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Проаналiзовано проблеми формування коре-ляцшних матриць в задачах идентифтаци матричних моделей динамши реальних вироб-ничих об'eктiв. Запропоновано узагальнен алго-ритми, що дозволяють звести щ матриц до ана-логiчних матрицям корисних сигналiв. При цьому врахован специфши реальних зашумлених тех-нологiчних nараметрiв, показана можлив^ть застосування даних алгоритмiв для випадтв вид-сутностi i присутностi кореляци мiж корисним сигналом i перешкодою
Ключовi слова: стохастичний процес, техно-логiчний параметр, зашумлений сигнал, кореля-
цшна матриця, модель динамти
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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 519.216
|DOI: 10.15587/1729-4061.2015.54098|
DEVELOPMENT OF THE ALGORITHMS OF CORRECTION OF CORRELATION MATRICES
T. Aliev
Doctor of Engineering, Full Member of the Academy of Sciences* Е-mail: telmancyber@rambler.ru N. Musayeva Doctor of Engineering, Professor* Е-mail: musanaila@gmail.com U. Sattarova PhD in Engineering* Е-mail: ulker.rzaeva@gmail.com N. Rzayeva Researcher* Е-mail: nikanel1@gmail.com *Institute of Control Systems, Azerbaijan National Academy of Sciences B. Vahabzade str., 9, Baku, Azerbaijan Republic, AZ1141
1. Introduction
It is known [1-6] 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 [1-6]. 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 [1-6]. In practice, such dynamic characteristics as impulsive admittance k(t) and transfer functions ^(s) 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 [6-8].
2. Analysis of published data and problem statement
The statistical correlation method for determining these dynamic characteristics is based on the solution of an inte-
gral equation that includes the correlation functions RXX (t) and Rxy (t) of the input X(t) and output Y(t) 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 e(t). 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.
©
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 [8-10]. The ones that allow for eliminating the error of the noises caused by external factors have found a wide application [10-12]. 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 ^(s) 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: r (m)«—Zg (iAt) g ((i
Rg (m)*
W (m)« [w (0) W (At) ... W ((N - 1)At)]T, 1 N
Rxx(m)« ^zX(iAt)x((i+m)At),
N k=1
1N
Rxy (m)« ^ Z X (iAt) y ((i+m)At).
n k=1
RXX (m) is the square symmetric matrix of the autocorrelation functions with dimension N x N of the centered input signal X(t); Rxy (m) is the column vector of the cross-correlation functions between the input X (t) and the output Y(t), W(m) is the column vector of the impulsive admittance functions.
For equation (1), matrices (2), (3) are formed from the estimates of the useful signals X (t) andY (t).
As previously stated, the real-life technological parameters g(At) and n(iAt) are the sum of the useful signals X(t), Y(t) and noises e(iAt), n(iAt), i. e.
g (t) = X (t) + e(t), n(t) = Y (t) + <Kt).
Therefore, matrix equation (1) and the correlation matrix of real technological processes can be represented as follows:
RCT(m)=Rgg (m)W(m),m=o, At, 2At, ..., (n-1)At
R« (0) R« (At)
Rgg (At) R« (0)
R«[(N- 1)At] R«[(N-2)At]
Rgg [(N - 1)At] Rgg [(N - 2)At]
Rgg (0)
,(4)
Rgn(m)« [Rgn(0) Rgn(At) •• Rgn[(N- ^^ (5)
Rxy (m)« Rxx (m) w(m),
N 1
i +
where
m) At) = n Z (X (iAt) + e(iAt)) (X ((i + m) At) + e ((i + m) At)),
Rgn(m)« N Z g (iAt)n((i+m)At)=N Z(Y (iAt)+e(iAt))(Y ((i+m)At)+^((i+m)At)),
(6)
m = 0, At, 2At,..., (N-1)At, (1)
where
Rxx (m)<
R, RX
X (0) £(At)
Rxx (At) Rxx (0)
Rxx [(N - 1)At ] Rxx [(N - 2) At]
R>
<[(N - 1)At] t[(N - 2) At]
Rxx (0)
Rxy (m) « [Rxy (0) Rxy (At) . R,y [(N - 1)At]]\ (3)
Dg « Rgg (0), Dn « Rgn(0) are the estimates of variances of the signals g(t), n(t) at m = 0; m , mn are the mathematical expectations of
g(fy n(t).
It is impossible to calculate the estimates of the correlation functions RXX (m),
Rxy (m)
(2) the useful signals X(t) and n(t) of the technological parameters g(t), n(t) in practice. For this reason, correlation matrices (4), (5) are formed based on the estimates of Rgg (m), R (m) correlation functions of the noisy signals
g (4 n(t).
