Lecture 8 & Tutorial 1: Experimental idention of low order model

Transcription

1 Eurother Advaced School Metti 5 Roscoff Jue 13-18, 211 Lecture 8 & utorial 1: Experietal idetio of low order odel Jea-Luc Battaglia, Laboratory I2M, Departet REFLE, I2M Bordeaux

2 Measureet iversio idetified ϕ (t) Τ (t) easured siulated with the idetified syste tie (sec) Τ ext h j Measured phi (W/²) ϕ = { } = M ϕ t t kow tie (sec) syste

3 Classical requireets for a good resolutio of the iverse proble Ipleetig theral sesors i the syste at strategic locatios ad akig easureets Havig a reliable ad accurate odel that describes well the experiet. he reliability of the direct odel rests o the accuracy o two sets of data: the theral properties the locatio of the sesor. Ucertaities o those data will lead to a very low cofidece doai for the estiated heat flux

4 Syste idetificatio Measured Τ (t) easured siulated with the idetified syste 1 kow ϕ (t) tie (sec) Τ ext h j phi (W/²) ϕ = { } = M ϕ t t idetified tie (sec) syste

5 Liear oovariablesystes he ipulse respose = = ( ) t h t ϕ t h t τ ϕ τ dτ For oovariableliear systes, the ipulse respose fully characterizes the theral behaviour. herefore, ay kid of iverse strategy ca be based o the direct odel expressed as the ipulse respose of the syste

6 Syste idetificatio approach Advatages he syste idetificatio approach will be first iterestig to obtai a reliable ad accurate low order odel that will require less coputatioal tie for siulatio. here is o eed to kow the theral properties of the syste (theral coductivity, desity, specific heat, heat exchage coefficiets, theral resistaces at the iterfaces, paraeters related to theral radiatio ). It is ot required to kow the sesor locatio iside the syste. It is ot required calibratig the sesor. he idetificatio procedure is fast (this will be viewed later with the descriptio of the differet techiques). Drawbacks he odel idetificatio ust be achieved i the exactly sae coditios as those ecoutered durig the iversio (heat exchages betwee the surroudig ad the syste ust reai the sae for the two cofiguratios). he predictio of the idetified odel rests o strog assuptios (i particular, it is better reachig the statioary behaviour durig the syste idetificatio process). I geeral, the idetified syste is oly valid for the tie duratio of the syste idetificatio process.

7 he decovolutiotechique 1/2 k ϕ ϕ ( ) k t = h k i t k t = h k t k i t i= i= k ϕ h 1 ϕ1 ϕ h 1 = M M O O M N ϕ N L ϕ1 ϕ hn Assuig a additive easureet error of oral distributio (zero ea ad costat stadard deviatio) ϕ i= k y k t = k t + e k t = h k t k i t + e k t

8 he decovolutiotechique 2/2 li h k k = y ϕ e y ϕ 1 1 ϕ h e 1 M M O O h 1 M = + y ϕ Q Q ϕ1 ϕ L M eq M M L M M hq { M y ϕ N N ϕ N Q+ 1 ϕ N Q H en { L Q { Y Φ E N N N ( E E N N ) H 1 = Φ Φ Φ Y Q N N N N

9 he correlatiotechique = ( τ ) ϕ ( τ ) dτ + y t h t e t y t ϕ t τ dt h t τ ϕ τ ϕ t τ dt dτ ϕ t τ e t dt ( ) = ( ) ( ) + ( ) ( τ ) = ( τ ) ( τ ) dτ + ( τ ) ( τ ) C h t C h t C yϕ yϕ eϕ C ϕϕ ( τ ) = δ ( τ ) C y = h ϕ τ τ

