Identification of thermal systems using fractional models

1
Identification of thermal systems using
fractional models
J-D. GABANO, T. POINOT Université de Poitiers
Laboratoire dAutomatique et dInformatique
Industrielle Ecole Supérieure dIngénieurs de
Poitiers 40, Avenue du Recteur Pineau, 86022
POITIERS cedex - FRANCE
2
Introduction
Thermal processes with different geometric shapes


3
Outline

Analysis of thermal impedances of materials with
two geometries and in
the framework of front-face thermal
characterization experiments

Wall and Sphere simulators
Input / ouput signals

Continuous fractional model

Fractional integrator state-space representation

Identification results using the simulators

Experimental results on a laboratory pilot

Conclusions
4
Wall heat transfer modelling
Back face
Front face
0
5
Wall heat transfer modelling Contd
Input
Output
Back face
Front face
0
6
Sphere heat transfer modelling
Outer face
0
Inner face
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Sphere heat transfer modelling Contd
Outer face
0
Inner face
The sphere behaves like a non integer integrator
whose order is equal to 0.5.
Input
Output
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Simulators parameters
Ball geometry
In order to get realistic input/output data,
the following physical parameters, corresponding
to brass, have been used
r1 1 cm r2 3 cm
Rth s 4.779 10-2 C W-1 Cth s 0.3573
kJ.C-1
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Thermal impedances Bode plots
Slope
Wall
Sphere
10
Simulators spatial discretization
Wall
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Simulators equations
Wall
Sphere
12
Simulators input/output data
Interest of the simulators
noiseless time data
test of high frequencies
performances of the investigated fractional model
Sampling time Ts 0.5 s
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Ideal fractional integrator
Integrator of non integer order
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Modelling using fractional integrator
The proposed fractional integrator is defined by
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Fractional integrator behaviour
Inside the frequency band wb, wh
Outside the frequency band wb, wh
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State-space representation of In(s)
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One nth order fractional derivative model
Consider the black-box fractional model
"Macro" state-space representation
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Fractional model state-space representation
State-space model using one fractional derivative
of order
Model parameters vector
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From the black-box modelling to the physics
Thermal impedances
Model
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Model with one integrator of order ½
Model parameters vector
Parsimonious model with only 3 parameters
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Marquardt algorithm
The parameters are estimated in an iterative
way using an OE technique

gradient

hessian

output sensitivity function

m monitoring parameter
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Identification results with noiseless data
Time error modelling
Sphere
Wall
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Identification results with noiseless data
We check the results on the estimated Bode plots
deduced from the estimated parameters obtained in
the time domain.
Identification of model
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Identification results with noiseless data
Thermal impedances frequency modelling errors
Estimated thermal impedances
Frequency modelling errors
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Identification results with noiseless data
Parameter estimates
Wall
Sphere
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Laboratory pilot
Heating immersion circulator
Thermally controlled enclosure
(30.7 C ? 0.1 C )
Transistor base control voltage
Heat flux
Thermal power control
Pin
Power transistor
Brass ball
(radius 3 cm)
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Experimental results
Measured and estimated inner temperature
Sampling time Ts 0.5 s
50 experiments
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Experimental results
Experimental statistical estimation results
Mean value
Standard deviation
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Experimental results
Bode plots dispersion around the mean estimated
model
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Conclusions

We presented an original continuous time
identification algorithm which yields, thanks to
a fractional model and a weak number of
parameters, a good frequency approximation of
heat diffusion in homogeneous media using time
data.
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Conclusions Contd

The model used is built with a unique fractional
integrator of order inside an
intermediate frequency band
which acts as a conventional first order
integrator outside this frequency band.
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Conclusions Contd

The lower frequency is one of the three
estimated parameters and allows the
identification algorithm to adapt the model to
the thermal system geometry.
This property has been evaluated by using
simulators of thermal front face experiments for
two different geometries the wall and the
sphere.
33
Identification of thermal systems using
fractional models

• J-D. GABANO, T. POINOT
• Université de Poitiers
• Laboratoire dAutomatique et dInformatique
  Industrielle
• Ecole Supérieure dIngénieurs de Poitiers
• 40, Avenue du Recteur Pineau, 86022 POITIERS
  cedex - FRANCE

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Wall simulator Bode plots
The simulator frequency validity depends on the
spatial discretization (number I of elementary
blocks).
Wall
I 200 blocks
I 1000 blocks
2500 rad.s-1
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Sphere simulator Bode plots
The simulator frequency validity depends on the
spatial discretization (number I of elementary
blocks).
Sphere
I 200 blocks
I 1000 blocks
2500 rad.s-1
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Identification results with noisy data
SNR 10
Wall
Sphere
1000 Monte-carlo simulations
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Identification results with noisy data
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Experimental results
Validation data
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Simulators equations
Wall