However, obvious inequalities emerge in this case:
RX
<(m)* Rgg (m)^
-(m)* Rgn(m),f
due to which the following inequalities take place
RX
R
<(m)* R gg (m),l '(m)* R ».1
(7)
the normalized cross-correlation functions between the input X(t) and the output Y(t), W(m) is the column vector of the impulsive admittance functions.
It is known that the normalized auto- and cross-correlation functions r^ (m), rgn (m) of the noisy signals consisting of the sum of the random useful signals X (t), Y (t) and the corresponding noises e(t), ^(t) are calculated from the following formulas:
Rgg (0)
D„
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 X (t), Y (t) are calculated from formulas [4, 6]:
rgg (m)» Rgg (m)/D r»» Rgn (m)
(10)
The corresponding normalized correlation matrices of the noisy signals g (t), n(t) are represented in the following form:
g (m);
Rgg (At)
D„
Rgg (At)
Rgg (0)
D„
D„
Rgg [(N - 1)At] Rgg [(N - 2)At]
D„
D„
Rgg [(N - 1)At] Dg
Rgg [(N - 2)At ] D„
Rgg (0) D„
,(11)
c(m)-
£(m) /Dx,
'(m) ^VDxDy
R» Rg,(At) Rg,[(N - 1)At]
Vw) (V00
.(12)
where DX - RXX (0), DY - RYY (0) - where RXX (m), RXY (m) are the estimates of the auto- and cross-correlation functions of the signals X(t), Y(t) at m = 0, m = At, m = 2At, m=3At, ... .
In this case, the normalized correlation matrices of the
Comparing matrices (8) and (11), substantial difference between their respective elements are obvious, i. e.
rgg (m)* rxx (m),|
rgn(m)^ rxY (m),J
therefore, the following inequalities take place.
c(m)»
Rxx (0)
Rxx (At)
RX
DY
DY
Rxx (at)
Rxx (0)
dy
dy
Rxx [(N - 1)At] Rxx [(N - 2) At ]
dy
dy
:[(N - 1)At] Dx
Rxx [(N - 2) At] Dy
Rxx (0) Dy
ä(m)*1
(13)
,(8)
useful signals are as follows:
,(0) Rxy (At )
-(m)<
R
Rxy [( N - 1)At]
VDD;) (VDD;) "' (^DD;)
■ (9)
Naturally, matrix equation (1) for this case can be represented in the following form:
?XY (m)° ?XX (m) W(m), m=o, At, 2At, ..., (n - 1)At,
where rxx (m) is the square symmetric matrix of the normalized autocorrelation functions with dimension N x N of the centered input signal X(t); rxY (m) is the column vector of
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 RR (m), RRn(m), rgg (m), rgRn(m), ensuring that the following equalities hold:
RRg (m). W rR (m)-
R
n(m);
(m),
RXY OA
xx (m), xy (m).
(14)
3. 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 RT^R»« Rxy(0) R (At)«
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.
4. Technologies for forming the robust correlation matrices in the absence of a correlation between X(t) and s(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 e(t) is zero, i. e.
Based on expressions (17), correlation matrix (5) can also be represented as follows:
Rxy (At) ... Rgn r( N - 1)At]« Rxy [(N - !)At]]T~RXY (4(19)
1 N
- g X (iAt )£((i + ^)At )«0, 1N
- g y (iAt )#+^)At H
(15)
expression (6) for calculating the estimates of the auto- and cross-correlation functions can be represented as follows:
Rgg N g g (iAt) g ((i+^)At);
N
[Rxx(0) + De at |^ = 0, 1 Rxx M at I-1 * 0 ,
(16)
N gg (iAt)n((i + ^)At)» N g ( x (iAt) + e(kAt)|x
xl Y ((i + ^)At ) + ,((i + ^)At )1«Rxy (^
(17)
Taking into account expression (16), the correlation matrix of the noisy signals g(t), R. „ (|i) from formula (4) can be transformed as follows: gg
RRg (4
Rgg (0)- De« Rxx(0) Rgg (At) ~ Rxx(At)
Experimental research has demonstrated that for those industrial objects, for which conditions (15) are met by determining the estimates of the elements of R(^) from expression (17), it is possible to form the robust matrices
RR>)
from formula (19), which would match the correlation matrix Rxy (|i) of the useful signals X(t), Y(t). At the same time, it follows from expression (18) that the correlation matrix Rgg (|i) of the noisy input signal g (t) differs from the correlation matrix
RXX M
(2) of the useful signal X(t) in the diagonal elements that represent the sum of estimates of the correlation function of the useful signals RXX (0) and the noise variance De.
