A control system reliability assessment method based on multi-level nested Copula functions

Abstract

The present invention discloses a control system reliability assessment method based on multi-level nested Copula functions, which belongs to the technical field of control system reliability assessment. The method comprises the following steps: first, analyzing life data of a control system and obtaining marginal distribution; second, calculating the Kendall rank correlation coefficient for a Copula model considering marginal distribution; third, determining the optimal Copula function of underlying variables and obtaining joint probability; fourth, establishing a Copula nested structure; and fifth, evaluating the reliability of a motor control system. The present invention provides a control system reliability assessment method based on multi-level nested Copula functions. Taking into account the fault correlation and coupling characteristics of components, subsystems and the entire system in the system, a complete bottom-up reliability assessment framework is established by combining the Kendall rank correlation coefficient with the nested Copula function, which effectively describes the fault dependency relationship between each level and improves the accuracy, computational efficiency and operability of reliability assessment of complex systems.

Description

Technical Field

[0001] The present invention relates to the technical field of control system reliability assessment, and in particular to a control system reliability assessment method based on multi-level nested Copula functions. Background Art

[0002] As motor control systems become more sophisticated, their internal structures become more complex. At the same time, fault correlation between system components (fault correlation refers to the fact that the failure of one component affects the failure rate of other components) is becoming increasingly apparent. Current control system reliability research primarily assumes that each component fails independently, without considering fault correlation within the multi-level component-subsystem-system structure. In actual engineering applications, dependencies exist between components and subsystems. Copula functions decompose a multidimensional joint distribution function into a joint expression of multiple one-dimensional marginal distribution functions, capable of describing nonlinear correlations between different variables.

[0003] Based on this, the present invention proposes a control system reliability assessment method based on multi-level nested Copula functions. The reliability of each subsystem and component is used as a low-dimensional marginal distribution function, and the reliability of the system is used as a joint distribution function. By establishing the rank correlation coefficient between different components as the basis for nested Copula analysis, the Copula function is used to comprehensively consider the fault correlation at different levels, and a reliability assessment method for a multi-level control system is given. Summary of the Invention

[0004] The purpose of the present invention is to provide a control system reliability assessment method based on multi-level nested Copula functions to solve the problem of reliability modeling and assessment of multi-level structures in current complex control systems existing in the above background technology.

[0005] To achieve the above object, the present invention provides a control system reliability assessment method based on multi-level nested Copula functions, comprising the following steps:

[0006] Step 1: Analyze the life data of the control system and calculate the marginal distribution;

[0007] Step 2: For the Copula model considering marginal distribution, calculate the Kendall rank correlation coefficient;

[0008] Step 3: Determine the optimal Copula function of the underlying variables and calculate the joint probability;

[0009] Step 4: Create a Copula nested structure;

[0010] Step 5: Evaluate the reliability of the motor control system.

[0011] Preferably, step one is specifically:

[0012] Life / pseudo-life data of components at the bottom layer of the three-tier control system , combined with the distribution forms of different components, the maximum likelihood estimation is used to estimate the parameters, which is:

[0013] ;

[0014] in, is the number of samples, is the parameter vector to be estimated for different distributions, It is the probability density function of different distributions, including exponential distribution, normal distribution and Weibull distribution. The component level includes degradation products and life products. The failure life of life products is The pseudo lifespan of a degraded product, which follows one of the exponential and Weibull distributions, reaches the failure threshold (the "pseudo lifespan" refers to the time it takes for the product's performance degradation to first reach the failure threshold). It obeys one of the Weibull distribution and the normal distribution, and the cumulative failure functions of each distribution are:

[0015] ;

[0016] ;

[0017] ;

[0018] in, is the mean life span of the exponential distribution, and are the scale parameter and shape parameter of the Weibull distribution, and are the location parameter and scale parameter of the normal distribution, is the cumulative distribution function of the standard normal distribution;

[0019] By solving the formula Get parameter estimates for the distribution ,in, It means to find partial derivatives.

[0020] Preferably, the three-level control system is a motor control system comprising a three-level system structure of system-subsystem-components.

