CN105320797A - Method for predicting service life of key system of rail transit vehicles - Google Patents

Abstract

The invention discloses a method for predicting the service life of a key system of rail transit vehicles. The method comprises the following steps of firstly, determining the key system of urban rail vehicles, and bringing in a proportional hazard model to carry out service life prediction on the key system; secondly, screening historical overhauling data of the key system to obtain concomitant variable factors, and processing concomitant variables by applying a principal component analysis method so as to obtain a concomitant variable matrix; thirdly, solving a parameter estimation value in the proportional hazard model by adopting a maximum likelihood estimation method and a Newton iteration method; finally, calculating the predicted service life of the key system of the rail transit vehicles by utilizing the obtained proportional hazard model. According to the method disclosed by the invention, the influence of the concomitant variable factors on the service life of the key system of the rail transit vehicles is considered, and the defect that the service life prediction is inaccurate as only system failure is considered in traditional service life prediction is overcome.

Description

The life-span prediction method of rail traffic vehicles critical system

Technical field

The invention belongs to traffic safety field of engineering technology, particularly a kind of life-span prediction method of rail traffic vehicles critical system.

Background technology

City railway vehicle is a complicated Mechatronic Systems, because its service condition is complicated, bad environments, can be aging gradually in During Process of Long-term Operation, and residual life can progressively decline, and easily causes serious accident to occur, and causes huge property loss and casualties; If carry out maintain and replace blindly, huge waste can be brought.So the residual life of correct Prediction city rail vehicle is for guarantee equipment safety operation, very large meaning of increasing economic efficiency.

The method of current lifetime of system prediction has the life-span prediction method based on mechanics and the life-span prediction method based on probability statistics, as the life-span prediction method based on stress, life-span prediction method based on strain, the life-span prediction method based on accumulation of fatigue damage, the life-span prediction method based on fracturing mechanics, the life-span prediction method based on damage mechanics, the life-span prediction method based on energy, based on the life-span prediction method of artificial intelligence technology and the life-span prediction method based on mechanical equipment state monitoring.Life-span prediction method based on mechanics predicts its residual life from the dynamics lost efficacy with failure mechanism, when the inefficacy of part be single inefficacy mechanism or play major control effect by a kind of inefficacy mechanism time, the prediction of its residual life seems comparatively simple, as fatigue life prediction, prediction creep life and wear-out life prediction etc., but due to city rail vehicle complex structure, the failure theory that various of failure should be adopted to be coupled carrys out bimetry.Life-span prediction method based on probability statistics sets up statistical model by the test figure of accumulation and field data, by determining life characteristic values distribution in time and failure probability, predicting and requiring the life-span under fiduciary level.From the meaning of probability statistics, the life prediction result based on probability statistics more can reflect universal law and the overall permanence in engineering goods life-span, but needs the accumulation of lot of experiments and data.Traditional life-span prediction method based on probability statistics is taken maintenance into account a mathematic(al) expectation over time and not on the impact in life-span.

Summary of the invention

The object of the present invention is to provide the life-span prediction method of a kind of simple and effective, accurately reliable rail traffic vehicles critical system.

The technical solution realizing the object of the invention is: a kind of life-span prediction method of rail traffic vehicles critical system, comprises the following steps:

Step 1, determines vehicle critical system: according to historical failure data and the system importance degree of vehicle, determine the critical system of rail traffic vehicles;

Step 2, chooses the substrate inefficacy function of proportional hazard model: according to the overhaul data of critical system in step 1, adopts fitting of distribution to obtain the substrate inefficacy function of critical system;

Step 3, covariant is screened: analyze the overhaul data of critical system described in step 1, calculates failure logging frequency ratio and obtains the covariant factor affecting the critical system life-span;

Step 4, covariant pre-service: according to the gained covariant factor in step 3, utilizes principal component analysis (PCA) to carry out process to the covariant factor and obtains covariant matrix;

Step 5, covariant is comprehensive: adopt linear regression fit to covariant multiple in step 4, finally obtain the integrated value of covariant;

Step 6, model parameter estimation: according to gained substrate inefficacy function and covariant integrated value in step 2 and step 5, adopts Maximum-likelihood estimation and Newton iteration method to try to achieve the parameter of proportional hazard model;

Step 7, life prediction: parameters obtained in step 6 is substituted into proportional hazard model, obtains the Reliability Function of critical system, utilizes Reliability Function to predict the in-service life-span of critical system.

Compared with prior art, its remarkable advantage is in the present invention: (1) introduces the covariant factor in influential system life-span in life prediction, adds the accuracy of lifetime of system prediction; (2) overhaul data is used as covariant, considers that the factor of maintenance increase lifetime of system makes life prediction more accurate; (3) adopt Newton iteration method to calculate the parameter of likelihood function, make calculating easier and general.

Accompanying drawing explanation

Fig. 1 is the process flow diagram of the life-span prediction method of rail traffic vehicles critical system of the present invention.

Fig. 2 is the reduced graph of time covariant in the embodiment of the present invention 1.

Fig. 3 is the Reliability Function figure of critical system in the embodiment of the present invention 1.

Fig. 4 is the Reliability Function figure that in the embodiment of the present invention 1, critical system PHM and weibull distributes.

