JP2009258068A - Apparatus for optimizing measurement point for measuring control object - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To optimize measurement points in measuring a control object. <P>SOLUTION: An apparatus is provided for optimizing measurement points for measuring performance data of the control object with respect to a predetermined control parameter. The apparatus predicts performance characteristics of the control object with respect to the control parameter using a model of the control object and computes predicted performance data (RSM_est). The predicted performance data is sampled at the set measurement points, and sampling performance data (RSM_mm) is computed based on the sampled performance data. Arrangement of the measurement points is optimized to minimize an error (Ersm_nm) between the predicted performance data and the sampling performance data. In a preferred form, the number of the measurement points is also optimized. <P>COPYRIGHT: (C)2010,JPO&INPIT

Description

  The present invention relates to an apparatus for optimizing measurement points for measuring a controlled object.

  A performance characteristic of a certain control target is automatically measured at a predetermined measurement point (a combination of control parameters). For example, when the control target is an internal combustion engine (hereinafter referred to as an engine), various data and tables necessary for engine control are stored in an in-vehicle control device (referred to as an ECU). In order to generate the table and the table, it is necessary to accurately grasp the performance characteristics of the engine, and therefore, automatic measurement of the engine is performed.

  Patent Document 1 below shows a hardware configuration for automatically measuring engine performance characteristics. However, this document only shows a system configuration that automatically measures the engine performance, and even though it can reduce human labor, the number of control parameters becomes large and the number of measurement points becomes enormous. . Therefore, it is impossible to answer the recent demand for measuring the engine performance characteristics in a shorter time.

  The CAMEO system of AVL List GmbH (Austria) uses an experimental design method to automatically measure engine performance. In this system, the measurement time is shortened by reducing the number of measurement points of engine performance by the experimental design method. However, when this is applied to recent engine measurement that shows a sudden change in the control parameter, if the number of measurement points is reduced too much, the unevenness in engine performance may not be accurately captured. Therefore, in practice, measurement points cannot be sufficiently reduced. In addition, when performing automatic measurement, it is necessary to input in advance the approximate position of the inflection point of engine performance, so it is difficult to perform automatic measurement of an engine that has not been measured in the past.

  The current engine has many variable devices such as a free-running valve system, a direct injection fuel system that can inject fuel several times in one cycle, and a variable geometry supercharger. The number of combinations is enormous.

  For this reason, in order to grasp the performance characteristics of the engine, it is necessary to measure the combination conditions of an enormous number of control parameters, and the measurement takes a lot of time. Moreover, in order to optimize a combination of a plurality of control parameters for a predetermined evaluation index (fuel consumption, output, emission, etc.), it is necessary to perform measurement under many conditions.

  From this, a combination of values of each control parameter is determined in a lattice pattern using an experimental design method as shown in FIG. 1A, and the control parameter is automatically set to each set value for each combination. Automatic measurement is performed to perform measurement automatically while maintaining a constant value.

  For example, in the conventional automatic measurement method shown in FIG. 1B, after the values of the control parameters A and B are changed from one combination to another (time t1), measurement is performed until the performance data is stabilized. A sequence is adopted in which measurement is stopped and measurement is performed in a predetermined period thereafter. For this reason, it takes several tens of seconds to several minutes to measure performance data for one measurement point. Therefore, even if the experimental design method is used, the effect of reducing the measurement points is not sufficient, and enormous time of several weeks or months is required to grasp the performance under all conditions of the engine.

Further, when the controlled object is an engine, the characteristic of the controlled object has a very complicated uneven curved surface characteristic with respect to the control parameter as shown in FIG. 2, and the change is very steep. For this reason, if the number of measurement points is excessively reduced by using the experimental design method, it becomes impossible to capture the unevenness characteristics and peak points of the actual engine performance characteristics. For example, in FIG. 3, the actual performance characteristic of the engine is indicated by reference numeral 101, and the measurement points by the experimental design method are three points where the control parameter A takes Ua1, Ua2 and Ua3. When the performance data measured at these three measurement points are connected, a performance characteristic as indicated by reference numeral 103 is obtained. As is clear from comparison between the actual performance characteristic 101 and the performance characteristic 103 obtained by the measurement, the performance characteristic obtained by the measurement does not follow the actual performance characteristic. Therefore, actual unevenness characteristics and peak points as indicated by reference numeral 105 cannot be obtained with performance characteristics obtained by measurement. If the unevenness characteristics and peak points that could not be obtained were the optimum values of the performance data that was originally desired, the engine performance could not be used to the maximum, and this measurement made sense. Will not. For these reasons, in the method based on the experimental design used by the current automatic measuring apparatus, the measurement points cannot actually be reduced so much and the measurement time cannot be shortened. In other words, when the number of measurement points is reduced, there is a high possibility that the optimum point and the unevenness characteristic cannot be measured.
JP2000-35379 Real-Time Optimization by Extremum Seeking Control, Kartik B. Ariyur, Miroslav Krstic, Wiley-Interscience, 2003/09

  Therefore, it is possible to optimize the arrangement of measurement points so that the optimum points and unevenness characteristics of the performance data as described above can be reliably captured for a control target having complicated performance characteristics such as an engine. An apparatus is desired. In addition, there is a demand for an apparatus that can optimize the number of measurement points so that the number of measurement points can be reduced as much as possible.

  In order to solve the above problems, the present invention provides an apparatus for optimizing a measurement point for measuring performance data for a predetermined control parameter with respect to a control target. The apparatus uses the model of the control target to predict performance characteristics for the control parameter of the control target and calculate predicted performance data (RSM_est); and a measurement point set for the predicted performance data Means for calculating sampling performance data (RSM_nm) based on the sampled performance data, and a measurement point so that an error (Ersm_nm) between the predicted performance data and the sampling performance data is minimized. Means for optimizing the arrangement of the.

  According to the present invention, it is possible to optimize the arrangement of measurement points even for a control target such as an engine whose performance characteristics with respect to control parameters show complicated increases and decreases. If the control object is measured at the optimized measurement point, a more accurate performance characteristic of the control object can be acquired. Therefore, various controls for the control object can be performed based on the acquired performance characteristic. By creating the necessary data and tables, it is possible to realize control that maximizes the performance of the control target. For example, if the object to be controlled is an engine, the engine performance includes emissions, fuel consumption, power, noise / vibration (NV) performance, and the like, and good performance can be realized.

