Correction for thermal lag in dynamic temperature measurements using resistancethermometersKrzysztof Tomczuk and Radoslaw Werszko Citation: Review of Scientific Instruments 84, 074903 (2013); doi: 10.1063/1.4816648 View online: http://dx.doi.org/10.1063/1.4816648 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/84/7?ver=pdfcov Published by the AIP Publishing

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REVIEW OF SCIENTIFIC INSTRUMENTS 84, 074903 (2013)

Correction for thermal lag in dynamic temperature measurementsusing resistance thermometers

Krzysztof Tomczuk1,a) and Radoslaw Werszko2

1Department of Mechanical and Power Engineering, Wroclaw University of Technology,Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland2Department of Mechanical Engineering, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27,50-370 Wroclaw, Poland

(Received 28 January 2013; accepted 11 July 2013; published online 26 July 2013)

Periodical changes of temperature in the autoclave for the purpose of automatic control are measuredwith the aid of an encased resistance thermometer. To minimize dynamic errors of this thermometer,two different correcting algorithms have been employed: a known single time-constant one and analgorithm proposed by the authors—two time-constant one. The verification and comparison of thetwo algorithms was done using a physical model of the autoclave and a real thermometer. Addition-ally, three methods for the determination of time constants of the second order model were comparedand factors influencing the algorithms accuracy, including time constants and sampling time, wereanalysed. The presented methods make possible to increase both the bandwidth of dynamical tem-perature measurements and its precision with relatively limited increase in computational complexityof the correction algorithm. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4816648]

I. INTRODUCTION

The measurement of time variable temperature is usu-ally inherent with a bigger or smaller dynamic error.1–5 Itdepends mainly upon dynamic properties of the instrumentand upon the speed and, possibly, acceleration of the mea-sured temperature fluctuations.6 To minimize dynamic errors,dedicated correctors for the modification of the thermometeroutput signal have been used. The structure of these correc-tors is based on the theory of dynamic systems:4, 5 the cor-rector transfer function shall be the inverse of the transferfunction defining the dynamic properties of the corrected sys-tem. That means in practice that those correctors shall be idealdifferential elements which are difficult in terms of hardwareimplementation.

As computers have been used more and more for the stor-age of measurement results, the compensation of dynamicalproperties can be done by an algorithmic method.7–13 A gen-eral scheme of a layout for the compensation of the dynamicalproperties of instruments is shown in Fig. 1.

An analogue output signal of instrument 1, mostly asvoltage, must be converted into a digital form in interface2 responsible for its discretisation: it is replaced by a se-ries of the signal values read at the moment of sampling.Next, the signal is input to computer 3 using, e.g., an USBport. The sampled values are stored and used for the calcula-tions of corrected values of the instrument output signal. Thisis done based upon a correction equation input to the com-puter, the variables of which are current values of the sampledsignal yn.

The objective of this paper is the elaboration and ex-perimental verification of an algorithmic compensation of aresistance thermometer designed for the autoclave automaticcontrol. To solve this problem, a few practical issues had

a)E-mail: krzysztof.tomczuk@pwr.wroc.pl

to be investigated, i.e., experimental determination of timeconstants of the thermometer, selection of sampling time, theaccuracy of readout of current signal values including theerror transfer analysis. The solutions of the abovementionedproblems were not found in any available publications.

II. DESCRIPTION OF THE COMPENSATION METHOD

Thermometers are modelled usually as first order ele-ments or, seldom, as second order elements due to the pres-ence of a inflexion point in the step response waveform ofthermometers. It means that relation between output and inputsignals of thermometers can be described using, respectively,first or second order differential equations. The second ordermodel is especially useful in situation when resistive elementof the thermometer is secured by massive encasing. Such acondition occurs inside an autoclave, where the thermometeris immersed in a toxic medium. Moreover, the second ordermodel allows more precisely to restore not only amplitude ofthe temperature runs but also its phase delay. It is especiallyimportant for the autoclave control system. We will return tothis problem at explanation of the experiment verification inchapter 4. The second order model was selected for our re-search. It contains two first order elements connected in series(Fig. 2).

