Experimental identification of low order model

Transcription

1 METTI METTI5 Tutorial T o Experimetal idetificatio of low order model Jea-Luc Battaglia, Laboratory IM, Departmet TREFLE. Abstract The goal of this tutorial is to apply the system idetificatio techique i order to obtai a accurate direct model devoted to measuremets iversio. This tutorial is closely related to Lesso L8. A simple experimet will be used i order to give the basic ideas of the optimal experimet i system idetificatio approach. It will be particularly emphasized o the choice of the excitatio sequece. Two methods will be used: the correlatio method ad the parametric method. A direct model will thus be obtaied ad it will be aalyzed i terms of reliability ad accuracy. As a coclusio, it will be poited out the advatages of this approach with respect to the classical oe based o the resolutio of the heat diffusio equatio. System descriptio. Real life cotext I high ethalpy plasma flows for aerospace applicatios but also i high power pulsed laser physics as well as i laser surface hardeig moitorig, very high heat fluxes o the order of several MW/m have to be measured (see Figure ). Oly trasiet measuremet techiques have bee developed so far. Recetly, fast trasiet heat flux measuremets have bee coducted usig a ovel calibratio approach. I priciple, these sesors ca stay a very short time, of the order of millisecods, i the harsh eviromet i order ot to reach meltig temperature. The trasiet respose of the sesor recorded durig this short time is used to estimate the heat flux.

2 Plasma torche MW/m² Pe fluxmeter Fluxmètre coique Figure : heat flux measuremet i a plasma flow usig a fluxmeter. Oe basic priciple of such a heat flux sesor is to measure the temperature, usually with a thermocouple embedded iside a appropriate material which is part of the sesor, ad to estimate the heat flux from iverse heat coductio calculatio. Cosequetly, highest reliability i terms of measuremet accuracy ad local resolutio withi the sesor is reached whe the distace betwee the temperature measuremet ad the heated surface is small (see Figure ). Solvig the iverse heat coductio problem requires a model, so-called direct model (DM) of the heat trasfer from the heated surface of the sesor to the locatio of the thermocouple iside the sesor. Model preseted i the literature geerally assume that the sesor behaves like a semi-ifiite wall. Figure : fluxmeter descriptio Cosiderig a particular sesor ad liear heat trasfer, i.e. costat material properties for the measuremet time, the Duhamel s theorem makes sure that the DM ca be viewed as the impulse respose. This is the trasiet temperature of the thermocouple due to a heat flux o the form of a Dirac fuctio. Assumig the sesor behaves as a semi ifiite wall, the impulse respose is aalytically expressed accordig to the thermal properties of the medium as well as the locatio of the thermocouple iside the sesor. However, i real cofiguratio, the heat flux sesors ivolve several types of materials arraged i a more or less complicated way. The kowledge of their thermophysical properties as well as the thermal cotact resistaces at the iterfaces have to be kow if oe wats to use the fiite elemet method to solve a detailed DM. Furthermore, it appears that the thermocouple rise time ca be greater tha the samplig time iterval durig the acquisitio process. This meas that the thermocouple juctio is ot at a uiform temperature for each acquisitio time. I other words,

