Adaptive curve fitting

Abstract

A method for adaptive curve fitting is provided. The method includes obtaining a relational data ordered sequence relating an inferred parameter to one or more measurable parameters, fitting a first function to the relational data ordered sequence over a first range of the relational data ordered sequence, determining a measured value of the one or more measurable parameters, using the first function to determine an estimated value of the inferred parameter based on the measured value of the one or more measurable parameters, selecting a second range of the relational data ordered sequence based on the estimated value of the inferred parameter, wherein the second range is shorter than the first range, and fitting a second function to the second range of the relational data ordered sequence over a second range of the relational data ordered sequence.

Description

[0001] ADAPTIVE CURVE FITTING

[0002] TECHNICAL FIELD

[0003] The embodiments described below relate to curve fitting and, more particularly, to adaptive curve fitting.

[0004] BACKGROUND

[0005] One of the applications available in Coriolis flow and density meters is to predict the concentration of one of the components (e.g., Component A) in a two-component mixture (e.g., Components A and B) using a matrix of density at different concentrations of Component A in the mixture and the temperature of the mixture. A two-dimensional curve fit of this ‘density matrix’ is used to calculate the concentration of Component A based on the measured mixture density (from the Coriolis meter) and mixture temperature (from the Coriolis meter or other sensors). The accuracy of the calculated concentration of Component A is strongly dependent on the accuracies of the mixture density and temperature measurement, the density matrix data for the curve fit, and the curve fit algorithm.

[0006] In most cases, the density matrices are empirically determined or are derived using complicated mathematical equations involving very high order polynomials and / or logarithmic, trigonometric, or exponential functions (or some combination thereof). So, finding the ‘inverse’ of this complicated mathematical equation - to calculate the concentration for a measured density and temperature - gets even more complicated. A possible solution is to fit a low order (e.g., 1 to 3) polynomial to the density matrix so that its inverse could be easily, from a computing resources perspective, calculated.

[0007] To improve accuracy of the calculated concentrations using these low order polynomial curve fits, different density matrices are used depending on the expected concentration (of Component A) and temperature of the mixture. The differences between these density matrices are the range of the concentration (of Component A) in the mixture and the mixture temperature. This requires advanced knowledge of the expected concentration (of Component A) in the mixture and temperature of the mixture. Accordingly, there is a need for accurate curve fitting without significant consumption of computing resources, which may be accomplished with adaptive curve fitting.

[0008] SUMMARY

[0009] A method for adaptive curve fitting is provided. According to an embodiment, the method comprises obtaining a relational data ordered sequence relating an inferred parameter to one or more measurable parameters, fitting a first function to the relational data ordered sequence over a first range of the relational data ordered sequence, determining a measured value of the one or more measurable parameters, using the first function to determine an estimated value of the inferred parameter based on the measured value of the one or more measurable parameters, selecting a second range of the relational data ordered sequence based on the estimated value of the inferred parameter, wherein the second range is shorter than the first range, and fitting a second function to the second range of the relational data ordered sequence over the second range of the relational data ordered sequence.

[0010] According to an aspect, an electronics for adaptive curve fitting comprises a processing system configured to perform one or more of the foregoing methods.

[0011] According to an aspect, a device having adaptive curve fitting comprises a transducer and the electronics of above, wherein the transducer is communicatively coupled to the electronics.

[0012] ASPECTS

[0013] According to an aspect, a method for adaptive curve fitting comprises obtaining a relational data ordered sequence relating an inferred parameter to one or more measurable parameters, fitting a first function to the relational data ordered sequence over a first range of the relational data ordered sequence, determining a measured value of the one or more measurable parameters, using the first function to determine an estimated value of the inferred parameter based on the measured value of the one or more measurable parameters, selecting a second range of the relational data ordered sequence based on the estimated value of the inferred parameter, wherein the second range is shorter than the first range, and fitting a second function to the second range of the relational data ordered sequence over the second range of the relational data ordered sequence.

[0014] Preferably, the method further comprises using the second function to determine a second estimated value of the inferred parameter based on the measured value of the one or more measurable parameters.

[0015] Preferably, the relational data ordered sequence is part of a multi-parameter data ordered sequence comprising a plurality of relational data ordered sequences indexed by at least one of the one or more measurable parameters.

[0016] Preferably, the method further comprises determining a central tendency value of the inferred parameter based on a plurality of values of the inferred parameter about a coordinate of the second estimated value of the inferred parameter and a measured value of an indexing measurable parameter.

[0017] Preferably, determining the central tendency value of the inferred parameter based on the plurality of values of the inferred parameter about the coordinate of the second estimated value of the inferred parameter and the measured value of the indexing measurable parameter comprises determining the central tendency value based on a subset of the relational data ordered sequence, the subset comprising a plurality of values of the inferred parameter and a plurality of values of the indexing measurable parameter.

[0018] Preferably, the relational data ordered sequence comprises the inferred parameter and two measurable parameters, wherein one of the two measurable parameters is the indexing measurable parameter.

[0019] Preferably, determining the central tendency value based on the plurality of values of the inferred parameter about the coordinate of the second estimated value of the inferred parameter and the measured value of the indexing measurable parameter comprises selecting a plurality of ordered pairs of the inferred parameter and the indexing measurable parameter about the coordinate of the second estimated value of the inferred parameter and the measured value of the indexing measurable parameter and determining the central tendency value of the inferred parameter based on values of the inferred parameter in the selected plurality of ordered pairs.

[0020] Preferably, determining the central tendency value based on the values of the inferred parameter in the selected plurality of ordered pairs comprises determining a plurality of central tendency values of the inferred parameter based on a plurality of subsets of the selected plurality of ordered pairs of the inferred parameter and the indexing measurable parameter and determining a central tendency value of the plurality of central tendency values of the inferred parameter.

[0021] Preferably, fitting the first function to the relational data ordered sequence over the first range of the relational data ordered sequence comprises fitting a high order polynomial function to the relational data ordered sequence over the first range of the two-parameter data ordered sequence.

[0022] Preferably, fitting the first function to the relational data ordered sequence comprises fitting the first function to the relational data ordered sequence over an entire range of the relational data ordered sequence.

[0023] Preferably, obtaining the relational data ordered sequence comprises obtaining a two-parameter data ordered sequence that relates the inferred parameter to one of the one or more measurable values.

[0024] Preferably, another of the one or more measurable values are indexing parameters used to obtain the two-parameter data ordered sequence that relates the inferred parameter to the one of the one or more measurable values.

[0025] Preferably, the inferred parameter is concentration and the one or more measurable parameters comprises density and temperature.

[0026] According to an aspect, an electronics for adaptive curve fitting comprises a processing system configured to perform one or more of the foregoing methods.

[0027] According to an aspect, a device having adaptive curve fitting comprises a transducer and the electronics of above, wherein the transducer is communicatively coupled to the electronics.

[0028] BRIEF DESCRIPTION OF THE DRAWINGS

[0029] The same reference number represents the same element on all drawings. It should be understood that the drawings are not necessarily to scale.

[0030] FIG. 1 shows a device 100 for adaptive curve fitting.

[0031] FIG. 2 shows a graph 200 showing a wide-range curve fit for an adaptive curve fit. FIG. 3 shows a graph 300 showing a narrow-range curve fit for an adaptive curve fit.

[0032] FIG. 4 shows a graph 400 shows another narrow-range curve fit for an adaptive curve fit.

[0033] FIG. 5 shows a graph 500 illustrating an averaging grid selection for an adaptive curve fit.

[0034] FIG. 6 shows a concentration-temperature grid 600 for adaptive curve fitting.

[0035] FIG. 7 shows the concentration-temperature grid 600 described with reference to FIG. 6 in nine depictions with different averaging configurations.

[0036] FIG. 8 shows a graph 800 illustrating another adaptive curve fit.

[0037] FIG. 9 shows a method 900 for an adaptive curve fit.

[0038] FIG. 10 shows a vibratory meter 5 configured to measure parameters for an adaptive curve fit.

[0039] FIG. 11 shows a meter electronics 20 configured to estimate and determine a steady state condition of a process.

[0040] DETAILED DESCRIPTION

[0041] FIGS. 1-11 and the following description depict specific examples to teach those skilled in the art how to make and use the best mode of embodiments of adaptive curve fitting. For the purpose of teaching inventive principles, some conventional aspects have been simplified or omitted. Those skilled in the art will appreciate variations from these examples that fall within the scope of the present description. Those skilled in the art will appreciate that the features described below can be combined in various ways to form multiple variations of adaptive curve fitting. As a result, the embodiments described below are not limited to the specific examples described below, but only by the claims and their equivalents.

[0042] FIG. 1 shows a device 100 for adaptive curve fitting. As shown in FIG. 1, the device 100 is comprised of a transducer 120 and an electronics 140. The transducer 120 is communicatively coupled to the electronics 140. The device 100 may be configured to measure and / or control one or more parameters of an object or event. Examples of an object include a mechanical device, a fluid flow, etc. An event may be a transitory change in an object that has, for example, event related parameters such as time- duration, intensity, etc. It may be desirable to determine inferred parameter values from the measured values. Accordingly, the device 100 may be configured to determine the inferred parameter values form the measured values.

[0043] In the example of FIG. 1, a fluid flow is shown as entering and exiting the device 100 with arrows. The device 100 may therefore be a fluid flow device, such as the vibratory meter 5 described with reference to FIGS. 10 and 11, although any suitable device may be employed. The device 100 is configured to measure a fluid flow parameter to obtain a time-series of fluid flow parameter values. The device 100 may additionally be configured to store and / or process the time-series of fluid flow parameter values.

