Lecture 8 & Tutorial 1: Experimental idention of low order model - - PowerPoint PPT Presentation

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Lecture 8 & Tutorial 1: Experimental idention of low order model Jean-Luc Battaglia, Laboratory I2M,

Department TREFLE, I2M Bordeaux

Eurotherm Advanced School – Metti 5 – Roscoff – June 13-18, 2011

SLIDE 2

Measurement inversion

ϕ (t)

Τm (t) Τext

j

h ϕ =

1 2 3 4 5 20 40 60 80 100

time (sec) phi (W/m²)

1 2 3 4 5

• 5

5 10 15 20 25 30 35

time (sec)

measured simulated with the identified system

( ) ( )

{ }

m

T t t ϕ = M

identified Measured known system

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Classical requirements for a good resolution of the inverse problem

• Implementing thermal sensors in the system at

strategic locations and making measurements

• Having a reliable and accurate model that describes

well the experiment. The reliability of the direct model rests on the accuracy on two sets of data:

– the thermal properties – the location of the sensor.

Uncertainties on those data will lead to a very low confidence domain for the estimated heat flux

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System identification

ϕ (t)

Τm (t) Τext

j

h ϕ =

1 2 3 4 5 20 40 60 80 100

time (sec) phi (W/m²)

1 2 3 4 5

• 5

5 10 15 20 25 30 35

time (sec)

measured simulated with the identified system

( ) ( )

{ }

m

T t t ϕ = M

known Measured identified system

SLIDE 5

Linear monovariable systems

• The impulse response
• For monovariable linear systems, the impulse

response fully characterizes the thermal

• behaviour. Therefore, any kind of inverse

strategy can be based on the direct model expressed as the impulse response of the system

( ) ( ) ( ) ( ) ( )

d

m m m

T t h t t h t ϕ τ ϕ τ τ

∞

= ∗ = −

∫

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System identification approach

• Advantages
• The system identification approach will be first interesting to obtain a reliable and

accurate low order model that will require less computational time for simulation.

• There is no need to know the thermal properties of the system (thermal

conductivity, density, specific heat, heat exchange coefficients, thermal resistances at the interfaces, parameters related to thermal radiation…).

• It is not required to know the sensor location inside the system.
• It is not required calibrating the sensor.
• The identification procedure is fast (this will be viewed later with the description
• f the different techniques).
• Drawbacks
• The model identification must be achieved in the exactly same conditions as those

encountered during the inversion (heat exchanges between the surrounding and the system must remain the same for the two configurations).

• The prediction of the identified model rests on strong assumptions (in particular, it

is better reaching the stationary behaviour during the system identification process). In general, the identified system is only valid for the time duration of the system identification process.

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The deconvolution technique 1/2

( ) ( )

( ) (

) ( ) ( )

( )

k k m m m i i

T k t h k i t k t h k t k i t ϕ ϕ

= =

∆ = − ∆ ∆ = ∆ − ∆

∑ ∑

1 1 1 1 m m N N mN

T h T h T h ϕ ϕ ϕ ϕ ϕ ϕ                   =                  

• (

) ( ) ( ) ( ) ( )

( )

( )

k m m m i

y k t T k t e k t h k t k i t e k t ϕ

=

∆ = ∆ + ∆ = ∆ − ∆ + ∆

∑

Assuming an additive measurement error of normal distribution (zero mean and constant standard deviation)

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The deconvolution technique 2/2

• 1

1 1 1 1 1

H Y E

m m Q Q Q mQ N N Q N Q N N Q N N N

y e h y e h y e h y e ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ

− + −

                              = +                                       Φ

• (

)

1

H Y

T T Q N N N N −

= Φ Φ Φ lim

k k

h

→∞

=

( )

E E

T N N

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The correlation technique

( ) ( ) ( ) ( )

d

m m

y t h t e t τ ϕ τ τ

∞

= − +

∫

( ) ( ) ( ) ( ) ( ) ( ) ( )

