Mathematical models
It is extremely important to describe the physical models constructed above using symbols in the form of mathematical formulas and equations. These symbols—parameters of objects (they also denote physical quantities)—are interconnected in the form of the above-formulated physical laws.
A set of formulas and equations that establish a connection between these parameters (physical quantities) based on the laws of physics and obtained within the framework of selected physical models will be called a mathematical model of an object or process.
Consequently, physical quantities can be spoken of as parameters that characterize both qualitatively and quantitatively constructed physical models.
The process of creating a mathematical model can also be divided into 3 stages:
Stage 1. Drawing up formulas and equations that describe the state, movement and interactions of objects within the framework of selected physical models.
Stage 2. Solution and research of purely mathematical problems formulated at the first stage. The main issue here is the solution of the so-called direct problem, ᴛ.ᴇ. obtaining theoretical consequences and numerical data. At this stage important role plays mathematical apparatus and computing technology (computer).
Stage 3. Determining whether the results of analysis and calculations agree with the results of measurements within the accuracy of the latter. The deviation of the calculation results from the measurement results indicates:
Or about the incorrectness of the applied mathematical methods;
Or about the incorrectness of the accepted physical model;
Or about the incorrectness of the measurement procedure.
Determining the sources of errors requires great skill and highly qualified researchers.
It happens that when constructing a mathematical model, some of its characteristics or relationships between parameters remain uncertain due to the limitations of our knowledge about physical properties object. For example: sometimes it turns out that the number of equations describing the properties of an object and the connections between objects is less than the number of parameters (physical quantities) characterizing the object. In these cases, it is necessary to introduce additional equations characterizing the object and its properties, sometimes even trying to guess these properties, so that the problem is solved and the results correspond to the experimental results within a given error. Problems of this type are called inverse.
The problem of the reliability of our ideas about the world around us, ᴛ.ᴇ. the problem of correspondence between a model of an object and a real object is a key problem in the theory of knowledge. Today it is generally accepted that the criterion of the truth of our knowledge is experience. The model is adequate to the object if the results of theoretical studies (calculations) coincide with the results of experiments (measurements) within the error limits of the latter.
Errors occur not only in measurements, but also in theoretical modeling. For theoretical models, in accordance with the nature of their occurrence, we will distinguish:
Errors arising during the development of a physical model;
Errors arising when compiling a mathematical model;
Errors arising when analyzing a mathematical model;
Errors associated with a finite number of digits in calculations.
In the latter case, for example, the number π within the symbolic notation as the ratio of the circumference to the diameter represents exact number, but an attempt to write it in numerical form (π=3.14159265...) causes an error associated with a finite number of digits.
The listed errors always occur. It is impossible to avoid them, and they are called methodical. During measurements, methodological errors manifest themselves as systematic.
Example: errors in the physical and mathematical model of a pendulum that arise when measuring the period of oscillation of a pendulum in the form of a body suspended on a thread.
Physical model of a pendulum:
The thread is weightless and inextensible;
Body is a material point;
There is no friction;
The body makes a plane motion;
The gravitational field is uniform (ᴛ.ᴇ. g=const at all points in space where the body is located);
There is no influence of other bodies and fields on the movement of the body.
Obviously, a real body should not be a material point; it has volume and shape; during movement or over time, the body is deformed. At the same time, the thread has mass, it has elasticity and is also deformed. The movement of the pendulum is influenced by the movement of the suspension point, caused by the action of vibrations that always take place. Also, the movement of the pendulum is affected by air resistance, friction in the thread and the method of its attachment, external magnetic and electric fields, inhomogeneity gravitational field Earth and even the influence of the gravitational field of the Moon, Sun and surrounding bodies.
The listed factors, in principle, can be taken into account, but this is quite difficult to do. To do this, you will need to involve almost all branches of physics. Ultimately, taking these factors into account will significantly complicate the physical model of the pendulum and its analysis. Not taking into account the listed, as well as many other factors not mentioned here, significantly simplifies the analysis, but leads to research errors.
Mathematical model of a pendulum:
within the framework of the selected simplest physical model, the mathematical model of the pendulum - the differential equation of motion of the pendulum - has the following form:
, (1), where L– thread length; φ – deviation of the body from the equilibrium position.
At φ<<1 обычно считают, что sinφʼʼφ, and then the equation of motion is written:.(2)
This is a linear differential equation that must be solved exactly. This solution has the form 
, Where . It follows that the period of oscillation of the pendulum T 0 =2p/w 0 does not depend on the amplitude φ 0 . At the same time, this solution cannot be considered an exact solution to the problem of pendulum oscillations, represented by the simplest physical model, since the original equation (1) was different.
You can clarify the solution. If you expand sinφ in a series and take into account at least the first two terms of the expansion, ᴛ.ᴇ. assume that sinφʼʼφ+φ 3 /6, then solving the differential equation will become significantly more complicated. It can be written approximately in the form 
, Where 
. It follows that in this approximation the period of oscillation of the pendulum T=2p/w depends on the amplitude of oscillations according to the parabolic law.
Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, error of the mathematical model (equation (2)) associated with the replacement sinφ by φ, leads to an error in the result of calculating the period of oscillation of the pendulum. An estimate of this error must be obtained from solving the problem in the second approximation.
The problem of constructing and analyzing a mathematical model of a research object with a given accuracy, as well as assessing the error of calculations in some cases is very difficult. A high mathematical culture of the researcher is required, a thorough mathematical analysis of both the model itself and the solution methods used is required.
For example, it makes no sense to require that equation (1) be solved with an accuracy significantly greater than the accuracy of constructing a physical model. In particular, in the previous example there is no point in replacing sinφʼʼφ+φ 3 /6 instead of sinφʼʼφ if the thread is noticeably deformed or the air resistance is high.
The use of computers has significantly increased the possibilities of constructing and studying mathematical models in technology, but one should not think that perfect knowledge of mathematics, numerical methods and programming languages will allow solving any physical and applied problem. The fact is that even the most elegant and accurate calculation methods cannot correct the errors made when constructing a physical model. Indeed, if the length L is not constant, or if the dimensions of the body are comparable to the length of the thread, or the friction is high and the oscillations of the pendulum quickly decay, then even an absolutely accurate solution of equation (1) will not allow one to obtain an exact solution to the problem of oscillations of the pendulum.
