System reliability evaluation and verification method based on life mixed distribution

Abstract

The invention provides a system reliability evaluation and verification method based on life mixed distribution aiming at the feasibility problem of prolonging the maintenance cycle interval of an equipment system, which is a complete complex system reliability evaluation and verification method comprising random truncation data processing, reliability modeling and parameter estimation. A system basic mixed distribution reliability evaluation model is established through a mixed distribution principle, a mixed distribution model point estimation value is solved through an EM algorithm of random truncation data, namely an Expectation Maximization Algorithm algorithm, also known as an expectation maximization algorithm, and the solved mixed distribution model is used for checking the rationality of the system for prolonging the maintenance interval. And finally, the rationality and feasibility of the hybrid distribution model applied to reliability evaluation of the equipment system are verified, and the change condition of the reliability of the system in the operation process is given.

Description

Technical Field

[0001] The present invention relates to a system reliability evaluation and verification method based on life mixed distribution, which is a system reliability evaluation and verification method that couples mixed distribution theory and the basic principle of maximum likelihood estimation. This method accurately analyzes the complete failure data and random truncated life test data of core failure components in multivariate distributed complex systems, uses maximum likelihood estimation and expectation maximization algorithms to jointly estimate the mixed distribution parameters of the overall life of the system, builds a system reliability evaluation model based on the parameter estimation values, and gives the law of change of system reliability with maintenance cycle. This method is suitable for reliability evaluation and verification of the service life improvement and maintenance cycle extension needs of high-reliability equipment in fields such as aviation and rail transportation. Background Art

[0002] The stable operation of large and complex systems, represented by complex equipment systems in the fields of aviation and rail transit, is the key to ensuring passenger safety, improving available efficiency and reducing long-term maintenance costs. These systems are often characterized by high value and long life, and during their long service life, they need to be regularly inspected, maintained and maintained to ensure that they continue to play their due value. However, as the service time increases, the reliability indicators of the components and subsystems inside the system will gradually decline due to the influence of various complex factors. In order to select a scientific and reasonable maintenance method that can effectively reduce maintenance costs and improve maintenance efficiency, it is necessary to combine effective reliability assessment and verification methods with the reform of equipment system maintenance procedures and systems, so as to achieve the evaluation and verification of the reliability status of complex equipment systems.

[0003] Usually, the reliability evaluation of equipment systems can be studied from two perspectives: components and the whole machine. However, the present invention evaluates the reliability of the whole machine from the perspective of components, considering the inference from components to the whole machine, that is, the sub-distribution of each component constitutes the overall distribution, and selects a mixed distribution model to describe the change of the reliability of the whole machine. The mixed distribution model can well describe a whole composed of multiple components or sub-totals.

[0004] The weight value in the mixed distribution model can reflect the proportion of samples in each sub-distribution in the overall distribution sample, so as to further understand the reliability change process of the equipment system from the perspective of data samples. Compared with other reliability assessment models, the mixed distribution model has higher fitting accuracy and can better reflect the overall situation of the data. At the same time, the mixed distribution model can obtain predictive results for future systems and flexibly and dynamically give the reliability change trend of the equipment system.

[0005] The probability density function of the mixed distribution is:

[0006]

[0007] Among them, t is a random variable, p j represents the mixing weight parameter, that is, the weight of each sub-distribution in the overall distribution. j represents the probability density function of the jth distribution in the mixed distribution model, Ω is the value space of t, j = 1, 2, ..., k. θ j ,j=1,2,...,k represents the jth distribution probability density function f in the mixed distribution model j Parameters. Θ=(p1,p2,...,p k ,θ1,θ2,...,θ k ) is the parameter space of the mixture distribution model.

[0008] The process of solving the parameters of the mixed distribution model is very complicated, especially when there is censored data in the sample. It is difficult to use the usual maximum likelihood method to directly solve the parameters of the mixed distribution model.

