
R Z =
2
Finite Medium (Heat Losses Effects) :
m
θ ~
m
Φ ~
1
~ θ
1
~ Φ
C R
c
R
Hot-Wire Finite Medium Heat Losses Contact Resistance
1
Z
2
Z
3
Z
hS 1
α Ideal Case
−”Hot-Wire" Thermal Conductivity −”Hot-Wire" Thermal Diffusivity −"Medium" Thermal Conductivity −"Medium" Thermal Diffusivity − Contact Resistance ”Hot-Wire / Medium” − ”Convective" resistance (Heat Losses) Parameters :
Sensitivities : Correlation Factor :
( )
i i
K t t X ∂ ∂ ) , F( = , K K
et
( ) ( )
i i i i i
K t K t X K t X ∂ ∂ ) , F( = =
*
K
« reduced »
( ) ( )
( )
( )
j i j i j i
K K K K K K Var Var , Cov = , ⋅ ρ
Semi-Infinite Medium
44
Liquid Walls T(t)
e e h
Flash Lamps Liquid Walls T(t)
e e h
Flash Lamps
requires to work in a “pseudo-conduction” regime
Liquid Walls
1
r
2
r
T(t) Liquid Walls
1
r
2
r
T(t)
“pseudo-conduction” regime (choose the aspect ratio of the measurement cell e/h<<1) Principle of the Measurement
45
Walls
h
Walls
h
hS 1
w w w w
D C B A
l l l l
D C B A
hS 1
( )
p φ
( )
p θ
w w w w
D C B A
hS 1
w w w w
D C B A
l l l l
D C B A
hS 1
( )
p φ
( )
p θ
w w w w
D C B A
Quadrupole Representation
hS 1
: Convective Heat Losses
Liquid T(t)
x h h
Impulsed Flux Liquid T(t)
x h h
Impulsed Flux
= =
i i i i
a pe D A
2
cosh =
i i i i i
a pe a p S B
2
sinh 1 λ =
i i i i i
a pe a p S C
2
sinh λ
: thickness of the material : thermal diffusivity : thermal conductivity
i
e
i
a
i
λ
and ,
46
The rear-face temperature θ(p) is given by:
2
) ( 2 hs hS p p B A + + =C φ θ
A, B and C represent the coefficients of the transfer matrix:
=
w w w w l l l l w w w w
A C B A A C B A A C B A D C B A
With: ( ) ( )
w l w l w w l w l w
C A B B A A C B A A + + + = A
( ) ( )
w l w l w w l w l w
A A B B A B C B A A + + + = B
( ) ( )
w l w l w w l w l w
C A A B C A C A A C + + + = C
( )
Q p = φ
For a Heat Pulse (Dirac of Flux) →
p t T θ
L =
47
el 5 , 4 =
1 2.
. 5
− −
= K m W h
1 1.
. 597 ,
− −
= K m W
l
λ
1 2 7
. 10 . 43 , 1
− −
= s m al
1 1.
. 132 ,
− −
= K m W
l
λ
1 2 8
. 10 . 33 , 7
− −
= s m al
1 1.
. 395
− −
= K m W
w
λ
1 2 4
. 10 . 15 , 1
− −
= s m aw mm
ew 2 5 , =
4
. 10 . 4
−
= m J S Q
48
4 Unknown Parameters:
h and S Q e a e
l l l l
= = = =
4 3 2 1
, , β β λ β β
β β β β β , , , , ,
4 3 2 1
t f t f T = =
Reduced Sensitivity Coefficient:
Xj
* maximum → small error
Xj
* proportional → parameters are correlated
4 3 2 1
β β β β , ,
*
t T t X
j j j
∂ ∂ =
49
being the noise at time t
( ) ( ) ( ) ( )
( ) ( )
t X X X
t n n t n n n
ε β β
1 1
ˆ
− +
+ =
t ε
: expected values of parameters (unbiased estimator)
( )
β β = ˆ E
( )
( )
( )
( ) ( ) ( )
= =
− j i i t
