Complex equipment elasticity reliability evaluation method considering fault and repair factors
Through dynamic Bayesian networks and multi-level topological directed graph models, combining fault and repair data, the elasticity and reliability of complex equipment is evaluated, and the problem of failure and repair factors in the existing technology is solved, achieving more accurate complexity Equipment reliability assessment.
Patent Information
- Application Number
- CN202411345827.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-09-26
- Publication Date
- 2025-01-03
AI Technical Summary
The prior art cannot accurately evaluate the elasticity and reliability of complex equipment, and does not consider the coupling relationship between failure and repair factors, resulting in inaccurate calculations and unreasonable decision-making.
Using dynamic Bayesian network and multi-level topological directed graph model, a fault propagation attribute and repair probability model is constructed through historical fault and repair data analysis, and combining dynamic propagation attributes and conditional probability tables to evaluate the elasticity and reliability of complex equipment.
Accurate quantitative evaluation of the elastic reliability of complex equipment is achieved, taking into account the impact of fault propagation mechanism and repair factors, and improving the accuracy and practicality of the evaluation results.
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Figure CN119249736A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of complex equipment reliability assessment, and in particular to a complex equipment elastic reliability assessment method taking failure and repair factors into consideration. Background Art
[0002] As an important carrier of high-end manufacturing, complex equipment is usually composed of many functional components. Its notable features are multi-level structures, complex causal relationships between components, and strong coupling. This will make complex equipment have the characteristics of nonlinearity, strong coupling, and high uncertainty, resulting in an increased probability of system failure, ultimately causing huge losses or even catastrophic consequences. At present, service-oriented manufacturing with equipment reliability assessment and maintenance as the core has become popular, and it is of great significance to study the elastic reliability assessment of complex equipment. However, the traditional reliability assessment method is mainly based on the historical failure data of components, combined with the basic method of probability theory, to statically calculate the reliability of components. Although it has certain significance for the reliability assessment of complex equipment, it has considerable limitations and often cannot quantitatively assess the elastic reliability of complex equipment based on historical data.
[0003] Resilient reliability refers to the ability of complex equipment to quickly recover and complete specified functions within a specified time and under specified conditions after being subjected to a destructive event. Existing scholars have not considered the coupling relationship between failure and repair factors in reliability assessment, and are unable to quantitatively assess the elastic reliability of complex equipment, which often leads to inaccurate calculations or assessments and unreasonable decisions. Therefore, a complex equipment elastic reliability assessment method that considers failure and repair factors is needed. Summary of the invention
[0004] The present invention aims to provide a complex equipment elastic reliability evaluation method taking failure and repair factors into consideration.
[0005] The technical solution adopted by the present invention is:
[0006] A method for evaluating the elastic reliability of complex equipment taking into account failure and repair factors, the method comprises the following steps: Step 1: prior parameter calculation, the purpose of which is that the prior parameters will be used as input parameters of a dynamic Bayesian network and dynamic propagation properties.
[0007] Step 1.1: Divide the complex equipment into modules according to the module division method, collect the historical failure time interval data and maintenance time data of the complex equipment, apply the probability density distribution to perform correlation analysis on the failure data and maintenance data, and preliminarily determine the failure probability density distribution function f(t) and the repair probability density distribution function r(t). The relationship between the probability density distribution function f(t) and the cumulative failure function F(t) and the reliability R(t) is shown in formula (1).
[0008]
[0009] Where t is the running time of the complex equipment.
[0010] Step 1.2: Use the Kolmogorov-Smirnov (KS) test method to test the goodness of fit of the distribution model. Formula (2) is the test formula. When the significance level γ = 0.05, the critical value D of the KS test is obtained by looking up the table. u,γ , if D u <D u,γ , then the test passes and the null hypothesis is accepted; otherwise, the null hypothesis is rejected.
[0011] D u =maxF u (t)-F(t)(2)
[0012] Where u is the number of fault data counted; F(t) is the corresponding distribution function calculated; F u (t) is the empirical distribution function,
[0013] Step 2: Establish a multi-level topological directed graph model. The purpose of this step is to use the model as a structural model of a dynamic Bayesian network.
[0014] Step 2.1 constructs the direct impact matrix Y based on the associated fault data, and further calculates the standardized direct impact matrix X, as described in formula (3) and formula (4).