[0009] The usual maximum likelihood estimation principle is:

[0010] Assume that the total sample is n, and a component fails r times in the test. Its life observation values ​​are t1, t2, ..., t r . Let the probability density function be f(t,θ), θ=(θ1,θ2,...,θ r ), which means there are r unknown parameters in its distribution function and probability density function. The likelihood equation of the maximum likelihood method is:

[0011]

[0012] In actual work, equipment system samples often contain a large amount of truncated data, which makes it impossible to directly use the general maximum likelihood estimation method for parameter estimation. At the same time, using the general maximum likelihood estimation method to estimate the parameters of the mixed distribution will lose the uniqueness of the life observation value. i It can only come from one of the k distributions of the mixed distribution. Therefore, the present invention introduces a random truncated sample indicator variable δ i And an indicator variable Z used to indicate that the life observation value specifically comes from a sub-distribution in the mixed distribution model. Based on the above principle, the present invention establishes an expectation maximization algorithm that introduces indicator variables to estimate the parameters of the mixed distribution model. After research, it is found that this method can accurately estimate the parameters of the mixed distribution model.

[0013] Based on the above background technology, the present invention proposes a system reliability evaluation and verification method based on life mixed distribution, and establishes an equipment system reliability evaluation model based on mixed distribution and expectation maximization algorithm. Taking into account the data type and evaluation method, the change process of the reliability index of the equipment system during service and the feasibility of extending the maintenance cycle interval of the complex equipment system are given. Summary of the invention

[0014] (1) The purpose of the present invention: To address the feasibility of extending the maintenance cycle interval of equipment systems, a system reliability evaluation and verification method based on mixed distribution is provided, which is a complete complex system reliability evaluation and verification method including random truncated data processing, reliability modeling, and parameter estimation. A basic mixed distribution reliability evaluation model of the system is established based on the mixed distribution principle, and the EM algorithm of random truncated data, i.e., the Expectation Maximization Algorithm, also known as the expectation maximization algorithm, is used to obtain the point estimate of the mixed distribution model, and the mixed distribution model obtained by the solution is used to verify the rationality of extending the maintenance interval of the system. Finally, the rationality and feasibility of applying the mixed distribution model to the reliability evaluation of the equipment system are verified, and the changes in the reliability of the system during operation are given.

[0015] (2) Technical solution:

[0016] The present invention needs to establish the following basic settings:

[0017] Setting 1: If any one of the k components in the equipment system fails, it is considered as a failure of the entire machine.

[0018] Setting 2: If any one of the k components in the equipment system fails, the equipment system will be shut down immediately to replace the component, that is, the component will be immediately repaired as good as new.

[0019] Setting 3: After the equipment system has been repaired, all parts are restored to be as good as new.

[0020] Data background setting of the present invention:

[0021] The data set selected for verification of the present invention is derived from actual data of a large-scale equipment system. It contains complete failure data and truncated data of four types of core failure components A, B, C, and D of a large-scale equipment system. Among them, components A and C obey lognormal distribution. Components B and D obey gamma distribution, and the distribution parameters are known. Complete failure data refers to the data generated by the failure of a core failure component of a certain equipment system before maintenance. Truncated data is generated when the component reaches the maintenance time or the current operating time, and the component has not failed, or the component that has failed and has been replaced has not failed when it reaches the maintenance time or the current operating time. Based on the above conditions and assumptions, the truncated data of key components of the equipment system in the present invention are randomly truncated data.

[0022] The life t of the core failure components of the equipment system obeys the log-normal distribution and the gamma distribution, and the probability density functions of each component are:

[0023] ①Part A:

[0024]

[0025] ②Part B:

[0026]

[0027] ③C component:

[0028]

[0029] ④D components:

[0030]

[0031] Among them, component A is equipped with an important axle box bearing, component B is equipped with a traction system component, component C is equipped with a braking system component, and component D is equipped with a bogie system component. A ,σ A ,k B ,θ B ,u C ,σ C ,k D ,θ D The known distribution parameters of each core failure component of the equipment system. A With σ A Respectively represent the mean and standard deviation of component A following the lognormal distribution; k B With θ B They represent the shape parameter and scale parameter of the gamma distribution of component B; u C With σ C Respectively represent the mean and standard deviation of the C component following the log-normal distribution; k D With θ D They represent the shape parameter and scale parameter of the gamma distribution of component D respectively.

[0032] The method proposed in the present invention mainly targets the complete failure data and randomly truncated data of the core failure components of the equipment system, establishes a mixed distribution model of the core failure components of the equipment system according to the mixed distribution theory, and uses the EM algorithm of randomly truncated data to evaluate the reliability parameters of the mixed distribution model of the equipment system, thereby evaluating the reliability status of the entire equipment system and obtaining the feasibility of extending the maintenance cycle interval.