Cov Var X X V β β β σ σ β , ˆ
2 1 2
ε ,β t T Y + =
− = ∑
= n i i i
,β t T Y S
1 2
β
( ) ( ) ( ) ( )
j n i i i j i
,β t T Y t T S β β β β β ∨ = − ∂ ∂ = = ∂ ∂
=
,
1
Good Estimation if
( )
( ) ( ) ( ) ( )
= =
j j i j i i n t n
Var Cov X X V β β β σ σ β , ˆ
2 2
: covariance matrix (σn : standard deviation of noise)
( ) ( )
( )
( )
j i j i j i
β β β β β β ρ Var Var , Cov = , ⋅
Large Sensitivities Correlation coefficients are far from unity
50
i i i
ε ,β t T Y + =
Variances are small
The estimation problem is non-linear → the estimation depends on the nominal values of the parameters → an optimal walls thickness exists
51
Water – 0,5 mm Water – 2 mm
0.3394
2.4913 -17.4179 18.4144 10.4120 1.4724 -9.4267 10.4120 9.7216 0,3218
0,7528 -2,0146 1,7770 -1,3092
Oil – 0,5 mm Oil – 2 mm
0,0649
0,2533 -1,1408 0,9958 0,4599 0,1216 -0,4388 0,4599 0,3979 0,1920
0,1500 -0,2825 0,1413 -0,0219
Variance-Covariance Matrix
52
New Parameters:
h and S Q e c a e
l l l l
= = = =
4 3 2 1
, , β β ρ β β
Water – 1 mm
4 parameters: β1, β2 , β3 and β4 3 parameters (β2 fixed): β1, β3 and β4
Covariance Covariance
0.2567 1.5697 1.0776 0.0993 1.5697 9.8171 6.6809 -0.2249 1.0776 6.6809 4.5673 0.1590 0.0993 -0.2249 0.1590 4.9007 0.0057 0.0094 0.1353 0.0094 0.0208 0.3121 0.1353 0.3121 4.8955
Correlation Correlation
1.0000 0.9888 0.9952 0.0886 0.9888 1.0000 0.9977
0.9952 0.9977 1.0000 0.0336 0.0886 -0.0324 0.0336 1.0000 1.0000 0.8596 0.8074 0.8596 1.0000 0.9777 0.8074 0.9777 1.0000
53
Fluid (Water) [ef =4,5 mm, λf=0,597 W.m-1.K-1, al=1,43.10-7 m2.s-1, ρcl=4,17.106 J.m-3.K-1 Fluid (Oil) [ef =4,5 mm, λf=0,132 W.m-1.K-1, al=7,33.10-7 m2.s-1, ρcl=1,8.106 J.m-3.K-1 ] Walls (Copper) [λw=395 W.m-1.K-1, aw=1,15.10-4 m2.s-1, ρcw=3,43.106 J.m-3.K-1 ] h=5 W.m-2.K-1 – Q/S=4.104 J.m-2
54
Estimation Program: Levenberg-Marquardt Algorithm with 4 parameters Standard deviation of the noise: σn= 0.005 K
h and S Q e c a e
l l l l
= = = =
4 3 2 1
, , β β ρ β β
Fluid (Water) [ef =4,5 mm, λf=0,597 W.m-1.K-1, al=1,43.10-7 m2.s-1, ρcl=4,17.106 J.m-3.K-1 Fluid (Oil) [ef =4,5 mm, λf=0,132 W.m-1.K-1, al=7,33.10-7 m2.s-1, ρcl=1,8.106 J.m-3.K-1 ] Walls (Copper) [λw=395 W.m-1.K-1, aw=1,15.10-4 m2.s-1, ρcw=3,43.106 J.m-3.K-1 ] h=5 W.m-2.K-1 – Q/S=4.104 J.m-2
55
4 parameters: , , and Water Oil Parameters Parameters
Nominal al=1,43.10-7 m2.s-1 ρcl=4,17.106 J.m-3.K-1 h=5 W.m-2.K-1 Q/S=4.104 J.m-2 Nominal al=7,33.10-7 m2.s-1 ρcl=1,8.106 J.m-3.K-1 h=5 W.m-2.K-1 Q/S=4.104 J.m-2 Estimated al=7,284.10-7 m2.s-1 ρcl=1,827.106 J.m-3.K-1 h=5,014 W.m-2.K-1 Q/S=4,026.104 J.m-2 Estimated al=1,417.10-7 m2.s-1 ρcl=4,276.106 J.m-3.K-1 h=5,083 W.m-2.K-1 Q/S=4,071.104 J.m-2
