CN1655082A - Non-linear fault diagnosis method based on core pivot element analysis - Google Patents

Abstract

A non-linear fault diagnosis method based on kernel pivot analysis carries out non-linear analysis with the monitor data of a normal state system to pick up the non-linear pivot information and applies the kernel pivot model of important non-linear structure system at normal state, images the new measured data of the system to the kernel pivot model to reconstruct the data to the character information picked up by the new test data, judges the working condition of the system by the difference between the computed new tested data and reconstruction to it by the kernel pivot model. When the difference exceeds the top confidence limit, the new test data is judged to be fault, and the system is at fault state.

Description

Method based on the non-linear fault diagnosis of core pivot element analysis

Technical field

What the present invention relates to is a kind of method of non-linear fault diagnosis, particularly a kind of method of the non-linear fault diagnosis based on core pivot element analysis.Be used for the Electronics and Information Engineering technical field.

Background technology

Because product quality, economic benefit, safety and environmental protection requirement, it is very complicated that industrial process and relevant control system become, and in order to guarantee the normal operation of industrial system, Fault Diagnosis is being played the part of very important role with detection in industrial process.In recent years, multivariate statistical analysis is applied to process monitoring and fault diagnosis has obtained extensive studies.Pivot analysis is one of widely used method in industrial process fault diagnosis and the detection.

Find through literature search prior art, " Computer-based monitoring and fault diagnosis:a chemical processcase study " (" computer monitoring and fault diagnosis: a chemical process case study " that people such as Ralston P deliver on " ISA Transactions ", " ISA meeting will ", 2001,40 (1): 85-98) in the literary composition, describe principle and methods for using them in detail based on the method for diagnosing faults of pivot analysis, its method is linear diagnostic method, mainly is the state of judging system by the surplus difference of calculating measurement data.Yet pivot analysis is mainly used in linear system, and when pivot analysis was applied to nonlinear system, the pivot analysis model of this moment is the essential information of reactive system really.Most industrial system all has nonlinear characteristic.Therefore when pivot analysis being applied to the fault diagnosis of non-linear industrial system is inappropriate.

Summary of the invention

The objective of the invention is to overcome deficiency of the prior art, a kind of method of the non-linear fault diagnosis based on core pivot element analysis is provided, make it solve deficiency and defective that the pivot analysis technology exists, by the nonlinear relationship of kernel function implementation procedure, the dirigibility of kernel function and convenience can be suitable for the fault detection and diagnosis of multiple non-linear process.

The present invention is achieved by the following technical solutions, Monitoring Data when the present invention at first uses system's normal condition (residing state during system safety operation) is carried out nonlinear analysis, extract nonlinear principal component information, the kernel pivot model (KPCA model) when using important nonlinear principal component tectonic system normal condition.Then the new data map of measuring of system is arrived kernel pivot model, data are reconstructed with the characteristic information of kernel pivot model to new survey data extract.Data by calculating new survey and the kernel pivot model working condition of the surplus difference between its reconstruct being judged system at last.When Monitoring Data and kernel pivot model surpass top confidence limit to the surplus difference between its reconstruct, can judge that the data of new measurement are fault data, system is in malfunction.

Below the present invention is further illustrated, comprise four steps:

1. the structure of kernel pivot model

At first use a Nonlinear Mapping Ψ () with l sample data: the x of n dimension when the normal condition of system ₁..., x _l∈ R ⁿ, from former space R ⁿBe mapped to high-dimensional feature space Ψ (x)=( (x ₁), (x ₂) ..., (x _l)), it is carried out pivot analysis.

Mapping (enum) data (x in the feature space _j) covariance matrix be:

Find the solution eigenvalue problem:

λV＝CV， (2)

Eigenvalue＞0 wherein, proper vector V ∈ Ψ ().(2) formula both sides premultiplication (x _k), can get:

λ((x _k)·V)＝((x _k)·CV) k＝1，…，l.

(3) owing to eigenvalue ≠ 0 characteristic of correspondence vector V is opened by the vector of feature space, so exist

α wherein _iBe coefficient. definition l * l matrix K: K _Ij=K (x _i, x _j)=( (x _j) (x _j)), by (3) formula and (4) Shi Kede:

L λ K α=K ²α equivalence and following formula:

lλα＝Kα (α＝(α ₁，…，α _l) ^T). (5)

Eigenvalue _kWith corresponding feature α _k(k=1 ..., l) can find the solution by following formula.