[0033] Based on the above assumptions and ideas, the system reliability evaluation and verification method based on hybrid distribution of the present invention is implemented through the following steps:

[0034] Step 1: Establish a mixed distribution model of the equipment system.

[0035] For the mixed distribution of the equipment system, which is composed of various components as sub-distributions, let the lifespans of k components be T1, T2, ..., T k Independent and identically distributed in a finite mixed distribution model. At the same time, it is assumed that each of the k types of components in the equipment system obeys the same distribution, that is, the equipment system mixed distribution model is composed of k sub-distributions, and the k types of components contain a total of n life values, including complete failure data and truncated data. All life values ​​are regarded as a whole, and the life value of any component is t i . Assume that the observed life span is t i ′. Assume that the observation deadline of the i-th life value is c i (The truncation time in the present invention is a random truncation), when the life value is t i ≤c i , life observation value t i ′=t i , the observed data is a failure data of the component; when t i >c i When the observed life span value t of the i-th life span value in the sample i ′=c i , the observed data at this time is a truncated data of the component.

[0036] According to the above principle, the equipment system life value t can be obtained i The probability density function f of the mixed distribution Θ (t i ) and mixed distribution reliability function R Θ (t i )for:

[0037]

[0038] Among them, t i Indicates the equipment system life value, p j Represents the mixing weight parameter of the jth sub-distribution, that is, the weight of each component's sub-distribution in the overall distribution. j (t i θ j ) represents the probability density function of the jth distribution in the mixed distribution model, Ω is the value space of t, j = 1, 2, ..., k. j (t i θ j) represents the reliability function of the jth distribution in the mixed distribution model. j ,j=1,2,...,k represents the jth distribution probability density function f in the mixed distribution model j (t i θ j ) and the parameters of the reliability function of the jth distribution. Θ=(p1,p2,...,p k ,θ1,θ2,...,θ k ) is the parameter space of the mixture distribution model.

[0039] Among them, t i >0,0≤p j ≤1. And:

[0040]

[0041] The parameter x in formula (9) is an intermediate variable in the integration process and has no specific meaning.

[0042] Step 2: Find the maximum likelihood estimate of the mixed distribution model introducing the indicator variable.

[0043] Under the condition of mixed distribution model, given a given life observation sample t=(t1′,t2′,...,t n ′) The likelihood function of the parameter vector θ is:

[0044]

[0045] Among them, δ i is a randomly censored sample indicator variable, that is:

[0046]

[0047] According to the relevant t in step 1 i and c i Description, when the life value is t i ≤c i , life observation value t i ′=t i When t i >c i When the life observation of the i-th life value in the sample is truncated at time c i Indicates that t i ′=c i .f Θ (t i ) represents the i ≤c i , that is, the observed life value is the failure time t i When the likelihood component in the likelihood function is the failure time t iThe probability density function f Θ (t i ) indicates. Θ (t i ) represents the i >c i When the observed life span is truncated at time c i When the likelihood component in the likelihood function is truncated at time c i The reliability function R Θ (t i )express.

[0048] Taking the logarithm of both sides of equation (11), we get the log-likelihood function of the mixed distribution:

[0049]

[0050] Since the lifetime observation t i It can only come from one of the k distributions in the mixed distribution. Introduce indicator variables Z = (Z1,...,Z n ), where Z = (Z i1 ,...,Z ik ),i=1,2,...,n. Then we have:

[0051]

[0052] For indicator variable Z i There are the following constraints: Z i There is only one 1 in the element, and the rest are all 0. According to the above process, the joint distribution of life expectancy and indicator variable is:

[0053]

[0054] where f j (t i θ j ) here represents R j (c i θ j ) represents the observed life value as failure time t i , that is, the observed life span t i ′=t i , t i Substitute the probability density function f of the jth distribution into the mixed distribution model j The obtained likelihood component f j (t i θ j );R j (c i θ j ) represents the life observation value as the censoring time c i When ti ′=c i When c i Substitute the reliability function R of the jth distribution into the mixed distribution model j The obtained likelihood component R j (c i θ j ).

