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Determination of optimum measurement points via A-optimality criterion for the calibration of measurement apparatus
Measurement 43 (2010) 563–569
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Determination of optimum measurement points via A-optimality criterion for the calibration of measurement apparatus Ch. Hajiyev * Istanbul Technical University, Faculty of Aeronautics and Astronautics, Maslak, 34469 Istanbul, Turkey
a r t i c l e
i n f o
Article history: Received 2 February 2007 Received in revised form 16 October 2009 Accepted 28 December 2009 Available online 11 January 2010 Keywords: Measuring apparatus Calibration curve Calibration design Least squares method
a b s t r a c t A procedure for optimal selection of the measurement points to get the best calibration characteristics (for the chosen optimality criterion) of measuring apparatus is proposed. The coefficients of the calibration characteristics are evaluated by the classical least squares method. For this work, the A-optimality criterion has been used as an optimality criteria. As an example, the problem of optimal selection of the standard pressure setters (the piston gauges) during calibration of the differential pressure gage is solved. Obtained values of the optimum measurement points for the calibration of the differential pressure gage are checked via actual experiments. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Accurate measurement is the basis of almost all engineering applications, since uncertainty inherently exists in the nature of any measuring apparatus. On the other hand, the cost of a measuring apparatus increases with its accuracy. Therefore, low cost accurate measurement devices are one of the main goal of metrology engineers. One way for decreasing the sensor uncertainties is the calibration process [1]. Therefore, this paper deals with the calibration of a low cost sensor using the corresponding values obtained by reference standards. From the practical point of view, the calibration characteristics should be in a polynomial form. The accuracy of this polynomial depends on the noise-free data which is used to obtain the characteristics [2]. To reduce the effect of the noise, excessive number of data should be used. However this requires more experiments and that increases the cost. Thus, the main question becomes the evaluation of accurate calibration characteristics with minimum number of experimental data. In [1–4] the equidistant measurement points are used for calibration of measurement apparatus. On the * Tel.: +90 2122853105; fax: +90 2122852926. E-mail address: [email protected] 0263-2241/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2009.12.029
other hand, even though it is paradoxical, the application of the equidistant measurement points to get the best calibration characteristic (for the chosen optimality criterion) is erroneous, because these points are not optimum in a sense that given performance criterion is minimum. The equidistant calibration points are obtained by simply dividing the range of the transducer into the appropriate number of samples. In [5,6] an approach to design sensor calibration is proposed with the aim of reducing the calibration curve uncertainty. This uncertainty reduction is achieved by minimizing either the standard deviations of the regression curve coefficients or the standard deviation of the whole estimated calibration curve. In particular, criteria for the choice of the number of calibration points and their optimum location are theoretically identified when the response characteristic is a polynomial and the uncertainties of the sensor outputs can be neglected. The proposed criteria is not directly applicable to non-linear sensors and complex apparatus with an indirect estimation of the measurand. The calibration of sensors, or in general, the measurement apparatus, can be considered as a typical problem of ‘‘model searching” from experimental data and then, it should be approached by applying experimental design
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Ch. Hajiyev / Measurement 43 (2010) 563–569
techniques. In this case, the calibration design should drive the operator in the choice of [7]: the experimental plan (e.g. the number and the location of the calibration points, the number of the repetitions); the main influence quantities; the regression curve; the regression technique; the standard references and their uncertainties. The calibration design using the experimental design techniques is proposed in [7], when the relationship between the indirectly calculated measurands and the sensor inputs is non-linear or more than one measurand has to be considered. In particular, the optimum calibration plan for measurement chain is identified by suitably elaborating the error propagation law suggested by the ISO Guide [8]. Sensor calibration and compensation using the artificial neural network are performed in [9]. The artificial neural network based inverse modeling technique is used for the sensor response linearization. The choice of the order of the model and the number of the calibration points are important design parameters in this technique. An intensive study about the effect of the order of the model and the number of the calibration points on the lowest asymptotic root-mean-square (RMS) error has been reported in this paper. In [10] a genetic algorithm is used to perform optimizations for both the computation of the optimal input to the sensor and the optimal constant feedback gain. A calibration method, which uses the neural network and genetic algorithms together, is presented in [11]. It uses the improved back-propagation neural network to model the characteristic curve of the vortex flowmeter. And then it applies the genetic algorithms to seek two additional optimum calibration points intelligently at the intervals where the curve are non-linear obviously. At last the vortex flowmeter is calibrated at the new calibration points. The methods based on the artificial neural networks and genetic algorithms do not have physical bases. Therefore according to the different data, which corresponds to the same event, the model gives different solutions. Thus, the model should be continuously trained by using the new data. In this study, optimal selection method of the sample measurement points via A-optimality criterion for the calibration of measurement apparatus, which takes the uncertainties on their outputs into account, is proposed. 2. Problem statement As it is known from practical considerations, the calibration curve should be in a polynomial form as follows:
yi ¼ a0 þ a1 pi þ a2 p2i þ þ am pm i ;
ð1Þ
where yi is the output of the low cost transducer, pi are the values of the reference standard and a0, a1, . . ., am are the calibration curve coefficients. Measurement contains random noises in Gaussian form
zi ¼ yi þ di ¼ a0 þ a1 pi þ a2 p2i þ þ am pm i þ di
ð2Þ
where zi is the measurement result, di is measurement error with zero mean and standard uncertainty r. Let the calibration curve coefficients be denoted as ~ h¼ ½a0 ; a1 ; . . . ; am T . The coefficients in these polynomials are evaluated in [12] by the least squares method. The expressions that are used to make the evaluation had the form:
e Þ1 ð X e T zÞ eT X ~h ¼ ð X e ^hÞ ¼ ð X eT X e Þ1 r2 Dð
ð3Þ
where z ¼ ½z1 ; z2 ; . . . ; zn ments;
2
1 p1 61 p 6 2 6 6 e 6 X ¼6 6 6 4
p21 p22
1 pn
p2n
T
is the vector of the measure-
3 pm 1 m7 p2 7 7 7 7 7 7 7 5
ð4Þ
pm n
is the matrix of the known coordinates (here, p1, p2, . . ., pn are the values that are producible by the reference stane ^ dard instrument), Dð hÞ is the dispersion matrix of the estimated coefficients. The values of the reference standard p1, p2, . . ., pn, should be such that the polynomial characteristics given by (1) approximates the real calibration characteristics in the best form. Thus the problem can be stated as follows: Find the values of p1, p2, . . ., pn such that the values of a1, a2, . . ., am are optimum in the sense that a given performance criterion is minimum. e can be used as a As mentioned above, the matrix D measure of the error between the low cost transducer and the high precision reference standard instrument. A e performance criteria for the minimum of the matrix D can be selected by the following ways: Minimize the trace (the sum of the diagonal elements) e (the A-optimality criterion). of the matrix D Minimize the generalize dispersion (the determinant) of e (the D-optimality criterion). the matrix D e (the Minimize the maximal eigenvalue of the matrix D E-optimality criterion). e Minimize the sum of the all elements of the matrix D and etc. Each of the above-mentioned measures characterizes one or another geometrical parameter of the correlation ellipsoid. The choice of one or another scalar measure as the optimality criteria for a solved particular problem depends on the basic indicators of optimization (the mathematical simplicity, convenience of obtaining analytical results, volume of computational expenditures, etc.). For this reason, A-optimality criterion is used in this work, i.e.