Embodiment

Below in conjunction with accompanying drawing, the present invention is described in further detail.

Composition graphs 1, the life-span prediction method of rail traffic vehicles critical system of the present invention, comprises the following steps:

Step 1, determines vehicle critical system: according to historical failure data and the system importance degree of vehicle, determine the critical system of rail traffic vehicles, using system high for vehicle trouble rate as critical system.

Step 2, chooses the substrate inefficacy function of proportional hazard model: according to the overhaul data of critical system in step 1, adopts fitting of distribution to obtain the substrate inefficacy function of critical system, specific as follows:

The functional form becoming the proportional hazard model of covariant during general area is as follows:

λ(t,Z(t))＝λ ₀(t)exp(g(Z(t)))(1)

In formula, λ (t, Z (t)) is failure rate estimation, and relevant with time and covariant; λ ₀t () is only relevant with time substrate failure rate estimation; Z (t)=(z1, z2, zn) be covariant vector at time t influential system failure probability, the covariant function that g (Z (t)) is influential system, and g (Z (t))=γ Z (t), wherein γ=(γ ₁, γ ₂..., γ _m) be regression parameter vector, m is the number of covariant;

The characteristic quantity of each state parameter of covariant Z (t) characterization device, maintenance, operation factor is the Other Concomitant Factors of influential system life-span behavior.When covariant Z (t) is for constant, λ (t) and λ ₀t () is proportional, so model is called proportional hazard model.

The failure rate estimation such as Weibull distribution, exponential distribution, log series model can be selected in proportional hazard model.Wherein Weibull distribution can portray the fail data that crash rate rises in time, also can portray the fail data that crash rate declines in time, and the life-span of Mechatronic Systems generally obeys two-factor Weibull distribution, therefore substrate failure rate estimation λ ₀t () adopts following two-factor Weibull distribution:

${\mathrm{\λ}}_0\left(t\right)={\frac{\mathrm{\β}}{\mathrm{\η}}\left(\frac{t}{\mathrm{\η}}\right)}^{\mathrm{\β}}---\left(2\right)$

In formula, β is form parameter, η is scale parameter;

Formula (2) is substituted into formula (1),

$\mathrm{\λ}\left(t\right|Z\left(t\right))=\frac{\mathrm{\β}}{\mathrm{\η}}\frac{t^{\mathrm{\β}}}{\mathrm{\η}}\exp \left(\mathrm{\γZ}\left(t\right)\right)---\left(3\right)$

The citation form of rail traffic vehicles critical system proportional hazard model used is shown in formula (3).

Step 3, covariant is screened: analyze the overhaul data of critical system described in step 1, calculates failure logging frequency ratio and obtains the covariant factor affecting the critical system life-span, specific as follows:

Covariant can be divided into outside covariant and inner covariant two class by its effect.Outside covariant refers to the variable that its change can cause system failure sign and changes, and as working load, environment temperature etc., but no matter whether event of failure occurs, and the value of outside covariant all can not change.Inner covariant refers to the variable that can reflect that system failure sign changes, and state monitoring information can be included into this type of, and usually, if event of failure occurs, larger change can occur the value of inner covariant.

Covariant can with time correlation, also can be incoherent.If covariant is uncorrelated with the time, that PHM model parameter estimation process will simplify, and does not also need in life prediction to predict covariant.If covariant and time correlation, PHM model parameter estimation meeting more complicated, also will consider the variation tendency of covariant during life prediction.According to the feature of door device overhaul data state parameter, selected covariant and time correlation.

Step 4, covariant pre-service: according to the gained covariant factor in step 3, utilizes principal component analysis (PCA) to carry out process to the covariant factor and obtains covariant matrix, specific as follows:

Proportional hazard model requires that the partial correlation coefficient between each covariant is as far as possible little, and principal component analysis (PCA) is a kind of method of effective elimination correlativity, so available principal component analysis (PCA) obtains the separate covariant parameter of door device.Suppose that rail traffic vehicles critical system covariant is X ₁, X ₂..., X _m, then standardization is done to covariant data, obtains correlation matrix R:

$R=\left[\begin{array}{cccc}{r}_{11}& r_{12}& ...& r_{1m}\\ r_{21}& r_{22}& ...& r_{2m}\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ r_{m1}& r_{m2}& ...& r_{\mathrm{mm}}\end{array}\right]---\left(4\right)$

In formula, matrix element r _ijfor covariant X _iwith X _jrelated coefficient, wherein i, j=1,2 ..., m;

Principal component analysis (PCA) is utilized to obtain the detailed process of covariant matrix as follows:

(1) secular equation is: | λ I-R|=0, obtains eigenvalue λ by solving, and makes its order arrangement by size, i.e. λ ₁>=λ ₂>=...>=λ _m>=0;

(2) obtain respectively corresponding to eigenvalue λ _iproper vector e _i, require || e _ij||=1, namely wherein e _ijrepresentation feature vector e _ia jth component;

(3) the major component load of covariant is calculated:

Finally obtaining covariant matrix is:

$Z\left(t\right)=\left|\begin{array}{c}{z}_1\\ z_2\\ .\\ .\\ .\\ z_m\end{array}\right|=\left[\begin{array}{cccc}{l}_{11}& l_{12}& ...& l_{1m}\\ l_{21}& l_{22}& ...& l_{2m}\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ l_{m1}& l_{m1}& ...& l_{\mathrm{mm}}\end{array}\right]\·\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ .\\ .\\ .\\ x_m\left(t\right)\end{array}\right]---\left(5\right)$

In formula, z _ifor the covariant factor, the x of influential system failure probability _jfor the initial value of covariant.