  In one form of this invention, the said optimization is performed also about the number of measurement points. In this way, not only the arrangement of measurement points, but also the number of measurement points is optimized, so the performance characteristics of the control target are measured even for a control target such as an engine whose performance characteristics with respect to the control parameters show complex fluctuations. It is possible to acquire more accurate performance characteristics of the control target while shortening the time required for the control.

  In one embodiment of the present invention, the actual measurement target is measured at the optimized measurement point, and the actual performance characteristic of the control target is acquired. Thus, as described above, by performing measurement at the optimized measurement points, it is possible to acquire more accurate performance characteristics for the actual control target.

  In one form of this invention, the arrangement and number of the measurement points are optimized by a genetic algorithm (GA).

  To optimize measurement points, it is necessary to solve complex optimization problems where local minimums (local optimum points) are likely to exist. With a general linear optimization algorithm, the number of measurement points can be sufficiently reduced or measured. It is difficult to optimize the arrangement of points. For example, the arrangement and number of optimized measurement points may differ depending on the initial value of the arrangement and number of measurement points to be optimized. By applying the genetic algorithm, it is possible to increase the possibility that the arrangement and number of measurement points can be searched globally without falling into the local minimum.

  In one embodiment of the present invention, a DNA element having a meaning of reducing the number of measurement points is incorporated into a DNA individual when a mutation is processed in a genetic algorithm. Thereby, the number of measurement points can be reduced more efficiently.

  In a general genetic algorithm, optimization that increases or decreases the number of elements of DAN cannot be performed. For this reason, it has been difficult to solve the problem of simultaneously optimizing the arrangement and number of measurement points with a genetic algorithm. However, according to the present invention, it is possible to optimize the arrangement and number of measurement points by incorporating a value (“99” in the embodiment) indicating that the measurement points are invalidated (source) in the DNA element. Can be performed simultaneously.

  In one form of this invention, when evaluating a DNA individual using an evaluation function in a genetic algorithm, the evaluation function is weighted so that the evaluation by the evaluation function decreases as the number of measurement points increases. I do.

  In general, the greater the number of measurement points, the more likely it is that the performance characteristics of the controlled object can be measured with better accuracy. It is difficult to reduce the amount efficiently. However, according to the present invention, since the probability that a solution with a small number of measurement points is derived can be improved by weighting the evaluation function, the number of measurement points can be efficiently reduced.

  In one form of this invention, in the selection process in the genetic algorithm, DNA individuals are selected in descending order of evaluation by the evaluation function for each number of measurement points of each DNA individual.

  As described above, in general, the greater the number of measurement points, the higher the possibility that the performance characteristics of the controlled object can be measured with better accuracy. However, it is difficult to efficiently reduce the number of measurement points. However, according to the present invention, the survival probability of a DNA individual whose number of measurement points has been reduced can be increased even in the process of DNA evolution, and therefore the number of measurement points can be reduced more efficiently.

  According to another embodiment of the present invention, the optimization of the arrangement of the measurement points is an optimization for calculating the arrangement of the measurement points for minimizing the error when the number of measurement points is a predetermined value. Performed by algorithm.

  As an optimization algorithm, for example, the steepest descent method and the least square method are known, but the arrangement of measurement points can be optimized even using such an optimization algorithm.

  According to another embodiment of the present invention, the optimization of the arrangement and the number of the measurement points further repeats the processing by the optimization algorithm while changing the number of measurement points, and the error results as a result of the repetition. This is done by determining the arrangement and number of measurement points that are minimized.

  In general, optimization algorithms such as the steepest descent method and the least squares method cannot simultaneously optimize both the number and arrangement of measurement points. However, by repeating the process using the optimization algorithm while changing the number of measurement points, not only the arrangement of the measurement points but also the number of measurement points can be optimized. Therefore, the performance characteristics of the actual control target can be acquired with a smaller number of measurement points.

  According to another embodiment of the present invention, a predetermined evaluation function is used to obtain the arrangement and number of measurement points at which the error is minimized, and the evaluation function corresponds to an increase in the number of measurement points. Weighting is performed so that the evaluation by the evaluation function is lowered.

  By using the weighted evaluation function, it is possible to improve the probability that a solution with a small number of measurement points is derived, and thus it is possible to efficiently reduce the number of measurement points.

  Embodiments of the present invention will be described below with reference to the drawings. FIG. 4 is a block diagram of an apparatus for optimizing measurement points according to one embodiment of the present invention. The apparatus can be realized in any computer, and the computer includes a CPU (central processing unit), a storage device (including a memory and a hard disk device), an input device such as a keyboard and a mouse, and an output device such as a display. It has. Each functional block shown in the figure can be realized by the CPU executing a computer program stored in the memory.

  In the following embodiments, an internal combustion engine (hereinafter referred to as an engine) is a control target. The controlled object exhibits performance characteristics corresponding to the control parameter. When the control target is an engine, the control parameters include, for example, the lift amount of the intake valve, the phase of the intake cam, the negative pressure of the intake pipe, the air-fuel ratio, the fuel injection amount, and the like. The performance characteristics include, for example, engine output torque, emissions (amount of HC and NOx delivered from the engine), and the like. For example, when the control parameters are the lift amount of the intake valve and the phase of the intake cam, output torque corresponding to the values of these control parameters is obtained as performance characteristics of the engine. However, the present invention can also be applied to other control objects.

  The predicted performance unit 11 uses a predetermined engine model to predict a predetermined performance characteristic for a predetermined control parameter of the engine, and generates a predicted performance model Rsm_est representing performance data for the control parameter.

  The engine model is a model of an actual engine, and an engine model generated by any appropriate method can be used. For example, an engine model of a chemical element reaction model (zero dimension) can be used. The chemical reaction model describes a chemical reaction in an engine by a combination of a number of elementary reactions. The engine model may be a one-dimensional or three-dimensional model. For example, the engine intake and exhaust flow can be modeled by one-dimensional or three-dimensional analysis.

  Using such an engine model, a control parameter and performance data corresponding to the control parameter can be acquired as predicted performance data, and the relationship between the two can be represented as a model. This is called a predicted performance model RSM_est. Such a model can be generated in any suitable manner. For example, the performance data for the control parameter can be represented by a response surface, which can be used as a predicted performance model. For example, a Kriging method, a quadratic function surface, or a neural network can be used to generate the response surface. Alternatively, the performance data value for each value (combination) of the control parameters can be represented by one grid point, and the prediction performance model can be represented as an interpolation map having a huge number of grid points.