[0055] Using the maximum likelihood method, we can get the random censored sample with the indicator variable δ i and Z i The mixed distribution likelihood function is:

[0056]

[0057] The parameter t represents the observed value of life span t i ′ parameter space. z represents the indicator variable Z i The parameter space formed.

[0058] Step 3: Use the EM algorithm to estimate the parameters of the mixture distribution model.

[0059] The maximum likelihood estimation of the indicator variable introduced into the mixed distribution model in step 2 is solved using the EM algorithm.

[0060] The specific steps of EM calculation are:

[0061] I. (E-step) Calculate the expected value of equation (15), given the initial value of the mixed weight p 0 and the initial value of the parameter θ 0 , (select the proportion of A, B, C, D parts samples in the total samples p j As the initial value, the distribution parameters of each component are selected as the initial values ​​of the parameters of each sub-distribution model of the mixed distribution). The calculation process of the mth iteration is as follows: The parameter value obtained at the m-1th iteration is θ (m-1) and p (m-1) , solve the log-likelihood function with parameter value θ (m-1) 、p (m-1) The expected value Q(θ, p; θ (m-1) , p (m-1) ):

[0062]

[0063] in represents the parameter value θ obtained when the value of the mixed distribution likelihood function in equation (15) is substituted into the m-1th iteration (m-1) 、p (m-1) The mathematical expectation of .

[0064] The indicator variable Z iThe expected value is solved. Indicator variable Z i The expected value of can be expressed by the proportion of each sub-distribution in the overall distribution. When the observed value is the complete failure data t i or randomly censored data c i The expected value is different when . Suppose you need to solve the mixed distribution weight value of the j-th sub-distribution at the m-th iteration and parameters When the indicator variable Z i The expected value of is expressed as:

[0065]

[0066] Among them, the parameters represents the known mixing weight value of the j-th sub-distribution obtained at the m-1-th iteration, Represents the known parameter value of the j-th subdistribution obtained at the m-1-th iteration. Parameter represents the known mixing weight value of the lth sub-distribution obtained at the m-1th iteration, represents the known parameter value of the lth subdistribution obtained at the m-1th iteration, and It is named the parameter of the lth subdistribution because it is in the summation term in the denominator, distinguishing it from the parameter that needs to be solved for the jth subdistribution. j and f l , R j and R l The naming rules are the same as above.

[0067] II. (M-step) Update the estimated parameters. Maximize the expected value function to find θ (m) and p (m) , so that

[0068] Q(θ (m) , p (m) θ (m-1) , p (m-1) ) = max Q(θ, p; θ (m-1) , p (m-1) ). Solve the expected value obtained in step E to obtain the mixed distribution weight value of the j-th sub-distribution at the m-th iteration. for:

[0069]

[0070] At the same time, the parameter value of the jth sub-distribution of the mixed distribution model at the mth iteration can be obtained for:

[0071]

[0072] in Include Parameters One iteration is completed after the E-step and the M-step. The E-step and the M-step are repeated until the parameters converge to obtain the point estimate of the parameters of the mixed distribution model.

[0073] Step 4: Calculate the equipment system reliability index and verify the feasibility of extending the maintenance cycle interval.

[0074] The ultimate goal of the present invention is to quantitatively verify whether the percentage value of extending the maintenance cycle of the equipment system by the original cycle is feasible. Reliability is selected as the verification index of whether the extended maintenance cycle is feasible and a fixed step length is set to calculate the reliability value during the operation of the equipment system. A reliability threshold W is set, that is, if the reliability of the equipment system after the extension does not reduce the original reliability threshold, then the maintenance extension plan is feasible.

[0075] In step 3, we get the point estimates of the mixed distribution model parameters, including the sub-distribution weights. And the parameters of each sub-distribution From this, the probability density function of the equipment system mixed distribution can be obtained as follows:

[0076]

[0077] Therefore, the reliability function can be determined for:

[0078]

[0079] Select a fixed step length S, calculate the reliability value of the equipment system at each step length node through the reliability function and make a reliability curve, and calculate whether the reliability value of the equipment system exceeds the specified threshold W after the specified extended maintenance cycle. For example: select S = 1 year, calculate the reliability value of the equipment at each node from 0 to 50 years. Assume that the original fixed maintenance cycle of the equipment is 10 years, and now the maintenance cycle needs to be extended to 12 years. Assume that the reliability of the equipment in the 10th year is R 10 , the reliability in the 12th year is R 12 .like If the maintenance period is extended, it is feasible, otherwise it is not feasible. The threshold W is usually determined by engineering experience.