l l
a e
h
S Q
l le
c ρ
Covariance Covariance
0.2604 1.6099 1.1144 0.1305 1.6099 10.1769 6.9852 -0.0272 1.1144 6.9852 4.8152 0.2932 0.1305 -0.0272 0.2932 4.8916 0.0885 0.4597 0.1814 -0.0129 0.4597 2.5109 0.9409 -0.2640 0.1814 0.9409 0.3747 -0.0099
Correlation Correlation
1.0000 0.9890 0.9952 0.1156 0.9890 1.0000 0.9978 -0.0038 0.9952 0.9978 1.0000 0.0604 0.1156 -0.0038 0.0604 1.0000 1.0000 0.9753 0.9962
0.9753 1.0000 0.9700 -0.2642 0.9962 0.9700 1.0000 -0.0257
% 5 , =
a
σ % 6 , 1 =
c ρ
σ % 8 , =
c ρ
σ % 3 , =
a
σ
56
4 parameters ( fixed): , and Water Oil Parameters (ρcl=4,17.106 J.m-3.K-1) Parameters (ρcl=1,8.106 J.m-3.K-1)
Nominal al=1,43.10-7 m2.s-1 h=5 W.m-2.K-1 Q/S=4.104 J.m-2 Nominal al=7,33.10-7 m2.s-1 h=5 W.m-2.K-1 Q/S=4.104 J.m-2 Estimated al=7,323.10-7 m2.s-1 h=5,022 W.m-2.K-1 Q/S=4,005.104 J.m-2 Estimated al=1,428.10-7 m2.s-1 h=5,084 W.m-2.K-1 Q/S=4,005.104 J.m-2
l l
a e
h
S Q
l le
c ρ
Q/S=4.10 J.m Q/S=4.10 J.m Q/S=4,005.10 J.m
Covariance Covariance
0.0058 0.0095 0.1347 0.0095 0.0211 0.3117 0.1347 0.3117 4.8196 0.0044 0.0092 0.0354 0.0092 0.0223 0.0888 0.0354 0.0888 0.3651
Correlation Correlation
1.0000 0.8609 0.8089 0.8609 1.0000 0.9779 0.8089 0.9779 1.0000 1.0000 0.9339 0.8879 0.9339 1.0000 0.9840 0.8879 0.9840 1.0000 Q/S=4,005.10 J.m
% 08 , =
a
σ % 06 , =
a
σ
57
4 parameters: , , and
Covariance
0.1453 0.6414 0.3682 0.0941 0.6414 2.9094 1.6528 0.1704
l l
a e
l le
c ρ s Q
h
0.6414 2.9094 1.6528 0.1704 0.3682 1.6528 0.9452 0.1949 0.0941 0.1704 0.1949 1.6610
Estimation on an Experimental Thermogram (Water) – 4 Parameters Model
Residuals x 10
Small bias Very little noise ± = ± =
−
K m J c s m a . / 10 . 170 , 38 , 4 / 10 . 025 , 42 , 1
3 6 2 7
ρ
58
3 parameters: , and
Covariance
l l
a e s Q h
C . J/m 4,15.10
3 6
° to fixed is
l le
c ρ
Estimation on an Experimental Thermogram (Water) – 3 Parameters Model
Residuals x 10 0.0039 0.0039 0.0557 0.0039 0.0064 0.0959 0.0557 0.0959 1.5606
s m a / 10 . 004 , 45 , 1
2 7 −
± = Small bias
59
Estimation by a non-linear O.L.S method (Levenberg-Marquardt)
Bias is reduced by using 4 thermocouples
60
61
Principle of the experiment
62
High Insulating Material : λ =0,02 W.m-1.K-1, ρc = 5000 J.m-3.K-1, a= 4.10-6 m2.s-1
( )
t T ∂
( )
λ λ ∂ ∂ t T 2
2D Model sensitivity Study – (T1 is assumed as being known) ensitivties
( )
a t T a ∂ ∂
2
( )
h t T h ∂ ∂
2
High Accurate Estimation of λ λ λ λ is theoretically possible 2D Model
2D Model Reduced Sen 63
