Genetic algorithm and least square method‐based calibration of measurement apparatus
2019; Institution of Engineering and Technology; Volume: 13; Issue: 5 Linguagem: Inglês
10.1049/iet-smt.2018.5030
ISSN1751-8830
Autores Tópico(s)Sensor Technology and Measurement Systems
ResumoIET Science, Measurement & TechnologyVolume 13, Issue 5 p. 715-721 Research ArticleFree Access Genetic algorithm and least square method-based calibration of measurement apparatus Ulviye Hacizade, Corresponding Author Ulviye Hacizade ulviyehacizade@halic.edu.tr orcid.org/0000-0002-0073-996X Department of Computer Engineering, Halic University, Sütlüce Mah., Imrahor Cad., No: 82, Beyoğlu, Istanbul, TurkeySearch for more papers by this author Ulviye Hacizade, Corresponding Author Ulviye Hacizade ulviyehacizade@halic.edu.tr orcid.org/0000-0002-0073-996X Department of Computer Engineering, Halic University, Sütlüce Mah., Imrahor Cad., No: 82, Beyoğlu, Istanbul, TurkeySearch for more papers by this author First published: 01 July 2019 https://doi.org/10.1049/iet-smt.2018.5030Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The calibration process is one of the methods for decreasing the measurement apparatus uncertainties. In this study, the sensor calibration is performed using the values reproduced by standard setting devices. Generally, the calibration characteristics of measurement apparatus are taken in a polynomial form. In this study, a 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. The calibration results, founded using the proposed GA, are compared with the results obtained via the classical least square method. As a result, the recommendations on how to get better calibration characteristics for the differential pressure gauge are given. 1 Introduction All the measuring apparatus have inherent uncertainty in their measurements. One of the methods for decreasing the measurement apparatus uncertainties is the calibration process [1]. Therefore, this study investigates the sensor calibration using the values reproduced by standard setting devices. In most of the cases, the calibration characteristics of measurement apparatus are taken in a polynomial form [2]. The calibration of measurement apparatus can be considered as a typical problem of 'model identification' from experimental data. For this purpose, the experimental design techniques can be applied. Ref. [3] proposes the experimental design technique-based calibration design when the relationship between the indirectly calculated measurands and the sensor inputs is non-linear or more than one measurand has to be considered. A procedure for the optimal selection of measurement points via the D-optimality criterion to get the best calibration characteristics of measurement sensors is proposed in [4]. The calibration curve coefficients are estimated by the classical least squares method (LSM). As an example, the problem of optimal selection of standard pressure setters is solved for the differential pressure sensor calibration. Ref. [5] uses the artificial neural network-based inverse modelling technique for the sensor response linearisation. In this technique, the selection of the model order and the number of the calibration points are important design parameters. The paper reports the effect of the model order and the number of the calibration points on the lowest asymptotic root mean square (RMS) error. For the calculation of the calibration parameters, Gauss–Newton repeating non-linear regression method is used in [6]. In the paper, accelerometers and gyroscopes are mathematically modelled based on the error factors including bias, sensitivity, coning angle, and azimuth angle. Calibration procedures for accelerometers and gyroscopes are formulated using non-linear Gauss–Newton regression logic. The effectiveness of the proposed calibration procedures are proven by simulation and experiment using high-accuracy two-axis rotational gimbal motion system. Ref. [7] 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 in the example of sixth-order low-pass system calibration. The properties of smart sensor error correction algorithms and calibration procedures are given in [8]. The results of the bivariate spline approximation algorithm are presented in this work. Ref. [9] shows that Tekscan I-Scan pressure sensors (Tekscan, Inc., Boston, MA, USA) force measurements may be accurate to be within 0.6% when calibration algorithms use a least squares minimisation technique. The three-point quadratic calibration decreases the error in force measurement from 2.7 to 1.5%. It is recommended that investigators design their own calibration curves. The paper [10] shows the performance of a progressive polynomial algorithm utilising different grades of relative non-linearity of an output sensor signal. It also presents an improvement to this algorithm, which obtains an optimal response with minimum non-linearity error, based on the number and selection sequence of the readjust points. One