  Generation of such a predictive performance model can be realized using, for example, a commercially available tool (software). In such a tool, an engine model is generated, and in the generated model, for example, a torque (performance data) with respect to an opening degree (control parameter) of the throttle valve is acquired, and thereby the control parameter and performance data corresponding thereto are obtained. Can be obtained by, for example, response surface methodology. FIG. 5 shows an example of a predicted performance model generated from control parameters A and B and performance data corresponding thereto in a predetermined engine model as an example.

  Alternatively, if there is performance data actually measured for an engine with specifications close to the engine to be measured, use this as prediction performance data, and use the response surface etc. from the prediction performance data A predicted performance model may be generated.

  The predicted performance model RSM_est generated in this way is stored in the storage device of the computer.

  The designated measurement point determination unit 12 designates a predetermined measurement point in advance. The designated measurement point is called a designated measurement point. The designated measurement point can be determined by an arbitrary standard. In this embodiment, a measurement point necessary or desired to control the engine is set as a designated measurement point.

As an example, the designated measurement point can be determined as follows.
1) The point where the predicted performance data is maximized and / or minimized.
2) The point where the value of the control parameter satisfies a predetermined condition. Or the point at which the value of the control parameter is maximum and / or minimum.
3) A point where the value of the predicted performance data satisfies a predetermined condition. Or the point where the predicted performance data is maximum and / or minimum.

  Referring now to FIG. 6, a predicted performance data value 111 (represented by a predicted performance model) with respect to the value of the control parameter A is shown, where an example of a designated measurement point is shown. Yes. The designated measurement point A (i) is a designated measurement point that satisfies the above 1), and is a control parameter value that maximizes the predicted performance data. A (ii) is a designated measurement point that satisfies the above 2), and is a designated measurement point having a minimum control parameter value. A (iii) is a designated measurement point where the value of the predicted performance data satisfies a predetermined condition (a condition that the value X (iii) is taken).

  By using the designated measurement points, the necessary or desired measurement points can be included in the optimum measurement points without being excluded.

  The initial population determination unit 13 determines an initial population (generation zero) in the genetic algorithm so as to include the designated measurement point as a chromosome (hereinafter referred to as a DNA element) in each DNA individual. The initial population includes a predetermined number of DNA individuals.

  The optimization unit 15 optimizes the arrangement of the measurement points (which value of the control parameter is to be combined) using a genetic algorithm. In a preferred form, the optimization unit 15 optimizes not only the arrangement of measurement points but also the number using a genetic algorithm.

  More specifically, the sampling model generation unit 21 reads the prediction performance model RSM_est from the storage device, samples the prediction performance model RSM_est at the control parameter value defined by each DNA individual, and the sampled prediction performance A sampling performance model RSM_nm is generated from the data. The evaluation unit 22 compares the prediction performance model RSM_est with the sampling performance model RSM_nm, and evaluates the sampling performance model RSM_nm using an evaluation function.

  The DNA update unit 23 performs selection, crossover, and mutation processing in the genetic algorithm so as to leave a DNA individual with a higher evaluation. The convergence determination unit 24 determines whether the optimization has converged based on the difference between the DNA individual with the highest evaluation among the DNA individuals of the current generation and the DNA individual with the highest evaluation among the DNA individuals of the previous generation. Judging. If it is determined that the target has converged, the value of the control parameter represented by the highest-evaluated DNA individual in the current generation is set as an optimized measurement point. In this way, the arrangement and number of measurement points are optimized so that the error between the prediction performance model RSM_est and the sampling performance model RSM_nm is minimized.

  The measurement unit 17 automatically measures the actual engine at the optimized measurement point. For the measurement, a known appropriate technique and apparatus can be used. For example, as is known as a bench test, there is known a device for measuring the performance of an engine by installing an engine to be tested on a gantry (bench), operating the engine via a connected test device. Yes. What kind of test is performed can be controlled by a computer connected to the test apparatus. For example, the computer controls the test apparatus to automatically acquire engine performance data for each optimum measurement point, and the acquired performance data is taken into a storage device of the computer or displayed on an output device such as a display. Can be made.

  Since the predicted performance model is based on the engine model, the predicted performance data obtained from the predicted performance model simulates the behavior of the actual engine performance, such as the unevenness characteristic and the increase / decrease characteristic. This has an error. According to the present invention, an arrangement and number of measurement points suitable for the behavior of the predicted performance model can be obtained. By measuring the actual engine at the measurement point, the performance that follows the actual behavior and size of the engine. Data can be acquired.

  The engine performance data actually measured at the optimum measurement point in this way can be used for generating data and tables to be stored in the ECU, for example. By executing engine control using these data, tables, and the like, the engine control can be realized so as to maximize the performance of the engine.

  Next, referring to FIG. 7 to FIG. 15, executed by the CPU of the computer according to one embodiment of the present invention, more specifically, executed by the initial group determination unit 13 and the optimization unit 15 of FIG. 4. A detailed algorithm will be described.

  In step S11, a generation parameter m indicating the generation of the population of DNA individuals in the genetic algorithm is initialized to zero.

  In step S12, an initial group (generation m = 0) is generated. Referring now to FIG. 8, an example of the initial population is shown. In this embodiment, two types of control parameters, that is, control parameters A and B are used. Therefore, the designated measurement point is determined for the control parameter B by the designated measurement point determination unit 12 described above. For the control parameter A, the three designated measurement points A (i) to A (iii) described with reference to FIG. 6 are used. For the control parameter B, for example, two designated measurement points B (i) and B (ii) that satisfy the above-described conditions 1) and 2) are used. The number of designated measurement points for the control parameters A and B is oa and ob, respectively. Therefore, in this example, oa = 3 and ob = 2. For each control parameter, the number of designated measurement points can be arbitrarily determined. In addition, the number of types of control parameters can be arbitrarily determined.

  The number of DNA individuals per generation is N. The maximum number of chromosomes (DNA elements) for the control parameter A is pa, and the maximum number of DNA elements for the control parameter B is pb. pa and pb indicate the maximum number of measurement points for the control parameters A and B, respectively, and these values can also be arbitrarily determined.

  Here, for each DNA element notation, the first digit indicates the type of control parameter, the second digit indicates the DNA element number, the third digit indicates the DNA individual number, and the fourth digit indicates the generation number. Indicates. For example, “A135” represents the first DNA element in the third DNA individual of the fifth generation of the control parameter A.

