Studying additional measurement errors from control tools using an integral functional method

Our research has established that under industrial conditions the correction to the result of current measurements when an influencing parameter deviates from the rated value is rarely introduced. In a general case, the procedure for determining an additional measurement error implies that the measured values for an influencing parameter are applied to determine the degree of its deviation while a correction to the current measurement result is calculated as the product of this degree by its rated value.In a general case, a procedure for determining an additional measurement error includes two stages. At the first stage, the measured values for an influencing parameter are used to determine the degree of its deviation from the rated value. At the second stage, correction is calculated as the product of this degree by the rated value for an additional error.Such a technique to calculate a correction is time consuming and insufficiently precise, as it does not take into consideration the non-linear dependence of the additional error on a change in the influencing parameter, as well as the current value for the output signal of control tool. To determine the actual value for an influencing parameter and the additional measurement error under industrial operation of control tools, an integral functional method has been proposed. The method implies determining the difference of areas under the nominal and actual acreage static characteristics, limited to a range of measurement. The difference of areas is a function of the output signal of a control tool, a measured parameter and a change in the influencing parameter. It has been shown that the proposed method makes it possible to calculate the actual values for a technological parameter based on its measured and influencing parameters only. We have established regularities between the actual value for a measured parameter, the current value for the output signal from a control tool, and the measured value for an influencing parameter. The proposed method is important and valuable in the operation of computer-integrated control systems of technological parameters, as it makes it possible to determine the actual values for a measured parameter based on relevant algorithms without calculating corrections.


Introduction
The efficiency of technological processes is defined by the precision in maintaining technological parameters at the assigned level, which is standardized by respective regulations. To meet this requirement, technological parameters are subject to control by using technical tools. The latter transmit the result from measurements in the form of an output signal for visualization, for example, by a real-time monitor at a computer-integrated control system. The output signal from such tools is standardized at the stage of industrial execution by establishing a range of measurement and an accuracy class. In addition, documentation on technical means specifies additional measurement errors (AME) that are caused by deviations of influencing parameters from the rated values. The basic influencing parameters typically include the ambient temperature, atmospheric pressure, air humidity, supplied voltage, current frequency, and some others. At constant measured parameter, the output signal from a control tool varies towards larger or lower values by a certain magnitude due to the deviation of an influencing parameter from the rated value. Thus, a change in output signal is a function of the change in the influencing parameter. The working range of change in the basic influencing parameters is specified in the technical documentation for a control tool, while AME − only for a certain change in it.
Research has found that the dependence of AME on deviation in an influencing parameter on the rated value in a general case is nonlinear, which increases with an increase in the coefficient sensitivity along the channel of this parameter's effect. In addition, it was established that at an increase in the influencing parameter from the rated, the AME value is considerably less than when it decreases by the same magnitude. In this case, there is an increase in the non-linearity of AME dependence on a decrease in the influencing parameter. Under conditions of industrial operation, control tools generate an output signal, which is used, at inverse calibration, to obtain the value for a technological parameter. Since the primary measuring converter is affected by both internal and external influencing parameters, then the output signal does not match the actual value for a measured technological parameter, but it is offset by the AME magnitude. As the actual value for a measured technological parameter is accepted to be the reading from a control tool based on its graded scale, then, in this case, AME is not taken into consideration, which is predetermined by some influencing parameter. Therefore, the operators always face the task on determining this error with the greatest precision in order to introduce the corresponding correction to the measurement result. Currently, AME is typically determined by two methods. The most accurate is the method of comparison with the metric at which a working control tool is under real operating conditions, while the reference one − under conditions of calibrating the working one. The second method is the estimation, based on determining AME based on the measured value for a current influencing parameter at a known conversion coefficient along the channel of its effect. Typically, this method is approximate and does not take into consideration the character of AME distribution both over the measurement range of a technological parameter and based on a change in the influencing parameter. As the operation of control tools at an industrial facility does not involve metrics and reference means, then, as a rule, the second method for determining AME is applied. In most cases, the second method is used to rate AME for the normative-technical documentation on a control tool. It implies that the technical conditions specify AME, which is predetermined at a deviation of the influencing parameter by a certain rated variable, for example, at a deviation of temperature for every 10 °C from the normal flat (20±5) °C. If the working range of change in an influencing parameter is large enough, for example, for a temperature change from minus 40 °С to plus 120 °C, then the temperature of AME dependence on the deviation in temperature for every 10 °C is nonlinear. That leads to that the correction to a measurement result is determined with an error, which may exceed its value. In this case, the calculated actual value for a measured parameter (AVMP) is underestimated (or overestimated), which leads to the incorrect assessment of the quality of a produced article or the operation of an automated control system. Thus, the relevance of this work is in the construction of a method, which would make it possible to determine AVMP employing an appropriate algorithm based on the current value for an output signal from a control tool and the measured current value for an influencing parameter.

