Thermal system identification for large temperature ...

Thermal system identification for large temperature variations using fractional Volterra series A. Maachou ∗ R. Malti ∗ P. Melchior ∗ J-L. Battaglia ∗∗ A. Oustaloup ∗ and B. Hay ∗∗∗ ∗

Université de Bordeaux 1, IPB, IMS UMR 5218 CNRS, 351 cours de la Libération, Bˆ at A4, 33405 Talence Cedex, France {asma.maachou, rachid.malti, pierre.melchior}@ims-bordeaux.fr ∗∗ Université de Bordeaux 1, ENSAM, TREFLE UMR 8508 CNRS, Esplanade des Arts et Métiers, 33405 Talence Cedex, France [email protected] ∗∗∗ LNE, Centre Métrologie et Instrumentation, 29 avenue Roger Hennequin, 78197 Trappes Cedex, France, [email protected]

Abstract: Linear fractional differentiation models have proven their efficacy in modeling thermal diffusive phenomena for small temperature variations with constant thermal parameters (thermal diffusivity and conductivity). However for large temperature variations, the thermal parameters are no longer constant but vary along with the temperature itself. Consequently, the thermal system is no longer linear and could be modeled by non linear fractional differentiation models. The extension of Volterra series to fractional systems is first recalled. Fractional orthogonal generating functions are used as kernels. As compared to a previous work, non linear parameters, such as sν -poles of the orthogonal functions and commensurate differentiation order, are estimated along with linear coefficient of Volterra series. Then, Volterra series are applied to model an ARMCO iron sample for large temperature variations in simulation with data generated using finite elements method and in a real life experiment using real data. Keywords: System identification, Fractional models, Non linear model, Volterra series. 1. INTRODUCTION The work presented in this paper is a part of the CONTROLTHERM project supported by the french “Agence Nationale de la Recherche”. The project aims to develop a new device for temperature measurement and follow-up in machining tools in severe conditions of materials with high added value. Models between the injected thermal flux at the cutting edge and the temperature measured at the probe are hence required. Machining in mild conditions involve fractional differentiation models Battaglia et al. (2004). Machining in severe conditions involve high temperature variations and hence require non linear models Maachou et al. (2010a). Different system identification methods based on fractional linear models were developed in the literature (Oustaloup (1995)). Cois et al. (2000) extended the output error model by using a modal representation. Cois et al. (2001) proposed to use a prediction error method combined to instrumental variable and state variable filters. This method was enhanced in Malti et al. (2008) by choosing optimal instrumental variables. Battaglia and Kusiak (2005) implemented recursive least squares for parameters estimation. Aoun et al. (2007); Malti et al. (2005) used fractional orthogonal basis for system approximation. ⋆ This work is supported by the french “Agence Nationale de la Recherche” in the scope of CONTROLTHERM project.

Some fractional non linear models have been proposed in the literature such as fractional Hammerstein models in Aoun et al. (2002). Fractional multimodels were first proposed in Malti et al. (2003) and then used in modeling of salamander muscle in Sommacal et al. (2008). Fractional neural networks were used for modeling diffusion interface in thermal systems in Benoˆıt-Marand (2007). For thermal systems Maachou et al. (2010b) recalls that fractional systems are well suited for the diffusive character of thermal systems but that, for large temperature variations, linear models are not accurate enough. Hence, they propose to use Volterra series which kernels are fractional orthogonal generating functions. Nevertheless, only linear coefficients of Volterra series were estimated in Maachou et al. (2010b). This paper completes the previous one by also estimating the non linear coefficients, such as the sν poles and the commensurate differentiation order of the generating functions. This method is set to identify the thermal behavior of an industrial cutting tool in severe conditions subject to high input flux, causing large temperature variations. Volterra series are described in section 2 as a generalization of linear models. The chosen structure for the Volterra kernels is explained in section 3. Non linear parameters estimation is developed in section 4. Finally, The system identification method using Volterra series is tested to model thermal diffusion in an ARMCO iron sample for large temperature

variations, first on simulated data generated using finite elements and then on data acquired in a real life experiment. 2. VOLTERRA SERIES: FROM LINEAR MODELS TO NON LINEAR MODELS Volterra series are relevant for slightly non linear systems Bard (2005) such as thermal ones Maachou et al. (2010a). This tool may be expressed in the time domain as well as in the frequency domain. Given a linear system characterized by its impulse response h1 (t) and an input signal u(t), then the output y1 (t) is expressed as the convolution product: Z +∞ y1 (t) = h1 (τ )u(t − τ )dτ. (1) −∞