[0044] The transducer 120 may be any suitable transducer that senses and / or controls a parameter of an object and / or event. As shown in FIG. 1, the transducer 120 is configured to sense and / or control a fluid flow parameter. For example, the transducer 120 may sense a density, pressure, temperature, viscosity, refractive index, flow rate, such as mass or volume flow rate, velocity, salinity, and / or the like, of the fluid flow. Additionally, or alternatively, the transducer 120 may control a pressure, temperature, flow rate, and / or the like, of the fluid flow. Accordingly, the transducer 120 may provide and / or send one or more signals to and / or from the electronics 140. The one or more signals may be indicative, such as proportional to, the sensed and / or controlled fluid flow parameter.

[0045] The electronics 140 is configured to send and / or receive the one or more signals to and / or from the transducer 120. Although not shown, the electronics 140 may include a processing system having one or more processors including and / or coupled to one or more memories. The electronics 140 may also include signal conditioning circuits configured to convert the one or more signals provided to the electronics 140 into digital data. Additionally, or alternatively, the electronics 140 may include signal generators, amplifiers, and / or the like that convert one or more digital signals, such as a digital set point value, to a signal that is provided to the transducer 120.

[0046] The one or more processors may be configured to process the one or more digital signals so as to determine the time- series of fluid flow parameter values. For example, where the transducer 120 is a density transducer, a voltage signal provided by the transducer 120 may be proportional to a density of the fluid flow. The signal conditioning circuit may convert the signal to a digital signal that is, for example, proportional to the density of the fluid flow. The one or more memories may include calibration factors that scale the digital signal to a density value. Accordingly, the electronics 140 may generate fluid flow parameter values of the fluid flow parameter.

[0047] The generated time fluid flow parameter values may be any suitable values, such as, for example, density, mass flow rate, viscosity, temperature, and / or the like. The generated time-series of fluid flow parameter values may be measured fluid flow parameter values. The time-series of fluid flow parameters may be used to determine parameter values, such as a concentration, gas volume ratio, volume flow rate, fluid flow velocity, etc. values. The parameter values may be fluid flow parameter values, or other values that are not necessarily fluid flow parameters. The apparent parameter values may be referred to as apparent parameter values. The apparent parameter values may be obtained from the generated fluid flow parameter values by using a relationship between a given fluid flow parameter and a given apparent parameter.

[0048] For example, if it is known that the device 100 is measuring a first parameter and it is known that the first parameter has a one-to-one relationship (e.g., an injective or bijective function) with a corresponding parameter, then a first parameter value can be used to determine a corresponding parameter value. One exemplary relationship or function is between density and concentration of a two-component fluid. For example, a water-alcohol mixture at a given temperature will have a one-to-one relationship between density and concentration. Other two-component fluids have similar one-to-one relationships.

[0049] It should be appreciated that as the temperature of the two-component fluid varies, the relationship between the density and the concentration will change. Accordingly, the various relationships between the density and concentration of a given two-component fluid can be stored as data ordered sequence of density, concentration, and temperature values. Additionally, as can be appreciated from the foregoing discussion, the data ordered sequence of density, concentration, and temperature values can be indexed by temperature such that a data ordered sequence of ordered pairs can be obtained that reflect a one-to-one relationship or function between density and concentration. Accordingly, if a density value and a temperature value of a two-component fluid are known and correspond to a particular ordered pair of density and concentration values in a two-parameter data ordered sequence corresponding to the two-component fluid at the measured temperature, then a concentration value of the two-component fluid can be determined. Additionally, if the density value lies between two density values of two ordered pairs of density and concentration values in the two-parameter data ordered sequence corresponding to the two-component fluid at the measured temperature, then various analytical techniques, such as, for example, a curve fit may be employed to estimate a concentration value from the density value. The following discusses a curve fitting technique that employs all of the ordered pairs of the two- parameter data ordered sequence for a given temperature.

[0050] Curve fitting over a wide range

[0051] FIG. 2 shows a graph 200 showing a wide-range curve fit for an adaptive curve fit. As shown in FIG. 2, the graph 200 includes a concentration axis 210 and a density axis 220 that are respectively shown as a unitless percentage and in units of grams-per- cubic centimeter (g / cc). The concentration axis 210 ranges from zero to 100 percent and the density axis ranges from 0.80 to 1.00 g / cc, although any suitable ranges may be employed. Also shown is a two-parameter data ordered sequence 230 that relate density values to concentration values, although any suitable parameter and / or values may be employed. Also shown is a curve 240 fit to the two-parameter data ordered sequence 230.

[0052] The two-parameter data ordered sequence 230 is shown as a series of dots representing coordinates of each ordered pair of the two-parameter data ordered sequence. It should be appreciated that the two-parameter data ordered sequence 230 may be a type of multi -parameter data ordered sequence or relational data ordered sequence. That is, a relational data ordered sequence that includes and relates only two parameters may be referred to as a type of relational data ordered sequence. A relational data ordered sequence relates data of a type to other data of a type via a relationship, such as a tendency, trend, function, etc. When more than two parameters are included and related, then the term multi-parameter data ordered sequence may be employed.

[0053] As can be appreciated, the two-parameter data ordered sequence 230 and the curve 240 extend over the entire range of concentration values from zero to 100 percent. Accordingly, the curve 240 may be referred to as a curve fit of a wide-range concentration-density matrix or, in more general terms, a curve fit of a wide-range two- parameter data ordered sequence. Regardless of terminology, the curve 240 can be used to estimate a concentration value from a density value, such as a measured density value.

[0054] To illustrate the estimation of the concentration value, also shown are two measured density values. The first measured density value is 0.975 g / cc and the second measured density value is 0.85 g / cc, although any density value or values can be employed. The two measured density values are proximate dashed lines that extend to and intersect with the curve 240 at two points of intersection. Two additional dashed lines extend from the two points of intersection to the concentration axis 210. Where these two additional dashed lines intersect with the concentration axis 210 correspond to the two measured density values. Accordingly, as shown in FIG. 2, the measured density value of 0.975 g / cc corresponds to an estimated concentration value of 20.576 percent and the measured density value of 0.85 percent corresponds to an estimated concentration value of 82.852 percent.

[0055] The curve 240 may be a low order polynomial fit. Accordingly, as can be appreciated from comparing the two-parameter data ordered sequence 230 and the curve 240, an estimated concentration value corresponding to a given density value may have an error that can be significant depending on the desired level of accuracy for an estimated concentration value. For example, the concentration-density ordered-pairs about a density value of 0.90 g / cc likely do not improve, and may decrease, the accuracy of the estimated concentration values of 82.852 and 20.576 percent. It is possible to improve an accuracy of the curve 240 by employing higher order polynomial, exponential fits, or the like, but these techniques can require a significant amount of computing resources that would be detrimental to other processes on, for example, legacy systems, embedded systems, real-time signal processing, etc. The following provides for more accurate curve fitting without consuming an undesirable amount of computing resources.

[0056] Adaptive curve fit

[0057] FIG. 3 shows a graph 300 showing a narrow-range curve fit for an adaptive curve fit. As shown in FIG. 3, the graph 300 includes the concentration axis 210 and the density axis 220 described with reference to FIG. 2. Also shown is a narrow-range two- parameter data ordered sequence 330 that comprises a portion of the two-parameter data ordered sequence 230 shown in FIG. 2. The narrow-range two-parameter data ordered sequence 330 have density values proximate the density value of 0.975 g / cc. As shown in FIG. 3, the narrow-range two-parameter data ordered sequence 330 is comprised of four concentration-density ordered pairs, which is significantly fewer than the number of density concentration ordered pairs in the two-parameter data ordered sequence 230 of FIG. 2. Accordingly, the narrow -range two-parameter data ordered sequence 330 may be referred to as a narrow-range concentration-density matrix or, in more general terms, a narrow-range two-parameter data ordered sequence.

[0058] A curve fit to the narrow-range two-parameter data ordered sequence 330 results in a curve 340 shown in FIG. 3. A dashed line associated with the density value of 0.975 g / cc extends from the density axis 220 to the curve 340. A dashed line from this intersection to the concentration axis 210 indicates that an estimated concentration value of 18.655 percent corresponds to the density value of 0.975 g / cc, which is different than the estimated concentration value of 20.576 percent determined from FIG. 2.

[0059] The curve 340 may be a low order polynomial curve, similar to the curve 240 shown in FIG. 2. However, because the curve 340 is fit to fewer density concentration ordered pairs, the curve 340 will provide more accurate estimated concentration values for density values within the narrow-range two-parameter data ordered sequence 330. Accordingly, it will be appreciated that the estimated concentration value of 18.655 percent shown in FIG. 3 is more accurate than the estimated concentration of 20.576 percent shown in FIG. 2. Additionally, because the curve 340 is a low order polynomial fitted to fewer concentration-density ordered pairs, the computing resources required to obtain the curve 340 are less than that to obtain the curve 240 described with reference to FIG. 2. Similar advantages can be obtained at other density values when a curve fit is made to other narrow-range concentration-density matrices or, in more general terms, narrow-range two-parameter data ordered sequence, as the following discussion illustrates.