Literature review and problem statement
As it is known [1], an additional measurement error of a control tool denotes a component of the error, which occurs as a result of deviation in any influencing magnitude from its normal value or because it leaves the region of normal values. A working range of values for every influencing magnitude, within which one regulates additional errors or change in the readings from a measurement tool (control) is indicated in the specifications separately for any tool. It is typically assumed that AME is distributed evenly along the entire working region of change in every influencing magnitude, and this error is rated over a certain predefined part of this region. At the maximal permitted deviation of an influencing magnitude, AME is determined from multiplying the rated AME by the degree of deviation in an influencing magnitude from the rated value. The main disadvantage of the method is that it does not take into consideration the current value for an output signal from a control tool and the non-linearity of its static characteristics both on a change in the influencing and measured parameters. Papers [2, 3] consider the impact of change in the coefficient of conversion for a control tool when it is exposed to the action of deviation in an influencing parameter on the rated value; the authors also describe an algorithm for determining a correction to the measurement result. The proposed algorithm is implemented in the following stages: -calibration of thermometers at a calibrating laboratory under normal conditions and determining its conversion function; -determining a conversion function under working conditions when a thermometer takes several values of temperature, reproduced a reference transmitter under working conditions; -computation of deviations in the values for temperature, determined by a conversion function, established under normal conditions and defined for actual temperature under working industrial conditions; determining corrections at several fixed points of temperature; -determining a temperature dependence of corrections for the output signal from a thermometer according to industrial conditions by interpolating the values for corrections; -introduction of corrections to temperature values determined by a calibrated thermometer.
A positive aspect of [2] is that the authors established a functional dependence of correction between the deviation in an influencing parameter on the rated value, which is nonlinear, as well as the dependence of the value for a correction on the measured parameter. The main disadvantage of work [3] is the complexity of the algorithm for computing a correction and its independence from the current value for a measured parameter.
Papers [4,5] proposed a CFD modeling method to study AME. In accordance with the method described in [4], a corrective factor is determined, which changes the conversion coefficient at the current value for the output signal from a control tool. A positive aspect of paper [5] is that based on the known influencing structural parameters, a corrective factor is determined for any converting link in a control tool. The value of corrective factor is introduced to an appropriate algorithm, which computes the actual value for a measured parameter.
The main disadvantage of the method is the complexity of computing a corrective factor for control tools with nonlinear static characteristics, while the correction is introduced not to the result of current measurement but to a conversion coefficient. Paper [6] shows that a variety of influencing parameters cause a change in conversion coefficients and it is proposed, to determine AVMP, to perform the calibration of such tools directly under working conditions. The advantage is a high accuracy of AME determination, its independence from the nonlinearity of the static characteristic. The disadvantages of the method include, first, the need to have a reference portable control tool, second, the method is periodic and inapplicable for the current introduction of a correction to the measurement result.
The method of test influences [7] is the most suitable for use in computer-integrated systems of control and management. According to this method, based on the measured current values for influencing parameters one can calculate corrections to a measurement result and correct test programs. The method of test influences is rather complicated, time-consuming when calculating corrections, and it generates only the multiplicative component of a correction to the measurement result and does not account for the influence of a nonlinear one.
In many cases, the output signal from a control tool is oscillatory in nature with different frequency and amplitude of oscillations, and its AME is predetermined by a deviation in the influencing parameter from the rated value. To assess the quality of accuracy and the stability of characteristics, which are used to control technological processes and are regulated by normative documents, determining the upper and lower bounds for an output signal is implied, based on formulae [8] -for the upper bound where μ оі , σ і are the average value and the root mean square deviation, determined based on the results from previous studies into the trend of an output signal; z (1-a/2) is the coefficient of significance. In this case, the spread region of permissible values for the resultant vector of the output signal is determined as difference Â=UCL i -LCL i and the coefficient of significancefrom formula where x o is the average value from a Shewhart control chart [9]; n is statistical sample. Since the task on determining AME refers to two-parametric problems, we suggest estimating stability of the technological process based on Hotelling cards [10], in accordance with which one calculates a correlation coefficient. The latter is counted as a result of the current values for an output signal. The method is complicated, it refers to statistical methods, requires a large volume of research and cannot be used for the automatic introduction of a correction to the result of current measurements. One can see from the results of analysis that AME are determined based on difference in the readings of control tools at normal and current values for influencing parameters. In this case, AME are examined, typically, for a particular control tool, while AVMP is determined from the algebraic sum of the current value for a measured signal and AME. If the current value for a measured parameter is known, then AME should be calculated based on a relevant procedure, which in many cases is imperfect. In almost all cases, AME for a working range of change in an influencing parameter is determined based on the linear principle. In most cases, the dependence of AME on change in an influencing parameter is nonlinear, which leads to the emergence of nonlinear components of an error, whose determining is a rather complex task. The issue on studying AME is typically associated with determining AVMP; when a control tool is exposed to the action of a variety of internal and external factors the normalized values for AME under operating conditions change in most cases. Therefore, there is a task on determining AVMP based only on the current value for an output signal from a control tool and the measured value for an influencing parameter.