~ 1 r2 ~ T XÞ minPi ½TrðX
ð5Þ
is sought. The values of p1, p2, . . ., pn, found by solving the above equations, should be in the range of 0–pmax. Otherwise the solution is invalid.
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Ch. Hajiyev / Measurement 43 (2010) 563–569
3. The solution algorithm The objective function is described in the following form
e ; p ; . . . ; p Þg: f ðp1 ; p2 ; . . . ; pn Þ ¼ Trf Dðp 1 2 n
ð6Þ
As explained above the problem is a constrained optimization problem. The objective function is multivariable, nonlinear, continuous and it has derivative in the considered interval. Assume that the minimum of f (p1, p2, . . ., pn) exists for the following values of p1, p2, . . ., pn:
p ¼ ½p1 ; p2 ; . . . ; pn T : In order to gain p* which is a minimum of (6), the following conditions should be satisfied [13]:
Df ðp Þ ¼ 0;
ð7Þ
D2 f ðp Þ is semi positive;
ð8Þ
where D denotes the gradient. The extremum condition given by (7) can be explicitly written as:
e ; p ; . . . ; p Þg=@p ¼ 0; @½Trf Dðp 1 2 n i
ði ¼ 1; 2; . . . ; nÞ:
ð9Þ
When the derivatives in (9) are calculated n algebraic equations with n unknowns are obtained.
Q i ðp1 ; p2 ; . . . ; pn Þ ¼ 0;
ði ¼ 1; 2; . . . ; nÞ;
ð10Þ
where n denotes the number of the measurements. Derivation of Eq. (10) for the case of m = 2 is given in detail in Appendix A. Numerical routines such as the gradient descent algorithms [14] can be used. It should be noted that the sign of D2f(p) should be calculated together with Df(p) in order to determine whether the found values correspond to a local minimum or a local maximum. Furthermore, those vale 1, p2, . . ., pn)} minimum should be in ues which make Tr{ D(p the range 0 pmax. The solution set which satisfies the above conditions can be used to calculate the polynomial coefficients given in (1). This polynomial approximates the calibration characteristics between the low cost and the reference standard equipment in the best way and it can be used to extract the accurate values from the outputs of the low cost transducer.
The calibration characteristics of the examined measuring device (in the present case, the differential pressure gage) is described by 2nd order polynomial as follows [12]:
yi ¼ a0 þ a1 pi þ a2 p2i : Measurement equation is written in the form
zi ¼ a0 þ a1 pi þ a2 p2i þ di ;
i ¼ 1; n;
ð12Þ
where di is the measurement error with zero mean and the standard uncertainty r. The optimum measurement points for the calibration of the above mentioned differential pressure gage are evaluated via Eq. (A-4), which is presented in Appendix A. The method is used to obtain the optimum coefficients of the characteristic polynomial for a selected criteria. Calculation is performed for n = 3 and n = 4. Closed form algebraic Eq. (A-4) is calculated to solve the equation given in (5). The software program MATHEMATICA is used to find the optimum values of pi (i – 1, i – n). The optimum calibrae ^ tion points and the corresponding Tr½ Dð hÞ values for the cases of n = 3 and n = 4 are tabulated in Tables 1 and 2, respectively. The equidistant calibration points are obtained by simply dividing the range of the transducer into the appropriate number of samples. For a comparison, the optimality criterion values, where pressure values pi, i = 1, . . ., n are equally spaced, are also shown in the tables. As it can be seen clearly from the presented tables, the optimality criterion values, where the coefficients are calculated by the proposed method, are smaller than the ones where pi, i = 1, . . ., n are equally spaced. As a result, the suggested method can be used to obtain the best (with respect to selection criteria) calibration characteristics polynomial. 5. Experimental checking of obtained results The obtained values of the optimum measurement points for the calibration of the differential pressure gage for the case of n = 4 are checked via actual experiments.
Table 1 e ^hÞ Optimum and equidistant calibration points and corresponding Tr½ Dð values for n = 3.
4. Computational results
Calibration points
The results for the determination of the optimum measurement points for the calibration of the measurement apparatus are given below for the cases of m = 2 and n = 3, n = 4. In the calculations the following data and initial conditions are taken:
Optimum points Equidistant points
Calculation of the optimum measurement points is made for the differential pressure gage Sapphir-22DD (‘‘Teplokontrol”, Kazan, Russia). The range of the transducer is 0 6 pi 6 1600 bar. The differential pressure gage errors are subjected to the normal distribution with zero mean and the standard uncertainty ri = 2,6 bar [15].
