Identification of Dynamic Nonlinear ThermalTransfers for Precise Correction of Bias induced by

Temperature VariationsCeline Casenave∗†, Gerard Montseny∗†, Henri Camon∗† and Francois Blard∗†

∗CNRS; LAAS; 7 avenue du colonel Roche, F-31077 Toulouse, France†Universite de Toulouse; UPS, INSA, INP, ISAE; LAAS; F-31077 Toulouse, FranceEmail: casenave@laas.fr, montseny@laas.fr, camon@laas.fr, francois.blard@laas.fr

Abstract—We present a least squares method for dynamiccorrection of biases induced by temperature variations when veryhigh precision is required. This method is based on a simpledynamic model allowing to take into account the macroscopiceffects of complex underlying thermal phenomena inside thedevice.

I. INTRODUCTION

In general, high precision devices, such as MEMS devotedto metrology applications for example, need to include algo-rithms allowing to correct biases resulting from variations ofthe ambient temperature. Most of time, static corrections areinvolved, which means that the ambient temperature (measuredby means of a sensor) is supposed to be also the temperatureat any point of the device itself.

This is of course not exactly true because the ambient tem-perature is never strictly constant and so, due to the complexdistributed nature of physical objects, the temperature insidethe device evolves following a complex diffusion equation.So, when measuring any physical quantity (for example avoltage) on the device under consideration, small residualfluctuations due to temperature variations remain in spite ofthe static correction, because dynamic thermal phenomena areinvolved, which cannot be described by means of a staticcorrespondence.

In some cases, when very high precision is required, it canbe judicious to envisage to build some dynamic corrections,elaborated from a suitable dynamic treatment of the tempera-ture data, in such a way that the macroscopic effects resultingfrom the distributed evolution of the temperature inside thedevice can be taken into account, and then corrected.

In this paper, we present a method devoted to such adynamic correction.

II. RECALLS ON STANDARD (STATIC) POLYNOMIALCORRECTION

Consider a quantity U (to be measured), linked to theambient temperature T through a static relation of the form:

U = f(T ). (1)

The aim is to correct the effect of the temperature T , thatis to get an (if possible optimal) estimation of the quantity

U0 = f(T0), where T0 is chosen a priori as the referencetemperature. If f is regular (which is the case in concrete sit-uations), U can be expressed following the Taylor developmentaround T0:

U = f(T0)+ f ′(T0)(T −T0)+1

2f ′′(T0)(T −T0)

2 + ..., (2)

that is, if we consider that terms beyond the nth order arenegligible:

U = U0 +n∑

i=1

ai(T − T0)i. (3)

So if the coefficients ai have been previously accuratelyestimated (from experimental measurements) and T is known,the bias induced by sufficiently slow temperature variationsaround T0 can be corrected thanks to the relation:

U0 = U −n∑

i=1

ai(T − T0)i. (4)

Indeed, when U is measured N times with some additive zeromean noise w, we then have Uk = Uk + wk, k = 1 : N andso, we get the correction relation:

U0,k = Uk −n∑

i=1

ai(Tk − T0)i

= Uk −n∑

i=1

ai(Tk − T0)i + wk (5)

= U0 + wk, k = 1 : N.

An estimate of the unknown quantity U0 can then be obtainedby filtering the data U0k, for example by:

U0 =1

N

N∑k=1

(Uk −

n∑i=1

ai(Tk − T0)i

). (6)

We then have from (5) and thanks to the law of large numbers:

E(U0) = U0, (7)

U0 =1

N

N∑k=1

(U0 + wk) → U0 when N → ∞; (8)

so, the estimate U0 is convergent and unbiased: the effectsof temperature variations have been corrected. Finally, if the

unknown quantity U0 is not exactly constant but presents somevery slow (and small) variations, it can be pursued for exampleby means of the following moving average estimate based onthe corrected data U0,k:

U0,k =1

N

N−1∑q=0

[Uk−q −

n∑i=1

ai(Tk−q − T0)i

], (9)

with the following residual estimation noise (inherited fromthe measurement noise):

εk =1

N

N−1∑q=0

wk−q. (10)

Preliminary estimation of the correction coefficients fromexperimental data The correction coefficients ai are esti-mated from experimental data, whose number must be suffi-ciently large to make the residual estimation error negligible.In practice, this step can be very long, particularly if measure-ment data are significantly corrupted by noise. The classicalestimation of coefficients ai from data Uk, k = 1 : K isin general obtained from the least squares method, that is bysolving the problem:

minai∈R

K∑k=1

(Uk − a0 −

n∑i=1

ai(Tk − T0)i

)2

; (11)

this problem is rewritten under the matrix form:

mina∈Rn+1

(Qa− U)T (Qa− U) (12)

and the optimal solution a∗ = (a∗i ) is classically given by:

a∗ = (QTQ+ εI)−1QT U (13)

with ε > 0 a small parameter devoted to numerical condition-ing of the matrix inversion.