General characteristics of the concept of “measurement” (information from metrology)
In metrology, the definition of the concept of “measurement” is given by GOST 16.263-70.
Measurement– scientifically based experience to obtain quantitative information with the required or possible accuracy about the parameters of the measured object.
Measurement includes the following concepts:
Object of measurement;
Purpose of measurement;
Measurement conditions (a set of influencing quantities that describe the state of the environment and objects);
Method of measurement, ᴛ.ᴇ. a set of techniques for using principles and measuring instruments (the principle of measurement is a set of physical phenomena that form the basis of measurement);
Measurement technique, ᴛ.ᴇ. an established set of operations and rules, the implementation of which ensures that the necessary results are obtained in accordance with a given method.
Measuring:
▪ measuring transducers,
▪ measuring instruments,
▪ measuring installations,
▪ measuring systems,
▪ measuring and information systems;
Measurement results;
Measurement error;
Concepts characterizing the quality of measurements:
▪ reliability(characterized by confidence probability, ᴛ.ᴇ. the probability that the true value of the measured quantity is within the specified limits);
▪ right(characterized by the value of systematic error);
▪ convergence(closeness to each other of the results of measurements of the same quantity, performed repeatedly using the same methods and means and under the same conditions; reflects the influence of random errors on the result);
▪ reproducibility(closeness to each other of the results of measurements of the same quantity, carried out in different places, by different methods and means, but brought to the same conditions).
Errors of theoretical models - concept and types. Classification and features of the category "Errors of theoretical models" 2017, 2018.
In general, measurement results and their errors should be considered as functions that vary randomly over time, i.e. random functions, or, as they say in mathematics, random processes. Therefore, the mathematical description of the results and measurement errors (i.e., their mathematical models) should be built on the basis of the theory of random processes. Let us outline the main points of the theory of random functions.
By a random process X(t) is a process (function), the value of which for any fixed value t = tQ is a random variable X(t). A specific type of process (function) obtained as a result of experience is called implementation.
Rice. 4. Type of random functions
Each realization is a non-random function of time. The family of implementations for any fixed value of time t (Fig. 4) is a random variable called cross section random function corresponding to time t. Consequently, a random function combines the characteristic features of a random variable and a deterministic function. With a fixed value of the argument, it turns into a random variable, and as a result of each individual experiment it becomes a deterministic function.
Mathematical expectation random function X(t) is a non-random function which, for each value of the argument t, is equal to the mathematical expectation of the corresponding section:
where p(x, t) is the one-dimensional distribution density of the random variable x in the corresponding section of the random process X(t).
Variance random function X(t) is a non-random function whose value for each moment of time is equal to the dispersion of the corresponding section, i.e. dispersion characterizes the spread of realizations relative to m (t).
Correlation function- non-random function R(t, t") of two arguments t and t", which for each pair of argument values is equal to the covariance of the corresponding sections of the random process:
The correlation function, sometimes called autocorrelation, describes the statistical relationship between the instantaneous values of a random function separated by a given time value t = t"-t. If the arguments are equal, the correlation function is equal to the variance of the random process. It is always non-negative.
Random processes that occur uniformly in time, partial implementations of which oscillate around the average function with a constant amplitude, are called stationary. Quantitatively, the properties of stationary processes are characterized by the following conditions:
The mathematical expectation is constant;
The cross-sectional dispersion is a constant value;
The correlation function does not depend on the value of the arguments, but only on the interval.
An important characteristic of a stationary random process is its spectral density S(w), which describes the frequency composition of the random process for w>O and expresses the average power of the random process per unit frequency band:
The spectral density of a stationary random process is a non-negative function of frequency. The correlation function can be expressed in terms of spectral density
When constructing a mathematical model of measurement error, all information about the measurement being carried out and its elements should be taken into account.
Each of them can be caused by the action of several different sources of errors and, in turn, also consist of a certain number of components.
Probability theory and mathematical statistics are used to describe errors, but first it is necessary to make a number of significant reservations:
The application of methods of mathematical statistics to the processing of measurement results is valid only under the assumption that the individual readings obtained are independent of each other;
Most of the probability theory formulas used in metrology are valid only for continuous distributions, while error distributions due to the inevitable quantization of samples, strictly speaking, are always discrete, i.e. the error can only take a countably many values.
Thus, the conditions of continuity and independence for measurement results and their errors are observed approximately, and sometimes are not observed. In mathematics, the term “continuous random variable” is understood as a significantly narrower concept, limited by a number of conditions, than “random error” in metrology.
In metrology, it is customary to distinguish three groups of characteristics and error parameters. The first group is the measurement errors (standards of errors) specified as required or permissible standards for characteristics of measurements. The second group of characteristics are errors attributed to the totality of measurements performed according to a certain technique. The characteristics of these two groups are used mainly in mass technical measurements and represent probabilistic characteristics of the measurement error. The third group of characteristics—statistical estimates of measurement errors—reflect the proximity of a separate, experimentally obtained measurement result to the true value of the measured quantity. They are used in the case of measurements carried out during scientific research and metrological work.
A set of formulas describing the state, movement and interaction of objects obtained within the framework of selected physical models based on the laws of physics will be called mathematical model of an object or process. The process of creating a mathematical model can be divided into a number of stages:
1) drawing up formulas and equations that describe the state, movement and interaction of objects within the framework of the constructed physical model. The stage includes recording in mathematical terms the formulated properties of objects, processes and connections between them;
2) study of mathematical problems that are approached at the first stage. The main issue here is the solution of the direct problem, i.e. obtaining numerical data and theoretical consequences. At this stage, mathematical apparatus and computing technology (computer) play an important role.
3) finding out whether the results of analysis and calculations or consequences from them are consistent with the results of observations within the accuracy of the latter, i.e. whether the adopted physical and (or) mathematical model satisfies the practice-the main criterion for the truth of our ideas about the world around us.
The deviation of the calculation results from the observation results indicates either the incorrectness of the applied mathematical methods of analysis and calculation, or the incorrectness of the adopted physical model. Determining the sources of errors requires great skill and highly qualified researchers.
