Experimental identification of transfer functions for diffusive - - PowerPoint PPT Presentation
Experimental identification of transfer functions for diffusive and/or advective heat transfer for linear time invariant dynamical systems Denis Maillet U niversity of L orraine & CNRS, Nancy, France U niversity of L orraine & CNRS,
Denis Maillet University of Lorraine & CNRS, Nancy, France
Experimental identification of transfer functions for diffusive and/or advective heat transfer for linear time invariant dynamical systems
University of Lorraine & CNRS, Nancy, France Laboratoire d'Energétique et de Mécanique Théorique et Appliquée (LEMTA)
New Trends in Parameter Identification for Mathematical Models IMPA, Rio de Janeiro, Brazil, October 30 – November 3, 2017 Contribution: Waseem Al Hadad
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
2
Experimental inverse problems in heat transfer and engineering METTI Group, SFT (French Heat Transfer Society) Recently: interest in convolutive models and associated inverse problems
* Pollutant source identification in a ventilated domain (turbulence, transient concentration measurements) * Transient thermal behaviour of heat exchanger (PhD W. Hadad, Fives Cryo postDoc) * Virtual sensor construction in a furnace under vacuum conditions (PhD T. Loussouar, Safran Group)
(LTI) coefficients
2.1 Case of a heat exchanger 2.2 Experimental Impedance/transmittance estimation for a half heat exchanger
3.1 Reference case: 1D transient conduction
3
3.1 Reference case: 1D transient conduction 3.2 Noisy matrix and Total Least Squares 3.3 Comparison of calibration methods
4.1. Point versus averaged values for input and unknown 4.2 Rectangular deconvolution (using « stairs » parameterizing) 4.3 Rectangular estimation with n < m non uniform (NU) time steps
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
solid - Ω3
Flowing fluid
Material multicomponent system = K solid or fluid domains solid fluid
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
4
Assumptions: time constant thermophysical properties and velocity field
solid - Ω1 solid - Ω4 solid - ΩK
fluid ΩK -1
solid - ΩK-2
Flowing fluid Ω2
Initial uniform state or steady state temperature field + one single separable thermal excitation
solid - Ω1 solid - Ω3 solid - Ω4
Flowing fluid ΩK -1 Flowing
) (t Qv
s
P
init s s
T t T t Q ≠ ) (
) (
init
T t T ≠
∞ )
(
s
P
(P) h
s
P
) ( ) ( t T c m t Q
in b in in in
& =
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
5
Fixed geometrical support:
solid - ΩK solid - ΩK-2
Flowing fluid Ω2
Time part of thermal excitation u (t) (starts at time t = 0) :
) (t T in
b
init
T t T ≠
∞ )
(
) (
) ( t T t Q
s s
) (t Qv
P
t coefficien transfer heat (P) v : field velocity (P) : heat volumetric (P) : ty conductivi thermal c r ρ λ
Change of perspective: one single heterogeneous fluid in one single domain (if solid part : zero velocity)
) (t Qv
s
P
init s s
T t T t Q ≠ ) (
) (
init
T t T ≠
∞ )
(
s
P
(P) h
) ( ) ( t T c m t Q
in b in in in
& & =
6 6
) (P, ) ( t T t y ≡
P
(P) : ) boundaries (external t coefficien transfer heat h
Transient separable thermal excitation :
u (t)
) (P, ) ( t T t y ≡
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
Point response at any point P :
Physical system: Set of solids AND fluid(s): 3D forced convection with constant velocities (in time but not in space) P = ANY point in the system
Recap:
One single thermal excitation defined by its support
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
( ) ( )
(P) ) ( ) (P (P) ) (P (P) P ) (P P
source
f V t Q t , T t , T u c t , t T c
v
+ ∇ ∇ = ∇ + ∂ ∂ r r r r λ ρ ρ . .
Transient Advection Conduction Internal source
Assumptions : Transient heat equation + boundary conditions with time-invariant coefficients + uniform initial temperature or steady state (the system is Linear and also Time-Invariant LITI)
One single thermal excitation defined by its support
7
Assumptions : Transient heat equation + boundary conditions with time-invariant coefficient + uniform initial temperature (the system is Linear and also Time-Invariant LITI)
(P)
(P ) (P
init
T t , T t , = θ
Temperature rise at any point P:
t t , t p p , d ) (P ) (- exp ) (P θ θ
∞
=
Its Laplace transform :
Laplace parameter
( ) ( )
(P) ) ( ) (P (P) ) (P (P) P ) (P P
source
f V p Q p , p , u c p , p c
v
+ ∇ ∇ = ∇ + θ λ θ ρ θ ρ r r r r . .
