Integrated system reliability evaluation method and system based on nested moment integral

Abstract

The invention belongs to the technical field of integrated system reliability evaluation, and relates to a nested moment integral-based integrated system reliability evaluation method and system, and the method comprises the steps: S1, generating an initial model parameter set of integrated system reliability parameters; s2, establishing an integrated system reliability parameter updating model based on nested moment integration, and outputting an integrated system reliability parameter set and a weight; and S3, establishing a response estimation model of the nested moment integral of the integrated system, and outputting a reliability evaluation result of the integrated system. According to the method, an objective function is constructed by using an orthogonal polynomial, the numerical stability of a solving process is enhanced, the algebraic precision is improved and the convergence of a current result is evaluated by increasing integral nodes, and the proposed nested moment integral quantization method allows model evaluation to be carried out from a small sample, so that the calculation accuracy is improved. The accuracy of a prediction result is improved by gradually increasing the number of samples; and model evaluation results can be reused.

Description

Technical Field

[0001] The present invention belongs to the technical field of integrated system reliability assessment, and in particular relates to an integrated system reliability assessment method and system based on nested moment integration. Background Art

[0002] The reliability assessment of an integrated system relies on the quantification of the uncertainty of each factor of its components. By effectively quantifying uncertainty, the reliability of the system can be more accurately assessed and improved. The uncertainty in an integrated system may come from a variety of factors, including uncertainty in material properties, environmental uncertainty, uncertainty caused by model simplification, and uncertainty caused by human factors. By properly considering and quantifying these uncertainties, the behavior of the system can be accurately predicted, thereby effectively monitoring the system; with the increasing complexity of modern integrated systems, the demand for computing resources for reliability assessment is gradually increasing. Therefore, in order to quickly obtain accurate reliability assessment using fewer model evaluation times in practical applications, this study proposes an integrated system reliability assessment method based on nested moment integration. This method uses nested moment integration rules to quantify the uncertainty in the integrated system, and through appropriate variable substitution, the points of the nested moment integration rules are transformed into the model input parameter set of the integrated system. The corresponding results are weighted according to the weights of the nested moment integration rules to obtain the estimated moments of each order of the system response, and thus calculate its corresponding reliability index.

[0003] Methods for quantifying uncertainties in integrated systems can generally be divided into three categories. The first category is uncertainty quantification methods based on asymptotic estimation, including second-order perturbation methods and Laplace approximation methods. These methods fit the original distribution to a Gaussian distribution for subsequent analysis by performing Taylor expansion at the peak of the distribution and ignoring higher-order terms. To ensure the effectiveness of the fit, the original distribution is usually required to have asymptotic symmetry and unimodal characteristics, so there are certain limitations when dealing with large skewness or monotonic distributions. Since the peak is often unknown, it is necessary to calculate the derivative of the model response to the uncertain variable to identify the peak. In addition, this method also requires the calculation of the Hessian matrix of the original distribution at the identified peak. For systems with 3 to 5 uncertain variables, the entire process may require dozens to hundreds of model evaluations. The second category is uncertainty quantification methods based on Monte Carlo, which generate a large number of random samples and use each group of samples to perform statistical analysis on the model. The accuracy of this method is proportional to the square root of the sample size, so in order to obtain acceptable accuracy, a large amount of calculation is usually required, which faces significant computational overhead limitations when dealing with large and complex models. Although there are several improved techniques, such as importance sampling, quasi-Monte Carlo and stratified sampling, which can improve the convergence speed by reducing the variance of the generated random samples, these methods still need to adjust the key parameters by trial and error, which still requires a large number of model running experiments, thus limiting their application in complex models. The third category is the uncertainty quantification method based on surrogate models, including response surface method, Kriging method, polynomial chaos method and metamodel method. The core idea of ​​this method is to establish a simplified model based on the relationship between uncertain input parameters and model response, and use the simplified model for subsequent analysis. Compared with the original model, the simplified model usually has lower computational overhead, but its accuracy is affected by the number of samples. Therefore, special care should be taken when selecting samples to ensure that the peaks and valleys of the model response in the parameter space can be effectively captured. In addition, the simplified model usually uses a polynomial regression model. As the order of the polynomial increases, the number of model evaluations required may increase significantly, resulting in an increase in computational overhead. So far, a large number of studies have focused on these three types of problems, and the aforementioned methods are only part of them.