III. PRECISE CORRECTION OF BIAS FROM IDENTIFIEDDYNAMIC THERMAL TRANSFER

A. Dynamic influence of the ambient temperature on a phys-ical quantity

In practice, a physical system such as a MEMS withits environment is a distributed system: when the ambienttemperature T is not constant, the temperature of the systemcannot be a simple scalar: it becomes a scalar field θ(t, x),where each (vector) value of the spatial variable x is associatedwith a particular physical point of the system. If the variationsof T remain small, the evolution of the field θ is theoreticallygoverned by a complex but linear partial differential equationwith input T (t), of the abstract form:

∂θ(t, x)

∂t= A(∇) θ(t, x) +B(x)T (t), x ∈ Ω ⊂ R3, (14)

where A(∇) is a second order linear differential operator,associated with suitable boundary conditions, and B is theinput operator describing how the exterior temperature T actson the evolution of θ. Because equation (14) is in general

defined on t > t0, the initial condition must be added in orderto get a well-posed problem:

θ(t0, x) = θ0(x), x ∈ Ω. (15)

Except in some ideal simple cases, such an equation isin practice impossible to formulate explicitly, due to thecomplexity of any physical system.

Similarly, a quantity U relating to the physical system underconsideration and depending at first sight on T only, dependsin general on the whole field θ. As in the ideal static cases,this dependence can of course be nonlinear, in such a way thatwe have formally:

U(t) = F(θ(t, x)), (16)

where F is a nonlinear spatial operator. So, the quantity Uappears as an output of (14) and the dynamic (nonlinear)transfer T 7−→ U is then resulting from the input-outputcorrespondence defined by (14,16).

In most of cases, the evolution of T (t) is slow and it can beconsidered that the difference |θ(t, x)−T (t)| can be neglectedand then, the static approach previously described can be used.In some cases however, namely when very high precision isrequired, small differences between θ(t, x) and T (t), whichresult from the dynamic nature of equation (14), can generatesome significant residual biases which cannot be suppressedby static correction. In such cases, the static model (3) is nomore sufficiently accurate and a dynamic correction is needed.

B. Dynamic correction of bias resulting from temperaturevariations

This correction method is based on an explicite and uni-versal differential model of the dynamic transfer (14,16), asdescribed here-after1. First, thanks to the linear and diffusivenature of equation (14), a generic 1D diffusive input-stateequation is used to model the underlying thermal dynamics.By assuming for simplicity that the mean ambient temperatureis T0 = 0, this model is:

∂ψ(t, ξ)

∂t= −ξ ψ(t, ξ) + T (t), ξ > 0, t > t0

ψ(t0, ξ) = 0.(17)

On the other hand, from linearity, the contribution of the initialcondition θ0 is separately expressed from the exponentialfamily:

ψ0(t, ξ) = e−ξt ψ0(ξ); (18)

note that the function (t, ξ) 7→ ψ(t, ξ)+ψ0(t, ξ) is solution of(17) with initial condition ψ0(ξ) in place of 0. We then definethe function:

Ψ(t, ξ) := ψ(t, ξ) + ψ0(t, ξ)− T (t) (19)

1This approach approach is relating to the so-called diffusive representation.More details about this theory and its applications in identification of nonlinearcomplex models or other various fields can be found in [3], [10].

which expresses the difference between the ambient ”static”temperature and the dynamic field ψ(t, ξ) + ψ0(t, ξ); thisfunction will be devoted to dynamic correction.