For a non linear system, a model structure of Volterra series is written as the infinite sum: ∞ X y(t) = yk (t) = y1 (t) + y2 (t) + ... + yk (t) + ..., (2) k=0

where yk is the generalization of the convolution product (1) to the order k: Z ∞ Z ∞ k Y yk (t) = ··· hk (τ1 , · · · , τk ) u(t − τi )dτi . (3) −∞

−∞

i=1

Therefore, by combining equations (2) and (3), the output y(t) of a non linear system may be expressed as: Z ∞ ∞ Z ∞ k X Y y(t) = ··· hk (τ1 , · · · , τk ) u(t−τi )dτi , (4) k=0

−∞

−∞

i=1

where hk is the k th order Volterra kernel and its Laplace transform Hk (s1 , · · · , sk ) is given by Crum and Heinen (1974): Hk (s1 , · · · , sk ) = Z ∞ Z ∞ ··· hk (τ1 , · · · , τk )e−s1 τ1 ···−sk τk dτ1 · · · dτk . (5) −∞

−∞

Moreover, the Laplace transform of the convolution integral (3) can be written as: Yk (s1 , · · · , sk ) = Hk (s1 , · · · , sk )U (s1 ) · · · U (sk ). (6) Therefore, the Laplace transform of the output signal y(t) in (2) is: Y (s) = Y1 (s1 )+Y2 (s1 , s2 )+· · ·+Yk (s1 , · · · , sk )+· · · , (7) where  Y1 (s1 ) = H1 (s1 )U (s1 )    Y2 (s1 , s2 ) = H2 (s1 , s2 )U (s1 )U (s2 ) (8) ..   .   Yk (s1 , · · · , sk ) = Hk (s1 , · · · , sk )U (s1 ) · · · U (sk ). Obviously, the first order kernel H1 (s1 ) corresponds to the linear part of the system. All other kernels contain system non linearities. For more simplicity, all kernels are supposed to be symmetric: hk (τ1 , τ2 , · · · , τk ) = hk (τi1 , τi2 , · · · , τik ), (9) where (i1 , i2 , · · · , ik ) is any permutation of (1, 2, · · · , k). 3. VOLTERRA SERIES WITH FRACTIONAL DIFFERENTIATION KERNELS As the behavior of a system modeled by Volterra series is described by its kernels, identification consists in first

choosing an expression for the kernels hk and then computing optimal parameters. In Aoun et al. (2007), Malti et al. (2005), orthonormal bases functions are extended to fractional differentiation order by applying a Gram Shmidt orthogonalisation procedure on a set of generating functions Fn . As a consequence their inverse Laplace transform fn (t) form a basis in L2 [0, ∞) Aoun et al. (2007), Ak¸cay (2008). Hence, the linear part of the Volterra series h1 (t) can always be approximated using orthogonal functions: M X h1 (t) = bn fn (t), (10) n=1

where bn is the Fourier coefficient associated to fn , and M the truncation order. The fractional Laguerre generating functions Fn (Aoun et al. (2007)), defined by 1 , (11) Fn (s) = ν (s + λ)n are characterized by the presence of a single real sν -pole at λ. Nevertheless, when system sν -poles are real or complex conjugate, fractional generalized orthonormal basis (Malti et al. (2005)) may be used. Its generating functions are iteratively defined as: 1 φn−1 if λn is real or Fn = ν s + λn  ν s φn−1 Fn′ = and (sν +λn )(sν +λn ) (12) φ n−1 Fn′′ = ν if λn is complex (s +λ )(sν +λ ) n

n

with  φ0 = 1 φ = Fn−1 if n > 1 and λn−1 is real or  n−1 ′ ′′ φn−1 = Fn−1 or Fn−1 if n > 1 and λn−1 is complex . (13)