[0004] Recently, a new moment integral quantization method has been proposed to efficiently solve the uncertainty quantization problem in complex computational models. The core idea of ​​this method is to calculate the origin moments of the model response of each order by designing a Gaussian integral rule for arbitrary input distribution. For low-dimensional complex problems, this method can obtain results comparable to thousands or even millions of model runs using the Monte Carlo method through a small number of model evaluations. For high-dimensional complex problems, the computational overhead can be reduced from exponential to polynomial by combining the sparse tensor product rule. However, whether it is the moment integral method or its sparse variant, when it is hoped to improve the algebraic accuracy and evaluate the convergence of the current results by adding integration nodes, the integration nodes of two consecutive Gaussian integral rules are staggered, resulting in the inability to reuse the running results of the current model, resulting in a waste of computing resources. The nested moment integral quantization method proposed in this study allows model evaluation to start from a small sample and improve the accuracy of the prediction results by gradually increasing the number of samples. The model evaluation results of each step can be reused and combined with the new model evaluation quantity to obtain a more accurate model estimate. In addition, for one-dimensional problems, this method is the least computationally expensive method to determine the convergence of the current model estimation results, which is particularly useful when evaluating the results of a certain dimension of the model. Summary of the invention

[0005] In view of the shortcomings of the prior art, the present invention provides an integrated system reliability assessment method and system based on nested moment integration, which uses orthogonal polynomials to construct the objective function, enhances the numerical stability of the solution process, and improves the algebraic precision and the convergence of the current evaluation results by adding integration nodes. The proposed nested moment integration quantization method allows model evaluation to start from a small sample, and improves the accuracy of the prediction results by gradually increasing the number of samples; the model evaluation results can be reused.

[0006] To achieve the above object, the present invention discloses the following technical solution:

[0007] An integrated system reliability assessment method based on nested moment integration includes:

[0008] S1: Generate the initial model parameter set of integrated system reliability parameters;

[0009] According to the distribution type of the random variable of the reliability parameter of the integrated system, the initial integration rule of n points under the corresponding probability density function is generated to obtain the initial model parameter set of the reliability parameter of the integrated system. The initial model parameter set includes the input parameter x j and the corresponding weight ω j , j is a constant;

[0010] S2: Establish an integrated system reliability parameter update model based on nested moment integration, and output the integrated system reliability parameter set and weight;

[0011] S21: Determine the n input parameters x1,…,x of the initial model parameter set n , n is the number of input parameters, the initialization moment integral algebraic precision m = 3n + 1, the moment integral nesting points are z1, ..., z n+1 and weight

[0012] S22: According to the distribution type of random variables, calculate the orthogonal polynomial φ of the reliability parameters of the first m-order integrated system i (x):

[0013] φ i (X z )=s 1,i+1 X z 0 +s 2,i+1 X z 1 +…+s i+1,i+1 X z i ,i=0,…,m (1)

[0014] Among them, φ i (X z ) represents the orthogonal polynomial of the integrated system reliability parameter; s represents the input parameter coefficient of the orthogonal polynomial; X z Represents the monomial basis of the orthogonal polynomial, including the input parameter x and the moment integral nesting point z; i is the input number of the orthogonal polynomial;

[0015] S23: Use the orthogonal polynomial of the integrated system reliability parameters to construct the first objective function r of the integrated system reliability parameter update model:

[0016] r=V·w-e1 (2)

[0017] Where r represents the first objective function; V represents the Vandermonde coefficient matrix; w represents the moment integral nesting point weight, and e1 represents the identity matrix, and

[0018] S24: Based on the penalty function method, the constrained optimization problem of the integrated system reliability parameters is transformed into an unconstrained problem, and the second objective function of the integrated system reliability parameter update model is obtained. for:

[0019]

[0020] in, represents the second objective function; P represents the penalty term of the second objective function; c represents the penalty factor;

[0021] S25: Solve the second objective function in step S24 to obtain the integrated system reliability parameter set x1,…,x n ,z1,…,z n+1 and weights ω1,…,ω 2n+1 ;

[0022] S3: Establish a response estimation model of nested moment integration of the integrated system and output the reliability assessment results of the integrated system;

[0023] Obtain the integrated system reliability parameter set obtained in step S2, build the response estimation model of the nested moment integral of the integrated system, and output the integrated system reliability assessment result as follows:

[0024]

[0025] Among them, I n ψ represents the reliability evaluation result of the integrated system; g(x) represents the response expression of the integrated system model; p(x) represents the distribution function of the reliability parameter of the integrated system; Ξ j represents the j-th response matrix of the response estimation model, and ω j represents the jth weight matrix of the response estimation model response quantity, and θ j represents the reliability assessment correction parameter; Ω represents the reliability parameter set of the integrated system.