150 s Theoretical and experimental thermograms Residuals x 4
64
70 s 300 s
Time Interval 70 s 150 s 300 s a (m²/s) 3.76.10-6 3.22.10-6 2.21.10-6 λ λ λ λ (W/m. C) 0.031 0.064 0.084
Rigid Foam : a=4,68 to 4,54.10-7 m²/s and λ λ λ λ=0,039 to 0,042 W/m. C
65
66
( ) (
− = − = + =
r r β c r c r c r
β β e β β b X X β , β t, F β , β t, F
r
ˆ ˆ ~
We have: (1)
c c b c r r r c r nom c r
X + X + , t, F = , t, F
nom
β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β
β β β β
− −
∧ ∧
( ) (
− = + =
c c b c r c r c r
β β e X X β , β t, F β , β t, F
nom c nom
ˆ
With:
r r β
β β b
r
− = ˆ
: « bias on estimated parameters »
c c β
β β e
nom c
− =
: « error on fixed parameters »
c r β
, β t, F
: Detailed model
nom
c r β
, β t, F ˆ ~
: Reduced modelor biased model (1)
67
Sensitivity expressions:
To “unknown” parameters :
( )
c r
t F X β β ∂ = , ,
To “known” parameters:
( )
r c r r
t F X β β β ∂ ∂ = , ,
c c r c
t F X β β β ∂ ∂ = , ,
Relation (1) shows that:
r r c r r
X t F X
nom
= ∂ ∂ = β β β ˆ , ˆ , ~ ~
68
“Unknown” Parameter Estimation: O.L.S Method
− =
nt c r i c r i
t F t F S
2
, ˆ , ~ , , β β β β
=
− =
i c r i c r i
nom
t F t F S
1
, ˆ , , , β β β β
, ˆ , ~ , , . ~ ˆ = − ⇒ = ∂ ∂
nom
c r c r r r
t F t F X S β β β β β
t r
With:
nom
c r c r
t F t F Y β β β β , ˆ , ~ , , − =
Matrix formulation:
69
− = − = + =
c c b r r c r c r c r
nom c r nom
e b X X t F t F β β β β β β β β
β
ˆ , , , ˆ , ~
. ~ = Y X
t r
c r
b c t r r t r t r
β
c r
b c t r r t r
1 −
β
Determination of bias is not possible because is unknown !!!
70
c
b c e
X .
c
b c t r r t r r
−1
Can be estimated from the Residuals curve:
c
b c e
X .
c
r r
Assumptions:
c
b c t r r t r r
−1
71
Linear System to Solve …
“Known” “Known”
Principle of the “Flash” method
t T a x T ∂ ∂ = ∂ ∂ 1
2 2
( )
= ∂ ∂ = − = ∂ ∂ = = =
in x 0, in x 0, at t
e
hT x T t hT x T T λ ϕ λ
Q h
e
BC et IC: Heat Eq.: Solution if given by:
= t e a he f T , ,
2
λ
Two parameters:
10 , = Fo 05 0, Bi =
Nominal Values :
h
72
β β β β β β β β ≈ ˆ
Unbiased Model
( )
β β β β β β β β β β β β β β β β
β β β β
X + t, T = t, T − ˆ ˆ
t X X X + =
t
t
ε ε ε ε β β β β β β β β ˆ
β β β β β β β β = E ˆ
β β β β β β β β = E
t 2 b
X X V σ σ σ σ β β β β = ˆ
( ) ( )
β β β β β β β β ˆ , t , t t
i i i
T Y r − =
( )
= r E
2 b
r V σ σ σ σ =
Residuals
73
( )
049 099 , , ˆ = E β β β β = 05 10 , , , , , , , , β β β β
β β β β β β β β = E ˆ ( )
=
− − 9 9
10 907 10 024 . , . , ˆ V β β β β
( )
=
− − − 9 9 1
10 900 10 024 . , . , X X
t 2 b
σ σ σ σ
t 2 b
X X V σ σ σ σ β β β β = ˆ
t X X X + =
t
t
ε ε ε ε β β β β β β β β ˆ
( )
5
10 989 9
−
= . , r V
4
10 1
−
= .