[0015]
[0016] Among them, y ij For node v i The node v is caused by a fault j The frequency of failure; n is the number of rows in the matrix; It directly affects the maximum value of the row sum of matrix Y.
[0017] Step 2.2 Construct the system comprehensive influence matrix T through formula (5), construct the overall influence matrix H through formula (6), and solve the system reachability matrix M through formula (7) r ,The purpose of this step is to reduce the complexity of calculating the ,reachable matrix in the multi-level reconstruction of the ,model.
[0018]
[0019] Among them, E is the k-order unit matrix.
[0020] H=T+E(6)
[0021]
[0022] Among them, m ij is the reachability matrix M r The value of each element in; n is the number of rows in the matrix.
[0023] Step 2.3 decomposes the reachability matrix and divides the complex equipment into a multi-level structure.
[0024] The reachability matrix M r All the columns with 1 in the i-th row are called functional components v i The reachable set R i , all rows with 1 in the i-th column are called the predecessor set S i , as described in formula (8). If the public set C i =R i ∩S i =R i If holds, then in the reachable matrix M r Remove row i and column i from the component v i It is a termination component and is located at the top layer L1.
[0025] Repeat this step until all components are removed to achieve multi-level partitioning of the system.
[0026]
[0027] Step 2.4 extracts the skeleton matrix according to the construction principle and introduces virtual nodes to appropriately adjust the model structure. The purpose of this step is to avoid non-standard hierarchical structures such as jump propagation between adjacent levels and obtain a standard multi-level topological model.
[0028] Construction principle 1: Form a lower triangular matrix based on the order of removal;
[0029] Construction principle 2: Remove parameters that have binary relationships that are beyond the level or that have binary relationships themselves (i.e., m ij =0).
[0030] Step 3: Calculation of dynamic fault propagation attributes. The purpose of this step is to use the dynamic propagation attributes as input parameters of the conditional probability table in the dynamic Bayesian network.
[0031] Step 3.1 Adjacency matrix A = [a ij ] n×n It is equivalent to the direct influence matrix Y in formula (3).
[0032] Step 3.2 constructs the transpose of the adjacency matrix and calculates the initial state transfer matrix Q = [q ij ] n×n .
[0033]
[0034] Among them, q ij is the value of each element in the initial state transfer matrix; a ij To directly affect the values of the corresponding elements in matrix Y.
[0035] Step 3.3 Use the vector (1 / n, 1 / n, ..., 1 / n) to replace the 0 element in the initial state transfer matrix, and establish a new state transfer matrix Q'=[q ij '] n×n .
[0036] q ij '=q ij +η(1 / n),(i=1,2,…n; j=1,2,…n) (10)
[0037] in, I(v i ) represents the incoming link set of a component.
[0038] Step 3.4 Based on the hierarchical division, define the n-dimensional vector pr = [v1, v2, ..., v n ] T , the sub-vector is the fault impact degree of each component, which is calculated by formula (11).
[0039]
[0040] Among them, pr x+1 and pr x Respectively represent the fault impact vectors of each system component obtained at the (x+1)th and xth iterations; d is the damping factor; e is the natural base, which is 2.71818. Given the initial value The initial value will not affect the final iteration and convergence results of formula (11). Set the iteration convergence threshold ε = 0.0001. When the iteration result satisfies pr x+1 -pr x When <ε, the iteration ends.
[0041] Step 3.5 calculates the fault propagation impact value of complex equipment according to formula (12).
[0042]
[0043] Among them, pr(v i ) and pr(v j ) represents the node failure impact value.
[0044] Step 3.6 calculates the dynamic propagation properties according to formula (13).
[0045]
[0046] in, is the cumulative probability density function of each node, which is calculated by formula (1).
[0047] Step 4: Apply the conditional probability table.
[0048] When a component of a complex system fails, it is often considered to be in a failed state, that is, a two-state system (normal = 0 and failure = 1). The parameters of the dynamic Bayesian network will be set according to the following rules:
[0049] Rule 1: If component n c If all parent nodes of are in a faulty state at time t, then the probability that the component is in a faulty state at the same time step = 1 (i.e., P(n c t=1)=1). If any component in the path fails, the path will be interrupted;
[0050] Rule 2: If component n c The probability that a component has been in a faulty state for k time slices before time t and that it recovers to an operational state is given by the repair probability density distribution function r(k);
[0051] Rule 3: If component n c If a component has been in normal state for k time slices before time t, then the probability that the component stops serving at time t is given by the failure probability density distribution function f(k).