ð11Þ
e ^ Tr½ Dð hÞ, bar2
Pressure values, bar p1
p2
p3
0 0
857.1 800
1600 1600
7.290072 7.290074
Table 2 e ^hÞ Optimum and equidistant calibration points and corresponding Tr½ Dð values for n = 4. Calibration points
Optimum points Equidistant points
e ^ Tr½ Dð hÞ, bar2
Pressure values, bar p1
p2
p3
p4
0 0
200 533
845 1066
1600 1600
5.319307 6.924947
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Ch. Hajiyev / Measurement 43 (2010) 563–569
In the experiments, the following data and the initial conditions are taken: Measurements are taken via the differential pressure gage Sapphir-22DD (‘‘Teplokontrol”, Kazan, Russia) with the measurement range 0 6 pi 6 1600 bar. The output signal of the mentioned gage is an electrical signal in the unit of mV. Calibration of the differential pressure gage is made by the help of the pressure standard (piston gage set) [16], which reproduces the pressure signals at the corresponding optimum and equidistant calibration points. The holding calibration experiment results are presented in Tables 3 and 4, respectively. ^1 and a ^2 found by the estimation ^0 ; a The coefficients a algorithm (3), are presented in Table 5, and their errors’ variances are given in Table 6. ^1 and a ^2 , the ^0 ; a After determination of the coefficients a polynomial
^0 þ a ^1 pi þ a ^2 p2i yi ¼ a
ð13Þ
can be used as the calibration curve of the differential pressure gage. The obtained calibration curves corresponding to the optimum and equidistant calibration points are shown in Figs. 1 and 2, respectively. As it is seen from Table 5, there is a very significant dif^0 between the optimum and equiference in the values of a ^2 ^1 and a distant calibration point cases, while the values a show little difference. In the calibration characteristic ^1 is multiplied by the value of the in(13) the coefficient a ^2 by p2i . As the input signal put signal pi and the coefficient a pi can get high values (up to 1600 bar), the effects of the coefficients to the calibration characteristic (13) are not very different from each other. As it is seen from the results e ^ given in Table 6, the values Tr½ Dð hÞ in the case of optimum calibration points are considerably less than the ones obtained in the case of equidistant calibration points. It proves that the calibration characteristic obtained through the proposed method is more accurate, than the conventional method.
Table 3 Calibration experiment results corresponding to the optimum calibration points. Experiment no.
pi, bar
zi, mV
1 2 3 4
0 200 845 1600
0.0051 1192.4 7867.2 21172.9
Table 4 Calibration experiment results corresponding to the equidistant calibration points. Experiment no.
pi, bar
zi, mV
1 2 3 4
0 533 1066 1600
0.0051 4097.9 11150 21172.9
In real conditions, after obtaining each measurements of zi, the inverse problem is solved in the microprocessor of differential pressure gage, i.e., the roots of Eq. (13) are found
^ð1;2Þi ¼ p
^1 a
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^21 4a ^2 ða ^ 0 zi Þ a ^2 2a
ð14Þ
^1i is assumed as the estimation of the meaand the root p sured pressure. The root
^ 2i ¼ p
^1 a
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^21 4a ^ 2 ða ^ 0 zi Þ a ^2 2a
ð15Þ
is negative or considerably different from the value zi. Experiment for the verification, based on the mentioned calibration characteristics, is realized for the above example. In the experiments the piston gages reproduce pressure signals of pi ; i ¼ 1; 17 in the measurement interval of 0 6 pi 6 1600 bar with the step of 100 bar and the output signals of the differential pressure gage zi are registered. The holding experiment results are presented in the Table B1 of Appendix B. Using calibration experiment results presented in the Table B1 (zi, i ¼ 1; 17), and the equations of the appropriate ^1i , i ¼ 1; 17 are eval^i ¼ p calibration curves, the values of p uated via formula (14). Then the appropriate absolute error values Dabsi and the relative error values Dreli are determined by the means of the following expressions:
^i pi ; Dabsi ¼ p
Dreli ¼
Dabsi 100%: pi
ð16Þ
Since a true value cannot be determined, in the expressions (15) a value obtained by a perfect measurement or a conventional true value may be used [8]. Therefore in (15) the values of the reference standard pi are accepted as a true value. ^i and the absolute and The obtained calibration values p relative calibration errors which corresponds to the optimum and equidistant calibration points, are presented in the Table B2 of Appendix B and the graphs of the relations Dabsi ¼ f ðpi Þ and Dreli ¼ f ðpi Þ are shown in the Figs. 3 and 4, respectively. As it is seen from the presented results, the calibration errors corresponding to the optimum calibration points are smaller than the ones corresponds to the case of the equidistant calibration points. The experiment results confirm the correctness of the obtained theoretical results. The piston gage set has a high accuracy. Furthermore, the calibration characteristic of the investigated differential pressure gage is simple (two order polynomial). Therefore in this case benefit of the proposed method is small. In some cases, for example, for in flight calibration of the measurement devices of flying vehicles, standard setting devices are generally absent and the ordinary measurement systems are used as the standard devices. In this situation and in the case where the calibration characteristic is described by a higher order polynomial, the proposed calibration method will give significantly better results.
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Ch. Hajiyev / Measurement 43 (2010) 563–569 Table 5 Calibration coefficient estimates. Using calibration method
^0 a
^1 a
^2 a
Optimum calibration points are used Equidistant calibration points are used
0.2271007284 0.2431230065
4.9199050638 4.9212484558
0.0051956150 0.0051950696
Table 6 Variances of the errors of the coefficient estimates. Using calibration method Optimum calibration points are used Equidistant calibration points are used
Da^0 , bar2 5.3192342 6.9248849
Fig. 1. Calibration curve (optimum calibration points are used).
Fig. 2. Calibration curve (equidistant calibration points are used).
Da^1 , bar2 0.0000733 0.0000628
Da^2 , bar2
e ^ Tr½ Dð hÞ bar2 10
0.27 10 0.23 1010
5.3193074 6.9249478
Fig. 3. Absolute calibration errors: (1) calibration is performed by using optimum calibration points and (2) calibration is performed by using equidistant calibration points.