Transient Advection Conduction Internal source
( ) ( )
(P) ) ( ) (P (P) ) (P (P) P ) (P P
source
f V t Q t , T t , T u c t , t T c
v
+ ∇ ∇ = ∇ + ∂ ∂ r r r r λ ρ ρ . .
Transient Advection Conduction Internal source
Consequences :Laplace transformed heat equation (no time derivative)
8
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
Linear system with a single excitation Temperature or flux response at any point P in the system ⇒
) ( ) , (P ) (P, p u p H p y = t' ' t u ' t t H t u * t , H t , y
t
d ) ( ) (P, ) ( ) (P ) (P
− = =
response excitation
= simple product (Laplace domain)
H (P , t)
init , in b in b init s init s init s s init v v
T t T T t T T t T Q t Q Q t Q t u t u
(
(
) (P
(
) (
) ( ) ( ) (
∞ ∞
− − = : Excitation
) (P, direction any in flux heat local
(P)
(P, ) (P, ) ( ) ( t T t T t t y t y
x init
ϕ θ = = : P point specific any in Response
Transfer function
response
9
« init »= initial steady state
t
H
t' ' t u ' t t H t u * t , H t , y
t
d ) ( ) (P, ) ( ) (P ) (P
− = =
response excitation
t
) (t u
t
) (t y
) (t H
ss
y
ss
u
Steady state (ss) version
t H u y H
ss ss ss
∞
= =
Time distribution asymptotic values
impedance) (thermal variation) re (temperatu (P) power) (thermal : case Z H T T y Q Q
Q u
init SS
≡ ⇒ − = ≡ − ≡ θ
Q Z T T
ss ss
= −
∞
Φ R T T
in
= −
10
Thermal resistance, flux pipe between 2 isothermal surfaces Generalized resistance, no flux pipe
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
T t , T = = ) (P
Constant thermo-physical properties (fluid and walls) and velocities (LTI heat equation) : Assumptions :
= ∂ ∂ t / β
.... , , , u , u
cold hot
ρ λ β ≡ Uniform initial conditions/initial steady state)
2.1 Case of a heat exchanger
(P) ) (P
ss
T t , T = =
init
T t , T = = ) (P
( ) ( )
init q q
T t T t − = θ
Transient/unsteady thermal regime with observed responses at any point q :
( ) ( )
and ≠ > = ≤ t t
q q
θ θ
One single heat source (inlet temperature increase) that starts at t = 0+ :
( ) (
1 1
≠ =
init
T t T t θ
Heat losses through convection/(linearized) radiation with environment through a uniform heat transfer coefficient h at temperature
init
T T =
∞
Cause Consequences
( ) (
4 4
= =
init
T t T t θ
11
(P) ) (P
ss init
T t , T = =
Calculation of convolution products (transmittance case) Parameterizing with piecewise constant functions, square case m / t t m i t i t t
final i
= = = = ∆ ; to 1 for ∆ ;
response unique pseudo source transmittance
(P) ∆ ) (P d ) (P, ) (P, d ) ( ) (P, ) ( ) (P ) (P
1
1 1 1 1 1 j , m j i t t
W t t , ' t ' t W ' t t t' ' t ' t t W t * t , W t ,
j i + −
=
≈ − = − = = θ θ θ θ θ θ
) (
1 t
θ
) (
P1 t
θ
t
1
θ
1
t
2
t
m
t
K t
q
θ
1
t
2
t
m
t
K
Heat exchanger
) ( ) (
3 2
t , t θ θ
1
P
3 2, q =
q
W1
12
( )
(P)
) ( for ) ( ) ( 2 1 d ) ( ∆ 1
1 1
1
W t z t z t z t t z t z
i i t t i
i i
θ = + ≈ =
−
∫
−
1 j =
sampled averaged over 1 time interval
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
( ) ( )
1 2 1
P
4 3 2 , , q , t t
q q q
= = θ θ M θ 1 1 1 1
) ( ) ( θ W W θ θ
q q q
M M = =
≡
− − 1 2 1 1 2 3 1 2 1
∆ ) ( z z z z z z z z z z t
m m m
L O M M M z M
M (z) Lower triangular Toeplitz matrix function
( )
(P)
) ( for ) ( ) ( 2 1 d ) ( ∆ 1
1 1
1
W t z t z t z t t z t z
i i t t i
i i
θ = + ≈ =
−
∫
−
= w w
2 1
M w
=
, , 2 1 1 1 1
θ θ M θ
13
( )
tm
q
θ M
First experiment:
instantaneous (sampled) values
m
w1 M
m , 1
θ M
time averaged values
Next experiments:
(same as source estimation problem) q
1 1)
(
−
θ M
q
ˆ
1
W
measured measured estimated
q
1 1 )
(
− q
W M
1
θ ˆ
measured previously estimated estimated
