temperature estimation of cooking vessel content via … · temperature estimation of cooking...

Temperature estimation of cooking vessel

content via EKF and sliding mode

observers in induction cooking systems

Antonio Boccuni ∗ Luca M. Capisani ∗∗ Francesco Del Bello ∗∗∗

Daniele De Vito ∗∗∗∗ Antonella Ferrara † Jurij Paderno ‡

∗ Department of Electrical Engineering, University of Pavia, Pavia,Italy, e-mail: boccuni [email protected].

∗∗ Department of Computer Engineering and Systems Science,University of Pavia, Pavia, Italy e-mail: [email protected].

∗∗∗ Altran Italia, Roma, Italy, e-mail: [email protected].∗∗∗∗ Whirlpool Europe s.r.l., Cassinetta di Biandronno (VA), Italy,

e-mail: Daniele De [email protected].† Department of Computer Engineering and Systems Science,

University of Pavia, Pavia, Italy e-mail: [email protected].‡ Whirlpool Europe s.r.l., Cassinetta di Biandronno (VA), Italy,

e-mail: Jurij [email protected].

Abstract: The problem of estimating the thermal state of an induction cooking system isanalyzed. The first step is to model the thermal system composed by the cooktop, the pot,and the pot content. Then, by relying on the formulated model, the aim is to design a suitablestate observer, so that an estimation of the temperature of the cooking vessel content can bemade online. Two kinds of observers are proposed. An Extended Kalman Filter (EKF) and aSliding Mode Observer (SMO). A comparison of the performances which can be obtained withthese two solutions is then made by relying on experimental results obtained on a real inductioncooktop heating a pot containing a certain quantity of water.

1. INTRODUCTION

The goal of saving energy and having as constant referencethe quality, the efficacy, as well as production costs isessential in order to market products on a large scale,see Acero et al. [2006], Acero et al. [2010], and Aceroet al. [2008]. To achieve this goal, companies are constantlycommitted to develop advanced research techniques anddesign hardware and software accordingly. The increasinguse of microcontrollers makes the integration of the newblocks easier because of their affordability and simplicityof implementation.

This paper discusses the possible implementation of ablock estimating the thermal state of an induction cook-top, see, for example, Lu et al. [2009]. To this purpose, thesignal of electrical power fed into the primary coil and thetemperature of the lower face of the glass of the cookingsystem, are acquired.

A state observer is then designed in order to estimatethe evolution of the system, see Paesa et al. [2009], Basinand Rodriguez-Ramirez [2010]. The observer is based onthe model of the system which includes the descriptionof the electromagnetic subsystem used to transmit theenergy to the bottom of the pot, and the description ofthe thermal interaction between the components of theinduction cooktop, the content of the pot (water, forinstance, as considered for the sake of simplicity in thispaper), and the environment.

To design the observer, two methodologies are exploited.First, the Extended Kalman Filter theory is suitablyconsidered, see Jilong et al. [2008], then the approachbased on the generation of the so-called sliding modes, (seeUtkin et al. [1999], Kachroo and Tomizuka [1996], Edwardsand Spurgeon [1998], and Basin et al. [2007]) is adopted.It must be emphasized that the reliability of the estimatedepends not only on type of observer implemented, butalso on the whole process. In the present case, a significantrole is obviously played by the power supply system (seeLlorente et al. [2002], Millan et al. [2007] and Beato et al.[2006]), by the sensors measuring the thermal variables onwhich, in fact, the observability of the system depends,and by the installation, and environmental conditions.

After the design step, to compare the performances of thetwo possible solutions, some simulations and experimen-tal tests are made. The experimental tests reported inthe paper have been performed on a real cooktop withdifferent initial conditions. The results are obtained andanalyzed relying on the use of Matlabr, Simulinkr, andLabVIEWr.

This paper is structured as follows: Section 2 describes theplant under consideration. In Section 3 the thermal modelof the plant is formulated. In Section 4 the observers aredescribed, while Section 5 reports the experimental results.Finally, in Section 7, some conclusions are gathered.