Often, when constructing a mathematical model, some of its characteristics or relationships between parameters remain uncertain due to the limited knowledge of our knowledge about the physical properties of the object. For example, it turns out that the number of equations describing the physical properties of an object or process and the connections between objects is less than the number of physical parameters characterizing the object. In these cases, it is necessary to introduce additional relationships characterizing the object of study and its properties, sometimes even trying to guess these properties, so that the problem can be solved and the results correspond to the experimental results within a given error.
Information correction of variable systematic errors of measuring instruments and measuring information systems
Reviewer: Tuz Yu.M.
Director of the Research Institute of AEI, Doctor of Technical Sciences, Prof., laureate of the State Prize of Ukraine in the field of science and technology
Introduction
Requirements for accuracy, correctness and convergence of measuring instruments are constantly increasing. Increasing the requirements was usually carried out by moving from the one used to a new physical measurement principle, which ensured higher quality measurements. At the same time, the methodology and technology for carrying out measurements were improved, and the requirements for a set of normal (standard) conditions accompanying the measurement process were tightened.
Any measuring device, system, channel “reacts” not only to the measured value, but also to the external environment, because inevitably connected with it.
A good illustration of this theoretical thesis can be the effect of tidal waves caused by the Moon in the earth's crust on the change in the energy of charged particles produced at the large ring accelerator at the Center for European Nuclear Research. The tidal wave deforms the 27-kilometer (2.7·10 7 mm) accelerator ring and changes the path length of particles along the ring by approximately 1 mm (!). This results in a change in the energy of the accelerated particle by almost ten million electron volts. These changes are very small, but exceed the possible measurement error by about ten times and have already led to a serious error in the measurement of the boson mass.
Formulation of the problem
Metrological support for radio-electronic measurements can be characterized by the following typical problems. The use of theoretical methods for analyzing the influence of environmental factors on the errors of measuring instruments is difficult. The nature of the influence is complex, unstable, difficult to interpret from the standpoint of logical and professional analysis by a specialist; changeable when moving from instance to instance of the same type of measuring instruments.
It is noted that it is methodologically difficult to obtain dependencies of an unknown type on several variables and that “... the possibilities of studying the dependencies of the error on environmental factors are very limited and not very reliable, especially with regard to the joint influences of factors and dynamic changes in their values.”
As a result of the above reasons and the significant diversity of their manifestation, it is concluded that for a group of measuring instruments of the same type, the most adequate description of the errors of measuring instruments from influencing environmental factors should be recognized as a zone of uncertainty, the boundaries of which are determined by the extreme dependencies of the instances.
The indicated difficulties in solving the problem of reducing the errors of measuring instruments are a consequence of the systemic properties of these instruments: emergence, integrity, uncertainty, complexity, stochasticity, etc. Attempts at a theoretical description at the level of nomographic sciences in the situations under consideration are often not effective. An experimental-statistical approach is necessary, since it allows for an idiographic description of the patterns of specific phenomena in detailed conditions of time and place.
Both in radio-electronic measurements and in ensuring the accuracy of assessing the results of quantitative chemical analysis, an important feature of errors is noted: the systematic errors of the result for most measuring instruments are significant in the sense that they exceed the random one, and the error of a given instance of a measuring instrument at each point of the factor space is determined by basically constant.
To further improve the quality of measurements, it is necessary to use not only physical - design, technological, operational - capabilities, but also information. They consist in implementing a systematic approach to obtaining information about all types of errors: instrumental, methodological, additional, systematic, progressive (drift), model and possibly others. Having such information in the form of a multifactor mathematical model and knowing the values of factors (conditions) accompanying the process measurements, you can obtain information about the given errors and, therefore, know the measured value more accurately.
Requirements for the methodology of mathematical modeling of systematic errors of measuring instruments
It is necessary to develop a technique for multifactor mathematical modeling of naturally varying systematic errors, taking into account the following requirements.
- A systematic approach to describing systematic errors, taking into account many factors and, if necessary, many criteria for the quality of a measuring instrument.
- The applied level of obtaining mathematical models when their structure is unknown to the researcher.
- The efficiency (in a statistical sense) of obtaining useful information from source data and reflecting it in mathematical models.
- Possibility of accessible and convenient meaningful interpretation of the obtained models in the subject area.
- The effectiveness of using mathematical models in a subject area compared to the cost of resources to obtain them.
Main stages of obtaining mathematical models
Let us consider the main stages of obtaining multifactor mathematical models that meet the above requirements.
Selecting a multifactorial experimental plan that provides the necessary properties of the resulting mathematical models
In the considered (metrological) class of experimental studies, it is possible to use a full and fractional factorial experiment. By a definable mathematical model we mean a model that is linear with respect to the parameters and, in the general case, nonlinear with respect to the factors, a model of arbitrarily high but finite complexity. The expanded effects matrix of a full factorial experiment will include a dummy factor column X 0 = 1, columns of all main effects and all possible main effects interactions. If the effects of factors and interactions of factors are expressed as a system of orthogonal normalized contrasts, then the variance-covariance matrix will take the form:
Where X
– matrix of effects of a full factorial experiment;
σ y 2 – dispersion of reproducibility of experimental results;
N– number of experiments in the experimental plan;
E
– identity matrix.
The mathematical model obtained according to the scheme of a full factorial experiment corresponds to many remarkable properties: the coefficients of the model are orthogonal to each other and independent in a statistical sense; most stable ( cond= 1); each coefficient carries semantic information about the influence of the corresponding effect on the modeled quality criterion; the experimental design meets the criteria D-, A-, E-, G-optimality, as well as the criterion of proportionality of frequencies of factor levels; the mathematical model is adequate at the response surface approximation points. We will consider such a model to be true and “best”.
In cases where the use of a full factorial experiment is impossible due to the large number of experiments, it should be recommended to use multifactorial regular (preferably uniform) experimental designs. With the correct choice of the number of necessary experiments, their properties are as close as possible to the given properties of a full factorial experiment.