The kernel pivot model of forming system by system eigenwert and corresponding proper vector when the normal condition.

2. data are to the mapping and the reconstruct of kernel pivot model

In order to extract the new information of nonlinear system, the data x that newly records is mapped to the kernel pivot model (also being on k the pivot) of system at the picture (x) of feature space:

The information that extracts with kernel pivot model is reconstructed data.During to the reconstruct of data, neglect unessential pivot, use the mapping β of (x) on q important pivot _k(k=1 ..., q) (x) is reconstructed (variance contribution ratio of q pivot reaches 85% before supposing in feature space) in feature space.Define a reconstruct operator P, the reconstruct that can get (x):

Wherein (x) is the reconstruct of (x).

To (x) when being reconstructed, directly finding the solution of following formula is very difficult in feature space.And (x) is unimportant to us, and we need is reconstruct data x in the former data space.Reconstruct data x in the former data space obtains by finding the solution the non-linear least square optimization problem equally, promptly finds the solution

$\min \underset{j=1}{\overset{l}{\mathrm{\Σ}}}{|K(\stackrel{\‾}{x},x_j)-2\underset{k=1}{\overset{q}{\mathrm{\Σ}}}{\mathrm{\β}}_k\underset{i=1}{\overset{l}{\mathrm{\Σ}}}{\mathrm{\α}}_i^{k}K(x_i,x_j)|}^2,---\left(8\right)$

Can obtain by the reconstruct data of kernel pivot model new survey data.

3. the detection of fault and diagnosis

Surplus poor between the kernel pivot model that new when surveying data x and system's normal condition, the deviation between data x and the kernel pivot model just, can estimate by square prediction error (SPE):

$\mathrm{SPE}=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{(x-\stackrel{\‾}{x})}^2.---\left(9\right)$

When a system was in normal condition, all the other differences mainly were made up of noise.When system is in abnomal condition, surplus difference will significantly increase.Therefore, as the SPE of sample during less than certain fiducial limit, can think that the data when new data are with system's normal condition are consistent, system is in normal state.When the SPE of data surpassed certain fiducial limit, the data when then thinking the data of new survey and system's normal condition were inconsistent, can conclude that system is in abnomal condition.Simultaneously, for the data of abnomal condition system, can also calculate the contribution of each dimension in each sample or variable by following formula to surplus difference:

${\mathrm{DSPE}}_i={({\stackrel{\‾}{x}}_i-x_i)}^2,i=1,\·\·\·,m.---\left(10\right)$

Wherein m is the dimension of sample, DSPE _iBe that each ties up the contribution to surplus difference in the sample.By calculating DSPE _i, can determine the dimension or the variable of error contribution maximum in the surplus difference.

4. determine the upper bound of surplus difference

Q-statistics provide a kind of test macro whether drift go out the effective ways of normal condition.This article detects the unusual variance that data depart from kernel pivot model with the Q statistics.The Q-statistical representation can not be pounced on the variable quantity of the data that obtain by kernel pivot model.Noise is represented in the variation of data in surplus poor degree of confidence scope, and when the variation of data exceeded fiducial limit (being the surplus poor upper bound), the variation of data can not be pounced on by kernel pivot model and be obtained.Therefore as the SPE of sample during, can conclude that system is in abnormality greater than the surplus poor upper bound.A kind of surplus poor upper bound Q of determining that adopts Jachson to propose _αMethod:

$Q_{\mathrm{\α}}={\mathrm{\θ}}_1{[\frac{c_{\mathrm{\α}}\sqrt{{2\mathrm{\θ}}_2{h}_0^2}}{{\mathrm{\θ}}_1}+\frac{{\mathrm{\θ}}_2{h}_0(h_0-1)}{{\mathrm{\θ}}_1^2}+1]}^{1/h_0}---\left(11\right)$

Wherein ${\mathrm{\θ}}_1=\underset{i=q+1}{\overset{l}{\mathrm{\Σ}}}{\mathrm{\λ}}_i,$ ${\mathrm{\θ}}_2=\underset{i=q+1}{\overset{l}{\mathrm{\Σ}}}{\mathrm{\λ}}_i^2,$ ${\mathrm{\θ}}_3=\underset{i=q+1}{\overset{l}{\mathrm{\Σ}}}{\mathrm{\λ}}_i^3,$ $h_0=1-\frac{2{\mathrm{\θ}}_1{\mathrm{\θ}}_3}{3{\mathrm{\θ}}_2^2},$ λ _iBe eigenwert, c _αIt is normal deviate corresponding to fiducial probability 1-α.