[0052] Step 5: Applying evidence information. The purpose of this step is to use the evidence information as an input parameter of the evidence node in the dynamic Bayesian network.
[0053] Evidence information is used to simulate actual scenarios and evaluate the resilience of the entire system. Evidence information, such as repair degree, component status, and fault information, is applied to a specific subset of dynamic Bayesian network components. The repair status is derived from the repair probability and expert experience information.
[0054] Step 6: Elastic reliability assessment.
[0055] Based on the steps 1 to 5, the elastic reliability curve is obtained by formula (14). The present invention mainly studies the elastic change of the reliability of complex systems due to fault factors, which can be quantified as t0 to t k The loss of system reliability over time is characterized by calculating the difference between the system reliability curve and its initial value.
[0056]
[0057] Where F0 is the cumulative probability of initial failure, t kIndicates the end time of the complex equipment operation.
[0058] The advantages and positive effects of the present invention are: the present invention fully considers the influence of the fault propagation mechanism and the repair factor, and according to the above method, the elastic reliability of any complex equipment can be evaluated. Therefore, the elastic reliability evaluation method of complex equipment considering fault and repair factors proposed by the present invention is more practical. BRIEF DESCRIPTION OF THE DRAWINGS
[0059] Figure 1 Reliability assessment process for complex equipment flexibility.
[0060] Figure 2 Dynamic Bayesian network structure.
[0061] Figure 3 Conditional probability tables are applied.
[0062] Figure 4 Schematic diagram of elastic reliability curve. DETAILED DESCRIPTION
[0063] In order to further understand the content, features and effects of the present invention, a typical complex equipment representative CNC machine tool is taken as an example and described in detail with the accompanying drawings as follows:
[0064] See also Figure 1 to Figure 4 The present invention provides a complex equipment elastic reliability evaluation method considering failure and repair factors, including the following steps: Step 1: Prior parameter calculation, see Figure 1 The purpose of this step is that the prior parameters will be used as input parameters of the dynamic Bayesian network and dynamic propagation properties.
[0065] 1.1 According to the module division method of complex equipment, CNC machine tools are divided into 9 functional components, namely electrical system (E), hydraulic system (H), feed system (F), CNC system (NC), cooling system (C), spindle system (S), lubrication system (L), basic structure (A), and tool magazine (M). The historical fault time interval data and maintenance time data of CNC machine tools are collected, and the probability density distribution is used to perform correlation analysis on the fault data and maintenance data, and the fault probability density distribution function f(t) and the repair probability density distribution function r(t) are preliminarily determined. Combined with practical applications and research, it is assumed that the components of CNC machine tools obey the Weibull distribution, and the relationship between its probability density distribution function f(t) and the cumulative failure function F(t) and reliability R(t) is shown in formula (1). Most patents have detailed descriptions of the parameter estimation method of the distribution function, and parameter estimation mostly uses the least squares method or the maximum likelihood method.
[0066]
[0067] Among them, t is the running time of the complex equipment; η is the scale parameter, and α is the shape parameter.
[0068] 1.2 The Kolmogorov-Smirnov (KS) test method is used to test the goodness of fit of the distribution model. Formula (2) is the test formula. When the significance level γ = 0.05, the critical value D of the KS test is obtained by looking up the table. u,γ , if D u <D u,γ , then the test passes and the null hypothesis is accepted; otherwise, the null hypothesis is rejected.
[0069] D u =maxF u (t)-F(t)(2)
[0070] Where u is the number of fault data counted; F(t) is the corresponding distribution function calculated; F u (t) is the empirical distribution function,
[0071] Step 2: Establish a multi-level topological directed graph model, see Figure 2 The purpose of this step is to use the multi-level topological directed graph model as a structural model of the dynamic Bayesian network.
[0072] 2.1 According to the fault data, combined with correlation analysis, the causal relationship of the associated faults of CNC machine tools can be obtained, and the direct influence matrix Y is constructed. On this basis, the standardized direct influence matrix X is constructed, as described in formulas (3) and (4).