[0060] FIG. 4 shows a graph 400 showing another narrow-range curve fit for an adaptive curve fit. As shown in FIG. 4, the graph 400 includes the concentration axis 210 and the density axis 220 described with reference to FIG. 2. Also shown is a narrow-range two-parameter data ordered sequence 430 that comprises a portion of the two-parameter data ordered sequence 230 shown in FIG. 2. The narrow-range two- parameter data ordered sequence 430 has density values proximate the density value of 0.85 g / cc. As shown in FIG. 4, the narrow-range two-parameter data ordered sequence 430 is comprised of four concentration-density ordered pairs, which is significantly fewer than the number of density concentration ordered pairs in the two-parameter data ordered sequence 230 of FIG. 2. Accordingly, the narrow-range two-parameter data ordered sequence 430 may be referred to as a narrow-range concentration-density matrix or, in more general terms, a narrow-range two-parameter data ordered sequence. A curve fit to the narrow-range two-parameter data ordered sequence 430 results in a curve 440 shown in FIG. 4. A dashed line associated with the density value of 0.85 g / cc extends from the density axis 220 to the curve 440. A dashed line from this intersection to the concentration axis 210 indicates that an estimated concentration value of 83.255 percent corresponds to the density value of 0.85 g / cc, which is different than the estimated concentration value of 82.852 percent determined from FIG. 2.

[0061] The curve 440 may be a low order polynomial curve, similar to the curve 240 shown in FIG. 2. However, because the curve 440 is fit to fewer density concentration ordered pairs, the curve 440 will provide more accurate estimated concentration values for density values in the range of the narrow-range two-parameter data ordered sequence 430. Accordingly, it will be appreciated that the estimated concentration value of 83.255 percent shown in FIG. 4 is more accurate than the estimated concentration of 82.852 percent shown in FIG. 2. Additionally, because the curve 440 is a low order polynomial fitted to fewer concentration-density ordered pairs, the computer resources required to obtain the curve 440 are less than that to obtain the curve 240.

[0062] Selection and iterations of the narrow range curve fitting The foregoing steps of selecting a narrow -range two-parameter data ordered sequence and fitting a function to the narrow-range two-parameter data ordered sequence may be performed iteratively. For example, in general terms, a first function may be fit to a wide-range two-parameter data ordered sequence. The first function may be referred to as a wide-range function that relates a dependent variable to an independent variable. An estimated dependent variable value may be determined from the wide-range function based on a measured density value. A first narrow-range two-parameter data ordered sequence may be selected based on the estimated dependent variable value, as will be described in more detail below. A function may be fit to the first narrow-range two-parameter data ordered sequence. This function may be referred to as a first narrow -range function. A second estimated dependent variable value may be determined based on the narrow-range function and the measured independent variable value. The estimated dependent variable value may be used to select a second narrow-range two-parameter data ordered sequence.

[0063] A function may be fit to the second narrow-range two-parameter data ordered sequence. This function may be referred to as a second narrow -range function. The selection, curve-fitting, and estimating based on the narrow-range two-parameter data ordered sequence may be performed iteratively until a change between dependent variable values in successive iterations is below an acceptable value (e.g., less than 0.01 or 1 percent). The same concept may be extended in two dimensions (e.g., concentration and temperature) to account for changes in, for example, in parameters, such as, for example, temperature, as the following discussion explains.

[0064] The foregoing discusses narrow -range two-parameter data ordered sequence that are of wide-range two-parameter data ordered sequence comprising ordered pairs of real numbers. It should be noted that the wide-range two-parameter data ordered sequence may be selected based on a third parameter. For example, in the above discussed concentration-density matrices, the two-parameter data ordered sequence 230 may be selected based on a measured temperature. By way of illustration, a measured temperature value may be compared to a plurality of concentration-density wide-range two-parameter data ordered sequence, each of which is associated with a temperature value. The wide-range two-parameter data ordered sequence associated with a temperature value most close to the measured temperature may be selected for curve fitting. As discussed above, the narrow-range two-parameter data ordered sequence is based only on the selected two-parameter data ordered sequence. A concentration value is estimated from a curve fit to the narrow-range two-parameter data ordered sequence of the selected two-parameter data ordered sequence.

[0065] As can be appreciated, the estimated concentration value obtained from only the selected two-parameter data ordered sequence may have an error that corresponds to a difference between the measured temperature and the temperature associated with the selected two-parameter data ordered sequence. In addition, each of a plurality of measured temperature values may have a measurement error that causes seemingly incongruent changes in the concentration values when a temperature changes during a process. Accordingly, it may be beneficial to accommodate for erroneous temperature measurements when the temperature changes by, for example, smoothing out any changes in the estimated concentration values.

[0066] An estimated concentration value based on a two-dimensional grid

[0067] FIG. 5 shows a graph 500 illustrating an averaging grid selection for an adaptive curve fit. As shown in FIG. 5, the graph 500 includes a concentration axis C, a density axis p, and a temperature axis T. The graph 500 is conceptual and therefore does not have a scale or units. Also shown in FIG. 5 is a concentration-density-temperature matrix 510 that may be comprised of a plurality of a concentration-density-temperature 3-tuplcs. A selection square 520 surrounds and encompasses a portion of the plurality of the concentration-density-temperature 3 -tuples of the concentration-density-temperature matrix 510. The selection square 520 is centered about a measured temperature- estimated concentration coordinate 530.

[0068] The measured temperature-estimated concentration coordinate 530 of FIG. 5 is obtained as described above with reference to FIGS. 2-4. That is, the measured temperature is used to select a concentration-density ordered pair data ordered sequence of the concentration-density-temperature matrix 510. A curve is fit to the selected concentration-density ordered pair data ordered sequence, including a wide-range fit and narrow-range fit. An estimated concentration value is obtained from the resulting curve fits. The measured temperature value and the estimated concentration value are then used as the measured temperature-estimated concentration coordinate 530 to center the selection square 520 about the portion of the plurality of the concentration-density- temperature 3-tuples of the concentration-density-temperature matrix 510.

[0069] It should be appreciated that the selected portion of the plurality of concentration-density-temperature 3-tuples may be a narrow-range concentrationdensity-temperature matrix that could be used to determine an aggregate estimated concentration value. The term “aggregate” may refer to an average, mean, mode, and / or the like of the concentration values or other aggregated concentration values in the narrow-range concentration-density-temperature matrix. Accordingly, the following explains that the selected portion of the plurality of the concentration-density- temperature 3-tuples of the concentration-density-temperature matrix 510 can be used to estimate a new concentration value, which may be referred to as the aggregate estimated concentration value to distinguish the above estimated concentration value during a given estimation iteration.

[0070] FIG. 6 shows a concentration-temperature grid 600 for adaptive curve fitting. As shown in FIG. 6, the concentration-temperature grid 600 includes a grid of concentrations and temperatures that range from Cl to C6 and T1 to T6 that correspond to the selection square 520 described with reference to FIG. 5. The concentrationtemperature grid 600 is comprised of concentration and temperature lines illustrated by grid lines. Each intersection of the lines represents a coordinate of a given concentration-density ordered pair within the selection square 520 described above with reference to FIG. 5. Also shown in FIG. 6 is the measured temperature-estimated concentration coordinate 530 described with reference to FIG. 5. An averaging square 630 is also illustrated, which surrounds a portion of the concentration-density ordered pairs of FIG. 6.

[0071] The averaging square 630 shows which of the concentration-density ordered pairs are used to calculate an average concentration value. As shown in FIG. 6, the averaging square 630 surrounds a 4x4 grid of concentration-density ordered pairs. The averaging square 630 is smaller (i.e., smaller dimensions) than the selection square 520 shown in FIG. 5. The total number of concentration-density ordered pairs to be averaged is sixteen. The concentration values of the sixteen concentration-density ordered pairs are summed and then divided by 16 to obtain the average estimated concentration value of the values in the averaging square 630 shown in FIG. 6.

[0072] As can be appreciated, a new average estimated concentration value may be obtained from a displaced averaging square immediately to the left or right or up or down of the averaging square 630 shown in FIG. 6 and would include 3x4 or 4x3 of the concentration-density ordered pairs in the averaging square 630 shown in FIG 6. Similarly, a displaced averaging square immediately left and up, right and up, right and down, or left and down will include 3x3 of the concentration-density ordered pairs in the averaging square 630 shown in FIG. 6. As a result, a new measured temperature- estimated concentration coordinate will result in a new average estimated concentration value that is weighted by a plurality of concentration values of the concentration-density ordered pairs in the averaging square 630.

[0073] As can also be appreciated, an average of these different “weighted” average values, or in more general terms an aggregate value of other aggregate values, could smooth out any noise induced by variations in the measured temperature and / or estimated concentration values during a changing process, as the following explains in more detail.

[0074] FIG. 7 shows the concentration-temperature grid 600 described with reference to FIG. 6 in nine depictions with different averaging configurations. As shown in FIG. 7, the concentration-temperature grid 600 includes an averaging square in nine different positions depicted by a dashed-line square. The nine different positions indicate the 4x4 concentration-temperature ordered pairs that are used to calculate an average estimated concentration value for each of the nine averaging configurations. Accordingly, nine average estimated concentration values are obtained, which may then be used to calculate an aggregate estimated concentration value of the narrow-range concentrationtemperature matrix represented by the concentration-temperature grid 600. The aggregate estimated concentration value can be used to determine a new narrow range two-parameter data ordered sequence as described above with reference to FIGS. 3 and 4.