of the basic problems of space measuring techniques, such as navigation and positioning and attitude measurement and control, is three-axis measuring technology. It is very difficult to avoid the orthoerror and gain error in the three-axis measuring device even if the manufacturing process is improved, and the measurement precision is tremendously limited by these errors. In [11], a GA is proposed for calibrating the orthoerror and gain error. The obtained simulation results confirm that the GA shows a better performance than other algorithms, and a triaxial measure system can be calibrated more efficiently by GA. A dynamic calibration system based on the shock tube is presented in [12] to find the dynamic performance parameters of the calibrated pressure sensor. Based on the features of the output signal of the calibrated pressure sensor and the interference noises observed in the calibration test, a processing algorithm is proposed to obtain the dynamic calibration signal of the pressure sensor. In ref. [13], based on least square linear regression technology, a linear calibration model is formed to calibrate the parameter of tire pressure sensor through fitting the data and continuous data analysis. Sensor parameter calibration is based on the least square method (LSM) to fit the values. Furthermore, a linear calibration model is obtained through continuous linear regression analysis. In ref. [14] the calibration method for the three-axis magnetometer calibration based on GA is proposed. 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. In [15], it is shown that the procedures for sensor collaboration are pre-defined and less adaptive to dynamic changes in the 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 [15] using GA. Sensors in wireless sensor networks are required to be self-calibrated periodically during their prolonged deployment periods. In calibration planning, employing intelligent algorithms are essential to optimise both the efficiency and the accuracy of calibration. The minimum-cost bounded-error calibration tree (MBCT) problem is a spanning tree problem with two objectives, minimising the spanning tree cost and bounding the maximum post-calibration skew. A GA-based solution to the optimisation version of the MBCT problem is proposed in [16]. The investigations show that LSM and GA are the most commonly used calibration methods in the measurement technique. However, we do not find a comparison of GA- and LSM-based sensor calibration results in the current literature. In this study, GA- and LSM-based differential pressure sensor calibration results are compared. GA is developed for the calibration of measurement devices using the reproduced values by standard setting devices. 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 LSM and the recommendations on how to get the better calibration characteristics for the differential pressure gauge are given. 2 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 (1) The measurement equation is written as (2) where is the Gaussian random measurement errors with zero mean and the standard uncertainty σ. It was assumed that values of arguments are generated by standard setting devices. It is required to design an algorithm for the calibration curve (1) coefficients identification via GA. 3 Operational principles of GA Our goal here is to find the calibration curve of the differential pressure gauge with the help of GA. As the calibration curve, the second-order polynomial is taken as (3) This formula shows the relationship between input and output of the transducer. As can be seen from the formula, we need to find the polynomial coefficients . The following steps must be performed to find the coefficients. Step 1: First, the GA parameters to be used must be determined. These are [17]: • Number of chromosomes in the population. • The number of chromosomes in the population in 'bit'. • Crossover probability. • Mutation probability. Step 2: After setting the parameters of the GA, we create the population. In this study, the binary GA is applied. Each chromosome of the population is a randomly generated sequence of zeros and ones (0 and 1). Each subsequence is an encoded representation of one of the coefficients of a given polynomial. By this way, each chromosome of the population contains the coded values of the coefficients , and . Thus, the population is created. Let each chromosome contains m bits. Accordingly, the formula is The following parameters necessary for coding should be found; the first m0 bits for the coefficient , the next bit group m1 for the coefficient , and the last bit group m2 for the coefficient . 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 the case of 10-based coding, the fourth step should be started directly (4) Here, Xi is the decoded value of the ith coefficient, and ai and bi are the upper and lower bounds of the values that the appropriate coefficient can take. It is clear that Xi . 