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 De 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.
RRg (4
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:
(^0)- Rgg(^0)
(20)
Naturally, the formula for calculating the estimates of normalized cross-correlation functions can also be represented as follows:
R»
^Dg - Dœ)(Dn- D,
(21)
Rgg (At) ~ Rxx(At)
Rgg (0)- De« Rxx(0)
Rgg [(N - 1)At]» Rxx [(N - 1)At] Rgg [(N - 2)At> Rxx [(N - 2)At]
Rgg [(N - 1)At]» Rxx [(N - 1)At] Rgg [(N - 2)At> Rxx [(N - 2)At]
Rgg (0)- De« Rxx(0)
.(18)
r
Therefore, normalized correlation matrix (11) of the noisy signals g (iAt) can be represented as follows:
rgg M
1 Rgg (At)« Rxx(At) Rgg (At)« Rxx(At) Rgg [(N - 1)At] « Rxx [(N - 1)At]
Dg - "De 1 ... Dg Rgg [(N - 2)At]« -De Rxx [(N - 2) At]
Dg - D ... Dg- -D
Rgg [(N - 1)At]« Rxx [(N-1)At] Rg g [(N - 2)At]« Rxx [(N - 2) At ] 1
Dg - D Dg -
The matrix of normalized cross-correlation function can be formed in a similar manner:
« Rgn(0) " RxY (0) Rgn(At) " Rxy (At) Rgn[(N - 1)At]° Rxy [(N - 1)At]
De and D, of the technological parameters g(t), n(t). The research has demonstrated that it is appropriate to use expressions [11, 12] for that purpose
Thus, after the correction of errors of the noise, the diagonal elements of the normalized correlation matrix rgg (|i) of the noisy signals g (t) match the diagonal elements of the normalized correlation matrix rXX (|i) of the useful signals X (t) and are equal to one. However, the other elements of the normalized correlation matrix i-ggDe« — £ g(iAt)g(iAt)-2g(iAt)g((i + 1)At) + g(iAt)g((i + 2)At) of the input signal, as well as all elements of the N ;
normalized cross-correlation matrix r^ (|i) of the noisy input and output signals contain in the radical
expression of the denominator the values of varianc- d, «—£ n(iAt)n(iAt)- 2n(iAt)n((i + 1)At) + n(iAt)n((i + 2)At) es D , D of the useful signals X (t), Y (t) and the ' ' ' u ' >
1
N
,(24)
,(25)
values of variances De, D, of the noises e(t), ,(t). 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
which allow for calculating the estimates De, D, of the variances of the noises e(t), ,(t) of the noisy input g (t) and output n(t) signals [11, 12, 16]. At that, taking into account formula (16) and using the obtained estimates R (At)«RXX(At), Rgg(2At)«RXX(2At),...,Rgg[(N- 1)At]«RXX[(N- 1)At] robust normalized correlation matrices:
Rgg(At) « Rxx (At)
Dg - D
Dg - D
Rgg(At) « Rxx (At) Rgg [(N - 1)At]« Rxx [(N - 1)At]
Dg -1 "De ' Dg -Rgg [(N - 2)At]« -D Rxx [(N - 2) At]
Dg -
Rgg [(N - 2)At]« 5 Rxx [(N - 2) At] 1
Dg - De "
, (26)
1
rR
gn
(,« Rgn(0)« Rxy (0) Rgn(At)« Rxy (At) Rgn [(N - 1)At]« Rxy [(N - 1)At] (2?)
J(Dg - D^)(Dn-D<P) J(Dg - D,)(Dn-D<P) J(Dg - D,)(Dn-D<P)
can be formed.
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 e(t), Y(t) and ^(t) one can assume that the following equalities take place between those matrices:
5. 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, 15-18]. It usually affects the signals received from the corresponding sensors as noise, which in most cases correlates with the useful signal X(t) [15-19]. For this reason, the sum noise in such cases forms from the noise e1 (t), which is caused by the external factors and the noise £2 (t) 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]:
Rgg (0)" N fg2 (iAt)°
" N f X2(iAt) + 2N f X(iAt)e(iAt) + 1
+n f £2(iAt)" Rxx(0) + 2RXe (0) + Dee.
The sum noise
e(iAt) = e1 (iAt) + £2(iAt)
has a correlation with the useful signal X(t) and its variance De is determined from the expression
De = 2Rxe (0) + Dee,
where RXe(0) is the cross-correlation function between the useful signal X(t) and the noise e(iAt) , DE£ is the estimate of the variance of the noise e1 (iAt).
