CN115047853A - Micro fault detection method based on recursion standard variable residual error and kernel principal component analysis - Google Patents

Micro fault detection method based on recursion standard variable residual error and kernel principal component analysis Download PDF

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CN115047853A
CN115047853A CN202210737859.5A CN202210737859A CN115047853A CN 115047853 A CN115047853 A CN 115047853A CN 202210737859 A CN202210737859 A CN 202210737859A CN 115047853 A CN115047853 A CN 115047853A
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CN115047853B (en
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史贤俊
秦玉峰
秦亮
聂新华
王朕
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Naval Aeronautical University
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    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
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Abstract

The invention relates to the technical field of fault diagnosis, in particular to a tiny fault detection method based on recursion standard variable residual error and kernel principal component analysis, which comprises the steps of firstly constructing the standard variable residual error, extracting linear characteristics of data from the standard variable residual error, carrying out recursion filtering treatment on the standard variable residual error by using an exponential weighted moving average method, improving the sensitivity degree of the standard variable residual error to tiny faults, then extracting nonlinear principal components in the standard variable residual error by using KPCA (kernel principal component analysis), and providing two new fault detection statistics according to the extracted characteristics; in addition, a control limit for the fault detection statistic is determined using the kernel density estimate. Because the linear and nonlinear characteristics of the process data are extracted simultaneously, the detectability of the tiny faults in the nonlinear dynamic process is effectively improved.

Description

Micro fault detection method based on recursion standard variable residual error and kernel principal component analysis
Technical Field
The invention relates to the technical field of fault diagnosis, in particular to a micro fault detection method based on recursion standard variable residual error and kernel principal component analysis.
Background
With the increasing size and complexity of systems, fault diagnosis technology has become an important means for ensuring the safe operation of the systems. According to the size of the symptom caused by the fault, the fault can be classified into a significant fault and a minor fault. The micro fault is mainly characterized in that the development and the change are slow, the fault symptom is not obvious, and the micro fault is easily submerged by noise; over time, the amplitude of the fault slowly increases and develops into a significant fault, which if not discovered early, may cause the system to fail with serious consequences.
The nature of the micro-fault makes it very difficult to detect the occurrence of the micro-fault at an early stage. The existing tiny fault diagnosis method mainly comprises a fault diagnosis method based on knowledge, a fault diagnosis method based on an analytical model and a fault diagnosis method based on data driving. In consideration of the complexity of the structural composition and the function of the system, the fault diagnosis method based on knowledge and the analytic model is difficult to implement. Therefore, a data-driven minor failure diagnosis method becomes a research hotspot. The method has the advantages that complete prior knowledge of structural functions and the like is not needed, and an accurate physical model is not needed to be constructed. A number of multivariate statistical analysis techniques are used in the field of minor fault detection. For example, Harmouche and the like propose a KL-Leiblerdcargence (KLD) -based tiny fault detection method aiming at the problem that the T2 statistic of the traditional PCA method is insensitive to tiny faults; zhang and the like combine PCA and KLD, and optimize a projection vector obtained by PCA, so that the projection vector is locally optimal for a KLD fault detection method; chen and the like detect early tiny faults of a non-Gaussian electric drive system based on KLD and analyze the robustness of the method in a wider signal-to-noise ratio range; cai and the like provide a micro fault diagnosis method based on KLD-KPCA in consideration that a nuclear principal component analysis model cannot sensitively detect the change of a micro fault initial stage and KLD is introduced to measure the change degree of a nuclear principal component; establishing a small fault amplitude estimation model based on covariance matrix characteristic value change and KLD change by Dow soldiers and the like; gautam et al establishes fault detection indicators and fault characteristics using an extended Kalman filter, and then designs fault decision statistics based on KLD. Similar to KLD, Zhang et al detect and estimate early minor faults based on JS divergence (JSensen-Shannendigergence, JSD). The above document uses the characteristic that the probability density function is sensitive to the micro fault, and realizes the detection of the micro fault by measuring the difference between the probability density functions of the detection data before and after the fault occurs. Such methods generally require that the detected data be assumed to follow a normal distribution, but practical systems may not meet this requirement, and thus the application range is limited.
Another class of methods aims to improve the sensitivity of the residual to minor faults. For example, Ruiz-Crcel et al propose a tiny fault detection method based on normative variable analysis (CVA), and verify that the fault detection effect of the method is superior to that of the traditional PCA method; wu and the like introduce a deep neural network into the CVA, and classify faults by using a Bayesian inference classifier; the quotient brightness and the like introduce a first-order interference theory in the CVA, so that the calculation load is obviously reduced; furthermore, Pilario et al constructs a normative variable residual based on the difference between past and future normative variables, and deals with early minor fault detection problems through normative variable residual analysis (CVDA); shang et al propose a weighted average statistic based on CVDA, which improves the fault detection rate; li and the like provide a new fault detection statistic based on CVDA, which is more sensitive to the development and change of tiny faults, and simultaneously keeps low false alarm rate, improves the traditional contribution diagram method and improves the fault identifiability; chen and Luo propose a new multivariate q-sigma rule to monitor the standard variable residual error, and set a control limit for each variable, thereby reducing the detection delay and the false alarm rate; shuzhujun populates the CVDA to a nonlinear process, and proposes a fault detection method based on kernel canonical variable residual analysis (KCVDA); pilario et al combine different kernel functions to provide a hybrid KCVDA-based early minor fault method for a nonlinear dynamic process.