Obtaining the structure of a multifactor mathematical model
The structure of the resulting multifactor mathematical model, which is generally unknown to the researcher, must be determined based on the possible set of effects corresponding to the set of effects of the design of a full factorial experiment. It is given by the expression:
Where X 1 ,..., X k – factors of the required mathematical model;
s 1 ,..., s k – number of factor levels X 1 ,..., X k ;
k– total number of factors;
N n – the number of experiments of a full factorial experiment, equal to the number of structural elements of its scheme.
The search for the necessary effects - main and interactions - in the form of orthogonal contrasts for the desired model is carried out as a multiple statistical test of hypotheses about the statistical significance of the effects. Statistically significant effects are introduced into the model.
Selecting the number of required experiments for a fractional factorial experiment
Usually the researcher knows (approximately) information about the expected complexity of the influence of factors on the modeled quality criterion. For each factor, the number of levels of its variation is selected, which should be 1 greater than the maximum degree of the polynomial required for an adequate description of the response surface by this factor. The required number of experiments will be:
Where s i – number of factor levels X i ; 1 ≤ i ≤ k.
A coefficient of 1.5 is chosen for the case when the number of necessary experiments is significant (about 50...64 or more). If the required number of experiments is smaller, factor 2 should be chosen.
Choosing the structure of a multifactor mathematical model
To select the structure of the resulting mathematical model, it is necessary to use the developed algorithm. The algorithm implements a sequential scheme for identifying the required structure based on the results of a planned multifactorial experiment.
Processing of experimental results
To comprehensively process the results of experiments and obtain the necessary information for interpreting the results in the subject area, the software tool “Planning, Regression and Analysis of Models” (PS PRIAM) has been developed. The developer is the Laboratory of Experimental and Statistical Methods of the Department of Mechanical Engineering Technology of the National Technical University of Ukraine “Kiev Polytechnic Institute”. Assessment of the quality of the resulting mathematical models includes the following criteria:
- obtaining an informative subset of the main effects and interactions of factors for adoption as the structure of the desired multifactor mathematical model;
- ensuring the highest theoretical efficiency (up to 100%) in extracting useful information from source data;
- testing for statistical significance of a potential mathematical model;
- testing various assumptions of multiple regression analysis;
- checking the adequacy of the resulting model;
- checking for information content, i.e. the presence of useful information and its statistical significance in the mathematical model;
- checking the stability of the coefficients of the mathematical model;
- checking the actual effectiveness of extracting useful information from source data;
- assessment of semantics (information) based on the obtained coefficients of the mathematical model;
- checking the properties of residues;
- general assessment of the properties of the resulting mathematical model and the possibility of using it to achieve the goal.
Interpretation of the results obtained
It is carried out by a specialist (or specialists) who well understand both the formal results in the resulting models and the applied goals for which the models should be used.
A mathematical method for obtaining useful information about systematic errors accompanying the process of measuring a physical quantity and a measuring instrument create a supersystem with interaction (otherwise emergence) with each other. The interaction effect - higher accuracy of the measured value - cannot in principle be achieved only through individual subsystems. This follows from the structure of the mathematical model Ŷ (ŷ 1 ,..., ŷ p) = f j (SI, MM) for experiment 2 2 //4 (the absence of a subsystem is set to “–1”, and the presence of “1”) of the indicated subsystems:
Where Ŷ (ŷ 1 ,..., ŷ p) – vector of operational efficiency of the measuring instrument, 1 ≤ j ≤ p;
1 – symbol of the average value of the result (conditional reference point);
SI – measurement result obtained only from the measuring instrument;
MM – information obtained from a multifactor mathematical model about the systematic errors of the measuring instrument used with knowledge of internal and external measurement conditions relative to it;
SI · MM – the effect of interaction (emergence) of the measuring instrument and the mathematical model, provided they are used together.
Increasing measurement accuracy is achieved by obtaining more information about the measurement conditions and the properties of the measuring instrument in interaction with its internal and external environment.
The combination of physical and information principles in practice means the intellectualization of known systems, in particular, the creation of intelligent measuring instruments. Combining physical and information principles into a single integrated system allows us to solve old problems in a fundamentally new way.
An example of improving the measurement accuracy of digital scales
Let us consider the possibilities of the proposed approach using the example of increasing the accuracy of digital scales with a weighing range of 0...100 kgf. Capacitive type scale sensor with autonomous power supply from a portable voltage source. The scales are intended for operation in the ambient (air) temperature range of 0...60°C. The voltage from an autonomous voltage source during operation of the scales can vary in the range of 12.3...11.7 V with a calculated (nominal) value of 12 V.
A preliminary study of digital scales showed that changes in ambient temperature and supply voltage in the above ranges have a relatively small effect on the readings of the capacitive sensor and, consequently, on the weighing results. However, it was not possible to stabilize these external and internal conditions with the required accuracy and maintain them during the operation of the scales due to the fact that the scales should not be operated in stationary (laboratory) conditions, but on board a moving object.
A study of the accuracy of the scales without taking into account the influence of changes in temperature and supply voltage showed that the average absolute error of approximation is 0.16%, and the root mean square error of the remainder (in units of measurement of the output weighing value) is 53.92.
To obtain a multifactor mathematical model, the following designations of factors and the values of their levels were adopted.
X 1 – hysteresis. Levels: 0 (load); 1 (unloading). Quality factor.
X 2 – ambient temperature. Levels: 0; 22; 60°C.
X 4 – measured weight. Levels: 0; 20; 40; 60; 80; 100 kgf.
Taking into account the accepted levels of factor variation and the relatively labor-intensive amount of testing, it was decided to conduct a full factorial experiment, i.e. 2 · 3 2 · 6//108. Initial test data were provided by Prof. P.V. Novitsky. Each experiment was repeated only once, which cannot be considered a good solution. It is advisable to repeat each experiment twice. A preliminary analysis of the source data showed that they are highly likely to contain gross errors. These experiments were repeated and their results were corrected.
Natural values of the levels of variation of factors were transformed into orthogonal contrasts, otherwise into a system of orthogonal Chebyshev polynomials.