Following the development in Bibes (2004) for rational orthonormal basis, the Volterra kernel hk (τ1 , · · · , τk ) is a multivariable function defined on L2 ([0, ∞)k ), which can be expanded as: hk (t1 , · · · , tk ) = ∞ ∞ X X ··· bm1 ,··· ,mi fm1 ,··· ,mk (t1 , · · · , tk ). m1 =1

(14)

mk =1

Therefore, yk in equation (2) can be approximated by: yk (t1 , · · · , tk ) = Z

0

M X

m1 =1

mk−1

···

X

cm1 ,··· ,mk ×

mk =1

t

fm1 (τ1 )u(t − τ1 )dτ1 · · · fmk (τk )u(t − τk )dτk

(15)

with M the number of fractional orthogonal generating functions used to approximated the kernels and mk the kernel order. The k th order kernel is presented in Fig. 1. The number of unknown linear coefficients cm1 ,··· ,mi associated to the k th order kernel equals k-combinations with repetitions among M generating functions: (M + k − 1)! . (16) Pk = (M − 1!)k! Then, the number of parameters of an N th order Volterra model equals:

f1 (t) f1 (t)

f1 (t) Q

c1,··· ,1

×

c1,1

×

c1,2

×

c1,3

f1 (t) f1 (t) f2 (t)

fm1 (t) U fmk (t)

Q

cm1 ,··· ,mk

+

Yk

f1 (t) f3 (t) U

f2 (t)

fM (t)

fM (t)

Q

P =

k=1

×

c2,2

×

c2,3

×

c3,3

f2 (t)

N X (M + k − 1)! . Pk = (M − 1!)k!

Y2

f2 (t)

cM,··· ,M

f3 (t)

Fig. 1. k th order Volterra kernel developed using M generating functions N X

+

f3 (t) f3 (t)

(17)

k=1

Moreover, the number of unknown non linear coefficients λ1,··· ,M and ν, corresponding to the M sν -poles associated to generating functions and the commensurate order, equals M + 1. Obviously, the number of parameters increases considerably with the truncation order M of the generating functions (12) and with the truncation order of Volterra kernels N . For example, if the number of generating functions, M , is set to 3 and the number of Volterra kernels, N , is set to 3, then the number of linear coefficients equals 19, according to (17) and the number of non linear coefficients equals 4 (3 sν -poles and a commensurate order), which yields a total number of 23 parameters. In case of using 3 generating functions and 2 Volterra kernels the total number of parameters reduces to 9 linear coefficients and four non linear parameters: Θ = [λ1 λ2 λ3 ν]T . (18) Consequently, this paper presents results for Volterra model truncated to a quadratic order kernel with three generating functions (hence three poles). A quadratic kernel developed on a basis of three generating functions is illustrated in Fig. 2. 4. PARAMETER ESTIMATION

Fig. 2. Quadratic Volterra kernel developed using 3 generating functions Y (s) = Y1 (s) + Y2 (s1 , s2 ) with  Y (s) = c1 F1 (s)U (s) + c2 F2 U (s) + c3 F3 U (s)  1     Y (s , s 2 1 2 ) = c11 F1 (s1 )U (s1 )F1 (s2 )U (s2 )+     c12 F1 (s1 )U (s1 )F2 (s2 )U (s2 )+  (19) c13 F1 (s1 )U (s1 )F3 (s2 )U (s2 )+    c22 F2 (s1 )U (s1 )F2 (s2 )U (s2 )+     c23 F2 (s1 )U (s1 )F3 (s2 )U (s2 )+    c33 F3 (s1 )U (s1 )F3 (s2 )U (s2 ). It may also be written as : Y (s) = XC, (20) with C T = [ c1 · · · c3 c1,1 · · · c3,3 ], X = [ x1 , · · · , x3 , x21 , x1 x2 , · · · , x23 ] , and xi = Fi (sk )U (sk ). Optimal coefficients are computed by minimizing the quadratic norm of the output error: K−1 100 X ∗ Jid = (yid (kh) − L −1 {G(s)} ∗ uid (kh))2 . (21) K k=0