Although the CVDA and the KCVDA have certain effectiveness in the aspect of tiny fault detection, the single adoption of one model is not the best choice, namely the CVDA only extracts linear features in data and cannot extract nonlinear features of the data, and the nonlinear features usually appear in a residual space of a linear model; the KCVDA maps the original data to a high-dimensional space, thereby ignoring information of the original space. Therefore, if the fault detection is performed only by the linear or nonlinear model, the detectability of the minor fault is relatively low.
Disclosure of Invention
Therefore, the invention provides a tiny fault detection method based on recursion specification variable residual error and kernel principal component analysis, which is used for overcoming the problem that the detectability of tiny faults is relatively low when fault detection is carried out only by using a linear or nonlinear model in the prior art.
In order to achieve the above object, the present invention provides a method for detecting a minor fault based on recursive canonical variable residuals and kernel principal component analysis, comprising:
step S1, off-line training,
step S11: acquiring detection data Y in normal operation state 0 Standardizing the data to obtain standardized detection data Y;
step S12: constructing a past observation matrix Yp and a future observation matrix Yf, and respectively calculating covariance and cross covariance matrixes of Yp and Yf;
step S13: performing singular value decomposition on the matrix H and determining the number q of leading singular values;
step S14: calculating a canonical variable residual d and EWMA filtering d to obtain a filtered canonical variable residual
Figure BDA0003714775950000031
Step S15: constructing a kernel matrix K, carrying out mean value centralization, and solving a characteristic value and a characteristic vector of the kernel matrix K;
step S16: calculating pivot score vectors
Figure BDA0003714775950000032
Step S17: formation fault detection statistics
Figure BDA0003714775950000033
And Q ck ；
Step S18: calculating corresponding control limits
Figure BDA0003714775950000034
Q UCL，ck ；
In step S2, the online detection,
step S21: acquiring actual detection data and using Y 0 Is normalized by the mean and covariance of the two to obtain normalized detection data Y test ；
Step S22: constructing a past observation matrix Y p,test And future observation matrix Y f,test ；
Step S23: calculating a specification variable residual d by adopting the method of the step S14 test And to d test EWMA filtering to obtain filtered canonical variable residuals
Figure BDA0003714775950000035
Step S24: by using
Figure BDA0003714775950000036
And
Figure BDA0003714775950000037
constructing a kernel matrix K test Mean value centralization is carried out, and the characteristic value and the characteristic vector of the mean value centralization are solved;
step S25: calculating the pivot score vector by the method of the step S16
Figure BDA0003714775950000038
Step S26: calculating a fault detection statistic using the method of step S17
Figure BDA0003714775950000039
And Q ck，test ；
Step S27: if it is
Figure BDA0003714775950000041
Or Q ck，test ＞Q UCL，ck Then a fault occurrence is detected.
Further, at the stepIn step S11, the method of normalizing the detection data in the normal operation state is to set
Figure BDA0003714775950000042
Is the raw inspection data, wherein,
Figure BDA0003714775950000043
n is the number of samples, m is the number of variables,
Figure BDA0003714775950000044
for the kth sample, the raw test data is normalized and set
Figure BDA0003714775950000045
In the formula (I), the compound is shown in the specification,
Figure BDA0003714775950000046
is the mean of the jth variable, s j The standard deviation of the jth variable, j ═ 1,2,. . . M, the original detection data matrix Y 0 Conversion to: y ═ Y 1 …y n ] T 。
Further, in the step S12, a past observation matrix Y is constructed p And future observation matrix Y f And separately calculate Y p And Y f The method of covariance and cross-covariance matrix is that when the original test data matrix Y is 0 Conversion to: y ═ Y 1 …y n ] T Then, for the kth detection sample, the past observation vector y is set p (k) And future observation vector y f (k) Are respectively:
Figure BDA0003714775950000047
Figure BDA0003714775950000048
wherein p and f represent the window lengths of the past observation vector and the future observation vector, respectively;
setting a past observation matrix
Figure BDA0003714775950000049
And future observation matrix
Figure BDA00037147759500000410
Are respectively:
Y p ＝[y p (p+1)y p (p+2)…y p (p+N)] (4)
Y f ＝[y f (p+1)y f (p+2)…y f (p+N)] (5)
wherein N is N-f-p +1,
setting of Y p The expression of covariance of (a) is:
Figure BDA0003714775950000051
setting of Y f The expression of covariance of (a) is:
Figure BDA0003714775950000052
setting of Y p And Y f The expression of cross-covariance of (a) is:
Figure BDA0003714775950000053
in the formula, parameters p and f satisfy { mp, mf } < N.