Step 5, covariant is comprehensive: adopt linear regression fit to covariant multiple in step 4, finally obtain the integrated value of covariant, specific as follows:

Because the covariant parameter in proportional hazard model has multiple, the multiple covariants in raw data are carried out comprehensively, obtain comprehensive covariant function such as formula (6):

$g\left(z\left(t\right)\right)=\mathrm{\γZ}\left(t\right)=\left[\begin{array}{cccc}{\mathrm{\γ}}_1& {\mathrm{\γ}}_2& ...& {\mathrm{\γ}}_m\end{array}\right]\·\left[\begin{array}{cccc}{l}_{11}& l_{12}& ...& l_{1m}\\ l_{21}& l_{22}& ...& l_{2m}\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ l_{m1}& l_{m1}& ...& l_{\mathrm{mm}}\end{array}\right]\·\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ .\\ .\\ .\\ x_m\left(t\right)\end{array}\right]---\left(6\right)$

Linear regression fit is carried out to original covariant, namely

x _i(t)＝a _i·t+b _i(7)

In formula, a _i, b _ifor linear fit coefficient;

Obtain the integrated value of covariant such as formula (8):

$g\left(Z\left(t\right)\right)=\mathrm{\γZ}=\left[\begin{array}{ccc}{\mathrm{\γ}}_1& {\mathrm{\γ}}_2& {\mathrm{\γ}}_3\end{array}\right]\·\left[\begin{array}{cccc}{l}_{11}& l_{12}& ...& l_{1m}\\ l_{21}& l_{22}& ...& l_{2m}\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ l_{m1}& l_{m1}& ...& l_{\mathrm{mm}}\end{array}\right]\·\left[\begin{array}{c}{a}_1\·t+b_1\\ a_2\·t+b_2\\ .\\ .\\ .\\ a_m\·t+b_m\end{array}\right]---\left(8\right)$

Step 6, model parameter estimation: according to gained substrate inefficacy function and covariant integrated value in step 2 and step 5, adopts Maximum-likelihood estimation and Newton iteration method to try to achieve the parameter of model, specific as follows:

Maximum-likelihood estimation has excellent statistical property and good APPROXIMATE DISTRIBUTION, considers containing censored data in sample data simultaneously, therefore adopts maximum likelihood method to obtain the estimated value of each relevant parameters in model.

Be provided with n sample data, then the likelihood function of model is:

$L(\mathrm{\β},\mathrm{\η},\mathrm{\γ})=\underset{p=1}{\overset{n}{\mathrm{\Π}}}\mathrm{\λ}{(t_p,\mathrm{\β},\mathrm{\η},\mathrm{\γ})}^{{\mathrm{\δ}}_p}S{(t_p,\mathrm{\β},\mathrm{\η},\mathrm{\γ})}^{{1-\mathrm{\δ}}_p}=\underset{q=1}{\overset{W}{\mathrm{\Π}}}\mathrm{\λ}(t_j,\mathrm{\β},\mathrm{\η},\mathrm{\γ})\underset{p=1}{\overset{n}{\mathrm{\Π}}}S(t_q,\mathrm{\β},\mathrm{\η},\mathrm{\γ})---\left(9\right)$

In formula, W is the total sample number lost efficacy, p=1,2 ..., n, q=1,2 ..., W; t _pfor system overhaul data record moment, δ _pfor the property shown variable, δ _pget 0 or 1, δ _p=1 represents t _pfail data, δ _p=0 represents t _pit is censored data; S (t _q, β, η, γ) and be Reliability Function, obtaining Reliability Function by the citation form of formula (3) proportional hazard model is:

$S\left(t\right|Z\left(t\right))=\exp (-\underset{0}{\overset{t}{\∫}}\mathrm{\λ}\left(s\right|Z\left(s\right))\mathrm{ds})=\exp (-\underset{0}{\overset{t}{\∫}}\frac{\mathrm{\β}}{\mathrm{\η}}{\left(\frac{s}{\mathrm{\η}}\right)}^{\mathrm{\β}-1}\exp \left(\mathrm{\γZ}\left(s\right)\mathrm{ds}\right))---\left(10\right)$

Due to covariant Z (t) and time correlation, when the expression of Z (t) is unknown, Reliability Function formula (10) is expressed as:

$U_q=\underset{0}{\overset{t_q}{\∫}}\frac{\mathrm{\β}}{\mathrm{\η}}{\left(\frac{s}{\mathrm{\η}}\right)}^{\mathrm{\β}-1}\exp \left(\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\γ}}_k{Z}_k\left(s\right)\right)\mathrm{ds}=\underset{0}{\overset{t_q}{\∫}}\exp \left(\underset{k=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\γ}}_k{Z}_k\left(s\right)\right)d{\left(\frac{s}{\mathrm{\η}}\right)}^{\mathrm{\β}}---\left(11\right)$