Fig. 4. Relative calibration errors: (1) calibration is performed by using optimum calibration points and (2) calibration is performed by using equidistant calibration points.
6. Conclusion The paper shows that the accuracy of the calibration characteristics of a measuring apparatus depends on the selection of the calibration points. A procedure for the
optimal selection of sample measurement points to get the best calibration characteristics (for the chosen optimality criterion) of a measuring apparatus is proposed. For simplicity A-optimality criterion has been used for this
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Ch. Hajiyev / Measurement 43 (2010) 563–569
work but also any other criterion (D-optimality, E-optimality, etc.) can be used. As an example, the problem of the optimal selection of the standard pressure setters (the piston gages) during calibration of the differential pressure measurer is solved. The holding experiment results confirm the correctness of the obtained theoretical results. Further work includes the calculation of the calibration characteristics where different optimality criterias are used.
det ¼ n
n1 X n X
p2i p2j ðpi pj Þ2
i¼1 j¼iþ1
ðpi pj Þ þ
n X
pi
i¼1 n X
p2i
n1 X n X
i¼1
pi pj ðpi pj Þ2
a11
e ¼ ðr2 = detÞ6 D 4 a12 a13
a21
a31
n1 X n r2 4X
e¼ Tr D
det þn
3
a22
7 a32 5
a23
a33
i¼1
n X
eT X e where a11, a12, . . ., a33 are the algebraic minors of X matrix;
p2i p2j ðpi pj Þ2 þ n
j¼1
p2i
i¼1
ðA-1Þ
ðA-2Þ
i¼1 j¼iþ1
eT X e matrix. is the determinant of X After proper mathematical transformations, we have e dispersion matrix in the following form: the trace of D
Open the dispersion matrix of the estimations errors (3). After multiplication and inverting the matrix, we have
2
p3i pj
i;j¼1 i–j
2
Appendix A. Derivation of Eq. (10) for m = 2
n X
n X
p4i
i¼1
!2 3 pi 5
n X
n X
!2 p2i
i¼1
ðA-3Þ
i¼1
By taking the corresponding derivatives one has a system of n algebraic equations with n variables: " # n n n X X X 2 pi p3j ðpi pj Þ þ 4np3i 4pi p2i þ 2npi 2 pi
Table B1 Experiment results.
j¼1
i¼1
2
i¼1
n1 n n n X X X 6 X 2 6 p p2i p2j ðpi pj Þ pi p3i pj ðpi pj Þ 4n
Experiment no
pi, bar
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
i¼1
zi, mV
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600
0.0051 545.0231 1192.3651 1942.8624 2798.7251 3758.5748 4821.0851 5990.1285 7261.6704 8635.0752 10113.8051 11697.1899 13384.1651 15181.0295 17076.9348 19073.5267 21172.8851
þ
n X
p2i
i¼1
j¼iþ1
n1 X
n X
i¼1
pi pj ðpi pj Þ2 2n
p3i pj ðpi pj Þ
i;j¼1 i–j
2 4
n X
pi
n X
i¼1
n1 X n X
n X
pi
pi p2j ðpi pj Þð2pi pj Þ
p2i pj ð4pi 3pj Þ
j¼1 i–j 2
pi pj ðpi pj Þ þ
n X
i¼1 j¼iþ1 n1 X n X
n X j¼iþ1
p2i
i¼1 2
p2i p2j ðpi pj Þ þ n
i¼1 j¼1
i;j¼1 i–j
i¼1 j¼iþ1
n X
þ2pi
"
#
!2 3 5 ¼ 0; ði ¼ 1; nÞ
n X
n X
# pj ðpi pj Þð3pi pj Þ
j¼iþ1
p4i
i¼1
n X i¼1
!2 p2i
þn
n X
p2i
i¼1
ðA-4Þ
i¼1
Table B2 Absolute and relative calibration errors corresponding to the optimum and equidistant calibration points. Input pressure, bar
^i , bar optimum p points
^i , bar equidistant p points
Dabs, bar optimum points
Dabs, bar equidistant points
Drel, % optimum points
Drel, % equidistant points
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600
0.0451 100.1352 200.0226 299.8340 399.8390 499.8417 599.7238 699.8465 799.8132 899.6875 999.5873 1099.5765 1199.5435 1299.8336 1399.8382 1499.6808 1599.4813
0.0504 100.1922 200.0526 299.8440 399.8335 499.8239 599.6961 699.8105 799.7702 899.6385 999.5331 1099.5178 1199.4809 1299.7675 1399.7689 1499.6086 1599.4066
0.0451 0.1352 0.0226 0.1660 0.1610 0.1583 0.2762 0.1535 0.1868 0.3125 0.4127 0.4235 0.4565 0.1664 0.1618 0.3192 0.5187
0.0504 0.1922 0.0526 0.1560 0.1665 0.1761 0.3039 0.1895 0.2298 0.3615 0.4669 0.4822 0.5191 0.2325 0.2311 0.3914 0.5934
1 0.1350 0.0113 0.0554 0.0403 0.0317 0.0461 0.0219 0.0234 0.0347 0.0413 0.0385 0.0381 0.0128 0.0116 0.0213 0.0324
1 0.1919 0.0263 0.0520 0.0416 0.0352 0.0507 0.0271 0.0287 0.0402 0.0467 0.0439 0.0433 0.0179 0.0165 0.0261 0.0371
Ch. Hajiyev / Measurement 43 (2010) 563–569
With a view to find all the possible solutions of the system, one can apply numeric methods of searching that can be simply realized by a computer.
[6] [7]
Appendix B [8]
See Tables B1 and B2.