2 i.i.d. and independent noises 1 1 1
and ε θ θ ε θ θ + = + =
exp q q exp q measured
Model for W identification calibration problem:
q q
W θ θ ) ( 1 M =
exact unknown
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
14
( )
exp q exp q
ˆ θ θ W
1 1 )
(
−
= M
Ill-posed problem: Inversion needs regularization Here: Truncated SVD or 0 order Tikhonov
Practical way of making the inlet temperature vary
3
T
Cold fuid Hot fluid
3
S
h h
1
T
2
T
4
T
x
2
S
1
S ∞
T
P W Z Q
transmittance impedance
1 1
q q
1 1)
h
∞
T ) (
1 t
θ
) ( to ) (
3 2
t t θ θ
P t
q
1
1
) (t Q
Pseudo Source transmittance impedance
15
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
2.2 – Experimental Impedance/transmittance estimation for a half heat exchanger
Response : thermocouples d = 50.8 µm
Identification of transmitance using experimental transient measurements (calibration)
16
) (t Tin ) (t Tout
Upper and lower tranquilization chambers
Comparison of identified Z : step or periodical heating
Identification of transfer function using experimental temperature recording:
step heating identified impedances
− Zin Step - Tikhonov − Zout Step - Tikhonov + Zin Periodical – Tikhonov + Zout Periodical – Tikhonov 1)
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
Square periodical heating
( )
in
in
θ Q Z
1
) (
−
= M 17 + Zout Periodical – Tikhonov
Z (K . J-1 time (s)
heating step
C T
20. =
∞
Comparison of identified transmittance W (outlet/inlet): step or periodical heating
b in b
θ θ W ) (
1 −
= M
Pseudo-source Response Transmittance
W (s-1)
− Step - TSVD − Step - Tikhonov − Periodical – TSVD − Periodical – Tikhonov
heating periodical square
s period C T
5 21 = =
∞
, .
18
time (s)
Oscillations past first peak and for long times, zero initial level hard to recover : Estimation of transmittance W (noisy output and input) more difficult than Z (noisy output only)
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
boundary Heat eq. {
t a x ∂ ∂ = ∂ ∂ θ θ 1
2 2
> = = ∂ ∂ − = > = = ∂ ∂ − = for at ; for at ) ( t x h t t x t q t l θ θ λ ϕ θ λ ϕ
∞
= T t T t
( ) ( θ
change of variables ) (t W
) (t q ) (
1 t
T
) (
2 t
T
a c and , ρ λ
3.1 Reference case: 1D transient conduction θ θ ≡ ≡ ≡ y , u , W H
19
) ( ) ( ) ( ) ( ) ( ) (
1 2 1 2
t t W t p p W p θ θ θ θ ∗ = ⇔ = + =
−
) ( / ) ( sinh ) ( cosh 1 ) (
1
β λ β β l l h L t W a p =
2
β with
Transmittance
ϕ θ ψ ψ ψ
for d ) (- exp ) ( ) ( = ≡ ∫ t t p t , x p , x
t
Laplace transform initial
{
l ≤ ≤ = = x t for at θ
2 1
θ θ ≡ ≡ ≡ y , u , W H
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
Comparison: analytical W and identified W from synthetic profiles (COMSOL)
Identified W by OLS and TLS Analytical Laplace W + numerical inversion of Laplace
) (t W
) (t q ) (
1 t
θ
) (
2 t
θ
a c and , ρ λ
=
∞
θ
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
20
( )
s . t C ; s e t
ss ss
t
5 ∆ and 30 30 with 1
1 1
1
= ° = = − =
−
θ τ θ θ
τ
2 1 and
from θ θ W
Identified W by OLS and TLS
+ =
−
) ( / ) ( sinh ) ( cosh 1 ) (
1
β λ β β l l h L t W
Analytical Laplace W + numerical inversion of Laplace
Numerical Inversion of Laplace Transforms by Hoog’s algorithm Input :
Comparison: analytical W and identified W from synthetic profiles (COMSOL) Comparison without noise:
Validation without committing an INVERSE CRIME
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
21
Temperature profiles (COMSOL)
Analytical and identified W (without regularization)
Calibration problem: how to get the best transfer function H (t) at a point P?