Further, in the step S13, the method of performing singular value decomposition on the matrix H and determining the number q of dominant singular values is to perform singular value decomposition calculation on the matrix H after the step S12 is completed, and to set:
Figure BDA0003714775950000054
in the formula, U and V are matrices composed of left and right singular vectors, respectively, a diagonal matrix S is composed of ordered singular values, and S is set to be diag (Σ) 1 ，…，∑ γ 0, …, 0), γ is the rank of matrix H, taking the first q columns of U and V with the maximum correlation, where q < mp, to obtain matrices Uq and Vq after dimensionality reduction, then the expressions of projection matrices J and L are:
Figure BDA0003714775950000055
Figure BDA0003714775950000056
further, in the step S14, a normalized variable residual d is calculated and the d is subjected to the EWMA filtering to obtain a filtered normalized variable residual
Figure BDA0003714775950000057
By performing the calculation of projection matrices J and L such that JY p (k) And LY f (k) Maximize the correlation between, wherein JY p (k) And LY f (k) For the k-th detection sample, the expressions of the state vector x (k) and the residual vector e (k) are set as follows:
x(k)＝JY p (k) (12)
Figure BDA0003714775950000058
wherein I is a dimensional unit array, and the following fault detection statistics are constructed by using x (k) and e (k):
T 2 (k)＝x(k) T x(k) (14)
Q(k)＝e(k) T e(k) (15)
in the formula, T 2 (k) Statistics to measure the change of the state vector x (k), Q (k) statistics measureThe variation of the residual vector e (k), the expression of the normalized variable residual d (k) of the kth detection sample is:
d(k)＝Ly f (k)-S q Jy p (k) (16)
in the formula, S q ＝diag(Σ 1 ，…，Σ q ) Keeping in mind that a matrix composed of the normative variable residuals of all samples is Yd, and the expression of the covariance matrix is:
Figure BDA0003714775950000061
constructing a fault detection statistic D based on the relevant definition of the Mahalanobis distance, and setting:
Figure BDA0003714775950000062
filtering the standard variable residual error d by adopting an exponential weighted moving average method EWMA (equal weighted average), and filtering the filtered data
Figure BDA0003714775950000063
The expression of (c) is:
Figure BDA0003714775950000064
in the formula (I), the compound is shown in the specification,
Figure BDA0003714775950000065
in order to be a weight factor, the weight factor,
Figure BDA0003714775950000066
the larger the value is,
Figure BDA0003714775950000067
the more it is possible to reflect the actual data information,
Figure BDA0003714775950000068
the smaller the value is,
Figure BDA0003714775950000069
the more sensitive it is to small changes in data.
Further, in the step S15, the method for constructing and mean-centering the kernel matrix K and solving the eigenvalue and the eigenvector thereof includes obtaining the filtered standard variable residual error
Figure BDA00037147759500000610
Then, adopting KPCA algorithm to further extract
Figure BDA00037147759500000611
Non-linear characteristics of (1), hypothesis
Figure BDA00037147759500000612
By means of non-linear functions
Figure BDA00037147759500000613
Implicit mapping to a high dimensional feature space
Figure BDA00037147759500000614
And is
Figure BDA00037147759500000615
Then solving the eigenvalue of the covariance matrix of the sample, and setting:
Figure BDA00037147759500000616
Cξ＝λξ (21)
wherein C is a space
Figure BDA00037147759500000617
Where λ is an eigenvalue and ξ is an eigenvector and is included in the sample covariance matrix
Figure BDA0003714775950000071
In the subspace spanned, there is therefore a vector η ═ η 1 …η N ] T So that xi is expressed as
Figure BDA0003714775950000072
The linear combination of (a) and (b) is set to:
Figure BDA0003714775950000073
equation (21) is rewritten as:
Figure BDA0003714775950000074
where <, > represents the inner product, and substitution of equations (20) and (22) into equation (23) yields:
Figure BDA0003714775950000075
defining a kernel matrix
Figure BDA0003714775950000076
The elements in K meet the following conditions:
Figure BDA0003714775950000077
where k (·, ·) is a kernel function, i, j ═ 1, …, N, and a gaussian radial basis function is selected as the kernel function, then:
Figure BDA0003714775950000078
where h is the kernel width, equation (24) is rewritten as:
Figure BDA0003714775950000079
namely:
λNη＝Kη (28)
according to the formula (28) The feature vector η can be determined 1 ，…，η N And its corresponding characteristic value lambda 1 ，…，λ N In addition, the matrix K needs to be mean-centered before calculation:
Figure BDA00037147759500000710
in the formula (I), the compound is shown in the specification,
Figure BDA0003714775950000081
and wherein each element is 1/N, taking r eigenvalues lambda 95% before the cumulative variance contribution rate 1 ，…，λ r And its corresponding feature vector η 1 ，…，η r 。
Further, in the step S16, a pivot score vector is calculated
Figure BDA0003714775950000082
Is that, after said step S15 is completed, the k-th normative variable residual is subjected to
Figure BDA0003714775950000083
Obtaining its pivot score vector by the following projection
Figure BDA0003714775950000084
Setting:
Figure BDA0003714775950000085
Figure BDA0003714775950000086
in the formula (I), the compound is shown in the specification,
Figure BDA0003714775950000087
is a feature vector eta i I-1, …, r.
Further, in said stepConstructing fault detection statistics in step S17
Figure BDA0003714775950000088
And Q ck By constructing fault detection statistics after completing said step S16
Figure BDA0003714775950000089
And Q ck Setting:
Figure BDA00037147759500000810
Figure BDA00037147759500000811
wherein Λ is diag (λ) 1 ，…，λ r ),
Figure BDA00037147759500000812
Further, the method for calculating the control limit in step S18 is to determine the control limit of the fault detection statistic by using the kernel density estimation KDE, assuming that x is a random variable and p (x) is a probability density function of x, then:
Figure BDA00037147759500000813
the following can be obtained by a gaussian kernel function:
Figure BDA00037147759500000814
estimating the probability density function of x by a gaussian kernel function:
Figure BDA00037147759500000815
where psi is the bandwidth, x (i), i ═1,2, …, where N is the ith sample in x, and the fault detection statistic is set to J, and its control limit is J UCL Given a significance level α, then J UCL The calculation expression of (a) is:
Figure BDA0003714775950000091
further, if J.ltoreq.J UCL If yes, judging that no fault occurs; if J > J UCL Then it is determined that a fault is detected.