[9]
References [10] [1] W.D. Cooper, Electronic Instrumentation and Measurement Techniques, Prentice Hall, Inc., Englewood Cliffs, NJ, 1978. [2] K.K. Vasilchenko, V.A. Leonov, I.M. Pashkovskiy, B.K. Poplavskiy, Flight Testing of Aircrafts, second ed., Masinostroyeniye, Moscow, 1996 (in Russian). [3] J.G. Moreno, J.L.M. Vigil-Escalera, R.S. Alvarez, Statistical measurement calibration based on state estimator results, in: Proceedings of the IEEE Power Engineering Society Transmission and Distribution Conference, vol. 1, New Orleans, LA, USA, 1999, pp. 184–189. [4] S.M.S. Rezeki, W. Chan, M.R. Haskard, D.E. Mulcahy, D.E. Davey, Realization of self diagnosis and self calibration strategies using conventional signal processing and fuzzy approach for distributed intelligent sensor systems, in: Proceedings of SPIE. – The International Society for Optical Engineering, Newport Beach, CA, USA, 1999, pp. 278–286. [5] A. Bernieri, G. Betta, M. Dell’Isola, Statistical problems in calibration design, in: P. Ciarlini, M.G. Cox, F. Pavese, D. Richter (Eds.), Advanced
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Determination of optimum measurement points via A-optimality criterion for the calibration of measurement apparatus Ch. Hajiyev * Istanbul Technical University, Faculty of Aeronautics and Astronautics, Maslak, 34469 Istanbul, Turkey
a r t i c l e
i n f o
Article history: Received 2 February 2007 Received in revised form 16 October 2009 Accepted 28 December 2009 Available online 11 January 2010 Keywords: Measuring apparatus Calibration curve Calibration design Least squares method
a b s t r a c t A procedure for optimal selection of the measurement points to get the best calibration characteristics (for the chosen optimality criterion) of measuring apparatus is proposed. The coefficients of the calibration characteristics are evaluated by the classical least squares method. For this work, the A-optimality criterion has been used as an optimality criteria. As an example, the problem of optimal selection of the standard pressure setters (the piston gauges) during calibration of the differential pressure gage is solved. Obtained values of the optimum measurement points for the calibration of the differential pressure gage are checked via actual experiments. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Accurate measurement is the basis of almost all engineering applications, since uncertainty inherently exists in the nature of any measuring apparatus. On the other hand, the cost of a measuring apparatus increases with its accuracy. Therefore, low cost accurate measurement devices are one of the main goal of metrology engineers. One way for decreasing the sensor uncertainties is the calibration process [1]. Therefore, this paper deals with the calibration of a low cost sensor using the corresponding values obtained by reference standards. From the practical point of view, the calibration characteristics should be in a polynomial form. The accuracy of this polynomial depends on the noise-free data which is used to obtain the characteristics [2]. To reduce the effect of the noise, excessive number of data should be used. However this requires more experiments and that increases the cost. Thus, the main question becomes the evaluation of accurate calibration characteristics with minimum number of experimental data. In [1–4] the equidistant measurement points are used for calibration of measurement apparatus. On the * Tel.: +90 2122853105; fax: +90 2122852926. E-mail address: [email protected] 0263-2241/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2009.12.029
other hand, even though it is paradoxical, the application of the equidistant measurement points to get the best calibration characteristic (for the chosen optimality criterion) is erroneous, because these points are not optimum in a sense that given performance criterion is minimum. The equidistant calibration points are obtained by simply dividing the range of the transducer into the appropriate number of samples. In [5,6] an approach to design sensor calibration is proposed with the aim of reducing the calibration curve uncertainty. This uncertainty reduction is achieved by minimizing either the standard deviations of the regression curve coefficients or the standard deviation of the whole estimated calibration curve. In particular, criteria for the choice of the number of calibration points and their optimum location are theoretically identified when the response characteristic is a polynomial and the uncertainties of the sensor outputs can be neglected. The proposed criteria is not directly applicable to non-linear sensors and complex apparatus with an indirect estimation of the measurand. The calibration of sensors, or in general, the measurement apparatus, can be considered as a typical problem of ‘‘model searching” from experimental data and then, it should be approached by applying experimental design
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techniques. In this case, the calibration design should drive the operator in the choice of [7]: the experimental plan (e.g. the number and the location of the calibration points, the number of the repetitions); the main influence quantities; the regression curve; the regression technique; the standard references and their uncertainties. The calibration design using the experimental design techniques is proposed in [7], when the relationship between the indirectly calculated measurands and the sensor inputs is non-linear or more than one measurand has to be considered. In particular, the optimum calibration plan for measurement chain is identified by suitably elaborating the error propagation law suggested by the ISO Guide [8]. Sensor calibration and compensation using the artificial neural network are performed in [9]. The artificial neural network based inverse modeling technique is used for the sensor response linearization. The choice of the order of the model and the number of the calibration points are important design parameters in this technique. An intensive study about the effect of the order of the model and the number of the calibration points on the lowest asymptotic root-mean-square (RMS) error has been reported in this paper. In [10] a genetic algorithm is used to perform optimizations for both the computation of the optimal input to the sensor and the optimal constant feedback gain. A calibration method, which uses the neural network and genetic algorithms together, is presented in [11]. It uses the improved back-propagation neural network to model the characteristic curve of the vortex flowmeter. And then it applies the genetic algorithms to seek two additional optimum calibration points intelligently at the intervals where the curve are non-linear obviously. At last the vortex flowmeter is calibrated at the new calibration points. The methods based on the artificial neural networks and genetic algorithms do not have physical bases. Therefore according to the different data, which corresponds to the same event, the model gives different solutions. Thus, the model should be continuously trained by using the new data. In this study, optimal selection method of the sample measurement points via A-optimality criterion for the calibration of measurement apparatus, which takes the uncertainties on their outputs into account, is proposed. 2. Problem statement As it is known from practical considerations, the calibration curve should be in a polynomial form as follows:
yi ¼ a0 þ a1 pi þ a2 p2i þ þ am pm i ;
ð1Þ
where yi is the output of the low cost transducer, pi are the values of the reference standard and a0, a1, . . ., am are the calibration curve coefficients. Measurement contains random noises in Gaussian form
zi ¼ yi þ di ¼ a0 þ a1 pi þ a2 p2i þ þ am pm i þ di
ð2Þ
where zi is the measurement result, di is measurement error with zero mean and standard uncertainty r. Let the calibration curve coefficients be denoted as ~ h¼ ½a0 ; a1 ; . . . ; am T . The coefficients in these polynomials are evaluated in [12] by the least squares method. The expressions that are used to make the evaluation had the form:
e Þ1 ð X e T zÞ eT X ~h ¼ ð X e ^hÞ ¼ ð X eT X e Þ1 r2 Dð
ð3Þ
where z ¼ ½z1 ; z2 ; . . . ; zn ments;
2
1 p1 61 p 6 2 6 6 e 6 X ¼6 6 6 4
p21 p22
1 pn
p2n
T
is the vector of the measure-
3 pm 1 m7 p2 7 7 7 7 7 7 7 5
ð4Þ
pm n
is the matrix of the known coordinates (here, p1, p2, . . ., pn are the values that are producible by the reference stane ^ dard instrument), Dð hÞ is the dispersion matrix of the estimated coefficients. The values of the reference standard p1, p2, . . ., pn, should be such that the polynomial characteristics given by (1) approximates the real calibration characteristics in the best form. Thus the problem can be stated as follows: Find the values of p1, p2, . . ., pn such that the values of a1, a2, . . ., am are optimum in the sense that a given performance criterion is minimum. e can be used as a As mentioned above, the matrix D measure of the error between the low cost transducer and the high precision reference standard instrument. A e performance criteria for the minimum of the matrix D can be selected by the following ways: Minimize the trace (the sum of the diagonal elements) e (the A-optimality criterion). of the matrix D Minimize the generalize dispersion (the determinant) of e (the D-optimality criterion). the matrix D e (the Minimize the maximal eigenvalue of the matrix D E-optimality criterion). e Minimize the sum of the all elements of the matrix D and etc. Each of the above-mentioned measures characterizes one or another geometrical parameter of the correlation ellipsoid. The choice of one or another scalar measure as the optimality criteria for a solved particular problem depends on the basic indicators of optimization (the mathematical simplicity, convenience of obtaining analytical results, volume of computational expenditures, etc.). For this reason, A-optimality criterion is used in this work, i.e.
~ 1 r2 ~ T XÞ minPi ½TrðX
ð5Þ
is sought. The values of p1, p2, . . ., pn, found by solving the above equations, should be in the range of 0–pmax. Otherwise the solution is invalid.
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3. The solution algorithm The objective function is described in the following form
e ; p ; . . . ; p Þg: f ðp1 ; p2 ; . . . ; pn Þ ¼ Trf Dðp 1 2 n
ð6Þ
As explained above the problem is a constrained optimization problem. The objective function is multivariable, nonlinear, continuous and it has derivative in the considered interval. Assume that the minimum of f (p1, p2, . . ., pn) exists for the following values of p1, p2, . . ., pn:
p ¼ ½p1 ; p2 ; . . . ; pn T : In order to gain p* which is a minimum of (6), the following conditions should be satisfied [13]:
Df ðp Þ ¼ 0;
ð7Þ
D2 f ðp Þ is semi positive;
ð8Þ
where D denotes the gradient. The extremum condition given by (7) can be explicitly written as:
e ; p ; . . . ; p Þg=@p ¼ 0; @½Trf Dðp 1 2 n i
ði ¼ 1; 2; . . . ; nÞ:
ð9Þ
When the derivatives in (9) are calculated n algebraic equations with n unknowns are obtained.
Q i ðp1 ; p2 ; . . . ; pn Þ ¼ 0;
ði ¼ 1; 2; . . . ; nÞ;
ð10Þ
where n denotes the number of the measurements. Derivation of Eq. (10) for the case of m = 2 is given in detail in Appendix A. Numerical routines such as the gradient descent algorithms [14] can be used. It should be noted that the sign of D2f(p) should be calculated together with Df(p) in order to determine whether the found values correspond to a local minimum or a local maximum. Furthermore, those vale 1, p2, . . ., pn)} minimum should be in ues which make Tr{ D(p the range 0 pmax. The solution set which satisfies the above conditions can be used to calculate the polynomial coefficients given in (1). This polynomial approximates the calibration characteristics between the low cost and the reference standard equipment in the best way and it can be used to extract the accurate values from the outputs of the low cost transducer.
The calibration characteristics of the examined measuring device (in the present case, the differential pressure gage) is described by 2nd order polynomial as follows [12]:
yi ¼ a0 þ a1 pi þ a2 p2i : Measurement equation is written in the form
zi ¼ a0 þ a1 pi þ a2 p2i þ di ;
i ¼ 1; n;
ð12Þ
where di is the measurement error with zero mean and the standard uncertainty r. The optimum measurement points for the calibration of the above mentioned differential pressure gage are evaluated via Eq. (A-4), which is presented in Appendix A. The method is used to obtain the optimum coefficients of the characteristic polynomial for a selected criteria. Calculation is performed for n = 3 and n = 4. Closed form algebraic Eq. (A-4) is calculated to solve the equation given in (5). The software program MATHEMATICA is used to find the optimum values of pi (i – 1, i – n). The optimum calibrae ^ tion points and the corresponding Tr½ Dð hÞ values for the cases of n = 3 and n = 4 are tabulated in Tables 1 and 2, respectively. The equidistant calibration points are obtained by simply dividing the range of the transducer into the appropriate number of samples. For a comparison, the optimality criterion values, where pressure values pi, i = 1, . . ., n are equally spaced, are also shown in the tables. As it can be seen clearly from the presented tables, the optimality criterion values, where the coefficients are calculated by the proposed method, are smaller than the ones where pi, i = 1, . . ., n are equally spaced. As a result, the suggested method can be used to obtain the best (with respect to selection criteria) calibration characteristics polynomial. 5. Experimental checking of obtained results The obtained values of the optimum measurement points for the calibration of the differential pressure gage for the case of n = 4 are checked via actual experiments.
Table 1 e ^hÞ Optimum and equidistant calibration points and corresponding Tr½ Dð values for n = 3.