It can be described by a set of two equations

T1du

dt+ u = x, (1)

T2dy

dt+ y = ku, (2)

or, after elimination of the variable u, with one equation

(T1 · T2)d2y

dt2+ (T1 + T2)

dy

dt+ y = kx, (3)

0034-6748/2013/84(7)/074903/7/$30.00 © 2013 AIP Publishing LLC84, 074903-1

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074903-2 K. Tomczuk and R. Werszko Rev. Sci. Instrum. 84, 074903 (2013)

FIG. 1. A general scheme of an algorithmic layout for the compensationof thermometers dynamic. 1—Thermometer, 2—data acquisition device, 3—computer, x—measured temperature, y—output voltage of thermometer, andyn—sampling value of voltage y.

where x is the measured temperature in ◦C, u is the temper-ature inside the casing in ◦C, y is the output signal of theinstrument in V, k is the static coefficient of the instrumentconversion in V/◦C, and T1, T2 are the time constants in s.

Equations (1) and (2) or (3) make it possible to find anunknown time dependence of the measured temperature x oncondition that the time dependence of the output signal of thethermometer y and constant coefficients k, T1, T2 are known.

As the differentiation of the signal y is carried out numer-ically Eqs. (1) and (2) are rewritten to the form

un−1/2 = yn−1/2 + T2

τ(yn − yn−1), (4)

xn = 1

k

[un + T1

τ

(un+1/2 − un−1/2

)], (5)

and similarly, in the case of an instrument defined by Eq. (3)

xn = 1

k

[yn + T1 + T2

2τ(yn+1 − yn−1)

+T1 · T2

τ 2(yn+1 − 2yn + yn−1)

], (6)

where xn is a sampling value of the restored temperature at amoment t = nτ in ◦C, yn, yn+1, yn−1, yn−1/2 are the sampledvalues of the output signal of the thermometer at moments:t = nτ , t = (n + 1)τ , t = (n − 1)τ , t = (n − 1/2)τ in V,un, un−1/2, un+1/2 are the sampled values of the intermediaryoutput signal of the thermometer at moments: t = nτ , t = (n− 1/2)τ , t = (n + 1/2)τ in ◦C, n is the sample number, and τ

is the sampling time in s.Fig. 3 shows the method of generation of un and xn values

from yn waveform. Two consecutive sampled values yn−1 andyn are used for evaluation of un−1/2 value according to Eq. (4).Additionally, next two values yn and yn+1 obtained from thesame equation are used for evaluation of un+1/2 value. These

FIG. 2. Thermometer model. 1—Encasing and 2—sensor (resistance wire).x—Measured temperature, u—temperature inside the casing, and y—outputvoltage of thermometer.

FIG. 3. Notation interpretation.

two values are substituted into Eq. (5) in order to produce xn

value. It should be noted that three consecutive readouts yn−1,yn, and yn+1 are necessary in order to compute the same valuefrom Eq. (6).

It is obvious that the proposed method of compensationdoes not require the knowledge regarding neither the mathe-matical description of the output signal nor the border or ini-tial conditions, therefore it is feasible that it can be used forcorrection of signal y of any degree of complexity.

Before the experimental verification of the proposedmethod of compensation, we can perform an additional sim-ulation, during which we will make an assumption that theoutput signal y is sinusoidal. If such a condition is met, thenthe measured value x can be reconstructed using the followinganalytic formulae:15

b

a= k√[

1 − T1T2 (2πf )2]2 + [

2πf (T1 + T2)]2

, (7)

ϕ = arctg2πf (T1 + T2)

T1T2 (2πf )2 − 1, (8)

where k, T1, and T2 are the thermometer parameters, a, b arethe amplitudes of sinusoidal waveforms, f is the frequency ofsinusoidal waves x and y, and ϕ is the phase shift betweensinusoidal waves x and y.

We assume that a sine waveform with a frequency f= 0.028 Hz and amplitude b = 1 (Fig. 4, line 1) is the out-put signal of thermometer, the time constants of which are T1

= 5 s, T2 = 3 s and static coefficient k = 1.After substituting relevant values to Eq. (7), the following

ratio was obtained:

b

a= 0.6678.

Hence

a = 1

0.6678= 1.4975.

On the other hand, the amplitude of the restored sinusoidalwaveform computed according to Eqs. (4)–(6) is equal to(Fig. 4)

a∗ = 1.4915.

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074903-3 K. Tomczuk and R. Werszko Rev. Sci. Instrum. 84, 074903 (2013)

FIG. 4. Runs of sine curves and comparison of the restoration accuracy ofthe measured value time dependence. 1—Output signal y(t) of the instrumentand 2—restored shape of the measured signal x*(t).