Compared with the prior art, the method has the beneficial effects that the method for detecting the micro fault based on the RCVD-KPCA is provided, the main contribution is that a mixed statistical modeling method is provided, the linear and nonlinear characteristics of process data are extracted simultaneously, the early micro fault detection performance of a nonlinear dynamic process is improved, and the detectability of the micro fault is improved.
Further, simulation results show that compared with the traditional CVDA and KCVDA methods, the fault detection statistic obtained by the method can detect the tiny fault more quickly, the false alarm rate and the omission factor are low, and the method has good fault detection performance.
Drawings
Fig. 1 is a flowchart of a minor fault detection method based on recursive canonical variable residuals and kernel principal component analysis according to an embodiment of the present invention.
Detailed Description
In order that the objects and advantages of the invention will be more clearly understood, the invention is further described below with reference to examples; it should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Preferred embodiments of the present invention are described below with reference to the accompanying drawings. It should be understood by those skilled in the art that these embodiments are only for explaining the technical principles of the present invention, and do not limit the scope of the present invention.
It should be noted that in the description of the present invention, the terms of direction or positional relationship indicated by the terms "upper", "lower", "left", "right", "inner", "outer", etc. are based on the directions or positional relationships shown in the drawings, which are only for convenience of description, and do not indicate or imply that the device or element must have a specific orientation, be constructed in a specific orientation, and be operated, and thus, should not be construed as limiting the present invention.
Furthermore, it should be noted that, in the description of the present invention, unless otherwise explicitly specified or limited, the terms "mounted," "connected," and "connected" are to be construed broadly, and may be, for example, fixedly connected, detachably connected, or integrally connected; can be mechanically or electrically connected; they may be connected directly or indirectly through intervening media, or they may be interconnected between two elements. The specific meanings of the above terms in the present invention can be understood by those skilled in the art according to specific situations.
Fig. 1 is a flowchart of a minor fault detection method based on recursive canonical variable residuals and kernel principal component analysis according to an embodiment of the present invention, including:
step S1, off-line training,
step S11: acquiring detection data Y in normal operation state 0 Standardizing the data to obtain standardized detection data Y;
step S12: constructing a past observation matrix Y p And future observation matrix Y f And separately calculate Y p And Y f Covariance and cross-covariance matrices of (a);
step S13: performing singular value decomposition on the matrix H and determining the number q of leading singular values;
step S14: calculating a canonical variable residual d and EWMA filtering d to obtain a filtered canonical variable residual
Figure BDA0003714775950000101
Step S15: constructing a kernel matrix K, carrying out mean value centralization, and solving a characteristic value and a characteristic vector of the kernel matrix K;
step S16: calculating pivot score vectors
Figure BDA0003714775950000102
Step S17: formation fault detection statistics
Figure BDA0003714775950000103
And Q ck ；
Step S18: calculating corresponding control limits
Figure BDA0003714775950000104
Q UCL，ck ；
In step S2, the online detection,
step S21: acquiring actual detection data and using Y 0 Is normalized by the mean and covariance of the two to obtain normalized detection data Y test ；
Step S22: constructing a past observation matrix Y p,test And future observation matrix Y f,test ；
Step S23: calculating a specification variable residual d by adopting the method of the step S14 test And to d test EWMA filtering to obtain filtered canonical variable residuals
Figure BDA0003714775950000105
Step S24: by using
Figure BDA0003714775950000111
And
Figure BDA0003714775950000112
constructing a kernel matrix K test Mean value centralization is carried out, and the characteristic value and the characteristic vector of the mean value centralization are solved;
step S25: calculating the pivot score vector by the method of the step S16
Figure BDA0003714775950000113
Step S26: by usingThe method of step S17 calculates a fault detection statistic
Figure BDA0003714775950000114
And Q ck，test ；
Step S27: if it is
Figure BDA0003714775950000115
Or Q ck，test ＞Q UCL，ck Then a fault occurrence is detected.
In this embodiment, the CVDA-based canonical variable residuals are constructed:
as a linear dimension reduction technique, CVAs can maximally correlate past and future data sets. Thus, data changes can be detected based on how predicted the past data set is for the future data set. The CVDA constructs a specification variable residual error by using the difference between the past specification variables and the future specification variables on the basis of the CVA, and detects the data change through the specification variable residual error. The specific implementation of CVDA is described below.
