The potential accuracy of measurements is a problem that both experimental physi- cists realizing high precision physical experiments and metrologists developing mea- surement standards of units of physical quantities to be measured constantly come across. However, until now, unlike other measurement problems, this problem has not received proper study and analysis in metrology literature. Moreover it was never pre- sented in the form of an separate part (section) in metrology reference books, training aids, and particularly manuals.
This circumstance has affected our text here since it required the development of a particular approach to the problem as well as the introduction of a number of additional concepts.
3.1.1 Concept of a system approach to the formalized description of a measurement task
The main point of the system approach in any study of some object lies, firstly, in an analysis of this object as asystemconsisting of interconnected elements. However, apart from individuating the elements into the object and the determining links be- tween them (i.e., determining the system structure), in this approach it is also very important to determine the boundaries of the system and its interaction with its “envi- ronment”, to study “inputs and outputs” of the system and the processes of transforma- tion between them. For systems connected with human activities (i.e., organizational and controlled activities) it is also very important to determine the aims and tasks of the system connected to its purpose.
As mentioned in Chapter 2 the most adequate mathematical apparatus used in form- ing a description and studying various systems is theset theory apparatus.
The language of the set theory is a universal mathematical language: any mathemat- ical statement can be formulated as a statement about a certain relationship between sets. Let us remember some basic elements of this “language” required to understand the material below.
The concept ofsetas the original, fundamental concept of set theory is indefinable.
Intuitively by a set one means a totality of certain quite distinguishable entities (ob- jects, elements), which is considered as a single whole. Practically it is the equivalent of the concept of “system”. Separate entities forming a set are called elements of the
Section 3.1 System approach to describing a measurement 165 set. The general notation of a set is a pair of figure bracesạ: : :º, within which the set elements are located.
Specific sets are denoted with capital letters of the alphabet, whereas set elements are denoted with small letters. Membership of an element “a” with respect to a set
“M” is denoted asa2M (“abelongs toM”).
Example 3.1.
M1D ạAlekseev, . . . , Ivanov, . . . , Yakovlevºis the set of students in a group;
M2D ạ0, 1, 2, 3,: : :ºis the set of all natural numbers;
M3D ạ0, 1, 2, 3,: : :,Nºis the set of all natural numbers not greater thanN; M4D ạx1,x2,: : :,xnºis the set ofnstates of any system ;
M5Dthe football team “Zenit” (i.e., the set of its football players);
M6Dthe set of all football teams of the premier league; it is seen thatM52M6. The last example shows that an element of a set can also be a set.
There are two methods of assigning sets: enumeration and description. Examples of the first method (enumeration of all elements belonging to the set) are the assignments of all setsM1–M6indicated above.
The description method consists in indicating a characteristic propertyp.x/, inher- ent in all elements of the setM D ạxjp.x/º. Thus, the setsM2,M3,M5in Example 3.1 can be assigned the following description:
M2D ạxjxintegerº; M3D ạx2M2jxNº;
M5D ạxjxfootball player of the team “Zenit”º.
Two sets are called equal if they consist of the same elements. whereby the order in the set is unessential:.3, 4, 5º D ạ4, 3, 5º. From this it also follows that a set may not have distinguishable (equal) elements: the notationạa,b,c,bºis not correct and has to be changed toạa,b,cº.
The setXis called asubsetof setY if any element ofXis the element ofY. In the notation:X Y the symbolis the sign of being contained (“Y containsX”). So, ifM1is the set of students of a group, andM10is the set of excellent students of the same group, thenM10M1, since an excellent student is simultaneously a student.
Sets can be finite (i.e., they consist of a finite number of elements) and infinite. The number of elements in a finite setMis called aset power. The set power is frequently denoted asjMj.
Very important is the concept of anordered set(or avector) as the totality of el- ements in which every element has a definite place, i.e., in which the elements form a sequence. At the same time the elements are calledcomponents(orcoordinates) of a vector. Some examples of ordered sets (vectors) are: a set of people standing in a queue; a set of words in a phrase; the numbers expressing the geographical longitude and latitude of a point on a site, etc.
To denote ordered sets (vectors) angle brackets h: : :i or round brackets.: : :/are used. The power of such sets is called the vector length. Unlike the usual sets, in or- dered sets (vectors) there can be similar elements: two similar words in a phrase, etc.
3.1.2 Formalized description of a measurement task
In previous works carried out earlier [37] the elements of the system approach have already been used. In particular some separate components of measurement were con- sidered, a general aim of any measurement was determined, and the stages of succes- sive transformations in the process of solving a measurement task were revealed, etc.
(see also Section 2.1.2.3).
Systemizing all previously used elements of the system approach, let us turn to mea- surement elements (as those of a system) arising at the stage of setting a measurement task.