The aim and objectives of the study
The aim of this work is to investigate additional errors from control tools using an integral functional method.
That would make it possible to determine AME over each polling cycle of a computer-integrated system of control and management and to introduce appropriate corrections to measurement result.
To accomplish the aim, the following tasks have been set: -to substantiate an integral functional method to study AME from control tools; -to define the effect of influencing parameters on change in the integrated functionality of a control tool; -to devise a procedure for calculating the actual values for a measured parameter based on the known value for an output signal from a control tool and an influencing parameter.

Substantiation of an integral functional method to study additional measurement errors from control tools
The systematization of results allows us to consider that existing approaches to solving a task on determining AME are individual and based on methods that are typical only for specific control tools. Such approaches make it possible to determine AME only when the metrological characteristics of control tools are known. As a rule, control tools are described by the experimental-statistical or deterministic methods, which do not make it possible to derive objective mathematical models of AME distribution both over a measurement range and over a working range of change in the influencing parameter. In addition, most methods do not take into consideration a correlation connection between the measured and influencing parameters, which can be significant and change the output signal from a control tool as shown in Fig. 1. The measuring range is limited to zero and end J k value for a measured parameter and is stable, established at calibration of the control tool. When an influencing parameter deviates from the rated value, the output signal from a control tool may change in proportion to a change in the measured parameter, or disproportionately. In this case, at the end of the measurement range one observes absolute errors, respectively, Dy 2 , Dy 3 and Dy 4 .
The deviation of the influencing parameter from the rated value may cause only the additive component of error Da. In this case, there is an issue on AME reduction and optimizing them along the measurement range. To study AME, an integral functional method is proposed, which implies determining a difference between planes under the normalized and current static characteristics for a control tool, which occur after the action of an influencing parameter. It is known that as a result of measuring the output signal from a control tool changes from preceding to new ones. Moreover, such a change may be caused both by the deviation in the measured and influencing parameters. If AVMP is known, the absolute value for AME equals where y( J , z) and y H ( J , z H ) are the current and normalized value for an output signal; J д , J are the actual and current value for a measured parameter, respectively; z H , z are the normalized and current value for an influencing parameter, respectively. When control tools are operated under industrial conditions, AVMP is unknown. Therefore, determining AME is a rather complex task, since a change in the output signal can be caused by both a change in the measured and influencing parameters. In this case, the current value for AME Dy( J, z) is a dependent function, simultaneously from two variablesthe current value for measured parameter J and for influencing parameter z. The actual values for deviation in the basic influencing parameters (for instance, temperature, atmospheric pressure, air humidity, supplied voltage, current frequency, and others) from their normalized values can be determined by measuring them directly. However, one needs to know the manner AME are distributed along the measurement range at a change in the measured parameter. If a change in the influencing parameter causes, along the measurement range, only the additive or multiplicative components of AME, it can be determined rather simply. To this end, it will suffice to know a conversion coefficient k z of the tool along a channel of the influencing parameter. Expand function y( J, z) into a Taylor's series based on variables J and z. Consequently, we obtain for AME where y H ( J , z H ) is the function of static characteristic under the rated value for an influencing parameter z H ; y z ( J , z H ) is the function of an output signal from a control tool when changing the measured and influencing parameters; J is the actual value for a measured parameter; Δz is the deviation of the influencing parameter from the rated value; ΔJ is the deviation in the measured parameter. AME in general is described by the following equation Derivatives in equation (3) are the conversion coefficients for a control tool. In order to reduce the conversion coefficients to the same dimensionality, we denote: where k μ Δz μ , k z are the multiplicative, nonlinear quadratic and cubic nonlinear components of AME, which are independent from the measured parameter; k Jzμ δJ Δz μ , J z are the multiplicative, nonlinear quadratic and cubic nonlinear components of AME, which are both dependent on a change in the measured and influencing parameters; δJ =ΔJ /J is the relative change in the measured parameter.
In coordinates J →z the product δJ Δz is a certain elemental plane ΔS, predetermined by a change in the measured and influencing parameters. Considering the predetermined, we denote ΔS μ =δJ Δz μ , ΔS v2 =δJ Δz v2 , and ΔS v3 =δJ Δz v3 in equation (4). Consequently, we obtain where ΔS μ , ΔS v2 and ΔS v3 are the multiplicative, nonlinear quadratic and cubic nonlinear components in an increment of areas, predetermined by a deviation of the corresponding components of the influencing parameter from the rated value. Denote in equation (5) ( ) µ µ ν ν ν ν D = D + D + D is AME component, predetermined by a simultaneous change in the measured and influencing parameters. Equation (5) then takes the following form , .
z Jz y J z y z y J z An important conclusion follows from equation (6) that suggests that AME is the sum of the component, predetermined by a change in an influencing parameter, and the component, proportional to plane ΔS μ (δJ , Δz), limited ton an increase in the measured and influencing parameters (Fig. 2). Since the output signal from a control tool changes both when changing the measured parameter and when the influencing parameter deviates from the rated value, then the AME component Δy Jz (δJ , Δz) exists always when Δz≠0.

