DEVELOPMENT OF THE ALGORITHMS OF CORRECTION OF CORRELATION MATRICES

 T. Aliev, N. Musayeva, U. Sattarova, N. Rzayeva, 2015 gral equation that includes the correlation functions ( ) XX R τ and ( ) XY R τ of the input ( ) X τ and output ( ) Y τ signals. It allows us to obtain the dynamic characteristics of an object without disturbing its normal operation mode. Therefore, statistical methods are widely used for determining the dynamic characteristics of objects during their normal operation [6–8]. However, the application of statistical methods for building mathematical models of real-life industrial objects presents the following difficulty. Interferences and noises are imposed upon the useful signal (that has to be obtained with the least possible amount of distortion), thus hindering the calculation of the estimates of their static characteristics. One should take into account that interferences and noises are also represented by random functions ( ) ε τ . The reasons behind the formation of interferences and noises can be very diverse [6–9]: a) thermal noises; b) noises caused by other machinery and equipment operating nearby; c) noises caused by power supply sources; d) noised caused by self-oscillations generated in feedback circuits, etc. For instance, for deep-water offshore platforms, noises are caused by waves, wind, etc. Another example is the radio detector of an antenna under a wind load, which also represents a random time function.

Y τ signals.It allows us to obtain the dynamic characteristics of an object without disturbing its normal operation mode.Therefore, statistical methods are widely used for determining the dynamic characteristics of objects during their normal operation [6][7][8].
However, the application of statistical methods for building mathematical models of real-life industrial objects presents the following difficulty.Interferences and noises are imposed upon the useful signal (that has to be obtained with the least possible amount of distortion), thus hindering the calculation of the estimates of their static characteristics.
One should take into account that interferences and noises are also represented by random functions ( ) ε τ .The reasons behind the formation of interferences and noises can be very diverse [6][7][8][9]: a) thermal noises; b) noises caused by other machinery and equipment operating nearby; c) noises caused by power supply sources; d) noised caused by self-oscillations generated in feedback circuits, etc.
For instance, for deep-water offshore platforms, noises are caused by waves, wind, etc.Another example is the radio detector of an antenna under a wind load, which also represents a random time function.

Introduction
It is known [1][2][3][4][5][6] that one of the main challenges in solving problems of automated control of industrial objects is establishing the quantitative interrelations between technological parameters characterizing the processes in those objects both in statics and dynamics.Such interrelations are called static and dynamic characteristics, respectively.These characteristics can be determined from differential equations of control objects [1][2][3][4][5][6].However, those differential equations are often unknown, which is why statistical methods are widely used -they make it possible to determine dynamic characteristics during normal operation of objects [1][2][3][4][5][6].In practice, such dynamic characteristics as impulsive admittance ( ) k t and transfer functions ( ) of linear systems are determined by applying to their input artificial stimulation of a certain type (impulse, step function, sinusoids) and measuring the response.However, in that case, random uncontrollable disturbances are superimposed on these impacts.As a result, it proves impossible to precisely determine dynamic characteristics based on typical input signals [6][7][8].

Analysis of published data and problem statement
The statistical correlation method for determining these dynamic characteristics is based on the solution of an inte-

DEVELOPMENT OF THE ALGORITHMS OF CORRECTION OF CORRELATION MATRICES T . A l i e v
Doctor of Engineering, Full Member of the Academy of Sciences* Е-mail: telmancyber@rambler.ru

N . M u s a y e v a
Doctor of Engineering, Professor* Е-mail: musanaila@gmail.com

U . S a t t a r o v a
PhD in Engineering* Е-mail: ulker.rzaeva@gmail.com

N . R z a y e v a
Researcher* Е-mail: nikanel1@gmail.com*Institute of Control Systems, Azerbaijan National Academy of Sciences B. Vahabzade str., 9, Baku, Azerbaijan Republic, AZ1141