Using a system of orthogonal contrasts, the structure of a full factorial experiment will have the following form:
(1 + x 1) (1 + x 2 + z 2) (1 + x 3 + z 3) (1 + x 4 + z 4 + u 4 + v 4 + ω 4) → N 108
Where x 1 ,..., x 4 ; z 2 ,..., z 4 ; u 4 , v 4 , ω 4 – linear, quadratic, cubic, fourth and fifth degree contrasts of factors, respectively X 1 ,..., X 4 ;
N 108 – the number of structural elements for the design of a full factorial experiment.
All effects (main and interactions) were normalized
where x iu (p) – value p-th orthogonal contrast i-th factor for the u-th row of the planning matrix, 1 ≤ u ≤ 108, 1 ≤ p ≤ s i – 1; 1 ≤ i ≤ 4.
A preliminary calculation of the mathematical model showed that the (approximately) value of 20.1 can be chosen as an estimate of the reproducibility variance.
The number of degrees of freedom (conditionally) is accepted V 2 = 108.
The variance was used to determine the standard error of the regression equation coefficients.
The calculation of the mathematical model and all its quality criteria was carried out using the PRIAM software. The resulting mathematical model has the form
| ŷ = 28968,9 – 3715,13x 4 + 45,2083x 3 – 37,5229z 2 + 23,1658x 2 – 19,0708z 4 – 19,6574z 3 – 9,0094x 2 z 3 – 9,27434z 2 x 4 + 1,43465x 1 x 2 + 1,65431z 2 x 3 , | (2) |
x 1 = 2 (X 1 – 0,5);
x 2 = 0,0306122 (X 2 – 27,3333);
z 2 = 1,96006 (x 2 2 – 0,237337x 2 – 0,575594);
x 3 = 3.33333 (X 3 – 12);
z 3 = 1,5 (x 2 3 – 0,666667);
x 4 = 0,02 (X 4 – 50);
z 4 = 1,875 (x 2 4 – 0,466667);
u 4 = 3,72024 (x 3 4 – 0,808x 4);
v 4 = 7,59549 (x 4 4 – 1,08571x 2 4 + 0,1296).
Table 1
Quality criteria for the resulting mathematical model
| Model adequacy analysis | |
| Residual variance | 21,1084 |
| Reproducibility variance | 20,1 |
| Estimated value F-criteria | 1,05017 |
| Significance level F-criterion for adequacy 0.05 for degrees of freedom V 1 = 97; V 2 = 108 | |
| Table value F-criteria for adequacy | 1,3844 |
| Table value F-criteria (in the absence of repeated experiments) | 1,02681 |
| Standard error of estimate | 4,59439 |
| Adjusted taking into account degrees of freedom | 4,80072 |
| Model | adequate |
| Note: Reproducibility variance is user defined |
|
| Analysis of the information content of the model | |
| Proportion of dispersion explained by the model | 0,999997 |
| Introduced regressors (effects) | 11 |
| Multiple correlation coefficient | 0,999999 |
| (adjusted for degrees of freedom) | 0,999998 |
| F attitude for R | 3.29697 10 6 |
| Significance level F-criterion for information content 0.01 for degrees of freedom V 1 = 10; V 2 = 97 | |
| Table value F-criteria for information content | 2,50915 |
| Model | informative |
| Box and Wetz criterion for information content | more than 49 |
| Information content of the model | very high |
table 2
Statistical characteristics of regression coefficients
| Name of the main effect or interaction of main effects | Regression coefficient | Standard error of regression coefficient | Calculated value t-Crete. | Share of participation in explaining the spread of the modeled value |
| x 4 | b 1 = –3715,13 | 0,431406 | 5882,9 | 0,999557 |
| x 3 | b 2 = 45,2083 | 0,431406 | 85,5631 | 0,000211445 |
| z 2 | b 3 = –37,5229 | 0,431406 | 62,2275 | 0,000111838 |
| x 2 | b 4 = 23,1658 | 0,431406 | 40,7398 | 4.79362·10 –5 |
| z 4 | b 5 = –19,0708 | 0,431406 | 33,0808 | 3.16065·10 –5 |
| z 3 | b 6 = –19,6574 | 0,431406 | 32,22 | 2.9983·10 –5 |
| x 2 z 3 | b 7 = –9,0094 | 0,431406 | 11,2035 | 3.62519·10 –6 |
| z 2 x 4 | b 8 = –9,27434 | 0,431406 | 10,5069 | 3.18838 10 –6 |
| x 1 x 2 | b 9 = 1,43465 | 0,431406 | 2,523 | 1.83848 10 –7 |
| z 2 x 3 | b 10 = 1,65431 | 0,431406 | 2,24004 | 1.44923·10 –7 |
b 0 = 28968,9
Significance level for t-criterion – 0.05
For degrees of freedom V 1 = 108. Table value t-criteria – 1.9821
In table Figure 1 shows a printout of the quality criteria of the resulting multifactor mathematical model. The model is adequate. The proportion of dispersion explained by the model is very high because the model is highly accurate, the variability of the response function is large, and its random variability is relatively small. Multiple correlation coefficient R is very close to 1 and stable, because being corrected for degrees of freedom, it practically does not change. Statistical significance R very large, i.e. the model is very informative. The high information content of the model is also confirmed by the value of the Box and Wetz criterion. The model coefficients are maximally stable: condition number cond= 1. The resulting model is semantic in an informational sense, since all its coefficients are orthonormal: they are statistically independent and can be compared in absolute value with each other. The sign of the coefficient shows the nature of the influence, and its absolute value indicates the strength of the influence. The resulting model is most convenient for interpretation in the subject area.
Taking into account the semantic properties of the resulting mathematical model and the share of participation of each of the model’s effects in the total share of dispersion explained by the model, it is possible to conduct a meaningful information analysis of the formation of the measurement result of the digital scales under study.
The dominant contribution to the simulation results of 0.999557 is generated by the linear main effect x 4 (with coefficient b 1 = –3715.13), i.e. measured weight (Table 2). Nonlinearity z 4 (with coefficient b 5 = –19.07) is relatively small (3.16·10 –5) and taking it into account in the model increases the measurement accuracy. Linear effect x 4 interacts relatively weakly (3.19 10 –6) with the quadratic effect z 2 ambient temperatures: interaction z 2 x 4 (b 8 = –9.27). Consequently, the mathematical model only depends on the factor measured weight X 4 should also include the effect of ambient temperature
ŷ 1 = 28968,90 – 3715,13x 4 – 19,07z 4 – 9,27z 2 x 4 ,
whose factor X 2 is uncontrollable.