For comparison purposes the following validation criterion is calculated on validation data: K−1 100 X ∗ Jval = (yval (kh) − L −1 {G(s)} ∗ uval (kh))2 . (22) K k=0

According to prior knowledge, a linear or non linear programming method can be employed for parameter estimation. The linear programming method has already been discussed in Maachou et al. (2010a): if the fractional order and the poles are chosen according to some prior knowledge, then only the coefficients cm1 ···mk have to be estimated. So Y (s), the Laplace transform of the output y(t), developed on three fractional generating functions is constituted of the first order kernel Y1 (s) (linear part) and the second order kernel Y2 (s1 , s2 ), illustrated in Fig. 2:

Parameter vector C can hence be computed using least squares: b = (X T X)−1 X T Y. C (23) The non linear programming (NLP) method is used when the fractional order and the poles are computed along with the linear coefficients. Hence, the NLP method is combined to linear programing as sketched in Fig. 3.

The first step consists in initializing Θ (18) to construct a first set of three generating functions F1 , F2 , F3 , and

Fig. 4. Thermal characteristics of ARMCO iron for different temperature in Kelvin Fig. 3. Parameter estimation method commensurate order ν. The three fractional orthogonal generating functions are initialized by fractional linear system identification to capture the dynamic behavior of the system whereas the commensurate order ν is set to 0.5 according to the physical model. In the second step, vector of linear coefficients C is estimated using least squares method (23). In step three, the vector Θ (18) is optimized using a non linear programming. The Levenberg-Marquard method implemented in the MATLAB function lsqnonlin is used. The second and the third steps are iterated till convergence or till the maximum number of iterations is reached. Identification an validation criteria of the linear model are compared to those of the non linear Volterra model to highlight the improvements.

Thermocouple

40

Sample 10

15

5. ARMCO SAMPLE IDENTIFICATION FOR LARGE TEMPERATURE VARIATIONS

d

S

The LNE, “Laboratoire National de métrologie et d’Essais”, the french national laboratory specialized in measurements and tests, has measured the thermal characteristics of the ARMCO iron used in the real experiment. They are given in Fig.4. Both thermal conductivity and specific heat vary along with the temperature, which induce non linearities in the thermal model. Moreover, around 1000K, the thermal characteristics variations are more important due to Curie temperature 1 . For large thermal characteristics variations, linearity hypothesis is not verified. Hence, a non linear fractional model using Volterra series is used. First the thermal system is identified using simulated data based on finite elements method and COMSOL software. Then the thermal system is identified using a real life data experiment. The former data allow to have high temperature variations (700K and more if necessary) whereas the latter are limited to 140K due to physical constraints . During machining in severe conditions, temperature variations go as high as 700K.

1

In physics and materials science, the Curie temperature (Tc), or Curie point, is the temperature at which a ferromagnetic or a ferrimagnetic material becomes paramagnetic.

Fig. 5. ARMCO sample pattern 5.1 Identification of the ARMCO sample from simulated data Fig. 5 shows the geometric pattern of the ARMCO iron sample. The geometric pattern is created in the application mode for heat transfer analysis by conduction and for a transient analysis in 2D axial symmetry. To amplify the sensitivity to the flux, the thermocouple of type K (diameter 0.5 mm) is placed as near as possible (d = 1 mm) from the surface S where the flux is injected. Nearer the thermocouple, more significant the s0.5 term (Maachou et al. (2010a)). Heat transfer is modeled by finite elements with the default quadratic Lagrange elements. Comsol software simulates an ideal configuration, that means with neither measurement noise nor separation between the thermocouple and the sample. Measurement noise and the air or the glue between the components may change system thermal characteristics. A white noise p(t) of 20 dB is added to the Comsol noise-free output y(t) : y ∗ (t) = y(t) + p(t).