[0021] Preferably, in step 2, for the Copula model considering marginal distribution, the variables are mostly nonlinearly correlated, and the Kendall rank correlation coefficient is often used to evaluate the correlation between two variables. It can measure the nonlinear correlation between two variables without assuming that the data follows a specific distribution; specifically:

[0022] Suppose there are two continuous random variables and The sample data is , for any two sample pairs and ,and , according to the order relationship, they are divided into the following three categories:

[0023] Consistent with:

[0024] ;

[0025] Inconsistent pairs:

[0026] ;

[0027] Tie Pair:

[0028] ;

[0029] The number of consistent pairs is recorded as , the number of inconsistent pairs is recorded as , then the Kendall rank correlation coefficient It is defined as the ratio of the difference between the number of consistent pairs and the number of inconsistent pairs to the number of all sample pairs. The formula is:

[0030] ;

[0031] in, It is from The number of random pairwise combinations selected from the samples.

[0032] Preferably, the Copula theory in step 3 represents a The joint distribution function of the dimension can be decomposed into a Copula function and Marginal distribution functions of different dimensions, where the Copula function is used to describe the correlation between variables. Specifically:

[0033] assumed is a one-dimensional continuous distribution function, let ,but All obey Uniform distribution in the interval, is a random variable The joint distribution function of for The joint distribution function of , then there exists a multivariate Copula function So that the joint distribution function and marginal distribution function Have the following relationship:

[0034] ;

[0035] Copula functions mainly include Archimedean Copula functions and elliptic Copula functions. Commonly used Archimedean Copula functions include Gumbel-Copula, Clayton-Copula, and Frank-Copula functions; commonly used elliptic Copula functions include Gaussian-Copula functions;

[0036] Different Copula functions are used to represent different correlations. The Akaike Information Criterion (AIC) is used to evaluate the fitting results. AIC is a standard for measuring the goodness of statistical model fitting. It evaluates the goodness of the probability density model fitting the data based on the maximum likelihood function. The calculation formula is:

[0037] ;

[0038] in, is the number of parameters in the model, is the likelihood function structure. The smaller the AIC value, the better the performance of the Copula function in fitting the original data.

[0039] Preferably, the distribution forms of the four types of Copula functions are:

[0040] ;

[0041] ;

[0042] ;

[0043] ;

[0044] in, and are two random variables, is the correlation coefficient, which is used to describe the linear correlation between two random variables. is the shape parameter.

[0045] Preferably, step four is specifically as follows:

[0046] Assume the system has subsystems, each of which contains multiple components with related relationships ;make Indicates the Subsystem For each marginal distribution, first calculate the rank correlation coefficient for all components of each subsystem, and select A pair of components Perform Copula fitting as the bottom-level variable:

[0047] ;

[0048] in, For the The joint probability distribution obtained by the penultimate layer of the subsystem is: Represents the optimal Copula function form selected by AIC. Repeat steps 2 and 3 recursively to obtain the final top-level failure probability of each subsystem. ;Finally, perform the same operation on different subsystems to obtain the system failure probability , and finally obtain the system's Copula nested structure.

[0049] Preferably, step five is specifically as follows:

[0050] Obtaining the system failure probability through recursive Copula nesting structure :

[0051] ;

[0052] The system reliability is expressed as .

[0053] Therefore, the present invention adopts the above-mentioned control system reliability assessment method based on multi-level nested Copula functions, which has the following beneficial effects:

[0054] (1) Based on the life / pseudo-life data of components in multi-level systems, a bottom-up reliability assessment method was established by introducing Copula functions to effectively describe the fault correlation between components in different levels of the system. This method fully considers the fault correlation relationship at different levels, accurately describes the overall fault structure of the product, and ensures the integrity and accuracy of the system reliability assessment;

[0055] (2) Compared with traditional multidimensional distribution, the nested Copula function method requires fewer parameters, the algorithm is simpler and more efficient, and is easy to operate and apply. In addition, this method still has strong adaptability and computational performance advantages in high-dimensional situations;

[0056] (3) The Kendall rank correlation coefficient is used to accurately capture the internal correlation structure of each hierarchical system, and the nested Copula model is used to complete the fault structure modeling of the multi-level system. This method can not only achieve a specific quantitative description of the correlation structure of complex systems, but also improve the scientific nature and interpretability of reliability assessment results.

[0057] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 This is an overall flow chart of a control system reliability assessment method based on multi-level nested Copula functions according to the present invention;

[0059] Figure 2 This is a schematic diagram of a multi-level structure of a motor control system according to an embodiment of the present invention;

[0060] Figure 3 This is a diagram of the copula nested structure of the system according to an embodiment of the present invention;

[0061] Figure 4 This is a graph showing how system reliability changes over time according to an embodiment of the present invention. DETAILED DESCRIPTION

[0062] The following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but rather merely represents selected embodiments of the present invention. All other embodiments derived by persons of ordinary skill in the art based on the embodiments of the present invention without inventive effort shall fall within the scope of protection of the present invention.

