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Calibration of the Pressure Gage via GA and LSM Based Combined Calibration Method
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51-30 (2018) 774–779 CalibrationIFAC of PapersOnLine the Pressure Gage via GA and LSM Based Calibration of the Pressure Gage GA Calibration Calibration ofCombined the Pressure Gage via viaMethod GA and and LSM LSM Based Based Calibration Method Calibration ofCombined the Pressure Gage via GA and LSM Based Combined Calibration Method Hacizade Combined U. Calibration Method
U. Hacizade U. Hacizade Department of Computer Engineering, Halic University, Sütlüce District, Imrahor Str., U. Hacizade No: 82, Beyoğlu-Istanbul, TURKEY Halic (e-mail: [email protected]) Department of Computer Engineering, University, Sütlüce District, Imrahor Str., Department of Computer Engineering, University, Sütlüce District, Imrahor Str., No: 82, Beyoğlu-Istanbul, TURKEY Halic (e-mail: [email protected]) Department of Computer Engineering, University, Sütlüce District, Imrahor Str., No: 82, Beyoğlu-Istanbul, TURKEY Halic (e-mail: [email protected]) No: 82, Beyoğlu-Istanbul, TURKEY [email protected]) Abstract: Low cost accurate measurement apparatus are(e-mail: one of the main goal of metrology engineers. One way Low for decreasing the measurement sensor uncertainties is the study dealsengineers. with the Abstract: cost accurate apparatus are calibration one of the process. main goalThis of metrology Abstract: cost accurate apparatus arevalues one of the process. main goalThis of metrology calibration of adecreasing low cost sensor using uncertainties the corresponding obtained by reference standards. From One way Low for the measurement sensor is the calibration study dealsengineers. with the Abstract: Low cost accurate measurement apparatus are one of main goalThis of form. metrology engineers. One way point for thecalibration sensor is the calibration study deals with practical of the should be the in aprocess. polynomial In this studythe calibration of adecreasing lowview, cost sensor using uncertainties thecharacteristics corresponding values obtained by reference standards. From thea One way for decreasing the sensor uncertainties is the calibration process. This study deals with the calibration of a low cost sensor using the corresponding values obtained by reference standards. From Genetic (GA) the is used for the measurement purpose.form. As anInexample, GA practicalAlgorithm point of view, calibration characteristicsapparatus should becalibration in a polynomial this study a calibration of a of low cost sensor using thecharacteristics corresponding values obtained by reference standards. From the practical point view, the calibration should becalibration in a polynomial study based of a differential gauge using standard pressure setting form. devices isthis examined. Geneticcalibration Algorithm (GA) is used forpressure the measurement apparatus purpose. As anInexample, GAa practical point ofresults, view, founded the calibration characteristics should becalibration in a with polynomial form. this study Genetic Algorithm is used for the measurement apparatus As obtained anInexample, The using proposed are compared the results via GA thea basedcalibration calibration of(GA) a differential pressure gauge GA, using standard pressure purpose. setting devices is examined. Genetic Algorithm (GA) is used for the measurement apparatus calibration purpose. As an example, GA based calibration of a differential pressure gauge using standard pressure setting devices is examined. classical Least Square As a result, how to get thevia better The calibration results, Method founded (LSM). using proposed GA, the are recommendations compared with theonresults obtained the based calibration of a differential pressure gauge GA, usingare standard pressure devices is examined. The calibration results, founded using proposed compared with setting theonresults obtained the calibration characteristics for the differential are given. classical Least Square Method (LSM). As pressure a result,gauge the recommendations how to get thevia better The calibration results, Method founded (LSM). using proposed GA, the are recommendations compared with theonresults obtained via the classical Least Square As a result, how to get the better calibration characteristics forinstrument, the differential pressure gauge are given. Calibration, Genetic algorithm, Keywords: Measurement Measurement uncertainty, © 2018, IFAC (International of Automatic Control) Hosting by Elsevieron Ltd.how All rights classical Least Square Method (LSM). As pressure a result, the recommendations to getreserved. the better calibration characteristics forFederation the differential gauge are given. Differential pressure gauge, Square Method. Keywords: characteristics Measurement instrument, Measurement uncertainty, calibration forLeast the differential pressure gauge are given. Calibration, Genetic algorithm, Keywords: Measurement instrument, Measurement uncertainty, Calibration, Genetic algorithm, Differential pressure gauge, Least Square Method. Keywords: Measurement instrument, Measurement uncertainty, Calibration, Genetic algorithm, Differential pressure gauge, Least Square Method. Differential pressure gauge, Least Square Method. 1. INTRODUCTION 1. INTRODUCTION Accurate measurement is the basis of almost all engineering 1. INTRODUCTION applications, since uncertainty inherently exists the nature 1. INTRODUCTION Accurate measurement is the basis of almost allinengineering Accurate measurement is theOn basis almost allin engineering of any measuring theof other hand, the cost of a applications, sinceapparatus. uncertainty inherently exists the nature Accurate measurement is the basis ofitsalmost all engineering applications, sinceapparatus. uncertainty exists the nature measuring apparatus increases with accuracy. of any measuring Oninherently the other hand, in theTherefore, cost of a applications, sinceapparatus. uncertainty exists the nature of any measuring Oninherently the other hand, the cost of a low cost accurate measurement devices are onein ofTherefore, the main measuring apparatus increases with its accuracy. of anyofmeasuring apparatus. OnOne the other hand, theTherefore, cost ofthea measuring apparatus increases with its accuracy. goal metrology engineers. way for decreasing low cost accurate measurement devices are one of the main measuring apparatus with its accuracy. low cost accurate measurement devices are ofTherefore, the main sensor isincreases the calibration process. Therefore, this goal ofuncertainties metrology engineers. One way forone decreasing the low cost accurate measurement devices are one of the main goal of metrology engineers. One way for decreasing the paper with the is calibration of a low cost sensor using the sensordeals uncertainties the calibration process. Therefore, this goal metrology engineers. One way for decreasing the sensorof uncertainties the calibration process. Therefore, this corresponding obtained by From paper deals withvalues the is calibration of areference low cost standards. sensor using the sensor uncertainties is the calibration process. Therefore, this paper deals with the calibration of a low cost sensor using the the practical point view, the calibrationstandards. characteristics corresponding valuesofobtained by reference From paper deals withvalues calibration of areference low costaccuracy sensor using the corresponding by standards. should be in a the polynomial The ofFrom this the practical point ofobtained view, form. the calibration characteristics corresponding values obtained by reference standards. From the practical point of view, the calibration characteristics polynomial depends on the noise-free data accuracy which is used to should be in a polynomial form. The of this the practical point of view, the calibration characteristics should be characteristics. in a polynomial form. The accuracy ofnoise, this obtain the reduce the effect of is theused polynomial depends on theTonoise-free data which to should be in a polynomial form. The accuracy of this polynomial depends on the data which to excessive of data should be used. obtain thenumber characteristics. Tonoise-free reduce the effect of is theused noise, polynomial depends on theTonoise-free data which is used to obtain the characteristics. reduce the effect of the noise, excessive numberofofsensors, data should used. the measurement The calibration inbe obtain thenumber characteristics. Toorreduce the effect of the noise, excessive of data should begeneral, used. apparatus, can beofof considered of “model excessive number data should used. problem The calibration sensors, oras inabetypical general, the measurement The calibration sensors, oras in general, the measurement searching” from data and problem then, it should be apparatus, can beofexperimental considered a typical of “model The calibration ofconsidered sensors, oras ina typical general, the measurement apparatus, can problem of “model approached by be applying Experimental Design In searching” from experimental data and then,techniques. it should be apparatus, can be experimental considered as data ashould typical problem of “model searching” from and then, it should be this case, the calibration design drive the operator in approached by applying Experimental Design techniques. In searching” experimental data and then,techniques. it should be approached by applying Experimental the choice offrom :calibration this case, the design shouldDesign drive the operator In in approached by applying Experimental Design techniques. In thischoice case, the the of :calibration design should drive the operator in the plan should (e.g. the and the this case, the calibration design drivenumber the operator in the choice of :experimental of the location of the calibration points, number the choice of :experimental the plan (e.g. the the number and location the experimental plan (e.g. the the number and repetitions); of the calibration points, number of the the the experimental plan (e.g. the the number and the location of the calibration points, number of repetitions); the main influence quantities; location of the calibration points, the number of the repetitions); regression curve; the main influence quantities; repetitions); the main influence quantities; technique; the regression curve; the main influence quantities; regression curve; the standard and their uncertainties. regressionreferences technique; the regression curve; technique; the standard references and their uncertainties. the standard regressionreferences technique; and their uncertainties. the standard references and their uncertainties.