Therefore, in that case, the variance of the sum noise De represents the sum of the variance DE£ of the noise e1 (iAt), which is caused by external factors and the cross-correlation function RXe(0) between the useful signal X(t) and the noise £2(iAt), 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 Rgg(m) can be represented as follows
1 N
Rgg (m)" N f g (iAt)g ((i+m)At)"
RXX(0) + De at m = 0, Rxx(m)+Rxe(m) atm*0.
It is essential to account for the correlation between X(t) and e(t) when forming the correlation matrices, because in real-life industrial objects a correlation between X(t) and e(iAt) often takes place even during several sampling intervals, i. e. at m = At, m = 2At, m = 3At, ... [16, 17].
Therefore, it is necessary to develop technologies for determining the estimates of the cross-correlation functions RXe(0), RX(At), RXe(2At), RXe(3At) .... 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 Rgg(0), Rgg(At), Rgg(2At), Rgg(3At),. 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 De from the estimates of Rgg (0), and the value of RXe(m) from the values of the estimates of R (m), i. e.
RR (m)" RXX (m)"
Rgg (0)-D
Rgg (At)-RXe(At)
Rgg (At)- RXE(At)
Rgg (0)-D
Rgg [(N - 1)At]-RX,[(N - 1)At] Rgg [(N - 2)At]-RXe[(N - 2)At] "
Rgg [(N - 1)At]-Rx[(N - 1)At] Rgg [(N- 2)At]-Rx[(N- 2)At]
Rgg (0)-D
1-gg (m)" rXX (m)s
1
Rgg (At)- R*(At)
Dg - De
Rgg [(N - 1)At] - Rxe[( N - 1)At]
Dg - De
Rgg (At)- Rxe(At) Rgg [(N - 1)At]- -Rxe[(N - 1)At]
Dg-D "' 1 " Dg Rgg [(N - 2)At ]- - Rxe [(N - 2) At]
Dg -
Rgg [(N - 2) At]- Rxe[(N - 2) At] Dg - D 1
In view of the above, alongside with determining the estimate De, it is also necessary to develop technologies for determining the estimate RXe(m^0) . To that end, let us first consider one of the possible ways to determine the estimate RXe(m) at m = 0, m = At, m = 2At,... by means of the estimates of the relay correlation functions Rgg (0) of the technological parameter g(iAt) . With this in mind, assuming the following notation
sgn g (iAt) = sgn X (iAt) =
+1 at g (iAt) > 0 0 at g (iAt) = 0 -1 atg (iAt) < 0
the formula for determining the estimates of the relay correlation function Rgg (0) of the noisy signal g(iAt) is rep -resented as follows:
1
R« (0)" N f sgng (iAt) g (iAt)"
1N
"-f sgng (iAt) ■ [X (iAt)+ e (iAt)] " N i=1
" N f [[sgng (iAt)-X(iAt)] + [sgng (iAt)e(iAt)]]»
" i fsgng (iAt) X (iAt) +1 fsgng (iAt) e (iAt)" N i=1 N i=1
" i ^sgnX (iAt) X (iAt) + N i=1
+N f sgnX (iAt)e(iAt)" RXx (0) + RXe(0), Rgg (0)" RXx (0) + RXe(0).
(28)
It is known from [16-19] that the estimate of RXe (0) can be determined from the expression
RXe(0)" N f[sgng (iAt) g (iAt) -
-2sgng (iAt) g ((i + 1)At) + sgng (iAt) g ((i + 2) At)]. (29)
Expanding the right-hand side of the formula with an allowance for expression (28), one can get
f [sgng (iAt) g (iAt)]-N f [2 sgng (iAt) g ((i + 1)At)] +
N1=1'
-f [sgng (iAt) g ((i + 2) At)] 5
Nt!'
" R« (0)-2R« (At) + R« (2At ) = = RXe (0) + RXx (0) - 2RXx (At) + RXx (2 At)" RXe (0).
Considering that the following equality holds for stationary technological parameters with the normal distribution law
RXx (0) + RXx (2 At)- 2RXx (At)" 0,
it can be assumed that the result of the calculations in formula (29) can be regarded as the estimate RXe(0) [19].
An analysis of expression (29) has demonstrated that considering the specifics of determining the estimate RXe (m) of the cross-correlation function between X(t) and e(t) can also be represented as follows:
RXe (At)" N fsgn [ g (iAt) g ((i + 1)At)]-
1 N "n
1N 'Ntf
.1 N !n
f 2sgn [g (iAt) g ((i + 2)At)] +
fsgn [g (iAt)(g (i + 3)At)]"
f [[sgn [X (iAt) + e(iAt)][x ((i + 1)At) + e((i + 1)At)]-
f 2sgn [X (iAt) + e(iAt)][x ((i + 2)At) + e((i + 2)At)] + 1 N -."