K=1 in formula, 2 ..., m;

Composition graphs 2, if the value of covariant Z (t) is at moment t _pknown, then γ Z (t) | and t>=0} can regard as right continuous step process [γ Z (t)] * | t>=0};

Suppose 0<t ₁<t ₂< ... <t _q-1<t _q, utilize integration by parts formula (11) to be expressed as:

$\begin{array}{c}{U}_q^{*}={\left(\frac{t_q}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\exp \left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_k{Z}_k\left(t_q\right)\right)-\underset{0}{\overset{t_q}{\&Integral;}}{\left(\frac{s}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\mathrm{dexp}\left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(s\right)\right)\\ ={\left(\frac{t_q}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\exp \left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(t_q\right)\right)-(\underset{0}{\overset{t_1}{\&Integral;}}+\underset{t_1}{\overset{t_q}{\&Integral;}}){\left(\frac{s}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\mathrm{dexp}\left[{\left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_{k\left(s\right)}\right)}^{*}\right]\end{array}---\left(12\right)$

Due to so two integral representations in formula (12) are formula (13) and formula (14):

$\underset{0}{\overset{t_1}{\&Integral;}}{\left(\frac{s}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\mathrm{dexp}\left[{\left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(s\right)\right)}^{*}\right]={\left(\frac{t_1}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\{\exp \left(\underset{k-1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(t_1\right)\right)-\exp [{\left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(0\right)\right)}^{*}\left]\right\}---\left(13\right)$ ${\underset{t_1}{\overset{t_q}{\&Integral;}}\left(\frac{s}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\mathrm{dexp}\left[{\left(\underset{k=1}{\overset{p}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(s\right)\right)}^{*}\right]=\underset{v=2}{\overset{q}{\mathrm{\&Sigma;}}}\left(\frac{t_v}{\mathrm{\&eta;}}\right)(\exp \left(\underset{k-1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(t_v\right)\right)-\exp \left(\underset{k-1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(t_{v-1}\right)\right))---\left(14\right)$

V=2 in formula, 3 ... q;

Formula (13) and formula (14) are brought into formula (12) to obtain:

$U_q^{*}={\left(\frac{t_1}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\exp {\left[\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(0\right)\right]}^{*}+\underset{v=2}{\overset{q}{\mathrm{\&Sigma;}}}[{\left(\frac{t_v}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}-{\left(\frac{t_{v-1}}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}]\&CenterDot;\exp \left[\left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(t_{v-1}\right)\right)\right]---\left(15\right)$

Formula (3) and formula (15) are substituted into formula (9) obtain basis function be the log-likelihood function of the proportional hazard model of two-factor Weibull distribution such as formula (16):

$\begin{array}{c}\ln L(\mathrm{\&beta;},\mathrm{\&eta;},\mathrm{\&gamma;})=\\ W\ln \left(\frac{\mathrm{\&beta;}}{\mathrm{\&eta;}}\right)+(\mathrm{\&beta;}-1)\underset{p=1}{\overset{W}{\mathrm{\&Sigma;}}}\ln \left(\frac{t_p}{\mathrm{\&eta;}}\right)+\underset{p=1}{\overset{W}{\mathrm{\&Sigma;}}}\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(t_p\right)-\underset{q-1}{\overset{n}{\mathrm{\&Sigma;}}}\{\left(\frac{t_1}{\mathrm{\&eta;}}\right)\exp \left(\right[{\left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(0\right)\right)}^{*}\left]\right)+\\ \underset{v-2}{\overset{q}{\mathrm{\&Sigma;}}}[{\left(\frac{t_v}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}-{\left(\frac{t_{v-1}}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}]\&CenterDot;\exp \left[\left(\underset{k-1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(t_{v-1}\right)\right)\right]\}\end{array}---\left(16\right)$

Adopt the gloomy N-R process of iteration of newton-pressgang to ask model parameter, suppose X=[β, η, γ]=[β, η, γ ₁, γ ₂..., γ _m], ask the first-order partial derivative of each parameter, second-order partial differential coefficient and mixed partial derivative to obtain matrix F (X) and matrix G (X) respectively to formula (15):

$F\left(X\right)=[\frac{\&PartialD;\ln L(\mathrm{\&beta;},\mathrm{\&eta;},\mathrm{\&gamma;})}{\&PartialD;\mathrm{\&beta;}},\frac{\&PartialD;\ln L(\mathrm{\&beta;},\mathrm{\&eta;},\mathrm{\&gamma;})}{\&PartialD;\mathrm{\&eta;}},\frac{\&PartialD;\ln L(\mathrm{\&beta;},\mathrm{\&eta;},\mathrm{\&gamma;})}{{\&PartialD;\mathrm{\&gamma;}}_1},...\frac{\&PartialD;\ln L(\mathrm{\&beta;},\mathrm{\&eta;},\mathrm{\&gamma;})}{{\&PartialD;\mathrm{\&gamma;}}_m}]$

Using matrix F (X) and matrix G (X) as iteration factor, newton iteration formula is utilized to obtain:

X ⁽ⁿ⁺¹⁾＝X ⁽ⁿ⁾-F(X ⁽ⁿ⁾)(G(X ⁽ⁿ⁾)) ^-1(17)

The initial value of given X is X ⁽⁰⁾, substitute into iterative formula (17), if | X ⁽ⁿ⁺¹⁾-X ⁽ⁿ⁾| during < ε=0.01, inequality is set up, then obtain the estimates of parameters X=X of proportional hazard model ⁽ⁿ⁺¹⁾.

