Robust Estimation of temperature fieldduring microwave tempering under

unknown dielectric properties

L. Boillereaux ∗ M. Alamir ∗∗ S. Curet ∗ O. Rouaud ∗P. Bellemain ∗∗

∗GEPEA, UMR CNRS 6144, ONIRIS, rue de la geraudiere, BP82225, Nantes. (Tel: +33 2 51785477 / Email:

lionel.boillereaux@oniris-nantes.fr)∗∗ CNRS-University of Grenoble

Gipsa-Lab, Control Systems Department.,Rue de la Houille Blanche, Domaine Universitaire,

Saint-Martin d’Heres, France(Tel: +33 4-76-82-63-26)/(Email:mazen.alamir@grenoble-inp.fr)

Abstract: In this paper, a framework is proposed for the estimation of internal temperatureswithin a foodstuff during microwave tempering under unknown dielectric properties. Thesolution is based on a partially known model issued from the closed form solutions of Maxwell’sequations coupled with the conduction heat equation solved by finite differences. The algorithmconverges towards acceptable dielectric properties functions whereas the temperature fieldis estimated simultaneously. This approach is carried out in simulation by considering thetempering of a block of raw beef located in a rectangular wave guide allowing to consider afundamental mode with perfectly known electromagnetic conditions.

Keywords: Nonlinear Model Predictive Control; Fast Systems; Real-Time Implementation;Control Parametrization; PVTOL.

1. INTRODUCTION

In the food industry, tempering consists in bringing afrozen foodstuff close to its transition temperature, with-out thawing. Such an operation permits to reach a suffi-cient malleability to allow mechanical operations, as mix-ing, slicing. The main objective is that the whole productreaches this target temperature homogeneously, by avoid-ing partial defrosting, to preserve it from a renewal ofmicro-organisms activity. However, microwave temperingpresents a major drawback, inherent to the use of mi-crowaves: the thermal runaway. This phenomenon is char-acterized by the apparition of hot points within the foodsample, next to areas still frozen (Liu et al. [2005]). Thesehot points are due to resonance phenomena occurringwhen the penetration depth of the electromagnetic wave islarger than the thickness of the product (Bhattacharya andBasak [2006]). This happens when the treated materialsare low dielectrics, as it is the case for frozen foodstuffs.

For this reason, despite its great interest, microwave tem-pering processes are still scarcely used in the industry. In-deed, the adjustment of microwave ovens is made difficultbecause the magnitude and location of these hot spots aredifficult to grasp, especially as the dielectric parametersare usually unknown in the frozen phase. Indeed, in theliterature, many studies are dedicated to dielectric prop-

? This work has been supported by the French National ResearchAgency Project ANR-CLPP

erties measurements of food sample by using the openended coaxial dielectric probe kit (Farag et al. [2008],Herve et al. [1998], Hu and Mallikarjunan [2005], Maoet al. [2003], Motwani et al. [2007], Nelson and Bartley[2000], Zhang et al. [2004]), but these measurements aremostly performed with defrosted food products (T > 0 ◦C)due to lower accuracy below freezing temperature. Themicrowave cavity perturbation technique is also widelyused to measure dielectric properties but few applicationsare dedicated to food products (Bengtsson and Risman[1971]). This method consists in measuring frequency shiftand change in cavity transmission characteristics when asmall dielectric object is inserted into a microwave reso-nant cavity (Kraszewski and Nelson [1996]). Recently, thismethod has also been employed for permittivity measure-ment of low loss dielectric materials (Li et al. [2007]). Thistechnique seems to be appropriate to frozen food due totheir low dielectric characteristics. However, the design ofthe cavity needs to be adapted to the sample dimensionand also to the frequency of the electromagnetic wave,i.e. 2.45 GHz. For those reasons, dimensions of the cavityare of crucial importance to obtain accurate dielectricmeasurements and, in the case of solid foods, the samplepreparation and the design of the cavity are also very time-consuming. It is important to underline at this stage thatthe objective of this work is not to obtain an accurate eval-uation of the dielectric properties versus temperature, butto reach a satisfying estimation of the temperature field inpresence of these parameter uncertainties. The monitoring

of temperatures within the foodstuffs in the presence oflarge uncertainties on the dielectric properties evolution isconsequently an interesting challenge. Obviously, in sucha context, invasive measurements are prohibited, limitingthe temperature measurement at the boundaries. To reachsuch an objective, the use of state observers is manda-tory. For nonlinear processes, observer design is a cumber-some task: indeed, on the first hand, analytic observers(Gauthier et al. [1992], Slotine et al. [1987]) are usuallyapplicable to a limited class of systems and, even whenpossible to use, requires specific skills in mathematics; onthe other hand, optimization based-observers, which canbe applied to complex systems, require some optimization-based approaches which are time-consuming.