+—f [X(iAt) + e(iAt)][x((i + 3)At) + e((i + 3)At)] "
" RXx (At) + RXe (At) + Rex (At) + R; (At) - 2RXx (2 At) --2RXe (2 At) - 2R*ex (2 At) - 2R; (2At) + RXx (3 At) +
+R L ( 3 At ) + R*x (3 At ) + R; (3 At ).
Considering that when
RXe (At )> 0,RXe(2At)" 0, Rxe(3 At)" 0
and the conditions of stationarity and normalcy of distribution law hold, the following equalities can be regarded as true:
RXx (At) + RXx (3At)- 2RXx (2At)" 0,
Ree (At)+ R*ee(3At)- 2R; (3At)" 0,
RXe(2At)"0, RXe(3At)"0, Rex (2At)" 0, Rex (3At)" 0,
in the right-hand side there is
RXe (At)" R*XB (At) + Rex (At)" 2RXe (At),
RXe (At)" ^RXe(At).
(30)
It can be shown that the formula for determining the estimate R** (2At) can also be represented in a similar form, i. e.
RXe(2At)" N f[sgng (iAt) g ((i + 1)At)--2sgng (iAt) g ((i + 2)At) + sgng (iAt) g ((i + 3)At)]
and the estimate RXe (2At) in that case will equal
RXe(2At) = ^RXe(2At).
(31)
Our analysis of literatures [15-19] and research have demonstrated that the following equalities take place between Rxe(0), ARgg (0) and RXe(0), ARgg (0); R^At), AR (At) and RXe(At), ARgg (At); RXe(2At), ARgg (2At) and RXe(2At), ARgg (2At), respectively:
Rxe(o) RXe(0)
ARg g(0)«" ARgg (0),
Rxe (At) Rxe (At)
ARg g (At) AR* (At), gg
Rxe (2At) « . Rxe(2At)
ARgg (2At) ARgg (2At)' from which, using the formulas
ARgg (0)RX,(0)
Rxe(o);
Rxe (At)« Rxe(2At)
AR* (0) '
gg x '
ARgg (At)RXe(At) ARgg (At) , ARgg (2At)RXe(2At)
« ARgg (2At) '
(32)
the estimates RXe(0), RXe(At), RXe(2At), ... are determined.
Thus, as determining the estimates De and RXe(0), Rxe(At), R^At),..., RXe(0), RXe(At), RXe(2At),..., it becomes possible to analyze the errors of the estimates of the correlation functions and the results of formation of the robust correlation matrices. It also becomes possible, depending on the presence or absence of a correlation between X(t) and e(iAt), to make a decision on the appropriate choice of a technology for identifying the models of control objects [12, 25, 28]. It should be noted that when RXe(0)> 0, RXe (At) = 0, RXe (2At) = 0 take place, the correlation matrix is formed in a similar way as in the absence ofa correlation between X(t) and e(iAt). At the same time, if a correlation is detected between X(t) and e(iAt) at time shifts |mAt = At, |m = 2At,..., the estimates RXe(At), RXe(2At) are determined, using expressions (32), and they are subtracted from the estimates of the elements in the respective lines and columns of correlation matrices (18), (22).
Since it is essential to ensure the robustness of the correlation matrices and adequacy of identification of the dynamics model, an alternative way to correct the errors of the corresponding elements of the correlation matrices is proposed below [19]. In this way, the estimates De, RXe(0), RXe(At), RXe(2At), etc. of the technological parameters g (iAt) are determined by means of the expressions developed on the basis of expressions (24), (25).
To that end, the results of decomposing the right-hand side of expression (24) in the presence of a correlation between X(t) and e(t) can be considered.
De= N E [ g (iAt) g (iAt)-2g (iAt) g ((i + l)At) +
+ g (iAt) g ((i + 2)At )] =
1 N
= — £[x (iAt) + e(iAt)][X (iAt) + e(iAt)]-N i=i 1 N
-^E 2 [X(iAt) + e(iAt)][X ((i + l)At)e((i + l)At)] +
1N
+—E[x(iAt) + e(iAt)][x ((i + 2)At) + e((i +2)At)] =
= RXXi (0) + R&(0)+ ReX (0)+ Rre(0)- 2Rxx (At)-
-2RX,(At)- 2ReX (At)- 2Rtt(At) +
+ Rxx (2At)+ RXe (2At)+ ReX (2At)+ Ree (2At). (33)
Considering that when
Rxe(0) > 0,Rxe(At) = 0,Rxe(2At)« 0
and the conditions of stationarity and normalcy of distribution of the technological parameters of the objects under investigation hold, the following equalities can be regarded as true
Rxx (0) + Rxx (2 At )- 2Rxx (At )« 0,
Ree (2At)« 0, Ree(At)« 0,
Rxe (At) « 0, Rxe (2At) « 0, Rex (At) « 0, Rex (2At) « 0 .
