CN112836366B - System reliability parameter estimation method based on component dependent life data - Google Patents

System reliability parameter estimation method based on component dependent life data Download PDF

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CN112836366B
CN112836366B CN202110120673.0A CN202110120673A CN112836366B CN 112836366 B CN112836366 B CN 112836366B CN 202110120673 A CN202110120673 A CN 202110120673A CN 112836366 B CN112836366 B CN 112836366B
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杨乐昌
王丕东
汪晨星
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University of Science and Technology Beijing USTB
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Abstract

The invention discloses a system reliability parameter estimation method based on component dependent life data, which comprises the following steps: dividing an original system reliability model into a plurality of mutually independent sub-models by adopting a model dividing technology; decoupling the whole likelihood function of the system into products of a plurality of mutually independent evidence likelihood functions based on the model segmentation result; constructing a general expression of each evidence likelihood function; drawing a evidence reasoning diagram according to the structural relation of the system reliability model, and compiling all evidence quantities by utilizing the evidence reasoning diagram; calculating the overall likelihood function of the system; and estimating and predicting the system reliability parameters. The method can solve the parameter estimation problem of the system reliability model containing the dependent data, and has universal applicability in the application range including fault trees, event trees, bayesian network models and the like.

Description

System reliability parameter estimation method based on component dependent life data
Technical Field
The invention relates to the technical field of system reliability research, in particular to a system reliability parameter estimation method based on component dependent life data.
Background
At present, a great challenge in estimating critical parameters of the reliability of a complex system is the lack of test data. In practical engineering, it is costly to perform multiple independent repeatability tests on the entire system. To reduce the cost of the test, and to reduce the number of tests, multiple sensors are often installed simultaneously on different parts of the system to collect test data, which is then non-independent. A difficulty in reliability studies using dependent lifetime data is the likelihood function analytical solution. In this case, the efficient and convenient processing of the correlation of the data is of great importance for the accurate estimation and prediction of critical parameters of the system reliability.
Aiming at the dependent service life data, a reliability parameter estimation method based on a cut set is provided, and the method uses a generalized cut set to determine the evidence quantity corresponding to the dependent data, so that the reliability analysis problem of a fault tree model containing the dependent data can be effectively processed. In addition, there is a likelihood function construction method based on non-independent fault time data, which lists all possible events consistent with the non-independent data based on the reliability structure of the system, calculates their contribution to the likelihood function, and further obtains the likelihood function of the whole system. However, these methods are only applicable to specific reliability models (e.g., fault trees), cannot be used for systems with common components, and have no general applicability.
Disclosure of Invention
In view of the above problems, the present invention provides a system reliability parameter estimation method based on component dependent lifetime data, which can be applied to parameter estimation problems of system reliability models including multiple dependent lifetime data (such as failure time data, left-cut tail data, right-cut tail data), and has universal applicability in a plurality of types of system reliability models including failure trees, event trees, bayesian networks, etc.
In order to solve the technical problems, the embodiment of the invention provides the following scheme:
a system reliability parameter estimation method based on component dependent life data comprises the following steps:
s1, dividing an original system reliability model into a plurality of mutually independent sub-models by adopting a model division technology;
s2, decoupling the whole likelihood function of the system into a product of a plurality of mutually independent evidence likelihood functions based on a model segmentation result;
s3, constructing a general expression of each evidence likelihood function;
s4, drawing a evidence reasoning diagram according to the structural relation of the system reliability model, and compiling all evidence by utilizing the evidence reasoning diagram;
s5, calculating a system overall likelihood function;
s6, estimating and predicting the system reliability parameters.
Preferably, the step S1 specifically includes:
dividing an original system reliability model into a plurality of mutually independent sub-models by adopting a model dividing technology based on d-separation according to the position distribution of the observed nodes;
the model without the shared node is divided according to the d-separation structure position in the graph model, and the model with the shared node is not divided.
Preferably, the step S2 specifically includes:
decoupling the overall likelihood function of the system into the product of a plurality of mutually independent evidence likelihood functions according to the mutual independence among the submodels:
where N is the node and w is the number of observed nodes in the system.
Preferably, the step S3 specifically includes:
the general expression for constructing each evidence likelihood function is as follows:
wherein N is j Is a inferable non-observation underlying node; n (N) k Is an observation node capable of reasoning;
the expression considers the situations of fault time data, left tail-cutting data and right tail-cutting data; wherein if the component fails before the first test, then a left-hand tail-out is considered to have occurred; if the component has not failed at the last test, then a right cut is considered to have occurred.
Preferably, the step S4 specifically includes:
drawing a evidence reasoning diagram based on a unit step function according to the structural relation of the system reliability model, and compiling all evidence by utilizing the evidence reasoning diagram;
wherein the unit step function is defined as follows:
for the observed node N, a step function H (t-t N ) When t is less than t N When H (t-t) N ) =0, the failure probability of the corresponding node at time t is 0; when t is greater than or equal to t N When H (t-t) N ) =1, the failure probability of the corresponding node at time t is 1; by a function H (t-t N ) I.e. the cumulative distribution function F of the node N can be replaced N (t)。
Preferably, the step S6 specifically includes:
the posterior distribution of unknown key parameters in the system reliability model is calculated using the following formula:
where pi (θ) is a priori distribution of the unknown key parameters, L (d|θ) is a likelihood function, and p (θ|d) is a posterior distribution of the unknown key parameters including the unknown model parameters and the unknown distribution parameters.
The technical scheme provided by the embodiment of the invention has the beneficial effects that at least:
in the embodiment of the invention, an original system reliability model is split into a plurality of sub-models by using a model segmentation technology based on d-separation, a system overall likelihood function is decoupled into products of a plurality of mutually independent evidence likelihood functions based on a model segmentation result, the evidence likelihood functions are further constructed, a evidence reasoning diagram is drawn, all evidence quantities are compiled, and the system overall likelihood function is calculated, so that effective estimation and prediction of system reliability key parameters are realized. The method can solve the parameter estimation problem of the system reliability model containing the dependent data, and has universal applicability in the application range including fault trees, event trees, bayesian network models and the like.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings required for the description of the embodiments will be briefly described below, and it is apparent that the drawings in the following description are only some embodiments of the present invention, and other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a flow chart of a system reliability parameter estimation method based on component dependent lifetime data provided by an embodiment of the invention;
FIG. 2 is a schematic diagram of model segmentation provided by an embodiment of the present invention;
FIG. 3 is a schematic diagram of a Bayesian network with shared nodes provided by an embodiment of the present invention;
FIG. 4 is a schematic diagram of a unit step function provided by an embodiment of the present invention;
FIG. 5 is a schematic diagram of a Bayesian network model provided by an embodiment of the present invention;
FIG. 6 is a schematic diagram of a d-separation sub-network model provided by an embodiment of the present invention;
fig. 7 is a schematic diagram of reasoning provided by an embodiment of the present invention.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the present invention more apparent, the embodiments of the present invention will be described in further detail with reference to the accompanying drawings.
The embodiment of the invention provides a system reliability parameter estimation method based on component dependent service life data, as shown in fig. 1, comprising the following steps:
s1, dividing an original system reliability model into a plurality of mutually independent sub-models by adopting a model division technology;
in the step, according to the position distribution of the observed node, an original system reliability model is divided into a plurality of mutually independent sub-models by adopting a model division technology based on d-separation; the model without the shared node is divided according to the d-separation structure position in the graph model, and the model with the shared node is not divided.
FIG. 2 shows an example of model segmentation, where the dashed box nodes are observed nodes with d-separation structure, based on graph theory d-separation theory, the original model can be decomposed into three sub-models. The common node is a node including a plurality of child nodes, and is exemplified in fig. 3, in which node S 1 And node C 1 Are all common nodes.
S2, decoupling the whole likelihood function of the system into a product of a plurality of mutually independent evidence likelihood functions based on a model segmentation result;
in the step, according to the mutual independence among the submodels, the whole likelihood function of the system is decoupled into the product of a plurality of mutually independent evidence likelihood functions:
where N is the node and w is the number of observed nodes in the system.
S3, constructing a general expression of each evidence likelihood function;
in this step, a general expression for each evidence likelihood function is constructed as follows:
wherein N is j Is a inferable non-observation underlying node; n (N) k Is an observation node capable of reasoning;
the expression considers the situations of life data such as fault time data, left tail cutting data, right tail cutting data and the like; wherein if the component fails before the first test, then a left-hand tail-out is considered to have occurred; if the component has not failed at the last test, then a right cut is considered to have occurred.
S4, drawing a evidence reasoning diagram according to the structural relation of the system reliability model, and compiling all evidence by utilizing the evidence reasoning diagram;
in this step, considering the structural relation of the system reliability model, drawing a evidence reasoning diagram based on the unit step function shown in fig. 4, and compiling all evidence quantities.
Wherein the unit step function is defined as follows:
for the observed node N, a step function H (t-t N ) When t is less than t N When H (t-t) N ) =0, the failure probability of the corresponding node at time t is 0; when t is greater than or equal to t N When H (t-t) N ) =1, the failure probability of the corresponding node at time t is 1; by a function H (t-t N ) I.e. the cumulative distribution function F of the node N can be replaced N (t)。
S5, calculating a system overall likelihood function;
s6, estimating and predicting the system reliability parameters.
In this step, the posterior distribution of the unknown key parameters in the system reliability model is calculated using the following formula:
where pi (θ) is a priori distribution of the unknown key parameters, L (d|θ) is a likelihood function, and p (θ|d) is a posterior distribution of the unknown key parameters including the unknown model parameters and the unknown distribution parameters.
The application of the invention in parameter estimation is described below in connection with the system reliability model shown in fig. 5. Wherein S is 1 ,S 2 ,C 6 Is unknown in structural relation, C 1 ,S 3 Is of parallel structure, C 2 ,C 3 Is of a series structure, C 4 ,C 5 Is in a parallel structure. There are three observed nodes S in this model 0 ,S 1 ,C 5 Suppose that a set of observations is obtained in a single test Left truncated data) and the failure time data distribution model of each component is known. Wherein the prior distribution pi (theta) of the target parameter, the conditional probability table parameter is set as follows:
Pr(S 0 ＝0|S 1 ＝0,S 2 ＝0,C 6 ＝0)＝p 000 Pr(S 0 ＝0|S 1 ＝1,S 2 ＝1,C 6 ＝0)＝p 110
Pr(S 0 ＝0|S 1 ＝1,S 2 ＝0,C 6 ＝0)＝p 100 Pr(S 0 ＝0|S 1 ＝0,S 2 ＝1,C 6 ＝1)＝p 011
Pr(S 0 ＝0|S 1 ＝0,S 2 ＝1,C 6 ＝0)＝p 010 Pr(S 0 ＝0|S 1 ＝1,S 2 ＝0,C 6 ＝1)＝p 101
Pr(S 0 ＝0|S 1 ＝0,S 2 ＝0,C 6 ＝1)＝p 001 Pr(S 0 ＝0|S 1 ＝1,S 2 ＝1,C 6 ＝1)＝p 111
the first step: the original system model is first decomposed into three subsystem models (as shown in fig. 6) by model segmentation.
And a second step of: the likelihood function of the system as a whole can be decomposed into equation (5):
L(S 0 ,S 1 ,C 5 )＝L(C 5 )L(S 1 )L(S 0 |S 1 ,C 5 ) (5)
and a third step of: constructing evidence likelihood functions of each sub-model:
fourth step: drawing an inference graph (as shown in fig. 7) based on the observed data, and compiling evidence amounts:
fifth step: bringing the above amounts of evidence into formulas (6), (7) and (8) yields:
further calculating the likelihood function of the whole system:
sixth step: from a given prior distribution of parameters, the posterior distribution of any unknown key parameters (including model parameters and distribution parameters) can be calculated using equation (14):
in summary, the invention provides a system reliability parameter estimation method based on a graph model d-separation structure and a Bayesian evidence likelihood function. The method considers the dependent service life data, expands the application range of the reliability method of the general system, and improves the reliability analysis and calculation efficiency.
The foregoing description of the preferred embodiments of the invention is not intended to limit the invention to the precise form disclosed, and any such modifications, equivalents, and alternatives falling within the spirit and scope of the invention are intended to be included within the scope of the invention.