[0080] (3) Advantages and effects: The present invention is a system reliability assessment and verification method based on hybrid distribution, which has the following advantages:

[0081] ① The present invention aims at the complete failure data of key components of the equipment system and the randomly truncated data, and establishes a basic mixed distribution reliability assessment model of the equipment system based on the mixed distribution principle. The point estimate of the mixed distribution model is obtained by solving the randomly truncated data with the EM algorithm, and the mixed distribution model obtained by solving the solution is used to test the rationality of extending the maintenance interval of the equipment system. Finally, the rationality and feasibility of using the mixed distribution model for reliability assessment of the equipment system are verified, and the changes in the reliability of the equipment system during operation are given, which scientifically and safely guarantees the requirements of extending the maintenance cycle of the equipment system and reducing the maintenance cost.

[0082] ② The algorithm of the present invention is better applicable to randomly truncated data, has low requirements on the initial values ​​of parameters, has simple steps, fast algorithm iteration and a convergent solution is certain to exist.

[0083] ③ The present invention works well when processing large sample data, and provides ideas for verifying the feasibility of extending the maintenance cycle interval of the equipment system from the perspective of reliability assessment. BRIEF DESCRIPTION OF THE DRAWINGS

[0084] Figure 1 The present invention is a flow chart of the method. DETAILED DESCRIPTION

[0085] The present invention will be further described in detail below with reference to an example of a large equipment system.

[0086] A certain type of EMU has been in service for 5 years under natural conditions. Fault data of more than 200 EMUs of this type have been collected. The fault data includes more than 1,300 complete failure data of four types of core failure components A, B, C, and D and random truncated failure data. The failure data in the data set are all expressed in terms of EMU running mileage. The basic data is shown in Table 1:

[0087] Table 1 Data of core failure components of EMU

[0088] A-Part B Part C-Component Other components Complete vehicle The number of completely failed data 26 25 19 116 186 Number of randomly censored data 280 280 280 280 1120 total 306 305 299 396 1306

[0089] Assume that a certain type of EMU has four types of core failure components, A, B, C, and D. In the EMU system, components A and C obey the log-normal distribution. Components B and D obey the gamma distribution, and the distribution parameters have been obtained by the maximum likelihood estimation method. According to the system reliability evaluation and verification method based on mixed distribution proposed in the present invention, a mixed distribution model of the EMU is established, and parameter evaluation is performed to evaluate the reliability status of the EMU vehicle and obtain the feasibility of extending the maintenance cycle interval.

[0090] The initial probability density functions of the core failure components A, B, C, and D are:

[0091]

[0092] See Figure 1 The present invention provides a system reliability evaluation and verification method based on hybrid distribution, which is implemented by the following steps:

[0093] Step 1: Establish a mixed distribution model of EMUs.

[0094] First, the basic expression of the EMU mixed distribution model is determined as:

[0095]

[0096] Since the mixed distribution model is composed of the distribution contained in the mixed sample, according to the results of the selection of the optimal distribution model of the core failure components of the EMU, the mixed distribution model is composed of four sub-distributions (two lognormal distributions and two gamma distributions), and its probability density function basic expression is:

[0097]

[0098] Step 2: Find the maximum likelihood estimate of the mixed distribution model introducing the indicator variable.

[0099] From formula (15), we can get the EMU system after introducing the indicator variable δ i and Z i The mixed distribution likelihood function is:

[0100]

[0101] This step is a prerequisite for step three. It is not necessary to calculate the specific value. It is necessary to substitute the mixed distribution likelihood function after introducing the indicator variable into step three to solve the final result.

[0102] Step 3: Use the EM algorithm to estimate the parameters of the mixture distribution model.

[0103] Select the proportion of A, B, C, and D parts in the total sample as p j Initial value, select the maximum likelihood estimation parameter as the initial value of each sub-distribution model parameter of the mixed distribution:

[0104]

[0105] The data is selected from the overall sample of the complete failure data of four key components and the truncated data. The iteration threshold is set to 10 -5 Substitute into equation (19) and equation (20) for iterative calculation. When the parameter difference before and after iteration is less than 10 -5 The EM algorithm is considered to converge at this time. After 6386 iterations, the parameters converge, and the estimated values ​​of the mixed distribution parameters are shown in Tables 2 and 3.

