THE FEATURES OFAPPLICATION OF NORMATIVE DOCUMENTS TO INVERSE PROBLEMS OF MEASUREMENT

The operation of many technically complex objects (TCO) involves the continuous automated control of most important parameters of technological processeswith the help of measuring systems. In many such systems, it is important to monitor the results of pressure measurements in different technological equipment. This is done by means of measuring pressure channels (MPC) consisting of a sensor or pressure sensors and a measuring line. The designed MPC are often linear channels with a small inertia of the sensors, which is described by their constant time. A measuring line for the pressure sensor O. Poliarus, Doctor of Technical Science, Professor, Head of the department of metrology and life safety, e-mail: poliarus.kharkov@ukr.net Ja. Brovko, assistant of the department, e-mail: yana.brovko@ukr.net Kharkiv National Automobile and Highway University, O. Maletska, candidate of technical sciences, assistant professor of the department of labor protection, standardization and certification, Ukrainian Engineering and Pedagogical Academy, Kharkiv, e-mail: maletska.olga@ukr.net

The most important parameters of technically complex objects are controlled by means of measuring channels, which in turn are calibrated in established terms.These procedures are sometimes economically unattractive and gradually, along with the classic calibration, online monitoring of channels is being introduced.It is promising to use online monitoring methods for solving inverse measurement problems that allow to obtain a slightly distorted input signal.Nowadays there are no normative documents regarding the inverse problems and the quality of the input signal restoration.The scientifi c basis must be created for their implementation.Some scientifi c theses that should be used when creating normative documents, terminology and recommendations are considered in the article.

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is a long tube filled with liquid or gas.The fluid is not compressed under pressure from a TCO.Putting dirt, air, etc. into the pipe leads to compression of the liquid and, as result, to transformation of the MPC into a nonlinear inertial system.The characteristics of the output random process of pressure in this case differ from the similar characteristics of the input process and the result of the diagnosis of the TCO state may become unreliable.Consequently, there is a need to spend significant funds during the object operation to eliminate all possible causes of nonlinearity MPC and reduce its inertia.
Another approach aimed at reducing operational costs is the partial reconciliation with the loss of linearity and the growth of inertia and the development of new methods for analyzing such channels based on the solution of the inverse measurement problem, which allows to determine the realization of inco ming random processes with acceptable measurement quality of results.However, at present there are no normative documents that regulate the requirements for methods of solving the inverse problems of measurement, which don't allow them to be widely used in practice.Therefore, the article deals with the issue of creating a relevant normative document, which will includerecommendations for the practical implementation of methods for solving the inverse problem of measurement.
There is a large number of publications on solving inverse measuring problems for linear inertial systems [1][2][3][4][5][6].They are based on the solution of the integral convolution equation using accurate data on the impulse response of the channel and the output signal realization.In some cases, the problem becomes poorattributed, and then various regularizing operators are used [7].Identification methods [8] are used to evaluate the dynamic characteristics of the measuring channels, which can't always ensure high accuracy of the impulse response determinationfor the channel.In addition, the measurement of the output signal realization is accompanied by errors, and the uncertainty of measurements can be high [9].In such conditions, the use of classical methods for solving inverse measuring problems is inappropriate due to the loss of stability and convergence of the solution, as well as the inadmissible errors in recovery of the input signal.In [10], the method of restoring the input signal in linear inertial measuring systems is described in detail.It is based on minimizing the distance in a functional space with a quadratic metric between the measured output signal and the convolution ofthe known impulse response and the input signal realization represented in the form of a series of orthogonal functions with unknown coefficients.As theresultof minimizingthis distance with the help of a genetic algorithm, the unknown coefficients are determined, and, hence, the input signal realization itself.For the qualitative solution of the inverse problem of measurements the requirements formulated in [11] are put forward to its metrological support.Nonlinear inertial measuring channels are described on basis of Volterra series with multidimensional pulse characteristics, and now there are no precise methods of solving inverse problems for such systems.A simplified approximate approach is formulated in [12].It involves the use of Hammerstein models for the analysis of nonlinear inertial MPCs.In these models, the function of nonlinearity and inertia are virtually separated and inverse problem can be solved in stages.The results of the solution and the requirement to ensure the unity of measurements for such problems are formulated in [13].
The purpose of the article is development of the proposals for using regulatory metrological documents in order to ensure a qualitative solution of the inverse measuring problem and recommendations for the new normative document creation.