The French ANR Project clpp aims to offer to processengineers and academic researchers a user-friendly toolwhich allows to perform online state and parameter es-timation, with limited skills in this field, provided thata model of the process is available. Details about thistool can be consulted in (Alamir et al. [2007]). Such atool seems particularly suited to the objective describedabove. Indeed, state estimation allows to reconstruct thelacking temperatures within the foodstuff, whereas pa-rameter estimation permits to evaluate in real-time thedielectric properties. This tool addresses systems that aredescribed by ordinary differential equations. Because heattransfer and electromagnetic wave are described by partialdifferential equations, a spatial meshing of the domain isproposed in order to reduce the distributed parameterssystem to a lumped parameters system. By consideringsmooth variations of the dielectric properties with temper-ature on the considered temperature range, and a constantand normal incident electric field, closed-form solutions areproposed to compute the electric field distribution withinthe food sample. The local electric field obtained fromclosed-form solutions is thus implemented in the sourceterm of a one-dimensional heat equation, solved using afinite difference scheme.

In this work, we propose to evaluate the clpp software insimulation, by considering the tempering of a block of rawbeef irradiated by a monochromatic microwave. To betterevaluate the relevance of the approach, the experimentaldevice, which has been used in previous works (Curet et al.[2008, 2009], Akkari et al. [2009]) to validate the modelingapproaches, is briefly described in section 2. The modeland its numerical resolution coupling closed-form solutionsof the Maxwell’s equations and heat transfer is developedin section 3, whereas section 4 presents briefly the clppsoftware. The results and related discussion are given insection 5.

2. SOME EXPERIMENTAL ISSUES

Although the validation of the proposed estimation ap-proach is based on simulations, the following section de-scribes some experimental issued that are worth under-standing when evaluating the relevance of the overallframework and more precisely of the adopted model. Sim-ulation has been preferred in this issue because it permitsa better evaluation of the estimation errors of the insidetemperatures. Indeed, temperature measurement is verydifficult due to the inherent approximation of the probe

location, especially in frozen foodstuff. However, the modelcoupling closed-form solutions of Maxwell’s equations withheat transfer, and described in the next section, has beenvalidated in previous works for different configurations(one-dimensional or two-dimensional problems, finite dif-ferences or orthogonal collocation concerning heat trans-fer), and it appears important to present briefly in thesequels the experimental device dedicated to the experi-mental validations.

2.1 Experimental device

A monochromatic wave is considered, in the fundamentalmode, denoted TE10, and operating at a frequency of 2.45GHz, delivering a half-sinusoidal electric field along thelarge dimension (86 mm) of a rectangular wave guide, anda constant one along the smallest dimension (43 mm). Inorder to consider a normal and constant incident electricfield at the top surface, the dimension of the sample issmall compared to the wave guide and this sample is in-serted at the centre. It is possible to measure the tempera-tures at the irradiated surface and at the opposite one withoptical fibre sensors (LUXTRON Fluroptic Thermometer,model 790, accurate to ±0.5 ◦C, Luxtron Corporation,Santa Clara, USA). These probes are usually inserted 1mm under the surface.

2.2 Simulation conditions

A piecewise decreasing power flux from 50000 to 12000W/m2 is considered in the clpp software. The sample issupposed to be homogeneous in temperature, at −20 ◦C.The simulation time is calculated in order to cover thewhole tempering range.

3. MODEL DERIVATION

In this section, we develop a one-dimensional finite volumescheme to solve the heat equation. The effect of microwavesis introduced in the heat equation through a source term,which results from the closed-form solution of Maxwell’sequations. In agreement with Fig. 1, the following assump-tions can be considered:Assumption 1: The product receives the electromagneticwave by the upper surface, and the propagation is normalto the surface.Assumption 2: The product is homogeneous and isotropic.Assumption 3: The dielectric properties are constant.Assumption 4: The mass transfer is negligible.Assumption 5: the sample is perfectly insulated on lat-eral and bottom faces.