  In each DNA individual, the designated measurement point is included as a fixed DNA element. In this example, the designated measurement points for the control parameter A are included in the first to third DNA elements in each DNA individual, and the designated measurement points for the control parameter B are 1 and 2 in each DNA individual. Included in the second DNA element.

  In each DNA individual, the designated measurement point is defined as a DNA element, and then the remaining DNA elements (pa-oa, pb-ob) are limited within the range that the control parameters A and B can take. Generate by random numbers. In this way, N DNA individuals are generated.

  Furthermore, a random value is used to extract an arbitrary DNA element, and the extracted element is a value (in this embodiment, meaning that the measurement point represented by the element is invalidated (that is, not used as a measurement point). , “99”, which is a value outside the possible range of the control parameters A and B, is used. However, the DNA element to be invalidated is extracted from DNA elements other than the DNA element for which the designated measurement point is defined. In this example, the DNA elements A420, B420, and A5n0 are invalidated. In each DNA individual, the number of DNA elements to be invalidated can be arbitrarily determined, and it is not necessary to equalize the number of DNA elements to be invalidated between the control parameters A and B. Thus, in each DNA individual, when the number of DNA elements replaced with “99” is zero, the maximum number of measurement points is pa and pb, and when the number of replaced DNA elements is 1, The number of points is (pa-1) and (pb-1). When the number of substituted DNA elements is 2, the maximum number of points is (pa-2) and (pb-2) . Here, the total number of measurement points not replaced in the nth DNA individual in the m generation is represented by pa_ef_nm and pb_ef_nm, respectively.

  Next, in step S13, the predicted performance model RSM_est stored in the storage device is read out, and the value Av_r (r = 1 to va) for the control parameter A and the value for the control parameter B with respect to the predicted performance model RSM_est. Sampling is performed in Bv_s (s = 1 to vb) to obtain predicted performance data Vest_rs. This is used for evaluation described later.

  va and vb are preferably large enough to satisfy va >> pa and vb >> pb (for example, 10 times or more of pa and pb, respectively). va and vb are generated by general experimental design or random numbers. You can do it.

  In step S14, a DNA individual evaluation process is executed. The evaluation process is shown in FIG.

  In step S21, the repetition parameter i (identifying the DNA individual number n) is set to zero.

In step S22, as described above, the total number of measurement points pa_ef_nm and pb_ef_nm that have not been invalidated by “99” is calculated, and the total number of measurement points Ns_nm (the number of combinations of the value of control parameter A and the value of control parameter B). Is calculated as follows.
Ns_nm = pa_ef_nm × pb_ef_nm

  In step S23, control parameters defined by each DNA element in the m-th generation n-th DNA individual are represented by Apnm (p = 1 to pa) and Bqnm (q = 1 to pb). Prediction performance data Yapn_bqn_m (p = 1 to pa, q = 1 to pb) is obtained by sampling the prediction performance model Rsm_est at Apnm and Bqnm.

  Here, referring to FIG. 10, an example of Yapn_bqn_m is shown. Note that when any of Apnm and Bqnm is invalidated (that is, replaced with “99”), the prediction performance data is not sampled. Therefore, since the total number of measurement points is Ns_nm, Ns_nm Yapn_bqn_m is acquired.

  In step S24, a sampling performance model RSM_nm is generated based on the sampled predicted performance data Yapn_bqn_m. As described above, the predicted performance model RSM_est is obtained by calculating a relationship between a control parameter and performance data corresponding thereto by a computer. Therefore, the sampling performance model RSM_nm can be generated from the control parameters Apnm and Bqnm and the performance data Yapb_bqn_m corresponding to the method similar to the method for generating the predicted performance model. The sampling performance model RSM_nm can be said to represent the performance characteristics of the engine that can be realized at the measurement point represented by the current DNA individual.

  In step S25, the sampling performance model RSM_nm is sampled with the same control parameter values Av_r and Bv_s as in step S13 described above, and sampling performance data Vnm_rs is obtained. The reason why the same control parameter values Av_r and Bv_s as in step S13 are used is to compare the predicted performance model RSM_est and the sampling performance model RSM_nm under the same conditions.

In general, the sampling performance model RSM_nm can be obtained with better accuracy by increasing the number of measurement points. However, in the present invention, the sampling performance model RSM_nm with higher accuracy can be obtained with a smaller number of measurement points. In order to search for the measurement point arrangement, the evaluation function is multiplied by a weight function (penalty) W corresponding to the number of measurement points as follows.

Alternatively, W = (Ns_nm) ² / Nmax ² may be used. Here, Nmax = pa × pb. Therefore, for a DNA individual having no invalidated DNA element, W = 1 and no weighting is performed.

  In step S26, a weight function W is calculated. The weighting function W may be calculated according to the above formula or may be obtained by referring to a table of the weighting function W. FIG. 11 shows an example of the table. As the number of measurement points Ns_nm is increased, the value of the weight W is increased and the evaluation is decreased. The corresponding W can be obtained by storing the table in the memory and referring to the table based on Ns_nm calculated in step S22.

In step S27, in order to evaluate the sampling performance model RSM_nm, the value Ersm_nm of the evaluation function including the weighting function W is calculated as follows.

  Since the evaluation function value Ersm_nm reflects an error of the sampling performance model RSM_nm with respect to the prediction performance model RSM_est, the smaller the evaluation function value Ersm_nm, the higher the evaluation of the DNA individual that is the basis of the sampling performance model RSM_nm. . Thus, it is evaluated how accurately the measurement point realized by the DNA individual can reproduce the performance data on the prediction performance model.

  In step S28, the repetition parameter i is incremented by one. In step S29, steps S22 to S28 are executed for each DNA individual until the value of i exceeds N indicating the number of DNA individuals. If the determination in step S29 is No, the process proceeds to step S15 in FIG. 7 to execute a DNA update process. The update process is shown in FIG.

  In step S31, N DNA individuals are arranged in ascending order of the evaluation function value Ersm_nm. An example of the arranged results is shown in FIG.

  In step S32, the smallest evaluation function value among the N DNA individuals is defined as Ersm_best. This is used for the optimization convergence determination described later.

  In step S33, DNA individuals are selected in ascending order of evaluation function value Ersm_nm for each number of measurement points. The maximum value Nmax of the total number of measurement points is pa × pb as described above. Further, as described above, the DNA elements at the designated measurement points are not invalidated, and the number thereof is oa and ob. Therefore, the minimum possible value Nmin of the total number of measurement points is oa × ob.