Core pivot element analysis is a kind of nonlinear stretch of pivot analysis, and it can be used for having between input variable the signature analysis of nonlinear relationship.The basic thought of core pivot element analysis is at first the input data to be mapped to feature space by Nonlinear Mapping, carries out pivot analysis then in feature space.Dexterously the dot product of the high-dimensional feature space of the unknown is converted into the calculation of assessing in former space at last by nuclear, carries out the nonlinear characteristic analysis.Core pivot element analysis has shown very outstanding performance when handling nonlinear characteristic analysis and feature extraction.

The present invention solves deficiency and the defective that the pivot analysis technology exists, nonlinear relationship by the kernel function implementation procedure, kernel function flexibly and the convenient fault detection and diagnosis that can be suitable for multiple non-linear process, for the fault detection and diagnosis of non-linear process provides a kind of new method.

Description of drawings

Fig. 1 principle of the invention and method figure

The surplus poor contribution plot of Fig. 2 data sample

The surplus poor contribution plot of single component in Fig. 3 data sample

Embodiment

Content in conjunction with the inventive method provides following embodiment:

As shown in Figure 1, the Monitoring Data during at first with system's normal condition is carried out core pivot element analysis, extracts nonlinear principal component information, the kernel pivot model during the tectonic system normal condition (KPCA model), and this is the basis of diagnosis; Utilize pivot information calculations top confidence limit; The new data map of measuring of system to kernel pivot model, is reconstructed data with the characteristic information of kernel pivot model to new survey data extract, obtains reconstruct data; By calculating the working condition of the surplus poor judgement system between new data of surveying and the reconstruct data, when the surplus difference between Monitoring Data and the reconstruct data surpasses top confidence limit, can judge that the data of new measurement are fault data at last, system is in malfunction.Concrete implementation step:

Step 1: image data is carried out the normalization pre-service, $x_i=\frac{x_i-\min \left(x_i\right)}{\max \left(x_i\right)-\min \left(x_i\right)}.$ Data configuration kernel pivot model during with nominal situation obtains eigenvalue according to (5) formula _kWith corresponding feature α _k, eigenvalue _kWith corresponding feature α _k

Step 2: carry out mapping and the reconstruct of data, obtain by the reconstruct data of kernel pivot model to new survey data according to (8) formula to kernel pivot model.

Step 3: determine the upper bound of surplus difference, according to the surplus poor upper bound of (11) formula to system;

Step 4: carry out the detection and the diagnosis of fault, calculate the new square prediction error of gathering sample data according to (9) formula and estimate (SPE), judge the state of system.And utilize (10) to calculate and find the source of trouble.

The data of gathering in the rolling bearing under certain actual industrial environment are carried out fault diagnosis research.Used vibrations and noise characteristic when the assessment technique state, each data sample comprises 12 feature s ₁-s ₁₂, feature s wherein ₁-s ₄The characteristic of expression noise, feature s ₅-s ₁₂The characteristic of expression vibrations.Each data sample all has a decision attribute D.Attribute D described bearing reality state (0 the expression bearing be in good state; 1 expression bearing is in malfunction).The data set that this article is used under the normal condition is set up the KPCA model.Use the state that pca model detected and diagnosed rolling bearing then.Because s ₁-s ₂With s ₃-s ₄The expression identical characteristics are so only select s for use in the research ₁-s ₂And s ₃-s ₄In one group.

Use the nuclear of radial basis function: $K(x,x_i)=\exp (-\frac{{\left|\right|x-x_i\left|\right|}^2}{2\·{\mathrm{\σ}}^2}),$ ‖ x-x wherein _i‖ ²Can by ${\left|\right|x-x_i\left|\right|}^2=\underset{k=1}{\overset{n}{\mathrm{\Σ}}}{(x^k-x_i^k)}^2$ Calculate, σ is the nuclear width, σ=0.6.