[0106] Table 2 EM algorithm weight p jCalculation results

[0107] <![CDATA[p1]]> <![CDATA[p2]]> <![CDATA[p3]]> <![CDATA[p4]]> Estimated value 0.01084 0.82975 0.05755 0.10186

[0108] Table 3 EM algorithm distribution parameter calculation results

[0109] <![CDATA[u A ]]> <![CDATA[σ A ]]> <![CDATA[k B ]]> <![CDATA[θ B ]]> <![CDATA[u C ]]> <![CDATA[σ C ]]> <![CDATA[k Other ]]> <![CDATA[θ Other ]]> Estimated value -0.9005 0.82941 109.55181 1.86059 2.61431 0.85778 4.69333 18.33670

[0110] Step 4: Calculate the EMU system reliability index and verify the feasibility of extending the maintenance cycle interval.

[0111] From Table 2 and Table 3, we can get the probability density function of the mixed distribution model:

[0112]

[0113] According to formula (22), its reliability function can be determined as:

[0114]

[0115] The reliability function of the mixed distribution model in formula (32) can calculate the reliability value of the EMU at any time. The ultimate goal of the present invention is to quantitatively verify whether it is feasible to extend the maintenance cycle of the EMU by 25% of the original cycle, that is, whether it is feasible to extend the maintenance time of the EMU from the original 1.32 million kilometers to 1.65 million kilometers. Therefore, S = 10,000 km is selected as the step size to calculate the reliability value of the EMU from 1.32 million km to 1.65 million km. The reliability of each node is shown in Table 4.

[0116] Table 4 Calculation results of reliability values ​​of each node

[0117]

[0118]

[0119] The reliability threshold W is set to 2% of the original reliability, that is, the reliability of the EMU after the extension does not decrease by 2% of the original reliability, then the maintenance extension plan is feasible. The calculation results of the node reliability reduction percentage are shown in Table 5.

[0120] Table 5 Calculation results of reliability values ​​of each node

[0121] Mileage / 10,000 km 132 133 134 135 136 percentage / % 0 0.05 0.09 0.14 0.18 Mileage / 10,000 km 137 138 139 140 141 percentage / % 0.23 0.27 0.31 0.36 0.40 Mileage / 10,000 km 142 143 144 145 146 percentage / % 0.44 0.48 0.52 0.56 0.61 Mileage / 10,000 km 147 148 149 150 151 percentage / % 0.65 0.70 0.75 0.80 0.86 Mileage / 10,000 km 152 153 154 155 156 percentage / % 0.92 0.99 1.06 1.14 1.24 Mileage / 10,000 km 157 158 159 160 161 percentage / % 1.34 1.45 1.58 1.73 1.89 Mileage / 10,000 km 162 163 164 165 percentage / % 2.07 2.28 2.51 2.77

[0122] After calculation, when the maintenance cycle is extended to 1.65 million km, its reliability is reduced by 2.77% of the original reliability at 1.32 million km, which is greater than the original threshold of 2%. Therefore, it is not feasible to extend the maintenance cycle by 25% of the original maintenance cycle. To be conservative, the maintenance cycle is extended to 1.60 million km at most, at which time the reliability is reduced by 1.73% of the original reliability.

[0123] The results show that the method of the present invention can be used to evaluate the reliability index of complex equipment systems and verify the feasibility of extending the maintenance cycle interval, thus achieving the expected purpose.

[0124] In summary, the present invention relates to a system reliability evaluation and verification method based on mixed distribution. Aiming at the feasibility problem of extending the maintenance cycle interval of the equipment system, it establishes a mixed distribution model of the core failure components of the equipment system through the mixed distribution theory, and performs maximum likelihood estimation on the parameters of the mixed distribution model of the equipment system by introducing the EM algorithm of the indicator variable, evaluates the reliable life, and verifies whether it is feasible to extend the maintenance cycle interval. The specific steps of the method are: firstly, establish a mixed distribution model of the equipment system, then introduce the truncated data indicator variable, find the maximum likelihood function of the mixed distribution model introducing the indicator variable, use the EM algorithm to estimate the parameters of the mixed distribution model, find the parameters of the mixed distribution model of the equipment system, and then calculate the reliability index of the equipment system, and finally verify the feasibility of extending the maintenance cycle interval. The present invention has broad application prospects in the fields of aviation and rail transportation, and can provide strong technical support for the maintenance process optimization of multi-component complex systems, helping the industry to achieve more efficient, safe and economical operation goals. At the same time, the present invention is also applicable to the fields of big data and truncated data evaluation in the reliability evaluation and verification of complex equipment systems, and has strong operability.