  One DNA individual with the smallest evaluation function value Ersm_nm is selected from the DNA individuals with Ns_nm = Nmin. Next, one DNA individual with the smallest evaluation function value Ersm_nm is selected from the DNA individuals with Ns_nm = Nmin + 1. Ns_nm = Nmin + 2, Nmin + 3,. . . Such a selection operation is repeated for Nmax. When Ns_nm = Nmax is reached, the process returns again to Ns_nm = Nmin, and Ns_nm = Nmin, Nmin + 1,. . . For each of Nmax, one DNA individual having the smallest evaluation function value Ersm_nm is selected from among the DNA solids not yet selected. When the number of selected DNAs reaches a predetermined value Nsel (<N), the selection operation is terminated.

  In a general genetic algorithm, such selection operation for each number of measurement points is not performed, and Nsel items are selected in ascending order of the evaluation function value Ersm_nm. In the present invention, in order to increase the survival rate of DNA individuals with a small number of measurement points, it is preferable to perform such a selection operation for each number of measurement points.

  In step S34, two DNA individuals are randomly selected from the selected Nsel DNA individuals, and the crossover position is determined by random numbers. Referring now to FIG. 14, an example of two selected DNA individuals and determined crossover positions is shown. In this example, the crossover position is determined between the DNA element related to the control parameter A and the DNA element related to the control parameter B, but is not limited to such a crossover position.

  Between the two DNA individuals, DNA elements existing on the right side of the crossover position are exchanged to generate two new DNA individuals. Thus, a predetermined number of Ncross (<N) DNA crossover individuals are generated.

  In step S35, one DNA individual is randomly selected from the selected Nsel DNA individuals, and the mutation position is determined by a random number. Mutation positions are determined so as to exclude DNA elements at designated measurement points. The value of the DNA element at the mutation position is replaced with a value obtained by random numbers or a value “99” indicating invalidation. Here, the value obtained by the random number is a value within a possible range for each of the control parameters A and B. Note that it is preferable to use a random number to determine which of the value obtained by random number and the value indicating invalidation is used for replacement. In this way, a predetermined number Nmut (<N) of new DNA individuals are generated.

  In step S36, numbers 1 to N are reassigned to N DNA individuals determined by selection, crossover, and mutation. Here, N = Nsel + Ncros + Nmut, and Nsel, Ncros, and Nmut are determined so as to have such a relationship. The order of assigning the numbers may be arbitrary.

  In step S37, the generation parameter m is incremented by 1, and the process proceeds to step S16 in FIG.

In step S16, as shown in the following formula, the absolute value D_ersm_best of the difference between the best evaluation function value Ersm_best in the current generation calculated in step S32 and the best evaluation function value Ersm_best in the previous generation is calculated. . In this equation, (m-1) indicates the current generation and (m-2) indicates the previous generation. This is because the generation parameter m is set to 1 only during the update process in step S15. This is because it is incremented (S37 in FIG. 12). When the best evaluation function value Ersm_best (m−1) is for the first generation, there is no previous generation, so a predetermined initial value is set in Ersm_best (m−2).
D_ersm_best = | Ersm_best (m-1) -Ersm_best (m-2) |

  If the difference D_ersm_best is equal to or smaller than the predetermined value D_ersm_conv in step S17, it is determined that the optimization has converged, and the process proceeds to step S19. Otherwise, go to step 18 to continue the optimization.

  If the number of generations m is equal to or less than the predetermined maximum number of generations Mmax in step S18, the process proceeds to step S14 to continue the optimization. If the generation number m is larger than Mmax, the process proceeds to step S19 in order to end the optimization.

  In step S19, the values of the control parameters A and B defined by the DNA individual that realizes the best evaluation function value Ersm_best (m-1) in the current generation are set as the optimum measurement points. Thus, the optimization ends. The reason why (m−1) indicates the current generation is that, as described above, the generation parameter m is incremented by 1 during the update process in step S15.

  As described above, actual engine measurement is performed at the optimum measurement point. The optimum measurement point is a measurement point determined so as to be able to follow the behavior such as unevenness and peak value (optimum value) of the engine performance obtained from an engine model obtained by modeling an actual engine with the best accuracy. Therefore, by measuring the actual engine at the optimum measurement point, the performance characteristics of the actual engine can be acquired with better accuracy.

  With reference to FIG. 16 and FIG. 17, the effect of the measurement point optimized according to the present invention will be described. In this example, performance data (for example, intake air amount, torque, combustion efficiency, etc.) indicating a relatively smooth convex characteristic with respect to the control parameter A (for example, throttle valve opening, intake valve lift amount, etc.) (A) shows the case where the arrangement of measurement points is determined according to the conventional experimental design, and (b) the case where the arrangement and number of measurement points are determined according to the technique of the present invention.

  In FIG. 16A, reference numeral 131 indicates the actual performance characteristic of the engine, and reference numeral 133 indicates the performance data measured at the measurement points A1 to A6 that are evenly arranged by the conventional experimental design method.

  In FIG. 16B, reference numeral 131 indicates the same actual engine performance characteristics as in FIG. 16A, reference numeral 135 indicates performance data based on the predicted performance model RSM_est, and reference numeral 137 indicates optimization according to the present invention. The performance data obtained by the measurement at the measured measurement points are shown. Here, the maximum number of measurement points pa = 6, and the number of optimum measurement points is reduced to five by the method of the present invention, and the arrangement of optimum measurement points is uneven. As described above, the predicted performance model simulates the behavior of the actual convex characteristics of the engine, but there is a difference in the size of the performance data.

  In (a), the actual engine performance characteristic 131 overlaps with the engine performance characteristic 133 measured at the measurement points A1 to A6. Therefore, the convex characteristics in the actual engine performance can be measured at the measurement points A1 to A6 with good accuracy by the six measurement points. That is, the engine performance characteristics calculated based on the measurement points A1 to A6 can accurately represent the actual engine performance characteristics.

  In (b), the actual engine performance characteristic 131 overlaps with the engine performance characteristic 137 measured at the measurement points A1 to A6. Thus, according to the present invention, it is possible to measure the performance characteristics of the engine with the same accuracy as (a), although the number of measurement points is reduced.

  FIG. 17 shows another example. The difference from FIG. 16 is that the performance characteristics of the engine have a relatively large increase / decrease (for example, the ignition timing of a gasoline engine, emissions such as Nox and HC, etc.). It is based on (if any).