Carry out l-G simulation test with 50 samples.On behalf of bearing, the 1-21 sample be in the data of normal state, and on behalf of bearing, the 22-50 sample be in the data of malfunction.Be in the 1-16 sample architecture KPCA model of normal state with the representative bearing.Remaining 18-50 sample is as test sample book.Whether be with confidence level that 95% Q statistics detects data consistent with the KPCA model.

Fig. 2 is that the Q of training sample and test sample book is surplus poor.Among Fig. 2, pitch the surplus poor of representative sample, horizontal dotted line is represented 95% fiducial limit.Surplus difference surpasses the point of this line and represents the trouble spot.Fig. 3 is the surplus poor contribution plot of each component (symptom) in the 30th sample.Among Fig. 3, the surplus difference contribution of each dimension (being each symptom) in the circle representative sample of little garden.

As can be seen from Figure 2, shown good performance, made diagnosis than high-accuracy based on the method for diagnosing faults of KPCA.Diagnosis to the 18-21 sample of normal condition is entirely true, and except that 24 and 48 samples, the diagnosis of all the other samples all is accurately to the 22-50 sample under the malfunction.Because each sample comprises 10 symptoms, therefore calculate the surplus poor most probable symptom that causes fault with detection of contributing of each symptom in each sample.As can be seen from Figure 2, No. 30 samples are the samples under the fault state.Find s by figure .3 ₁₂Surplus difference contribution maximum, so s in the 30th data sample ₁₂The relation of feature and bearing fault may be maximum.

Claims (6)

1, a kind of method of the non-linear fault diagnosis based on core pivot element analysis, it is characterized in that, Monitoring Data when at first being in normal condition with system is carried out nonlinear analysis, extract nonlinear principal component information, kernel pivot model when using nonlinear principal component tectonic system normal condition, then the new data map of measuring of system is arrived kernel pivot model, with the characteristic information of kernel pivot model data are reconstructed new survey data extract, data by calculating new survey and the kernel pivot model working condition of the surplus difference between its reconstruct being judged system at last, when Monitoring Data and kernel pivot model surpass top confidence limit to the surplus difference between its reconstruct, think that the data of new measurement are fault data, system is in malfunction.

2, according to the method for the described non-linear fault diagnosis based on core pivot element analysis of right 1, it is characterized in that, by following steps to its further qualification:

(1) structure of kernel pivot model: at first use a Nonlinear Mapping with the sample data of system when the normal condition from former spatial mappings to high-dimensional feature space, it is carried out pivot analysis, by finding the solution the nonlinear characteristic equation, obtain the NONLINEAR EIGENVALUE and the corresponding proper vector of nonlinear system, form the kernel pivot model of system by system eigenwert and corresponding proper vector when the normal condition;

(2) data are to the mapping and the reconstruct of kernel pivot model: the data that will newly record are mapped to the kernel pivot model of system at the picture of feature space, extract the new information of nonlinear system, neglect unessential pivot, utilize important pivot that data are reconstructed;

(3) detection of fault and diagnosis: estimate surplus poor between the new kernel pivot model when surveying data and system's normal condition by square prediction error, the deviation between data and the kernel pivot model just, when a system is in normal condition, all the other differences mainly are made up of noise, when system is in abnomal condition, surplus difference will significantly increase, and when surplus difference during greater than fiducial limit, then judgement system is in malfunction;

(4) determine the upper bound of surplus difference: the fiducial limit of adding up to determine surplus difference by Q-.