The calibration design using the experimental design techniques is proposed (Bettathe et al., 2001), when the The calibration design inusing experimental design The calibration design using the experimental design relationship between the indirectly calculated measurands techniques is proposed in (Betta et al., 2001), when the The calibration design inisusing the experimental techniques isbetween proposed (Betta et calculated al., when the and the sensor inputs nonlinear or 2001), moremeasurands thandesign one relationship the indirectly techniques is proposed in (Betta et al., 2001), when the relationship between the indirectly calculated measurands measurand has to be considered. and the sensor inputs is nonlinear or more than one relationship between the isindirectly and the sensor inputs nonlinearcalculated or moremeasurands than one measurand has to be considered. A procedure fortooptimal selection of measurement via and the sensor inputs is nonlinear or more points than one measurand has be considered. D-optimality criterion to get the best calibration measurand has to be considered. A procedure for optimal selection of measurement points via A procedure forcriterion optimal selection measurement points via characteristics of measurement sensors is proposed in D-optimality to getof the best calibration A procedure for optimal selection of measurement points via D-optimality criterion to get the best calibration (Hajiyev, 2016).ofThemeasurement coefficients ofsensors the calibration curve are characteristics is proposed in D-optimality to squares get sensors the best calibration characteristics of is (LSM). proposed in evaluated by thecriterion classical least Asare an (Hajiyev, 2016). Themeasurement coefficients of themethod calibration curve characteristics of measurement sensors is proposed in (Hajiyev, 2016). The coefficients of the calibration curve are example, by thetheproblem of standard evaluated classical of leastoptimal squares selection method (LSM). As an (Hajiyev, 2016). The coefficients of themethod calibration curve evaluatedsetters by classical leastoptimal squares (LSM). Asare an pressure when calibrating differential pressure sensor example, thetheproblem of selection of standard evaluated by the classical least squares method (LSM). As an example, the problem of optimal selection of standard is solved.setters when calibrating differential pressure sensor pressure example, the problemcalibrating of optimal selection of standard pressure differential pressure sensor is solved.setters whenand Sensor compensation usingpressure the artificial pressure setters when calibrating differential sensor is solved.calibration neural network are and performed in (Khanusing et al.,the 2003). The is solved. Sensor calibration compensation artificial Sensor network calibration compensation using artificial artificial neural network based inverse modeling technique is neural are and performed in (Khan et al.,the 2003). The Sensor calibration and compensation using the artificial neuralfornetwork are response performed in (Khan etThe al.,choice 2003). used the sensor ofThe the artificial neural network basedlinearization. inverse modeling technique is neural are and performed in (Khan al., technique 2003). The artificial neural network based inverse is order ofnetwork the model the number of modeling theetThe calibration used for sensor response linearization. choice points of the artificial neural network based inverse modeling technique is used for the sensor response linearization. The choice of the are important design parameters in this technique. An order of the model and the number of the calibration points used for sensor response linearization. The choice of and the orderimportant of the model and the number the points intensive study about theparameters effect of theof oftechnique. the model are design inorder thiscalibration An order of the model and the number of the calibration points are important design parameters in this technique. An the number of the calibration points onorder the lowest intensive study about the effect of the of the asymptotic model and are important design parameters in this technique. An intensive study about the effect of the order of the model and root-mean-square (RMS) error hasonbeen reported in this the number of the calibration points the lowest asymptotic intensive study about the effect of the order of the asymptotic model and the number of the calibration points the lowest paper. root-mean-square (RMS) error hasonbeen reported in this the number of the calibration points the lowest asymptotic root-mean-square (RMS) error hasonbeen reported in this paper. For the calculation of the calibration parameters, Gaussroot-mean-square (RMS) error has been reported in this paper. Newton regression parameters, method is used in paper. For the repeating calculationnonlinear of the calibration GaussFor theet repeating calculation of Inthethe calibration parameters, Gauss(Kim al., 2015). paper, accelerometers and Newton nonlinear regression method is used in For the repeating calculation of the calibration parameters, GaussNewton nonlinear method used in gyroscopes are 2015). mathematically modelled based onis the error (Kim et al., In theregression paper, accelerometers and Newton repeating nonlinear regression method is used in (Kim et al., 2015). In the paper, accelerometers and factors including bias, sensitivity, coning based angle and azimuth gyroscopes are mathematically modelled on the error (Kim etCalibration al., In the modelled paper, accelerometers and gyroscopes are 2015). mathematically based on the error angle. for accelerometers and factors including bias,procedures sensitivity, coning angle and azimuth gyroscopes are mathematically modelled based on the error factors including bias,procedures sensitivity, angle and azimuth gyroscopes are formulated using coning nonlinear Gauss-Newton angle. Calibration for accelerometers and factors bias, sensitivity, coningaccelerometers angletheandproposed azimuth angle. including Calibration procedures and regression logic. The effectiveness of gyroscopes are formulated using for nonlinear Gauss-Newton angle. Calibration procedures for accelerometers and gyroscopes are formulated using nonlinear Gauss-Newton regression logic. The effectiveness of the proposed gyroscopes formulated using nonlinear regression are logic. The effectiveness of Gauss-Newton the proposed regression logic. The effectiveness of the proposed 774
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calibration curve (1) coefficients identification via Genetic Algorithm.
calibration procedures are proven by simulation and experiment using high accuracy 2-axis rotational gimbal motion system.
3.
Tomczyk (2006) presents an application of a Genetic Algorithm(GA) for the calibration of a measurement system intended for dynamic measurements. The process of calibration is based on the determination of the maximum value of a chosen error criterion. The solutions presented in the paper refer to the integral-square error if the magnitude and rate of change constraints are imposed simultaneously on the calibrating signal. The practical application of the presented algorithm has been illustrated on the example of sixth order low-pass system calibration.
