DYNAMIC OBSERVERS
BASED ON GREEN’S
FUNCTION APPLIED
TO 3D INVERSE
THERMAL MODELS
Federal University of Uberlândia
School of Mechanical Engineering
INTRODUCTION
Based on Blum & Marquardt (1997) work focused on the one-dimensional linear case
Here it is extended to solve an inverse
multidimensional heat conduction problem.
That means: to apply the technique to solve 2D
and 3D thermal problems.
The Thermal Model: 1D
q(t) = ?
L
Isolated surface
x
0t,Lx0x
1
t 2
2
0t?)t(qx
k0x
0t0x Lx
Lx00)0,x(
Heat difusion equation:
Boundary condition:
Initial condition:
Fundamentals
Taking the Laplace transform of the spatially
discretized system :
The heat transfer function defined by an input/output
system obtains:
equationdDiscretizesQ
sa
sb
sxTnh
i
i
i
mh
i
i
i
j
)(),(
0
0
functiontransferheatsQ
sT
sa
sb
Gnh
i
i
i
mh
i
i
i
H
)(
)(
0
0
Fundamentals
GH TM q N
Gc
ĜH
T*M
+
+ +
TM ^
q ̂
q ^
NqGNTT HM
*
M
*
1ˆ
M
Hc
c TGG
Gq
The thermal model can be represented by
a dynamic system given by a block diagram
It can be observed from the block diagram that:
i) the unknown heat flux is applied to the conductor (reference model), GH,
ii) and results in a measurement signal T*M corrupted by noise N
q̂
Fundamentals
NQ GG
Hc
c
Hc
Hc NGG1
Gq
GG1
GGq̂
the estimate value is computed from the input data . Thus, the estimator can be represented in a closed-loop transfer function of the feedback loop as
H
Q
NHQNG
GGorGGG 1It can be observed that:
01 NQ GandG
It can be observed that if the algorithm estimates the heat flux correctly, GQ is equal to unity or GQ = 1
Fundamentals
The heat transfer function is known (GH) :
Frequência (rad/s)
Fas
e (
°)M
agni
tud
e (
dB
)
-600
-400
-200
0
200
10-8
10-6
10-4
10-2
100
102
104
-1080
-900
-720
-540
-360
-180
0
Phase
Modulu
s
Frequency, hz
Fundamentals
1- If , the noise can be amplified and this
depends of the choice of GQ:
10-8
10-6
10-4
10-2
100
102
104
-400
-300
-200
-100
0
100
200
300
400
Frequencia (rad/s)
Mag
nitu
de (
dB)
GH
GQ
GN
1
HNQ jGjG,1G
1-
01 NQ GandG
Modulu
s
Frequency, hz
Fundamentals
.0GQ
.000 HQN GthanfasterGG
2- To determine the C (cut frequency)
10-8
10-6
10-4
10-2
100
102
104
-1200
-1000
-800
-600
-400
-200
0
200
400
Mag
nitu
de (
dB)
Frequencia (rad/s)
GH
GQ
GN
C
Modulu
s
Frequency, hz
Fundamentals
)II()ik(q̂a)ik(q̂b)k(q̂nn
1ii
nm
0ii
In this case, any on-line estimator involves a phase
shift or lag. To remove this lag Blum and Marquardt
proposes a filtering procedure that can be resumed
in the use of two discrete-time difference
equations:
)I()ik(q̂a)ik(Tb)k(q̂nn
1ii
nm
0i
*Mi
The technique proposes to obtain GH by using
Green’s Function
The solution of a 3D-transient heat conduction
equation can be given in terms of Green’s function
as follows
)(),(,x qtxGtT h
The equation above, in the Laplace domain, can be written as the convolution product
d)(q)/t,z,y,x(Gt,z,y,Tt
H
0
x
Obtaining of Gh by using Green’s function
)s(q)s,x(Gs,T 1hx
This dynamic system can be represented as shown in Fig. Convolution equation can also be evaluated in the Laplace domain as a single product
q(s)
T(xi,s)
)s,x(Gh
The heat transfer function can, then, be obtained through the auxiliary problem which is a homogenous version of the original problem for the same region with a zero initial temperature and unit impulsive source located at the region of the original heating
Obtaining of Gh by using Green’s Function
X(t) = 1
Yi(t) = T+ (x,t)
),( txGh 1)t,x(Gt,T h x
s,Ts)s,x(Gh x
If the dynamic system is linear and physically invariable the response function
is the same, independently of the pairs input/output and can be obtained by
s
1)s,x(Gs,T h x
Results: GH comparison – 1D case
Frequencia(rad/sec)
Fas
e (
°)M
agni
tud
e (
dB
)
-600
-400
-200
0
200
Método Clássico
Este trabalho
10-8
10-6
10-4
10-2
100
102
104
-1080
-900
-720
-540
-360
-180
0
Phase
Modulu
s
Frequency, hz
Blum & Marquardt
this work
RESULTS AND DISCUSSION Heat flux estimation using temperature data without noise error.
a) sinusoidal;
0 20 40 60
0
2000
4000
6000Heat flux imposed
Ref. [8]
This work
hea
t flu
x [
W/m
2]
time [s]
RESULTS AND DISCUSSION Heat flux estimation using temperature data without noise error.
b) triangular;
0 20 40 60
0
2000
4000
6000Heat flux imposed
Ref. [8]
This work
hea
t flu
x [
W/m
2]
time [s]
Heat flux estimation using simulated temperatures with
i = 1.50C: a) flux sinusoidal (test 3D)
Results: 3D simulated case
0 10 20 30 40 50
0
20000
40000
60000
Heat flux imposed
Sensor 1
Sensor 2
Sensor 3
Sensor 4
hea
t flu
x [W
/m2]
time [s]
Heat flux estimation using simulated temperatures with
i = 1.50C: b) flux triangular (test 3D)
Results: 3D simulated case
0 20 40
0
20000
40000
60000
80000Heat flux imposed
Sensor 1
Sensor 2
Sensor 3
Sensor 4
hea
t flu
x [
W/m
2]
time [s]
Experimental Results: 3D
z
x
y
12.7 mm
12.7 mm
4.7 mm
10mm
10mm
T2(3.5,8.9,4.7)
T1(4.3,3.5,4.7)
q(t)=?
Two thermocouples were brazed on the bottom surface of the sample
opposite to the surface that receives the heat, at the points T1(t) and T2(t)
Experimental Results: 3D Comparison between the measured and
estimated heat flux using Y1(t)
0 10 20 30
0
10000
20000
30000
Observer
Beck
Heat flux imposed
time [s]
Hea
t fl
ux
[W
/m2]
Experimental Results: 3D
0 10 20 30
0
2000
4000
6000
8000
Observer error
Beck error
time [s]
Hea
t fl
ux r
esid
ual
s [W
/m2]
Residuals of heat flux
estimated and measured
Conclusions
The observers method was shown efficient for the
inverse heat conduction problem.
The proposal of obtaining numerically the heat
function transfer based on Green´s function gives
a great flexibility to the technique allowing to deal
with 3D transient problems both simulated and
experimental.
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