Therefore, in the right-hand side of formula (33) is
Ree (0) + Rxe (0) + Rex (0) « 2Rxe(0) + Dee « De.
This demonstrates that the estimate obtained from formula (33) actually is the estimate of the variance De of the sum noise.
Now the possibility of calculating the estimate Rxe (At) in the presence of a correlation between x (t) and e(t) at m = At can be considered from the following expression:
1
R xe (m)«N" f[ g (iAt) g ((i+l)At)-
1 N
- N
+N f [g (iAt) g((i+3)At)>
¿2 [g (iAt) g ((i + 2)At)]-
« N f [x (iAt) + e(iAt)][x ((i + 1)At) + e((i + 1)At)]^
1N
f 2 [x (iAt ) + e(iAt )][x ((i + 2)At) + e((i + 2)At )+] N i=1 1N
+—f [x(iAt) + e(iAt)][x((i + 3)At) + e((i + 3)At)]« N i=1
« Rxx (At) + Rxe(At) + R ex (At) + R œ(At)- 2Rxx (2 At)-
-2Rxe (2At) - 2Rex (2At) - 2Ree (2At) +
+ Rxx (3At)+ Rxe (3At) + Rex (3At)+ Ree(3At).
Considering that when the conditions of stationarity and normalcy of distribution law hold at
Rxe (At)> 0, Rxe (2At) « 0, Rxe (3At) « 0,
the following equalities can be regarded as true Rxx (At) + Rxx (3At)- 2 Rxx (2At) « 0,
Ree (At)+ Ree(3At)- 2Rœ(2At)« 0,
Rxe (2At) « 0, Rxe (3At) « 0 , Rex (2At) « 0, Rex (3At) « 0.
So it is obvious that
Rxe(m) « Rxe (At) + Rex (At) « 2Rxe (At) .
Therefore, the estimate Rxe (At) can be determined from the expression
RXe(At)« —RXe (At).
(34)
It is possible to show that in the presence of a correlation between X(t) and e(t) at | = 2At, the estimate RXe(2At) can be determined in a similar way, using the expression
RX; (2 At) « N £ [ g (iAt) g ((i + 2)At)-
-2g(iAt)g((i + 3)At)+g(iAt)g((i + 4)At)] , (35)
RX'e (2 At)« 2Rxe (2At), Rxe(2At) ~ ^RX'e (2At).
(36)
(37)
In the presence of a correlation between X(t) and e(t) at || = 3At, | = 4At,... the formulas for determining RXe (||) can be similarly represented as follows:
Rxe(3 At)« — RXe (3At)e,
Rxe(4At)« ^RXe(4At), etc.
-3g ((i + i + X + 1) At ) +
+2g ((i + [ + X)At) + g ((i + [ + X + 2)At)],
Rxe(i)« — RXe (i)
RX'e ( At) « - £ g (i At) [g ((i +1+1) At) - g ((i +1) At) -N i=1
-3g ((i +1+ 3 + 1)At) +
+2g ((i +1+3)At) + g ((i +1 + 3 + 2)At)],
RX,(At)« |RXe (At).
It is natural that in determining the estimates of the relay cross-correlation functions RXe(0),RXe (At), RXe(2At),..., errors related to the length of the correlation between X(t) and e(t) also emerge. To eliminate them, it is also appropriate to use similar generalized expressions that can be represented as follows:
1 N
RXe(l) sgng (iAt) [g ((i + | + 1)At) - g((i + |) At) -N i=1
-3g ((i + | + X + 1) At) +
+2g ((i + | + X)At) + g ((i + | + X+2)At)]. (42)
Taking into account formulas (30), (31), one can get:
RXe(i)« 2RXe(i).
(43)
(38)
(39)
Therefore, expression (32) can also be represented as follows:
However, according to the experimental research, in that case the accuracy of the estimate RXe (||) changes depending on the duration of the time shift | between X(t) and e(t). For instance, when RXe(At) > 0, RXe(2At) > 0, RXe(3At) « 0, the estimate RXe(2At) has a lesser error than RXe(At), because the error of the estimate RXe(At) is affected by the correlation between X(t) and e(t) at | = 2At.
To eliminate this shortcoming, generalized expressions eliminating the impact of length of the distance of correlation between X(t) and e(t) on the errors of the sought-for estimates RXe (||) are proposed below.