The relative error of the amplitude restoration expressed inpercents is therefore equal to

δ = a − a∗

a100% = 0.39%.

The error of the phase shift was verified in a similar way usingEq. (8). We have

ϕ = −68.74◦

and from shift of sinusoid in Fig. 4

ϕ∗ = −70.00◦.

Finally, the relative error of restoration of the phase shift com-pared against the sine period can be expressed by equation

δ = ϕ − ϕ∗

360◦ 100% = 0.35%.

The simulations were performed using LabView package andthe block diagram showed in Fig. 5.

These simulations proved that in spite of different struc-ture, equation sets (4) and (5) and Eq. (6) produce practicallyidentical results, and the level of reproduction errors – bothregarding amplitude and phase – remains notably below 1%.

TABLE I. Time constants T1 and T2 and restoration errors δ of the stepresponse curve.

Method T1 (s) T2 (s) δ (%)

Graphical 31.00 4.00 0.40Analytical 29.50 5.77 0.10Iteration algorithm 30.00 5.50 0.05

III. DETERMINATION OF TIME CONSTANTSFOR THE Pt100 THERMOMETER

Before programming Eqs. (4) and (5) or, alternatively, (6)for the restoration of the measured value x, the thermome-ter time constants T1 and T2 included in these equations shallbe experimentally determined. The method of inverted stepcharacteristic of an inertial system of the second order, knownboth in the graphical5 and analytical4 versions, was used forthis purpose.

Numerical values of time constants calculated with theaid of this method lacked accuracy. To make them more pre-cise, an additional iterative algorithm (trial and error) hasbeen employed. An equation describing the step characteris-tic of the inertia system of the second order was used for thispurpose

ϑ(t) = ϑ1 + �ϑ

[1 − 1

T1 − T2

(T1e

−t/T1 − T2e−t/T2

)],

(9)where ϑ(t) is the current value of the characteristic at time t,ϑ1 is the initial value of the step, �ϑ is the step amplitude,and T1, T2 are the time constants.

Within the framework of the proposed algorithm, valuesϑ(t) calculated for a given pairs of constants T1 and T2, wereconfronted with values ϑ(t) read from the experimental stepcharacteristic. In the next stage, time constants were modi-fied by a small value and again, the results of calculations ofvalues ϑ(t) were confronted with experimental data. As a re-sult of few dozens of iterations, a clear improvement of therestoration accuracy of the step characteristic of the instru-ment was obtained.

Finally, in the further analysis, those values of T1 andT2 were accepted for which the average discrepancy betweenvalues calculated ϑc(t) and read from characteristic ϑ r(t) andreferenced to the step value, do not exceed 0.05%. Table I

FIG. 5. Simplified flow chart for computing restored temperature. 1—Sine wave generator, 2—function blocks with Eqs. (4) and (5) implemented, 3—functionblock with Eq. (6) implemented, and 4—multi-signal chart display.

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074903-4 K. Tomczuk and R. Werszko Rev. Sci. Instrum. 84, 074903 (2013)

FIG. 6. A test stand for the determination of step characteristics of the ther-mometer. 1—Vessel with water of temperature ϑ1, 2—vessel with water oftemperature ϑ2, 3—heater, 4—tested thermometer, 5—reference thermome-ter, 6—data acquisition device, and 7—computer.

summarises the results of calculations

δ = ϑc(t) − ϑr (t)

�ϑ100% ≤ 0.05%.

The step characteristic of the thermometer was obtained in thetraditional way, i.e., by transferring the tested thermometer 4(part number T-109G, manufactured by Termo-Precyzja plc,Poland) from a vessel 1 (Fig. 6) filled with water of a fixedtemperature ϑ1 to a vessel 2 of an elevated but constant tem-perature ϑ2, which was equipped with a heater plate 3. Yet an-other miniature Pt100 thermometer 5 (part number PT106051by Termo-Precyzja plc) without encasing was mechanicallyand thermally coupled with the tested thermometer. It wasused during testing as a reference sensor with a time constantnearly two orders lower than that of the tested instrument andwith very accurate indications of the measured temperature.Resistance output signals of the two thermometers were con-nected with the inputs of a universal data acquisition device 6(NI9219, National Instruments).

After the analog/digital conversion the signals were inputvia a universal serial bus (USB) port to computer 7 with theLabView software (National Instruments) installed.