Y a 0 a1 p a 2 p 2
Step 1: First, the GA parameters to be used must be determined. These are (Jacobson and Kanber, 2015);
Number of chromosomes in population The number of chromosomes in the population in “bit” Crossover probability Mutation probability
Step 2: After setting the parameters of the genetic algorithm, we create the population. Each chromosome of the population contains the coded values of the coefficients a0 , a1 and a2 . In order to find these coefficients, the chromosome and the bits for each chromosome should be set to the random value that the user will enter. Thus, the population is created. Let the each chromosome contains m bits. Accordingly, the formula is
In this study a Genetic Algorithm is developed for the calibration of measurement devices using the corresponding values obtained by reference standards. On the basis of this algorithm, the calibration curve for the differential pressure gauge is found. The calibration results founded using proposed GA are compared with the results obtained via the classical Least Square Method (LSM) and the recommendations on how to get the better calibration characteristics for the differential pressure gauge are given.
k
m mi
(4)
i 1
The following parameters necessary for coding should be found; the first m0 bits for the coefficient a0 , the next bit group m1 for the coefficient a1 , and the last bit group m2 for the coefficient a2 . Step 3: After the population is created, the chromosomes need to be decoded. The following conversion formula should be used for Binary Base Encoding. In case of tenbased coding, the 4th step should be started directly
PROBLEM STATEMENT
Consider calibrating a measurement instrument by means of standard setting devices. The calibration characteristic of the measurement instrument is described adequately by an l-order polynomial
�� � �� � ������������� � ����� �
�� � ��
��� ��
(5)
Here, Xi is the decoded value of the ith coefficient, ai and bi are the upper and lower bounds of the values that the appropriate coefficient can take. It is clear that Xi � ��� � �� �.
(1)
The measurement equation is written as Z a 0 a1 p a 2 p 2 ... a l p l ,
(3)
This formula shows the relationship between input and output of the transducer. As it can be seen from the formula, we need to find the polynomial coefficients a0 , a1 , a 2 . The following steps must be performed to find the coefficients.
In (Yoon, 2017), it is shown that the procedures for sensor collaboration are pre-defined and less adaptive to dynamic changes in environment, or it is because the procedures are developed one time and deployed for many times. They are not satisfactorily adaptive to environmental changes, or are less efficient to work the sensors collaborative to cope with abrupt changes. The adaptation of the measurement sensor to the changing environment is provided in (Yoon, 2017) using GA.
Y a 0 a1 p a 2 p 2 ... a l p l
OPERATIONAL PRINCIPLES OF GA
Our goal here is to find the calibration curve of the differential pressure gauge with the help of Genetic Algorithm. As the calibration curve, the 2nd order polynomial is taken as,
Xueliang et al. (2015) propose the calibration method for the three-axis magnetometer calibration based on GA. The algorithm put the parameters of the calibration model as the volutionary population; According to the fitness, the poor individual is eliminated step by step and the optimal individual is obtained after crossing and mutation; The optimal parameter estimation corresponding to optimal individuals is achieved. The obtained simulation and experimental results show that the method based on GA can converge steadily and can achieve high estimation accuracy.
2.
775
Binary Base Encoding is preferred in this application. For each chromosome, m0, m1, m2 bit groups will be decoded by applying a conversion formula. Depending on this method, the polynomial coefficients a0 , a1 , a2 contain the coded values of the coefficients, in which each chromosome of the population has. Each chromosome contains m bits of each.
(2)
where is the Gaussian random measurement errors with zero mean and the standard uncertainty . It was assumed that values of arguments pi , i 1,..., n are generated by standard setting devices. It is required to design algorithm for the 775
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The first m0 bits are the number of bits required to encode the coefficient a0, the coefficient a1 if the next bit group m1, and
to create a new population for getting new coefficient values again.
the coefficient a2 , which is the last bit group, m2. Using this formula, the values of the coefficients are found after the chromosome is decoded.
4.
Calibration with GA and LSM is made for the differential pressure gage. The range of the transducer is 0 pi 9 bar. The differential pressure gage errors are subjected to the normal distribution with zero mean and the standard uncertainty i = 0.03 bar.
Step 4: After the coefficients are found, using the polynomial formula (3) for each input X an output value for Y is obtained. The error assessment between the output values obtained as a result of the experiment and via GA should be performed. ∑������� � � �� �� �����
SENSOR CALIBRATION RESULTS
The calibration characteristics of the examined measuring device (in the present case, the differential pressure gage) is described by 2nd order polynomial as follows:
(6)
Yi a 0 a1 p i a 2 p i 2 .
The above error is expressed as the square of the difference between the output values measured as a result of the experiments and calculated via GA. To get the maximization problem, the error (6) should be removed from a large number. This number is taken as 1500 and as a result the fitness function can be determined in the following form:
(8)
Measurement equation is written in the form Z i a 0 a1 p i a 2 p i2 i ,
i 1, n ,
(9)
(7)
where i is the measurement error with zero mean and the standard uncertainty .
Among the values obtained by the fitness function, the chromosome with high fitness value has a higher chance for approaching to the optimal solution. The greater the fitness value means the smaller difference between the outputs. This means that the chromosome consisting of those coefficients is approaching to the solution of problem (Cortes et al., 2004).
In the experiments the standard pressure setting devices reproduce pressure signals of p i , i 1,19 in the measurement interval of 0 pi 9 bar with the step of 0,5 bar and the output signals of the differential pressure gage zi are registered. The holding experiment results are presented in the Table 1.
���� �
∑������� � � �� �� �����
If there is any fitness value close for solving, this means that the desired solution is approached and the process can be terminated. Otherwise, it should continue through step 5.
Table 1. Experiment Results Experi-ment No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Step 5: The selection process is applied to create the second population. Selection process provides the elimination of chromosomes with low fitness values and transmission of chromosomes with high fitness values to the next population. The selection process is implemented in certain ways. Roulette Wheel Method is preferred in this application. In other words, a method that the possibility of choosing a larger fitness value is high and the possibility of selecting a smaller fitness value is low is preferred. Step 6: Cross the population obtained by the selection process. We determine the number of chromosomes to be crossed by multiplying the number of chromosomes by the probability of crossing that it is mentioned in step 1. Then the chromosomes are chosen randomly for crossovering. Any crossover technique can be used here. Step 7: Once the crossover is complete, the mutation process will be performed. In order to perform the mutation process, the total number of bits to be mutated must be found by multiplying the number of chromosomes by the probability of mutation. Then the chromosomes that will be mutated are randomly selected. In case of binary coding, the Bit Inverting Method, but in the case of decimal coding, a Small Number Addition Method should be used.
pi (bar) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9
Zi (V) 0.07935 0.48096 0.99671 1.52954 2.04468 2.55005 3.01941 3.58521 4.07837 4.63196 5.13733 5.66949 6.17919 6.69861 7.21924 7.67761 8.22509 8.74634 9.19739
4.1. Sensor Calibration via GA Steps to follow to define the model (8) coefficients are (Jacobson and Kanber, 2015 );
Step 8: After all the steps that have been performed, the final state of the population is obtained, and the step 3 is returned
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Determining GA parameters, Forming a primitive population out of stochastically selected chromosomes (these chromosomes are the coded versions of calibration coefficients),
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Table 2. Absolute and relative calibration errors corresponding to the GA
Process of decoding of chromosomes inside this formed population (valid for binary coding), Calculation of chromosomes’ target values, Process of Reproduction suitable to the solution, Crossover process, Mutation, Final form of population and returning to decoding of chromosomes.