10 Spectral techique 1/2 FF C y FF h d ϕ = t Cϕϕ = Y f Φ f = S y f ϕ ( τ ) ( τ ) ( τ ) τ FF Cϕϕ = FF t d = Φ f = Sϕϕ f 2 ( τ ) ϕ ( τ ) ϕ ( τ ) τ = + S f H f S f S f yϕ ϕϕ ϕe H ( f ) S = S y ϕ ϕϕ ( f ) ( f ) ( t) ϕ ( t) ( t) ϕπ = Π τ

11 Spectral techique 2/2 ( π τ f ) si Φ Π ( f ) = Φ( f ) τ π τ f ϕ ( t) ϕ ( t) g ( t) Π = τ g τ ( t) 2π t =.5 1 cos τ

12 he paraetricapproach ϕ 2 2 d t d t dϕ t d t ( t) + α1 + α2 + L = β 2 ϕ ( t) + β1 + β2 + L 2 dt dt dt dt (, ) 2 ( x, t) x t t = a (, ) x x t k = ϕ x (, ) x t x 2 ( t) = < x < e, t > x =, t > x = e, t > ( x, t ) =

13 he paraetricapproach 1 1 L{ ( t) } = θ ( s) = L{ ϕ ( t) } = Φ ( s) β = s a k β sih β e k β sih β e 2+ 1 z sih ( z) =, z 2 1! = ( + ) θ 1 1 s = Φ ( s) = Φ s e s k β k ! = a ( )! ( β e) ( + ) = f t L = s F s s 1 k d k 1 d k dt k= dt f = d α + 1 dt ( t) = ϕ ( t)

14 Output errorethod ϕ ϕ ( 1) ϕ ( 2) L ( 1) ( 2) k = b k + b k + b k + a k a k L ϕ (k) syste y (k) ε (k) output error odel (k) ( k ), 1, K, b i = = Sa k i i ai ( k ),,, = = b S k i i K L S k + a1 S k a S k = k i, i = 1, K, ai ai ai

15 Output errorethod L S k + a1 S k a S k = k i, i = 1, K, ai ai ai L S = S 1 = = S 1 = ai ai ai L ϕ b S k + b1 S k b S k = k i, i =, K, bi bi bi L S = S 1 = = S 1 = bi bi bi ε k = y k k = S k a + S k b ai i bi i i= 1 i= a1 ε M ε ( + 1) a E = = S = S Θ M b ε ( N ) M b L L Sa S 1 a S b S b S = M M M M Sa ( N ) L S 1 a N S b N L S b N Θ = Θ + Θ ν ν 1 ν 1 1 Θ = S S S E

16 Predictoerrorethod ( k ) b ϕ ( k ) b ϕ ( k 1) b ϕ ( k 2) L a y ( k 1) a y ( k 2) = L ϕ (k) syste y (k) e (k) predictive odel (k) = H Θ + y k k e k Θ = [ a1 L a b L b ] H( k ) = y ( k 1) L y ( k ) ϕ ( k ) L ϕ ( k )

17 Predictoerrorethod Y E YN = y L y ( N + ) N = Ψ N Θ + N L H Ψ = H + E N N L = + N e e N 1 Θ = Ψ Ψ Ψ Y N N N N 1 Θ = Θ + Ψ Ψ Ψ E N N N N ( { }) { } 1 E{ Θ } = Θ + E H( k ) H( k ) E H k e k { } It thus appears that if e( k ) is correlated with H ( k ) or if ad E { Θ} Θ. E e k i ot zero, the estiatio is biased

18 he use of recursivity Θ ( k ) = Θ ( k 1) + L( k ) y ( k ) H( k ) Θ ( k 1) L ( k ) = λ P( k 1) H( k ) ( k ) + H( k ) P( k 1) H( k ) P ( k ) P( k 1) ( k 1) ( k ) ( k ) ( k 1) ( k ) + H( k ) P( k 1) H( k ) P H H P = λ Θ = P D 6 = 1 I D