Preprints of the 18th IFAC World CongressMilano (Italy) August 28 - September 2, 2011

Copyright by theInternational Federation of Automatic Control (IFAC)

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φind

Icoil

Power

Fig. 1. The induction cooktop

2. THE INDUCTION HEATING COOK-TOP

Induction heating is the process of heating an electri-cally conducting object (usually a metal) by means ofelectromagnetic induction. Such a phenomenon designatesthe generation of eddy current in a circuit due to thefluctuation of the current flowing in another circuit placedclose to the former. Coherently, induction heating, whichis an applied form of Faraday’s discovery, stems from theAC current flowing through a primary circuit and affectingthe magnetic movement of a secondary circuit located nearit and composed of the cooking pot. The basic rationaleunderlying induction cook-tops is depicted in Fig. 1, wherethe layout of the system under consideration is schemati-cally portrayed (see also Boer et al. [2010]). In particular,the excitation circuit is a coil fed by a power electronicdriving circuit and positioned under a glass ceramic plate,whose top side allows for positioning the cooking potwithin the corresponding serigraphy. The current flowingthrough the coil generates a magnetic field crossing theglass plate and heading towards the bottom layer of thepot. The magnetic nature of the latter creates a couplingeffect between the coil and the pot, which, in turn, behavesas an inductive-resistive short circuit and thus gives riseto the generation of eddy current. The resistive part ofsuch a circuit becomes the heating source because of theJoule effect. Induction cookers are faster and more energyefficient than traditional hobs. Additionally, the risk ofaccidental burning is diminished since the hob itself onlygets marginally hot (due to heat conduction down from thecookware), allowing direct contact with a reduced chanceof harm. Moreover, it only works whenever a pot lies onit, since, if this is not the case, the magnetic flux producedby the coil has no effect. Nevertheless, non-ferrous pots,such as copper, aluminum and glass, cannot be used on aninduction cook-top.

3. DYNAMIC MODEL OF THE SYSTEM

Energy balance equations applied to the system describedso far lead to the dynamic model,

CCOILTCOIL = (1 − k1)P − (hCA + hGC)TCOIL+hGCTGLASS + hCATAIR

CGLASSTGLASS = −(hGA + hGC + hPG)TGLASS+hPGTPOT + hGCTCOIL + hGATAIR

CPOT TPOT = k1P − (hPA + hPG + hPW )TPOT +hpwTwater + hPGTGLASS + hPATAIR

mwatercW Twater = −(hWA+hPW )Twater + hPW TPOT +hWATAIR + mwaterHvs(Pest)

mwater = −Pevap

λ(Pest)−

σ(k(Twater − TSAT (Pest) + Tsigma))[−(hWA + hPW )Twater + hPW TPOT +

hWATAIR −Pevap

λ(Pest)Hvs

]

/Hvs

Pevap = φ(PTV (TW ) − η)φ = const; η = const; T0 = const;Tsigma = const; TAIR = const; k1 = const

(1)

where:

• CCOIL is the equivalent thermal capacity of the coil;• CGLASS is the equivalent thermal capacity of the

glass;• CPOT is the equivalent thermal capacity of the pot;• CW is the water specific thermal capacity;• TCOIL is the coil temperature;• TGLASS is the glass temperature;• TPOT is the pot temperature;• Twater is the water temperature;• mwater is the water mass;• P is the total active power absorbed at the coil;• Pest is the pressure of the athmosphere (external

pressure);• hCA represents the heat transfer coefficient coil to air

multiplied by the relative surface;• hGA represents the heat transfer coefficient glass to

air multiplied by the relative surface;• hPA is the heat transfer coefficient pot to air multi-

plied by the relative surface;• hWA is the heat transfer coefficient water to air

multiplied by the relative surface;• hGC is the heat transfer coefficient glass to coil

multiplied by the relative surface;• hPG is the heat transfer coefficient pot to glass

multiplied by the relative surface; hPW is the heattransfer coefficient pot to water multiplied by therelative surface;

• PTV (TW ) is the surface tension at temperature TW ;• is the λ(Pest) is the water evaporation latent heat at

the pressure Pest;• Hvs(Pest) is the saturated vapor enthalpy at the

pressure Pest;• σ(k) is the sigmoid function.