calculated calibration coefficients via GA are given below: aˆ 0 =-0.02932551; aˆ 1 =0.9002932; aˆ 2 =0.01173021
4.2 .Sensor Calibration via LSM The least squares method (LSM) can be used for estimation
a0 a1 a2 of polynomial in (8) (Abdullayev and Gadzhiyev, 1993). The LSM expressions in this case have the form: T
ˆ X T X
1
X Z;
D ˆ X T X T where Z z1
T
1
(10)
2 ,
(11)
z2 . . . zl
Yi , GA
abs
rel
V
bar
%
-0.029325510 0.4237536425 0.8826979000 1.3475072625 1.8181817300 2.2947213025 2.7771259800 3.2653957625 3.7595306500 4.2595306425 4.7653957400 5.2771259425 5.7947212500 6.3181816625 6.8475071800 7.3826978025 7.9237535300 8.4706743625 9.0234603000
Polynomial coefficient values are between: [-1, 2 ]. The
of coefficients
777
is the vector of the
0.0293 0.0762 0.1173 0.1525 0.1818 0.2053 0.2229 0.2346 0.2405 0.2405 0.2346 0.2229 0.2053 0.1818 0.1525 0.1173 0.0762 0.0293 -0.0235
15.24 11.73 10.17 9.09 8.21 7.43 6.70 6.10 5.34 4.69 4.05 3.42 2.80 2.18 1.56 0.95 0.34 -0.26
measurements; 1 x1 1 x2 . . X . . . . 1 xl
is the
matrix of the
x12 x22 . . . xl2
Table 3. Absolute and relative calibration errors corresponding to the LSM
known coordinates,
D ˆ is the
dispersion matrix of the calibration coefficients’ estimation errors. The estimated calibration coefficients are:
Yi , LSM V
abs
rel
bar
%
-0.00005 0.5119 1.0239 1.5362 2.0488 2.5615 3.0744 3.5875 4.1008 4.6143 5.1280 5.6419 6.1560 6.6703 7.1848 7.6995 8.2144 8.7295 9.2448
0.00005 -0.0119 -0.0239 -0.0362 -0.0488 -0.0615 -0.0744 -0.0875 -0.1008 -0.1143 -0.1280 -0.1419 -0.1560 -0.1703 -0.1848 -0.1995 -0.2144 -0.2295 -0.2448
(12)
aˆ0 , aˆ1 and aˆ2 via LSM
aˆ 0 0.00005; aˆ1 1.0236; aˆ 2 0.0004 4.3. Calibration Errors Values of the absolute abs and relative rel calibration errors corresponding to the GA and LSM are given in Table 2 and Table 3 respectively.
777
-2.38 -2.39 -2.41 -2.44 -2.46 -2.48 -2.50 -2.52 -2.54 -2.56 -2.58 -2.60 -2.62 -2.64 -2.66 -2.68 -2.70 -2.72
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4.4. Combined Calibration Method
The graphs of the calibration curves obtained using GA and LSM are presented in Fig.1. The graphs of the absolute values of absolute calibration errors corresponding to the GA and LSM are shown in Fig.2.
Taking into account the obtained calibration results for the differential pressure gage, we can draw the following recommendation. During the operation of the examined differential pressure gage, a calibration characteristic obtained on the basis of LSM is recommended to be used on the measuring range 0-6.7 bar, and the characteristic obtained via GA should be used on the range of 6.7-9 bar. The obtained experimental results confirm that, the two examined methods (GA and LSM) can be combined to provide good metrological characteristics for wide ranges in the pressure difference, namely in the measurement range 0-6.7 bar, where GA fitting gives an error greater than the permissible value, where one uses LSM, while GA fitting is used for the other parts. The graph of the combined calibration curve obtained using GA and LSM are presented in Fig.3.
Fig.1. Calibration curves based on the GA and LSM: GA-red line; LSM-blue line.
Fig.3. Combined calibration curve The graphs of the absolute values of the absolute calibration errors corresponding to the GA and LSM based combined calibration method are shown in Fig.4.
Fig.2. Absolute values of the absolute errors: GA-red line; LSM-blue line Comparison of the results given in Fig. 2 shows that in the measuring range 0-6.7 bar the calibration characteristics obtained on the basis that LSM gives better results than GA, and on the range of 6.7-9 bar the calibration characteristics obtained via GA gives significantly more accurate results than the LSM.
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REFERENCES Betta, G., Dell’Isola, M. Frattolillo, A. (2001). Experimental design techniques for optimising measurement chain calibration. Measurement, 30, 115-127. Hajiyev, C. (2016). Sensor calibration design based on Doptimality criterion. Metrology and Measurement Systems, 23(3), 413–424. Khan, S.A., Shabani, D.T., Agarwala, A.K. (2003). Sensor calibration and compensation using artificial neural network. ISA Transactions. 42, 337-352. Kim M.-S., Yu S.-B., Lee K.-S. (2014). Development of a high-precision calibration method for inertial measurement unit. International Journal of Precision Engineering and Manufacturing, 15, 567-575. Tomczyk, K. (2006). Application of genetic algorithm to measurement system calibration intended for dynamic measurement. Metrology and Measurement Systems, XIII(1), 93-103. Xueliang P., Pei J., Chunsheng, L. (2015). The calibration method of three-axis magnetometer based on genetic algorithm. Applied Mechanics and Materials, 722, 373378. Yoon, J. (2017). ANN-based collaborative sensor calibration and GA-approach to sensor mutation management. Proc. of the 6th IIAI International Congress on Advanced Applied Informatics (IIAI-AAI), Hamamatsu, Shizuoka,Japan’ 2017, pp. 897-902. Cortes, P., Larraneta, J., Onieva, L. (2004). Genetic algorithm for controllers in elevator groups: analysis and simulation during lunch peak traffic. Applied Soft Computing, 4(2), 59-174. Jacobson L., Kanber B. (2015). Genetic Algorithms in Java Basics. Apress, New York. Abdullayev, A.A., Gadzhiyev, Ch.M. (1993). Metrological support to data-acquisition systems for oil-product accounting. Measurement Techniques, 36(9), 977-981.
Fig.4. Absolute values of the absolute errors of the combined calibration The obtained experimental results show that the proposed combined approach to the calibration is an efficient instrument in calibrating of measurement apparatus. 5.
CONCLUSION
In this study a Genetic Algorithm is used for the measurement devices calibration purpose. The genetic algorithms are modelled on the processes of natural genetics. Their main advantage is that, we need only the information of the optimized function regardless of its derivatives, and no restriction is put on the shape of the function. As an example, GA based calibration of a differential pressure gauge using standard pressure setting devices is examined. The calibration characteristic of the differential pressure gauge is described adequately by the polynomial. The calibration results founded using proposed GA are compared with the results obtained via the classical Least Square Method (LSM) and the recommendations on how to get the better calibration characteristics for the differential pressure gauge are given. The obtained experimental results confirm that, the two examined methods (GA and LSM) can be combined to provide good metrological characteristics for wide ranges in the pressure difference, namely in the initial part of the measurement range , where GA fitting gives an error greater than the permissible value, where one uses LSM, while GA fitting is used for the other parts. The experimental results show that the proposed combined calibration approach is an efficient instrument in calibrating of measurement instruments. That approach can be easily realized and widely used in metrological support to measuring instruments in various branches of industry.
