INFORMATION MEASUREMENT SYSTEMS

I N I T I A L P R E R E Q U I S I T E S FOR C O M P U T I N G ERRORS

OF I N F O R M A T I O N M E A S U R E M E N T S Y S T E M S

M. A . Z e m e l ' m a n UDC 681.2.088.001.24

Information measurement systems (IMS) are being used on an increasingly wider scale. Their design and application entail the problem of computing their errors under working conditions and normal operation. In this connection it is important to know what requirements should be met by the applied computation methods, what should be the working conditions, and what properties of the system and its components should be taken into con- sideration in these methods. An IMS normally comprises an aggregate of primary and intermediate measuring transducers, secondary measuring, recording, and regulating instruments, computers, switching devices, and com- munication channels among the system components. An IM system is intended for

direct perception of primary data on the values of given parameters of the investigated objects or processes;

transformation of the primary data into a form suitable for remote transmission, recording, computer process- ing, human perception by an operator, reaction on controlling and regulating devices (if the IMS forms part of a control and regulation system);

remote transmission, recording, and indication of the transformed (intermediate) data;

processing on a computer intermediate and injected external data according to a given algorithm;

recording and display of computer processing results, transmission of signals to the controlling and regulating devices.

The IMS functional circuits are extremely varied. It is probably impossible to represent them in a general form. In Fig. 1 we show as an example one of the possible IMS circuits with n inputs.

This IMS circuit's output signals,which are denoted by y (indicating instruments' readings and recording results), represent the values of the signals x; the signals z (indicating instruments' readings and recording results)represent the values of certain given functions of the signals x and parameters p; thus z = f(x I . . . . Xnplpz)l the signals y cor- respond to the deviations of the signals x and functions z from values determined by the given programs yp and Zp. Therefore, this IM system comprises four types of conversion characteristics (functions) [1]:

yz = h (xi); i z = [2 (xl . . . . Xn, p~p~);

"r = [8 (xi, yp);

'Tz = f4 (xl . . . . Xn, plpezp)

(1)

The precision of an IMS is determined by the degree to which the actual transformation characteristics (1) occurring inthe course of operation under IMS working conditions correspond to the nominal transformation charac- teristics. The difference between the actual and nominal transformation characteristics corresponds to the IMS error of the respective outputs (A, B, C, and D)[2].

The IMS error is in a general case determined by the following factors: a) the components' own properties; b) the effect of external factors reacting on the components and the noninformative input signals' parameters; c) the difference between the input signals' frequency spectra and the nominal frequencies of input signals for which the nominal IMS conversion characteristics were adopted. Thus, an IMS error can be considered as consisting of three components corresponding to the above factors, namely the basic, additional, and dynamic errors. Moreover, the

Translated from Izmeritel 'naya Tekhnika, No. 4, pp. 14-16. April, 1973.

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Fig. 1

Z eeo

(t}

yp(t)

latter two depend directly on the external affecting quantities and on the input signals' noninformative parameters and their frequency spectrum. Therefore, both these quantities and the resulting error change substantially under the effect of the above factors, which can vary over a wide range in the course of an IMS utilization. Various com- ponents of a given IMS can then be subjected to the effect of different quantities to varying extents.

In selecting an IMS for utilizing it in a specific manner, it is necessary to know its resulting error when work- ing with given external effects under known operating conditions (nature of the processes whose parameters are being measured, method of feeding to the IMS various signals subject to transformation and pmcessessing, etc.). The resulting IMS error can be determined by means of three methods, comprising the computation, experimental, and combined methods.

The computation method is used when the error evaluations and other metrological characteristics [1] of all the IMS components either are known or can be calculated and when the algorithm (formula) for computing the re- suiting error of the IMS from the metrological characteristics of the components and from known effects and operat- ing conditions is known.

For the application of the experimental method it is necessary to know the investigation technique and to have an equipment suitable for reproducing the actual effects and operating conditions of an IMS as well as reference equipment for measuring with the required precision the input and output signals of an IMS under working conditions. Moreover, it should be borne in mind that the IMS error varies with time in a random manner,and therefore, the error parameters in the course of utilization and at the time of certification of an IMS can differ.

In using the combined method, the resulting error is determined by calculation, but the computation formula includes both given (known) quantities and quantities whose values are determined experimentally.