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 contain the coded values of the coefficients, in which each chromosome of the population has. Each chromosome contains m bits of each. The first m0 bits are the number of bits required to encode the coefficient a0, the coefficient if the next bit group m1, and the coefficient , which is the last bit group, m2. Using this formula, the values of the coefficients are found after the chromosome is decoded. 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 (5) The above-mentioned 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 maximisation problem, the error (5) 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: (6) Among the values obtained by the fitness function, the chromosome with high fitness value has a higher chance of 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 the solution of the problem [18]. 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. Step 5: The selection process is applied to create a 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 crossover. 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 the case of binary coding, the bit inverting method, but in the case of decimal coding, a small number addition method should be used. Step 8: After all the steps that have been performed, the final state of the population is obtained, and step 3 is returned to create a new population for getting new coefficient values again. The flow diagram of sensor calibration with GA is presented in Fig. 1. Fig 1Open in figure viewerPowerPoint Flow diagram of sensor calibration with GA 4 Sensor calibration results Calibration with GA and LSM is made for the differential pressure gage. The operational range of the investigated transducer is 0 ≤ pi ≤ 9 bar. The transducer errors are subjected to the normal distribution with zero mean and the standard uncertainty σi = 0.03 bar. The calibration characteristics of the examined differential pressure gage are described by second-order polynomial as: (7) The measurement equation is (8) where δi is the measurement error with zero mean and the standard uncertainty σ. In the experiments, the standard pressure setting devices reproduce pressure signals of in the measurement interval of bar with the step of 0.5 bar and the output signals of the transducer zi are registered. The results of the conducted experiment are presented in Table 1. Table 1. Experiment results Experiment no pi, bar Zi, V 1 0 0.07935 2 0.5 0.48096 3 1 0.99671 4 1.5 1.52954 5 2 2.04468 6 2.5 2.55005 7 3 3.01941 8 3.5 3.58521 9 4 4.07837 10 4.5 4.63196 11 5 5.13733 12 5.5 5.66949 13 6 6.17919 14 6.5 6.69861 15 7 7.21924 16 7.5 7.67761 17 8 8.22509 18 8.5 8.74634 19 9 9.19739 A block diagram of the experimental setup is given in Fig. 2. Fig 2Open in figure viewerPowerPoint Block diagram of the experimental setup 4.1 Sensor calibration via GA Steps to follow to define the model (7) coefficients are [17, 19]: Determining GA parameters. Forming a primitive population out of stochastically selected chromosomes (these chromosomes are the coded versions of calibration coefficients). 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 the population and returning to the decoding of chromosomes. Polynomial coefficient values are between: [–1, 2]. The calculated calibration coefficients via GA are given below: = –0.02932551; = 0.9002932; = 0.01173021 The repeated GA runs do not necessarily give consistent estimates for the calibration coefficients. Reason for lower consistency of repeated results from GA is heuristics (stochastic nature of optimisation algorithm). One way to handle this kind of stochastic results is to run optimisation procedure many times and then apply the average of the obtained estimates of the calibration coefficients. By running GA 10 times and calculating average values for the coefficients, we obtain = -0.02821; = 0.9109; = 0.0123 4.2 Sensor calibration via LSM The LSM can be used for the estimation of coefficients of a polynomial in (7) [20]. The LSM expressions, in this case, have the form (9) (10) where is the vector of the measurements; (11) is the matrix of the known coordinates, is the covariance matrix of the calibration coefficients' estimation errors. The estimated calibration coefficients and via LSM are: Running LSM several times with the same input data gave the same estimated coefficients. However, as pointed out above, repeated GA runs do not necessarily produce the same estimated coefficients. A better term for describing the difference between LSM and GA is that LSM has a closed form solution (9) while GA does not. 4.3 Calibration errors Values of the absolute and relative calibration errors corresponding to the GA and LSM are given in Tables 2 and 3, respectively. Table 2. Absolute calibration errors corresponding to the GA and LSM bar Yi GA – single run, V Yi GA – average, V Yi LSM V 0 −0.03 0.030 −0.028 0.028 −0.00 0.00 0.5 0.424 0.076 0.431 0.069 0.512 −0.012 1 0.883 0.117 0.895 0.105 1.024 −0.024 1.5 1.348 0.152 