t' ' t t H ' t t u t , H * t u t , y
t
d ) ( ) (, ) (P ) ( ) (P − − = =
sampling
) ( with
) ( u A H A y H u y M M = = =
parameterization
Available information: Model Measurements 3.2 Noisy matrix and Total Least Squares
u
p
u ex
u
r
22
Available information: discrete noisy values
i i i exp
t y t y ε + = ) ( ) (
i i i exp
t u t u τ + = ) ( ) (
noise
Measurements
y
exp
y
y
r
noisy matrix
) ( with τ M = + = ⇒
u u exp
ε ε A A
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
H Identification from u and y measurements :
H u y ) ( M =
ˆ r r H and residuals minimum as such
u
y
exp
y
y
r
p
u ex
u
r
Total Least Squares (TLS) solution : augmented matrix
( )
y u G M =
concatenation
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
23
u y
ˆ r r H and residuals minimum as such
G G H r H r H − = = =
= + = exp G m i m j j i F G TLS
g J ) ( with ) ( ) (
1 1 1 2 2
Frobenius norm: SVD Form of G Regularized form: Truncated TLS ill-posed
Comparison: analytical W and identified W from synthetic profiles (COMSOL) Comparison with noise:
noised
1
θ
3.3 Comparison of calibration methods
24
Temperature profiles (COMSOL)
Analytical and identified W (with regularization)
K , K 1
2 1
= = σ σ
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
Comparison: analytical W and identified W from synthetic profiles (COMSOL) Comparison with noise:
noised
2
θ
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
25
Temperature profiles (COMSOL)
Analytical and identified W (with regularization)
K , K 1
2 1
= = σ σ
% . /
noised noised
90 3
2 exact 2 2
= − θ θ θ
Noise over signal ratio:
Comparison: analytical W and identified W from synthetic profiles (COMSOL) Comparison with noise:
noised
2 1
θ θ ,
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
26
Temperature profiles (COMSOL)
Analytical and identified W (with regularization)
2 1
Conclusion of this comparison of deconvolution techniques
noised
1
θ K , K 1
2 1
= = σ σ noised
2
θ
K , K 1
2 1
= = σ σ
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
27
noise on the source θ1.
noised
2 1
θ θ ,
K 1
2 1
= = σ σ
into account the convolutive structure of the matrix M (θ1) no improvement of the estimate
identified TF in a calibration experiment: largest impact on posterior inverse input estimation experiments.
) (t W
) (t q ) (
1 t
θ
) (
2 t
θ
a c and , ρ λ
=
∞
θ
Calibration problem : Calibration = model identification for a given structure
) ( with
1 2
θ A W A θ M = =
m output data n =m unknowns m input data
Can the ill-posed problem be more parsimonious ? Less many unknowns n than output data: n << m
Idea: tayloring the definition
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
28
2 paths deserve to be investigated BEFORE regularization: Rectangular parametrization using piecewise constant parametrization for W ARX model construction (perspective) Calibration = model identification for a given structure
m input noisy data
exp 2
θ
m output noisy data
exp 1
θ
Parameterization/sampling :
n parameters each
Reduced model structure Regularization
reg
W ˆ
Regularized estimate
noise its
and
zation parameteri
type
depending ) (
values Singular
1 1
θ θ M
10
10 10
5
lar values (1/s)
(red)
1 sampled
θ (black)
1 stairs
θ
with noise no noise
T
V S U θ = ) ( 1 M
4.1. Point versus averaged values for input and unknown
200 400 600 800 1000 1200 1400 10
10
10
singular value order singular
no noise with noise
) ( cond ) ( cond
1 1
ampled, 1 , 1 σ α σ α s , stairs ,
θ θ M M ≥
α : truncation order - σ1 = noise level
) ( ) (
1 1
ampled, 1 1 , 1 1 σ α σ α s , stairs ,
s s θ θ M M ≈
( )
m s / s
,
≤ ≤ = α
α α
1 for ) ( cond
1 1
θ M
Sampled = parameters = instantaneous value (parameterization over a basis of « hat » functions) : zi = z(ti) Parameterized = averaged value (parameterization over a base of « doors » functions) : zi = 0.5 (z(ti) + z(ti-1))
8 10 x 10
(1/s)
ZOOM
Exact W: sampled (continuous) and parameterized (stairs) Estimated W: sampled (continuous) and parameterized
W z
for
1
θ =
Effect of the type of parameterization: noise on θ 2 only
30
100 200 300 400 500 600 700
2 4 6 time t (s) Estimated transmittance W (1
: noise no
2 1
= = ε ε
K 1 :
the
noise i.i.d.