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51-30 (2018) 774–779 CalibrationIFAC of PapersOnLine the Pressure Gage via GA and LSM Based Calibration of the Pressure Gage GA Calibration Calibration ofCombined the Pressure Gage via viaMethod GA and and LSM LSM Based Based Calibration Method Calibration ofCombined the Pressure Gage via GA and LSM Based Combined Calibration Method Hacizade Combined U. Calibration Method
U. Hacizade U. Hacizade Department of Computer Engineering, Halic University, Sütlüce District, Imrahor Str., U. Hacizade No: 82, Beyoğlu-Istanbul, TURKEY Halic (e-mail: [email protected]) Department of Computer Engineering, University, Sütlüce District, Imrahor Str., Department of Computer Engineering, University, Sütlüce District, Imrahor Str., No: 82, Beyoğlu-Istanbul, TURKEY Halic (e-mail: [email protected]) Department of Computer Engineering, University, Sütlüce District, Imrahor Str., No: 82, Beyoğlu-Istanbul, TURKEY Halic (e-mail: [email protected]) No: 82, Beyoğlu-Istanbul, TURKEY [email protected]) Abstract: Low cost accurate measurement apparatus are(e-mail: one of the main goal of metrology engineers. One way Low for decreasing the measurement sensor uncertainties is the study dealsengineers. with the Abstract: cost accurate apparatus are calibration one of the process. main goalThis of metrology Abstract: cost accurate apparatus arevalues one of the process. main goalThis of metrology calibration of adecreasing low cost sensor using uncertainties the corresponding obtained by reference standards. From One way Low for the measurement sensor is the calibration study dealsengineers. with the Abstract: Low cost accurate measurement apparatus are one of main goalThis of form. metrology engineers. One way point for thecalibration sensor is the calibration study deals with practical of the should be the in aprocess. polynomial In this studythe calibration of adecreasing lowview, cost sensor using uncertainties thecharacteristics corresponding values obtained by reference standards. From thea One way for decreasing the sensor uncertainties is the calibration process. This study deals with the calibration of a low cost sensor using the corresponding values obtained by reference standards. From Genetic (GA) the is used for the measurement purpose.form. As anInexample, GA practicalAlgorithm point of view, calibration characteristicsapparatus should becalibration in a polynomial this study a calibration of a of low cost sensor using thecharacteristics corresponding values obtained by reference standards. From the practical point view, the calibration should becalibration in a polynomial study based of a differential gauge using standard pressure setting form. devices isthis examined. Geneticcalibration Algorithm (GA) is used forpressure the measurement apparatus purpose. As anInexample, GAa practical point ofresults, view, founded the calibration characteristics should becalibration in a with polynomial form. this study Genetic Algorithm is used for the measurement apparatus As obtained anInexample, The using proposed are compared the results via GA thea basedcalibration calibration of(GA) a differential pressure gauge GA, using standard pressure purpose. setting devices is examined. Genetic Algorithm (GA) is used for the measurement apparatus calibration purpose. As an example, GA based calibration of a differential pressure gauge using standard pressure setting devices is examined. classical Least Square As a result, how to get thevia better The calibration results, Method founded (LSM). using proposed GA, the are recommendations compared with theonresults obtained the based calibration of a differential pressure gauge GA, usingare standard pressure devices is examined. The calibration results, founded using proposed compared with setting theonresults obtained the calibration characteristics for the differential are given. classical Least Square Method (LSM). As pressure a result,gauge the recommendations how to get thevia better The calibration results, Method founded (LSM). using proposed GA, the are recommendations compared with theonresults obtained via the classical Least Square As a result, how to get the better calibration characteristics forinstrument, the differential pressure gauge are given. Calibration, Genetic algorithm, Keywords: Measurement Measurement uncertainty, © 2018, IFAC (International of Automatic Control) Hosting by Elsevieron Ltd.how All rights classical Least Square Method (LSM). As pressure a result, the recommendations to getreserved. the better calibration characteristics forFederation the differential gauge are given. Differential pressure gauge, Square Method. Keywords: characteristics Measurement instrument, Measurement uncertainty, calibration forLeast the differential pressure gauge are given. Calibration, Genetic algorithm, Keywords: Measurement instrument, Measurement uncertainty, Calibration, Genetic algorithm, Differential pressure gauge, Least Square Method. Keywords: Measurement instrument, Measurement uncertainty, Calibration, Genetic algorithm, Differential pressure gauge, Least Square Method. Differential pressure gauge, Least Square Method. 1. INTRODUCTION 1. INTRODUCTION Accurate measurement is the basis of almost all engineering 1. INTRODUCTION applications, since uncertainty inherently exists the nature 1. INTRODUCTION Accurate measurement is the basis of almost allinengineering Accurate measurement is theOn basis almost allin engineering of any measuring theof other hand, the cost of a applications, sinceapparatus. uncertainty inherently exists the nature Accurate measurement is the basis ofitsalmost all engineering applications, sinceapparatus. uncertainty exists the nature measuring apparatus increases with accuracy. of any measuring Oninherently the other hand, in theTherefore, cost of a applications, sinceapparatus. uncertainty exists the nature of any measuring Oninherently the other hand, the cost of a low cost accurate measurement devices are onein ofTherefore, the main measuring apparatus increases with its accuracy. of anyofmeasuring apparatus. OnOne the other hand, theTherefore, cost ofthea measuring apparatus increases with its accuracy. goal metrology engineers. way for decreasing low cost accurate measurement devices are one of the main measuring apparatus with its accuracy. low cost accurate measurement devices are ofTherefore, the main sensor isincreases the calibration process. Therefore, this goal ofuncertainties metrology engineers. One way forone decreasing the low cost accurate measurement devices are one of the main goal of metrology engineers. One way for decreasing the paper with the is calibration of a low cost sensor using the sensordeals uncertainties the calibration process. Therefore, this goal metrology engineers. One way for decreasing the sensorof uncertainties the calibration process. Therefore, this corresponding obtained by From paper deals withvalues the is calibration of areference low cost standards. sensor using the sensor uncertainties is the calibration process. Therefore, this paper deals with the calibration of a low cost sensor using the the practical point view, the calibrationstandards. characteristics corresponding valuesofobtained by reference From paper deals withvalues calibration of areference low costaccuracy sensor using the corresponding by standards. should be in a the polynomial The ofFrom this the practical point ofobtained view, form. the calibration characteristics corresponding values obtained by reference standards. From the practical point of view, the calibration characteristics polynomial depends on the noise-free data accuracy which is used to should be in a polynomial form. The of this the practical point of view, the calibration characteristics should be characteristics. in a polynomial form. The accuracy ofnoise, this obtain the reduce the effect of is theused polynomial depends on theTonoise-free data which to should be in a polynomial form. The accuracy of this polynomial depends on the data which to excessive of data should be used. obtain thenumber characteristics. Tonoise-free reduce the effect of is theused noise, polynomial depends on theTonoise-free data which is used to obtain the characteristics. reduce the effect of the noise, excessive numberofofsensors, data should used. the measurement The calibration inbe obtain thenumber characteristics. Toorreduce the effect of the noise, excessive of data should begeneral, used. apparatus, can beofof considered of “model excessive number data should used. problem The calibration sensors, oras inabetypical general, the measurement The calibration sensors, oras in general, the measurement searching” from data and problem then, it should be apparatus, can beofexperimental considered a typical of “model The calibration ofconsidered sensors, oras ina typical general, the measurement apparatus, can problem of “model approached by be applying Experimental Design In searching” from experimental data and then,techniques. it should be apparatus, can be experimental considered as data ashould typical problem of “model searching” from and then, it should be this case, the calibration design drive the operator in approached by applying Experimental Design techniques. In searching” experimental data and then,techniques. it should be approached by applying Experimental the choice offrom :calibration this case, the design shouldDesign drive the operator In in approached by applying Experimental Design techniques. In thischoice case, the the of :calibration design should drive the operator in the plan should (e.g. the and the this case, the calibration design drivenumber the operator in the choice of :experimental of the location of the calibration points, number the choice of :experimental the plan (e.g. the the number and location the experimental plan (e.g. the the number and repetitions); of the calibration points, number of the the the experimental plan (e.g. the the number and the location of the calibration points, number of repetitions); the main influence quantities; location of the calibration points, the number of the repetitions); regression curve; the main influence quantities; repetitions); the main influence quantities; technique; the regression curve; the main influence quantities; regression curve; the standard and their uncertainties. regressionreferences technique; the regression curve; technique; the standard references and their uncertainties. the standard regressionreferences technique; and their uncertainties. the standard references and their uncertainties.