1.366 0.134 1.536 −0.036 2 1.818 0.182 1.843 0.157 2.049 −0.049 2.5 2.295 0.205 2.326 0.174 2.562 −0.062 3 2.777 0.223 2.815 0.185 3.074 −0.074 3.5 3.265 0.235 3.311 0.189 3.588 −0.088 4 3.760 0.240 3.812 0.188 4.101 −0.101 4.5 4.260 0.240 4.320 0.180 4.614 −0.114 5 4.765 0.235 4.834 0.166 5.128 −0.128 5.5 5.277 0.223 5.354 0.146 5.642 −0.142 6 5.795 0.205 5.880 0.120 6.156 −0.156 6.5 6.318 0.182 6.412 0.088 6.670 −0.170 7 6.848 0.152 6.951 0.049 7.185 −0.185 7.5 7.383 0.117 7.496 0.004 7.700 −0.200 8 7.924 0.076 8.046 −0.046 8.214 −0.214 8.5 8.471 0.029 8.603 −0.103 8.730 −0.230 9 9.024 −0.02 9.166 −0.166 9.245 −0.245 Table 3. Relative calibration errors corresponding to the GA and LSM bar GA – single run GA – average LSM 0 0.5 15.24 13.80 −2.38 1 11.73 10.50 −2.39 1.5 10.17 8.93 −2.41 2 9.09 7.85 −2.44 2.5 8.21 6.96 −2.46 3 7.43 6.16 −2.48 3.5 6.70 5.40 −2.50 4 6.10 4.68 −2.52 4.5 5.34 4.00 −2.54 5 4.69 3.32 −2.56 5.5 4.05 2.65 −2.58 6 3.42 2.00 −2.60 6.5 2.80 1.35 −2.62 7 2.18 0.70 −2.64 7.5 1.56 0.05 −2.66 8 0.95 0.58 −2.68 8.5 0.34 1.21 −2.70 9 −0.26 1.84 −2.72 The RMS errors, sum of absolute values of absolute error, sum of squared errors, and maximal absolute value of absolute error for the different calibration methods (GA-single run, G-average and LSM) are given in Table 4. Table 4. RMS Errors for the different calibration methods Method GA – single run GA – average LSM sum of absolute values of , bar 2.944 2.297 2.247 sum of squared errors, bar2 0.564 0.3408 0.374 RMS errors, bar 0.1723 0.134 0.140 maximal absolute value of , bar 0.240 0.189 0.245 As seen from Table 4, running GA many times and taking an average from the obtained estimates significantly increase the calibration accuracy. The graphs of the calibration curves obtained using GA-average algorithm (in future GA) and LSM are presented in Fig. 3. The graphs of the absolute values of absolute and relative calibration errors corresponding to the GA and LSM are shown in Figs. 4 and 5, respectively. Fig 3Open in figure viewerPowerPoint Calibration curves based on the GA and LSM (GA – red line; LSM – blue line) Fig 4Open in figure viewerPowerPoint Absolute values of the absolute errors (GA – red line; LSM – blue line) Fig 5Open in figure viewerPowerPoint Absolute values of the relative errors (GA– red line; LSM – blue line) A comparison of the results given in Figs. 4 and 5 shows that in the measuring range 0–5.7 bar, the calibration characteristics obtained on the basis that LSM gives better results than GA, and on the range of 5.7–9 bar, the calibration characteristics obtained via GA gives significantly more accurate results than the LSM. 4.4 Combined calibration method 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 0–5.7 bar, and the characteristic obtained via GA should be used on the range of 5.7–9 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 0–5.7 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 is presented in Fig. 6. Fig 6Open in figure viewerPowerPoint Combined calibration curve The graphs of the absolute values of the absolute and relative calibration errors corresponding to the GA- and LSM-based combined calibration method are shown in Figs. 7 and 8, respectively. Fig 7Open in figure viewerPowerPoint Absolute values of the absolute errors of the combined calibration Fig 8Open in figure viewerPowerPoint Absolute values of relative 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 and discussion In this study, the sensor calibration results, founded using GA, are compared with the results obtained via the classical LSM. As a result, the recommendations on how to get better calibration characteristics for the differential pressure gauge are given. GA is used for the measurement devices calibration purpose. The GAs are modelled on the processes of natural genetics. Their main advantage is that we need only the information of the optimised 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 LSM. Running LSM several times with the same input data gave the same estimated coefficients. Repeatedly running GA does not necessarily give consistent estimates for the calibration coefficients. Reason for lower consistency of repeated results from GA is heuristics (stochastic nature of optimisation algorithm). Therefore, GA optimisation procedure was run many times and the average of the obtained estimates is taken as the calibration coefficients estimates. This procedure significantly increased the calibration accuracy via GA. The RMS errors, sum of absolute values of absolute error, sum of squared errors, and maximal absolute value of absolute error corresponding to the calibration curves obtained by LSM and two GA algorithms (GA-single run GA-average) are also presented and compared. In this GA-developed application, C # was used as a programming language and Microsoft Visual Studio was used as a development environment. 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. 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