1 N
RX'e (l) »^ £ g (iAt) [g ((i + | + 1)At) - g ((i + |) At) -
N i=1
Rxe (0) Rxe (At)
.ARgg (0)RXe(0)
AR* (0) ,
gg x '
ARgg (At)RXe(At) ARgg (At) ,
R (2At)«ARgg (2At) RXe (2At) Rxe(2At)~ ARgg(2At) ,
Rxe (i)
ARgg (i)RXe(i) AR^ (i) .
(44)
(40)
where X is the length of the distance of correlation between X(t) and e(t).
In that case, after the estimate RXe(|) has been determined, using the formula
(41)
it is possible to determine the sought-for estimate similar to expressions (34)-(39). For instance, when
RXe (At) > 0, RXe (2At) > 0, RXe (3At) > 0, RXe (4At)« 0 ,
in determining the estimate RXe (|At), it can be considered that X = 3.
In that case, the expressions for determining RXe(At) and RXe (At) will have the following form:
It should be noted that the value X is determined on the basis of the estimate RXe(|), at which RXe(|) ~ 0. It is easy to implement by alternatively determining the estimates RXe(l) by means of expression (42) at X = 0,1,2,3,4,.... For instance, if RXe(3At) = 0, then X = 3.
The use of generalized expressions (40)-(44) makes it possible to correct the corresponding elements of the correlation matrices by determining the estimates RXe(0), RXe(At), RXe(2At), RXe(3At), etc. To that end, first of all determination of the presence or absence of a correlation between X(t) and e(t) in the elements of the matrix from expression (42) using the estimate RXe(|) take place. After that, for the elements with a correlation, the estimates RXe (|) are determined from expressions (40)-(44) and they are corrected. For instance, in the presence of a correlation between X (t) and e(t) in the elements Rgg(At),Rgg(2At), Rgg(3At),..., they are corrected by subtracting from them the corresponding estimates RXe(At),RXe(2At), RXe(3At),... and the value De in the columns and lines of the correlation matrices, in which they are located. For clarity the correction procedure at RXe(At)> 0,RXe(2At) = 0, RXe(3At) = 0,...., is demonstrated below. Here the estimate RXe (At) > 0 is used to correct the second column of the first line and the second line of the first column of matrices (18) and (26)
RR (m)« Rxx (m)«
Rgg(0)-De «Rxx(0) Rgg (At)- Rxe(At)« Rxx (At)
1
Rgg (At)-Rxe (At) « Rxx(At) Rgg [(N - 1)At]« Rxx [(N - 1)At]
Rgg (At)- Rxe (At) « Rxx (At)
Rgg (0)- De «Rxx(0)
Rgg (At)-Rxe(At)« Rxx(At) 1
Rgg [(N - 2)At]« RXX [(N - 2)At]
- Rgg [(N - 1)At] «RXX [(N - 1)At]
- Rgg [(N - 2)At] «RXX [(N - 2)At]
Rgg (0)-De «Rxx(0)
Rgg [(N - 1)At]« RXX [(N - 1)At]
- Rgg [(N - 2)At]« RXX [(N - 2)At]
- 1
Rgg [(N - 1)At] « Rxx [(N - 2)At] Rgg [(N - 2) At] « Rxx [(N - 2) At]
rgg (m)« rxx (m)s
Rgg (At)- Rxe(At)« Rxx (At)
Rgg (At)- Rxe(At)« Rxx (At)
D - D„
D - De
Rgg [(N - 1)At]« Rxx [(N - 1)At] Rgg [( N - 2)At]« Rxx [(N - 2)At]
D - De
D - D
Rgg [(N - 1)At]= = Rxx [( N - 1)At]
Dg - De
Rgg [(N - 2)At]« Rxx [(N - 2) At]
Dg- "De
] 1
In this case, the result of formation of correlation matrices is regarded as valid only when the estimates RXe (m) at m = 0, m = At, m = 2At, m = 3At — obtained from expressions (40)-(44) match, i.e. adequacy of the obtained results is achieved by their duplication. Therefore, after such correction, the obtained matrix can be considered equivalent to the matrix of the useful signals.