3 there exists a temperature gradiet i the juctio ad oe must also cosider heat diffusio iside the juctio. Obviously, it is ot allowed applyig the electrothermal coversio betwee the voltage drop ad the temperature of the thermocouple sice it rests o the assumptio of a uiform temperature of the juctio. I order to overcome these problems, the basic idea rests o the system idetificatio of the heat flux sesor. It cosists i calibratig the sesor by applyig a measurable trasiet heat flux i the time domai of iterest usig a modulated laser source. Give that the idetified system has to be accurate for all the swept frequecies, it is searched a heat flux waveform whose Power Spectral Desity is comparable to that of a white oise. This calibratio thus ivolves the realizatio of a specific experimetal setup i laboratory which allows applyig the heat flux o the form of a radom or Pseudo Radom Biary Sigal (PRBS) ad measurig it precisely as well as the voltage drop at the thermocouple. This approach does ot require kowig the thermophysical properties of materials as well as the exact locatio of the thermocouple. Also the kowledge of the thermocouple iertia is circumveted sice it is take ito accout withi the calibratio. From a theoretical poit of view, the idetified system is the direct model whe solvig the iverse heat coductio problem. It meas that the idetified system ad the voltage drop measuremet at the sesor durig the use of the sesor for a give applicatio is sufficiet to estimate the heat flux. This approach does ot deped o a particular sesor geometry which the allows maufacturig also particular sesor geometries for particular applicatios. Major drawback of this approach is that the calibratio must be realized i the same coditios tha those ecoutered o the process durig the use of the heat flux sesor otherwise a liear behavior has to be assumed. I other words, oe must reproduces i the laboratory the same boudary coditios i terms of trasiet ad magitude of heat flux.. System idetificatio hardware The sesor cosists of a cylidrical copper tube, where a thermocouple is itegrated. The tip is of spherical form. The thermocouple is of type K with a juctio diameter of.8 mm. As specified by the maufacturer, the rise time for the thermocouple is about msec. However, owadays the heat flux sesor has to be used o comparable time duratio, but with. msec samplig iterval. The Figure 3 shows the schematic view of the scale experimetal setup for these calibratio measuremets. Sice the ull-poit calorimeter is used to measure very high heat flux desities up to MW/m², the laser pulse eergy has to be high i order to achieve a sufficietly resolvable sigal of the thermocouple. The laser i use is a laser diode that ca provide W at 98 m wavelegth. It is kow i system idetificatio, that the best results are achieved whe laser pulses of variable legth are geerated i order to better distiguish each of the characteristic times of the system: the diffusio i the tip, the respose of the thermocouple ad the ifluece of the iterfaces. The laser is drive usig a fuctio geerator that ca geerate a Pseudo Radom Biary Sigal (PRBS) for example. O the Figure 4, the Power Spectral Desity (PSD) of such a sigal is compared to that of a

4 Dirac fuctio ad that of the Heaviside fuctio. It is obviously foud that the best represetatio of the frequecies is give with a perfect Dirac fuctio which is very difficult to implemet experimetally, particularly i terms of reproducibility. Cocerig the two others, the best oe is the PRBS because its PSD is superior to the Heaviside fuctio PSD i the studied frequecy rage. Fast Photodiode for laser trasiet behaviour Optical device for laser beam desig IR Pyrometerfor temperature measuremeto the heated area laser diode W Camera for measurigthe heated area Figure 3: system idetificatio hardware at scale Power spectral desity 5 5 PSD of PRBS PSD of a Heaviside fuctio PSD of a Dirac fuctio 4 6 Frequecy (Hz) Figure 4: Power Spectral Desity of a Dirac fuctio, a Heaviside fuctio ad a PRBS. A very small part (less tha %) of the heat flux from the laser is recorded usig a fast photodiode,. All data are recorded usig a fast oscilloscope. Moreover, the thermocouple data is amplified with a costat gai of 5. The icidet heat flux desity for the laser is deduced from the calibratio curve of the maufacturer based o blackbody absorptio. The sesor is made off copper which has bee oxidized i a furace at 4 C durig four hours i order to achieve high surface catalycity ad to reach high absorptivity. Its emissivity has the bee measured to.7 at the laser wavelegth.

5 The oise measuremet at the thermocouple is recorded without heatig the sesor by the laser ad results (samplig time is µsec) ad the oise histogram taht shows that the oise has a Gaussia distributio are preseted i Figure 5. O the other had, the computatio of the oise auto-correlatio fuctio is represeted i Figure 6. This fuctio is close to from the secod poit ad is thus equivalet to a Dirac fuctio. I coclusio, all the assumptios cocerig oise measuremet have bee checked ad the applicatio of the least square algorithm i order to estimate the parameters of the o iteger system is fully cosistet. Measuremet oise (V) Time (sec) Number of poits Sigal (V) Figure 5: Noise measuremet ad oise distributio fuctio Number of shifts Figure 6: Autocorrelatio fuctio applied to the oise measuremet. The ucertaity o the emissivity measuremet is about 6%. Cocerig the laser heat flux measured with a fa cooled broad bad sesor, the costructor gives a calibratio certificate ad a relative precisio of about %. Fially, the radius of the beam laser is measured maually ad the ucertaity is estimated to be 5%. I coclusio, the ucertaities of the measured heat flux desity absorbed by