19 Liitof the d /dt odels ( β e) ( β e) cosh cosh L{ ( t) } = θ ( s) = L{ ϕ ( t) } = Φ ( s) β = s a k β sih β e k β sih β e z z cosh ( z) = ad sih ( z) =, z 2! 2 1! θ = = ( β e) ( β e) ( + ) ( + ) 2 2 e s 2! a ( 2 )! s = Φ s = Φ s e s k β k ! = a ( )! = = = + 1 α θ β = = = Φ s s s s α = k a + e 2+ 1 ( + ) 1 2 1! 2 e β = a 2! d + 1 α = = dt = d ϕ t t β dt

20 Asyptoticbehaviours li ( e s a ) sih cosh 1 s = k e s a e s a k a s li β s = β s s α s α s = = = ek a s

21 he use of d α /dt α f t L = s F s s ν 1 ν d ν k 1 d ν ν dt k = dt ν { } { } { } ( ν ) ( ν ) ν D f t = D I f t N, Re >, 1 Re < f I ν 1 ν 1 f t = ( t u) f ( u) du Γ { } Γ ( ν ) t ( ν ) = u ν ( u) 1 exp du L 1 1 s 1 t k a s k a 1 Φ = I 1 2 { ϕ } D = = 2 { ( t) } = D 2 { ( t) } α β ϕ

22 explaatio cosh 2z ( 1 ) z z z e + e e e ( z) = = sih ( z) 2z ( 1 ) z z z e e e + e = = 2 2 e z z =, z! = θ ( s) 2β e = e k β Φ 2β e ( e 1) θ ( s) ( s) 2 β ' s = = Φ ( + 1) 2 α ' s = ( s) α ' 1 1 k e e = ad β ' ( 1) 2 = ( 2) 2 a! a! ( + ) ' D = = = 1 2 { t } ' D 2 { ( t) } α β ϕ

23 Heat flux estiatio i achiig Heat flux estiatio i fast drillig process, ool coatig ifluece

24 Drillig: the pheoeology A(s) Y 1 V c B 2 (s) B 1 (s) Y 2 φ 1 V f Workpiece φ 2 F( s) 1 s F1,1 s F1,2 s φ1 s = F 1 s 2,1 s F2,2 s φ2 s Multivariable syste : two sesors ad two heat fluxes. 24

25 eperature easureet i the drill Aplificatio AD595CQ herocouples type K 4 split rigs Acquisitio NI-Labview 25

26 he coplete device 26

27 No iteger syste idetificatio Electric power geerator F i, j ( s) = [ ij L ] [ ïj] k = L [ ij M ] [ ij k = M ] β [ ij] k α [ ij] k s ξ k s ξ k 7 [ ij] α = 1 M ξ = 1 2 switch drill φ(t) Y 1 (t) Y 2 (t) Aplificatio easured heat flux (W) easured teperature sesor 1 ( C) easured teperature sesor 2 ( C) 27 rotatig collector Acquisitio tie (s)

28 Applicatio: Aluiu achiig Idustrial parters: DASSAUL, CEIM, SLCA Project MEDOC ANR CONROLHER 1 tr -1 28

29 Aluiiu drillig Perçage Aluiiu 12 tepérature ( C) C Y1 essai 1 Y2 essai 1 Y1 essai 2 Y2 essai 2 Y1 essai 3 Y2 essai teps (s) 29

30 Copariso of the perforaces of the various coatigs φ (W) Al2O3 in ialn ialnwc ialn+mos2 Raisig operatio of the surface of a disc fro the periphery towards the ceter. Oe represets the differece betwee the flux estiated for the tool without coatig ad that for the covered tool tie (s) φ = φ φ r rev 3

31 Heat flux estiatio i severe theral coditios Heat flux estiatio i a plasa tuel

32 Estiatio de flux das u jet supersoique de plasa Idustrial parters: EADS-ASRIUM, CEA-CESA Applicatios: re-etry siulatio Degradatio of aterials uder high heat flux desity ad high teperature 32

33 Syste idetificatio

34 Syste idetificatio

35 35 he «pe»sesor