Проанализированы трудности формирования корреляционных матриц в задачах идентификации матричных моделей динамики реальных производственных объектов. Предложены обобщенные алгоритмы, позволяющие свести эти матрицы к аналогичным матрицам полезных сигналов. При этом учтены специфики реальных зашумленных технологических параметров, показана возможность применения данных алгоритмов для случаев отсутствия и присутствия корреляции между полезным сигналом и помехой
Ключевые слова: стохастический процесс, технологический параметр, зашумленный сигнал, корреляционная матрица, модель динамики UDC 519.216DOI: 10.15587/1729-4061.2015.54098 In view of the above, many algorithms and technologies of filtration have been proposed with the aim of eliminating the effects of the noise on the result of identification of statistical models of the dynamics of control objects over a long period of time [8][9][10].The ones that allow for eliminating the error of the noises caused by external factors have found a wide application [10][11][12].However, in real-life objects, noises of technological processes form under the influence of various factors.Some of them reflect indirectly certain processes that cause defects in the objects under investigation.For this reason, the range of the noise spectrum frequently overlaps the spectrum of the useful signal.Besides, the spectra of the noise and the useful signal in real-life technological parameters are not strictly stable.Therefore, filtration does not always yield the desired result.Sometimes, the spectrum of the useful signal is even distorted from the filtration [11,12].
Taking into account the above, the paper considers one possible option of creating alternative digital methods and technologies for eliminating the error induced by noise during the formation of correlation matrices in the process of identification of the dynamic model of industrial objects.
As stated above [6,7], the main dynamic characteristics of linear objects are their impulsive admittance ( ) k t and transfer ( ) functions.The differential equations of those objects are often unknown, and the methods based on the application of artificial stimulation are inapplicable, usually due to the following reasons: -it is undesirable or impossible to apply a special kind of stimulation to the object's input, as it disturbs the normal running of the process; -random uncontrollable disturbances are imposed on that stimulation, and their effects are impossible to separate from the effect of the artificial stimulation.
In this regard, in creating systems for automated control of continuous stochastic processes, the statistical method is widely used, allowing one to determine the dynamic characteristics of complex objects during their normal operation.In practice, the solving of this problem comes to solving the problem of identification of the mathematical model of object's dynamics by methods of theory of stochastic processes [6-8, 13, 14].Object's state in the general case is described by matrix equations of the following type: where ( ) Therefore, matrix equation ( 1) and the correlation matrix of real technological processes can be represented as follows: where ( ) due to which the following inequalities take place As a result, in practice, adequacy of identification of the model of the dynamics (1) of technological processes fails in many cases.
At the same time, in many real-life industrial objects, various sensors are used, in which signals often represent various physical quantities (such as temperature, pressure, displacement, vibration, etc.).In such cases, the estimates of correlation function of the signals ( ) X t , ( ) Y t are reduced to dimensionless values [8].To that end, the estimates of the normalized auto-and cross-correlation functions of the useful signals ( ) In this case, the normalized correlation matrices of the useful signals are as follows: Naturally, matrix equation (1) for this case can be represented in the following form: The corresponding normalized correlation matrices of the noisy signals ( ) g t , ( ) are represented in the following form: Comparing matrices ( 8) and (11), substantial difference between their respective elements are obvious, i. e.
From inequalities ( 7) and ( 13), it follows that correlation matrices (4), ( 5) and ( 11), (12) differ from original matrices (2), ( 3) and ( 8), (9).Therefore, in many cases, ensuring adequacy of identification of the dynamic model of an object by means of these matrices in actual practice is impossible [11].Accordingly, to ensure adequate identification of matrix models of the dynamics of industrial objects, it is necessary to develop technologies for forming the robust correlation matrices , ensuring that the following equalities hold: )