6 the sesor is approximately 7% which is high compared to the ucertaities ivolved by the parameter estimatio method. For this tutorial it has bee realized a experimetal setup at scale /4. Ideed, the laser diode is 5W maximum power. It is drive through Natioal Istrumet card uder Labview software. The other parts of the experimet remai idetical to those of the experimet scale..3 Heat trasfer model i the fluxmeter Thermal properties of the fi are λ for the thermal coductivity, α for the thermal diffusivity, ρ for the desity ad C p for the specific heat. Heat losses betwee the fi ad the surroudig fluid are characterized by the heat exchage coefficiet h. The sectio of the thi is deoted S ad the circumferece is deoted s. If the temperature depeds oly o the logitudial directio, (Bi = h d/λ <<.) carryig out a heat flux balace o a slice of the fi of width x, we get: dt φe φs φ p = ρ Cp S x () dt φe φ p φs with: x x+ x φ s φ e d T ( x, t) = λ S () dx d T ( x + x, t) = λ S (3) dx ( (, ) ) φ p = h s x T x t T (4) Substitutig these relatios ito () we obtai: ( + ) d T x x, t d T x, t λ S h s x( T ( x, t) T ) = dx dx d T x, t ρ Cp S x dt (5) that is, Puttig ' d T x, t d, T x t λ S x h s x ( T ( x, t) T ) = ρ C p S x (6) dx dt T x = T x T, the above expressio becomes:

7 Applyig the Laplace trasform to with '(, ) = ρ (7) d T ' x, t d T ' x, t λ S h st x t C p S dx dt T ', we obtai: k ( x p) k θ ( x, p) = (8) d θ, dx ρ C S p + h s p h s = = + (9) λ S α λ S p The solutio is: Whe x we fid A=. The: θ ( x, p) = Aexp( k x) + B exp( k x) () ( ) θ = L T, t T = B () ad thus: The heat dissipated by the semi-ifiite fi is: that is, ψ θ ( x, p) = θ exp( k x) () d θ ( x, p) ( p) = λ S (3) θ dx x= ψ ( p) ( x, p) = (4) λ S k Relatio (4) shows that thermal impedace of the fi is: Z a exp( k x) =, k λ S k p h s α λ S = + (5) ψ θ Z a Figure 7: thermal impedace of the fi.

8 Correlatio ad spectral techiques. Theoretical backgroud Based o the liearity assumptio, the measured temperature y ( t ) is related to the heat flux ( t ) from the covolutio product: = ( τ ) ( τ ) dτ + y t h t e t (6) h ( t ) is the impulse respose ad e( t ) deotes the measuremet error. Now, let us multiply the two members of this equality by ( τ ) It is the obtaied: t ad itegrate from t= to ifiity. y ( t) ( t τ ) dt = h ( t τ ) ( τ ) ( t τ ) dt dτ + ( t τ ) e( t) dt (7) Oe recogizes the cross ad auto correlatio fuctios, C y ( τ ), C e ( τ ) ad is thus writte o the followig form: By choosig the heat flux t y e C τ. Relatio (7) C ( τ ) = h ( t τ ) C ( τ ) dτ + C ( τ ) (8) as a white oise lead to: C ( τ ) = δ ( τ ) (9) Fially, if it is admitted that the oise measuremet is ot correlated to the heat flux ( Ce = ), relatio (8) is summarized to: C h τ = τ () y Oe sees that the impulse respose is equal to the cross correlatio fuctio betwee the temperature of the sesor ad the heat flux. This approach is very sesitive to oise measuremet magitude, so oe would rather use the power spectral desity (PSD) istead of the correlatio fuctio. I practice, it cosists i applyig the Fourier trasform o the cross correlatio ad auto correlatio fuctios, i. e.: ad FFT Cy FFT τ = h t τ C τ dτ = Y f Φ f = S y f ()

9 Y ( f ) ad ( f ) FFT C = FFT t d = Φ f = S f () ( τ ) ( τ ) ( τ ) τ Φ are the Fourier trasforms of the temperature ad the heat flux respectively as well as S ( f ) ad S ( f ) y are the auto ad cross PSD. The, by applyig the Fourier trasform o relatio (8) it is immediately obtaied: S f = H f S f + S f (3) y e Fially, assumig that the oise measuremet is ot correlated with the heat flux ( S ( f ) expressio of the trasfer fuctio is: H ( f ) S ( f ) ( f ) e = ), the y = (4) Sice the legth of the experimet is set to a fixed value τ, the real iput sigal is: I this relatio, the heat flux leads to: S ( t) ( t) ( t) Π = Π τ (5) Π t = whe t τ ad elsewhere. The applyig the Fourier trasform o τ ( f ) ( f ) ( π τ f ) si τ π τ f Φ Π = Φ (6) It appears that the Fourier trasform of the heat flux is covoluted by the sius cardial fuctio. Usually, the heat flux is pre widowed by a specific fuctio g ( t) the fuctio ( t) Π as: τ ( t) ( t) g ( t) For example, it is ofte used of the Haig widow defied by: τ which decreases the ifluece of Π = τ (7) g τ ( t) π t =.5 cos τ (8) It is used a improved estimatio of S ( f ) ad S ( f ) y proposed by Welch. The method cosists i dividig the time series data ito possible overlappig segmets, computig the auto ad cross power spectral desities ad averagig the estimates.. Applicatio Durig the tutorial we will geerate kids of heat flux sequece:

10 .. liear swept-frequecy cosie sigal We will cosider first a heat flux o the form of a liear swept-frequecy cosie sigal: The frequecy varies liearly with time as: ( t) ( π f ( t) t) = si (9) f f f t f t = +, t t t (3) f is the istataeous frequecy at time, ad f is the istataeous frequecy at time t (see Figure 8) Figure 8: example of liear swept-frequecy cosie sigal with f =.Hz ad f = Hz. It will be used the Welch techique. The swept-frequecy cosie heat flux waveform has two major iterestig features. The first is that the offset must be easily removed from the experimetal heat flux i order to fully satisfy the relatio (9). The secod feature is that the explored frequetial domai, defied from the sesitivity aalysis, is perfectly swept. Furthermore, the use of auto ad cross power spectral desity fuctios allows defiig the so called coherece fuctio as: C y ( f ) y y y f S = (3) S f S f This fuctio ca be viewed as the correlatio coefficiet betwee the temperature ad the heat flux ad lies betwee ad. If it is at a certai frequecy, the there is perfect correlatio betwee the two sigals at this frequecy. I other words, there is cosequetly o oise iterferig at this frequecy, what lead to: ( ) S f = S f C f (3) e y y y

11 .. PRBS sigal I a secod stage we will cosider the heat flux sequece as a Pseudo Radom Biary Sigal (PRBS). White oise is the term give to completely radom upredictable oise, such as the hiss you hear o a utued radio. It has the property of havig compoets at every frequecy. A pseudo-radom biary sequece (PRBS) ca also have this property, but is etirely predictable. A PRBS is rather like a log recurrig decimal umber- it looks radom if you examie a short piece of the sequece, but it actually repeats itself every m bits. Of course, the larger m is, the more radom it looks. You ca geerate a PRBS with a shift register ad a XOR gate. Coectig the outputs of two stages of the shift register to the XOR gate, ad the feedig the result back ito the iput of the shift register will geerate a PRBS of some sort. Some combiatios of outputs produce loger PRBSs tha others- the logest oes are called m-sequeces (where m meas maximum legth ). A biary sequece (BS) is a sequece of N bits, a j for j =,,...,N, i.e. m oes ad N m zeros. A BS is pseudo-radom (PRBS) if its autocorrelatio fuctio: has oly two values: C v N = a j a j+ v (33) j= C v m, if v mod N = m c,otherwise (34) where c = m N (35) is called the duty cycle of the PRBS Figure 9: example of a PRBS sequece (X axis is the umber of samples).

12 A PRBS is radom i a sese that the value of a a j elemet is idepedet of the values of ay of the other elemets, similar to real radom sequeces. It is 'pseudo' because it is determiistic ad after N elemets it starts to repeat itself, ulike real radom sequeces, such as sequeces geerated by radioactive decay or by white oise. The PRBS is more geeral tha the -sequece, which is a special pseudo-radom biary sequece of bits geerated as the output of a liear shift register. A -sequece always has a / duty cycle ad its umber of elemets N = k. 3 Parametric idetificatio 3. AR models, theoretical backgroud As expressed by relatio (5) heat trasfer i a fi is modelled as: θ exp( k x) ( x, p) = ψ ( p), with λ S k k p h s α λ S = + (36) We ca write that: p h s h s λ S k = + = p + (37) α λ S λ S α h s Usig the series it is foud: O the other had oe has: h s λ S h s S k = p + = p λ S α h s λ S α h s e z = ( )! λ = ( )! (38) z =, z (39)! Replacig k i the expoetial fuctio ad this former with its series, it is foud that o obtai a equivalet expressio of relatio (36) o the followig form: Where α ad + α s θm ( s) = β s Φ ( s) (4) = = β have complex but aalytical expressios. Give to the iitial coditio (temperature is zero at each poit of the domai) ad usig the property: it thus appear that relatio (4) is equivalet to: d f t k d f L = s F ( s) s (4) dt k= dt