3, the method for the non-linear fault diagnosis based on core pivot element analysis according to claim 2 is characterized in that the structure of described kernel pivot model is implemented as follows:

At first use a Nonlinear Mapping ψ () with l sample data: the x of n dimension when the normal condition of system ₁..., x _l∈ R ⁿ, from former space R ⁿBe mapped to high-dimensional feature space ψ (x)=( (x ₁), (x ₂) ..., (x _l)), it is carried out pivot analysis, mapping (enum) data (x in the feature space _j) covariance matrix be:

Find the solution eigenvalue problem: λ V=CV, eigenvalue＞0 wherein, proper vector V ∈ ψ (), following formula both sides premultiplication (x _k): λ ( (x _k) V)=( (x _k) CV) k=1 ..., l

Because eigenvalue ≠ 0 characteristic of correspondence vector V is opened by the vector of feature space, so exist

α wherein _iBe coefficient, definition l * l matrix K: K _Ij=K (x _i, x _j)=( (x _i) (x _j))

Get by top several formulas: l λ K α=K ²α, equivalence and following formula:

lλα＝Kα(α＝(α ₁，…，α _l) ^T)

Eigenvalue _kWith corresponding feature a _k, k=1 ..., l is found the solution by following formula, is made of the kernel pivot model of system system eigenwert and corresponding proper vector when the normal condition.

4, the method for the non-linear fault diagnosis based on core pivot element analysis according to claim 2 is characterized in that, described data are implemented as follows to the mapping and the reconstruct of kernel pivot model:

Is on k the pivot with the data x that newly records at the kernel pivot model that the picture (x) of feature space is mapped to system:

The information that extracts with kernel pivot model is reconstructed data, during to the reconstruct of data, utilizes important pivot, uses the mapping β of (x) on q important pivot _k (x) is reconstructed in feature space, k=1 ..., q supposes that the variance contribution ratio of preceding q pivot in feature space reaches 85%, defines a reconstruct operator P, gets the reconstruct of (x):

Wherein (x) is the reconstruct of (x);

To (x) when being reconstructed, the reconstruct data x in the former data space obtains by finding the solution the non-linear least square optimization problem equally, promptly in feature space $\min \underset{j=1}{\overset{l}{\mathrm{\Σ}}}{|K(\stackrel{\‾}{x},x_j)-2\underset{k=1}{\overset{q}{\mathrm{\Σ}}}{\mathrm{\β}}_k\underset{i=1}{\overset{l}{\mathrm{\Σ}}}{\mathrm{\α}}_i^{k}K(x_i,x_j)|}^2,$ Acquisition is by the reconstruct data of kernel pivot model to new survey data.

5, the method for the non-linear fault diagnosis based on core pivot element analysis according to claim 2 is characterized in that the detection of described fault and diagnosis are implemented as follows:

Estimate by square prediction error: $\mathrm{SPE}=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{(x-\stackrel{\‾}{x})}^2,$ As the SPE of sample during less than fiducial limit, think that the data when new data are with system's normal condition are consistent, system is in normal condition, when the SPE of data surpasses fiducial limit, data when then thinking the data of new survey and system's normal condition are inconsistent, conclude that system is in abnomal condition; For the data of abnomal condition system, calculate each dimension in each sample or variable contribution: DSPE to surplus difference by following formula _i=(x _i-x _i) ², i=1 ... m, wherein m is the dimension of sample, DSPE _iBe that each ties up the contribution to surplus difference in the sample.

6, the method for the non-linear fault diagnosis based on core pivot element analysis according to claim 2 is characterized in that, the described upper bound of determining surplus difference is implemented as follows:

Adopt Q _αMethod determine the surplus poor upper bound: $Q_{\mathrm{\α}}={\mathrm{\θ}}_1{[\frac{c_{\mathrm{\α}}\sqrt{2{\mathrm{\θ}}_2{h}_0^2}}{{\mathrm{\θ}}_1}+\frac{{\mathrm{\θ}}_2{h}_0(h_0-1)}{{\mathrm{\θ}}_1^2}+1]}^{1/h_0}$

Wherein ${\mathrm{\θ}}_1=\underset{i=q+1}{\overset{l}{\mathrm{\Σ}}}{\mathrm{\λ}}_i,{\mathrm{\θ}}_2=\underset{i=q+1}{\overset{l}{\mathrm{\Σ}}}{\mathrm{\λ}}_i^2,{\mathrm{\θ}}_3=\underset{i=q+1}{\overset{l}{\mathrm{\Σ}}}{\mathrm{\λ}}_i^3,h_0=1-\frac{2{\mathrm{\θ}}_1{\mathrm{\θ}}_3}{3{\mathrm{\θ}}_2^2},$ λ _iBe eigenwert, c _αIt is normal deviate corresponding to fiducial probability 1-α.

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