FEATURES OF INVERSE PROBLEMS OF MEASUREMENT IN TERMS OF METROLOGY
The purpose of the inverse measurement problems is recovery the input action.For MPC the input action is pressure.At first sight, it is possible to assume that the inverse problem is reduced to the recovery of the input pressure, but from a practical point of view it makes no sense, because the consumer is interested in the numerical values of the input pressure, not pressure itself.Due to the change of the input process over time, the inverse problem of measurement allows us to obtain a mathematical description of the input pressure or the input signal.According to [14], in this case, the measurement process can be considered as a set of operations necessary to obtain the value of the input signal, that is, this process is the inverse measurement.There are no normative documents concerning the inverse measurement, including terminological ones.There is no difference between the direct and inverse measurements if the measuring equipment, which includes the MPC, has an output signal that is not mathematically different from the input one.Physically, they may be different, for example, there is an alternating pressure at the input of the measuring channel, and at the

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outputa voltage proportional to the pressure value.This is possible if the transfer of unit sizes is carried out not only by pressure, but also by the time that is characteristic of dynamic measurements [15].
Really, the output signals may differ from the input ones due to the distortion of the input signal in the MPC, which has a limited bandwidth and exhibits nonlinear propertiesin some situations.In this case, the input signal can be obtained by solving the inverse problem of measurement, which in fact must be carried out in the reverse time and such procedure is physically impossible.That is why it is better to call the inverse problem of measurement by inverse modeling.Thus, in [16] the inverse problem is treated as the use of the results of measuring the parameters of the process for obtaining the actual values of the parameters of the model.Let's consider, first of all, the model of an input signal, as well as the model of the MPC.Consequently, the mathematical description of the input signal is an estimate ofits model parameters.

INVERSE PROBLEM AS THE PROBLEM OF MODELING
The analysis demonstrates the impossibility of obtaining the real input signal by hardware method and therefore the inverse problem becomes a problem of modeling.Since it requires output measurement results, the inverse problem remains the problems of measurement.All measuring problems require the presence of a true or reference value according to DSTU GOST ISO 5725 [21].The true value can be replaced by the standard.However, due to significant technical problems and high cost of the pressure standards their using is not feasible.In addition, in this case, it would be necessary to restore not the input signal model, but the variable pressure itself and this is a very difficult task.
Consequently, for a model inverse problem, the standard must also be the model not in the physical sense, but in the mathematical one.Unlike the artifact, which is difficult for creation, the model standard is easily implemented by software way using computers.The model standard, as a rule, is the input theoretical signal, which must be restored with the help of the developed method using the accepted model of MPC.The choice of the reference signal should be made in the possible ranges of the real input signal variation in amplitude, spectrum width, and so on.This requirement is simply performed by creating many modeling standards.The examples of such standards are deterministic harmonic signals with a given amplitude and frequency, and, as a special case of a harmo nic signal, constant pressure,frequency of which is zero.
Another example of a deterministic model is the sum of several deterministic harmonic signals with different amplitudes and frequencies.It is easy to detect the errors of the inverse measuring problem solution, because the signal value of the model standard is the reference value of the pressure in the MPC.
The more complicated problem is evaluation of the errors of recovery of the input signal, which is the sum of the deterministic signal and noise, most often, white noise.In this case, the standard has the known statistical characteristics.In metrology, the characteristics of all artifacts used as a standard are also random, for example, the characteristics of meters (length) and kilogram (mass) vary with time randomly, thoughslightly.These model standards are the example of signals describing the realizations of stationary input processes.
Model standards can be used for describing nonstationary input processes, in which the mean value or variance changes over time.To do this, a linear signal is added to the deterministic signal, whose amplitude increases (or decreases) linearly over time.A white noise with time-varying spectral density may also be used.Consequently, unlike the artifacts, the number of model standards may be large.It, in the absence of relevant normative documents, should be determined by the designers and researchers of the MPC, which is rather difficult.The main task of these standards is checking the possibility of the input signal restorationin the inverse problem for the different modes of TCO operation in different operating conditions.Therefore, it is advisable to create a normative document that will regulate the methodology for developing a model standard.