[0073]
[0074]
[0075] Among them, y ij For node v i The node v is caused by a fault j The frequency of failure; n is the number of rows in the matrix; It directly affects the maximum value of the row sum of matrix Y.
[0076] 2.2 The comprehensive impact matrix is constructed by formula (5), the overall impact matrix is constructed by formula (6), and the system reachability matrix M is solved by formula (7) r ,The purpose of this step is to reduce the complexity of calculating the ,reachable matrix in the multi-level reconstruction of the ,model.
[0077]
[0078] Among them, E is the n-order unit matrix.
[0079] H=T+E(6)
[0080]
[0081] 2.3 Decompose the reachable matrix and divide the complex equipment into a multi-level structure.
[0082] The reachability matrix M r All the columns with 1 in the i-th row are called functional components v i The reachable set R i , all rows with 1 in the i-th column are called the predecessor set S i , as described in formula (8). If the public set C i =R i ∩S i =R i If holds, then in the reachable matrix M r Remove row i and column i from the component v i It is a termination component and is located at the top layer L1.
[0083] Repeat this step until all components are removed to achieve multi-level partitioning of the system.
[0084]
[0085] 2.4 Extract the skeleton matrix according to the construction principle, introduce virtual nodes to appropriately adjust the model structure. The purpose of this step is to avoid non-standard hierarchical structures such as jump propagation between adjacent levels and obtain a standard multi-level topological model.
[0086] Construction principle 1: Form a lower triangular matrix based on the order of removal;
[0087] Construction principle 2: Remove parameters that have binary relationships that are beyond the level or that have binary relationships themselves (i.e., m ij =0).
[0088] Step 3: Calculation of dynamic fault propagation attributes. The purpose of this step is to use the dynamic propagation attributes as input parameters of the conditional probability table in the dynamic Bayesian network.
[0089] 3.1 Adjacency matrix A = [a ij ] n×n It is equivalent to the direct impact matrix in formula (3).
[0090] 3.2 Construct the transpose of the adjacency matrix and calculate the initial state transfer matrix Q = [q ij ] n×n .
[0091]
[0092] Among them, qij is the value of each element in the initial state transfer matrix; a ij To directly affect the values of the corresponding elements in matrix Y.
[0093] 3.3 Use the vector (1 / n, 1 / n, ..., 1 / n) to replace the 0 element in the initial state transfer matrix, and establish a new state transfer matrix Q' = [q ij '] n×n .
[0094] q ij '=q ij +η(1 / n),(i=1,2,…n; j=1,2,…n) (10)
[0095] in, I(v i ) represents the incoming link set of a component.
[0096] 3.4 According to the hierarchical division, define the n-dimensional vector pr = [v1, v2, ..., v n ] T , the sub-vector is the fault impact degree of each component, which is calculated by formula (11).
[0097]
[0098] Among them, pr x+1 and pr x Respectively represent the fault influence vectors of each component obtained at the (x+1)th and xth iterations; d is the damping factor; e is the natural base, which is 2.71818. Given the initial value The initial value will not affect the final iteration and convergence results of formula (11). Set the iteration convergence threshold ε = 0.0001. When the iteration result satisfies pr x+1 -pr x When <ε, the iteration ends.
[0099] 3.5 Calculate the fault propagation impact value of each subsystem of the CNC machine tool according to formula (12).
[0100]
[0101] Among them, pr(v i ) and pr(v j ) represents the node failure impact value.
[0102] 3.6 Calculate the dynamic propagation properties according to formula (13).
[0103]
[0104] in, is the cumulative probability density function of each node, which is calculated by formula (1).
[0105] Step 4: Apply the conditional probability table.
[0106] When a component of a complex system fails, it is often considered to be in a failed state, that is, a two-state system (normal = 0 and failure = 1). The parameters of the dynamic Bayesian network will be set according to the following rules:
[0107] Rule 1: If component n c If all parent nodes of are in a faulty state at time t, then the probability that the component is in a faulty state at the same time step = 1 (i.e., P(n c t=1)=1). If any component in the path fails, the path will be interrupted;
[0108] Rule 2: If component n c The probability that a component has been in a faulty state for k time slices before time t and that it recovers to an operational state is given by the repair probability density distribution function r(k);
[0109] Rule 3: If component n c If a component has been in normal state for k time slices before time t, then the probability that the component stops serving at time t is given by the failure probability density distribution function f(k).