[0063] A motor control system in this embodiment Driven by a motor , power supply and control and Sensing and Monitoring The system consists of three subsystems, each of which consists of four typical components. The components are mainly divided into two categories: life-type products that can directly collect failure life and degradation products that need to indirectly calculate pseudo life. The specific system multi-level structure is as follows: Figure 2 As shown. The life data of a batch of component products was recorded in an operating environment with an average temperature of 25°C. For life-type products, the natural time from operation to failure was recorded. For degradation-type products, the time when their performance first degraded to the specified threshold was recorded. The failure life time and degradation pseudo-life time were calculated, and 10 valid data points for each component were taken as component-level test data, as shown in Table 1:

[0064] Table 1 Lifespan / pseudo-lifespan data of components in a motor control system

[0065] ;

[0066] in, Indicates sky.

[0067] See also Figure 1, a control system reliability evaluation method based on multi-level nested Copula functions, comprising the following steps:

[0068] Step 1: Analyze the life span data and find the marginal distribution.

[0069] The edge distribution forms of different component products are as follows Figure 2 As shown in Table 2, the parameter estimation results of the marginal distribution of parts obtained by maximum likelihood estimation are shown in Table 2:

[0070] Table 2 Component edge distribution parameters

[0071] ;

[0072] Accordingly, the edge distribution of each component is recorded as

[0073] .

[0074] Step 2: Calculate the Kendall rank correlation coefficient.

[0075] For the fault correlation between components, we first calculate the Kendall rank correlation coefficient between each component in each subsystem. The component with the smallest rank correlation coefficient is used as the bottom-level variable. The results of the bottom-level correlation coefficient at the component level are shown in Table 3:

[0076] Table 3 Correlation coefficient results of component bottom layer

[0077] ;

[0078] The minimum Kendall rank correlation coefficients of the three subsystems were obtained as 0.6220, 0.8312 and 0.4204 respectively. These three sets of variable pairs were used as underlying variables to perform Copula fitting and determine the optimal Copula function.

[0079] Step 3: Determine the optimal Copula function of the underlying variables and calculate the joint probability.

[0080] The parameters of the Copula function are estimated for the two components at the bottom of each subsystem. A joint distribution is constructed for the two variables. The optimal Copula function is determined by performing a goodness-of-fit test based on the AIC. The AIC test values are shown in Table 4.

[0081] Table 4 Bottom-level AIC test values

[0082] ;

[0083] Determine the optimal Copula function of the underlying variables of the three subsystems in turn, and calculate the actual joint probability of the underlying components of different subsystems based on the Copula distribution form:

[0084] ;

[0085] Then, the joint probability is used as a new variable in the second-to-last Copula function at the component level, and the rank correlation coefficient is calculated with the remaining variables and the optimal Copula function is determined.

[0086] Step 4: Create a Copula nested structure.

[0087] Subsystem and There are three components that can create a two-layer nested structure. There are four components, and a three-layer nested structure can be established. Correspondingly, the analysis results of the second-to-last layer and the top-most layer of the component-level variables are shown in Tables 5 and 6:

[0088] Table 5 Analysis results of the second-to-last level variables at the component level

[0089] ;

[0090] The second to last layer also gets , and .

[0091] Table 6 Analysis results of the top-level variables at the component level

[0092] ;

[0093] Get the subsystem through the top level The joint probability The final subsystem failure probability is obtained as , and Afterwards, a subsystem-system level Copula nested structure was also established. The analysis results of the bottom-level and top-level variables at the subsystem level are shown in Tables 7 and 8:

[0094] Table 7 Analysis results of the lowest level variables at the subsystem level

[0095] ;

[0096] The joint probability of the subsystems is also obtained through the bottom layer .

[0097] Table 8 Analysis results of top-level variables at the subsystem level

[0098] ;

[0099] Finally, the three-layer Copula nested structure of motor control system components-subsystem-system is obtained as follows Figure 3shown.

[0100] Step 5: Evaluate the reliability of the motor control system.

[0101] The system nested structure is built, with a total of 9 Copula functions, which are used to calculate the failure probability of different subsystems. , and The system reliability can be obtained , where the computing system is at time Reliability under :

[0102] ;

[0103] The system reliability curve over time is obtained by nested Copula method. Figure 4 When the reliability is 0.90, the reliable life of the system is 210 days.