Step 7, life prediction: parameters obtained in step 6 is substituted into proportional hazard model, obtains the Reliability Function of critical system, utilizes Reliability Function to predict the in-service life-span of critical system, specific as follows:

The failure rate estimation being distributed as the proportional hazard model of basis function by Weibull obtains Reliability Function and is:

$S<t|Z\left(t\right)>=\exp (-\underset{0}{\overset{t}{\&Integral;}}\frac{\mathrm{\&beta;}}{\mathrm{\&eta;}}{\left(\frac{s}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}-1}\exp \left(g\left(s\right)\right)\mathrm{ds})---\left(18\right)$

Obtained by Reliability Function, the non-failure operation time of equipment after u month:

T(S ₀)＝inf{t:S(t,Z(T))＝S ₀,t＞0}(19)

Calculate fiduciary level when rail traffic vehicles critical system runs to u month by formula (18) and (19), in fiduciary level θ exit door non-failure operation time, wherein θ ∈ (0,1), u are natural number.

Below in conjunction with specific embodiment, the present invention is described in further detail.

Embodiment 1

Choose the overhaul data of certain MTR's No. 2 line herein, after supposing annual test or un-wheeling repair, then think that door device reparation is as newly.By analyzing No. 2 lines overhaul data of 2011 ~ 2012 years, the maintenance frequency of known main track fault, vehicle in vehicle life cycle is the principal element affecting vehicle ages, the factor that human factor causes car door normally to work also being occurrence number more, therefore the main track number of stoppages, vehicle maintenance number of times and the car door number of stoppages that causes due to the passenger covariant as door device is chosen, as table 1.

Table 1 door device repair history record

In his-and-hers watches 1, original covariant data carry out standardization, obtain correlation matrix

$R=\left[\begin{array}{ccc}1.0000& 0.6822& 0.1636\\ 0.6822& 1.0000& 0.1463\\ 0.1636& 0.1463& 1.0000\end{array}\right]$

Obtain eigenwert and the proper vector of matrix R, obtain covariant matrix

$Z=\left[\begin{array}{ccc}-0.8989& -0.8944& -0.3722\\ -0.3996& 0.3972& 0.0107\\ -0.1796& -0.2057& 0.9281\end{array}\right]\&CenterDot;\left[\begin{array}{c}{x}_1\\ x_2\\ .\\ .\\ .\\ x_m\end{array}\right]$

By linear regression model (LRM), multiple covariant is comprehensively become comprehensive covariant factor of influence, obtain the fitting result of covariant linear regression as table 2.

The linear fit result of table 2 covariant parameter

Data in associative list 1 and formula (6 ~ 17), select initializaing variable X ⁽⁰⁾=[2.3,500 ,-0.12 ,-0.289,0.03], can obtain the parameter estimation result of proportional hazard model as table 3.

The parameter estimation result of table 3 proportional hazard model

Can be able to according to parameter in table 2 and table 3 proportional hazard model that weibull is distributed as basis function is

$\mathrm{\&lambda;}<t|Z>=\frac{1.988}{637.143}{\left(\frac{t}{637.143}\right)}^{0.988}\&CenterDot;\exp (-0.0023t-0.0955)$

The Reliability Function obtaining door device according to failure rate estimation is

$S<t|Z\left(t\right)>=\exp \{-\underset{0}{\overset{t}{\&Integral;}}\frac{1.988}{637.143}{\left(\frac{s}{637.143}\right)}^{0.988}\&CenterDot;\exp (-0.0023s-0.0955)\mathrm{ds}\}$

Composition graphs 3, can be obtained the Reliability Function figure of door device by the Reliability Function of door device.

By the Reliability Function of door device calculate its fiduciary level reduce to 0.95 time, Q-percentile life is t _0.95=170 days.

Composition graphs 4, the Reliability Function of contrast weibull distributed model and proportional hazard model gained door device.As shown in Figure 4, less on the impact of reliability in the stage maintenance of initially coming into operation, but maintenance can extend the Q-percentile life of door device after two months, so the Q-percentile life of proportional hazard model prediction door device is more reasonable than weibull distributed model.