The plots of the step characteristics of the two thermome-ters are shown in Fig. 7.

IV. VERIFICATION OF THE COMPENSATION METHOD

The method verification was carried out using a physicalmodel of an autoclave. The autoclave consists of a vessel 1

FIG. 7. Step characteristics of Pt100 thermometers. 1—Encased and 2—without encasing.

FIG. 8. Scheme of the test stand: a simplified model of the autoclave witha measuring system. 1, 2—Vessels with water, 3—tested thermometer, 4—reference thermometer, 5—data acquisition device, and 6—computer.

filled with water (Fig. 8) and immersed in a vessel 2 providedwith heating or cooling medium (water was also used for thispurpose).

The temperature of water in vessel 1 was controlled withthe tested resistance thermometer 3 and with a reference, notencased, thermometer 4. Both thermometers were of a plat-inum Pt100 type. The outputs of the thermometers were con-nected to the inputs of a data acquisition device 5. Digitallyconverted signals were fed to computer 6 for the purpose ofvisualisation and further processing.

The purpose of the autoclave is a periodical, fast heatingof the medium (water) in vessel 1 and then equally fast cool-ing. When heating, the autoclave is fed with steam and whencooling, with brine. In the case of the laboratory experimentin question, vessel 1 was periodically transferred from a ves-sel with boiling water to a vessel containing coldwater andvice versa.

Fig. 9 (plot 1) illustrates the actual temperature variationin vessel 1 and plot 2, the temperature indicated by the en-cased thermometer.

Plots 3 and 4 correspond to corrected temperature plotsbased upon two time-constants algorithm and single time-constant algorithm. Plots 1 and 3 are very similar and a dif-ference in their amplitude is equal to ε1 = 1.5 ◦C on av-erage. In relation to the amplitude of plot 1 the error is

FIG. 9. Temperature variation in the autoclave. 1—Actual temperaturemeasured by reference thermometer, 2—temperature indicated by testedthermometer, 3—corrected temperature based upon two time-constants al-gorithm, and 4—corrected temperature based upon single time-constantalgorithm.

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074903-5 K. Tomczuk and R. Werszko Rev. Sci. Instrum. 84, 074903 (2013)

FIG. 10. Temperature variation in the autoclave. 1—The actual temperaturemeasured by reference thermometer, 2—the temperature indicated by testedthermometer, 3—plot of the corrected temperature based upon two time-constants algorithm, and 4—plot of the corrected temperature based uponsingle time-constant algorithm.

equal to δ = 4.2% whereas an analogous error between theamplitudes of plots 1 and 2 is equal to δ = 43%. On the otherhand, a difference between amplitudes of plots 1 and 4 isslightly bigger and is equal to ε2 = 3.5 ◦C (δ = 9.8%). Thephase shift is also bigger, which means additional errors inthe case of automatic control. As an example, at t = 200 s,ε1 = 3.8 ◦C and ε2 = 10.8 ◦C. The ratio ε2/ε1 = 2.8 is herebigger than in the previous case (ε2/ε1 = 2.3). These valuesconfirm to the usefulness of the proposed method of compen-sation of dynamic errors of resistance thermometers.

In the final stage of the research, a hypothetical assump-tion was considered, i.e., the frequency of temperature varia-tions in the autoclave is bigger (1.3 counts per minute (cpm))than assumed (0.4 cpm) and the amplitude of theses fluctu-ations is smaller (which usually goes in a pair). The relevanttemperature plots are shown in Fig. 10. As is seen, in this casethe two time-constants algorithm proves to be better.

The applicability range of both algorithms needs furtherresearch with different objects and different time constants.As the authors were unable to found such data in the availableliterature, they decided to carry out this type of research withreference to a real industrial problem (autoclave).

V. ANALYSIS OF MEASUREMENT ERRORS

Sampled values yn, yn−1, and yn+1, present in Eq. (6) areread by a conversion card and computer with a certain error.The same is valid for the calculations of time constants T1 andT2. To check if these errors (assuming quasi-static conditions)have a certain impact on the uncertainty of sampled values xn,Eq. (6) in the following form was used:

xn = yn + T1

2τ�y1 + T2

2τ�y1 + T1 · T2

τ 2�y2, (10)

where �y1 and �y2 are the sampled values differences duringsampling, where

�y1 = yn+1 − yn−1,

�y2 = yn+1 − 2yn + yn−1.