[0075] The averaging routine may be performed by dropping two exterior values from each of the concentration and temperature range of the concentration-temperature grid 600. For example, the center averaging square in the concentration-temperature grid 600 may be obtained by indexing ±2 indices, or number of rows and columns, about the measured temperature-estimated concentration coordinate 530 (shown in FIG. 5) rather than the ±2 indices of the concentration-temperature grid 600. The indices may be incremented by ±1 to obtain the other averaging squares. The concentration values in each averaging square may be averaged to obtain an averaged concentration value of a given averaging square, which may be referred to as an averaging square value. Other shapes may be employed, and therefore a more generic term may be employed, such as a grid subset average may be employed. An average of the grid subset average may be determined. Although an average value is determined, any suitable central tendency value may be employed, including a mean, median, mode, etc. It should be appreciated that the foregoing determines a sequence of ordered pairs or ordered pair data ordered sequence that relates a concentration of a two- component fluid to a measured density of the two-component fluid. A concentration value can therefore be determined from a corresponding density value. Accordingly, the concentration parameter may be referred to as an inferred parameter whereas the density parameter may be referred to as a measurable parameter. In mathematical function terms, the concentration may be a dependent variable or parameter whereas the density may be an independent variable or parameter. As can also be appreciated, in the concentration-temperature grid 600, both the concentration and temperature axes served as indexing parameters. Accordingly, concentration may be referred to as an indexing inferred parameter and the temperature may be referred to as indexing measurable parameter.

[0076] Additionally, other relational data ordered sequences may be employed. For example, an inferred parameter may be dependent on more than one measurable parameter. Accordingly, as discussed above, a two-parameter data ordered sequence may be a type of relational data ordered sequence within a larger set of data ordered sequences. By way of illustration, the concentration parameter may be dependent on density and temperature. Therefore, the ordered pairs of concentration and density may, for example, be a subset at a constant temperature value of a larger set of data ordered sequences or source data (e.g., contemporaneous calibrated measurements of a plurality of parameters of a fluid and / or a fluid control device and / or system in a given state) of concentration, density, and temperature values.

[0077] It should also be appreciated that the foregoing examples can be generalized to any suitable form of data, expressions, etc. For example, tables, lists, matrices, vectors, sequences, arrays, and / or the like, can be employed. Additionally, any suitable algorithm, mathematical, data, etc., method can be employed to determine the narrowrange relational data ordered sequence and / or an estimated inferred value. As a starting point, as the foregoing indicates, it is understood that a source data, which may be a source relational data ordered sequence, may be a complete high-resolution data sequence or matrix relating various parameter values. From the source data, various narrow range multi -parameter data ordered sequences, such as two-parameter data ordered sequences, may be selected for adaptive curve fitting, as the following explains. Matrices

[0078] The foregoing FIGS. 2 through 7 are directed to data having three parameters from which a two-parameter data ordered sequence is obtained. As can be appreciated, a source data may have more than three parameters. That is, in the context of metrology, the source data may include measured values, such as calibrated measured values, of two or more parameters, including those with greater than three parameters. Although the source data may be comprised of calibrated measured values, the two or more parameters may be characterized as either an inferred and measurable parameter depending on the measurement capabilities of a measurement device or system performing adaptive curve fitting using measured parameter values and the source data. For example, referring to the concentration-density-temperature matrix 510 described with reference to FIG. 5, a measurement device may only be capable of measuring a density and temperature of a two-component fluid. Accordingly, the concentration may be referred to as an inferred parameter and the density and temperature may be referred to as a measurable parameter.

[0079] That said, the foregoing discussed examples can be generalized to a multiparameter source data of unknown parameters. For example, it can be said that a source data can be expressed as a source matrix A of dimension D : Equation [1] where:

[0080] D is the number of dimensions or parameters of the source matrix A. i,j, k, ... , D are indexes of each of the parameters; n, m, o, NDare the number of elements in each dimension; and aijk...D is anelement of the source matrix A.

[0081] As can be appreciated, because each of the parameters in the source data correspond to a dimension of the source matrix, each matrix element aij ...Dcanbe treated as an n-tuplc having D number of elements comprising ordered parameter values.

[0082] By way of illustration, where three parameters arc related to each other, a three- dimensional source matrix A may be used: im,n,o

[0083] 71Luijk i ,k=l- Equation [2] In the case of the concentration-density-temperature source data depicted as the concentration-density-temperature matrix 510 shown in FIG. 5, a concentration-density- temperature source matrix A may be expressed in 7?-tuple form as: Equation [3]

[0084] C are concentration values; p are density values; and

[0085] T are temperature values.

[0086] Accordingly, each element of the concentration-density-temperature source matrix A is a 3-tuple: C, p, T)ijkwhere parameter values are elements of the 3-tuple.

[0087] When the concentration-density-temperature matrix 510 shown in FIG. 5 is considered, it can be appreciated that, in a two-component fluid at a given temperature, the concentration C and density p have a one-to-one relationship. That is, there is a functional relationship between concentration C and density p of a two-component fluid: C = f(p). Accordingly, for a given temperature Tk, the concentration-density relationship may be represented by a sequence of coordinates or, in data and discrete math terminology, a sequence of ordered pairs. Said another way, because there is a one-to-one correspondence between density and concentration for a given temperature as shown in FIG. 5, a 2 X 2 matrix of concentration rows and density columns for a given temperature will be a diagonal matrix.

[0088] Therefore, in the situation where the temperature remains constant over concentration-density ordered pair data ordered sequence, the concentration-density- temperature source matrix A having dimensions of m X n x o can be filtered into sifted source matrix A' having m X o dimensions where each element of the sifted source matrix A' is an ordered pair of concentration and density values. The sifted source matrix A' may be expressed as:

[0089] A' = [(c< )ifc] Equation [4] where: i is an index of rows; and k is an index of columns, each of which corresponds to a temperature value Tk. It should be appreciated that this assumes that the temperature remains constant over the concentration-density ordered pairs. Where the temperature varies over the concentration-density ordered pairs, there may be a functional relationship between the concentration C and the density p and temperature T of a two-component fluid: C = P.D.

[0090] As can be appreciated, for the constant temperature situation, the foregoing sifted source matrix ' can be visualized as the following table:

[0091] Table 1. Temperature and concentration-density ordered pairs matrix

[0092] For the varying temperature situation, 3-tuples of concentration, density, and temperature values (C, p, T') may be employed where the columns are not temperatures but instead are indexed.

[0093] The foregoing table can be “flattened” into a concentration-temperature grid, such as the concentration-temperature grid 600 as shown in FIG. 6, by discarding the density values: Af' = [6^]”^°1(where the superscript prime and subscript f indicates a sifted and flattened version of the source matrix A. Assuming that, for a given temperature value Tk, the concentration remains constant, the foregoing table may be further simplified to the square concentration-temperature grid 600 shown in FIG. 6. It should be appreciated that the concentration-temperature grid of FIG. 6 is a simplification or a special case where a concentration value remains constant over a range of temperatures and a given temperature value remains constant over a range of concentration values.

[0094] For a matrix of concentration-density-temperature 3-tuples, a similar grid may be obtained where each line is indexed by row and columns instead of representing a constant concentration or temperature value. In this latter scenario, an ordered pair of concentration and temperature values would be located at each intersection in the grid. Accordingly, alternative concentration-temperature grids may represent varying concentrations and temperatures by subscripting the numerical indices of the concentration-temperature grid 600 shown in FIG. 6 to indicate that the lines of the concentration-temperature grid represent concentration and temperature indices rather than concentration and temperature values. For example, instead of temperature value lines Tl, T2, an alternative concentration-temperature grid may have lines associated with concentration and temperature indices: TvT2, .... Accordingly, where the lines intersect, the coordinates may have different, for example, temperature values along a temperature line.

[0095] As discussed above, from a metrological standpoint, it can be appreciated that the source matrix A may include a relationship between an inferred parameter and two or more measurable parameters. A relationship between an inferred parameter value and two measurable parameters may be expressed in functional form as afe). In addition, filtering the source matrix A to obtain a sifted source matrix A' can include assigning a constant value to measurable parameters that are not part of the inferred- measurable parameter relationship. For example, for the foregoing relationship — ak^, a sifted source matrix A' may have a parameter with index of I where any parameters having an index greater than I may be assigned a constant value.

[0096] It should also be appreciated that a sifted source matrix B’ may have rows of n- tuples rather than ordered pair. For example, the source matrix B may have four dimensions for four parameters of a concentration C, density p, refractive index RI, and temperature 7 of a three-component fluid. In this example, the concentration C may have an infcrrcd-mcasurablc relationship to density p and refractive index RI for a given temperature I). Therefore, for a given temperature T, the inferred-measurable relationship may be expressed as a 3-tuple (C, p, RI). Accordingly, a sifted matrix B' of the source matrix A may have elements of 3-tuples comprising concentration C, density p, and refractive index RI that includes columns of temperature Tr. Equation [5]

[0097] In addition, if the source matrix A has additional parameters, then the same sifted matrix A' may be obtained by assigning constant values to those additional parameters.

[0098] The foregoing discussion began with an assumption that the source data is represented by a multi-dimensional matrix. The data represented by a multi-dimensional matrix may be generated, stored, searched, analyzed in any suitable form. For example, the multi-dimensional matrix can be stored in a memory as a series of tables, each relating two parameters, where each table is indexed by another parameter value. For example, in the above discussion, a plurality of tables relating concentration and density is indexed by temperature. Therefore, a given table may be associated with a given temperature value. Accordingly, each table may be considered a “slice” of a concentration-density-temperature matrix.