Heat transfer is based on the generalized heat equa-tion which depends on thermophysical properties of theproduct. With respect to assumption 5, a one-dimensionalproblem can be considered by taking:

ρCp(T )∂T

∂t=

∂

∂z(k(T )

∂T

∂z) +Qabs(z) (1)

Concerning raw beef, analytical devices (differential scan-ning calorimetry and hot wire probe), available in thelaboratory, have permitted to establish the following equa-tions for the specific heat and thermal conductivity on thetempering range (Akkari et al. [2005], Curet et al. [2008]):

Fig. 1. Some loss factor evolutions for food products(Bengtsson and Risman [1971])

Cp(T ) ={

52.9T + 3784 if T < −10 ◦C31785 · e0.231T if T ∈ [−10 ◦C,−2.6 ◦C] (2)

k(T ) =1.4325

1 + e1.3794T+5.7945+ 0.5187 (3)

The source term Qabs quantifies the amount of powerwhich is dissipated into the product by dielectric losses.The source term is computed from the knowledge of thelocal electric field strength as follows (Lanz [1998]):

Qabs(z) =12ω · ε0 · ε”r · |E(z)|2 (4)

This term depends explicitly on the loss factor ε”r andimplicitly on the dielectric permittivity ε

′

r through thelocal field E(z) [see equation (6) below].Let us denote by Dp the penetration depth of the wave inthe material, namely:

Dp =C0

π · f

[2 · ε

′

r ·((

1 +(ε”rε′r

)2)1/2

− 1)]− 1

2

(5)

where ε′

r and ε”r represent respectively the dielectric per-mittivity and the loss factor of the material, C0 = 3 ×108 m/s, f = 2.45 GHz. In the sequel, κ0 =

2πfC0

=

51.3 m−1 is the propagation constant.In a thin layer of frozen material (5 cm in this work),Dp � L, authors in (Bhattacharya and Basak [2006])proposed closed-form solution provided that assumptions1 and 3 mentioned above are satisfied:

Qabs(z) =8πfε”rκ

20

C0× C−

Cd× F0 (6)

where F0 represents the incident power flux in W/m2.

C− = c1 cos(κ2aL(1− z′)) + c2 cosh(κ2bL(1− z

′)) +

+c3 sin(κ2aL(1− z′)) + c4 sinh(κ2bL(1− z

′))

Cd = (c23 − c21) cos(2κ2aL) + (c22 + c24) cosh(2κ2bL)−−2c1c3 sin(2κ2aL) + 2c2c4 sinh(2κ2bL)

c1 = κ22a + κ2

2b − κ20 ; c2 = κ2

2a + κ22b + κ2

0

c3 =−2κ2aκ2b ; c4 = 2κ0κ2a

z′=

2zL

; κ2a =2πλm

; κ2b =1Dp

λm =C0

√2

f

[ε′

r

(√1 +

(ε”rε′r

)2 + 1)]−1/2

On the considered temperature range, during food temper-ing, it is obvious that the dielectric properties are not con-stant. In order to meet assumption 3, we will consider inthe sequel piecewise variable dielectric properties. By ob-serving the evolution of the dielectric properties publishedin the food science literature (Fig. 1), the dielectric proper-ties can be approximated as functions of the temperatureas shown on Figure 2. For tempering application, we aremainly interested in negative temperature. Therefore, theTi’s that are used in the definition of the parametrizationof ε

′

r and ε”r (Figure 2) are chosen as follows:

T1 = −40◦C ; T2 = −10◦C ; T3 = −5◦C ; T4 = −1◦ Conce these temperatures are fixed, the temperature depen-dent profile of ε

′

r and ε”r are respectively defined by theparameters v1, . . . , v4 and v5, . . . , v8.