THE PROBLEM OF ESTIMATING THE QUALITY OF THE INPUT SIGNAL RECOVERY
Reliability of an input signal restoration characterizes its quality and expediency of application for a particular MPC.The main factors influencing the accuracy of the input signal recovery are the inertia of the measuring channel (constant time or bandwidth) and the type of the conversion function for the MPC (linear, nonlinear).Other factors will not be taken into accounts.The first set of indices (inertia) can be adequately evaluated only in linear systems.To create the possibility of using them in nonlinear inertial measuring systems, it is necessary to create an acceptable model of MPC for inverse problem.A known model of such systems is the Hammerstein model in which the properties of nonlinearity and inertia of the system are virtually separated, which creates the possibility for a separate analysis of the influence of these properties on the quality of solution of the inverse problem of measurement.The first block of the Hammerstein model is non-linear non-inertial, and the second one is linear inertial.The known test methods for determining the time constant at which a specific test signal in the form of a very short impulse or step enters a system input is inappropriate for a nonlinear inertial channel, since the nonlinear unit substantially distorts the spectrum of the test signal and, consequently, the output signal.If for a linear system the output signal as a response to a short pulse will be considered a pulse characteristic, then in a nonlinear system this signal is no longer a pulse characteristic and it has no practical significance.The constant time of the MPC can be determined by the method of noise analysis [17], the level of which corresponds to the linear part of the conversion function of the nonlinear block.The test signal in this case is the noise that always exists when the fluid pulses in the pipe, which transmits pressure from the TCO to the pressure sensor.If the bandwidth of the linear inertial block is much smaller than the width of the spectrum of noise, then the latter can be considered as white.The conversion function of a nonlinear block can be determined in the static mode when the various values are fed to the MPC input.
Let's consider the quality estimation of the restored input signal.It is advisable to compare this signal with the reference one and, of course, these signals will not completely coincide.The difference between them is an error of recovery, although such a concept in normative metrological documents does not exist.According to [18], the direct aim of measurements is to determine the reliable values of the constant or variable measured values.Such a goal coincides with thegoal of solving the inverse problem of measurement, the final result of which is to obtain the necessary information about the quantitative properties of processes that can be obtained on the basis of measurements.Measurement is the process of experimentally determining one or more values that can be reasonably attributed to the value [20].To solve the inverse problem, the output signal of the measuring channel, obtained experimentally, is used.This indicates that the inverse problem is a measuring problem and the error of the input signal recovery can be considered as measuring errors, which are well described in normative documents.Moreover, it is important for consumers to know precisely the errors of measuring the input signal, and not the output.In practice, things are often the opposite: the measurement errors of the output signal are estimated, and they can differ significantly from the errors of the measurement of the input signalin the nonlinear inertial channels.That is why the inverse problem of measurement is relevant for these measuring systems.
For each technicalobject the requirements for measuring information and measuring systems are established.For example, this may be a range of values that must contain a physical value, a maximum deviation from the nominal value, and so on.Statistical estimations of random and systematic errors of measurements of MPCs coincide with those set in [19]: mean square deviation (SCR) of measurement errors, the limits within which the measurement error is with a given probability.In [21] accuracy of measurements is estimated by correctness and precision.Indicator of the correctness is the significance of the systematic error of measurements, and the precision expresses the degree of proximity to each other of independent measurements obtained in specific conditions, that is, it depends on random factors.These factors are determined by the state of the environment and are similar to many measuring systems.However, there are random factors that are specific in solving the inverse problem of measurement.They are due to the application of the method of the global extremum random search for the aim function (genetic algorithm).Hence, there are additional requirements regarding the repeatability and reproducibility of results that are regulated by regulatory documents on control and testing methods, for example [22].
The static and dynamic properties of the measuring systems [15] are not ideal.The static properties of the measuring channel are determined by its conversion function and they change if the function slope changes or in general it becomes non-linear one during some time of exploitation.In such situations, a static error of measurement appears in the system, which is nonlinearly dependent on the input action in the nonlinear conversion function of the MPC.Since the instantaneous values of the input signal are changed, the static error will change during the measurement process, but the law of its change is difficult to predict due to the dependence not only on the input signal value, but also on the type of conversion function and thedynamic properties of the system, which are determined by its bandwidth.The systematic static error of measurements depends on both the static errors of the MPC and the measurement method.When solving the inverse problem of measurement, the static error may additionally «deform».