[0026] Preferably, in step S1, according to the distribution type of the integrated system reliability parameter random variable, an n-point initial integration rule under the corresponding probability density function is generated to obtain an initial model parameter set, specifically:

[0027] S11: When the probability density function of the random variable X of the integrated system reliability parameter is p(x), the k-order origin moment μ of the integrated system reliability parameter k It is expressed as:

[0028] μ k =E[X k ]=∫ x∈Ω x k p(x)dx (5)

[0029] Among them, μ k represents the k-order origin moment of the reliability parameter of the integrated system; E represents the mathematical expectation of the random variable X of the reliability parameter of the integrated system; p(x) represents the distribution function of the reliability parameter of the integrated system; Ω represents the set of reliability parameters of the integrated system; X, x, k represent the random variable, random variable value and origin moment order of the reliability parameter of the integrated system respectively;

[0030] When the probability density function of the random variable of the integrated system reliability parameter is unknown, the k-order origin moment μ of the integrated system reliability parameter is k Equal to the sample moment of the integrated system reliability parameter

[0031]

[0032] in, p n represents the sample moment and sample number of the reliability parameter of the integrated system; x i represents the sample value of the reliability parameter of the i-th integrated system;

[0033] S12: According to the k-order origin moment μ of the integrated system reliability parameter k , construct the Hankel matrix H of the integrated system reliability parameters k :

[0034]

[0035] Among them, H k Hankel matrix representing the reliability parameters of the integrated system;

[0036] S13: Hankel matrix H for integrated system reliability parameters k Perform Cholesky decomposition to obtain the upper triangular matrix R k :

[0037]

[0038] Where r represents the upper triangular matrix R k Elements of

[0039] S14: Construct the moment characteristic matrix J of the integrated system reliability parameters:

[0040]

[0041] Where J represents the moment characteristic matrix of the integrated system reliability parameters; α j represents the jth first characteristic element of the moment characteristic matrix, and β j represents the jth second characteristic element of the moment characteristic matrix, and

[0042] S15: Calculate the eigenvalue λ of the moment characteristic matrix J of the integrated system reliability parameter j With the feature vector v j =(v j1 ,…,v jn ) T, j = 1, ..., n; get the initial model parameter set and corresponding weights, and the model input parameter x j and its corresponding weight ω j It consists of two parts, where x j =λ j ,

[0043] Preferably, the input parameter coefficient s of the orthogonal polynomial in step S22 is an m-order upper triangular matrix R m The elements of the inverse matrix are:

[0044] Calculate the first m-order origin moments and use the Hankel matrix H of the integrated system reliability parameters k Get the m-order upper triangular matrix R m :

[0045]

[0046] Calculate the upper triangular matrix R m The inverse matrix of :

[0047]

[0048] in, Represents the upper triangular matrix R m The inverse matrix of .

[0049] Preferably, the Vandermonde coefficient matrix V in step S23 is obtained according to the orthogonal polynomial of the integrated system reliability parameter in step S22, specifically:

[0050] The n input parameters x1,…,x n and moment integral nested points z1,…,z n+1 Substituting the orthogonal polynomial of the integrated system reliability parameter, we get the Vandermonde coefficient matrix V:

[0051]

[0052] Where V represents the Vandermonde coefficient matrix.

[0053] Preferably, in step S24, the constrained optimization problem of the reliability parameters of the integrated system is converted into an unconstrained problem based on the penalty function method to obtain the second objective function Specifically:

[0054] Calculate the penalty term P of the second objective function, and set the constraint of the penalty function method as follows: the integral node needs to be in the set domain, the weight needs to be greater than 0, and the conditions are:

[0055]

[0056] Among them, P j represents the jth penalty term of the iteration termination condition; a and b represent the upper and lower bounds of the set domain respectively;

[0057] Get the second objective function penalty term P:

[0058] P=[P1 … P 3n+2 ] (14)

[0059] Where P represents the penalty term of the second objective function;

[0060] Calculate the penalty factor c of the penalty term:

[0061] c=max{10 3 ,1 / ||r|| 2} (15)

[0062] Among them, max represents the maximum value function;

[0063] Construct the second objective function

[0064] Preferably, the iterative termination condition of the penalty function method in step S24 is for:

[0065]

[0066] in, represents the iterative termination condition of the penalty function method; P j The j-th penalty term representing the iteration termination condition.

[0067] Preferably, the iterative optimization process of the penalty function method in step S24 is specifically as follows:

[0068] Step S241: When m=3n+1, if the iteration termination condition Less than the preset value∈, all integration nodes fall within the set domain, the negative weight ratio of the model input parameter weight is less than 50%, and the minimum value is greater than -10 -4 , then the condition for continuing the iteration upward is satisfied, and the algebraic precision m of the moment integral is increased by 1, and the next round of iterative optimization of the penalty function method is carried out; if the above requirements are still not met at the maximum number of iteration steps, the condition for continuing the iteration downward is satisfied, and the algebraic precision m of the moment integral is reduced by 1, and the next round of iterative optimization of the penalty function method is carried out;

[0069] Step S242: If the iteration termination condition And m<3n+1, or the iteration termination condition is reached at the maximum number of iterations If m>3n+1, the iteration termination condition is met. In the first case, m remains unchanged. In the second case, the algebraic precision of the moment integral m is reduced by 1, and the iterative optimization process of the penalty function method is terminated.