In order to introduce the constraints on the monotonicityof the profiles over the range of negative temperatures, weconsider the normalized parameter

q ∈ [0, 1]8 ⊂ R8

such that:

v1 = ε′

r,min + q1 · (ε′

r,max − ε′

r,min) (7)

v2 = v1 + q2 · (ε′

r,min − v1) (8)

v3 = v2 + q3 · (ε′

r,min − v2) (9)

v4 = v3 + q4 · (ε′

r,min − v3) (10)

v5 = ε”r,min + q5 · (ε”r,max − ε”r,min) (11)

v6 = v5 + q6 · (ε”r,min − v5) (12)

v7 = v6 + q7 · (ε”r,min − v6) (13)

v8 = v7 + q8 · (ε”r,min − v7) (14)

(15)

where ε′

r,min, ε”r,min, ε′

r,max and ε”r,max are a priori givenlower and upper bounds. Based on the above definition andprovided that the system of partial differential equationrepresented by (1) is transformed to a system of ordinarydifferential equation using appropriate numerical scheme(finite differences for instance), the system model that isused in the estimation algorithm can be described shortlyby the following set of ordinary differential equations ofthe form:

x= f(x, u, q) ; y = h(x) (16)

Fig. 2. Approximation functions for dielectric propertiesin the tempering area

where x is the state representing the temperature at thespacial discretization nodes, u is the vector of exogenoussignals (power profile), q is a set of model parameters whiley is the vector of sensors output that may be used inthe reconstruction of the temperature profile inside thefoodstuff. The principle and the tool used to perform thisinverse problem solution are described in the next section.

4. RECALL ON MOVING HORIZON OBSERVERS

The French National Research project anr 1 -clpp 2 hasbeen initiated based on the observation according towhich, many process researchers, when dealing with theirown works on the control, the supervision and/or theoptimal design of their processes, are quite frequently facedwith the problem of observer design. For non special-ists, this task is cumbersome because analytic observersrarely apply while optimization-based approaches needsome technicalities that may be time-consuming and dif-ficult to master. A user-friendly software (Alamir et al.[2007]) has been developed to facilitate the design of thestate/parameter reconstruction.

To achieve the estimation task for a system given by (16),clpp uses moving-horizon strategy (Michalska and Mayne[1995]) in which the measurements collected during thepast observation horizon [t − T, t] (where t is the currenttime while T is called the observation horizon) are usedto recover the values of the unknown variables. Typically,the latter gather the value of the state vector at instantt− T and the value of the parameter vector q, namely:

p(t) =(x(t− T )

q

)More generally, clpp enables a flexible definition of theparametrization by allowing the user to define its ownparametrization map according to:

[x(t− T ), q]|tt−T

= param(p(t)) (17)

1 Agence Nationale pour la Recherche2 from the french ”Capteurs Logiciels Plug & Play”

Namely, the user define how at each instant t, the currentvalue of the parameter is used to compute the correspond-ing estimation of the past state x(t−T ) and the vector ofphysically meaningful parameter vector q. More precisely,the resulting unknown p(t) is obtained by minimizing thefollowing cost function:

J (t)(p) :=ny∑i=1

[ ∑k∈κi(t)

|yi(tk)− yi(tk|p)|]

(18)

in which ny is the number of sensors, κi(t) is the setof indices of past instants tk ∈ [t − T, t] at which ameasurement yi(tk) of sensor i is available. The notationyi(tk|p) denotes the output component that would beexpected if the state x(t−T ) and the vector of parametersq are those given by p in accordance with (17). During eachobserver updating period [τj , τj+1] (where τj = j · τo), afinite number neval of function evaluations are allowed inorder to look for a minimum of the cost function J (τj)(·)starting from an initial guess p+(τj−1) that is compatiblewith the past estimate p(τj−1) leading to the followingupdating process for the dynamic variable p:

p(τj) := Sneval(p+(τj−1)

)(19)

where S denotes an iteration of some optimization al-gorithm and Sq denotes successive iterations of S thatinvolves neval function evaluations. In its current ver-sion, clpp already implements several Gradient-free di-rect search algorithms (Simplex, Torcszon, Trust-region,etc.). Such algorithms enable non smooth inverse problemsto be tackled, avoid asking the user to provide analyti-cal gradient or the numerical troubles associated to thecomputation of the sensitivity matrices. Moreover, recentresults (Alamir et al. [2009]) on singularities avoidanceare implemented in order to enhance global convergencein situations where the cost function shows strong nonconvexity.