Fig. 2. Graphical explanation of the method of planes
At present, AME is determined provided that the measurement parameter J =const. Under industrial operation of control tools, for example, when measuring the rate of material flows (for instance, a gas flow rate), both the measured and influencing parameters may change at the same time. In such cases, the AME component y Jx (δJ , Δz) can greatly exceed the component Δy z (Δz). Especially important is to measure the rate of turbulent gas flows. Hence, it follows that for industrial conditions for the measurement of rate of material flows, the most effective is the method of plane, which makes it possible to simultaneously take into consideration both a change in the measured and influencing parameters. Since the static characteristic is a dependence of output signal y( J , z) y (J , z) from a control tool on the actual value for measured parameter J , then the plane under this characteristic is where J 1 , J 2 are the initial and final value for the measured parameter; z 1 , z 2 are the initial and final value for the influencing parameter. If one assumes that for industrial conditions of operating a control tool the range of change in an output signal relative to the actual value for the measured and influencing parameters is linear, then for a single power function, at f( J ,z)=1, equation (7) takes the following form Taking into consideration that a deviation in the output signal from a control tool Δy( J , z)=k H ΔJ , equation (8) is reduced to the following form where k H is the conversion coefficient along a channel of the measured parameter; Δy z ( J , z) is the additional measurement error, caused by a change in influencing parameter z. It follows from equation (9) that AME is fully defined by the plane, which is equal to the AME product multiplied by a deviation Δz in the influencing parameter from the rated value. Since Δz=z-z H , where z, z H are the current, to be measured, and the normalized values for an influencing parameter, respectively, then in order to determine AME, it will suffice to know an increment in the plane ΔS y [y( J , z), z]. In order to solve this problem, in this work we have proposed an integral functional method.

Studying additional measurement errors from control tools using an integral functional method
Integral functional is widely used to optimize the movement of systems of automated control and is described by the following equation where x(t) is the function of movement over time t; F[x(t), x'(t)] is some function that is associated with a motion function over time interval from t 0 to t 1 ; x'(t) is the derivative from a motion function. Accept that variable t is the current measured technological parameter J і . Then the output signal from a control tool is a function of both the measured parameter J і and the magnitude of deviation Dz of the influencing parameter from its normalized value z H .
Let the function of the rated output signal y H =y( J і , z H ) be a nominal static characteristic for a control tool, and y z =y( J і , z) -current. Variation Since the integrand function F[Δz, y( J і , Δz)] at z=z H equals the plane under the curve y H =y( J і , z H ), and when z=z H +Δz -the plane under the curve y z =y( J і , z z ), then their difference is nothing but an increment in the plane between them. A measurement range is stable, while deviations in the influencing parameter from the rated value can vary widely enough. Then a change in the plane between the nominal and current static characteristics would be defined only by the deviation in the influencing parameter. Moreover, for the same measured and influencing parameters the increment in the plane will be the same. Thus, an increment in the plane ΔS z (δJ і , Δz) will be the same both at a change in the actual value for the measured and influencing parameters and will be described by the following equations: -when an influencing parameter deviates from the rated value where Δz is the deviation in the influencing parameter when it decreases and increases from the normalized value, respectively; k П is the coefficient, which is equal to the power of the characteristic equation of the integrand function; -when a measured parameter changes along the measurement range where J 0 , J і are the beginning and end of the measurement range, respectively. In a general case, the increment in a plane is determined from multiplying a change in the output signal from a control tool by a change in the actual value for the measured or influencing parameter, that is