Note that, in most of the cook-tops on the market, the onlymeasurable quantity is the temperature of the glass andthat this measure is somewhat complicated to be acquireddue to the position of the temperature sensor, which mayalso sense the effect of the air surrounding it. Furthermore,


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the power P can be derived from electrical measurements.Finally, it stands to reason that model (1) is nonlinearand includes many parameters that are difficult to identify,making the problem under investigation very challenging.

4. OBSERVER DESIGN

As previously stated, this paper proposes the design oftwo observer schemes that are in charge of estimating thestate variables of system (1) by exploiting the availableinformation coming from the power absorbed by the coil,the current flowing inside it and the temperature of theglass. Hence both the observers that will be described inthe following subsections rely on the architecture shownin Fig. 2. Among all the state entries, the temperature ofthe water stands for the performance index considered inthis paper.

Pind (t)

Icoil (t)

Tglassmis

u (t)OBSERVER Twest

Fig. 2. Architecture of the proposed observation scheme.

4.1 The Extended Kalman Filter (EKF)

As a first proposal, in order to estimate the temperature ofthe vessel content, in the induction cooking system underconsideration, an EKF is designed (see Ljung [1979]). Tothis end, consider the following nonlinear dynamic system

{

x (t + 1) = f (x (t) , u (t) , t) + g (x (t) , t)w (t)y (t) = h (x (t) , t) + v (t)

(2)

where w (t) e v (t) are white stochastic Gaussian process,with null media and covariance matrix Q and R respec-tively, being u(t) the input to the system (in this casegiven by the power signals P in model (1)), while y(t)is the output of the system (TGLASS in model (1)), andg(x(t), t) is a nonlinear function which collects unmodelledeffects.

Let the term x (t|t − 1) and x(t|t) be, respectively, thepredicted and filtered estimation of the state variable,defined in the following algorithm. By linearizing, one has

A (t) =∂f (x, u, t)

∂x x=x(t|t),u=u(t)(3)

C (t) =∂h (x, t)

∂x x=x(t|t−1)(4)

D (t) = g (x(t|t), t) (5)

(6)

Relying on (3)-(5), the EKF is given by the followingrecursions:

Measurement update:

x(t|t) = x(t|t − 1) + K (t) [y (t) − h (x(t|t − 1), t)] (7)

K (t) = P (t|t − 1)Ct (t)(

R + C (t)P (t − 1|t)Ct (t))−1 (8)

Time update:

x (t + 1|t) = f (x (t) , u (t) , t) (9)

Measurement update:

P (t|t) = P (t|t − 1) − P (t|t − 1)Ct (t)(

R + C (t)P (t|t − 1)Ct (t))−1

C (t)P (t|t − 1)(10)

Time update:

P (t + 1|t) = A (t)P (t|t)At (t) + D (t)QDt (t) (11)

Note that, depending on the specific application, matricesP and Q are selected on the basis of experimental tests.

4.2 The Sliding Mode Observer

To design the sliding mode observer, we rely on theformulation proposed by Utkin et al. [1999], adopting suchan approach to the specific applicative case in question.The observer structure is

{

˙x1 (t) = A11x1 (t) + A12y (t) + B1u (t) + Lν˙y (t) = A21x1 (t) + A22y (t) + B2u (t) − ν

(12)

where x1 is an estimate of the state variables vector[TCOIL, TGLASS, TPOT , Twater]

T and y is an estimate ofthe output variable TGLASS, respectively, L ∈ R

(n−p)×p isa constant gain matrix and ν is a vector, the componentsof which are defined as follows

νi = Msign (yi − yi) (13)

where M ∈ R+. If (A, C) is observable, also (A11, A21) isobservable.