4. Computational results
Calibration points
The results for the determination of the optimum measurement points for the calibration of the measurement apparatus are given below for the cases of m = 2 and n = 3, n = 4. In the calculations the following data and initial conditions are taken:
Optimum points Equidistant points
Calculation of the optimum measurement points is made for the differential pressure gage Sapphir-22DD (‘‘Teplokontrol”, Kazan, Russia). The range of the transducer is 0 6 pi 6 1600 bar. The differential pressure gage errors are subjected to the normal distribution with zero mean and the standard uncertainty ri = 2,6 bar [15].
ð11Þ
e ^ Tr½ Dð hÞ, bar2
Pressure values, bar p1
p2
p3
0 0
857.1 800
1600 1600
7.290072 7.290074
Table 2 e ^hÞ Optimum and equidistant calibration points and corresponding Tr½ Dð values for n = 4. Calibration points
Optimum points Equidistant points
e ^ Tr½ Dð hÞ, bar2
Pressure values, bar p1
p2
p3
p4
0 0
200 533
845 1066
1600 1600
5.319307 6.924947
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In the experiments, the following data and the initial conditions are taken: Measurements are taken via the differential pressure gage Sapphir-22DD (‘‘Teplokontrol”, Kazan, Russia) with the measurement range 0 6 pi 6 1600 bar. The output signal of the mentioned gage is an electrical signal in the unit of mV. Calibration of the differential pressure gage is made by the help of the pressure standard (piston gage set) [16], which reproduces the pressure signals at the corresponding optimum and equidistant calibration points. The holding calibration experiment results are presented in Tables 3 and 4, respectively. ^1 and a ^2 found by the estimation ^0 ; a The coefficients a algorithm (3), are presented in Table 5, and their errors’ variances are given in Table 6. ^1 and a ^2 , the ^0 ; a After determination of the coefficients a polynomial
^0 þ a ^1 pi þ a ^2 p2i yi ¼ a
ð13Þ
can be used as the calibration curve of the differential pressure gage. The obtained calibration curves corresponding to the optimum and equidistant calibration points are shown in Figs. 1 and 2, respectively. As it is seen from Table 5, there is a very significant dif^0 between the optimum and equiference in the values of a ^2 ^1 and a distant calibration point cases, while the values a show little difference. In the calibration characteristic ^1 is multiplied by the value of the in(13) the coefficient a ^2 by p2i . As the input signal put signal pi and the coefficient a pi can get high values (up to 1600 bar), the effects of the coefficients to the calibration characteristic (13) are not very different from each other. As it is seen from the results e ^ given in Table 6, the values Tr½ Dð hÞ in the case of optimum calibration points are considerably less than the ones obtained in the case of equidistant calibration points. It proves that the calibration characteristic obtained through the proposed method is more accurate, than the conventional method.
Table 3 Calibration experiment results corresponding to the optimum calibration points. Experiment no.
pi, bar
zi, mV
1 2 3 4
0 200 845 1600
0.0051 1192.4 7867.2 21172.9
Table 4 Calibration experiment results corresponding to the equidistant calibration points. Experiment no.
pi, bar
zi, mV
1 2 3 4
0 533 1066 1600
0.0051 4097.9 11150 21172.9
In real conditions, after obtaining each measurements of zi, the inverse problem is solved in the microprocessor of differential pressure gage, i.e., the roots of Eq. (13) are found
^ð1;2Þi ¼ p
^1 a
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^21 4a ^2 ða ^ 0 zi Þ a ^2 2a
ð14Þ
^1i is assumed as the estimation of the meaand the root p sured pressure. The root
^ 2i ¼ p
^1 a
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^21 4a ^ 2 ða ^ 0 zi Þ a ^2 2a
ð15Þ
is negative or considerably different from the value zi. Experiment for the verification, based on the mentioned calibration characteristics, is realized for the above example. In the experiments the piston gages reproduce pressure signals of pi ; i ¼ 1; 17 in the measurement interval of 0 6 pi 6 1600 bar with the step of 100 bar and the output signals of the differential pressure gage zi are registered. The holding experiment results are presented in the Table B1 of Appendix B. Using calibration experiment results presented in the Table B1 (zi, i ¼ 1; 17), and the equations of the appropriate ^1i , i ¼ 1; 17 are eval^i ¼ p calibration curves, the values of p uated via formula (14). Then the appropriate absolute error values Dabsi and the relative error values Dreli are determined by the means of the following expressions:
^i pi ; Dabsi ¼ p
Dreli ¼
Dabsi 100%: pi
ð16Þ
Since a true value cannot be determined, in the expressions (15) a value obtained by a perfect measurement or a conventional true value may be used [8]. Therefore in (15) the values of the reference standard pi are accepted as a true value. ^i and the absolute and The obtained calibration values p relative calibration errors which corresponds to the optimum and equidistant calibration points, are presented in the Table B2 of Appendix B and the graphs of the relations Dabsi ¼ f ðpi Þ and Dreli ¼ f ðpi Þ are shown in the Figs. 3 and 4, respectively. As it is seen from the presented results, the calibration errors corresponding to the optimum calibration points are smaller than the ones corresponds to the case of the equidistant calibration points. The experiment results confirm the correctness of the obtained theoretical results. The piston gage set has a high accuracy. Furthermore, the calibration characteristic of the investigated differential pressure gage is simple (two order polynomial). Therefore in this case benefit of the proposed method is small. In some cases, for example, for in flight calibration of the measurement devices of flying vehicles, standard setting devices are generally absent and the ordinary measurement systems are used as the standard devices. In this situation and in the case where the calibration characteristic is described by a higher order polynomial, the proposed calibration method will give significantly better results.
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Ch. Hajiyev / Measurement 43 (2010) 563–569 Table 5 Calibration coefficient estimates. Using calibration method
^0 a
^1 a
^2 a
Optimum calibration points are used Equidistant calibration points are used
0.2271007284 0.2431230065
4.9199050638 4.9212484558
0.0051956150 0.0051950696
Table 6 Variances of the errors of the coefficient estimates. Using calibration method Optimum calibration points are used Equidistant calibration points are used
Da^0 , bar2 5.3192342 6.9248849
Fig. 1. Calibration curve (optimum calibration points are used).