The calibration design using the experimental design techniques is proposed (Bettathe et al., 2001), when the The calibration design inusing experimental design The calibration design using the experimental design relationship between the indirectly calculated measurands techniques is proposed in (Betta et al., 2001), when the The calibration design inisusing the experimental techniques isbetween proposed (Betta et calculated al., when the and the sensor inputs nonlinear or 2001), moremeasurands thandesign one relationship the indirectly techniques is proposed in (Betta et al., 2001), when the relationship between the indirectly calculated measurands measurand has to be considered. and the sensor inputs is nonlinear or more than one relationship between the isindirectly and the sensor inputs nonlinearcalculated or moremeasurands than one measurand has to be considered. A procedure fortooptimal selection of measurement via and the sensor inputs is nonlinear or more points than one measurand has be considered. D-optimality criterion to get the best calibration measurand has to be considered. A procedure for optimal selection of measurement points via A procedure forcriterion optimal selection measurement points via characteristics of measurement sensors is proposed in D-optimality to getof the best calibration A procedure for optimal selection of measurement points via D-optimality criterion to get the best calibration (Hajiyev, 2016).ofThemeasurement coefficients ofsensors the calibration curve are characteristics is proposed in D-optimality to squares get sensors the best calibration characteristics of is (LSM). proposed in evaluated by thecriterion classical least Asare an (Hajiyev, 2016). Themeasurement coefficients of themethod calibration curve characteristics of measurement sensors is proposed in (Hajiyev, 2016). The coefficients of the calibration curve are example, by thetheproblem of standard evaluated classical of leastoptimal squares selection method (LSM). As an (Hajiyev, 2016). The coefficients of themethod calibration curve evaluatedsetters by classical leastoptimal squares (LSM). Asare an pressure when calibrating differential pressure sensor example, thetheproblem of selection of standard evaluated by the classical least squares method (LSM). As an example, the problem of optimal selection of standard is solved.setters when calibrating differential pressure sensor pressure example, the problemcalibrating of optimal selection of standard pressure differential pressure sensor is solved.setters whenand Sensor compensation usingpressure the artificial pressure setters when calibrating differential sensor is solved.calibration neural network are and performed in (Khanusing et al.,the 2003). The is solved. Sensor calibration compensation artificial Sensor network calibration compensation using artificial artificial neural network based inverse modeling technique is neural are and performed in (Khan et al.,the 2003). The Sensor calibration and compensation using the artificial neuralfornetwork are response performed in (Khan etThe al.,choice 2003). used the sensor ofThe the artificial neural network basedlinearization. inverse modeling technique is neural are and performed in (Khan al., technique 2003). The artificial neural network based inverse is order ofnetwork the model the number of modeling theetThe calibration used for sensor response linearization. choice points of the artificial neural network based inverse modeling technique is used for the sensor response linearization. The choice of the are important design parameters in this technique. An order of the model and the number of the calibration points used for sensor response linearization. The choice of and the orderimportant of the model and the number the points intensive study about theparameters effect of theof oftechnique. the model are design inorder thiscalibration An order of the model and the number of the calibration points are important design parameters in this technique. An the number of the calibration points onorder the lowest intensive study about the effect of the of the asymptotic model and are important design parameters in this technique. An intensive study about the effect of the order of the model and root-mean-square (RMS) error hasonbeen reported in this the number of the calibration points the lowest asymptotic intensive study about the effect of the order of the asymptotic model and the number of the calibration points the lowest paper. root-mean-square (RMS) error hasonbeen reported in this the number of the calibration points the lowest asymptotic root-mean-square (RMS) error hasonbeen reported in this paper. For the calculation of the calibration parameters, Gaussroot-mean-square (RMS) error has been reported in this paper. Newton regression parameters, method is used in paper. For the repeating calculationnonlinear of the calibration GaussFor theet repeating calculation of Inthethe calibration parameters, Gauss(Kim al., 2015). paper, accelerometers and Newton nonlinear regression method is used in For the repeating calculation of the calibration parameters, GaussNewton nonlinear method used in gyroscopes are 2015). mathematically modelled based onis the error (Kim et al., In theregression paper, accelerometers and Newton repeating nonlinear regression method is used in (Kim et al., 2015). In the paper, accelerometers and factors including bias, sensitivity, coning based angle and azimuth gyroscopes are mathematically modelled on the error (Kim etCalibration al., In the modelled paper, accelerometers and gyroscopes are 2015). mathematically based on the error angle. for accelerometers and factors including bias,procedures sensitivity, coning angle and azimuth gyroscopes are mathematically modelled based on the error factors including bias,procedures sensitivity, angle and azimuth gyroscopes are formulated using coning nonlinear Gauss-Newton angle. Calibration for accelerometers and factors bias, sensitivity, coningaccelerometers angletheandproposed azimuth angle. including Calibration procedures and regression logic. The effectiveness of gyroscopes are formulated using for nonlinear Gauss-Newton angle. Calibration procedures for accelerometers and gyroscopes are formulated using nonlinear Gauss-Newton regression logic. The effectiveness of the proposed gyroscopes formulated using nonlinear regression are logic. The effectiveness of Gauss-Newton the proposed regression logic. The effectiveness of the proposed 774
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calibration curve (1) coefficients identification via Genetic Algorithm.
calibration procedures are proven by simulation and experiment using high accuracy 2-axis rotational gimbal motion system.
3.
Tomczyk (2006) presents an application of a Genetic Algorithm(GA) for the calibration of a measurement system intended for dynamic measurements. The process of calibration is based on the determination of the maximum value of a chosen error criterion. The solutions presented in the paper refer to the integral-square error if the magnitude and rate of change constraints are imposed simultaneously on the calibrating signal. The practical application of the presented algorithm has been illustrated on the example of sixth order low-pass system calibration.