1 2
= = ε ; σ
Truncated SVD: α = 19 for m = 1400 times
parameterized (stairs)
( )
exact
ˆ β β β − = E ) ( bias
( ) ( )
2 2
, ,
bias bias
σ α σ α sampled stairs
ˆ ˆ W W ≤
4.2 Rectangular deconvolution (using « stairs » parameterizing)
1
W z =
31
Less many parameters than measurement times: n << m Simplest method = use of a basis of n piecewise constant functions : m/n = c (integer) t0 =0 time
Same time step for measurements and parameterization Parameterization time step larger than measurement one
t’0 =0
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
Square model :
square square square
W A W θ θ = = ) (
1 2
M
m x m m x 1 m x 1 m x m
[ ]
n
A A A A L
2 1
=
sensitivity matrix sensitivity vectors
Rectangular model :
W X W θ θ = =
square gular tan rec
) ( 1
2
M c = 3
t’j
z’j+1 z’j z’j+2 z’j+3 z’j+4z’j+5 z (t) zk zk+1
t’j-1 t’j+1 t’j+5
t
tk tk +2 tk +1
1
W z =
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
32
sensitivity matrix sensitivity vectors
[ ]
n
X X X X L
2 1
=
m x 1 m x n m x 1 n x 1 m x n
repetition of some lines
=
+ − + 1 1 1 1 1
∆
j m , j , j , j
t θ θ θ M M A
+ − =
=
c k c k j j k
c
1 ) 1 (
1 A X
The n rectangular sensitivity vectors are simply the averaged values of the m square sones
local averaging m x 1 n x 1 m x n
W G W =
square
W X W G A θ = =
gular tan rec 2 m x 1 m x n n x 1 m x n
Effect of rectangular inversion: 1 unknown every c =16 time steps
2 4 6 8 10 x 10
: noise no , estimation r rectangula & square
2 1
= = ε ε K 1 :
the
noise i.i.d.
1 2
= = ε ; σ
Square Truncated SVD: α = 19 for n = m = 1400 times
Rectangular TSVD, for n = 87
ated transmittance W (1/s)
33
Rectangular case with n = 87 unknowns
2530 ) ( cond = A
100 200 300 400 500 600 700
Estimate time t (1/s) No gain in term of estimation bias % . / ˆ
ed recalculat , square
88 3 ) (
1 2
2 2 2
= −
σ α σ
θ W θ θ
% . / ˆ
OLS ed recalculat , gular tan rec
88 3 ) (
1 2
2 2 2
= −
σ σ
θ W θ θ
Same relative RMS residual
5000 6000 7000 8000 9000
ensitivity vector (1/s)
k
X
(rectangular case, n = 87 sensitivity vectors) low sensitivity
Study of the norm (length) of the sensitivity vectors
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
34 100 200 300 400 500 600 700 1000 2000 3000 4000 5000
time relative to each sensitivity vector (s) Norm of each sen
j
A
(square case, m = 1400 sensitivity vectors) low sensitivity to long times estimates
4.3 Rectangular estimation with n < m non uniform (NU) time steps
W X θ
NU NU , gular tan rec
=
NU
X X X X where ≡
∑ = =
− = +
−
1 1 1
where 1
1
k ' k k k a a j k NU k
c a c
k k
A X
=
+ − + 1 1 1 1 1
∆
j m , j , j , j
t θ θ θ M M A
Each of the n rectangular sensitivity vectors are simply the averaged values of ck square ones
Different number of elementary time steps
35
local averaging lower triangular Toeplitz matrix (square) m x n m x n m x m
) ( with
1 2
θ G A G X W X θ M
NU NU NU NU NU , gular tan rec
= = =
m x n n x 1 m x 1
n NU
X X X X L
2 1
where ≡
Question: how to chose the n limits of the n different time steps ?