The basic deficiency of the computation and combined methods consistsof the lack of sufficiently substan- tiated and practical means for calculating the resultant IMS error. Moreover, the normally known initial data are insufficient for a substantiated computation. However, these methods are very necessary since in many cases they are the only possible ones. For instance, they are required in designing technological processes, when it is neces- sary to select the IMS structural schematic and its components.

Since any method for computing errors is to a certain extent approximate, it is sometimes preferable to use the experimental method (individual certification of IM systems). However, in many cases the experimental evalu- tion of the Ibis error only appears to be highly precise since it can be attained only if the above-mentioned condi- tions are met. This means that during certification all the IMS components must be maintained under conditions identical to those prevailing in the course of the system's operation. This includes the noninformative parameters and the input signal's frequency spectrum. Moreover, the IMS error characteristics should not change substantially during the time from the instant of their evaluation up to the system's application. If the above conditions cannot be met, the experimental investigations will serve to evaluate only the resulting error components of a working system, which depend only on the properties of the IMS components and do not change with time, in particular, the systematic component of the basic error.

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It is known from the experience in applying IM systems that it is often impossible to produce the conditions required for the experimental method. Therefore, it is advisable to use this method when it is known that the pre- dominant part of the IMS resulting error consists under utilization conditions of the basic error's systematic com- ponent which is invariable with time or when it is necessary to evaluate this component for subsequent application of corrections of the measurement results, In the latter case it becomes necessary to evaluate by means of other methods the IMS error which remains under working conditions after the application of corrections.

From the above it is possible to arrive at the conclusion that computation methods for evaluating the IM8 error are exceptionally important.

A large number of works describing methods for computing the error of an IMS from known errors of its com- ponents have appeared in the last decade. Among them it is possible to distinguish two extreme groups, one of which comprises the simpler elementary but rougher methods, and the other the more complex computations which are assumed to be precise.

The first group comprises [3, 4]. In [3] the error components are considered as independent random quanti- ties with a uniform distribution law within the tolerance specified for each of them and the mathematical expecta- tions equal to zero. A formula is provided for the interval within which the system's error lies with a probability of 0.99. The interval is determined as the square root of the weighted (with the transformation coefficient of each component taken into consideration) summation of error components' variances multiplied by 2.58. The variances are determined by means of the given error component tolerances and the well-known formula for the uniform dis- tribution law.

In [4] the error components are also considered as independent random quantities, but no limitations are speci- fied for their distribution law. It is assumed that their mathematical expectation and variances or root-mean- square deviations are known. It is recommended to find the mathematical expectation and variance of the result- ing IMS relative error. The former is determined as the algebraic sum of the components' relative-errormathemati- col expectations, and the latter as the summation of their variances.

An example of a more complicated way of computing the IMS error consists of the method described in [5]. The IMS error components are considered in this method as random quantities. For computation purposes three problems are considered: 1) when a single measured quantity is fed to the IMS input; the conditional (for a given input signal value) distribution laws of the error components are given; it is required to determine the IM8 error's distribution law, mathematical expectation, and variance; 2) similar to the first case, but with several measured quantities fed to the IMS inputs; 3) only tolerances (intervals) for the error components are given; it is required to determine the IMS error's interval.

In solving the first problem the IMS distribution law is determined as the composition of the error components' conditional distribution laws and the input signal distribution law. A general formula is provided (when the given distribution laws are represented analytically in a general form) which is of little use for practical calculations. Two particular cases are then examined:

a) when components consist of linear links; b) when the components' transformation characteristics consist of differentiable functions and the components' errors are small. In these two cases the IMS error's mathematical expectation and variance are determined from a general formula as weighted summations of the component errors ~ mathematical expectations and variances respectively.

The second problem (several measured quantities) is solved by representing the resultant error distribution by means of a general formula which can then be simplified provided the error components are independent. This condition can hold for basic errors, but it is unlikely to be met in a general case of comt~onent errors under working conditions, when external quantities have a considerable effect on the error.

In solving the third problem (when only toIerances for the components' errors are provided) the possible parti- cular cases are considered whose given initial data may be adequate for the problem's solution. This material is based on the recommendations [6].

It will be seen from the above that the IMS components' errors are considered as random quantities and the calculatio~ of the error is reduced to given methods of adding random quantities. Virtually all the works known to us in this field are characterized, irrespective of the problems' degree of generality, by a given abstraction of the IMS components' working properties and conditions. In solving any engineering problem certain idealization of the object under consideration is inevitable. However, in this case important IMS features which cannot be neglected have not been taken into consideration.