Y a 0 a1 p a 2 p 2
Step 1: First, the GA parameters to be used must be determined. These are (Jacobson and Kanber, 2015);
Number of chromosomes in population The number of chromosomes in the population in “bit” Crossover probability Mutation probability
Step 2: After setting the parameters of the genetic algorithm, we create the population. Each chromosome of the population contains the coded values of the coefficients a0 , a1 and a2 . In order to find these coefficients, the chromosome and the bits for each chromosome should be set to the random value that the user will enter. Thus, the population is created. Let the each chromosome contains m bits. Accordingly, the formula is
In this study a Genetic Algorithm is developed for the calibration of measurement devices using the corresponding values obtained by reference standards. On the basis of this algorithm, the calibration curve for the differential pressure gauge is found. The calibration results founded using proposed GA are compared with the results obtained via the classical Least Square Method (LSM) and the recommendations on how to get the better calibration characteristics for the differential pressure gauge are given.
k
m mi
(4)
i 1
The following parameters necessary for coding should be found; the first m0 bits for the coefficient a0 , the next bit group m1 for the coefficient a1 , and the last bit group m2 for the coefficient a2 . Step 3: After the population is created, the chromosomes need to be decoded. The following conversion formula should be used for Binary Base Encoding. In case of tenbased coding, the 4th step should be started directly
PROBLEM STATEMENT
Consider calibrating a measurement instrument by means of standard setting devices. The calibration characteristic of the measurement instrument is described adequately by an l-order polynomial
�� � �� � ������������� � ����� �
�� � ��
��� ��
(5)
Here, Xi is the decoded value of the ith coefficient, ai and bi are the upper and lower bounds of the values that the appropriate coefficient can take. It is clear that Xi � ��� � �� �.
(1)
The measurement equation is written as Z a 0 a1 p a 2 p 2 ... a l p l ,
(3)
This formula shows the relationship between input and output of the transducer. As it can be seen from the formula, we need to find the polynomial coefficients a0 , a1 , a 2 . The following steps must be performed to find the coefficients.
In (Yoon, 2017), it is shown that the procedures for sensor collaboration are pre-defined and less adaptive to dynamic changes in environment, or it is because the procedures are developed one time and deployed for many times. They are not satisfactorily adaptive to environmental changes, or are less efficient to work the sensors collaborative to cope with abrupt changes. The adaptation of the measurement sensor to the changing environment is provided in (Yoon, 2017) using GA.
Y a 0 a1 p a 2 p 2 ... a l p l
OPERATIONAL PRINCIPLES OF GA
Our goal here is to find the calibration curve of the differential pressure gauge with the help of Genetic Algorithm. As the calibration curve, the 2nd order polynomial is taken as,
Xueliang et al. (2015) propose the calibration method for the three-axis magnetometer calibration based on GA. The algorithm put the parameters of the calibration model as the volutionary population; According to the fitness, the poor individual is eliminated step by step and the optimal individual is obtained after crossing and mutation; The optimal parameter estimation corresponding to optimal individuals is achieved. The obtained simulation and experimental results show that the method based on GA can converge steadily and can achieve high estimation accuracy.
2.
775
Binary Base Encoding is preferred in this application. For each chromosome, m0, m1, m2 bit groups will be decoded by applying a conversion formula. Depending on this method, the polynomial coefficients a0 , a1 , a2 contain the coded values of the coefficients, in which each chromosome of the population has. Each chromosome contains m bits of each.
(2)
where is the Gaussian random measurement errors with zero mean and the standard uncertainty . It was assumed that values of arguments pi , i 1,..., n are generated by standard setting devices. It is required to design algorithm for the 775
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The first m0 bits are the number of bits required to encode the coefficient a0, the coefficient a1 if the next bit group m1, and
to create a new population for getting new coefficient values again.
the coefficient a2 , which is the last bit group, m2. Using this formula, the values of the coefficients are found after the chromosome is decoded.
4.
Calibration with GA and LSM is made for the differential pressure gage. The range of the transducer is 0 pi 9 bar. The differential pressure gage errors are subjected to the normal distribution with zero mean and the standard uncertainty i = 0.03 bar.
Step 4: After the coefficients are found, using the polynomial formula (3) for each input X an output value for Y is obtained. The error assessment between the output values obtained as a result of the experiment and via GA should be performed. ∑������� � � �� �� �����
SENSOR CALIBRATION RESULTS
The calibration characteristics of the examined measuring device (in the present case, the differential pressure gage) is described by 2nd order polynomial as follows:
(6)
Yi a 0 a1 p i a 2 p i 2 .
The above error is expressed as the square of the difference between the output values measured as a result of the experiments and calculated via GA. To get the maximization problem, the error (6) should be removed from a large number. This number is taken as 1500 and as a result the fitness function can be determined in the following form:
(8)
Measurement equation is written in the form Z i a 0 a1 p i a 2 p i2 i ,
i 1, n ,
(9)
(7)
where i is the measurement error with zero mean and the standard uncertainty .
Among the values obtained by the fitness function, the chromosome with high fitness value has a higher chance for approaching to the optimal solution. The greater the fitness value means the smaller difference between the outputs. This means that the chromosome consisting of those coefficients is approaching to the solution of problem (Cortes et al., 2004).
In the experiments the standard pressure setting devices reproduce pressure signals of p i , i 1,19 in the measurement interval of 0 pi 9 bar with the step of 0,5 bar and the output signals of the differential pressure gage zi are registered. The holding experiment results are presented in the Table 1.
���� �
∑������� � � �� �� �����
If there is any fitness value close for solving, this means that the desired solution is approached and the process can be terminated. Otherwise, it should continue through step 5.
Table 1. Experiment Results Experi-ment No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Step 5: The selection process is applied to create the second population. Selection process provides the elimination of chromosomes with low fitness values and transmission of chromosomes with high fitness values to the next population. The selection process is implemented in certain ways. Roulette Wheel Method is preferred in this application. In other words, a method that the possibility of choosing a larger fitness value is high and the possibility of selecting a smaller fitness value is low is preferred. Step 6: Cross the population obtained by the selection process. We determine the number of chromosomes to be crossed by multiplying the number of chromosomes by the probability of crossing that it is mentioned in step 1. Then the chromosomes are chosen randomly for crossovering. Any crossover technique can be used here. Step 7: Once the crossover is complete, the mutation process will be performed. In order to perform the mutation process, the total number of bits to be mutated must be found by multiplying the number of chromosomes by the probability of mutation. Then the chromosomes that will be mutated are randomly selected. In case of binary coding, the Bit Inverting Method, but in the case of decimal coding, a Small Number Addition Method should be used.
pi (bar) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9
Zi (V) 0.07935 0.48096 0.99671 1.52954 2.04468 2.55005 3.01941 3.58521 4.07837 4.63196 5.13733 5.66949 6.17919 6.69861 7.21924 7.67761 8.22509 8.74634 9.19739
4.1. Sensor Calibration via GA Steps to follow to define the model (8) coefficients are (Jacobson and Kanber, 2015 );
Step 8: After all the steps that have been performed, the final state of the population is obtained, and the step 3 is returned
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Determining GA parameters, Forming a primitive population out of stochastically selected chromosomes (these chromosomes are the coded versions of calibration coefficients),
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Table 2. Absolute and relative calibration errors corresponding to the GA
Process of decoding of chromosomes inside this formed population (valid for binary coding), Calculation of chromosomes’ target values, Process of Reproduction suitable to the solution, Crossover process, Mutation, Final form of population and returning to decoding of chromosomes.