  In (a), since the six measurement points are evenly arranged by the conventional experimental design method, the increase / decrease characteristic of the performance data in the region where the value of the control parameter A is small cannot be measured accurately. . For this reason, the engine performance characteristic 133 obtained based on the measurement point has a large error with respect to the actual engine performance characteristic 131 in the region where the value of the control parameter A is small. In order to eliminate this error, it is necessary to arrange more measurement points.

  In (b), the number of measurement points is reduced from six to five according to the method of the present invention, but these measurement points are unevenly arranged at appropriate positions. Therefore, in spite of the fact that the number of measurement points is reduced, in the region where the value of the control parameter A is small, without causing a large error between the measured engine performance characteristic 137 and the actual engine performance characteristic 131, The engine performance characteristics can be measured with good accuracy.

  In the above-described embodiments, a mode has been described in which not only the arrangement of measurement points but also the number is optimized using a genetic algorithm. Alternatively, only the arrangement of measurement points may be optimized. In this case, when the initial population in step S12 of FIG. 7 is generated, a process for inserting a value indicating the invalidation of the measurement point into the DNA element is not required. Control parameter values may be entered for all DNA individuals. As for the weighting function W in step S26 of FIG. 9, since each DNA individual does not have an invalidated DNA element, W = 1, and the evaluation function is not weighted. Therefore, the calculation of the weight function may be omitted. Furthermore, since the number of measurement points of all the DNA individuals is the same in the selection process in step S33 of FIG. 12, it is not necessary to select every number of measurement points, and all the DNA individuals are arranged in ascending order of the evaluation function value Ersm_nm. Nsel DNA individuals may be selected from among them. In the mutation process in step S35, replacement with a value indicating invalidation is not performed. Even in the case of such an alternative form, measurement points having an arrangement suitable for the behavior of the prediction performance model can be obtained through the genetic algorithm. Therefore, by measuring the actual engine at the measurement points, the actual engine It is possible to obtain performance data that follows the behavior and size of the.

  In the above embodiment, optimization is performed using a genetic algorithm. According to the genetic algorithm, the arrangement and number of measurement points can be optimized simultaneously. However, in an alternative form, a genetic algorithm may not be used. This form will be described below.

  FIG. 18 is a block diagram of an apparatus for optimizing measurement points according to another embodiment of the present invention. This apparatus optimizes the arrangement and number of measurement points using another optimization algorithm different from the genetic algorithm.

  The device according to the other embodiment has a configuration similar to that of FIG. The same components as those in FIG. 1 are assigned the same reference numerals, and description thereof will be omitted.

  The optimization unit 115 optimizes the arrangement of the measurement points using an optimization algorithm different from the genetic algorithm. In a preferred embodiment, the optimization unit 115 optimizes not only the arrangement of measurement points but also the number.

  More specifically, the measurement point set initialization unit 113 sets initial values for a set of measurement points of each control parameter (hereinafter referred to as measurement point set). The sampling performance model generation unit 121 reads the prediction performance model RSM_est from the storage device, samples the prediction performance model RSM_est at the control parameter value defined by the measurement point set set by the measurement point set initialization unit 113, and A sampling performance model RSM_nm is generated from the sampled predicted performance data. The evaluation unit 122 evaluates the sampling performance model RSM_nm by calculating an error between the prediction performance model RSM_est and the sampling performance model RSM_nm. The measurement point set updating unit 123 uses an optimization algorithm to update the measurement point set by executing an optimization algorithm that calculates the measurement point set so as to minimize the error. Unlike the above-described embodiment using the genetic algorithm, the number of measurement points is not changed by updating the measurement point set. As the optimization algorithm, a known steepest descent method, least square method, simplex method, car marker method, or the like can be used.

  The convergence determination unit 124 determines whether or not the error has converged. If not converged, a sampling performance model is again generated for the updated measurement point set, and the measurement point set is evaluated. If it has converged, the measurement point number updating unit 125 updates the number of measurement points included in the measurement point set. For the measurement point set including the updated number of measurement points, the processing by the initial measurement point set setting unit 113, the sampling performance model generation unit 121, the evaluation unit 122, the measurement point set update unit 123, and the convergence determination unit 124 is repeated. .

  When the number of measurement points included in the measurement point set reaches a predetermined value in the measurement point number update unit 125, the optimum measurement point selection unit 126 sets a minimum value for the error in the measured measurement point set. A measurement point set is selected, and the value of the control parameter defined by the selected measurement point set is set as an optimized measurement point.

  In the above-described embodiment using the genetic algorithm, the arrangement and number of measurement points can be optimized simultaneously by incorporating a DNA element indicating invalidation of measurement points into a DNA individual. Accordingly, the optimum measurement point can be determined at the same time as the evaluation of the DNA individuals of a predetermined number of generations is completed or the optimization is judged to have converged. On the other hand, with the optimization algorithm used in the embodiment shown in FIG. 18, the arrangement and number of measurement points cannot be optimized simultaneously. The optimization algorithm obtains the arrangement of measurement points that minimizes the error for a predetermined number of measurement points. However, by repeating such an optimization algorithm while changing the number of measurement points, not only the arrangement of measurement points but also the number can be optimized.

  A detailed algorithm executed by the CPU of the computer, more specifically, executed by the optimization unit 115 of FIG. 18 according to the embodiment shown in FIG. 18 will be described with reference to FIGS. 19 and 20.

  Step S111 is the same as step S13 of FIG. The aforementioned predicted performance model RSM_est is read from the storage device, and the value Av_r (r = 1 to va) for the control parameter A and the value Bv_s (s = 1 to vb) for the control parameter B with respect to the predicted performance model RSM_est. Sampling is performed to obtain predicted performance data Vest_rs. This is used for evaluation described later.

  In step S112, an initial value pa_s is set to ia indicating the number of measurement points for the control parameter A, and in step S113, an initial value pb_s is set to ib indicating the number of measurement points for the control parameter B. The initial value can be set to an arbitrary value (for example, 3). In step S114, ia measurement points Aia_ib_n (n = 1 to ia) for the control parameter A and ib measurement points Bia_ib_m (m = 1 to ib) for the control parameter B are initialized with random numbers. As described above, Aia_ib_n and Bia_ib_m represent a set of measurement points (measurement point set) for the control parameters A and B, respectively. Aia_ib_n (0) and Bia_ib_m (0) are shown in FIG. 21. In this example, three measurement points Aia_ib_1 to Aia_ib_3 (ia = 3) are included in the measurement point set Aia_ib_n (0) for the control parameter A. Values representing points are set, and these are initialized with random numbers. The same applies to the control parameter B.