Claims

1. A system reliability assessment and verification method based on life mixed distribution requires the following settings: Setting 1: If any one of the k components in the equipment system fails, it is considered as a failure of the whole machine; Setting 2: If any one of the k components in the equipment system fails, the equipment system will be shut down immediately to replace the component, that is, the component will be immediately repaired to be as good as new; Setting 3: After the equipment system is repaired, all parts are restored to be as good as new; Features: The steps include: Step 1: Establish a mixed distribution model of equipment system; For the mixed distribution of the equipment system, which is composed of various components as sub-distributions, let the lifespans of k components be T1, T2, ..., T k Independent and identically distributed in a finite mixed distribution model; at the same time, assume that each of the k types of components in the equipment system obeys the same distribution, that is, the equipment system mixed distribution model is composed of k sub-distributions, and the k types of components contain a total of n life values, including complete failure data and truncated data; Step 2: Find the maximum likelihood estimate of the indicator variable introduced into the mixed distribution model; Step 3: Use the EM algorithm to estimate the parameters of the mixed distribution model; The maximum likelihood estimation of the indicator variable introduced into the mixed distribution model is solved by using the EM algorithm; Step 4: Calculate the reliability index of the equipment system and verify the feasibility of extending the maintenance cycle interval; Select reliability as the verification index of whether the extension of maintenance cycle is feasible and set a fixed step length to calculate the reliability value during the operation of the equipment system; A reliability threshold W is set, that is, if the reliability of the equipment system after the extension does not decrease the original reliability threshold, then the maintenance extension plan is feasible.

2. The system reliability evaluation and verification method based on lifetime mixed distribution according to claim 1 is characterized in that: In step 1, all life values ​​are considered as a whole, where the life value of any component is t i ; Assume that the observed life span is t i ′; Let the observation deadline of the i-th life value be c i , when the life value is t i ≤c i , life observation value t i ′=t i , the observed data is a failure data of the component; when t i >c i When the observed life span value t of the i-th life span value in the sample i ′=c i , the observed data at this time is a truncated data of the component.

3. The system reliability evaluation and verification method based on lifetime mixed distribution according to claim 2 is characterized in that: Equipment system life value t i The probability density function f of the mixed distribution Θ (t i ) and mixed distribution reliability function R Θ (t i )for: Among them, t i Indicates the equipment system life value, p j represents the mixing weight parameter of the jth sub-distribution, that is, the weight of each component's sub-distribution in the overall distribution; f j (t i θ j ) represents the probability density function of the jth distribution in the mixed distribution model, Ω is the value space of t, j = 1, 2, ..., k; R j (t i θ j ) represents the reliability function of the jth distribution in the mixed distribution model; θ j ,j=1,2,...,k represents the jth distribution probability density function f in the mixed distribution model j (t i θ j ) and the parameters of the reliability function of the jth distribution; Θ = (p1, p2, ..., p k ,θ1,θ2,...,θ k ) is the parameter space of the mixed distribution model; Among them, t i >0,0≤p j ≤1; and: The parameter x in formula (9) is an intermediate variable in the integration process and has no specific meaning.

4. The system reliability evaluation and verification method based on lifetime mixed distribution according to claim 1 is characterized in that: In step 2, under the condition of the mixed distribution model, given a given life observation sample t = (t1′, t2′, ..., t n ′) The likelihood function of the parameter vector θ is: Among them, δ i is a randomly censored sample indicator variable, that is: When the life value is t i ≤c i , life observation value t i ′=t i When t i >c i When the life observation of the i-th life value in the sample is truncated at time c i Indicates that t i ′=c i ;f Θ (t i ) represents the i ≤c i , that is, the observed life value is the failure time t i When the likelihood component in the likelihood function is the failure time t i The probability density function f Θ (t i ) indicates; R Θ (t i ) represents the i >c i When the observed life span is truncated at time c i When the likelihood component in the likelihood function is truncated at time c i The reliability function R Θ (t i )express.