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The following are the basic conditions which must be taken into account in developing methods for computing

IMS errors.

1. The IMS components consists of dynamic links with given nominal transformation characterist ics, dynamic characterist ics, influence functions, as well as input and output impedances [1].

2. The IMS components ' errors are determined by the characterist ics normal ized for the given type of com- ponent. They must represent the properties of the error as a random process with its frequency spectrum.

For calcula t ing the IMS basic error as a metrological character is t ic which represents cer ta in properties of the IMS proper, i rrespective of the specific effects and operating conditions, i t is sufficient to take into account the two above-ment ioned conditions [7]. The basic IMS error can serve as one of the cri ter ia in comparing different IM systems to one another. However, in developing computat ion methods for evaluat ing IMS errors under actual effects and operating conditions, it is necessary also to take into account the conditions described beIow.

3. It is necessary to set for each IMS component the conditions of its u t i l iza t ion in the form of an aggregate of affecting quantities with an indicat ion of their values (if they can be considered constant), the max imum dev ia - tions from normal values ( if it is required to evaluate only the maximum IMS errors instead of their s ta t is t ical char-

acteristics), or the distr ibution-law pamameters , for instance, the affecting quanti t ies ' ma themat i ca l expecta t ion and variance (if it is necessary to determine the IMS stat is t ical characterist ics under u t i l iza t ion conditions). If ce r - ta in of the affecting quantities can change in the course of the IMS operation at a sufficiently rapid rate to bring into play the inertia of the IMS component e lements which react in turn on the given affecting quantity, then its rate of change character is t ic should be provided.

4. If the IMS input signal 's frequency spectrum differs in the course of the IMS operation from the frequency

taken as nominal (corresponding to the nominal components ' transformation character is t ic) to such an extent that

these deviations produce not iceable changes in the IMS output signal, then the IMS error in the operating condition must be ca lcu la ted as a dynamic one. For this purpose the IMS input signal 's frequency spectrum must be given.

It will be seen from clauses 1 and 2 that it is impossible to ca lcu la te in a substantiated manner the MS error

on the basis of the normal izat ion methods of the measuring equipment ' s metrologica l character is t ics provided in the

existing normal iz ing and technica l documents, since the normal ized characteris t ics do not represent the IMS compo-

nents' properties which affect substantially the IMS error. The necessary prerequisites for computing the IMS error

from the components ' normal ized characterist ics are provided in the All-Unin State Standard (GOST) 8.009-72 [1], which comes into force on January 1, 1974, since it specifies the normal iza t ion of the required characterist ics. How-

ever, it should be noted that the characterist ics specif ied in GOST 8.009-72 refer direct ly only to measuring instru- ments which can be considered as l inear dynamic links. The specif icat ion of nonlinear components is considerabIy

more compl ica ted . The above-ment ioned considerations, factors, and conditions apply both to the nonlinear and l inear components , however there do not as yet exist general methods for non-l inear c i rcui t computations which could be used as a basis for general IMS nonlinear error calculat ions suitable for pract ical appl icat ion. It would appear that each specific nonlinear IMS should be considered separately a iming at the appl ica t ion of known approx- imate computat ion methods or digi tal methods for s imulat ing nonlinear circuits and evaIuating their errors.

Thus, the methods for computing the IMS error should be suitable for evaluat ing i t under the u t i l iza t ion effects in the operating conditions on the basis of the above-ment ioned given working conditions and properties of the IMS components. The next a r t ic le will deal with l inear IMS errors' computations which meet the above-ment ioned requireme nts.

L I T E R A T U R E C I T E D

1. GOST 8.0009-72

2. M . A . Zeme l 'man , Automatic Correction of Measuring-Device Errors [in Russian], Izd. Standartov, Moscow (1972).

3. T.B. Jawar, "Estimating total errors in large control systems," Control Engineering, No. 10 (1963). 4. "Die Fehlerfertplf ianzug in der Messkette. Eine kleine Fehlertheorie filr Praktiker," Archiv flit Technisches

Messen, No. 437 (1972).

5. I . M . Shenbrot and M. Ya. Ginzburg, Calculat ing the Precision of a Centxalized Testing System [in Russian], nerglya, Moscow (1970).

6. B.E. Rabinovich, "Technique for adding frequency errors in the field of radiotechnical measurements," Trans- actions of the Commit tee of Standards, Measures and Measuring Instruments, No. 57 (117), Standartgiz (1962).

7. GOST 16263-70.

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