From ψ (the solution of (17)) and ψ0, we can then modelthe unknown dynamic transfer T 7→ U under the formU = G(T,Ψ) with G a suitable nonlinear operator; a genericformulation of G is the following:

U(t) = U0 +∑i

ai T (t)i +

∫µ1(ξ)Ψ(t, ξ) dξ

+

∫∫µ2(ξ1, ξ2)Ψ(t, ξ1)Ψ(t, ξ2) dξ1 dξ2 (20)

+

∫∫∫µ3(ξ1, ξ2, ξ3)Ψ(t, ξ1)Ψ(t, ξ2) Ψ(t, ξ3) dξ1 dξ2 dξ3

+

∫∫∫∫...,

where the coefficient ai and the functions µi allow to synthe-size from T (t) and the field Ψ(t, ξ) a wide class of nonlineartransfers. Thanks to the nice properties of equation (17), anapproximated synthesis with only a few Ψj(t) := Ψ(t, ξj),ξj > 0, of the form:

U(t) = U0 +n∑

i=1

ai T (t)i +

J1∑j=1

µ1,j Ψj(t)

+

J2∑j=1

J′2∑

j′=1

µ2,j,j′ Ψj(t)Ψj′(t)

+

J3∑j=1

J′3∑

j′=1

J′′3∑

j′′=1

µ3,j,j′ Ψj(t)Ψj′(t)Ψj′′(t)

+∑j

∑j′

∑j′′

∑j′′′

..., (21)

will be in practice sufficient to get accurate approximations ofthe dynamic nonlinear transfer T 7→ U .

Finally, if the temperature variations are sufficiently small(this will be the case in the problem studied in the nextsection), the cross terms Ψj Ψj′ , Ψj Ψj′ Ψj′′ , etc., can beneglected and the following simplified synthesis will be yetsufficient to get significant improvement compared to the staticmodel (3):

U(t) = U0 +n∑

i=1

ai T (t)i +

J∑j=1

µj (ψj(t) + ψ0,j(t)− T (t)),

(22)in such a way that (22) can be rewritten, from (18) and withνj := µj ψ0(ξj), under the following form in which all thecorrection parameters ai, µj , νj are linearly involved:

U(t) = U0+

n∑i=1

ai T (t)i+

J∑j=1

µj (ψj(t)−T (t))+J∑

j=1

νj e−ξjt.

(23)Algorithm for dynamic correction Similarly to the staticcase, when U is measured at times tk = k∆t with someadditive zero mean noise wk that is Uk = Uk+wk, and thanks

to the fact that e−ξjtk → 0 when k → +∞, we finally getthe asymptotic dynamic correction formula, suitable for largek (NB: the correction coefficients are ai, µj):

U0,k = Uk −∑n

i=1 ai (Tk)i −∑J

j=1 µj (ψj,k − Tk), (24)

where the quantities ψj,k, on which is based the dynamiccorrection, are computed from the temperature data Tk viathe following discrete-time dynamic relation deduced fromintegration of (17):

ψj,k+1 = e−ξj∆t ψj,k + 1−e−ξj∆t

ξjTk. (25)

Finally, exactly as in the static case, U0,k = U0 + wk andso, estimates U0 or U0,k of the quantity U0 can yet be takenas:

U0 =1

N

N∑k=1

U0,k, U0,k =1

N

N−1∑q=0

U0,k−q; (26)

we have yet, from (24) and thanks to the law of large numbers:

E(U0) = U0, (27)

U0 =1

N

N∑k=1

(U0 + wk) → U0 when N → ∞, (28)

εk =1

N

N−1∑q=0

wk−q. (29)

Preliminary estimation of the correction coefficients fromexperimental data The correction coefficients ai, µj , νj areestimated from experimental data whose number must besufficiently great to get negligible estimation errors. Thisestimation from data (with noise) Uk, k = 1 : K is obtainedagain by means of the least squares method, by minimizingwith respect to ai, µj , νj ∈ R the quantity:

J =

K∑k=1

(Uk − a0 −

n∑i=1

ai(Tk)i

−J∑

j=1

µj (ψj,k − Tk)−J∑

j=1

νj e−ξtk

2

, (30)

which can be rewritten under matrix form, with (a, µ, ν) ∈R(n+1) × RJ × RJ :

min(a,µ,ν)

Q a

µν

− U

T

·

Q a

µν

− U

, (31)

the solution (a∗, µ∗, ν∗) of which is given by: a∗

µ∗

ν∗

=[QTQ+ εI

]−1QT U . (32)

IV. APPLICATION TO A HIGH PRECISION VOLTAGEREFERENCE BASED ON AN ELECTROSTATICALLY

ACTUATED MEMS

In the sequel, we present some experimental results obtainedby use of the method described above. The physical systemunder consideration is an electrostatically actuated MEMSdevoted to the construction of a voltage reference of highprecision and stability. From the electrical point of view, it issimply a variable capacitance in which the electrostatic forceis opposed to the one of a mechanical spring between the twoelectrodes. So, there exists a so-called pull-in voltage, beyondwhich the mechanical force is no more able to balance theelectrostical one: this defines the reference voltage, which isdetermined by the only mechanical design of the MEMS [11],[8], [9].