Claims (6)

1. a life-span prediction method for rail traffic vehicles critical system, is characterized in that, comprises the following steps:

Step 1, determines vehicle critical system: according to historical failure data and the system importance degree of vehicle, determine the critical system of rail traffic vehicles;

Step 2, chooses the substrate inefficacy function of proportional hazard model: according to the overhaul data of critical system in step 1, adopts fitting of distribution to obtain the substrate inefficacy function of critical system;

Step 3, covariant is screened: analyze the overhaul data of critical system described in step 1, calculates failure logging frequency ratio and obtains the covariant factor affecting the critical system life-span;

Step 4, covariant pre-service: according to the gained covariant factor in step 3, utilizes principal component analysis (PCA) to carry out process to the covariant factor and obtains covariant matrix;

Step 5, covariant is comprehensive: adopt linear regression fit to covariant multiple in step 4, finally obtain the integrated value of covariant;

Step 6, model parameter estimation: according to gained substrate inefficacy function and covariant integrated value in step 2 and step 5, adopts Maximum-likelihood estimation and Newton iteration method to try to achieve the parameter of proportional hazard model;

Step 7, life prediction: parameters obtained in step 6 is substituted into proportional hazard model, obtains the Reliability Function of critical system, utilizes Reliability Function to predict the in-service life-span of critical system.

2. the life-span prediction method of rail traffic vehicles critical system according to claim 1, is characterized in that, the substrate inefficacy function of the proportional hazard model described in step 2 is chosen, specific as follows:

With time to become the functional form of proportional hazard model of covariant as follows:

λ(t,Z(t))＝λ ₀(t)exp(g(Z(t)))(1)

In formula, λ (t, Z (t)) is failure rate estimation, and relevant with time and covariant; λ ₀t () is only relevant with time substrate failure rate estimation; Z (t)=(z1, z2, zn) be covariant vector at time t influential system failure probability, the covariant function that g (Z (t)) is influential system, and g (Z (t))=γ Z (t), wherein γ=(γ ₁, γ ₂..., γ _m) be regression parameter vector, m is the number of covariant;

Substrate failure rate estimation λ ₀t () adopts following two-factor Weibull distribution:

${\mathrm{\&lambda;}}_0\left(t\right)={\frac{\mathrm{\&beta;}}{\mathrm{\&eta;}}\left(\frac{t}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}---\left(2\right)$

In formula, β is form parameter, η is scale parameter;

Formula (2) is substituted into formula (1),

$\mathrm{\&lambda;}\left(t\right|Z\left(t\right))=\frac{\mathrm{\&beta;}}{\mathrm{\&eta;}}\frac{t^{\mathrm{\&beta;}}}{\mathrm{\&eta;}}\exp \left(\mathrm{\&gamma;Z}\left(t\right)\right)---\left(3\right)$

The citation form of rail traffic vehicles critical system proportional hazard model used is shown in formula (3).

3. the life-span prediction method of rail traffic vehicles critical system according to claim 1, is characterized in that, utilizes principal component analysis (PCA) to carry out process to the covariant factor and obtain covariant matrix described in step 4, specific as follows:

Suppose that rail traffic vehicles critical system covariant is X ₁, X ₂..., X _m, then standardization is done to covariant data, obtains correlation matrix R:

$R=\left[\begin{array}{cccc}{r}_{11}& r_{12}& ...& r_{1m}\\ r_{21}& r_{22}& ...& r_{2m}\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ r_{m1}& r_{m2}& ...& r_{\mathrm{mm}}\end{array}\right]---\left(4\right)$

In formula, matrix element r _ijfor covariant X _iwith X _jrelated coefficient, wherein i, j=1,2 ..., m;

Principal component analysis (PCA) is utilized to obtain the detailed process of covariant matrix as follows:

(1) secular equation is: | λ I-R|=0, obtains eigenvalue λ by solving, and makes its order arrangement by size, i.e. λ ₁>=λ ₂>=...>=λ _m>=0;

(2) obtain respectively corresponding to eigenvalue λ _iproper vector e _i, require || e _ij||=1, namely wherein e _ijrepresentation feature vector e _ia jth component;

(3) the major component load of covariant is calculated:

Finally obtaining covariant matrix is:

$Z\left(t\right)=\left|\begin{array}{c}{z}_1\\ z_2\\ .\\ .\\ .\\ z_m\end{array}\right|=\left[\begin{array}{cccc}{l}_{11}& l_{12}& ...& l_{1m}\\ l_{21}& l_{22}& ...& l_{2m}\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ l_{m1}& l_{m1}& ...& l_{\mathrm{mm}}\end{array}\right]\&CenterDot;\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ .\\ .\\ .\\ x_m\left(t\right)\end{array}\right]---\left(5\right)$

In formula, z _ifor the covariant factor, the x of influential system failure probability _jfor the initial value of covariant.

4. the life-span prediction method of rail traffic vehicles critical system according to claim 1, is characterized in that, the covariant described in step 5 is comprehensive, specific as follows:

Because the covariant parameter in proportional hazard model has multiple, the multiple covariants in raw data are carried out comprehensively, obtain comprehensive covariant function such as formula (6):

$g\left(z\left(t\right)\right)=\mathrm{\&gamma;Z}\left(t\right)=\left[\begin{array}{cccc}{\mathrm{\&gamma;}}_1& {\mathrm{\&gamma;}}_2& ...& {\mathrm{\&gamma;}}_m\end{array}\right]\&CenterDot;\left[\begin{array}{cccc}{l}_{11}& l_{12}& ...& l_{1m}\\ l_{21}& l_{22}& ...& l_{2m}\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ l_{m1}& l_{m1}& ...& l_{\mathrm{mm}}\end{array}\right]\&CenterDot;\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ .\\ .\\ .\\ x_m\left(t\right)\end{array}\right]---\left(6\right)$