To avoid additional calculation related errors, values of theoutput voltage of the thermometer converter bridge were en-tered to Eq. (10).

The absolute error of the calculation results ex, basedupon the total differential of Eq. (10), is expressed with thefollowing equation:14

ex =∣∣∣∣ ∂x

∂yn

∣∣∣∣ · eyn+

∣∣∣∣ ∂x

∂�y1

∣∣∣∣ · e�y1 +∣∣∣∣ ∂x

∂�y2

∣∣∣∣ · e�y2

+∣∣∣∣ ∂x

∂T1

∣∣∣∣ · eT1 +∣∣∣∣ ∂x

∂T2

∣∣∣∣ · eT2 , (11)

where | ∂x∂yn

|, | ∂x∂�y1

|, | ∂x∂�y2

|, | ∂x∂T1

|, | ∂x∂T2

| are the partial deriva-tives with respect to all variables, i.e., yn, �y1, �y2, T1, T2.

Hence, after determination of respective partialderivatives

ex = eyn+ T1 + T2

2τe�y1+

T1 · T2

τ 2e�y2+

(�y1

2τ+T2�y2

τ 2

)eT1

+(

�y1

2τ+ T1�y2

τ 2

)eT2 , (12)

where

τ = 2 (s), T1 = 30 (s), eT1 = 0.35 (s), T2 = 5.5 (s),

eT2 = 0.26 (s),

�y1 = 1.79 − 1.69 = 0.1 (V),

�y2 = 2.5697 − 2 · 2.4599 + 2.3485 = 0.0016 (V),

eyn= 0.0001 (V), e�y1 = 0.0002 (V), e�y2 = 0.0004 (V).

After substituting the above data to Eq. (12), the absolute erroris equal to

ex = 0.0001 + 35.5

4· 0.0002 + 165

4· 0.0004

+(

0.1

4+ 5.5 · 0.0016

4

)· 0.35

+(

0.1

4+ 30 · 0.0016

4

)· 0.26 = 0.0001

+ 0.001775 + 0.0165 + 0.00952 + 0.00338,

hence

ex = 0.03128 (V),

ex = 0.03128 · 19.95 = 0.6 (◦C),

where k = 19.95 ◦C/V is the static conversion coefficient.When comparing the obtained error value to the ampli-

tude of actual temperature and expressing the result in per-cents, we can express the relative error is equal to

δ = ex

ymax

· 100% = 0.031

1.75· 100% = 1.8%

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074903-6 K. Tomczuk and R. Werszko Rev. Sci. Instrum. 84, 074903 (2013)

or

δ = 0.031 · 19.95

1.75 · 19.95· 100% = 1.8%.

As can be seen, the error obtained from actual temperaturevariations (δ = 4.2%) is of the same order.

VI. SUMMARY AND COMMENTS

The measurement of variable temperatures carries usu-ally a bigger or smaller dynamic error. This error can be min-imized by correcting a recorded temperature variation with arelatively simple algorithm: single time-constant one or twotime-constants one.

The experiments done confirm that the dynamic error canbe reduced about ten times, i.e., from 43% to 4.2% (in case oftwo time-constants algorithm) and about four times, to 9.8%(in case of single time-constant algorithm).

Similarly, if we take into account phase shift (importantin automatic control), much better is two time-constants algo-rithm (see Figs. 9 and 10).

When analysing errors, guided by intuition, the samplingtime was considered as error-free. On the other hand, it is ob-vious that the selection of τ values is crucial to the compen-sation process.

Research done was based upon the followingrecommendations:

τ ′ ≈ T

10(13)

or

τ ′′ ≈ To

5, (14)

where τ ′ and τ ′′ are the alternative sampling times, T is theequivalent time constant of the object (thermometer),15 andTo is the delay,16 Fig. 11.

From the characteristic of the tested thermometer wehave T = 28.8 s, To = 8.2 s. A mean value of τ was assumedas the sampling time

τ = τ ′ + τ ′′

2= (28.8/10) + (8.2/5)

2≈ 2 s.

It was experimentally confirmed that the sampling time equalto 2 s provides the best result in the context of minimisationof the dynamic error.

FIG. 11. Step characteristic of the object (thermometer).