Fig. 2. Calibration curve (equidistant calibration points are used).
Da^1 , bar2 0.0000733 0.0000628
Da^2 , bar2
e ^ Tr½ Dð hÞ bar2 10
0.27 10 0.23 1010
5.3193074 6.9249478
Fig. 3. Absolute calibration errors: (1) calibration is performed by using optimum calibration points and (2) calibration is performed by using equidistant calibration points.
Fig. 4. Relative calibration errors: (1) calibration is performed by using optimum calibration points and (2) calibration is performed by using equidistant calibration points.
6. Conclusion The paper shows that the accuracy of the calibration characteristics of a measuring apparatus depends on the selection of the calibration points. A procedure for the
optimal selection of sample measurement points to get the best calibration characteristics (for the chosen optimality criterion) of a measuring apparatus is proposed. For simplicity A-optimality criterion has been used for this
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Ch. Hajiyev / Measurement 43 (2010) 563–569
work but also any other criterion (D-optimality, E-optimality, etc.) can be used. As an example, the problem of the optimal selection of the standard pressure setters (the piston gages) during calibration of the differential pressure measurer is solved. The holding experiment results confirm the correctness of the obtained theoretical results. Further work includes the calculation of the calibration characteristics where different optimality criterias are used.
det ¼ n
n1 X n X
p2i p2j ðpi pj Þ2
i¼1 j¼iþ1
ðpi pj Þ þ
n X
pi
i¼1 n X
p2i
n1 X n X
i¼1
pi pj ðpi pj Þ2
a11
e ¼ ðr2 = detÞ6 D 4 a12 a13
a21
a31
n1 X n r2 4X
e¼ Tr D
det þn
3
a22
7 a32 5
a23
a33
i¼1
n X
eT X e where a11, a12, . . ., a33 are the algebraic minors of X matrix;
p2i p2j ðpi pj Þ2 þ n
j¼1
p2i
i¼1
ðA-1Þ
ðA-2Þ
i¼1 j¼iþ1
eT X e matrix. is the determinant of X After proper mathematical transformations, we have e dispersion matrix in the following form: the trace of D
Open the dispersion matrix of the estimations errors (3). After multiplication and inverting the matrix, we have
2
p3i pj
i;j¼1 i–j
2
Appendix A. Derivation of Eq. (10) for m = 2
n X
n X
p4i
i¼1
!2 3 pi 5
n X
n X
!2 p2i
i¼1
ðA-3Þ
i¼1
By taking the corresponding derivatives one has a system of n algebraic equations with n variables: " # n n n X X X 2 pi p3j ðpi pj Þ þ 4np3i 4pi p2i þ 2npi 2 pi
Table B1 Experiment results.
j¼1
i¼1
2
i¼1
n1 n n n X X X 6 X 2 6 p p2i p2j ðpi pj Þ pi p3i pj ðpi pj Þ 4n
Experiment no
pi, bar
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
i¼1
zi, mV
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600
0.0051 545.0231 1192.3651 1942.8624 2798.7251 3758.5748 4821.0851 5990.1285 7261.6704 8635.0752 10113.8051 11697.1899 13384.1651 15181.0295 17076.9348 19073.5267 21172.8851
þ
n X
p2i
i¼1
j¼iþ1
n1 X
n X
i¼1
pi pj ðpi pj Þ2 2n
p3i pj ðpi pj Þ
i;j¼1 i–j
2 4
n X
pi
n X
i¼1
n1 X n X
n X
pi
pi p2j ðpi pj Þð2pi pj Þ
p2i pj ð4pi 3pj Þ
j¼1 i–j 2
pi pj ðpi pj Þ þ
n X
i¼1 j¼iþ1 n1 X n X
n X j¼iþ1
p2i
i¼1 2
p2i p2j ðpi pj Þ þ n
i¼1 j¼1
i;j¼1 i–j
i¼1 j¼iþ1
n X
þ2pi
"
#
!2 3 5 ¼ 0; ði ¼ 1; nÞ
n X
n X
# pj ðpi pj Þð3pi pj Þ
j¼iþ1
p4i
i¼1
n X i¼1
!2 p2i
þn
n X
p2i
i¼1
ðA-4Þ
i¼1
Table B2 Absolute and relative calibration errors corresponding to the optimum and equidistant calibration points. Input pressure, bar
^i , bar optimum p points
^i , bar equidistant p points
Dabs, bar optimum points
Dabs, bar equidistant points
Drel, % optimum points
Drel, % equidistant points
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600
0.0451 100.1352 200.0226 299.8340 399.8390 499.8417 599.7238 699.8465 799.8132 899.6875 999.5873 1099.5765 1199.5435 1299.8336 1399.8382 1499.6808 1599.4813
0.0504 100.1922 200.0526 299.8440 399.8335 499.8239 599.6961 699.8105 799.7702 899.6385 999.5331 1099.5178 1199.4809 1299.7675 1399.7689 1499.6086 1599.4066
0.0451 0.1352 0.0226 0.1660 0.1610 0.1583 0.2762 0.1535 0.1868 0.3125 0.4127 0.4235 0.4565 0.1664 0.1618 0.3192 0.5187
0.0504 0.1922 0.0526 0.1560 0.1665 0.1761 0.3039 0.1895 0.2298 0.3615 0.4669 0.4822 0.5191 0.2325 0.2311 0.3914 0.5934
1 0.1350 0.0113 0.0554 0.0403 0.0317 0.0461 0.0219 0.0234 0.0347 0.0413 0.0385 0.0381 0.0128 0.0116 0.0213 0.0324
1 0.1919 0.0263 0.0520 0.0416 0.0352 0.0507 0.0271 0.0287 0.0402 0.0467 0.0439 0.0433 0.0179 0.0165 0.0261 0.0371
Ch. Hajiyev / Measurement 43 (2010) 563–569
With a view to find all the possible solutions of the system, one can apply numeric methods of searching that can be simply realized by a computer.
[6] [7]
Appendix B [8]
See Tables B1 and B2.
[9]
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