[0110] See also Figure 3 , combined with the probability density function obtained in step one and the dynamic fault propagation attribute obtained in step three, a conditional probability table is formulated according to the three rules and applied to the dynamic Bayesian network structure.
[0111] Step 5: Applying evidence information. The purpose of this step is to use the evidence information as an input parameter of the evidence node in the dynamic Bayesian network.
[0112] Evidence information is used to simulate actual scenarios and evaluate the resilience of the entire system. Evidence information is applied to a specific subset of dynamic Bayesian network components, such as repair degree, component status, and fault information. The repair status is derived from the repair probability and expert experience information. Evidence propagation schemes for various scenarios are set according to the on-site maintenance strategy, as shown in Table 1.
[0113] Table 1 Settings for evidence propagation in multiple scenarios
[0114]
[0115] Step 6: Elastic reliability assessment.
[0116] See also Figure 4, based on the steps 1 to 5, the elastic reliability curve is obtained by formula (14). Figure 4 The reliability curve of CNC machine tools evaluated by the traditional method is compared with the elastic reliability curve obtained by the method described in the present invention. The present invention mainly studies the elastic change of the reliability of complex systems due to fault factors, which can be quantified as t0 to t k The loss of system reliability over time is characterized by calculating the difference between the system reliability curve and its initial value.
[0117]
[0118] Where F0 is the cumulative probability of initial failure, t k Indicates the end time of the complex equipment operation.
[0119] It can be seen from the above technical solutions that the present invention provides a method for evaluating the elastic reliability of complex equipment that takes into account fault and repair factors. The method obtains the historical fault information and repair information of the complex equipment, calculates the fault accumulation function, reliability function, and dynamic fault propagation properties, considers the fault propagation mechanism and evidence propagation technology, and establishes a dynamic Bayesian network based on a multi-level fault topology directed graph model to simulate actual scenarios to obtain the elastic reliability evaluation curve of the complex equipment. The method for evaluating the elastic reliability of complex equipment that takes into account fault and repair factors provided by the present invention has smaller errors than traditional reliability evaluation methods, more accurate reliability evaluation results, and can more accurately and quantitatively evaluate the elastic reliability of complex equipment.
[0120] The specific implementation provided above is only an example under the general concept of the present invention and does not constitute a limitation on the protection scope of the present application. For those skilled in the art, any other implementation expanded based on the scheme of the present invention without creative work belongs to the protection scope of the present invention.
Claims
1. A method for evaluating the elastic reliability of complex equipment considering failure and repair factors, the method comprising the following steps: Step 1: calculation of a priori parameters, the a priori parameters will be used as input parameters of a dynamic Bayesian network and a dynamic propagation attribute; Step 1.1: Divide the complex equipment into modules according to the module division method, collect the historical failure time interval data and maintenance time data of the complex equipment, apply the probability density distribution to perform correlation analysis on the failure data and maintenance data, and preliminarily determine the failure probability density distribution function f(t) and the repair probability density distribution function r(t). The relationship between the probability density distribution function f(t) and the cumulative failure function F(t) and the reliability R(t) is shown in formula (1); in, t is the running time of the complex equipment; Step 1.2: Use the Kolmonov-Smirnov KS test method to test the goodness of fit of the distribution model. Formula (2) is the test formula. When the significance level γ = 0.05, the critical value D of the KS test is obtained by looking up the table. u,γ , if D u <D u,γ , then the test passes and the null hypothesis is accepted, otherwise the null hypothesis is rejected; D u =max|F u (t)-F(t)| (2) Where u is the number of fault data counted; F(t) is the corresponding distribution function calculated; F u (t) is the empirical distribution function, Step 2: Establishing a multi-level topological directed graph model, wherein the multi-level topological directed graph model is used as a structural model of a dynamic Bayesian network; Step 2.1, construct the direct impact matrix Y based on the associated fault data, and calculate the standardized direct impact matrix X, as described in formula (3) and formula (4); Among them, y ij For node v i The node v is caused by a fault j The frequency of failure; n is the number of rows in the matrix; To directly affect the maximum value of the row sum of matrix Y; Step 2.2, construct the system comprehensive influence