Linear regression fit is carried out to original covariant, namely

x _i(t)＝a _i·t+b _i(7)

In formula, a _i, b _ifor linear fit coefficient;

Obtain the integrated value of covariant such as formula (8):

$g\left(Z\left(t\right)\right)=\mathrm{\&gamma;Z}=\left[\begin{array}{ccc}{\mathrm{\&gamma;}}_1& {\mathrm{\&gamma;}}_2& {\mathrm{\&gamma;}}_3\end{array}\right]\&CenterDot;\left[\begin{array}{cccc}{l}_{11}& l_{12}& ...& l_{1m}\\ l_{21}& l_{22}& ...& l_{2m}\\ .& .& .& .\\ .& .& .& .\\ .& .& .& .\\ l_{m1}& l_{m1}& ...& l_{\mathrm{mm}}\end{array}\right]\&CenterDot;\left[\begin{array}{c}{a}_1\&CenterDot;t+b_1\\ a_2\&CenterDot;t+b_2\\ .\\ .\\ .\\ a_m\&CenterDot;t+b_m\end{array}\right]---\left(8\right)$

5. the life-span prediction method of rail traffic vehicles critical system according to claim 1, is characterized in that, the model parameter estimation described in step 6 is specific as follows:

Be provided with n sample data, then the likelihood function of model is:

$L(\mathrm{\&beta;},\mathrm{\&eta;},\mathrm{\&gamma;})=\underset{p=1}{\overset{n}{\mathrm{\&Pi;}}}\mathrm{\&lambda;}{(t_p,\mathrm{\&beta;},\mathrm{\&eta;},\mathrm{\&gamma;})}^{{\mathrm{\&delta;}}_p}S{(t_p,\mathrm{\&beta;},\mathrm{\&eta;},\mathrm{\&gamma;})}^{{1-\mathrm{\&delta;}}_p}=\underset{q=1}{\overset{W}{\mathrm{\&Pi;}}}\mathrm{\&lambda;}(t_j,\mathrm{\&beta;},\mathrm{\&eta;},\mathrm{\&gamma;})\underset{p=1}{\overset{n}{\mathrm{\&Pi;}}}S(t_q,\mathrm{\&beta;},\mathrm{\&eta;},\mathrm{\&gamma;})---\left(9\right)$

In formula, W is the total sample number lost efficacy, p=1,2 ..., n, q=1,2 ..., W; t _pfor the system overhaul data record moment; δ _pfor the property shown variable, δ _pget 0 or 1, δ _p=1 represents t _pfail data, δ _p=0 represents t _pit is censored data; S (t _q, β, η, γ) and be Reliability Function, obtaining Reliability Function by the citation form of formula (3) proportional hazard model is:

$S\left(t\right|Z\left(t\right))=\exp (-\underset{0}{\overset{t}{\&Integral;}}\mathrm{\&lambda;}\left(s\right|Z\left(s\right))\mathrm{ds})=\exp (-\underset{0}{\overset{t}{\&Integral;}}\frac{\mathrm{\&beta;}}{\mathrm{\&eta;}}{\left(\frac{s}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}-1}\exp \left(\mathrm{\&gamma;Z}\left(s\right)\mathrm{ds}\right))---\left(10\right)$

Due to covariant Z (t) and time correlation, when the expression of Z (t) is unknown, Reliability Function formula (10) is expressed as:

$U_q=\underset{0}{\overset{t_q}{\&Integral;}}\frac{\mathrm{\&beta;}}{\mathrm{\&eta;}}{\left(\frac{s}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}-1}\exp \left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(s\right)\right)\mathrm{ds}=\underset{0}{\overset{t_q}{\&Integral;}}\exp \left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(s\right)\right)d{\left(\frac{s}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}---\left(11\right)$

K=1 in formula, 2 ..., m;

If the value of covariant Z (t) is at moment t _pknown, then γ Z (t) | and t>=0} can regard as right continuous step process [γ Z (t)] * | t>=0};

Suppose 0<t ₁<t ₂< ... <t _q-1<t _q, utilize integration by parts formula (11) to be expressed as:

$\begin{array}{c}{U}_q^{*}={\left(\frac{t_q}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\exp \left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_k{Z}_k\left(t_q\right)\right)-\underset{0}{\overset{t_q}{\&Integral;}}{\left(\frac{s}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\mathrm{dexp}\left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(s\right)\right)\\ ={\left(\frac{t_q}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\exp \left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(t_q\right)\right)-(\underset{0}{\overset{t_1}{\&Integral;}}+\underset{t_1}{\overset{t_q}{\&Integral;}}){\left(\frac{s}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\mathrm{dexp}\left[{\left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_{k\left(s\right)}\right)}^{*}\right]\end{array}---\left(12\right)$

Due to so two integral representations in formula (12) are formula (13) and formula (14):

$\underset{0}{\overset{t_1}{\&Integral;}}{\left(\frac{s}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\mathrm{dexp}\left[{\left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(s\right)\right)}^{*}\right]={\left(\frac{t_1}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\{\exp \left(\underset{k-1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(t_1\right)\right)-\exp [{\left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(0\right)\right)}^{*}\left]\right\}---\left(13\right)$