[0099] As can be appreciated, the tables are arranged by rows and columns and therefore the ordered pairs of parameter values may be indexed by row and column numbers. It is appreciated that selection by indices represented by sequential integer values, rather than parameter values, may be faster than filtering multiple rows by values. In addition, selection of individual two-parameter tables for processing and / or analysis may be less resource consuming compared to processing and / or analyzing multiple tables. Accordingly, selection of individual two-parameter tables by a third parameter index may be preferential. Regardless, it may be preferential to utilize a single table or array that includes all of the parameters, which is discussed in the following.

[0100] Single table or array arranged as a matrix expressed as follows:

[0101] Equation [6] where: m is a number of rows of values for parameters; and n is a number of columns of the parameters.

[0102] For example, column 1 may be a concentration value, column 2 may be a density value, and column 3 may be a temperature value. Each row relates parameter values in that the density is an independent variable of a functional relationship that relates a concentration to density. In metrological terms, density may be a measurable parameter and concentration may be an inferred parameter that is related to density. Temperature may also be a measurable parameter and concentration may depend on temperature, but the relationship may be accounted for by selecting a suitable two-parameter data ordered sequence of concentration and density. Accordingly, temperature may be a measurable parameter and a selection parameter to choose a particular concentration-density relationship. By way of illustration, all concentration and density values may be selected where temperature is, for example, 20 °C. That is, all rows of the matrix X that include a temperature value of 20 °C may be selected. In matrix and vector terms, the selection may be based on a comparison of a given row of the matrix X to a vector B of measured values where, in the case of the concentration-density-temperature matrix X, vector V is a temperature value Tmconcatenated to two zeros: V = [0 0 Tm. The zeros of the vector V may be considered null values that are treated as equal to any value in the corresponding column of the matrix X. After a comparison results in p matching rows, the following sifted matrix may be obtained: Equation [7] where: p is the number of rows that matched the vector V and n the number of columns of parameters.

[0103] It should be noted that all of the columns of the matrix X are retained in the sifted matrix X' although all of the rows are not retained. Accordingly, all of the columns of the sifted matrix X' that correspond to measured values (i.e., non-null value columns) of the vector V are also included in the sifted matrix X' .

[0104] The vector V may be viewed as identifying columns that have some meaningful relationship (i.e., relational columns) that is of import and those that are used as selection parameters for indexing relational columns to obtain a particular relational data ordered sequence. For example, the null values of the vector V can be viewed as identifying relational columns and the columns with measured values as identifying non-relational or selection columns. As can be appreciated, the sifted matrix X' may be characterized as having one or more independent variable or measurable parameter columns and a dependent variable or inferred parameter column. In the concentration- den sity-temperature of a two-component fluid example, concentration may be a dependent variable or inferred parameter and the density may be an independent variable or measurable variable.

[0105] Accordingly, a column may be viewed as having a functional relationship with one or more other columns. For example, as explained above, the concentration column x(1is in a functional relationship with the density column xi2where a concentration value depends on a density value. Therefore, a function that is fit to a relational data ordered sequence obtained from the inferred parameter column and one or more measurable parameter columns of the sifted matrix X' may accurately relate a dependent variable such as concentration to an independent variable such as density.

[0106] In addition, subsets of data may be selected from the sifted matrix X' . The selection may be from a sorted version of the sifted matrix X' . For example, the independent variable column of the sifted matrix X' may be sorted by ascending value. Subsequently, a narrow-range relational data ordered sequence may be selected from the sifted matrix X' . The narrow-range relational data ordered sequence may be selected by selecting dependent variable values about an estimated value determined from the function fit to a wide-range relational data ordered sequence. Accordingly, the estimated value of the dependent variable may be used to select values from the dependent variable column of the sorted version of the sifted matrix X' . The independent variable values corresponding to the selected concentration values can also be selected so as to form a sequence of ordered pairs of a dependent variable value and an independent variable value.

[0107] In another example, it may be desirable to form a grid of values to determine, for example, an average of parameter values. By way of illustration, referring to the concentration-density-temperature example, it may be desirable to form from the sifted matrix X' a concentration-temperature grid that bounds an estimated concentration value and measured temperature coordinate. The grid may be formed by selecting other coordinates that are indexed from the sorted version of the sifted matrix X' . For example, the sifted matrix X' may be sorted by the independent parameter column (e.g., density values) in ascending values. Subsequently, values of the dependent variable column proximate the estimated dependent variable value may be selected. Corresponding values from a selection parameter column may also be selected to form a sequence of dependent variable value and selection parameter value coordinates.

[0108] It should also be appreciated that the sequence of ordered pairs relating an inferred parameter or dependent variable to a measurable parameter or independent variable can be obtained from a table of related values. More specifically, the sequence of the ordered pairs of concentration and density values can be obtained from a matrix of related concentration, density, and temperature values. Each list or group of associated concentration, density, and temperature values may be referred to as a 3-tuple of concentration, density, and temperature values. Accordingly, a matrix of concentration, density, and temperature values may alternatively be considered as a sequence of 3-tuples represented by (xltxs, %3), where x±is a concentration value, x2is a density value, and x3is a temperature value.

[0109] To obtain the ordered pair of concentration and density values for a given temperature value, a measured temperature may be compared to the temperature values of the sequence of 3-tuples to select a temperature value for selecting a sub-sequence 3- tuples having a temperature value the same as or close to the measured temperature, a range of temperature values about the measured temperature for a 2-D grid, etc. Once the relational data ordered sequence is selected, the foregoing described curve fitting and estimating of a first parameter value may be employed. For example, if the selected relational data ordered sequence is a wide-range concentration-density data series of a common temperature, then an initial well-fitting function may be employed.

[0110] If the selected two-element data sequence is a narrow-range 2-D grid of temperature and concentrations about a coordinate of an estimated concentration value and a measured temperature value, then the concentration-temperature grid 600 described with reference to FIG. 6 may be employed. For example, the selected two- element data sequence may be a two-element data sequence corresponding to a plurality of temperature values. Accordingly, it should be appreciated that when the concentration-temperature grid 600 is selected from a temperature range of concentration-density ordered pairs illustrated by the concentration-density-temperature matrix 510 as shown in FIGS. 5 and 6, the concentration is also used as a selection parameter. However, since the concentration and temperature are used to obtain a narrow range 2-D grid of ordered pairs, the concentration and temperature may be referred to as narrow range selection parameters, individually or as an ordered pair or parameters.

[0111] Obtaining narrow ranges and grids

[0112] As can be appreciated from the foregoing discussion, a range of a source data may be a number of values of a parameter. For example, a range of concentration could be a number of concentration values in the source data. A wide range of the source data may be a total number of values of a parameter in the source data. Accordingly, a range of the source data may be a number of elements in a dimension of the source data if the source data is expressed in multi-dimensional matrix form. By contrast, a narrow range may be shorter than the wide range of a parameter. For example, if the concentration values are filtered for a given temperature or a range of temperatures, indexed by values or indices, then the filtered concentration values may be referred to as narrow -range concentration values or, in more general terms, a narrow-range of a parameter.

[0113] As can also be appreciated from the foregoing discussion, a parameter may have a bijective relationship with two or more other parameters. That is, a given parameter value must map to a single value of a related parameter. Accordingly, a range of, for example, an ordered pair data sequence may be the number of ordered pairs in the data sequence. If the ordered pair data sequence is all of the ordered pairs in the source data, then the term “wide range ordered pair data ordered sequence” may be used. An example of a wide range ordered pair data ordered sequence is the two-parameter data ordered sequence 230 shown in FIG. 2. If the ordered pair data sequence is a narrow range subset of the wide range ordered pair data ordered sequence, then the term “narrow range ordered pair data ordered sequence” may be used. Examples of a narrow range ordered pair data ordered sequence are the narrow range two-parameter data ordered sequences 330, 430 shown in FIGS. 3 and 4. Similarly, the terms “narrow range n- tuple data ordered sequence” and “wide range n-tuple data ordered sequence” may be used. It should be appreciated that the term narrow range data ordered sequence may be used for the foregoing examples, as well the concentration-temperature grid 600 of FIG. 6.

[0114] From the foregoing discussion, it is apparent that the source data may be a relational data ordered sequence. For example, the source data may include values of various inferred and measurable parameters. The relationships between the inferred and measurable parameters may be one-to-one, although other relationships may be employed. For example, a range of some parameters may be greater than a range of other parameters. Accordingly, some parameters may have injective or surjective relationships. It should be appreciated that the term relational data ordered sequence does not require that all parameters have a known relationship with all other parameters. The term relational data ordered sequence can be interpreted as contemporaneous values of a system, such as fluid or mechanical systems, where the parameters may have some relationship. A relational data ordered sequence that relates an inferred parameter to one or more measurable parameters may be obtained from the source data by selecting values for indexing measurable parameters. For example, a value for temperature, RI, and / or VOS may be chosen to obtain an ordered pair of concentration and density values. It should be appreciated that one or more of the indexing measurable values may be set to a range of values or indices. For example, referring to the concentration-density- temperature matrix 510 discussed above, the temperature may be set to a range, either by temperature values or indices, about a coordinate of the estimated concentration value and the measured temperature value. It should be appreciated that an inferred parameter may be treated as an indexing inferred parameter to form a grid of inferred and measurable parameter values.