matrix T through formula (5), construct the overall influence matrix H through formula (6), and solve the system reachability matrix M through formula (7) r , reducing the complexity of calculating the reachable matrix in the multi-level reconstruction of the model; Among them, E is the k-order unit matrix; H=T+E (6) Among them, m ij is the reachability matrix M r The value of each element in; n is the number of rows in the matrix; Step 2.3 decomposes the reachable matrix and divides the complex equipment into a multi-level structure; The reachability matrix M r All the columns with 1 in the i-th row are called functional components v i The reachable set R i , all rows with 1 in the i-th column are called the predecessor set S i , as described in formula (8); if the public set C i =R i ∩S i =R i If holds, then in the reachable matrix M r Remove row i and column i from the component v i It is a termination component and is located at the top layer L1. Repeat this step until all components are removed to achieve multi-level division of the system. Step 2.4 extracts the skeleton matrix according to the construction principle, introduces virtual nodes to properly adjust the model structure, avoids non-standard hierarchical structures such as jump propagation between adjacent levels, and obtains a standard multi-level topological model; Construction principle 1: Form a lower triangular matrix based on the order of removal; Construction principle 2: Remove parameters that have binary relationships that are beyond the level and that have binary relationships themselves, i.e. m ij =0; Step 3: Calculate dynamic fault propagation attributes. The purpose of this step is to use the dynamic propagation attributes as input parameters of the conditional probability table in the dynamic Bayesian network. Step 3.1 Adjacency matrix A = [a ij ] n×n Equivalent to the direct impact matrix Y in formula (3); Step 3.2 constructs the transpose of the adjacency matrix and calculates the initial state transfer matrix Q = [q ij ] n×n ; Among them, q ij is the value of each element in the initial state transfer matrix; a ij To directly affect the values of the corresponding elements in the matrix Y; Step 3.3 Use the vector (1 / n, 1 / n, ..., 1 / n) to replace the 0 element in the initial state transfer matrix, and establish a new state transfer matrix Q'=[q ij '] n×n ; q ij '=q ij +η(1 / n),(i=1,2,…n;j=1,2,…n) (10) in, I(v i ) represents the incoming link set of the component; Step 3.4 Based on the hierarchical division, define the n-dimensional vector pr = [v1, v2, ..., v n ] T , the sub-vector is the fault impact degree of each component, which is calculated by formula (11); Among them, pr x+1 and pr x Respectively represent the fault impact vectors of each system component obtained at the (x+1)th and xth iterations; d is the damping factor; e is the natural base; given initial value The initial value will not affect the final iteration and convergence results of formula (11); the iteration convergence threshold ε is set to 0.0001. When the iteration result satisfies |pr x+1 -pr x When |<ε, the iteration ends; Step 3.5: Calculate the fault propagation impact value of complex equipment according to formula (12); Among them, pr(v i ) and pr(v j ) represents the node failure impact value; Step 3.6 Calculate the dynamic propagation properties according to formula (13); in, is the cumulative probability density function of each node, which is calculated by formula (1); Step 4: Apply the conditional probability table; When a component of a complex system fails, it is often considered to be in a failed state, that is, a two-state system: normal = 0 and fault = 1. The parameters of the dynamic Bayesian network will be set according to the following rules: Rule 1: If component n c If all the parent nodes of are in a faulty state at time t, then the probability that the component is in a faulty state at the same time step = 1; if any component in the path fails, the path will be interrupted; Rule 2: If component n c The probability that a component has been in a faulty state for k time slices before time t and that it recovers to an operational state is given by the repair probability density distribution function r(k); Rule 3: If component n c If a component has been in normal state for k time slices before time t, then the probability of the component being out of service at time t is given by the failure probability density distribution function f(k); Step 5: applying evidence information, which will be used as input parameters of evidence nodes in the dynamic Bayesian network; Evidence information is used to simulate actual scenarios and evaluate the resilience of the entire system. Evidence information is applied to a specific subset of dynamic Bayesian network components, including repair degree, component status, and fault information. The repair status is derived from the repair probability and expert experience information. Step 6: Elastic reliability assessment; The elastic reliability curve is obtained by formula (14); based on the elastic change of the failure factor on the reliability of the complex system, it is quantified as t0 to t k The loss of system reliability over time is characterized by calculating the difference between the system reliability curve and its initial value; Where F0 is the cumulative probability of initial failure, t k Indicates the end time of the complex equipment operation.