0 0

d d d d

m m

y t t t h t t t t e t t ϕ τ τ ϕ τ ϕ τ τ ϕ τ

∞ ∞ ∞ ∞

− = − − + −

∫ ∫∫ ∫

( ) ( ) ( ) ( ) ( )

d

m m

y m y m e

C h t C h t C

ϕ ϕ ϕ

τ τ τ τ τ τ

∞

= − + −

∫

( ) ( )

Cϕϕ τ δ τ =

( ) ( )

m

y

C h

ϕ τ

τ =

SLIDE 10

Spectral technique 1/2

( ) ( ) ( ) ( ) ( ) ( )

FFT FFT d

m m

y m m y

C h t C Y f f S f

ϕ ϕϕ ϕ

τ τ τ τ

∞

    = − = Φ =      

∫

( ) ( ) ( ) ( ) ( )

2

FFT FFT d C t f S f

ϕϕ ϕϕ

τ ϕ τ ϕ τ τ

∞

    = − = Φ =      

∫

( ) ( ) ( ) ( )

m

y e

S f H f S f S f

ϕ ϕϕ ϕ

= +

( ) ( ) ( )

m

y

S f H f S f

ϕ ϕϕ

=

( ) ( ) ( )

t t t

τ

ϕ ϕ

Π

= Π

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Spectral technique 2/2

( ) ( ) ( )

sin f f f f π τ τ π τ

Π

  Φ = Φ ∗   

( ) ( ) ( )

t t g t

τ

ϕ ϕ

Π

=

( )

2 0.5 1 cos t g t

τ

π τ     = −        

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The parametric approach

( ) ( ) ( ) ( ) ( ) ( )

2 2 1 2 1 2 2 2

d d d d d d d d

m m m

T t T t t t T t t t t t t ϕ ϕ α α β ϕ β β + + + = + + +

• (

) ( )

2 2

, , T x t T x t a t x ∂ ∂ = ∂ ∂ , x e t < < >

( ) ( )

, T x t k t x ϕ ∂ − = ∂ 0, x t = >

( )

, T x t x ∂ = ∂ , x e t = >

( )

, T x t =

SLIDE 13

The parametric approach

( )

{ }

( ) ( ) ( )

{ }

( ) ( )

1 1 sinh sinh

m m

L T t s L t s k e k e θ ϕ β β β β = = = Φ s a β =

( ) ( )

2 1

sinh , 2 1 !

n n

z z z n

+ ∞ =

= ∀ ≥ +

∑

( ) ( ) ( ) ( ) ( ) ( )

2 1 2 1 1 1

1 1 2 1 ! 2 1 !

m n n n n n n

s s s e s e k k a n n θ β β

+ + + ∞ ∞ + = =

= Φ = Φ + +

∑ ∑

( ) ( ) ( )

1 1

d d d d

n k n n n k n k k

f t f L s F s s t t

− − − =

  = −    

∑

( ) ( )

1

d d

n m n n

T t t t α ϕ

+ ∞ =

=

∑

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Output error method

( ) ( ) ( ) ( ) ( ) ( )

1 2 1 2

1 2 1 2

m m m

T k b k b k b k a T k a T k ϕ ϕ ϕ = + − + − + − − − − −

• system
• utput error

model ε (k) ϕ (k) Tm (k) ym (k)

( ) ( ),

1, ,

i

m a i

T k S k i n a ∂ = = ∂

• ( )

( ) ,

0, ,

i

m b i

T k S k i n b ∂ = = ∂

• ( )

( ) ( ) ( )

1

1 , 1, ,

i i i

a a n a m

S k a S k a S k n T k i i n + − + + − = − − =

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Output error method

( ) ( ) ( )

1 1

i i i

a a a

S S S n = = = − =

• ( )

( ) ( ) ( )

1

1 , 1, ,

i i i

a a n a m

S k a S k a S k n T k i i n + − + + − = − − =

• ( )

( ) ( ) ( )

1

1 , 0, ,

i i i

b b n b

b S k b S k b S k n k i i n ϕ + − + + − = − =

• ( )

( ) ( )