Specifically, in step S11, the method of normalizing the detection data in the normal operation state is to provide
Figure BDA0003714775950000116
Is the raw inspection data, wherein,
Figure BDA0003714775950000117
n is the number of samples, m is the number of variables,
Figure BDA0003714775950000118
for the kth sample, the raw test data is normalized and set
Figure BDA0003714775950000119
In the formula (I), the compound is shown in the specification,
Figure BDA00037147759500001110
is the mean of the jth variable, s j The standard deviation of the j-th variable, j is 1,2,. . . M, the original detection data matrix Y 0 Conversion to: y ═ Y 1 …y n ] T 。
Specifically, in the step S12, the past observation matrix Y is constructed p And future observation matrix Y f And separately calculate Y p And Y f The method of covariance and cross-covariance matrix is that when the original test data matrix Y is 0 Conversion to: y ═ Y 1 …y n ] T Then, for the kth detection sample, the past observation vector y is set p (k) And future observation vector y f (k) Are respectively:
Figure BDA00037147759500001111
Figure BDA0003714775950000121
wherein p and f represent the window lengths of the past observation vector and the future observation vector, respectively;
setting a past observation matrix
Figure BDA0003714775950000122
And future observation matrix
Figure BDA0003714775950000123
Are respectively:
Y p ＝[y p (p+1)y p (p+2)…y p (p+N)] (4)
Y f ＝[y f (p+1)y f (p+2)…y f (p+N)] (5)
wherein N is N-f-p +1,
setting of Y p The expression of covariance of (a) is:
Figure BDA0003714775950000124
setting of Y f The expression of covariance of (a) is:
Figure BDA0003714775950000125
setting of Y p And Y f The expression of cross-covariance of (a) is:
Figure BDA0003714775950000126
in the formula, to avoid Σ pp Sum-sigma ff Being singular, the parameters p and f need to satisfy: { mp, mf } < N.
Specifically, in step S13, the method of performing singular value decomposition on the matrix H and determining the number q of dominant singular values is to perform singular value decomposition calculation on the matrix H after completion of step S12, and to set:
Figure BDA0003714775950000127
in the formula, U and V are matrices composed of left and right singular vectors, respectively, a diagonal matrix S is composed of ordered singular values, and S is set to be diag (Σ) 1 ，…，∑ γ 0, …, 0), γ is the rank of matrix H, and since only q (q < mp) dominant singular values describe the dynamic characteristics of the system, the first q columns of U and V with the maximum correlation are taken to obtain matrices Uq and Vq after dimensionality reduction, and then the expressions of projection matrices J and L are respectively:
Figure BDA0003714775950000131
Figure BDA0003714775950000132
specifically, in the step S14Calculating a canonical variable residual d and EWMA filtering d to obtain a filtered canonical variable residual
Figure BDA0003714775950000133
In such a way that the CVA aims to obtain the projection matrices J and L, so that JY p (k) And LY f (k) Maximize the correlation between, wherein JY p (k) And LY f (k) Called a canonical variable, for the kth detection sample, the expressions of the state vector x (k) and the residual vector e (k) are set as follows:
x(k)＝JY p (k) (12)
Figure BDA0003714775950000134
wherein I is a dimensional unit array, and the following fault detection statistics are constructed by using x (k) and e (k):
T 2 (k)＝x(k) T x(k) (14)
Q(k)＝e(k) T e(k) (15)
in the formula, T 2 (k) Statistics to measure the change of state vector x (k), q (k) statistics to measure the change of residual vector e (k), the predictability of future observation vectors from past observation vectors can effectively detect the slight change of data, so the expression of the normalized variable residual d (k) of the k-th detection sample is:
d(k)＝Ly f (k)-S q Jy p (k) (16)
in the formula, S q ＝diag(∑ 1 ，…，∑ q ) Keeping in mind that a matrix composed of the normative variable residuals of all samples is Yd, and the expression of the covariance matrix is:
Figure BDA0003714775950000135
constructing a fault detection statistic D based on the relevant definition of the Mahalanobis distance, and setting:
Figure BDA0003714775950000136
the validity of the statistic d (k) for minor fault detection is demonstrated in the prior art, but it can only evaluate the change of linear characteristics in the process data. Since the residual error of the linear model usually has nonlinear characteristics, the influence of which cannot be separated from other uncertainties, so that the model has a higher control limit, thereby reducing the detectability of minor faults, in order to improve the detectability of faults, it is necessary to further extract the nonlinear characteristics in the normative variable residual error d. Considering that the current kernel method is technically mature, the present embodiment applies the KPCA method to realize the nonlinear feature extraction.
In the embodiment, fault detection based on RCVD-KPCA:
in order to improve the sensitivity of the normalized variable residual d to data changes, an Exponential Weighted Moving Average (EWMA) method is firstly adopted to filter the normalized variable residual d, the EWMA method is a commonly used data processing method in engineering system process measurement, the solving process is actually a recursion process, and the filtered data is subjected to filtering
Figure BDA0003714775950000141
The expression of (a) is:
Figure BDA0003714775950000142
in the formula (I), the compound is shown in the specification,
Figure BDA0003714775950000143
in order to be a weight factor, the weight factor,
Figure BDA0003714775950000144
the larger the value is,
Figure BDA0003714775950000145
the more it is possible to reflect the actual data information,
Figure BDA0003714775950000146
the smaller the value is,
Figure BDA0003714775950000147
the more sensitive it is to small changes in data.