Bya measurement taskwe mean the description (formulation) of a problem mea- surement situation precedeing the preparations for carrying out a measurement, i.e., formulation of a task: “what do we wish to make?” It is obvious that for this purpose it is necessary to know (to indicate, to determine): what quantity has to be measured, on which object should be done, under what environmental conditions, at what accuracy, and within which space–time boundaries all this should be realized.
Using the formalism of set theory, the measurement task can be written in the form of the following ordered set (vector):
Zi h'i,oi, i,'i,gi,ti,ti,: : :i, (3.1) where
'iis the measurand as a quality;
oiis the object of study (the measurand carrier);
iis the conditions of measurements (the totality of external influencing factors);
'i is the confidence interval within which it is necessary to get a required value of the measurand (at a given confidence probability);
giis the form of presenting a measurement result;
ti is the moment of time when the measurement is carried out;
ti is the time interval within which it is necessary to carry out the measurement.
These are all the most important elements which are significant for setting any mea- surement task. It is important to note that a complete and correct formulation of the measurement task in itself is complicated, which will be seen from examples given below. At the same time, setting a measurement task in an incorrect and incomplete manner can result in either the impossibility of performing the measurement or its incorrect results.
The setting of a particular measurement task in its turn depends on the practical goal which can be reached for making a decision only on the basis of obtained measure-
Section 3.1 System approach to describing a measurement 167 ment information. This practical goal has to be connected with the measurement task with a chain (sometimes a very complicated one) of logical arguments of measure- ment information consumers. Correctness and completeness of the measurement task formulation entirely depend on the experience and skill of a measurement informa- tion consumer. However in any case this logical chain should be finished with a set of parameters concretizing the elements of a set (3.1).
Example 3.2. Let the final practical goal be the delivery of a cargo to point A within a given time interval (t0). To do this, it is necessary to make a decision about when (at what moment of time) (tn) has to go from a resting position (point B) to a railway station. Making the measurement problem more concrete, let us specify first of all that the measurement task has to be the time, or more precisely, the time intervalTn, which allows the distance B to A to be covered.
In this case an object of study is the process (the process of moving from B to A).
It is evident that this process should be detailed according to the method of moving (the kind of transportation and the route). The accuracy at which it is necessary to determine T depends on the deficiency of time. Provided the conditions are such, that it is possible to reach the railway station with five minutes to spare (T0TN D TnC5 min), then it is enough to assume a confidence interval equal to˙1 min at a confidence probability of 0.997.
Consequently, the root mean square deviation of a measurement result should be T Š20 s. A digital form for presenting the result is preferable. The time moment before which the result (Tr) should be obtained has to precedeTn:Tr Š Tn(hour), nbeing determined by the daily routine of a goer.
It is obvious that even for one quantity (measurand) there exists an infinite set of different settings of measurement tasks:
ạZi.'/º D hạoiº,ạ iº,ạ'iº,ạgiº,ạtiº,: : :i,
oi 2 ạo1,o2,: : :º, i 2 ạ 1, 2,: : :º, etc., (3.2) determined by a variety of realizations of the elements entering (3.2). In practice, how- ever, only individual groups of measurement tasks are allocated, which are close not only in the parameters of their elements, but also in the methods used for solving these tasks.
3.1.3 Measurement as a process of solving a measurement task
After introducing the measurement problem, a further process as a whole, connected with measurements (more precisely, with the achievement of a final goal of measure- ment, i.e., finding a value of a measurand) has to be considered as a process of solving this measurement task. In this process three stages are usually distinguished.
In the first stage, in accordance with the measurement task, reflecting the ques- tion “what should be done?”,a plan of the measuring experimentis developed which
makes the question more exact: “How should it be done?” At this stage one chooses the measurand unit, method and the required measuring instruments, the procedure for using them is defined more exactly, and then the methods and means of experimental data processing are specified. Moreover, at this stage an operator (experimenter) who can perform the experiment is determined.
Information corresponding to the measuring experiment plan is usually provided in the form ofa procedure for performing measurements. By analogy with (3.1) it can be structurally formalized in the form of an ordered set:
Uz hŒ' ,m,s,v,w,: : :i, (3.3) where
Œ' is the unit of the measured physical quantity;
mis the selected method of measurements (or an algorithm of using the selected tech- nical meanssandw);
sis the measuring instruments used for solving a given measurement task;
vis the operator (experimenter) realizing the plan of a measurement experiment;
w is the means of processing results of the measurement experiment (or some other auxiliary means);
(here everywhere the index “I”, meaning a membership of the setUzand that of each of its elements to the given measurement taskzi, is omitted).
During the same stage, when using the plan which has been set up, a preliminary estimation of the uncertainty of an expected result'O (an interval of measurand un- certainty), is made, and then this estimate is compared with the required accuracy of measurement (').
In the second stage of the measurement process the measurement experiment is realized, i.e., the process of real (physical) transformations connected with a physical interaction of the selected measuring instruments with the object and environmental conditions (with participation of the operator) is carried out which results in a response of the measuring instrument to the quantity measured (indicationxs).