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
5000 6000 7000 8000 9000
sensity vectors (1/s)
First try: constant level past time t = 400 s (steady state reached)
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
Study of the norm (length) of the sensitivity vectors
k
X
(rectangular case, n = 87 sensitivity vectors)
NU k
X
36
100 200 300 400 500 600 700 1000 2000 3000 4000
time interval relative to each sensitivity vector (s) Norms of the se
sensitivity vectors) (NU rectangular case, n = 51 sensitivity vectors)
j
A
(square case, m = 1400 sensitivity vectors)
2 4 6 8 10 x 10
mated transmittance W (1/s) Rectangular TSVD, NU case, n = 51, 18 singular values kept Exact W, m = n = 1400
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
37
100 200 300 400 500 600 700
time t (1/s) Estimate Rectangular TSVD, for n = 87, 19 singular values kept zoom Improvement of zero level and of the transmittance for short times
Importance and applicability of transfer functions (impedances, transmittances, …) in (exact) reduced convolutive model structures for Linear Time Invariant physical systems (detailled model = PDE, integro-differential equations, …) Convolution products can be given a commutative vector/matrix form in discrete time
Denis Maillet, Waseem Al Hadad, New trends in parameter identification, IMPA, Rio de Janeiro, Brasil , oct.-nov.20 17
38
ill-posed inverse problems: identification problem (calibration) first, inverse input problem (source estimation) or inverse (or direct) virtual sensor, or use in a Non Destructive Testing procedure next Path to improve the quality of estimation of transmittance: rectangular deconvolution and pertinent way of tayloring its unknown parameters Perspective: use of ARX structures (AutoRegressive models with eXternal inputs) for better estimation of transfer functions
39
2 i.i.d. and independent noises 1 1 1
and ε θ θ ε θ θ + = + =
exp q q exp q measured
Model for W identification calibration problem:
q q
W θ θ ) ( 1 M =
exact unknown
( )
exp q exp q
ˆ θ θ W
1 1 )
(
−
= M
( )
m T exp
s s s θ L
2 1 1
diag with ) ( = = S V S U M
( )
1 1 1 diag with
1 1
L L s / s / s / ˆ
exp T q
= =
− −
S θ U S V W
singular values
Ill-posed problem: Inversion needs regularization Here: Truncated SVD or 0 order Tikhonov
40
( )
1 1 1 diag with
2 1 1 1
L L
α α α α
s / s / s / ˆ
exp q T q
= =
− −
S θ U S V W
( )
W θ θ W r W W r W
W
) ( ) ( where ) ( min Arg
1 2 2 2 2 exp exp q q
ˆ M − ≡ + = µ
µ
+ + + = = =
− − 2 2 2 2 2 2 2 2 2 1 2 2 1 1 1
diag where with
m m exp q T q
s s s s s s ˆ µ µ µ
µ µ µ µ µ
L F S F S θ U S V W
2 2 2
) ( σ
γ
m ˆ ≈ W r
least squares sum residual vector standard deviation of
q
ε
Total Least Squares (TLS) solution : augmented matrix
u y
ˆ r r H and residuals minimum as such
G G H r H r H − = = =
= + = exp G m i m j j i F G TLS
g J ) ( with ) ( ) (
1 1 1 2 2
−
ˆ
( )
y u G M =
concatenation
u
exp
y
p
u ex
u
r
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= − =
−
2 2 2 2 2 2 2
22 21 12 11 2 22 22 12
with
α α α α α α α
V V V V V V V V H
T TLS T
ˆ
SVD form Regularized form: T-TSVD
= ∑ = − =
− 22 21 12 11 exp 1 12
and with
22
v v ˆ
T TLS
V V V V V U G V H
ill-posed
y
y
r
z (t) = W (t) or θ 1 (t) : defined on a basis of m piecewise constant functions θ 2 : vector of m sampled values
≡
− − 1 2 1 1 2 3 1 2 1
∆ ) ( z z z z z z z z z z t
m m m
L O M M M z M
Back to phe parameterization problem
1 2
[ ]
) ( H
( H ) (
1 1 j j m j j
t t t t z t z − − =
− =
Heaviside function
4.1. Point versus averaged values for input and unknown
42
W z t z z
i i
for ) (
1
θ = =
Other choice (sampled values)
( )
5 1
10 05 2 ) ( cond .
sampled
= ⇒ θ M
W z t z t z t t z t z
i i t t i
i i
for ) ( ) ( 2 1 d ) ( ∆ 1
1 1
1
θ = + ≈ =
−
−
( )
8 1
10 64 3 ) ( cond .
stairs
= ⇒ θ M