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The dynamic component of the measurement error appears because of the limited bandwidth of the measuring channel [15].If this width does not exceed the width of the input spectrum, part of the measurement information is lost and the input signal restoration errors are increased.To solve the inverse problem of measurements it is necessary to normalize as a conversion function of a nonlinear non-inertial block of the MPC, and the pulse characteristic (constant time or bandwidth) of the linear inertial block of the channel.The normalization consists in setting the nominal values and the limits of the permissible values of the conversion function and the constant time, for which the inverse problem is solved correctly with the errors of restoration that do not exceed the permissible values.The recovery errors are calculated as the maximum difference between the values of the reference and the restored signals.In some cases, it is still necessary to save the shape of the restored signal so that it most closely coincides with the reference signal, which is implemented by the model standard.Since in practice the input signals are random processes, the similarity of the recovered and real signal is estimated using correlation properties, for example, the correlation coefficient.

SETTING MAXIMUM ACCEPTABLE ERROR OF SIGNAL RECOVERY
In order to provide the required quality of measurements, a risk management system is introduced in [23].The risk relates to unlikely measurements that adversely affect the decisions made after receivingthe measurement data.Risk assessment can be obtained based on the technique used in technical diagnosis [24].
Let's consider the risk of restoring the input signal with a maximum error that is more than acceptable one (the risk of bad recovery).We will consider two situations: 1) normal conditions under which there is no significant influence of the external environment on the measurement results, and the method of solving the inverse problem works qualitatively; 2) abnormal conditions in which the external environment, for example, noise significantly affects the measurement process, and the method shows its vulnerabilities, for example, determines a local minimum of the target function, and not the global one.For each case, the laws of errors distribution of the input signal restoration are normal and the average error values and mean square deviations of pressure for this example are:  5 kPa,  10 kPa,  2 kPa,  6 kPa.
Normal conditions correspond to the case of a li-near inertial channel with a low value of the constant time, as well as a nonlinear inertial channel whose conversion function is described by a polynomial of degree no more than two.In these conditions, the inverse problem of measurement is solved, as a rule, with errors that do not exceed the established values of maximum permissible errors, if the input pressure process has a non-wide spectrum.If these conditions are not fulfilled and the noise and errors level of the output signal measurement are large, the input signal recovery errors often exceed the permissible ones or even the whole process of solving the inverse problem diverges and has a high probability of phantom solutions.
To estimate the risk of poor recovery of the input signal, the threshold value of the error is selected.This value helps to make decisions about the quality of signal reconstruction.If , we consider that the solution of the inverse problem is carried out qualitatively (state of measurement ), and when , we assume that the results of the input signal recovery have a low confidence (state ).Let's formulate these conditions mathematically by creation a decision rule: when , . In fact, the Gaussian curves , cover a wide range and under both conditions (1) there are two curves at the same time.Let's denote the possible solutions for the rule (1) as The first index i is the number of the decision: -it is believed that the normal restoration of the input signal is carried out, and when , it is abnormal one.The second index j characterizes the actual state of recovery: -the actual recovery is normal, and when , the recovery is abnormal.Consequently, the hypotheses , the correct decisions (the solution corresponds to the state).The hypothesis that there was an abnormal restoration of the input signal, but it was decided that this recovery was normal.This will result in the omission of an invalid signal recovery error.The hypothesis on the contrary, indicates that in the normal restoration a decision is made about inadmissible error (false alarm).The probability of correct and incorrect decisions is defined as the area under the corresponding parts of the curves .The greatest interest in risk assessment is the wrong decision to restore the input signal.Probability of false alarms (2) where is a priori probability of anomalous restoration of the input signal.
Figure 1 shows the dependence of probability of presence the inadmissible error of signal recovery (solid line) and the probability of false alarms (dotted line) from the threshold error value at .If the probability of a normal state ( ) much higher than the probability of an abnormal state ( ), then the probability of presence the inadmissible error of signal recovery at the input of the MPC significantly decreases (Fig. 2), although the probability of false alarms increases.
Similar graphs are obtained for the case of imposing fines for every wrong decision [24].

CONCLUSIONS
In modern metrology there are no normative documents for the solution of inverse measuring problems.
The main provisions of the article allow to solve the problems of ensuring the unity of measurements for the inverse measurement problems, namely: * to form a model standard which forms the reference value of pressure; * on basis of minimum risk method application to determine the value of the maximum permissible error of the input signal MPC recovery; * to develop a normative document on the solution of the inverse problems for the measuring channels of the TCO, which will allow in practice to carry out the metrological confirmation of the measuring channels for their application in specific measuring tasks.

Fig. 2 .Fig. 1 .
Fig. 2. The dependence of probability of omission the inadmissible error of signal recovery (solid line) and the probability of false alarms (dotted line) from the threshold value of error at unequal probabilities of normal and abnormal states