Matrix L can be chosen so as to correctly place theeigenvalues of A11 + LA21. In this paper a Sliding Modeobserver with a linear model of the plant is used toavoid stability problems. Clearly, the effectiveness of thisobserver needs to be verified in simulation on the nonlinearmodel of the plant, and, accordingly, in practice, on theexperimental set-up.

The matrices required to calculate matrix L, on the basisof the plant model, in order to solve the eigenvaluesplacement problem, are

A11 =

−

hCA + hGC

Ccoil

0 0

0hPA + hPG + hPW

Cpot

hPW

Cpot

0hPW

cwm−

hWA + hPW

cwm

A21 =

[

hGC

Cglass

hPG

Cglass

0

]

(14)

A well-known limit to the applicability of the slidingmode design methodology in practical application, is thegeneration of the so-called chattering (see Fridman [2001],Boiko and Fridman [2005]). So, also in the consideredapplication, there is the necessity of designing the observer


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so as to alleviate, as much as possible, this undesirableeffect. To this end, the sliding mode observer input law isregularized by replacing the sign function with a saturationfunction, i.e.

νi =

Msign (e (t)) ||yi − yi|| > ξM

ξe (t) ||yi − yi|| 6 ξ

(15)

The value selected for ξ, in our case, is 0.5, and M = 1.

Fig. 3. Water temperature sensor during the tests.

5. EXPERIMENTAL RESULTS

To test the proposal, 16 tests have been performed in orderto properly cover a wide range of operating conditionswith different pots, coils, quantity of water and startingtemperatures of the of the glass and the pot.

The tests planning is given in Table 1. Note that in Table1, “cold” indicates a temperature ≤ 30 ◦C, while “hot”indicates a temperature ≥ 60 ◦C. Moreover, in Table 1,the first column indicates the test identification number;the second column indicates the coil diameter in [mm]; thethird column indicates the pot filling; the fourth and fifthcolumn indicate the glass and the pot initial temperatureconditions; the last two columns indicate the pot size andthe water quantity.

A first set of acquired data of temperature, power andcurrent of the power system have been used to recalibratethe designed observers. For both algorithms, the newcalibration has been made on the test ID05 (see Table 1).The other tests have been used to validate the observers.

6. TEST ANALYSIS

6.1 The Extended Kalman Filter

The calibration of the Kalman filter has allowed to adjustappropriately the coefficients Qij,i=j

of the covariancematrix of the noise on the state, in order to increase thespeed of convergence. In fact, it has been noted that smallchanges in the measurements cause high fluctuations ofthe estimated temperature. So, by increasing the cutoff

0 500 1000 1500 20000

20

40

60

80

100

120Extended Kalman Filter

Time [s]

[°C

]

Tw

meas

Tw

estim

Tglass

Fig. 4. Comparisons between the measured and the es-timated water temperatures when the EKF is used(calibration test, ID05). The glass temperature is alsoreported.

0 50 100 150 200 25020

30

40

50

60

70

80

90

100Extended Kalman Filter

Tempo [s]

[°C

]

Tw

meas

Tw

estim

Tglass

Fig. 5. Comparisons between the measured and the esti-mated water temperatures when the EKF is used (oneof the validation tests, ID13). The glass temperatureis also reported.

frequencies of the filter, it is possible to partially mitigatethis effect.