  In step S115, an optimization algorithm is executed. The optimization algorithm is shown in FIG. 20. In step S121, the iteration parameter i (indicating a value for identifying the measurement point set) is initialized to zero.

  Steps S122 to S126 are processes similar to steps S23 to S27 of FIG. In step S122, the predicted performance data Yan_bm is obtained by sampling the predicted performance model Rsm_est in Aia_ib_n (n = 1 to ia) and Bia_ib_m (m = 1 to ib). Here, the total number of measurement points is ia × ib, and this is represented by Ns_nm. Therefore, Ns_nm pieces of predicted performance data Yan_bm are obtained.

  In step S123, a sampling performance model RSM_nm is generated based on the sampled predicted performance data Yan_bm. As described above, the predicted performance model RSM_est is obtained by calculating a relationship between a control parameter and performance data corresponding thereto by a computer. Therefore, the sampling performance model RSM_nm can be generated from the control parameters Aia_ib_n and Bia_ib_m and the corresponding performance data Yan_bm by a method similar to the method for generating the predicted performance model. Sampling performance model RSM_mn can be said to represent engine performance characteristics that can be realized at measurement points represented by the values of control parameters A and B of the current (i-th) measurement point set.

  In step S124, the sampling performance model RSM_nm is sampled using the same control parameter values Av_r and Bv_s as in step S111 described above, and sampling performance data Vnm_rs is obtained. The reason why the same control parameter values Av_r and Bv_s as in step S111 described above are used is to compare the predicted performance model RSM_est and the sampling performance model RSM_nm under the same conditions.

In step S125, a weight function (penalty) W is calculated. This can be done according to the previously mentioned equation W = Ns_nm / Nmax, alternatively W = (Ns_nm) ² / Nmax ² may be used. Here, since the maximum number of measurement points for the control parameter A is pa and the maximum number of measurement points for the control parameter B is pb (these are predetermined), Nmax = pa × pb.

  Instead of calculating the weight function W according to the above formula, a table of the weight function W as shown in FIG. 11 is stored in the storage device, and the weight function W is obtained by referring to the table based on Ns_nm. Also good. Thus, as the number of measurement points Ns_nm increases, the value of the weight W is increased so that the evaluation is lowered.

In step S126, using the calculated weight function W, the evaluation function value Ersm_nm of the sampling performance model is calculated according to the following equation.

  This equation is the same as the equation for the evaluation function described above, and the evaluation function value Ersm_nm (i) is a sampling obtained from the values Aia_ib_n (i) and Bia_ib_m (i) of the control parameters A and B of the current measurement point set. The error of the performance model RSM_nm with respect to the predicted performance model RSM_est is reflected. The smaller the error value, the higher the evaluation of the measurement points Aia_ib_n (i) and Bia_ib_m (i) that are the basis of the sampling performance model Rsm_nm. The evaluation function value Ersm_nm (i) calculated in this way is stored in a storage device such as a memory in association with the corresponding measurement point sets Aia_ib_n (i) and Bia_ib_m (i).

  In step S127, Aia_ib_n (i + 1) and Bia_ib_m (i + 1) are calculated using a predetermined optimization algorithm, and these are used as Aia_ib_n (i) and Bia_ib_m (i) in the next evaluation process (ie, Aia_ib_ (I) and Bia_ib_m (i) are updated).

  As the optimization algorithm, any known algorithm capable of calculating Aia_ib_n (i + 1) and Bia_ib_m (i + 1) that minimize the evaluation function value (error) Ersm_nm can be used, for example, the steepest descent method Algorithms based on least square method, simplex method, Karmarkar method, and the like can be used. For control parameter A, Aia_ib_n (i + 1) is generally determined by Aia_ib_n (i) + K × Ersm_nm (i), but the difference between these algorithms is how the coefficient K is calculated. However, both algorithms are the same in that the coefficient K is calculated so as to minimize the error Ersm_nm, and Aia_ib_n (i + 1) is calculated using the coefficient K. Therefore, any algorithm may be used.

Here, the case where the steepest descent method and the least square method are used will be described. The following shows an arithmetic expression when the steepest descent method is used. Expression (1-1) represents an expression for calculating Aia_ib_n (i + 1) for the control parameter A, and Expression (1-2) represents an expression for calculating Bia_ib_m (i + 1) for the control parameter B. Here, KP _A and KP _B correspond to the coefficient K described above. μ _A and μ _B indicate update gains and can be set in advance.

Next, an arithmetic expression in the case of using the sequential least square method is shown. Here, KPn (i) and KPm (i) correspond to the coefficients described above and are calculated sequentially. P represents an update gain and can be set in advance. Λ represents a weight parameter (0 <λ <1).

  In step S128, convergence determination is performed. That is, if the difference between the current value Ersm_nm (i) of the error and the previous value Ersm_nm (i−1) is equal to or greater than the predetermined value Drsm_conv, it is determined that the optimization has not converged, and the parameter i is repeatedly incremented by 1 in step S129. Then, steps S122 to S128 are repeated for Aia_ib_n (i + 1) and Bia_ib_m (i + 1) (that is, updated Aia_ib_n (i) and Bia_ib_m (i)) calculated in step S127. This repetition is performed until the repetition parameter i reaches a predetermined value N in step S130. In this way, evaluation is performed for each of the N measurement point sets while maintaining the predetermined number of measurement points.

  However, even before i reaches N, if the difference between the current value Ersm_nm (i) and the previous value Ersm_nm (i−1) of the error is smaller than the predetermined value Drsm_conv in step S128, it is determined that the optimization has converged, Returning to step S116 of the process of FIG.

  In step S116, the number of measurement points ib of the control parameter B is incremented by one. If the number of measurement points ib has not reached pb (the maximum number of measurement points for the control parameter B as described above) in step S117, the measurement points of steps S114 and S115 are again maintained while maintaining the number of measurement points of the control parameter A. Set initialization and optimization takes place. That is, only for the control parameter B, the number of measurement points is increased by one, the value of the increased number of measurement points is initialized with a random number (S114), and then optimized (S115). The control parameter A is initialized again with the same number of measurement points (S114) and then optimized (S115).