The supply voltage changes the weighing results as a linear effect x 3 (b 2 = 45.21) and quadratic effect z 3 (b 6 = –19.66). Their total share of participation is 2.41·10 –4.
Ambient temperature influences in the form of a quadratic z 2 (b 3 = –37.52) and linear x 2 (b 4 = 23.17) effects with a total participation share of 1.60·10 –4.
Ambient temperature and supply voltage form a pair interaction x 2 z 3 (b 7 = –9.01) with a participation share of 3.63·10 –6.
Evidence of statistical significance of the last two effects x 1 x 2 and z 2 x 3 cannot be justified, since they are significantly less than the effects x 2 z 3 and z 2 x 4, and the justified value of the reproducibility variance based on the results of repeated experiments was, unfortunately, absent in the presented initial data.
In table 2 shows the statistical characteristics of the regression coefficients. Note that the values of the regression coefficients are divided into normalization coefficients of orthogonal contrasts, which are not included in the given formulas for orthogonal contrasts. This explains the fact that when dividing the values of regression coefficients by their standard error, the resulting values t-criteria differ from the given correctly calculated values of this criterion in table. 2.
Rice. 1. Histogram of residuals
In Fig. 1 shows a histogram of residuals 
. It is relatively close to the normal distribution law. In table Figure 3 presents the numerical values of the residuals and their percentage deviations. The time graph of the residuals (Fig. 2) indicates the random nature of the change in the residuals depending on the time (sequence) of the experiments. Further improvement of the accuracy of the model is not possible. Analysis of the dependence of residuals on ŷ
(calculated value) shows that the greatest dispersion of residuals is observed for X 4 = 0 kgf ( y= 32581...32730) and X 4 = 100 kgf ( y= 25124...25309). The smallest spread at X 4 = 40 kgf. However, the statistical significance of such a conclusion requires knowledge of the reasonable value of the reproducibility variance.
Rice. 2. Time chart of balances
Taking into account various systematic errors, nonlinearities, and interactions of uncontrollable factors in the mathematical model made it possible to increase the accuracy of the measuring instrument according to the criterion of the average absolute error of approximation to 0.012% - by 13.3 times, and according to the criterion of the root-mean-square error of approximation to 4.80 (Table 1) - 11.2 times.
Experimental plan 2 2 //4 for the average absolute approximation error in % and the results obtained when using only a measuring instrument and a measuring instrument with a mathematical model of systematic errors are presented in Table. 4.
The mathematical model for the average absolute approximation error, obtained from experiment 2 2 //4, with the structure of the model (1) and the results of the functioning of the measuring instrument without a mathematical model and with its use, has the form
ŷ = 0,043 + 0,043x 1 ...0,037x 2 ...0,037x 1 x 2
Where x 1 – orthogonal contrast factor X 1 (SI) – measuring instrument;
x 2 – orthogonal contrast factor X 2 (MM) – mathematical model of systematic errors of the measuring instrument used;
x 1 x 2 – interaction of factors X 1 (SI) and X 2 (MM).
Table 3
Residues and their deviation percentages
1
– Experience number; 2
– Feedback from the experiment; 3
– Response according to the model; 4
- Remainder;
5
– Percentage of deviation; 6
– Experience number; 7
– Feedback from the experiment;
8
– Response according to the model; 9
- Remainder; 10
– Deviation percentage
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 32581 | 32574,2 | 6,832 | 0,0210 | 55 | 32581 | 32576,6 | 4,431 | 0,0136 |
| 2 | 31115 | 31108,7 | 6,349 | 0,0204 | 56 | 31115 | 31111,1 | 3,948 | 0,0127 |
| 3 | 29635 | 29631,7 | 3,308 | 0,0112 | 57 | 29633 | 29634,1 | –1,092 | –0,0037 |
| 4 | 28144 | 28143,3 | 0,710 | 0,0025 | 58 | 28141 | 28145,7 | –4,691 | –0,0167 |
| 5 | 26640 | 26643,4 | –3,445 | –0,0129 | 59 | 26637 | 26645,8 | –8,846 | –0,0332 |
| 6 | 25128 | 25132,2 | –4,159 | –0,0165 | 60 | 25124 | 25134,6 | –10,559 | –0,0420 |
| 7 | 32625 | 32638,6 | –13,602 | –0,0417 | 61 | 32649 | 32641 | 7,997 | 0,0245 |
| 8 | 31175 | 31173,1 | 1,915 | 0,0061 | 62 | 31179 | 31175,5 | 3,514 | 0,0113 |
| 9 | 29694 | 29696,1 | –2,126 | –0,0072 | 63 | 29699 | 29698,5 | 0,473 | 0,0016 |