${\underset{t_1}{\overset{t_q}{\&Integral;}}\left(\frac{s}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\mathrm{dexp}\left[{\left(\underset{k=1}{\overset{p}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(s\right)\right)}^{*}\right]=\underset{v=2}{\overset{q}{\mathrm{\&Sigma;}}}\left(\frac{t_v}{\mathrm{\&eta;}}\right)(\exp \left(\underset{k-1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(t_v\right)\right)-\exp \left(\underset{k-1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(t_{v-1}\right)\right))---\left(14\right)$

V=2 in formula, 3 ... q;

Formula (13) and formula (14) are brought into formula (12) to obtain:

$U_q^{*}={\left(\frac{t_1}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}\exp {\left[\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(0\right)\right]}^{*}+\underset{v=2}{\overset{q}{\mathrm{\&Sigma;}}}[{\left(\frac{t_v}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}-{\left(\frac{t_{v-1}}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}]\&CenterDot;\exp \left[\left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(t_{v-1}\right)\right)\right]---\left(15\right)$

Formula (3) and formula (15) are substituted into formula (9) obtain basis function be the log-likelihood function of the proportional hazard model of two-factor Weibull distribution such as formula (16):

$\begin{array}{c}\ln L(\mathrm{\&beta;},\mathrm{\&eta;},\mathrm{\&gamma;})=\\ W\ln \left(\frac{\mathrm{\&beta;}}{\mathrm{\&eta;}}\right)+(\mathrm{\&beta;}-1)\underset{p=1}{\overset{W}{\mathrm{\&Sigma;}}}\ln \left(\frac{t_p}{\mathrm{\&eta;}}\right)+\underset{p=1}{\overset{W}{\mathrm{\&Sigma;}}}\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(t_p\right)-\underset{q-1}{\overset{n}{\mathrm{\&Sigma;}}}\{\left(\frac{t_1}{\mathrm{\&eta;}}\right)\exp \left(\right[{\left(\underset{k=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(0\right)\right)}^{*}\left]\right)+\\ \underset{v-2}{\overset{q}{\mathrm{\&Sigma;}}}[{\left(\frac{t_v}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}-{\left(\frac{t_{v-1}}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}}]\&CenterDot;\exp \left[\left(\underset{k-1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&gamma;}}_k{Z}_k\left(t_{v-1}\right)\right)\right]\}\end{array}---\left(16\right)$

Adopt the gloomy N-R process of iteration of newton-pressgang to ask model parameter, suppose X=[β, η, γ]=[β, η, γ ₁, γ ₂..., γ _m], ask the first-order partial derivative of each parameter, second-order partial differential coefficient and mixed partial derivative to obtain matrix F (X) and matrix G (X) respectively to formula (15):

$F\left(X\right)=[\frac{\&PartialD;\ln L(\mathrm{\&beta;},\mathrm{\&eta;},\mathrm{\&gamma;})}{\&PartialD;\mathrm{\&beta;}},\frac{\&PartialD;\ln L(\mathrm{\&beta;},\mathrm{\&eta;},\mathrm{\&gamma;})}{\&PartialD;\mathrm{\&eta;}},\frac{\&PartialD;\ln L(\mathrm{\&beta;},\mathrm{\&eta;},\mathrm{\&gamma;})}{{\&PartialD;\mathrm{\&gamma;}}_1},...\frac{\&PartialD;\ln L(\mathrm{\&beta;},\mathrm{\&eta;},\mathrm{\&gamma;})}{{\&PartialD;\mathrm{\&gamma;}}_m}]$

Using matrix F (X) and matrix G (X) as iteration factor, newton iteration formula is utilized to obtain:

X ⁽ⁿ⁺¹⁾＝X ⁽ⁿ⁾-F(X ⁽ⁿ⁾)(G(X ⁽ⁿ⁾)) ^-1(17)

The initial value of given X is X ⁽⁰⁾, substitute into iterative formula (17), if | X ⁽ⁿ⁺¹⁾-X ⁽ⁿ⁾| during < ε=0.01, inequality is set up, then obtain the estimates of parameters X=X of proportional hazard model ⁽ⁿ⁺¹⁾.

6. the life-span prediction method of rail traffic vehicles critical system according to claim 1, is characterized in that, the critical system life prediction described in step 7, specific as follows:

The failure rate estimation being distributed as the proportional hazard model of basis function by Weibull obtains Reliability Function and is:

$S<t|Z\left(t\right)>=\exp (-\underset{0}{\overset{t}{\&Integral;}}\frac{\mathrm{\&beta;}}{\mathrm{\&eta;}}{\left(\frac{s}{\mathrm{\&eta;}}\right)}^{\mathrm{\&beta;}-1}\exp \left(g\left(s\right)\right)\mathrm{ds})---\left(18\right)$

Obtained by Reliability Function, the non-failure operation time of equipment after u month:

T(S ₀)＝inf{t:S(t,Z(T))＝S ₀,t＞0}(19)

Calculate fiduciary level when rail traffic vehicles critical system runs to u month by formula (18) and (19), in fiduciary level θ exit door non-failure operation time, wherein θ ∈ (0,1), u are natural number.