2 b
σ σ σ σ
2 b
r V σ σ σ σ =
β β β β β β β β ˆ , t , t t
i i i
T Y r − =
Residuals
( )
21
10 902 5
−
= . , r E
= r E
74
( )
( )
1 2
ˆ
−
= X X V
t r
σ β
( ) ( )
t c c r t r r
b b X X V
β β
σ β + =
−1 2
ˆ
Parameter Estimation Improvement Measurement noise Reduction Number of parameters Reduction Introduction of a bias But
Variance calculated from true values
( )
r r
E β β = ˆ
( )
c r r
b E
β
β β + = ˆ
2
ˆ
c c
t t r m r r
V X X b b
β β
σ
β
= +
r t r 2 b r
X X V σ σ σ σ β β β β
= = = =
∧ ∧ ∧ ∧
t 1
t r 2 b r m
c c b
b X X V
β β β β β β β β
σ σ σ σ β β β β + = ∧
Variance calculated from the biased parameters
75
c r β
β β β β β β β β β β β =
( )
c r X
X X =
= − T Y X
t r
( )
( )
e X X + , t, T = , t, T
c nom
b r r c r c r c r
−
∧ ∧
β β β β β β β β β β β β β β β β β β β β β β β β
c cnom
c
e β β β β β β β β
β β β β
− =
c
e X X X X X X X + =
c t r
r t r t r
r t r r r β β β β
ε ε ε ε β β β β β β β β − ˆ
Errror on the assumed “known” parameters:
76
c
b c t r r t r r
e . X . Id X ~ . X ~ X ~ . X ~ r − =
−1
Matrix determinant versus Matrix Rank Matrix inversion can not be implemented (Determinant = 0)
77
c r
b c t r r t r
1 −
β
We have previously shown:
78
In this case,
In this case,
=
c
b
e
=
r
bβ
≠
c
b
e
≠ r
r
bβ
. ~ =
c t r X
X
r
bβ
. ~ ≠
c t r X
X
c
b c e
X r . − =
= r
r c
b X e X r
r b c β
. . − − =
] [
1
t −
[ ]
2
t − Time intervals truncated to time t1 and t2 will be denoted :
2 1 1 2 2 1
/ 1 1 1 1 1 1 2 2
~ ~ ~ ~ ~
t t t t r t r t t t r t r r t r r t r
X X dt X X X X X X
→ → → =
− + =
Bias: Approximation:
5000 6000 7000 8000
c r c r
b c t r r t r r r b c t r r t r r r
e X X X X b e X X X X b . ~ . ~ ˆ . ~ . ~ ˆ
2 2 1 2 2 1 1 1 1 1
2 2 1 1
− −
− = − = − = − = β β β β
β β
c c r
b c t r r t r b c t r r t r r r
e X X X X e X X X X b . ~ . ~ . ~ . ~ ˆ ˆ
1 1 1 1 1 2 2 1 2 2
1 2 1 2
− −
+ − = − = ∆
−
β β
β
2 1 1 2
→
c r
b c t r r t r r r
e t X t X X X b . ~ . ~ ˆ ˆ
2 2 1 1 1
1 2 1 2
−
− = − = ∆
−
β β
β
Bias: Bias variation:
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100 200 300 400 500 600 700 800 900 1000 1000 2000 3000 4000 5000
Cumulative Norm
2 1
1
. ~ . ~ ˆ ˆ n n e t X t X X X b
t t
− − = − = ∆
−
β β
Setting: Bias difference can be written:
1 2 1 1 1
. ~ . ~ ˆ ˆ
1 2 1 2
n n e t X t X X X b
c r
b m c m t r r t r r r
− − = − = ∆
−
−
β β
β
1 2 1 1
~ ~ . ˆ ˆ .
1 2
n n t X X X e t X
m t r r t r r r b m c
c
− − − = β β
m r b m c m
t X e t X t Y b
c r
. − − =
β
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r c
r c β β
7
Bias Estimation using a time varying estimation
1.2 0.1021 1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 Time t,s yref Xr Xc 1 2 3 4 5 6 7 8 9 10
0.2 0.4 0.6 0.8 1 1.2 Time t,s Experimental Thermogram Simulated Thermogram Residuals x 10
1021 . ˆ =
r
β 03 . =
c
β
Estimated /fixes values: Nominal Values:
05 . =
c
β 10 . =
r
β
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Bias Estimation using a time varying estimation
β ˆ
0.1025
βr = F (Time Interval)
r
β ˆ
100 200 300 400 500 600 700 800 900 1000 0.1 0.1005 0.101 0.1015 0.102 # points
is a function of the time interval length It exhibits the presence of a bias
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r
β
b
Bias Estimation using a time varying estimation
0.02 Bias (Estimated Value) - bβr = 0.0021073 Xc*e 100 200 300 400 500 600 700 800 900 1000
0.005 0.01 0.015 # points Xc*ebc Xc(tm)*ebc Xc*ebc calculé
Estimated Bias = 0.0021073 Efficient technique Expected Bias = 0.0021
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How to chose the optimal length of the time interval ?
No General Rules in the Case of a Non-Linear Problems but a
Different Aspects specific to Non-Linear Problems must be
The Using of a Reduced Model is a Solution but a particular
Bias can be estimated from “Known” Quantities (Sensitivity
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