For stability tests, the electrical environment of the MEMSis shown (in a simplified form) in Fig. 2. The DC voltageapplied between the MEMS electrodes is chosen slightly lessthan the pull-in voltage. Variations of the ambient temperaturegenerate variations of the electrical capacitance of the MEMS(around 10pF) and therefore variations of the measured ACvoltage.

In order to evaluate the long time stability of the electricalcharacteristics of the device and its electrical environment,it is necessary to correct these voltage variations. However,due to the particular geometry of the MEMS whose desiredelectromechanical properties result from a complex designwith very thin spring suspension of the mobile electrode (seeFig. 1) and also to the imperfections of the other associateddevices (see Fig. 2), we can expect that large time constantsare possibly involved in the influence of temperature on the ACvoltage measurement. In such a case, the temperature cannotbe uniform in the whole system if the ambient temperaturevariations are not sufficiently slow. Taking into account thatvery high precision is required, the dynamic correction pre-sented above has therefore been implemented for a bettercorrection. The efficiency of the method is illustrated bycomparing the results with those obtained from standard staticcorrection.

The measured data are shown in Fig. 3; the sample period is∆t = 15s. Significant variations of the AC voltages are visible;they mainly result from ambient temperature variations. Forstatic correction, we have taken n = 1 (linear correction) orn = 2. For dynamic correction, J = 6 time constants 1/ξjhave been used.

In Fig. 4, we can see both the measured data and a dynamiccorrection a0 −

∑ni=1 ai (Tk)

i −∑J

j=1 µj (ψj,k − Tk), wherea0 is the estimate of the exact (unknown) AC voltage U0. Thecorrection is then subtracted to the measured data to get the so-called residual variations. These residual variations are shownin Fig. 5, 6, either non smoothed, or smoothed by means ofa standard moving average filter. We can clearly see that asignificant improvement is obtained from dynamic correction.

Evolutions of the temperature T and the ”auxiliary tempera-tures” ψj+ψ0,j are shown in Fig. 7,8,9. In Fig. 10, we can seethe contribution to these evolutions of the terms inherited frominitial conditions, that is: ψ0,j(tk) =

νj

µje−ξjtk . Finally, Fig.

11 shows the differences between the auxiliary temperaturesand T , that is the functions Ψj = ψj + ψ0,j − T .

From a more quantitative point of view, we have thefollowing results:

Mean measured AC voltage: 0.1040mV,Standard deviation before correction of the AC voltage:1.114× 10−3mV.

Standard deviation of the AC voltage after correction (mV):

non smoothed smoothedstat. lin. 9.985× 10−5 4.911× 10−5

stat. nonlin. 9.780× 10−5 4.471× 10−5

dyn. lin. 6.769× 10−5 1.331× 10−5

dyn. nonlin. 6.725× 10−5 1.173× 10−5.

Reduction of the residual variations of the AC voltage(smoothed corrected data):

stat. lin. −27, 3 dBstat. nonlin. −27.9 dBdyn. lin. −38.6 dBdyn. nonlin. −39.5 dB.

Improvement by use of dynamic correction versus static one:

linear −11.3 dBnonlinear −11, 6 dB.

Fig. 1. The electrostatically actuated MEMS under consideration

Fig. 2. Electrical environment of the MEMS for data acquisition

0 100 200 3000.102

0.1025

0.103

0.1035

0.104

0.1045

0.105

0.1055

Time (hours)

AC voltage (mV)

Fig. 3. Measured AC voltage

V. CONCLUSION

As clearly highlighted by these quantitative results, dynamiccorrection can in some cases generate significant improve-ments. Some residual variations remain after correction; theyprobably cannot be entirely induced by noises only. Indeed,great deviations of Ψj(t) sometimes appear (see Fig. 11),which suggest that the only linear part of the dynamic correc-tion is possibly no more sufficient in such cases. So, it couldbe judicious to involve higher order (nonlinear) terms of theseries (21), in particular quadratic ones: µ2,j,j′ Ψj(t)Ψj′(t).This will be studied in a further work, with a new set of datadevoted specifically to this problem.

REFERENCES

[1] H. Camon, C. Ganibal, N. Raphoz, M. Trzmiel, C. Pisella, C. Martinez,S. Valette, Solving functional reliability issue for an optical electrostaticswitch, Microsystem Technologies, Vol. 14, N. 7, July 2008.