Specifically, in step S15, the method for constructing and mean-centering the kernel matrix K and solving its eigenvalues and eigenvectors is to obtain the filtered normalized variable residual error
Figure BDA0003714775950000148
Then, adopting KPCA algorithm to further extract
Figure BDA0003714775950000149
Non-linear characteristics of (1), hypothesis
Figure BDA00037147759500001410
By means of non-linear functions
Figure BDA00037147759500001411
Implicit mapping to high dimensional feature space
Figure BDA00037147759500001412
And is
Figure BDA00037147759500001413
Then solving the eigenvalue of the covariance matrix of the sample, and setting:
Figure BDA00037147759500001414
Cξ＝λξ (21)
wherein C is a space
Figure BDA00037147759500001415
In (3), the covariance matrix of samples, λ is the eigenvalue, ξ is the eigenvector, since
Figure BDA00037147759500001416
Cannot be expressed explicitly, so the formula (21) cannot be expressed straightforwardlyThen, a solution is made, taking into account that xi is contained in
Figure BDA00037147759500001417
In the spanned subspace, there is therefore a vector: eta ═ eta 1 …η N ] T So that xi is expressed as
Figure BDA00037147759500001418
The linear combination of (a) and (b) is set to:
Figure BDA00037147759500001419
equation (21) is rewritten as:
Figure BDA00037147759500001420
where <, > represents the inner product, and substitution of equations (20) and (22) into equation (23) yields:
Figure BDA0003714775950000151
defining a kernel matrix
Figure BDA0003714775950000152
The elements in K satisfy:
Figure BDA0003714775950000153
where k (·, ·) is a kernel function, i, j ═ 1, …, N, and the gaussian radial basis function is selected as the kernel function in this embodiment, then:
Figure BDA0003714775950000154
where h is the kernel width, equation (24) is rewritten as:
Figure BDA0003714775950000155
namely:
λNη＝Kη (28)
the feature vector η can be determined according to the formula (28) 1 ，…，η N And its corresponding characteristic value lambda 1 ，…，λ N In addition, the matrix K needs to be mean-centered before calculation:
Figure BDA0003714775950000156
in the formula (I), the compound is shown in the specification,
Figure BDA0003714775950000157
and wherein each element is 1/N, taking r eigenvalues lambda 95% before the cumulative variance contribution rate 1 ，…，λ r And its corresponding feature vector η 1 ，…，η r 。
Specifically, in step S16, a pivot score vector is calculated
Figure BDA0003714775950000158
Is that, after said step S15 is completed, the k-th normative variable residual is subjected to
Figure BDA0003714775950000159
Obtaining its pivot score vector by the following projection
Figure BDA00037147759500001510
Setting:
Figure BDA00037147759500001511
Figure BDA0003714775950000161
in the formula (I), the compound is shown in the specification,
Figure BDA0003714775950000162
is a feature vector eta i I-1, …, r.
Specifically, the fault detection statistic is constructed in the step S17
Figure BDA0003714775950000163
And Q ck By constructing fault detection statistics after completing said step S16
Figure BDA0003714775950000164
And Q ck Setting:
Figure BDA0003714775950000165
Figure BDA0003714775950000166
wherein Λ is diag (λ) 1 ，…，λ r ),
Figure BDA0003714775950000167
In this embodiment, the control limit based on the kernel density estimation is designed:
kernel Density Estimation (KDE) is a common method of determining an upper control limit, particularly for non-linear or non-gaussian distributed process data. Considering that the actual detection data does not necessarily follow a normal distribution, the present embodiment utilizes a KDE to determine the control limits for the fault detection statistics.
Specifically, in the step S18, the control limit is calculated by assuming that x is a random variable and p (x) is a probability density function of x:
Figure BDA0003714775950000168
the following can be obtained by a gaussian kernel function:
Figure BDA0003714775950000169
estimating the probability density function of x by a gaussian kernel function:
Figure BDA00037147759500001610
where psi is the bandwidth, x (i), i is 1,2, …, N is the ith sample in x, the fault detection statistic is set to J, and the control limit is J UCL Given a significance level α, then J UCL The calculation expression of (a) is:
Figure BDA00037147759500001611
if J is less than or equal to J UCL If yes, judging that no fault occurs; if J > J UCL Then it is determined that a fault is detected.
Calculating the corresponding control limits according to equation (37)
Figure BDA0003714775950000171
And Q UCL，ck In the step S27, the method of the step S17 is adopted to calculate the fault detection statistic
Figure BDA0003714775950000172
And Q ck，test ，
If it is
Figure BDA0003714775950000173
Or Q ck，test ＞Q UCL，ck Then a fault occurrence is detected.