| 10 | 28208 | 28207,7 | 0,276 | 0,0010 | 64 | 28209 | 28210,1 | –1,125 | –0,0040 |
| 11 | 26709 | 26707,9 | 1,120 | 0,0042 | 65 | 26711 | 26710,3 | 0,719 | 0,0027 |
| 12 | 25198 | 25196,6 | 1,407 | 0,0056 | 66 | 25199 | 25199 | 0,006 | 0,0000 |
| 13 | 32659 | 32666,7 | –7,680 | –0,0235 | 67 | 32660 | 32669,1 | –9,081 | –0,0278 |
| 14 | 31199 | 31201,2 | –2,163 | –0,0069 | 68 | 31200 | 31203,6 | –3,564 | –0,0114 |
| 15 | 29723 | 29724,2 | –1,204 | –0,0040 | 69 | 29726 | 29726,6 | –0,605 | –0,0020 |
| 16 | 28241 | 28235,8 | 5,198 | 0,0184 | 70 | 28242 | 28238,2 | 3,797 | 0,0134 |
| 17 | 26741 | 26736 | 5,042 | 0,0189 | 71 | 26742 | 26738,4 | 3,642 | 0,0136 |
| 18 | 25232 | 25224,7 | 7,329 | 0,0290 | 72 | 25233 | 25227,1 | 5,928 | 0,0235 |
| 19 | 32632 | 32636,5 | –4,543 | –0,0139 | 73 | 32630 | 32637 | –7,012 | –0,0215 |
| 20 | 31175 | 31177,1 | –2,086 | –0,0067 | 74 | 31173 | 31177,6 | –4,554 | –0,0146 |
| 21 | 29705 | 29706,2 | –1,185 | –0,0040 | 75 | 29703 | 29706,7 | –3,654 | –0,0123 |
| 22 | 28225 | 28223,8 | 1,157 | 0,0041 | 76 | 28223 | 28224,3 | –1,311 | –0,0046 |
| 23 | 26734 | 26730,1 | 3,942 | 0,0147 | 77 | 26733 | 26730,5 | 2,474 | 0,0093 |
| 24 | 25233 | 25224,8 | 8,170 | 0,0324 | 78 | 25233 | 25225,3 | 7,702 | 0,0305 |
| 25 | 32710 | 32707,4 | 2,623 | 0,0080 | 79 | 32710 | 32707,8 | 2,155 | 0,0066 |
| 26 | 31251 | 31247,9 | 3,081 | 0,0099 | 80 | 31249 | 31248,4 | 0,612 | 0,0020 |
| 27 | 29777 | 29777 | –0,019 | –0,0001 | 81 | 29775 | 29777,5 | –2,488 | –0,0084 |
| 28 | 28294 | 28294,7 | –0,676 | –0,0024 | 82 | 28292 | 28295,1 | –3,145 | –0,0111 |
| 29 | 26799 | 26800,9 | –1,891 | –0,0071 | 83 | 26799 | 26801,4 | –2,360 | –0,0088 |
| 30 | 25297 | 25295,7 | 1,336 | 0,0053 | 84 | 25296 | 25296,1 | –0,132 | –0,0005 |
| 31 | 32730 | 32723,7 | 6,349 | 0,0194 | 85 | 32729 | 32724,1 | 4,880 | 0,0149 |
| 32 | 31269 | 31264,2 | 4,806 | 0,0154 | 86 | 31267 | 31264,7 | 2,338 | 0,0075 |
| 33 | 29794 | 29793,3 | 0,707 | 0,0024 | 87 | 29793 | 29793,8 | –0,762 | –0,0026 |
| 34 | 28310 | 28311 | –0,951 | –0,0034 | 88 | 28309 | 28311,4 | –2,419 | –0,0085 |
| 35 | 26814 | 26817,2 | –3,166 | –0,0118 | 89 | 26814 | 26817,6 | –3,634 | –0,0136 |
| 36 | 25309 | 25311,9 | –2,938 | –0,0116 | 90 | 25309 | 25312,4 | –3,407 | –0,0135 |
| 37 | 32616 | 32619,1 | –3,053 | –0,0094 | 91 | 32608 | 32616,2 | –8,183 | –0,0251 |
| 38 | 31152 | 31154,5 | –2,525 | –0,0081 | 92 | 31148 | 31151,7 | –3,656 | –0,0117 |
| 39 | 29677 | 29678,6 | –1,555 | –0,0052 | 93 | 29675 | 29675,7 | –0,686 | –0,0023 |
| 40 | 28192 | 28191,1 | 0,858 | 0,0030 | 94 | 28192 | 28188,3 | 3,727 | 0,0132 |
| 41 | 26696 | 26692,3 | 3,713 | 0,0139 | 95 | 26692 | 26689,4 | 2,582 | 0,0097 |
| 42 | 25189 | 25182 | 7,010 | 0,0278 | 96 | 25189 | 25179,1 | 9,880 | 0,0392 |
| 43 | 32713 | 32707,9 | 5,132 | 0,0157 | 97 | 32704 | 32705 | –0,998 | –0,0031 |
| 44 | 31244 | 31243,3 | 0,660 | 0,0021 | 98 | 31240 | 31240,5 | –0,471 | –0,0015 |
| 45 | 29770 | 29767,4 | 2,630 | 0,0088 | 99 | 29764 | 29764,5 | –0,501 | –0,0017 |
| 46 | 28285 | 28280 | 5,043 | 0,0178 | 100 | 28278 | 28277,1 | 0,912 | 0,0032 |
| 47 | 26784 | 26781,1 | 2,898 | 0,0108 | 101 | 26778 | 26778,2 | –0,233 | –0,0009 |
| 48 | 25262 | 25270,8 | –8,805 | –0,0349 | 102 | 25262 | 25267,9 | –5,935 | –0,0235 |
| 49 | 32717 | 32710,7 | 6,318 | 0,0193 | 103 | 32710 | 32707,8 | 2,187 | 0,0067 |
| 50 | 31249 | 31246,2 | 2,845 | 0,0091 | 104 | 31245 | 31243,3 | 1,715 | 0,0055 |
| 51 | 29770 | 29770,2 | –0,185 | –0,0006 | 105 | 29767 | 29767,3 | –0,315 | –0,0011 |
| 52 | 28280 | 28282,8 | –2,772 | –0,0098 | 106 | 28279 | 28279,9 | –0,903 | –0,0032 |
| 53 | 26779 | 26783,9 | –4,917 | –0,0184 | 107 | 26779 | 26781 | –2,048 | –0,0076 |
| 54 | 25267 | 25273,6 | –6,619 | –0,0262 | 108 | 25267 | 25270,8 | –3,750 | –0,0148 |
| The average absolute relative error in percentage is 0.0119. | |||||||||
Table 4
Experimental plan 2 2 //4
Analysis of the model coefficients shows that factor X 2 (MM) reduces systematic error not only in the form of the main effect x 2 (coefficient b 2 = –0.037), but also due to the interaction (emergence) of factors X 1 (SI) X 2 ( MM) (coefficient b 12 = –0.037).
A similar model can be obtained for the criterion of root-mean-square error of approximation.
To actually implement the resulting model (2), it is necessary to measure and use information about the ambient temperature and supply voltage using sensors and calculate the result using a microprocessor.