The initial conditions of the calibrated EKF are

• mEKF0= 1kg

• R = 10−3

• Q2200 =

75 0 0 0 00 20 0 0 00 0 2000 0 00 0 0 2000 00 0 0 0 9 · 10−4

The comparison between the measured and the estimatedwater temperature during the calibration test is reportedin Fig. 4. Fig. 5 refers instead to a validation test selectedas an example. It is test ID13. During this test, theobserver estimates the temperature of the water ratherefficiently, by reducing the observation error practically


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Test ID Coil mm %Qmax Tglass0Tpot0 Pot Size Water Quantity l

1 240 75% hot cold big 4.8

2 240 25% hot cold small 2.7

3 180 75% cold hot big 4.7

4 180 75% hot hot small 2.3

5 240 75% cold cold small 4.7

6 240 25% hot hot small 1.7

7 240 25% cold hot big 2

8 240 25% cold cold big 1.8

9 180 75% cold cold big 7.5

10 240 75% hot hot big 5.2

11 180 25% hot cold big 1.7

12 240 75% cold hot small 7.5

13 180 25% cold hot small 0.8

14 180 25% cold cold small 0.8

15 180 25% hot hot big 2.7

16 180 75% hot cold small 2.3

Table 1. Details of the experimental setup for the 16 considered tests.

to zero (the residual error is about 3 degrees over 100degrees).

6.2 The Sliding Mode Observer

The calibration of the Sliding Mode observer consists infinding suitable coefficients ki, defined as

pobs = [k1λcoil, k2λpot, k3λwater]T (16)

where vector pobs contains the values of the poles of theobserver. In our case, the coefficients ki are selected as

k1 = 4.7, k2 = 1.9, k3 = 3

thus, matrix L in (12) is given by

L =

1 · 10−7

76.9090.80

(17)

with M = 1 and ξ = 0.5. Note that the last componentof L is left to zero because we assume that the massof the water does not change during time. In fact, thevariation of the mass of the water is not significant whenthe temperature inside the pot is lower than the boilingpoint.

The comparison between the measured and the estimatedwater temperature during the calibration test is reportedin Fig. 6. Fig. 7 refers instead to a validation test selectedas an example. It is test ID10. During this test, the slidingmode observer provides satisfactory performance. In thiscase, in particular, the observation error turns out to bereduced with respect to that obtained via the EKF (about1 degrees over 100 degrees).

7. CONCLUSIONS

The problem of designing state observers suitable to esti-mate the thermal state of an induction cooking system isanalyzed. The proposed observers are based on the thermalmodel of the system. Two approaches to design the inputlaw of the observers have been exploited: extended kalmanfiltering and sliding modes. It is assumed that the inputpower to the thermal system and the temperature of the

0 200 400 600 800 1000 1200 14000

20

40

60

80

100

120Sliding Mode Observer

Time [s]

[°C

]

Tw

meas

Tw

estim

Tglass

Fig. 6. Comparisons between the measured and the es-timated water temperatures when the sliding is used(calibration test, ID05). The glass temperature is alsoreported.

0 50 100 150 200 25020

30

40

50

60

70

80

90

100

110

120Sliding Mode Observer

Time [s]

[°C

]

Tw

meas

Tw

estim

Tglass

Fig. 7. Comparisons between the measured and the esti-mated water temperatures when the sliding observeris used (one of the validation tests, ID10). The glasstemperature is also reported.


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glass are available. Then, the input laws of the observersare determined on the basis of the measured state. Acomparison of the performances which can be obtainedwith these two solutions is then made by relying on exper-imental results. Sixteen experimental tests have been madeon a real induction cooktop heating a pot containing acertain quantity of water. The EKF has given acceptableresults. Yet, the sliding mode observer allows to obtainsuperior estimation results with reduced computationalcomplexity, due to the robustness of the input law versusnoise and unmodelled effects. Moreover, the sliding modeobserver showed good capabilities in tracking the mea-sured state. Thus, at this stage of the system verificationand validation, we can conclude that the designed slidingmode observer presents some intrinsic features that makeit outperform the designed EKF, even if also this latter hasdemonstrated to be a satisfactory solution for this kind ofindustrial application.

REFERENCES

J. Acero, J.M. Burdio, L.A. Barragan, J.I. Artigas, andR. Alonso. An electromagnetic-based model for cal-culating the efficiency in domestic induction heatingappliances. In Proc. IEEE Power Electronics SpecialistsConference, pages 1–6, Jun. 2006.