[0115] Accordingly, a source data may be stored and subsequently accessed in approximately real time to obtain a relational data ordered sequence that relates an inferred parameter to one or more measurable parameters. For example, a relationship between two parameters may be known, such as a concentration and density parameter of a two-component fluid. Additionally, or alternatively, a relationship may be determined by, for example, statistical means, detecting diagonal matrices for a constant value for indexing measurable parameters, etc. Narrow range relational ordered sequencies may be obtained by setting a value for the measurable parameter to sequence of ordered pairs or n-tuples of an inferred parameter and one or more measurable parameters to obtain a narrow-range curve for curve fitting. To obtain a grid, the inferred parameter and one or more parameters may be set to a range of values.

[0116] The above discusses ordered pairs of a single relationship and generally related concentration to density and temperature. The foregoing can be applied to various applications, such as other inferred and measurable parameters, multiple different ordered pair ordered data series, and / or the like, as the following discussion illustrates. Exemplary alternative parameters

[0117] FIG. 8 shows a graph 800 illustrating another adaptive curve fit. As shown in FIG. 8, the graph 800 includes a Reynolds number axis 810 and a percentage flow error axis 820 that are unitless. The Reynolds number axis 810 ranges from 101to IO7and the percentage flow error axis 820 ranges from -1.2 to 0.0 percent, although any suitable ranges may be employed. The Reynolds number axis 810 corresponds to a ratio of inertial forces to viscous forces and can indicate a degree of laminar flow or turbulent flow. Generally, a relatively low Reynolds number indicates laminar flow and a relatively high number indicates turbulent flow. The percentage flow error axis 820 indicates a difference between an actual mass flow rate and a measured mass flow rate measured by, for example, the vibratory meter 5 described below with reference to FIGS. 10 and 11. A negative percentage flow error value indicates a measured mass flow rate that is less than the actual mass flow rate. As shown in FIG. 8, as the Reynolds number increases, an absolute value of the percentage flow error decreases.

[0118] Also shown is a curve 830 fit to a wide-range of a two-parameter data ordered sequence that relates Reynolds number values to flow rate error values. The wide-range two-parameter data ordered sequence is not shown for clarity. The wide-range two- parameter data ordered sequence may have been obtained from a larger relational data ordered sequence. The curve 830 is shown as having a wave-shaped profile that may be the result of a high-order polynomial fit. As can be appreciated, a high-order polynomial fit may require more resources than a low-order polynomial fit. The wave-shaped profile may also indicate that the high-order polynomial fit, or other computing resource intensive curves, may be required to accurately fit the wide-range two-parameter data ordered sequence selected from the multiple-parameter matrix. For example, a given two-parameter data ordered sequence of ordered pairs relating Reynolds number values to percentage flow rate error values may not have a trend with a simple shape, such as a simple polynomial curve.

[0119] To reduce the computing resources for accurate fits, an adaptive curve fitting approach may be utilized. Accordingly, the graph 800 also includes a plurality of narrow-range two-parameter data ordered sequence 840 comprising a first through fifth narrow-range two-parameter data ordered sequence 840a-840e, although any suitable number of narrow-range two-parameter data ordered sequence may be employed. The plurality of narrow-range two-parameter data ordered sequence may be selected from a multi-parameter matrix that includes a plurality of two-parameter data ordered sequence comprising a series of ordered pairs that relate Reynolds number values to percent flow rate error values. As can be appreciated from FIG. 8, each of the first through fifth narrow-range two-parameter data ordered sequence 840a-840e is offset from the curve 830. Accordingly, it can be seen that a curve fit to each of the first through fifth narrow- range two-parameter data ordered sequence 840a- 840e may provide a more accurate estimated Reynolds number-percentage flow rate error coordinate than one obtained from the curve 830.

[0120] To obtain the more accurate estimated Reynolds number-percentage flow rate error coordinate, the graph 800 also includes a plurality of curves 850 comprising a first through fifth curve 850a-850e that are respectively fit to the first through fifth narrowrange two-parameter data ordered sequence 840a- 840e. As can be appreciated, each of the first through fifth curve 850a-850e are respectively offset from the curve 830 fit to the wide-range two-parameter data ordered sequence. Therefore, an estimated Reynolds number-percentage flow rate error coordinate obtained from, for example, the third curve 850c may be more accurate than one obtained from the curve 830 fit to the wide- range two-parameter data ordered sequence.

[0121] Accordingly, it is possible to accurately compute a percentage flow rate error as a function of the Reynolds number at, for example, relatively small values of Reynolds number (e.g., below 100000 or le5). A complex mathematical function is derived using empirical data collected over a wide range of Reynolds number values (e.g., Ie2 to le7), as shown in the illustrative example in FIG. 8. By using, for example, the vibratory meter 5 described below with reference to FIGS. 10 and 11, a Reynolds number calculated from measured parameters could be used to automatically select different ranges of the empirical data. Then, a lower order (e.g., 1 to 3) polynomial could be fit on the narrow range of the empirical data as shown in FIG. 8, and the percentage flow rate error estimated with much less computational complexity than with the function derived by using the full range of the Reynolds number-percentage flow rate error data series. In addition, the estimated percentage flow rate error may also match the empirical data much better than when using the function, represented by the curve 830, derived using the full range.

[0122] Methods

[0123] FIG. 9 shows a method 900 for an adaptive curve fit. As shown in FIG. 9, the method 900 obtains a relational data ordered sequence relating an inferred parameter to one or more measurable parameters in step 910. The method 900, in step 920, fits a first function to the relational data ordered sequence over a first range of the relational data ordered sequence. In step 930, the method determines a measured value of the one or more measurable parameters. The method 900 uses the first function to determine an estimated value of the inferred parameter based on the measured value of the one or more measurable parameters in step 940. In step 950, the method 900 selects a second range of the relational data ordered sequence based on the estimated value of the inferred parameter, wherein the second range is shorter than the first range. The method also fits a second function to the relational data ordered sequence over the second range of the relational data ordered sequence.

[0124] The method may further comprise using the second function to determine a second estimated value of the inferred parameter based on the measured value of the one or more measurable parameters. As can be appreciated, the second function may provide a more accurate estimation of the inferred parameter within the second range of the relational data ordered sequence. That is, because the second function is fit to a smaller range than the first function, the fit to the values of the relational data ordered sequence within the second range may be tighter. It should also be appreciated that because the second range is smaller than the first range, the amount of computing resources required to obtain the second function is less than that required to obtain the first function. Accordingly, the second function may be a more accurate function (e.g.., higher order polynomial) than the first function without significantly increasing use of the computing resources.

[0125] The relational data ordered sequence may be part of a multi-parameter data ordered sequence comprising a plurality of relational data ordered sequences indexed by at least one of the one or more measurable parameters. For example, the multi-parameter data ordered sequence may be comprised of a plurality of parameter values that were obtained during a calibrated measurement of an object or event, such as a system undergoing a change in the various parameters. By way of illustration, a concentration and temperature of a two-component fluid may be varied while the concentration, density, and temperature are measured using calibrated equipment. As can be appreciated, more than three parameters and / or other objects or events may be measured.

[0126] The method 900 may further comprise determining a central tendency value of the inferred parameter based on a plurality of values of the inferred parameter about a coordinate of the second estimated value of the inferred parameter and a measured value of an indexing measurable parameter. As discussed above, the central tendency value may be a mean, median, mode, etc. of the inferred parameter. Determining the central tendency value of the inferred parameter based on the plurality of values of the inferred parameter about the coordinate of the second estimated value of the inferred parameter and the measured value of the indexing measurable parameter may comprise determining the central tendency value based on a subset of the relational data ordered sequence, the subset comprising a plurality of values of the inferred parameter and a plurality of values of the indexing measurable parameter. For example, the relational data ordered sequence may comprise the inferred parameter and two measurable parameters, wherein one of the two measurable parameters is the indexing measurable parameter.

[0127] Determining the central tendency value based on the plurality of values of the inferred parameter about the coordinate of the second estimated value of the inferred parameter and the value of the indexing measurable parameter may comprise selecting a plurality of ordered pairs of the inferred parameter and the indexing measurable parameter about the coordinate of the second estimated value of the inferred parameter and the measured value of the indexing measurable parameter and determining the central tendency value of the inferred parameter based on values of the inferred parameter in the selected plurality of ordered pairs. Determining the central tendency value based on the values of the inferred parameter in the selected plurality of ordered pairs may comprise determining a plurality of central tendency values of the inferred parameter based on a plurality of subsets of the selected plurality of ordered pairs of the inferred parameter and the indexing measurable parameter and determining a central tendency value of the plurality of central tendency values of the inferred parameter.

[0128] Fitting the first function to the relational data ordered sequence over the first range of the relational data ordered sequence may comprise fitting a high order polynomial function to the relational data ordered sequence over the first range of the two-parameter data ordered sequence. Although the first function may be a high order polynomial function, other functions may be employed such as a low order polynomial function. That is, the first function does not necessarily need to be a high order or of a higher order than the second function. It should also be appreciated that fitting the first function to the relational data ordered sequence may comprise fitting the first function to the relational data ordered sequence over an entire range of the relational data ordered sequence.

[0129] Obtaining the relational data ordered sequence comprises obtaining a two- parameter data ordered sequence that relates the inferred parameter to one of the one or more measurable values. The other or another of the one or more measurable values may be indexing parameters used to obtain the two-parameter data ordered sequence that relates the inferred parameter to the one of the one or more measurable values. For example, measured values of the indexing parameters may be used to obtain a smaller dimensioned matrix from a larger dimensioned matrix. By way of illustration, a three- dimensional concentration-density-temperature matrix may be filtered to a sifted two- dimensional concentration-density matrix. The two-dimensional concentration-density matrix may be a diagonal matrix and therefore could be represented by a two-parameter data ordered sequence comprised of, for example, a density-concentration ordered pair data sequence. Accordingly, in this example, the inferred parameter may be concentration and the one or more measurable parameters may comprise density and temperature.