Purpose and objectives of the study
The key purpose of this paper is to develop algorithms that allow for correcting the elements of the correlation  matrices of technological processes with the purpose of reducing them to the matrices of useful signals.
In accordance with the set goal the following research objectives are identified: 1. To avoid errors in the elements of correlation matrices, which emerge during the application of traditional methods of their formation due to the effects of the noise of technological parameters, and to ensure the robustness of the estimates of the elements of the correlation matrices.
2. To create technologies of noise analysis, with regard to the effects of the noise on the estimates of elements of the correlation matrices as a consequence of the noise emerging in real-life objects at the onset of various faults during operation.
3. To avoid the effects of the additional errors emerging during the normalization of the elements of correlation matrices of dynamics models, because the input and output technological parameters in many real-life industrial objects are physical quantities, such as consumption, pressure, temperature, speed, etc. 4. To create generalized robust technologies that enable one, with regard to all of the above, to reduce the correlation matrices of noise technological processes to the matrices of their useful signal, both in the absence of a correlation between the useful signal and the noise and in the presence of such.

Technologies for forming the robust correlation matrices in the absence of a correlation between X(t) and ε(t)
The research in [11] has demonstrated that the conditions of stationarity and normalcy of distribution law hold for technological parameters of many industrial objects.
When the correlation between the useful signals ( ) X t , ( ) Y t and the noise ( ) ) expression (6) for calculating the estimates of the auto-and cross-correlation functions can be represented as follows: 16) , Taking into account expression (16), the correlation matrix of the noisy signals ( ) from formula (4) can be transformed as follows: Based on expressions (17), correlation matrix (5) can also be represented as follows: Experimental research has demonstrated that for those industrial objects, for which conditions (15) are met by determining the estimates of the elements of ( ) g R η µ from expression (17), it is possible to form the robust matrices (19), which would match the correlation matrix It is obvious that by eliminating the errors of noise from the diagonal elements of matrix (18), it can be reduced to the form similar to matrix (2), whose elements contain no noise-induced error.Therefore, to form such matrices for real-life objects, it is necessary to determine the estimates of the noise variance D ε of the noisy technological parameters [16].In this case it is possible to form a matrix, for which equalities (13), (14) will hold, i. e.

( ) ( ) ( ) ( )
However, as discussed, solving identification problems for real-life objects often requires normalizing the estimates of correlation functions.It is clear that given expressions (16), formula (10) for determining the normalized estimate of the autocorrelation function can be transformed as follows: Naturally, the formula for calculating the estimates of normalized cross-correlation functions can also be represented as follows: Therefore, normalized correlation matrix (11) of the noisy signals ( ) g i t ∆ can be represented as follows: The matrix of normalized cross-correlation function can be formed in a similar manner: It follows that normalization leads to additional errors in the elements of correlation matrices.It is obvious that by eliminating said errors with the use of formulas (20), (21), normalized correlation matrices ( 22), (23) equivalent to matrices (8), (9) of the useful signals [15] can be formed.However, that requires determining the estimates of the noise variances D ε and D φ of the technological parameters ( ) g t , ( ) t η .The research has demonstrated that it is appropriate to use expressions [11,12] for that purpose which allow for calculating the estimates D ε , D φ of the variances of the noises ( ) t ε , ( ) t φ of the noisy input ( ) g t and output ( ) signals [11,12,16].At that, taking into account formula (16) and using the obtained estimates Comparing matrices (26), (27) with matrices ( 8), (9), one can see that the effects of the noise-induced errors on the elements have been eliminated and matrices (26), ( 27) can be regarded as equivalent to matrices (8), (9) of the useful signals.Therefore, in the absence of a correlation between ( ) X t and ( ) t ε , ( ) Y t and ( ) one can assume that the following equalities take place between those matrices:

Technology for forming the correlation matrix in the presence of a correlation between the useful signal and the noise
It should be noted that it is characteristic of real-life industrial objects to go into the latent period of origin of various defects, such as wear, microcracks, carbon deposition, fatigue strain, etc. [12,[15][16][17][18].It usually affects the signals received from the corresponding sensors as noise, which in most cases correlates with the useful signal ( ) X t [15][16][17][18][19].For this reason, the sum noise in such cases forms from the noise ( ) , which is caused by the external factors and the noise ( ) that emerge as a result of origin of various defects.The variance of the noisy signal in that case takes the following form [12,16,19 has a correlation with the useful signal X(t) and its variance D ε is determined from the expression where X R (0) ε is the cross-correlation function between the useful signal X(t) and the noise (i t) ε ∆ , D εε is the estimate of the variance of the noise ( ) Therefore, in that case, the variance of the sum noise D ε represents the sum of the variance D εε of the noise ( ) which is caused by external factors and the cross-correlation function X R (0) ε between the useful signal X(t) and the noise 2 (i t) ε ∆ , which is caused by various processes originating in the object itself [12,16,19].
In view of the above, the formula for determining the estimate gg R ( ) m can be represented as follows ( ) ( ) ( )

∑
It is essential to account for the correlation between X(t) and (t) ε when forming the correlation matrices, because in real-life industrial objects a correlation between X(t) and (i t) ε ∆ often takes place even during several sampling intervals, i. e. at t m = ∆ , 2 t m = ∆ , 3 t m = ∆ , … [16,17].Therefore, it is necessary to develop technologies for determining the estimates of the cross-correlation functions ( In view of the above, alongside with determining the estimate D ε , it is also necessary to develop technologies for determining the estimate X R ( 0) ε m ¹ .To that end, let us first consider one of the possible ways to determine the estimate X R ( ) by means of the estimates of the relay correlation functions ( ) * gg R 0 of the technological parameter g(i t) ∆ .With this in mind, assuming the following notation 1 at g i t 0 sgng i t sgnX i t 0 at g i t 0 1 at g i t 0 , the formula for determining the estimates of the relay correlation function

( )
* gg R 0 of the noisy signal g(i t) ∆ is represented as follows: It is known from [16][17][18][19] that the estimate of ( ) Expanding the right-hand side of the formula with an allowance for expression (28), one can get Considering that the following equality holds for stationary technological parameters with the normal distribution law it can be assumed that the result of the calculations in formula (29) can be regarded as the estimate ( ) An analysis of expression (29) has demonstrated that considering the specifics of determining the estimate * X R ( ) ε m of the cross-correlation function between ( ) X t and ( ) t ε can also be represented as follows: in the right-hand side there is It can be shown that the formula for determining the estimate  1 R 2 t sgn g i t g i 1 t N 2sgn g i t g i 2 t sgn g i t g i 3 t Our analysis of literatures [15][16][17][18][19] and research have demonstrated that the following equalities take place between

(
the estimates of the auto-and cross-correlation functions of the signals ( ) X t , ( ) of the useful signal ( ) X t in the diagonal elements that represent the sum of estimates of the correlation function of the useful signals ( ) XX R 0 and the noise variance D ε .
Thus, after the correction of errors of the noise, the diagonal elements of the normalized correlation matrix ( ) are equal to one.However, the other elements of the normalized correlation matrix ( ) gg r µ  of the input signal, as well as all elements of the normalized cross-correlation matrix ( ) g r η µ  of the noisy input and output signals contain in the radical expression of the denominator the values of variances X D , Y D of the useful signals ( ) X t , ( ) Y t and the values of variances D ε , D φ of the noises ( ) t ε , ( ) t φ .

ε∆
….During forming the correlation matrices, this will allow for ensuring that they are equivalent to the matrix of the useful signals by compensating for the errors of the elements gg R in the corresponding lines and columns of the correlation matrices (18), (22).Thus, to ensure that the correlation matrices are equivalent to the matrices of the useful signals, it is necessary to subtract the value of D ε from the estimates of ( ) gg R 0 , and the value of X R ( ) ε m from the values of the estimates of gg R ( ) m , i. e.

∆
can also be represented in a similar form, i. e.

(
R 2 t R 2 t R 2 t .