Finally, in thethirdand finalstageof the measurement procedure the obtained re- sponse of the measuring instrument is transformed into a value of the measurand, and an a posteriori degree of its uncertainty (in case of multiple measurements) is esti- mated, thereby achieving the final goal of measurement, i.e., getting a result corre- sponding to the solution of the posed measurement problem.
All transformations and operations in the second and third stages are realized in accordance with the measurement experiment plan developed at the first stage.
A general scheme of the measurement process (the structure of a measuring chain) is shown in Figure 3.1.
Section 3.1 System approach to describing a measurement 169
U2
Zi Iz
In
Jv(z)
{S}i
Oi Vi I2
{Ψ}i
h3
Js(Iz)
Vi
Iz
hs Wi Ip
Jw(Iz)
Figure 3.1.Stages of a measurement as those of the process of solving a measurement task.
Here:JOv.z/,JOs.Iz/,JOw.Iz/are the operators of a successive transformation of a priori infor- mationIa into a posteriori information at different stages of the measurement process; the remaining notations are explained in the text.
3.1.4 Formalization of a measurement as a system
From the above description of the measurement chain structure it is seen that it con- tains both the components of the measurement task (3.1) and the components of the measurement plan (3.3). Combining these sets gives a new set:
IZ hZi,Uzi, (3.4)
whereZi Iz,Uz Iz, which corresponds to a particular singular measurement, related to the system as a whole.
In equation (3.4) the elements contained in Zi are uncontrollable measurement components (i.e., given in advance in the given measurement task), and elements con- tained inUi arecontrollable(selected in planning the measurement experiment).
With regard to formal description (3.1), (3.3), and (3.4) we note the following.
1) None of the indicated sets replaces a detailed substantial analysis of a particular measurement situation which makes necessary the carrying out of a particular mea-
surement. The sets indicated serve as theguideline for this analysis, underlining those components of measurement which are present in each of them, and for each of them it is necessary to have quite sufficient information.
2) All components of measurement in equation (3.4) appear (from the point of view of the information about them) at the stage of posing the measurement task and at the stage of planning the measurement experiment, i.e., before carrying out the measure- ment procedure itself (the experiment). This once again underlines the great – as a matter of fact deciding – role of the a priori information in measurements.
The statement according to whichany particular measurement requires the avail- ability of definite (final) a priori informationabout components of measurement (3.4) is so universal that it also plays a part of one otherpostulate of metrology. An ade- quately convincing illustration of this statement is given in the preceding sections of this volume and will be supported further on.
3) As a consequence of the measurement problem analysis, at the stage of planning the measurement experiment it is understood that the experiment as such is not needed.
An amount of a priori information can occur which is sufficient to find a value of measurement information at a required accuracyby calculation.
4) In planning the measurement experiment it is possible to reveal more than one version of the solution for the measurement task which satisfies its conditions. In this case, the choice of a final version is based on economic considerations (the version requiring minimal cost is chosen).
3.1.5 Target function of a system
The goal functionof a systemIz is determined by the general destination (goal) of measurement, i.e., getting information about a value of the measurand'with the re- quired accuracy characterized by an interval'. Therefore, the goal of measurement will be achieved when the condition given below will be met:
'norm 'meas Š j'hmeas'lmeasj, (3.5) where'measis the value obtained as a result of measurement, and'measis the esti- mate of the uncertainty of the value obtained for the measurand.
Example 3.3. The measurement task described in Example 3.2 can be solved by a number of methods, depending on the availability of different a priori information.
1) If a clear-cut train schedule for the delivery of transported goods from point B to point A is available (accurate within 1 min), then, using the schedule, a train departure timeTnis calculated from the conditionTnDT0Tn5 min.
2) If there is no train schedule, then having chosen the kind of transportation (bus, taxi or pedestrian method of moving) based on economic considerations, a method, measuring instrument, and operator are chosen. As a measuring instrument a simple
Section 3.2 Potential and limit accuracies of measurements 171 chronometer can be used (electronic wristwatch) which provides the required accuracy of measurements. At the same time the method of measurement is the method of direct readout performed by the operator, who is the person who wishes to leave from the railway station.
3) If is a question of whether or not the external conditions in moving from point B to point A accidentally influence the time of moving, then multiple measurements are carried out under different conditions , and a dispersion or root-mean-square deviation of the values measuredTn.i /are found.
4) In the above various extreme situations are not taken into consideration (traf- fic congestions, unexpected ice-covered ground, absenteeism or nonfunction of trans- portation, etc.). These anomalies can be neglected if their probability.10.997/D 0.003. In the opposite case this probability has to be taken into account within a confi- dence interval, and when that increases it is necessary to correspondingly increase the time reserve for the departure from point B.