[0130] The method 900 can be performed by the device 100 or, more particularly the electronics 140, a computer communicatively coupled with the device 100, and / or other computing resources. An exemplary device is described in the following.

[0131] Exemplary device

[0132] FIG. 10 shows a vibratory meter 5 configured to measure parameters for an adaptive curve fit. As shown in FIG. 10, the vibratory meter 5 is a Coriolis flow meter that comprises a sensor assembly 1 and meter electronics 20. The sensor assembly 1 responds to mass flow rate and density of a process material. The meter electronics 20 is connected to the sensor assembly 1 via leads 10 to provide density, mass flow rate, and temperature information over path 26, as well as other information.

[0133] The sensor assembly 1 includes a pair of manifolds 15 and 15', flanges 11 and 111having flange necks, a pair of parallel conduits 13 and 13', driver 18, resistive temperature detector (RTD) 19, and a pair of pick-off sensors 171 and 17r. Conduits 13 and 13' have two essentially straight inlet legs and outlet legs, which converge towards each other at conduit mounting blocks 12 and 12'. The conduits 13, 13' bend at two symmetrical locations along their length and are essentially parallel throughout their length. Brace bars 14 and 14' serve to define the axis W and W' about which each conduit 13, 13’ oscillates. The legs and of the conduits 13, 13' are fixedly attached to conduit mounting blocks 12 and 12' and these blocks, in turn, are fixedly attached to manifolds 15 and 15'. This provides a continuous closed material path through sensor assembly 1.

[0134] When flanges 11 and IT are connected into a process line (not shown) which carries the process material that is being measured, material enters the inlet end of the vibratory meter through an orifice in the flange 11 and is conducted through the manifold 15 to the conduit mounting block 12 having a surface. Within the manifold 15 the material is divided and routed through the conduits 13, 13'. Upon exiting the conduits 13, 13', the process material is recombined in a single stream within the block 12’ having a surface and the manifold 15' and is thereafter routed to the outlet end connected by the flange IT to the process line (not shown).

[0135] The conduits 13, 13' are selected and appropriately mounted to the conduit mounting blocks 12, 12' so as to have substantially the same mass distribution, moments of inertia and Young's modulus about bending axes W— W and W'— W', respectively. These bending axes go through the brace bars 14, 14'. Inasmuch as the Young's modulus of the conduits change with temperature, and this change affects the calculation of flow and density, RTD 19 is mounted to conduit 13' to continuously measure the temperature of the conduit 13’. The temperature of the conduit 13’ and hence the voltage appearing across the RTD 19 for a given current passing therethrough is governed by the temperature of the material passing through the conduit 13’. The temperature dependent voltage appearing across the RTD 19 is used in a well-known method by the meter electronics 20 to compensate for the change in elastic modulus of the conduits 13, 13' due to any changes in conduit temperature. The RTD 19 is connected to the meter electronics 20 by a lead.

[0136] Both of the conduits 13, 13' are driven by driver 18 in opposite directions about their respective bending axes W and W' and at what is termed the first out-of-phase bending mode of the vibratory meter. This driver 18 may comprise any one of many well-known arrangements, such as a magnet mounted to the conduit 13' and an opposing coil mounted to the conduit 13 and through which an alternating current is passed for vibrating both conduits 13, 13’. A suitable drive signal is applied by the meter electronics 20, via a lead, to the driver 18.

[0137] The meter electronics 20 receives the RTD 19, and sensor signals appearing on leads 10 carrying left and right sensor signals, respectively. The meter electronics 20 produces the drive signal appearing on the lead to driver 18 and vibrate conduits 13, 13'. The meter electronics 20 processes the left and right sensor signals and the RTD 19 to compute the mass flow rate and the density of the material passing through sensor assembly 1. This information, along with other information, is applied by meter electronics 20 over path 26 as a signal.

[0138] A mass flow rate measurement can be generated according to the equation: m = FCF[ t — At0]; Equation [8] where: m is a measured mass flow rate;

[0139] FCF is a flow calibration factor;

[0140] At is a measured time-difference; and

[0141] At0is a zero-flow time-difference.

[0142] The measured time-difference At comprises an operationally derived (i.e., measured) time-difference value comprising the time-difference existing between the pickoff sensor signals, such as where the time-difference is due to Coriolis effects related to mass flow rate through the vibratory meter 5. The measured time-difference At is a direct measurement of a mass flow rate of the flow material as it flows through the vibratory meter 5. The zero-flow time-difference Ato comprises a time-difference at a zero flow. The zero-flow time-difference Ato is a zero-flow value that may be determined at the factory and programmed into the vibratory meter 5. The zero-flow time-difference Ato is an exemplary zero-flow value. Other zero-flow values may be employed, such as a phase difference, time-difference, or the like, that are determined at zero flow conditions. A value of the zero-flow time-difference Ato may not change, even where flow conditions are changing. A mass flow rate value of the material flowing through the vibratory meter 5 is determined by multiplying a difference between measured time-difference At and a reference zero-flow value Ato by the flow calibration factor FCF. The flow calibration factor FCF is proportional to a physical stiffness of the vibratory meter.

[0143] As to density, a resonance frequency at which each conduit 13, 13’ vibrates may be a function of the square root of a spring constant of the conduit 13, 13’ divided by the total mass of the conduit 13, 13’ having a material. The total mass of the conduit 13, 13’ having the material may be a mass of the conduit 13, 13’ plus a mass of a material inside the conduit 13, 13’. The mass of the material in the conduit 13, 13’ is directly proportional to the density of the material. Therefore, the density of this material may be proportional to the square of a period at which the conduit 13, 13’ containing the material oscillates multiplied by the spring constant of the conduit 13, 13’. Hence, by determining the period at which the conduit 13, 13’ oscillates and by appropriately scaling the result, an accurate measure of the density of the material contained by the conduit 13, 13’ can be obtained. The meter electronics 20 can determine the period or resonance frequency using the sensor signals and / or the drive signal. The conduits 13, 13’ may oscillate with more than one vibration mode.

[0144] The vibratory meter 5 may be calibrated with a factory zero-flow value while the vibratory meter 5 is in a no or zero-flow condition. A user, at any time, may additionally, and optionally, perform a push-button calibration to obtain a push-button zero-flow value. Additionally, or alternatively, the vibratory meter may automatically perform a calibration to obtain an automatic zero-flow value. The zero-flow value used to measure a flow rate of a fluid may be the factory zero- flow value, a push-button zeroflow value, the automatic zero-flow value, or any other suitable zero-flow value.

[0145] Measurements, saved values / constants, user settings, saved tables, etc., may be employed during the zero calibration of the vibratory meter 5. The calibration may monitor the vibratory meter 5 for conditions of the vibratory meter 5 and compensate for those conditions. The conditions may include user-input conditions, measured conditions, inferred conditions, or the like, without limitation. The conditions may include temperature, fluid density, flow rate, meter specifications, viscosity, Reynold’s number, post calibration compensation, etc. In addition, different constants, such as a flow calibration factor (FCF), for example without limitation, may be applied based on operating conditions or user preference. An initial zero-flow value may be determined during a calibration conducted as part of the initial factory setup of the vibratory meter 5. This may entail placing the vibratory meter 5 in a no or zero-flow condition and determining a time-difference, phase difference, or the like, between the left and right sensor signals. The determined value is stored in one or more memories as the initial zero-flow value and used as a reference zero-flow value. By way of example, for Equation [8] discussed above, the reference zero-flow value may be the ATo term, which may be a no or zero-flow timedifference between the left and right sensor signals. Once the reference zero-flow value is determined, the flow calibration factor (FCF) may be established, which, as can be appreciated from above Equation [8], may be a slope of a line that dictates the relationship between the measured time-difference Atmeasured and the mass flow rate m. The FCF may be stored in the one or more memories.

[0146] FIG. 11 shows a meter electronics 20 configured to estimate and determine a steady state condition of a process. As shown in FIG. 11, the meter electronics 20 includes an interface 21 and a processing system 22. The meter electronics 20 receives a vibrational response from a sensor assembly, such as the sensor assembly 1, for example. The meter electronics 20 processes the vibrational response in order to obtain flow properties of the flow material flowing through the sensor assembly 1. The meter electronics 20 may also perform checks, verifications, calibration routines, or the like, to ensure the flow properties of the flow material are accurately measured.

[0147] The interface 21 may receive the sensor signals from one of the pick-off sensors 171, 17r shown in FIG. 10. The interface 21 can perform any necessary or desired signal conditioning, such as any manner of formatting, amplification, buffering, etc. Alternatively, some or all of the signal conditioning can be performed in the processing system 22. In addition, the interface 21 can enable communications between the meter electronics 20 and external devices. The interface 21 can be capable of any manner of electronic, optical, or wireless communication. The interface 21 can provide information based on the vibrational response. The interface 21 may be coupled with a digitizer, such as a coder / decoder (CODEC), wherein the sensor signal comprises an analog sensor signal. The digitizer samples and digitizes an analog sensor signal and produces a digitized sensor signal. The processing system 22 conducts operations of the meter electronics 20 and processes flow measurements from the sensor assembly 1. The processing system 22 executes one or more processing routines and thereby processes the flow measurements in order to produce one or more flow properties. The processing system 22 is communicatively coupled to the interface 21 and is configured to receive the information from the interface 21.