  If the number of measurement points ib reaches pb in step S117, the process proceeds to step S118, and the number of measurement points ia of the control parameter A is incremented by one. In step S119, if the number of measurement points ia has not reached pa (the maximum number of measurement points for the control parameter A as described above), the control parameter B is maintained for the control parameter A while maintaining the incremented number of measurement points ia. While the number of measurement points is changed from the initial value pb_s to pb (S113, S116), the measurement point set in steps S114 and S115 is initialized and optimized.

  If the number of measurement points ia reaches pa in step S119, the optimum measurement point is determined in step S120. In step S120, Aia_ib_n (i) corresponding to the evaluation function value Ersm_nm (i) having the smallest value among the evaluation function values Ersm_nm (i) calculated in step S126 of FIG. 20 and stored in the storage device. ) And Bia_ib_m (i) are the optimum measurement points. Thus, the optimization is finished.

  In the above-described embodiment, a mode has been described in which not only the arrangement of measurement points but also the number is optimized using known optimization algorithms such as the steepest descent method and the least square method. Alternatively, only the arrangement of measurement points may be optimized. In this case, the number of measurement points as shown in steps S116 and S118 is not required in the process of FIG. For a predetermined number of measurement points, initialization of the measurement point set in steps S114 and S115 and an optimization algorithm therefor may be executed.

  In the embodiment shown in FIG. 18, the designated measurement point determination unit 12 may be provided between the prediction performance unit 11 and the measurement point set initialization unit 113 as in FIG. 4. In this case, as described with reference to FIG. 6, the designated measurement point is determined, and the measurement point set initialization unit 113 sets the value of the measurement point set so as to include the designated measurement point. More specifically, pa_s and pb_s in steps S112 and S113 in FIG. 19 are set to values greater than or equal to the number of designated measurement points, and in step S114, the values of measurement points other than the designated measurement points are initialized with random numbers. Is done. For example, in FIG. 21, Aia_ib_1 to Aia_ib_3 and Bia_ib_1 to Bia_ib_2 are used as values representing designated measurement points, and other Bia_ib_3 is initialized with random numbers. In addition, the measurement point set updating unit 123 updates the measurement point set so as to include the designated measurement point. More specifically, the optimization algorithm used in step S127 of FIG. 20 may calculate Aia_ib_n (i + 1) and Bia_ib_m (i + 1) for measurement points other than the designated measurement point.

  Although the present invention has been described with respect to specific embodiments, the present invention is not limited to such embodiments.

(A) The figure which shows the control parameter condition combination by an experiment design method, (b) The figure which shows the conventional automatic measurement method. The figure which shows the characteristic of engine performance. The figure which shows the bad influence of the measurement point reduction using an experiment design method. The block diagram of the apparatus for calculating the optimal measurement point according to one Example of this invention. The figure which shows an example of the prediction performance model according to one Example of this invention. The figure which shows an example of the designated measurement point according to one Example of this invention. Figure 5 is an optimization process flow according to one embodiment of the present invention. The figure which shows an example of the initial population in a genetic algorithm according to one Example of this invention. Fig. 5 is a flow of an evaluation process in a genetic algorithm according to an embodiment of the present invention. The figure which shows the combination of the value of the control parameter A and the value of the control parameter B in each DNA solid according to one Example of this invention. The figure which shows an example of the table for determining the weighting about an evaluation function according to one Example of this invention. Fig. 4 is a flow of a DNA update process in a genetic algorithm according to an embodiment of the present invention. The figure which shows the rearrangement of the DNA solid according to an evaluation function value in the genetic algorithm according to one Example of this invention. The figure which shows the cross process in a genetic algorithm according to one Example of this invention. The figure which shows the mutation process in a genetic algorithm according to one Example of this invention. The figure for demonstrating the effect of the optimal measurement point determined according to the method of this invention. The figure for demonstrating the effect of the optimal measurement point determined according to the method of this invention. The block diagram of the apparatus for calculating the optimal measurement point according to the other Example of this invention. Fig. 5 is a flow of an optimization process according to another embodiment of the present invention. 6 is a flow diagram illustrating an optimization algorithm of an optimization process according to another embodiment of the present invention. The figure which shows schematically the structure of the measurement group | group according to the other Example of this invention.

Explanation of symbols

11 Prediction performance unit 15, 115 Optimum unit

Claims (12)

A device for optimizing measurement points for measuring performance data for a predetermined control parameter for a control target,
Means for predicting performance characteristics of the control object with respect to the control parameter and calculating predicted performance data using the model of the control object;
Sampling the predicted performance data at a set measurement point, and calculating sampling performance data based on the sampled performance data;
Optimization means for optimizing the arrangement of the measurement points such that an error between the predicted performance data and the sampling performance data is minimized;
A device comprising: The optimization means further optimizes the number of measurement points;
The apparatus of claim 1. further,
A means for performing measurement on the actual control target at the optimized measurement point and obtaining an actual performance characteristic of the control target;
The apparatus according to claim 1 or 2. The optimization of the arrangement of the measurement points is performed by a genetic algorithm.
The apparatus according to claim 1. The optimization of the number of measurement points is performed by a genetic algorithm.
The apparatus according to claim 2. In the process of mutation in the genetic algorithm, a DNA element having a meaning of reducing the number of measurement points is incorporated into a DNA individual.
The apparatus according to claim 5. In the genetic algorithm, when evaluating a DNA individual using an evaluation function, weighting is performed on the evaluation function so that evaluation by the evaluation function decreases as the number of measurement points increases.
Apparatus according to claim 5 or 6. During the selection process in the genetic algorithm, DNA individuals are selected in descending order of evaluation by the evaluation function for each number of measurement points of each DNA individual.
The apparatus according to claim 7. The control object is an internal combustion engine.
The device according to claim 1. Optimization of the arrangement of the measurement points is performed by an optimization algorithm that calculates the arrangement of measurement points for minimizing the error when the number of measurement points is a predetermined value.
The apparatus according to claim 1 or 2. The optimization of the arrangement and the number of the measurement points is performed by further repeating the process according to the optimization algorithm while changing the number of the measurement points, and the arrangement and the number of measurement points that minimize the error as a result of the repetition. Done by asking for
The apparatus according to claim 10. A predetermined evaluation function is used to obtain the arrangement and number of measurement points that minimize the error, and the evaluation function is such that the evaluation by the evaluation function decreases as the number of measurement points increases. Weighted,
The apparatus of claim 11.

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