6. The robust technology for eliminating the errors of
calculation of the estimates of correlation functions
An analysis of the specifics of forming correlation matrices shows that during determining the estimates Rgg (m), Rgn (m), errors emerge in the calculations, which affect validity of the robustness conditions [11, 12]. For instance, during calculating the estimate Rgg (0), all paired products g (iAt) and g ((i + m)At) have the positive sign. Therefore, the errors of these products are combined and the error of the calculation turns out to be maximum. However, as the time shift m between g (iAt) and g ((i + m)At), as well as between gn(iAt) and gn((i + m)At) increases, the obtained estimates turn out to be equal to zero at some point. In this case, the sums of errors of the products g (iAt) g ((i + m)At) with the positive and negative signs in the amount of N+, N-, from which the sum error Rgg (m) forms, turn out equal and the equality N+ = N- takes place. As a result, the positive and negative errors of the products practically balance each other. Therefore, in determining the estimates Rgg(m), the calculation errors depend on the difference in the number of the paired products N + - N- with the positive and negative signs. That difference changes depending on the change of the time shift m between them. Therefore, to ensure equalities (38), there is a need to eliminate the errors of calculating the estimates
of elements of matrices (27), (28) and (36), (37). This issue is considered in detail in [11], and the following expressions are recommended to compensate the error from the difference of the positive and negative products of the estimates of the auto- and cross-correlation functions:
1
RgRg(m)« TfZ g(iAt)g((i + m)At) -
N ¡=1
-[N+(m) - N - (m)]( AV(0));
RRn(m)« g(iAt)n((i+m)At) -N i=1
-[N + (m)- n-(m)](AV(At)).
(45)
(46)
In that case, the error from the difference of the product y(At) is determined from the expressions
Rgg(At)-Rgg(At) « V(At),|
Rgn (At) -Rgn(At) «v(At),l
where
(AV(At)) = [1/n-(At)MAt).
(47)
(48)
Here Rgg (At), Rgg (At), Rg^(At), Rg^(At) are the estimates of the auto- and cross-correlation functions of the centered and non-centered signals g(iAt), n(iAt), respectively; n- is the number of negative products that emerges from the difference of the number of the products g (iAt) g (i + m)At or g(iAt) n(iAt) with the positive and negative signs, respectively, N+ - N-. It is obvious from expressions (41), (42) that when expressions (39), (40) are applied, the errors that arise
1
due to the difference of the number of the paired products g (iAt) g ((i + m)At) with the positive N+ and negative N-signs compensate one another. Therefore, when expressions (39), (40) are applied, the condition of robustness of the elements of correlation matrices [11] is ensured by eliminating the effects of the error on the calculations.
To sum up, the procedure for eliminating the error of calculation of the estimates R^ (m) is presented below
1. The estimate R^ (At) is determined from the expression
6. Conclusion
1 N
Rgg (m)=NI g(iAt)g (0+m)At).
N
2. The error of the estimate at the unit time shift mAt = 1At is determined:
V(At) = |Rgg(At) - Rgg(At),
(AV(At)) = [1/n-(At)]V(At),
where n- is the number of the negative products at mAt = 1At due to the difference of N+ - N-.
3. The error is determined:
vXx (m)"[n +(m)- n -(m)](AV(At^.
4. The variance is determined:
De =1 f (g(iAt)g(iAt) + g(iAt) x N i=1
x(g(i + 2)At)- 2g(iAt)(g(i + 1)At)).
5. Finally, the robust estimates are determined:
Rgg(m)=jRgg(mH^(m)+D] atm=°, (49)
^(m) at
Thus the formula (49) can be used to eliminate the errors occurring in the process of calculating and ensure fulfilment of the condition of robustness.
The paper considers the problems related to identification of the model of dynamics of real-life industrial objects. When traditional methods of formation of the correlation matrix are used, because of substantial errors of the estimates of its elements, the conditions of robustness are violated from the effects of the noise in the technological parameters; therefore, adequacy of the obtained results is not achieved in most cases. It is well known there are many filtration methods that eliminate various errors caused by effects of the noise. However, in real-life objects, noises of technological processes are caused by various faults during operation and affect the signals in the form of noise. The range of their spectrum often overlaps the spectrum of the useful signal. Moreover, their spectra are not strictly stable. For these reasons, filtration does not always yield the desired result. Filtration even causes distortion of the spectrum of the useful signal sometimes.
Besides in many real-life industrial objects, the input and output technological parameters are usually represented by such physical quantities as consumption, pressure, temperature, velocity, etc. Therefore, in identifying mathematical models of dynamics, in forming the correlation matrices, it is necessary to apply the procedure of normalization of their elements. This leads to an additional error, which also leads to the disruption of adequacy of the results. That is why methods and technologies for eliminating that error, which can also be widely used in systems of control and management of technological processes in various industries are proposed.
Taking into account above-mentioned problems two alternative robust generalized technologies that enable one to reduce the correlation matrices of noisy technological processes to the matrices of their useful signals both in the absence of a correlation between the useful signal and the noise and in the presence of such are proposed. The validity of the result is achieved through duplication of the obtained estimates of the elements of matrices by both methods.
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