5. RESULTS & DISCUSSION

The derivation of the ODE model (16) is done using 21-nodes finite difference scheme. The simulation is performedusing the dielectric properties of raw beef (Bengtsson andRisman [1971]) that are depicted on Figure 3 as functionsof the temperature. These profiles are used in the PDE’s(1) in order to produce the measurement that are fed to thesoftware in order to estimate the inner temperature profile.By doing so, (1) emulates the experimental device. Thebounds of the dielectric properties are taken as follows:

ε′

r,min = 2.0 , ε”r,min = 0.1 (20)

ε′

r,max = 60 , ε”r,max = 25 (21)The measurements consist of the two surface tempera-tures, namely:

y = h(x) =(x1

x21

)∈ R2 (22)

that are assumed to be acquired at some sampling periodτs (20 seconds). As for the parametrization map (17),specific choices have been made to end up with a tractableoptimization problem. Namely, the length of the prediction

-40 -20 0 20 400

10

20

30

40

50

Temperature T

evolution of εr' (black-solid) / ε

r" (blue-dashed)

Fig. 3. Experimental evolution of the dielectric proper-ties ε

′

r and ε”r according to (Bengtsson and Risman[1971]). These profiles are used when integrating (1)in order to produce the measurements that are fedto the clpp software which uses (16) to perform theestimation of the temperature profile.

horizon has been taken equal to the batch duration (inorder to acquire identifiability). Moreover, since at thestarting instant, the foodstuff is likely to have a uniformtemperature T0, the unknowns of the problem are repre-sented by the vector p given by:

p :=(T0

q

)∈ R9 (23)

Figure 4 shows evolution of the incident power flux usedin the validation scenario.

The results are shown on Figures 5-7. More precisely,Figure 5 shows the temperature profiles at several instantsduring the batch. The simulated profile is given using blue-square-marked lines, the estimated profiles using clppare plotted in red-star-marked line while the open-loopevolution without measurement based correction is givenin black-circle-marked line. Note that the correction beginsat instant t = 20 sec when at least 4 measurementsare available (2 for each sensors). This explain why theestimated and the open-loop profile coincide at instantt = 12.5 sec. The evolution of the parameter vector p dur-ing the scenario is shown in Figure 6 while the computationtime 3 needed to perform the neval = 100 function evalua-tion is given in Figure 7. Note that the computation timeis almost always lower than the sampling measurementacquisition τs = 20 sec which means that the proposedsolution is real-time implementable. Note also that thecomputation time increases since the observation horizonincreases and so do the computational burden.

Figure 8-9 shows the same scenario when a shorter ac-quisition period τs = 10 sec is used. The improvementof the estimation quality is noticeable, however, for thecomputational facility used here, the scheme would have

3 The computation has been performed using a Dell-LatitudeE6500 / 2.66GHz

Fig. 4. Power flux (in W/m2) used in the validatingsimulation

been only theoretical since the computation time largelyexceeds the real-time.

6. CONCLUSION

In this paper a method for the estimation of temperaturefield during microwave tempering is proposed for badlyknown dielectric properties. The proposed method enablesthe reconstruction of the temperature profile using onlythe two surface temperatures. Moreover, the estimationscheme is based on discrete-time measurement. It has beenin particular shown that rather long inter-sampling periodof 20 sec still enables a good and real-time implementablereconstruction to be performed. The validation of theproposed approach has been obtained using the genericsoftware clpp that enables simultaneous estimation of thestate and the parameter of a dynamical system.

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Fig. 5. Performance of the estimation scheme with a measurement sampling period of τs = 20 sec and using a maximumnumber of function evaluation neval = 100. Simulated temperatures (blue-squares) / clpp-estimated temperatures(red-stars) / open-loop estimation without parameter updating (black-circles). Note the initial error on the initialuniform temperature. The corresponding evolution of the parameter during this scenario is given in Figure 6

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Fig. 8. Performance of the estimation scheme with a measurement sampling period of τs = 10 sec and using a maximumnumber of function evaluation neval = 100. Simulated temperatures (blue-squares) / clpp-estimated temperatures(red-stars) / open-loop estimation without parameter updating (black-circles). Note the initial error on the initialuniform temperature. The corresponding evolution of the parameter during this scenario is given in Figure 9

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Fig. 10. The time needed to perform the neval =100 function evaluation for the validatingscenario depicted on Figure 5 where anacquisition period of τs = 10 sec is used.The abscissa is the period number duringwhich the computation is performed.

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