(24)

Table 1. Identification and validation criteria for a linear model versus the number of poles M for simulated data 1 6.16 5.65

2 3.86 3.42

3 1.51 2.59

4 1.28 2.65

6

15

6 1.32 2.57

600

6

x 10

15

10

10

5

5

0

0

0

5 1.39 2.40

700

Temperature variations (K)

M Jid × 104 Jval × 104

−Comparison between NLP Volterra model output, linear model output and system output− 800

2

4

6

8

10

800

800

600

600

400

400

200

200

0

0

x 10

NLP Volterra series model output first kernel output linear model quadratic kernel output Noise free system output

500

400

300

200

0

2

4

6

8

10

100

0

−200

0

2

4

6

8

10

−200

−100 0

0

2

4

6

8

Fig. 6. Left: identification data, Right: validation data Table 2. Identification and validation criteria according to the model for simulated data model Jid × 104 Jval × 104

Linear 1.51 2.60

1

2

3

4

5 Time (s)

6

7

8

9

10

10

Fig. 7. Comparison between system output, linear model output, Volterra series model output and its first two kernels on validation data. Ambiant temperature of about 300K was subtracted.

2nd order Volterra 0.99 1.61

Moreover, the sampling time is taken as h = 0.01 second which is reachable by the acquisition system of the real life experiment. This sampling time allows to settle the highest input signal frequency. The input signal, with a magnitude of 0 to 12 × 106 M W.m−2 and a total duration of 10 seconds, is plotted in Fig.6. Two sets of data are generated, one for system identification and the other for model validation as shown in Fig. 6. For comparison purposes, the system is identified with a linear model too. It also allows to initialize Θ in (18) for the non linear model. Table 1 shows that the truncation at M = 3 is a good choice because for M ≥ 3 the identification and the validation criteria do not significantly improve. The three estimated poles and the commensurate order ν = 0.5 are used for the initialization of Θ in (18) for the non linear model: Θ = [0.5344 + 2.3154i, 0.5344 − 2.3154i, −0.0144, 0.5]T . (25) The Θ vector converges after optimisation to: b = [0.4665 + 2.2673i, 0.4665 + 2.2673i, −0.0192, 0.4904]T . Θ (26) System and Volterra model outputs are plotted on validation data in Fig. 7. Moreover, contribution of each kernel of the Volterra series is highlighted which allows to see the contribution of the the linear part and the non linear part of the model. The linear model is also plotted for comparison purposes. The Volterra model fits better system output as compared to the linear model. As shown in table 2, the identification criterion reduces from 1.51 to 0.99 and the validation criterion from 2.60 to 1.61. Moreover, the estimated parameters of the linear model are biased due to the presence of nonlinearities which are better captured in the second order Volterra series model.

Fig. 8. Test bench at the TREFLE (Bordeaux) 5.2 Identification of the ARMCO sample from real life experiment Secondly, Input output data are generated using a real life experiment set at Bordeaux in the TREFLE laboratory. This experiment presented in Fig.8 consists in injecting a thermal flux generated by a 200 Watts laser, with a wavelength of 1µm and controlled by an internal function generator modulated with a maximum frequency of 10 kHz, to an ARMCO iron sample. A thermocouple is positioned inside the sample at a distance d from the surface S where the flux is injected. The excited surface S is a 1.25 mm radius disc. The aim is to provide enough energy to induce large temperature variations. As the thermal characteristics variations are more important around the Curie point (1000K) , the ARMCO sample is heated to 950◦ C. A conductor thread is rolled up around the ARMCO iron, and a current is applied to heat the sample. The acquisition is done with a sampling period h = 0.01 second. The input signal, a voltage from 0 to 5 V which

Input signal for the validation data 5

4

4 Flux command (V)

3 2

3

140

2 1

1 0 0

Comparison between NLP Volterra output, linear model output and system output 160

5

10

15

20

0 0

25

10

20

25

Output signal for the validation data

Output data for the identification data

200 Temperature variations (K)

Temperature (K)

15 time (s)

150

100

50

0 0

5

10

NLP Volterra model output linear model system output quadratic kernel output first kernel output

120 5

time (s)

15

20

25

150

100

50

0 0

time (s)

5

10

15

20

25

time (s)

Fig. 9. Left: identification data, Right: validation data Table 3. Identification and validation quadratic criteria according to the model model Jid × 105 Jval × 105