[0148] The processing system 22 can comprise a general-purpose computer, a microprocessing system, a logic circuit, or some other general purpose or customized processing device. Additionally, or alternatively, the processing system 22 can be distributed among multiple processing devices. The processing system 22 can also include any manner of integral or independent electronic storage medium, such as the storage system 24.

[0149] The storage system 24 can store vibratory meter parameters and data, software routines, constant values, and variable values. In one embodiment, the storage system 24 includes routines that are executed by the processing system 22, such as an operational routine 24a, calibration routine 24b and mass flow rate routine 24c of the vibratory meter 5. The storage system 24 can store values that are used in the routines, such as temperature correction, calibration, zero offset, etc. The storage system 24 can also store other types of values, such as statistical values, such as a mean, standard deviation, confidence interval, etc., or the like of the various values discussed herein. The storage system 24 can also store values in table format where each row in the table corresponds to a sample of time-series, such as a time-series of parameter data.

[0150] For the vibratory meter 5 described above, the calibration routine 24b may perform a zero verification, a flow calibration factor determination, and / or a mass flow rate error relationship determination and / or correction described above, although any suitable calibration routines may be employed. A density routine may employ the sensor signals received by the interface 21 to calculate a density value. For example, the sensor signals from one of the pick-off sensors 171, 17r may be employed to determine the resonance frequency at each conduit 13, 13’. The resonance frequency can be used to determine a density value 24d.

[0151] The mass flow rate routine 24c may determine a mass flow rate value 24e based on the sensor signals received by the interface 21. The mass flow rate value 24e may be determined from the sensor signals, such as a time-difference 24f between a left pickoff sensor signal and a right pickoff sensor signal, and a flow calibration factor (FCF) value 24g.

[0152] With respect to adaptive curve fitting, the storage system 24 is also shown as including source data 24h, an obtained relational data ordered sequence 24i, a narrow range 24j algorithm, and a curve fitting 24k algorithm. The source data 24h may be any relational data ordered sequence, such as the source data discussed above. It should be appreciated that the relational data ordered sequence can be stored in any particular order as long as an order of the sequence can be determined - such as by using indices, sorting by values, etc.

[0153] The source data 24h may be filtered with, for example, indexing parameters to determine the obtained relational data ordered sequence 24i. As an example, examples of the obtained relational data ordered sequence 24i includes the sifted matrices A' , B' , X' described above, although any suitable relational data ordered sequence may be employed. It should be appreciated that the obtained relational data ordered sequence can also include the flattened data, where one or more parameters are dropped. However, the obtained relational data sequence is typically referring to data that is not yet prepared for curve fitting, estimating, averaging, etc. That is, the obtained relational data ordered sequence is not yet a narrow-range data ordered sequence ready for analysis.

[0154] The narrow range 24j algorithm may be employed to determine a narrow range data ordered sequence. For example, a narrow range ordered pair data ordered sequence, such as the narrow-range two-parameter data ordered sequences 330, 430 or the concentration-temperature grid 600 described above, may be determined from the obtained relational data ordered sequence 24i. For example, values or a range of values may be set for indexing parameters of inferred and / or measurable values. The value or values of the indexing parameters may be of the indices and / or the parameter values.

[0155] The curve fitting 24k algorithm may perform curve fitting that are the same as or similar to the curve fitting shown in FIGS. 2 through 4, although any suitable curve fitting may be employed. The curve fitting 24k may be applied to the obtained relational data ordered sequence 24i and / or a narrow-range data ordered sequence determined by the narrow range 24j algorithm. Accordingly, the curve fitting 24k algorithm may be a high or low accuracy fit such as a high or low order polynomial fit, although any suitable curve fitting may be employed. One or more functions may be determined by the curve fitting 24k algorithm. An estimated value, such as an estimated inferred parameter value may be determined using the functions.

[0156] The central tendency 241 algorithm can determine, for example, an average value of a narrow-range data ordered sequence, such as the concentration-temperature grid 600 described with reference to FIG. 6. The central tendency 241 algorithm can determine an average value of a grid and / or an aggregated average value and a plurality of average values of sub-grids of a grid used to determine the aggregated average value. As discussed above, any suitable central tendency value may be employed.

[0157] The method 900 and device 100 described above may perform an adaptive curve fit. While the source data may be large, the use of computing resources may be reduced with adaptive curve fitting. In addition, the resulting estimated value of an inferred parameter may be more accurate due to selecting a smaller range of data for the curve fitting about an estimated inferred parameter value associated with a measured value. Tire curve fitting may be performed iteratively with or without determining a central tendency value of the inferred parameter. For example, some processes or measured values of a process may smoothly change and therefore the low pass filtering aspect of determining the central tendency value may not be needed. However, other processes or measured values of a process may be relatively noisy and therefore determining a central tendency value of the inferred parameter may help filter out any noise.

[0158] The detailed descriptions of the above embodiments are not exhaustive descriptions of all embodiments contemplated by the inventors to be within the scope of the present description. Indeed, persons skilled in the art will recognize that certain elements of the above-described embodiments may variously be combined or eliminated to create further embodiments, and such further embodiments fall within the scope and teachings of the present description. It will also be apparent to those of ordinary skill in the art that the above-described embodiments may be combined in whole or in part to create additional embodiments within the scope and teachings of the present description.

[0159] Thus, although specific embodiments are described herein for illustrative purposes, various equivalent modifications are possible within the scope of the present description, as those skilled in the relevant art will recognize. The teachings provided herein can be applied to other adaptive curve fits not just to the embodiments described above and shown in the accompanying figures. Accordingly, the scope of the embodiments described above should be determined from the following claims.

Claims

We claim:

1. A method for adaptive curve fitting, the method comprising: obtaining a relational data ordered sequence relating an inferred parameter to one or more measurable parameters; fitting a first function to the relational data ordered sequence over a first range of the relational data ordered sequence; determining a measured value of the one or more measurable parameters; using the first function to determine an estimated value of the inferred parameter based on the measured value of the one or more measurable parameters; selecting a second range of the relational data ordered sequence based on the estimated value of the inferred parameter, wherein the second range is shorter than the first range; and fitting a second function to the second range of the relational data ordered sequence over the second range of the relational data ordered sequence.

2. The method of claim 1, further comprising using the second function to determine a second estimated value of the inferred parameter based on the measured value of the one or more measurable parameters.

3. The method of claim 1 , wherein the relational data ordered sequence is part of a multi -parameter data ordered sequence comprising a plurality of relational data ordered sequences indexed by at least one of the one or more measurable parameters.

4. The method of claim 2, further comprising determining a central tendency value of the inferred parameter based on a plurality of values of the inferred parameter about a coordinate of the second estimated value of the inferred parameter and a measured value of an indexing measurable parameter.

5. The method of claim 4, wherein determining the central tendency value of the inferred parameter based on the plurality of values of the inferred parameter about the coordinate of the second estimated value of the inferred parameter and the measured value of the indexing measurable parameter comprises determining the central tendencyvalue based on a subset of the relational data ordered sequence, the subset comprising a plurality of values of the inferred parameter and a plurality of values of the indexing measurable parameter.

6. The method of claim 5, wherein the relational data ordered sequence comprises the inferred parameter and two measurable parameters, wherein one of the two measurable parameters is the indexing measurable parameter.

7. The method of claim 4, wherein determining the central tendency value based on the plurality of values of the inferred parameter about the coordinate of the second estimated value of the inferred parameter and the measured value of the indexing measurable parameter comprises: selecting a plurality of ordered pairs of the inferred parameter and the indexing measurable parameter about the coordinate of the second estimated value of the inferred parameter and the measured value of the indexing measurable parameter; and determining the central tendency value of the inferred parameter based on values of the inferred parameter in the selected plurality of ordered pairs.

8. The method of claim 7, wherein determining the central tendency value based on the values of the inferred parameter in the selected plurality of ordered pairs comprises: determining a plurality of central tendency values of the inferred parameter based on a plurality of subsets of the selected plurality of ordered pairs of the inferred parameter and the indexing measurable parameter; and determining a central tendency value of the plurality of central tendency values of the inferred parameter.

9. The method of claim 1, wherein fitting the first function to the relational data ordered sequence over the first range of the relational data ordered sequence comprises fitting a high order polynomial function to the relational data ordered sequence over the first range of the two-parameter data ordered sequence.

10. The method of claim 1, wherein fitting the first function to the relational data ordered sequence comprises fitting the first function to the relational data ordered sequence over an entire range of the relational data ordered sequence.

11. The method of claim 1 , wherein obtaining the relational data ordered sequence comprises obtaining a two-parameter data ordered sequence that relates the inferred parameter to one of the one or more measurable values.

12. The method of claim 11, wherein another of the one or more measurable values are indexing parameters used to obtain the two-parameter data ordered sequence that relates the inferred parameter to the one of the one or more measurable values.

13. The method of claim 1, wherein the inferred parameter is concentration and the one or more measurable parameters comprises density and temperature.

14. An electronics (140) for adaptive curve fitting, the electronics (140) comprising a processing system (22) configured to perform methods of one of the foregoing claims 1 through 13.

15. A device (100) having adaptive curve fitting, the device (100) comprising a transducer (120) and the electronics (140) of claim 14, wherein the transducer (120) is communicatively coupled to the electronics (140).

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