13 ( t) + d T M, t d α = β (4) = dt = dt Usig the discrete form of the derivatives a equivalet form of relatio (4) that lead to express the temperature at time k t from the heat flux ad the temperature at previous times as: T = b + b + + b a T a T L L, k (,, N ) k k k k k b k k b k a k a I this equatio the cotiuous time t have bee discretized i N samplig itervals of legth as T N t =, where T is the fial time. Parameters ( a, a,, a, b, b, b ) a b = L (43) t, such L L are real umbers. The iteger k correspods to the lag time that depeds o the distace betwee the output ad the iput. This laggig effect oly depeds o the distace betwee the tip of the isert ad the poit M. Sice the white oise term e( k ) defied by ek = Tk Yk (44) eters as a direct error i the differece equatio, the model (43) is ofte called a equatio error model. By substitutig equatio (44) i equatio (43) oe obtais where is the regressio matrix ad is the vector of ukows. Y = H Θ + e (45) k k k [ Y Y Y ] H = L L (46) k k k k a k k k k k k b [ a a a b b ] = L L (47) Θ a b By performig N successive measuremets, relatio (45) ca be writte o a matrix form as where Y = ΗΘ + E (48) [ Y Y Y ] T Y = d d + L N (49) [ e e e ] T E = d d + L N (5)

14 Η H H M H d d + = N (5) ad Resolutio of equatio (48) i the least square sese leads to The choice of = [ a, b, k] d = max a +, b + k + (5) ˆ T T Θ = Η Η Η Y (53) Λ is made by collectig i a matrix all the values of Λ to be ivestigated ad lookig o the value of the Aikake criterio defied by + N Ψ = V, = a + b + (54) N where is the total umber of estimated parameters ad V is the loss fuctio defied by V = (55) N ek k= Stadard errors of the estimates are calculated from the covariace matrix of ˆΘ. If the assumptios of additive, zeros mea, costat variace is expressed as A estimate of the variace σ, deoted σ ad ucorrelated errors are verified, the covariace matrix cov ( ˆ T ) = σ s, is Θ H H (56) s = N E E (57) T From equatio (53), by substitutig the estimated vector ˆΘ i equatio (48), oe fids ˆ T T Thereby, asymptotic estimatio of the mea of ˆΘ is Θ = Θ + H H H E (58) ˆ T T = + k k k k E E E e Θ Θ H H H (59) From this last relatio it is clear that the estimatio is ubiased if e k is ucorrelated with E[ e k ] =. Thereby, the typical whiteess test is to determie the covariace estimate k or if

15 N ek r k r N r= Idepedece betwee e ad q is tested usig the sample covariace R = e e (6) R N e k er k r N r= = (6) It may be oted that whe usig a model as equatio (43), the least square procedure automatically makes the correlatio betwee e k ad k δ zero for δ = k, k +, L, k + b. I practice, relatios (6) ad (6) are computed usig the Fourier Trasform of the covolutio product. 3. Applicatio The method will be applied durig the tutorial startig from the respose to a PRBS sequece for the heat flux. 3.3 NI models, theoretical backgroud At very short times, heat losses are egligible face to the heat diffusio i the fi. Therefore: Sice: p h s p ad k α λ S α (6) θ ( x, p) = ( k x) λ S k =! ψ ( p), with k p h s α λ S = + (63) Replacig k i this relatio leads to: θ x p / α! / = ( x, p) = ψ ( p) λ ρ C S p p / (64) That ca be also writte as: Sice we have show i L8 course that relatio: x λ ρ C S p θ x p p ψ p p (, ) = (65) / / / = α! ν ν d f t ν k d f L s F = ( s) s (66) ν dt k = dt remais true eve if ν is a real umber or more geerally complex, we ca express relatio (65) i the time domai as:

16 α D = β (67) { T } D { } m t t = With: x α = λ ρ Cp S ad β = (68) / α! It appears thus tha model (4) will ot be accurate eough to describe the trasiet behaviour at the short times. As said i lesso 8, a optimal structure of a low order model for heat trasfer problem by diffusio must be of the followig form: D m = = { T ( t) } = D { ( t) } α β (69) 3.4 Applicatio Durig the tutorial, we will apply the NISI method to estimate parameters i relatio (69) i order to fit temperature measuremets at the sesor startig from a PRBS of the heat flux.