Linear 1.18 2.56

2nd order Volterra 0.74 2.20

commands the laser from 0 to 200 Watts, is plotted in Fig.9. Two sets of data are generated, one for system identification and the other for model validation as shown in Fig. 9. For comparison purposes, the system is identified with a linear model too. It also allows to initialize Θ in (18) for the non linear model. It shows that the truncation at M = 3 is a good choice. The three estimated poles and the commensurate order ν = 0.5 are used for the initialization of Θ in (18) for the non linear model: Θ = [0, 0132 + 1.4144i, 0.0132 − 1.4144i, −0.0013, 0.5]T . (27) The Θ vector converges after optimisation to: b = [−0.0617+1.3137i, −0.0617−1.3137i, −0.0244, 0.4918]T . Θ (28) System and Volterra model outputs are plotted on validation data in Fig. 10. Moreover, contribution of each kernel of the Volterra series is highlighted which allows to see the contribution of the the linear part and the non linear part of the model. The linear model is also plotted for comparison purposes. The Volterra model fits better system output as compared to the linear model. As shown in table 3, the identification criterion reduces from 1.18 to 0.74 and the validation criterion from 2.56 to 2.20. Moreover, the estimated parameters of the linear model are biased due to the presence of nonlinearities which are better captured in the second order Volterra series model. Non linearities are even more obvious for a cutting tool in severe conditions undergoing large temperature variations, not reachable here because of the limited power of the laser. 6. CONCLUSION Linear fractional models have proven their efficiency in modeling thermal diffusive phenomena for small temperature variations. However for large temperature variations, thermal systems are no longer linear. Thermal diffusion in an ARMCO sample is identified, in this paper, for high temperature variations using Volterra series with kernels defined as fractional orthogonal generating functions. Model parameters are estimated using a non lin-

Temperature Variations (K)

flux command (V)

Input signal for the identification data 5

100

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−20 0

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25

time (s)

Fig. 10. Comparison between system output, linear model output, Volterra series model output and its first two kernels on validation data. Static temperature of about 950K was subtracted. ear programming for a Volterra model truncated to the second kernel. All the linear and non linear parameters are estimated, first on simulated data and, then in a real life experiment. The identified non linear models fit better system output as compared to the linear model. REFERENCES Ak¸cay, H. (2008). Synthesis of complete orthonormal fractional basis functions with prescribed poles. IEEE Transactions on signal processing, 56(10), 4716 – 4728. doi:10.1109/TSP.2008.928163. Aoun, M., Malti, R., Cois, O., and Oustaloup, A. (2002). System identification using fractional Hammerstein models. In 15th IFAC World Congress. Barcelona Spain. Aoun, M., Malti, R., Levron, F., and Oustaloup, A. (2007). Synthesis of fractional Laguerre basis for system approximation. Automatica, 43(9), 1640–1648. doi:DOI: 10.1016/j.automatica.2007.02.013. Bard, D. (2005). Compensation des non linéarités des systèmes haut-parleurs ` a pavillon. Ph.D. thesis, Ecole polytechnique fédérale de Lausanne. Battaglia, J.L. and Kusiak, A. (2005). Estimation of heat fluxes during high-speed driling. Int. J. of Adv. Manuf. Technol., 39, 750–758. Battaglia, J.L., Puigsegur, L., and Kusiak, A. (2004). Représentation non entière du transfert de chaleur par diffusion. Utilité pour la caractérisation et le contrˆ ole non destructif thermique. Int. J. of Thermal Science, 43, 69–85. Benoˆıt-Marand, F. (2007). Modélisation et identification des systèmes non linéaires par réseaux de neurones a temps continu. Apllication ` ` a la modélisation des interfaces de diffusion non linéaires. Ph.D. thesis, Université de Poitiers. Bibes, G. (2004). Modélisation de procédés de traitements des eaux et reconstruction de grandeurs physicochimiques. Ph.D. thesis, Université de Poitiers. Cois, O., Oustaloup, A., Battaglia, E., and Battaglia, J.L. (2000). Non integer model from modal decomposition

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