1 1

i i i

b b b

S S S n = = = − =

• ( )

( ) ( ) ( ) ( )

1

i i

n n m m a i b i i i

k y k T k S k a S k b ε

= =

= − = ∆ + ∆

∑ ∑

( ) ( ) ( )

1

1

n n

a n n a b N b ε ε ε ∆             + ∆   = = = ∆Θ     ∆           ∆   E S S

• ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 n n n n

a a b b a a b b

S n S n S n S n S N S N S N S N     =       S

• (

)

1 T T −

∆Θ = S S S E

1 1 ν ν ν − −

Θ = Θ + ∆Θ

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Predicton error method

( )

( ) ( ) ( ) ( ) ( )

1 2 1 2

1 2 1 2

m m m

T k b k b k b k a y k a y k ϕ ϕ ϕ = + − + − + − − − − −

• ( )

( ) ( )

m

y k k e k = Θ + H

system predictive model e (k) ϕ (k) Tm (k) ym (k)

[ ]

1 T n n

a a b b Θ =

• ( )

( ) ( ) ( ) ( )

1

m m

k y k y k n k k n ϕ ϕ = − − − − −     H

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Predicton error method

N N N

= Ψ Θ + Y E

( ) ( )

T N m m

y n y N n = +     Y

• ( )

( )

T N

n N n Ψ = +     H H

• ( )

( )

T N

e n e N n = +     E

• (

)

1 T T N N N N −

Θ = Ψ Ψ Ψ Y

• (

)

1 T T N N N N −

Θ = Θ + Ψ Ψ Ψ E

• { }

( ) ( )

{ }

( )

( ) ( )

{ }

1 T T

E E k k E k e k

−

Θ = Θ + H H H

It thus appears that if ( ) e k is correlated with

( )

k H

• r if

( )

{ }

E e k in not zero, the estimation is biased and

• { }

E Θ ≠ Θ .

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The use of recursivity

( ) (

) ( ) ( ) ( ) ( )

1 1

m

k k k y k k k   Θ = Θ − + − Θ −   L H

( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1

T T

k k k k k k k λ − = + − P H L H P H

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

T T

k k k k k k k k k k λ − − = − − + − P H H P P P H P H ( )

D

Θ = 0

( )

6

10

D

= P I

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Limit of the dn/dtn models

( )

{ }

( ) ( ) ( ) ( )

{ }

( ) ( ) ( )

cosh cosh sinh sinh

m m

e e L T t s L t s k e k e β β θ ϕ β β β β = = = Φ s a β =

( ) ( ) ( ) ( )

2 2 1

cosh and sinh , 2 ! 2 1 !

n n n n

z z z z z n n

+ ∞ ∞ = =

= = ∀ +

∑ ∑

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 2 2 1 2 1 1 1

2 ! 2 ! 2 1 ! 2 1 !

n n n n n n m n n n n n n

e e s n a n s s s e s e k k a n n β θ β β

∞ ∞ = = + + + ∞ ∞ + = =

= Φ = Φ + +

∑ ∑ ∑ ∑

( ) ( )

1 n n n m n n n

s s s s α θ β

∞ ∞ + = =

= Φ

∑ ∑

( ) ( )

1

d d d d

n n m n n n n

T t t t t ϕ α β

+ ∞ ∞ = =

=

∑ ∑

( )

2 1 1 2

1 !

n n n

e k a n α

+ +

= +

( )

2

2 !

n n n

e a n β =

SLIDE 20

Asymptotic behaviours

( ) ( )

cosh 1 lim sinh

s

e s a k a s k e s a e s a

→∞

=

1 1

2 1 1 lim

n n n n n s n n n n n

s s n s ek a s s β β α α

∞ = →∞ ∞ + + =

+ = =

∑ ∑

SLIDE 21

The use of dα/dtα

( ) ( ) ( )

1 1

d d d d

n n k k

f t f L s F s s t t

ν ν ν ν ν − − − =

  = −      

∑

( )

{ }

( )

{ }

{ }

( ) ( )