So far, the technical solutions of the present invention have been described in connection with the preferred embodiments shown in the drawings, but it is easily understood by those skilled in the art that the scope of the present invention is obviously not limited to these specific embodiments. Equivalent changes or substitutions of related technical features can be made by those skilled in the art without departing from the principle of the invention, and the technical scheme after the changes or substitutions can fall into the protection scope of the invention.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention; various modifications and alterations to this invention will become apparent to those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (10)

1. A tiny fault detection method based on recursion specification variable residual error and kernel principal component analysis is characterized by comprising the following steps:
step S1, off-line training,
step S11: acquiring detection data Y in normal operation state 0 Standardizing the data to obtain standardized detection data Y;
step S12: constructing a past observation matrix Y p And future observation matrix Y f And separately calculate Y p And Y f Covariance and cross-covariance matrices of (a);
step S13: performing singular value decomposition on the matrix H and determining the number q of leading singular values;
step S14: calculating a canonical variable residual d and EWMA filtering d to obtain a filtered canonical variable residual
Figure FDA0003714775940000011
Step S15: constructing a kernel matrix K, carrying out mean value centralization, and solving a characteristic value and a characteristic vector of the kernel matrix K;
step S16: computing pivot score vectors
Figure FDA0003714775940000012
Step S17: formation fault detection statistics
Figure FDA0003714775940000013
And Q ck ；
Step S18: calculating corresponding control limits
Figure FDA0003714775940000014
Q UCL，ck ；
In step S2, the online detection,
step S21: acquiring actual detection data and using Y 0 Is normalized by the mean and covariance of the two to obtain normalized detection data Y test ；
Step S22: constructing a past observation matrix Y p,test And future observation matrix Y f,test ；
Step S23: calculating a specification variable residual d by adopting the method of the step S14 test And to d test EWMA filtering to obtain filtered canonical variable residuals
Figure FDA0003714775940000015
Step S24: by using
Figure FDA0003714775940000016
And
Figure FDA0003714775940000017
constructing a kernel matrix K test Mean value centralization is carried out, and the characteristic value and the characteristic vector of the mean value centralization are solved;
step S25: calculating the pivot score vector by the method of the step S16
Figure FDA0003714775940000018
Step S26: calculating a fault detection statistic using the method of step S17
Figure FDA0003714775940000019
And Q ck，test ；
Step S27: if it is
Figure FDA0003714775940000021
Or Q ck，test ＞Q UCL，ck Then a fault occurrence is detected.
2. The method for detecting minor faults based on recursion norm variable residual error and kernel principal component analysis of claim 1, wherein in step S11, the method for normalizing the detection data in normal operation state is to set
Figure FDA0003714775940000022
Is the raw inspection data, wherein,
Figure FDA0003714775940000023
n is the number of samples, m is the number of variables,
Figure FDA0003714775940000024
for the kth sample, the raw test data is normalized and set
Figure FDA0003714775940000025
In the formula (I), the compound is shown in the specification,
Figure FDA0003714775940000026
is the mean of the jth variable, s j The standard deviation of the jth variable, j ═ 1,2,. . . M, the original detection data matrix Y 0 Conversion to: y ═ Y 1 …y n ] T 。
3. The minor fault detection method based on recursive canonical variable residual and kernel principal component analysis according to claim 2, wherein in the step S12, a past observation matrix Y is constructed p And future observation matrix Y f And separately calculate Y p And Y f The method of covariance and cross-covariance matrix is that when the original test data matrix Y is 0 Conversion to: y ═ Y 1 …y n ] T Then, for the kth detection sample, the past observation vector y is set p (k) And future observation vector y f (k) Are respectively:
Figure FDA0003714775940000027
Figure FDA0003714775940000028
wherein p and f represent the window lengths of the past observation vector and the future observation vector, respectively;
setting a past observation matrix
Figure FDA0003714775940000029
And future observation matrix
Figure FDA00037147759400000210
Are respectively:
Y p ＝[y p (p+1) y p (p+2)…y p (p+N)] (4)
Y f ＝[y f (p+1) y f (p+2)…y f (p+N)] (5)
wherein N is N-f-p +1,
setting of Y p The expression of covariance of (a) is:
Figure FDA0003714775940000031
setting of Y f The expression of covariance of (a) is:
Figure FDA0003714775940000032
setting of Y p And Y f The expression of cross-covariance of (a) is:
Figure FDA0003714775940000036
in the formula, parameters p and f satisfy { mp, mf } < N.
4. The method for detecting minor faults based on recursive canonical variable residual and kernel principal component analysis according to claim 3, wherein in the step S13, the singular value decomposition is performed on the matrix H and the number q of dominant singular values is determined by performing singular value decomposition calculation on the matrix H after the step S12 is completed, and setting:
Figure FDA0003714775940000033
in the formula, U and V are matrices composed of left and right singular vectors, respectively, a diagonal matrix S is composed of ordered singular values, and S is set to be diag (Σ) 1 ，…，∑ γ 0, …, 0), γ is the rank of matrix H, taking the first q columns of U and V with the maximum correlation, where q < mp, to obtain matrices Uq and Vq after dimensionality reduction, then the expressions of projection matrices J and L are:
Figure FDA0003714775940000034
Figure FDA0003714775940000035
5. the minor fault detection method based on recursive canonical variable residual and kernel principal component analysis according to claim 4Method, characterized in that in said step S14 a canonical variable residual d is calculated and d is EWMA filtered to obtain a filtered canonical variable residual
Figure FDA0003714775940000041
By performing the calculation of projection matrices J and L such that JY p (k) And LY f (k) Maximize the correlation between, wherein JY p (k) And LY f (k) For the k-th detection sample, the expressions of the state vector x (k) and the residual vector e (k) are set as follows:
x(k)＝JY p (k) (12)
Figure FDA0003714775940000042
wherein I is a dimensional unit array, and the following fault detection statistics are constructed by using x (k) and e (k):
T 2 (k)＝x(k) T x(k) (14)
Q(k)＝e(k) T e(k) (15)
in the formula, T 2 (k) The statistics measure the change of the state vector x (k), and q (k) the statistics measure the change of the residual vector e (k), so the expression of the normalized variable residual d (k) of the kth detection sample is:
d(k)＝Ly f (k)-S q Jy p (k) (16)
in the formula, S q ＝diag(∑ 1 ，…，∑ q ) Keeping in mind that a matrix composed of the normative variable residuals of all samples is Yd, and the expression of the covariance matrix is:
Figure FDA0003714775940000043
constructing a fault detection statistic D based on the relevant definition of the Mahalanobis distance, and setting:
Figure FDA0003714775940000044
filtering the standard variable residual error d by adopting an exponential weighted moving average method EWMA (equal weighted average), and filtering the filtered data
Figure FDA0003714775940000045
The expression of (c) is:
Figure FDA0003714775940000046
in the formula (I), the compound is shown in the specification,
Figure FDA0003714775940000047
in order to be a weight factor, the weight factor,
Figure FDA0003714775940000048
the larger the value is,
Figure FDA0003714775940000049
the more it is possible to reflect the actual data information,
Figure FDA00037147759400000410
the smaller the value is,
Figure FDA00037147759400000411
the more sensitive it is to small changes in data.