[0104] The results show that the reliability of the control system can be accurately evaluated by the method of the present invention, achieving the expected purpose.

[0105] Therefore, the present invention adopts the aforementioned control system reliability assessment method based on multi-level nested Copula functions. Based on the lifetime / pseudo-lifetime data of components in a multi-level control system, the Kendall rank correlation coefficient and nested Copula functions are used to evaluate the correlation between each component, thereby establishing a system nested Copula structure. Furthermore, the AIC is used to select the optimal Copula function for comprehensive system reliability assessment. This method is suitable for applications such as control system reliability assessment involving multiple component failures and is highly operational.

[0106] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit the same. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solutions of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A control system reliability assessment method based on multi-level nested Copula functions, characterized in that: The following steps are involved: Step 1: Analyze the life data of the control system and calculate the marginal distribution; Step 2: For the Copula model considering marginal distribution, calculate the Kendall rank correlation coefficient; Step 3: Determine the optimal Copula function of the underlying variables and calculate the joint probability; Step 4: Create a Copula nested structure; Step 5: Evaluate the reliability of the motor control system; Step three is as follows: assumed is a one-dimensional continuous distribution function, let ,but All obey Uniform distribution in the interval, is a random variable The joint distribution function of for The joint distribution function of , then there exists a multivariate Copula function So that the joint distribution function and marginal distribution function Have the following relationship: ； Copula functions include Gumbel-Copula, Clayton-Copula, Frank-Copula, and Gaussian-Copula functions; Different Copula functions are used to characterize different correlations. The Akaike Information Criterion (AIC) is used to evaluate the fitting results. The calculation formula is: ； Where w is the number of parameters in the model, is the likelihood function structure; The distribution forms of the four types of Copula functions are: ； ； ； ； in, and are two random variables, is the correlation coefficient, which is used to describe the linear correlation between two random variables. is the shape parameter.

2. A control system reliability assessment method based on multi-level nested Copula functions according to claim 1, characterized in that: Step 1 is as follows: Life / pseudo-life data of components at the bottom layer of the three-tier control system , combined with the distribution forms of different components, the maximum likelihood estimation is used to estimate the parameters, which is: ； Where k is the number of samples, is the parameter vector to be estimated for different distributions, is the probability density function of different distributions, including exponential distribution, normal distribution and Weibull distribution. The cumulative failure functions of each distribution are: ； ； ； in, is the mean life span of the exponential distribution, and are the scale parameter and shape parameter of the Weibull distribution, and are the location parameter and scale parameter of the normal distribution, is the cumulative distribution function of the standard normal distribution; By solving the formula Get parameter estimates for the distribution ,in, It means to find partial derivatives.

3. The control system reliability assessment method based on multi-level nested Copula functions according to claim 2, characterized in that: The three-level control system is a motor control system that includes a three-level system structure of system-subsystem-components.

4. A control system reliability assessment method based on multi-level nested Copula functions according to claim 3, characterized in that: In step 2, the nonlinear correlation between two variables is measured by the Kendall rank correlation coefficient, specifically: Suppose there are two continuous random variables and The sample data is , for any two sample pairs and ,and , according to the order relationship, they are divided into the following three categories: Consistent with: ； Inconsistent pairs: ； Tie Pair: ； The number of consistent pairs is recorded as , the number of inconsistent pairs is recorded as , then the Kendall rank correlation coefficient is defined as It is the ratio of the difference between the number of consistent pairs and the number of inconsistent pairs to the number of all sample pairs. The formula is: ； in , is from The number of random pairwise combinations selected from the samples.

5. The control system reliability assessment method based on multi-level nested Copula functions according to claim 1 is characterized in that: Step 4 is as follows: Assume the system has subsystems, each of which contains multiple components with related relationships ;make Indicates the Subsystem For each marginal distribution, first calculate the rank correlation coefficient for all components of each subsystem, and select A pair of components Perform Copula fitting as the bottom-level variable: ； in, For the The joint probability distribution obtained by the penultimate layer of the subsystem is: Represents the optimal Copula function form selected by AIC. Repeat steps 2 and 3 recursively to obtain the final top-level failure probability of each subsystem. ;Finally, perform the same operation on different subsystems to obtain the system failure probability , and finally obtain the system's Copula nested structure.

6. A control system reliability assessment method based on multi-level nested Copula functions according to claim 5, characterized in that: Step 5 is as follows: Obtaining the system failure probability through recursive Copula nesting structure : ； The system reliability is expressed as .

Images