J. Acero, J.M. Burdio, L.A. Barragan, D. Navarro,R. Alonso, J.R. Garcia, F. Monterde, P. Hernandez,S. Llorente, and I. Garde. The domestic inductionheating appliance: An overview of recent research. InProc. IEEE Applied Power Electronics Conference andExposition, pages 651–657, Feb. 2008.

J. Acero, J.M. Burdio, L.A. Barragan, D. Navarro,R. Alonso, J. Ramon, F. Monterde, P. Hernandez,S. Llorente, and I. Garde. Domestic induction appli-ances. IEEE Industry Applications Magazine, 16(2):39–47, Mar. 2010.

M. V. Basin and P. Rodriguez-Ramirez. Sliding mode filterdesign for linear systems with unmeasured states. IEEETransactions on Industrial Electronics, To appear. 2010.

M. V. Basin, A. Ferreira de Loza, and L. M. Fridman. Slid-ing mode identification and control for linear uncertainstochastic systems. International Journal of SystemsScience, 38(11):861–869, Nov. 2007.

A. Beato, C. Bocchiola, and S. Frattesi. Modelling anddesign of the half-bridge resonant inverter for inductioncooking application. In Proc. IEEE MediterraneanConference on Control and Automation, pages 1–6, Jun.2006.

A. Boer, F. Del Bello, D. N. Gutierrez, J. Paderno,D. Parachini, and G. Santacatterina. A method forcontrolling the induction heating system of a cookingapparatus. Patent. Applicant: Whirlpool CO, Teka indSA. Date: 2010-06-02, 2010.

I. Boiko and L. M. Fridman. Analysis of chattering incontinuous sliding-mode controllers. IEEE Transactionson Automatic Control, 50(9):1442–1446, Sep. 2005.

C. Edwards and S. K. Spurgeon. Sliding Mode Control:Theory and Applications. Taylor & Francis, London,UK, 1998.

L. M. Fridman. An averaging approach to chattering.IEEE Transactions on Automatic Control, 46(8):1260–1264, Aug. 2001.

Q. Jilong, T. Yantao, G. Yimin, and Zhucheng. A sen-sorless initial rotor position estimation scheme and anextended kalman filter observer for the direct torquecontrolled permanent magnet synchronous motor drive.In Proc. IEEE Electrical Machines and Systems, pages3945–3950, Oct. 2008.

P. Kachroo and M. Tomizuka. Chattering reduction anderror convergence in the sliding-mode control of a classof nonlinear systems. IEEE Transactions on AutomaticControl, 41(7):1063–1068, Jul. 1996.

L. Ljung. Asymptotic behavior of the extended kalmanfilter as a parameter estimator for linear systems. IEEETransactions on Automatic Control, 24(1):36–50, Feb.1979.

S. Llorente, F. Monterde, J.M. Burdio, and J. Acero. Acomparative study of resonant inverter topologies usedin induction cookers. In Proc. IEEE Applied PowerElectronics Conference and Exposition, volume 2, pages1168–1174, 2002.

Y. Lu, K. Cheng, K.W. Chan, Z.G. Sun, and S.W. Zhao.Development of a commercial induction cooker. InProc. IEEE Power Electronics Systems and Applica-tions, pages 1–3, May. 2009.

I. Millan, D. Puyal, J.M. Burdio, J. Acero, and S. Llorente.Resonant inverter topology for all-metal domestic induc-tion heating. In Proc. IEEE International Symposiumon Industrial Electronics, pages 913–918, Jun. 2007.

D. Paesa, S. Llorente, C. Sagues, and O. Aldana. Adaptiveobservers applied to pan temperature control of induc-tion hobs. IEEE Transactions on Industry Applications,45(3):1116–1125, May-Jun. 2009.

V. I. Utkin, J. Guldner, and J. Shi. Sliding Mode Control inElectromechanical Systems. Taylor & Francis, London,UK, 1999.


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temperature estimation of cooking vessel content via … · temperature estimation of cooking...

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