5. The system reliability evaluation and verification method based on lifetime mixed distribution according to claim 4 is characterized in that: Taking the logarithm of both sides of equation (11), we get the log-likelihood function of the mixed distribution: Since the lifetime observation t i It can only come from one of the k distributions in the mixed distribution; introduce the indicator variable Z = (Z1,...,Z n ), where Z = (Z i1 ,...,Z ik ),i=1,2,...,n; then: For indicator variable Z i There are the following constraints: Z i There is only one 1 in the element, and the rest are all 0; the joint distribution of life span and indicator variable is: where f j (t i θ j ) here represents R j (c i θ j ) represents the observed life value as failure time t i , that is, the observed life span t i ′=t i , t i Substitute the probability density function f of the jth distribution into the mixed distribution model j The obtained likelihood component f j (t i θ j );R j (c i θ j ) represents the life observation value as the censoring time c i When t i ′=c i When c i Substitute the reliability function R of the jth distribution into the mixed distribution model j The obtained likelihood component R j (c i θ j ).

6. The system reliability evaluation and verification method based on lifetime mixed distribution according to claim 5 is characterized in that: Using the maximum likelihood method, we can get the random censored sample with the indicator variable δ i and Z i The mixed distribution likelihood function is: Among them, the parameter t represents the observed value of life t i ′ parameter space; z represents the indicator variable Z i The parameter space formed.

7. A system reliability assessment and verification method based on lifetime mixed distribution according to claim 1 or 6, characterized in that: In step 3, the initial value p of the mixing weight is given 0 and the initial value of the parameter θ 0 ,; The calculation process of the mth iteration is as follows: The parameter value obtained at the m-1th iteration is θ (m-1) and p (m-1) , solve the log-likelihood function with parameter value θ (m-1) 、p ( m- 1) The expected value Q(θ, p; θ (m-1) , p (m-1) ): in Represents the parameter value θ obtained when the value of the mixed distribution likelihood function is substituted into the m-1th iteration (m-1) 、p (m-1) The mathematical expectation of The indicator variable Z i The expected value is solved; the indicator variable Z i The expected value of is expressed by the proportion of each sub-distribution in the overall distribution. When the observed value is the complete failure data t i or randomly censored data c i The expected value is different when the jth sub-distribution is solved at the mth iteration. and parameters When the indicator variable Z i The expected value of is expressed as: Among them, the parameters represents the known mixing weight value of the j-th sub-distribution obtained at the m-1-th iteration, represents the known parameter value of the jth subdistribution obtained at the m-1th iteration; parameter represents the known mixing weight value of the lth sub-distribution obtained at the m-1th iteration, θ l (m-1) represents the known parameter value of the lth subdistribution obtained at the m-1th iteration, and θ l (m-1) It is named the parameter of the lth subdistribution because it is in the summation term in the denominator, distinguishing it from the parameter that needs to be solved for the jth subdistribution; f j and f l , R j and R l The naming rules are the same as above.

8. The system reliability evaluation and verification method based on lifetime mixed distribution according to claim 7 is characterized in that: In step 3, we maximize the expected value function to find θ (m) and p (m) , so that Q(θ (m) , p (m) θ (m-1) , p (m-1) ) = maxQ(θ, p; θ (m-1) , p (m-1) ); Solve the expected value to obtain the mixed distribution weight value of the j-th sub-distribution at the m-th iteration for: At the same time, we get the parameter value of the jth sub-distribution of the mixed distribution model at the mth iteration for: in Include Parameters 9. The system reliability evaluation and verification method based on lifetime mixed distribution according to claim 1 is characterized in that: In step 4, the probability density function of the equipment system mixed distribution is:

10. The system reliability evaluation and verification method based on lifetime mixed distribution according to claim 9, characterized in that: Determine the reliability function for: A fixed step size S is selected, and the reliability value of the equipment system at each step node is calculated through the reliability function and a reliability curve is drawn. It is also calculated whether the reliability value of the equipment system exceeds the specified threshold after the specified extended maintenance cycle.

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