FIG. 12. The influence of sampling time (τ = 0.5 s) on the restoration of thetemperature variation with the aid of both algorithms. 1—The actual tempera-ture variation in the vessel, 2—temperature indicated the encased thermome-ter, 3—plot of the temperature obtained based upon the two time-constantsalgorithm, and 4—plot of the temperature obtained based upon the singletime-constant algorithm.

With longer sampling times, the restored plots weresmoother but their amplitudes, smaller. With shorter samplingtimes, the discussed temperature variations were more dis-torted most likely as a result of differentiation related noise(see Fig. 12).

As is seen, the single time-constant algorithm in this casehas proved to be more robust.

Further research shall be focused on a better method forthe determination of time constants of the instrument and de-termination of a relationship between the time constant, tem-perature variation period, and allowable dynamic error. Tem-perature variations with a 25 s duration period analysed in thepaper were satisfactorily corrected with an approximately 30 stime constant thermometer.

A problem that shall also be solved during further studiesis the elaboration of a criterion of sampling time selection orthe refinement of the compensation algorithm with the inter-ference filtration problem.

ACKNOWLEDGMENTS

The work was financed by a statutory activity subsidyfrom the Polish Ministry of Science and Higher Education forthe Faculty of Mechanical and Power Engineering of Wro-claw University of Technology.

1J. P. Bentley, Principles of Measurement Systems (Pearson, 2005).2R. P. Benedict, Fundamentals of Temperature, Pressure and Flow Measure-ments, 3rd ed. (John Wiley and Sons Inc., 1984), pp. 255–279.

3R. E. Bentley, Temperature and Humidity Measurement—Handbook ofTemperature Measurement (Springer, 1998), Vol. 1.

4L. Michalski, K. Eckersdorf, J. Kucharski, and J. McGhee, TemperatureMeasurement (John Wiley and Sons, 2001), pp. 279–329.

5E. O. Doebelin, Measurement Systems: Application and Design (McGraw-Hill Publishing Company, 1990), pp. 612–706.

6F. Cimerman, B. Blagojevic, and I. Bajsic, “Identification of the dynamicproperties of temperature-sensors in natural and petroleum gas,” Sens. Ac-tuators, A 96(1), 1–13 (2002).

7L. Qingyang, Z. Yuanliang, and X. Wei, “Dynamic compensation of Pt100temperature sensor in petroleum products testing based on a third ordermodel,” in IEEE International Workshop on Intelligent Systems and Appli-cations (IEEE, 2009), pp. 1–4.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

108.6.47.237 On: Fri, 14 Mar 2014 11:45:48

074903-7 K. Tomczuk and R. Werszko Rev. Sci. Instrum. 84, 074903 (2013)

8M. Tagawa, K. Kato, and Y. Ohta, “Response compensation of temperaturesensors: Frequency-domain estimation of thermal time,” Rev. Sci. Instrum.74, 3171–3174 (2003).

9M. Tagawa, K. Kato, and Y. Ohta, “Response compensation of fine-wiretemperature sensors,” Rev. Sci. Instrum. 76, 094904 (2005).

10C. Wang, H. Ding, Q. Liu, and H. Wang, “The dynamic compensation oftemperature sensors in sonic nozzle airflow standard facilities based onmethod of positive pressure,” in 2012 IEEE International on Instrumen-tation and Measurement Technology Conference (I2MTC) (IEEE, 2012),pp. 2005–2009.

11C. Zhang, W. Zhao, and G. Wang, “Research on dynamic temperaturemeasurement method based on transient forward and inverse heat transfermodel,” Chin. J. Sci. Instrum., Issue 11, 2499–2505 (2011).

12G. Milano, F. Scarpa, and M. Cartesegna, “Adaptive correction of dynamictemperature measurements to improve estimation of thermophysical prop-erties,” High Temp. - High Press. 32(3), 293–303 (2000).

13L. Jackowska-Strumillo, “ANN based modelling and correction in dynamictemperature measurements,” in International Conference on Artificial In-telligence and Soft Computing (ICAISC 2004) (Springer, 2004), pp. 1124–1129.

14J. R. Taylor, An Introduction to Error Analysis (University Science Books,1997), pp. 45–93.

15R. C. Dorf and R. H. Bishop, Modern Control Systems (Pearson PrenticeHall, 2008), pp. 493–497.

16P. de Larminat and Y. Thomas, Automatique des Systems Lineaires (Flam-marion Sciences, 1975).

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