[2] H. Camon, F. Larnaudie, Fabrication, simulation and experiment ofa rotating electrostatic silicon mirror with large angular deflection,13th Int. Micro Electro Mechanical Systems (MEMS 2000), Miyazaki(Japan), Jan. 23-27, 2000, pp. 645-650.

[3] C. Casenave, Representation diffusive et inversion operatorielle pourl’analyse et la resolution de problemes dynamiques non locaux, PhDthesis, Toulouse (France), dec. 2009.

240 260 280 300 320 3400.102

0.1022

0.1024

0.1026

0.1028

0.103

Time (hours)

AC voltage (mV)

Fig. 4. Measured AC voltage (:), dynamic correction (−)

0 50 100 150 200 250 300 350−4

−2

0

2

4x 10

−4

Time (hours)

AC

vo

ltag

e (

mV

)

0 50 100 150 200 250 300 350−4

−2

0

2

4x 10

−4

Time (hours)

AC

vo

ltag

e (

mV

)

Fig. 5. Residual variations: with static correction (top), with dynamiccorrection (bottom); non smoothed (:), smoothed (−)

[4] C. Casenave, G. Montseny, Diffusive Identification of Volterra Models byCancellation of the Nonlinear Term, 15th IFAC Symposium on SystemIdentification, SYSID 2009, Saint-Malo (France), July 6-8 2009, pp 640-645.

[5] C. Casenave, E. Montseny, H. Camon, Identification of a NonlinearDynamic Model of MEMS with Unstable Switching Zone, IEEE Trans-actions on Control Systems Technology, accepted for publication, 2009.

[6] C. Casenave, G. Montseny, Identification of Nonlinear Volterra Modelsby means of Diffusive Representation, 17th IFAC World Congress, Seoul(Korea), July 6-11, 2008, pp.4024-4029.

[7] C. Casenave, G. Montseny, Diffusive Representation and Applications,submitted to 4th IFAC Symposium on System, Structure and Control,SSSC 2010, Ancona (Italy), 2010.

[8] A. Karkkainen, N. Pesonen, M. Suhonen, A. Oja, A. Manninen, N.Tisnek, H. Seppa, MEMS based AC voltage reference, IEEE Trans.Instrum. Meas. 54, pp. 595–599, 2005

[9] J. Kyynarainen, A.S. Oja and H. Seppa, Stability of microelectrome-chanical devices for electrical metrology, IEEE Trans. Instrum. Meas.50, pp. 1499–1503, 2001.

[10] G. Montseny, Representation diffusive, Hermes Science, 2004.[11] M. Suhonen, H. Seppa, A.S. Oja, M. Heinila, I. Nakki, AC and DC

voltage standards based on silicon micromechanics, CPEM’98 Conf.Dig, pp. 23–24, 1998.

0 100 200 300−1.5

−1

−0.5

0

0.5

1

1.5x 10

−4

Time (hours)

AC

vo

ltag

e (

mV

)

Fig. 6. Smoothed residual variations: static correction (- -), dynamiccorrection (−)

0 100 200 300 40024.5

24.6

24.7

24.8

24.9

25

25.1

25.2

25.3

25.4

25.5

Time (hours)

Temperature (Celsius)

Fig. 7. Temperatures: T (t) and ψj(t) + ψ0,j(t), j = 1 : 6

0 20 40 6025

25.05

25.1

25.15

25.2

25.25

25.3

Time (hours)

Temperature (Celsius)

Fig. 8. Temperatures: T (t) and ψj(t) + ψ0,j(t), j = 1 : 6 (zoom)

220 240 260 280 300

24.6

24.7

24.8

24.9

25

25.1

25.2

25.3

25.4

Time (hours)

Temperature (Celsius)

Fig. 9. Temperatures: T (t) and ψj(t) + ψ0,j(t), j = 1 : 6 (zoom)

2 4 6 8 10 12

25.02

25.04

25.06

25.08

25.1

25.12

Time (hours)

Temperature (Celsius)

Fig. 10. Temperatures: T (t) and ψj(t) + ψ0,j(t), j = 1 : 6 (zoom)

50 100 150 200 250 300 350

−0.1

0

0.1

0.2

0.3

0.4

Time (hours)

Te

mp

era

ture

(C

els

ius)

Fig. 11. Dynamic differences of temperatures: Ψj(t) = ψj(t) +ψ0,j(t)−T (t), j = 1 : 6