Claims (1)

1. A system reliability parameter estimation method based on component dependent lifetime data, which is characterized in that the method is used for parameter estimation of a system reliability model containing various component dependent lifetime data; the non-independent life data are experimental data collected by installing a plurality of sensors on different parts of the system, and the experimental data comprise fault time data, left tail cutting data and right tail cutting data;
the method comprises the following steps:
s1, dividing an original system reliability model into a plurality of mutually independent sub-models by adopting a model division technology;
the step S1 specifically includes:
dividing an original system reliability model into a plurality of mutually independent sub-models by adopting a model dividing technology based on d-separation according to the position distribution of the observed nodes;
for the model without the shared node, the segmentation is carried out according to the d-separation structure position in the graph model, and for the model with the shared node, the segmentation is not carried out;
s2, decoupling the whole likelihood function of the system into a product of a plurality of mutually independent evidence likelihood functions based on a model segmentation result;
the step S2 specifically includes:
decoupling the overall likelihood function of the system into the product of a plurality of mutually independent evidence likelihood functions according to the mutual independence among the submodels:
where w is the number of observed nodes in the system;
s3, constructing a general expression of each evidence likelihood function;
the step S3 specifically includes:
the general expression for constructing each evidence likelihood function is as follows:
wherein N is j Is a inferable non-observation underlying node; n (N) k Is an observation node capable of reasoning;
the expression considers the situations of fault time data, left tail-cutting data and right tail-cutting data; wherein if the component fails before the first test, then a left-hand tail-out is considered to have occurred; if the component has not failed at the last test, then a right-cut is considered to have occurred;
s4, drawing a evidence reasoning diagram according to the structural relation of the system reliability model, and compiling all evidence by utilizing the evidence reasoning diagram;
the step S4 specifically includes:
drawing a evidence reasoning diagram based on a unit step function according to the structural relation of the system reliability model, and compiling all evidence by utilizing the evidence reasoning diagram;
wherein the unit step function is defined as follows:
for the observed node N, isIt constructs a step function H (t-t) N ) When t is less than t N When H (t-t) N ) =0, the failure probability of the corresponding node at time t is 0; when t is greater than or equal to t N When H (t-t) N ) =1, the failure probability of the corresponding node at time t is 1; by a function H (t-t N ) I.e. the cumulative distribution function F of the node N can be replaced N (t)；
S5, calculating a system overall likelihood function;
s6, estimating and predicting the system reliability parameters;
the step S6 specifically includes:
the posterior distribution of unknown key parameters in the system reliability model is calculated using the following formula:
where pi (θ) is a priori distribution of the unknown key parameters, L (d|θ) is a likelihood function, and p (θ|d) is a posterior distribution of the unknown key parameters including the unknown model parameters and the unknown distribution parameters.
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