D D I N, Re 0, 1 Re

n n

f t f t n n n

ν ν

ν ν

−

= ∈ > − ≤ <

( )

{ }

( ) ( ) ( )

1

1 I d

t

f t t u f u u

ν ν

ν

−

= − Γ

∫

( ) ( )

∫

∞ −

− = Γ

1 exp

du u uν ν

( ) ( )

{ }

1 1 2

1 1 1 I L s t k a s k a ϕ

− 

 Φ =    

( )

{ }

( )

{ }

2 2

D D

n n n m n n n

T t t α β ϕ

∞ ∞ = =

=

∑ ∑

SLIDE 22

explanation

( )

( )

2

1 cosh 2 2

z z z z

e e e e z

− −

+ + = =

( )

( )

2

1 sinh 2 2

z z z z

e e e e z

− −

− + − = =

( )

( )

( )

2 2

1

e m e

e s s k e

β β

θ β = Φ − , !

n z n

z e z n

∞ =

= ∀

∑

( )

( )

( )

2 1 2

' '

n n n m n n n

s s s s β θ α

∞ = ∞ + =

= Φ

∑ ∑

( ) ( )

1 1 1 2 2 2

' ' ! !

n n n n n n

k e e and a n a n α β

− − − −

= =

( )

( )

{ }

( )

{ }

1 2 2

' D ' D

n n n m n n n

T t t α β ϕ

∞ ∞ + = =

=

∑ ∑

SLIDE 23

Heat flux estimation in machining

Heat flux estimation in fast drilling process, Tool coating influence

SLIDE 24

24

Vf Vc Workpiece

φ1 φ2 Y1 Y2 A(s) B1(s) B2(s)

Multivariable system : two sensors and two heat fluxes.

Drilling: the phenomenology

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 1,1 1,2 2,1 2,2 1 2

T s s F s F s F s F s T s s s φ φ       =                 F

SLIDE 25

25

Thermocouples type K 4 split rings Amplification AD595CQ Acquisition

NI-Labview

Temperature measurement in the drill

SLIDE 26

26

The complete device

SLIDE 27

27

Electric power generator

φ(t)

switch Amplification rotating collector Acquisition Y2(t) Y1(t) drill

( )

[ ]

[ ] [ ]

[ ]

[ ] [ ]

,

ij ïj ij ij

L ij k k k L i j M ij k k k M

s F s s

ξ ξ

β α

= =

=

∑ ∑

Non integer system identification

1 2 ξ =

5 10 15 20 25 30 10 20 30 40 50 60 70 time (s) measured heat flux (W) measured temperature sensor 1 (° C) measured temperature sensor 2 (° C)

[ ]

1

ij M

α =

SLIDE 28

28

Application: Aluminum machining

10000 tr mn-1 Industrial partners: DASSAULT, CETIM, SLCA Project MEDOC ANR CONTROLTHER

SLIDE 29

29

Aluminium drilling

Perçage Aluminium 20 40 60 80 100 120 5 10 15 20 25 30

temps (s) température (° C)

Y1 essai 1 Y2 essai 1 Y1 essai 2 Y2 essai 2 Y1 essai 3 Y2 essai 3

70 ° C

SLIDE 30

30

2 4 6 8 10 12 14 16 18

• 1

1 2 3 4 5 6 time (s)

∆ φ (W)

Al2O3 TiN TiAlN TiAlNwc TiAlN+MoS2

n r rev

φ φ φ ∆ = −

Comparison of the performances of the various coatings

Raising operation of the surface of a disc from the periphery towards the center. One represents the difference between the flux estimated for the tool without coating and that for the covered tool.

SLIDE 31

Heat flux estimation in a plasma tunnel

Heat flux estimation in severe thermal conditions

SLIDE 32

32

Estimation de flux dans un jet supersonique de plasma

Applications :

• re-entry

simulation

• Degradation of

materials under high heat flux density and high temperature Industrial partners: EADS-ASTRIUM, CEA-CESTA

SLIDE 33

System identification

SLIDE 34

System identification

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35

The « pen » sensor