6. The method for detecting minor faults based on recursive canonical variable residual and kernel principal component analysis according to claim 5, wherein the kernel matrix K is constructed and mean-centered in the step S15, and the eigenvalues and eigenvectors thereof are solved by obtaining filtered canonical variable residual
Figure FDA0003714775940000051
Then, adopting KPCA algorithm to further extract
Figure FDA0003714775940000052
Non-linear characteristics of (1), hypothesis
Figure FDA0003714775940000053
By means of non-linear functions
Figure FDA0003714775940000054
Implicit mapping to a high dimensional feature space
Figure FDA0003714775940000055
And is
Figure FDA0003714775940000056
Then solving the eigenvalue of the covariance matrix of the sample, and setting:
Figure FDA0003714775940000057
Cξ＝λξ (21)
wherein C is a space
Figure FDA0003714775940000058
Where λ is an eigenvalue and ξ is an eigenvector and is included in the sample covariance matrix
Figure FDA0003714775940000059
In the subspace spanned, there is therefore a vector η ═ η 1 …η N ] T So that xi is expressed as
Figure FDA00037147759400000510
The linear combination of (a) and (b) is set to:
Figure FDA00037147759400000511
equation (21) is rewritten as:
Figure FDA00037147759400000512
where <, > represents an inner product, and the substitution of the equations (20) and (22) into the equation (23) yields:
Figure FDA00037147759400000513
defining a kernel matrix
Figure FDA00037147759400000514
The elements in K satisfy:
Figure FDA00037147759400000515
where κ (·, ·) is the kernel function, i, j ═ 1, …, N, and the gaussian radial basis function is chosen as the kernel function, then:
Figure FDA00037147759400000516
where h is the kernel width, equation (24) is rewritten as:
Figure FDA0003714775940000061
namely:
λNη＝Kη(28)
the feature vector η can be determined according to the formula (28) 1 ，…，η N And its corresponding characteristic value lambda 1 ，…，λ N In addition, the matrix K needs to be mean-centered before calculation:
Figure FDA0003714775940000062
in the formula (I), the compound is shown in the specification,
Figure FDA0003714775940000063
and wherein each element is 1/N, taking r eigenvalues lambda 95% before the cumulative variance contribution rate 1 ，…，λ r And its corresponding feature vector η 1 ，…，η r 。
7. The method for detecting minor faults based on recursive canonical variable residual and kernel principal component analysis according to claim 6, wherein in the step S16, principal component score vector is calculated
Figure FDA0003714775940000064
Is that, after said step S15 is completed, the k-th normative variable residual is subjected to
Figure FDA0003714775940000065
Obtaining its pivot score vector by the following projection
Figure FDA0003714775940000066
Setting:
Figure FDA0003714775940000067
Figure FDA0003714775940000068
in the formula (I), the compound is shown in the specification,
Figure FDA0003714775940000069
is a feature vector eta i I-1, …, r.
8. Recursion-based gauge as claimed in claim 7The method for detecting minor faults by norm variable residual and kernel principal component analysis is characterized in that, in the step S17, fault detection statistic is constructed
Figure FDA00037147759400000610
And Q ck By constructing fault detection statistics after completing said step S16
Figure FDA00037147759400000611
And Q ck Setting:
Figure FDA00037147759400000612
Figure FDA00037147759400000613
wherein Λ is diag (λ) 1 ，…，λ r ),
Figure FDA00037147759400000614
9. The method for detecting minor faults based on recursion norm variable residuals and kernel principal component analysis (KDE) of claim 8, wherein the step S18 is calculating the control limit by using Kernel Density Estimation (KDE) to determine the control limit of the fault detection statistic, and assuming that x is a random variable and p (x) is a probability density function of x:
Figure FDA0003714775940000071
the following can be obtained by a gaussian kernel function:
Figure FDA0003714775940000072
estimating the probability density function of x by a gaussian kernel function:
Figure FDA0003714775940000073
where psi is the bandwidth, x (i), i is 1,2, …, N is the ith sample in x, the fault detection statistic is set to J, and the control limit is J UCL Given a significance level α, then J UCL The calculation expression of (a) is:
Figure FDA0003714775940000074
10. the minor fault detection method based on recursive canonical variable residual and kernel principal component analysis according to claim 9,
if J is less than or equal to J UCL If yes, judging that no fault occurs;
if J > J UCL Then it is determined that a fault is detected.
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