calculated calibration coefficients via GA are given below: aˆ 0 =-0.02932551; aˆ 1 =0.9002932; aˆ 2 =0.01173021
4.2 .Sensor Calibration via LSM The least squares method (LSM) can be used for estimation
a0 a1 a2 of polynomial in (8) (Abdullayev and Gadzhiyev, 1993). The LSM expressions in this case have the form: T
ˆ X T X
1
X Z;
D ˆ X T X T where Z z1
T
1
(10)
2 ,
(11)
z2 . . . zl
Yi , GA
abs
rel
V
bar
%
-0.029325510 0.4237536425 0.8826979000 1.3475072625 1.8181817300 2.2947213025 2.7771259800 3.2653957625 3.7595306500 4.2595306425 4.7653957400 5.2771259425 5.7947212500 6.3181816625 6.8475071800 7.3826978025 7.9237535300 8.4706743625 9.0234603000
Polynomial coefficient values are between: [-1, 2 ]. The
of coefficients
777
is the vector of the
0.0293 0.0762 0.1173 0.1525 0.1818 0.2053 0.2229 0.2346 0.2405 0.2405 0.2346 0.2229 0.2053 0.1818 0.1525 0.1173 0.0762 0.0293 -0.0235
15.24 11.73 10.17 9.09 8.21 7.43 6.70 6.10 5.34 4.69 4.05 3.42 2.80 2.18 1.56 0.95 0.34 -0.26
measurements; 1 x1 1 x2 . . X . . . . 1 xl
is the
matrix of the
x12 x22 . . . xl2
Table 3. Absolute and relative calibration errors corresponding to the LSM
known coordinates,
D ˆ is the
dispersion matrix of the calibration coefficients’ estimation errors. The estimated calibration coefficients are:
Yi , LSM V
abs
rel
bar
%
-0.00005 0.5119 1.0239 1.5362 2.0488 2.5615 3.0744 3.5875 4.1008 4.6143 5.1280 5.6419 6.1560 6.6703 7.1848 7.6995 8.2144 8.7295 9.2448
0.00005 -0.0119 -0.0239 -0.0362 -0.0488 -0.0615 -0.0744 -0.0875 -0.1008 -0.1143 -0.1280 -0.1419 -0.1560 -0.1703 -0.1848 -0.1995 -0.2144 -0.2295 -0.2448
(12)
aˆ0 , aˆ1 and aˆ2 via LSM
aˆ 0 0.00005; aˆ1 1.0236; aˆ 2 0.0004 4.3. Calibration Errors Values of the absolute abs and relative rel calibration errors corresponding to the GA and LSM are given in Table 2 and Table 3 respectively.
777
-2.38 -2.39 -2.41 -2.44 -2.46 -2.48 -2.50 -2.52 -2.54 -2.56 -2.58 -2.60 -2.62 -2.64 -2.66 -2.68 -2.70 -2.72
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4.4. Combined Calibration Method
The graphs of the calibration curves obtained using GA and LSM are presented in Fig.1. The graphs of the absolute values of absolute calibration errors corresponding to the GA and LSM are shown in Fig.2.
Taking into account the obtained calibration results for the differential pressure gage, we can draw the following recommendation. During the operation of the examined differential pressure gage, a calibration characteristic obtained on the basis of LSM is recommended to be used on the measuring range 0-6.7 bar, and the characteristic obtained via GA should be used on the range of 6.7-9 bar. The obtained experimental results confirm that, the two examined methods (GA and LSM) can be combined to provide good metrological characteristics for wide ranges in the pressure difference, namely in the measurement range 0-6.7 bar, where GA fitting gives an error greater than the permissible value, where one uses LSM, while GA fitting is used for the other parts. The graph of the combined calibration curve obtained using GA and LSM are presented in Fig.3.
Fig.1. Calibration curves based on the GA and LSM: GA-red line; LSM-blue line.
Fig.3. Combined calibration curve The graphs of the absolute values of the absolute calibration errors corresponding to the GA and LSM based combined calibration method are shown in Fig.4.
Fig.2. Absolute values of the absolute errors: GA-red line; LSM-blue line Comparison of the results given in Fig. 2 shows that in the measuring range 0-6.7 bar the calibration characteristics obtained on the basis that LSM gives better results than GA, and on the range of 6.7-9 bar the calibration characteristics obtained via GA gives significantly more accurate results than the LSM.
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REFERENCES Betta, G., Dell’Isola, M. Frattolillo, A. (2001). Experimental design techniques for optimising measurement chain calibration. Measurement, 30, 115-127. Hajiyev, C. (2016). Sensor calibration design based on Doptimality criterion. Metrology and Measurement Systems, 23(3), 413–424. Khan, S.A., Shabani, D.T., Agarwala, A.K. (2003). Sensor calibration and compensation using artificial neural network. ISA Transactions. 42, 337-352. Kim M.-S., Yu S.-B., Lee K.-S. (2014). Development of a high-precision calibration method for inertial measurement unit. International Journal of Precision Engineering and Manufacturing, 15, 567-575. Tomczyk, K. (2006). Application of genetic algorithm to measurement system calibration intended for dynamic measurement. Metrology and Measurement Systems, XIII(1), 93-103. Xueliang P., Pei J., Chunsheng, L. (2015). The calibration method of three-axis magnetometer based on genetic algorithm. Applied Mechanics and Materials, 722, 373378. Yoon, J. (2017). ANN-based collaborative sensor calibration and GA-approach to sensor mutation management. Proc. of the 6th IIAI International Congress on Advanced Applied Informatics (IIAI-AAI), Hamamatsu, Shizuoka,Japan’ 2017, pp. 897-902. Cortes, P., Larraneta, J., Onieva, L. (2004). Genetic algorithm for controllers in elevator groups: analysis and simulation during lunch peak traffic. Applied Soft Computing, 4(2), 59-174. Jacobson L., Kanber B. (2015). Genetic Algorithms in Java Basics. Apress, New York. Abdullayev, A.A., Gadzhiyev, Ch.M. (1993). Metrological support to data-acquisition systems for oil-product accounting. Measurement Techniques, 36(9), 977-981.
Fig.4. Absolute values of the absolute errors of the combined calibration The obtained experimental results show that the proposed combined approach to the calibration is an efficient instrument in calibrating of measurement apparatus. 5.
CONCLUSION
In this study a Genetic Algorithm is used for the measurement devices calibration purpose. The genetic algorithms are modelled on the processes of natural genetics. Their main advantage is that, we need only the information of the optimized function regardless of its derivatives, and no restriction is put on the shape of the function. As an example, GA based calibration of a differential pressure gauge using standard pressure setting devices is examined. The calibration characteristic of the differential pressure gauge is described adequately by the polynomial. The calibration results founded using proposed GA are compared with the results obtained via the classical Least Square Method (LSM) and the recommendations on how to get the better calibration characteristics for the differential pressure gauge are given. The obtained experimental results confirm that, the two examined methods (GA and LSM) can be combined to provide good metrological characteristics for wide ranges in the pressure difference, namely in the initial part of the measurement range , where GA fitting gives an error greater than the permissible value, where one uses LSM, while GA fitting is used for the other parts. The experimental results show that the proposed combined calibration approach is an efficient instrument in calibrating of measurement instruments. That approach can be easily realized and widely used in metrological support to measuring instruments in various branches of industry.
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