Results of mathematical modeling of six-component strain gauge measuring systems
The mathematical modeling of six-component strain gauge measuring systems is considered. The proposed method was introduced at the Kiev Mechanical Plant (now the O.K. Antonov Aviation Scientific and Technical Complex). For the first time in the practice of carrying out similar measurements, this method has largely made it possible to eliminate the consequences of physical imperfections of measuring systems, manifested in the form of interaction between channels, the influence of other channels on the channel in question, nonlinearities, and to study the structural relationships of various channels.
The use of the mathematical modeling method in real enterprise conditions showed that the time for conducting experiments is reduced by 10...15 times; the efficiency of processing measurement information increases significantly (up to 60 times); the number of performers involved in measurement experiments is reduced by 2...3 times.
The final conclusion about the advisability of using the outlined approach depends on the economic efficiency of the following compared options.
A high-precision measuring instrument and, therefore, more expensive, used in standardized (standard) conditions that must be created and maintained.
Means of measuring less high accuracy, used in non-standardized (non-standard) conditions using the resulting mathematical model.
Main conclusions
1) The successfully implemented system approach in mathematical modeling of the measuring instrument made it possible to take into account the influence of external factors - ambient temperature - and internal environment - supply voltage. The efficiency of extracting useful information from the source data was 100%.
2) In the resulting multifactor mathematical model, the structure of which was not known a priori to the researcher, the nonlinearity of the measuring instrument and the systemic influence of factors (emergence) of the external and internal environment are revealed in a form convenient for interpretation in the subject area. In real operating conditions, stabilizing these factors with the required accuracy is not possible.
3) Taking into account the mathematical model of systematic errors made it possible to increase the accuracy of measurements according to the criterion of average absolute error by 13.3 times and according to the criterion of root-mean-square error by 11.2 times.
Our offers
The Laboratory of Experimental and Statistical Methods and Research is ready to provide algorithmic software for obtaining multifactor mathematical models, their analysis and interpretation, and transfer the accumulated experience for use in solving specific industrial and scientific problems.
We are ready to solve your problems in these and many other areas by using algorithms, software, and know-how created over many years; study and transfer of experience to your specialists.
Literature:
- Rybakov I.N. Fundamentals of accuracy and metrological support of radio-electronic measurements. – M.: Standards Publishing House, 1990. – 180 p.
- Radchenko S.G. Mathematical modeling of technological processes in mechanical engineering. – K.: JSC “Ukrspetsmontazhproekt”, 1998. – 274 p.
- Alimov Yu.I., Shaevich A.B. Methodological features of assessing the results of quantitative chemical analysis // Journal of Analytical Chemistry. – 1988. – Issue. 10. – T. XLIII. – S. 1893...1916.
- Planning, regression and analysis of PRIAM models (PRIAM). SCMC–90; 325, 660, 668 // Catalog. Ukrainian software products. Catalog. Software of Ukraine. – K.: JV “Teknor”. – 1993. – pp. 24...27.
- Zinchenko V.P., Radchenko S.G. Method for modeling multicomponent strain gauge measuring systems. – K.: 1993. – 17 p. (Prev. / Academy of Sciences of Ukraine. Institute of Cybernetics named after V.M. Glushkov; 93...31).
In general, the error model 0.95(t) can be represented as 0.95(t) = 0 + F(t), where D0 is the initial SI error; F(t) is a random function of time for a set of SIs of a given type, caused by physical and chemical processes of gradual wear and aging of elements and blocks. It is practically impossible to obtain an exact expression for the function F(t) based on physical models of aging processes. Therefore, based on data from experimental studies of changes in errors over time, the function F(t) is approximated by one or another mathematical dependence.
The simplest model of error change is linear:
where v is the rate of change of error. As the studies have shown, this model satisfactorily describes the aging of the SI at the age of one to five years. Its use in other time ranges is impossible due to the obvious contradiction between the failure rates determined by this formula and the experimental values.
Metrological failures occur periodically. The mechanism of their periodicity is illustrated in Fig. 1, a, where straight line 1 shows the change in the 95% quantile under a linear law.
Rice. 2.
In the event of a metrological failure, the error D0.95(t) exceeds the value Dpr=D0+nD3, where D3 is the value of the margin of the normalized error limit necessary to ensure the long-term operability of the measuring instrument. With each such failure, the device is repaired and its error returns to the original value D0. After time Tr = ti - ti-1, a failure occurs again (moments tt, t2, t3, etc.), after which repairs are made again. Consequently, the process of changing the SI error is described by broken line 2 in Fig. 1, a, which can be represented by the equation
where n is the number of failures (or repairs) of the SI. If the number of failures is considered to be an integer, then this equation describes discrete points on straight line 1 (Fig. 2, a). If we conditionally assume that n can also take fractional values, then formula (2) will describe the entire straight line 1 of the change in error D0.95(t) in the absence of failures.
The frequency of metrological failures increases with increasing speed v. It just as strongly depends on the margin of the normalized error value D3 in relation to the actual value of the measuring instrument error D0 at the time of manufacture or completion of repair of the device. The practical possibilities for influencing the rate of change v and the margin of error D3 are completely different. The rate of aging is determined by the existing production technology. The margin of error for the first overhaul interval is determined by the decisions made by the measuring instrument manufacturer, and for all subsequent overhaul intervals - by the level of culture of the user's repair service.
If the enterprise's metrological service ensures during repairs an SI error equal to the D0 error at the time of manufacture, then the frequency of metrological failures will be low. If, during repairs, only the fulfillment of the condition D0 (0.9... 0.95) Dpr is ensured, then the error may go beyond the permissible values in the coming months of operation of the SI and for most of the verification interval it will be operated with an error exceeding its class accuracy. Therefore, the main practical means of achieving long-term metrological serviceability of a measuring instrument is to ensure a sufficiently large reserve D3, normalized in relation to the limit Dpr.
The gradual continuous consumption of this reserve ensures the metrologically sound state of the SI for a certain period of time. Leading instrument-making plants provide D3 = (0.4...0.5) Dpr, which at an average aging rate v = = 0.05AP/year allows us to obtain a repair interval Tp = D3 = 1/T/v = 8... 10 years and failure rate p= 0.1... 0.125 year-1.
When the SI error changes in accordance with formula (1), all repair intervals Тр = 1/Т will be equal to each other, and the metrological failure rate p will be constant throughout the entire service life. However, experimental studies have shown that this is not true in practice.