[0070] Step S243: If m<3n+1 and the iteration is terminated at the maximum number of iterations If the condition for continuing iteration is met, the algebraic precision m of the moment integral is reduced by 1, and the next round of iterative optimization of the penalty function method is performed; if the iteration termination condition If m>3n+1, the condition for continuing to iterate upward is met. After the moment integral algebraic precision m is increased by 1, the next round of iterative optimization of the penalty function method is carried out.

[0071] The second aspect of the present invention proposes an integrated system reliability assessment system based on a nested moment integral integrated system reliability assessment method, which accurately assesses the reliability of the integrated system using a small sample based on the nested moment integral method, and comprises: a model starting unit, a multi-order orthogonal construction unit, a target optimization generation unit, a comprehensive regularization solution unit, an adaptive precision adjustment unit and a global optimization adjustment unit;

[0072] The model startup unit is used to generate an initial model parameter set of the integrated system reliability parameters, and obtain a moment characteristic matrix based on a moment integration method, wherein the eigenvalue of the moment characteristic matrix is ​​the initial model parameter set;

[0073] The multi-order orthogonal construction unit is used to generate an arbitrary order orthogonal polynomial corresponding to an arbitrary probability density function, generate a set of orthogonal polynomial systems based on the Schakowsky theorem, and obtain three recursive formulas corresponding to the orthogonal polynomials;

[0074] The target optimization generation unit is used to generate an unconstrained target optimization function. The initial target optimization function is obtained by multiplying the weight vector by the Vandermonde matrix constructed based on the orthogonal polynomial and performing a difference operation with the unit vector. The penalty function method is used to impose a cost for violating the constraint to convert it into an unconstrained final target optimization function.

[0075] The comprehensive regularization solving unit is used to solve the target optimization function, and minimize the norm of the target optimization function based on the Gauss-Newton method and the regularization technique, so as to obtain a solution that satisfies the accuracy;

[0076] The adaptive precision adjustment unit is used to generate a nested integration rule with the highest moment integral algebraic precision, and dynamically adjust the value of the moment integral algebraic precision according to the solution of the nested integration rule under the moment integral algebraic precision, so as to obtain the nested integration rule with the highest moment integral algebraic precision;

[0077] The global optimization adjustment unit is used to output an accurate integrated system reliability estimation, calculate the relative change rate of the estimation result based on the model response estimation result obtained based on the initial model parameter set and the updated model parameter set, thereby dynamically adjusting the number of nesting times, and finally outputting an estimation result that meets the accuracy requirements.

[0078] Compared with the prior art, the present invention has the following beneficial effects:

[0079] (1) The present invention designs a universal nested integration rule for arbitrary input distribution, aiming to calculate the origin moments of each order of the model response, so as to evaluate the reliability of the integrated system. The nested integration rule is based on the Gaussian integration rule, and realizes the nested integration rule with the smallest calculation increment with improved algebraic precision. This innovation enables random analysis in an efficient and accurate manner when dealing with complex distributions, significantly improving the ability to evaluate the reliability of the integrated system.

[0080] (2) The present invention uses orthogonal polynomials to construct the objective function, which greatly enhances the numerical stability in the solution process. At the same time, it provides a simple method for solving the orthogonal polynomial expression corresponding to any probability density function. This method not only improves the accuracy and stability of the calculation, but also provides an effective solution for processing different probability density functions, and broadens the scope of application of this technology in practical applications.

[0081] (3) The nested moment integral quantization method proposed in the present invention allows model evaluation to start from a small sample, and the accuracy of the prediction results can be improved by gradually increasing the number of samples. The model evaluation results of each step can be reused and combined with the new model evaluation quantity to obtain a more accurate model estimation. BRIEF DESCRIPTION OF THE DRAWINGS

[0082] Figure 1 It is a flow chart of the integrated system reliability assessment method based on nested moment integration of the present invention;

[0083] Figure 2 A schematic diagram of an integrated system for controlling water temperature in a boiler according to a specific embodiment of the present invention;

[0084] Figure 3 It is a result diagram of each level of integration rules under the uniform distribution standard form of the present invention;

[0085] Figure 4 A probability density diagram of the integrated circuit system response of the present invention;

[0086] Figure 5 It is a schematic diagram of a redundant flight control integrated system according to another specific embodiment of the present invention. DETAILED DESCRIPTION

[0087] The exemplary embodiments, features and aspects of the present invention will be described in detail below with reference to the accompanying drawings. The same reference numerals in the accompanying drawings represent elements with the same or similar functions. Although various aspects of the embodiments are shown in the accompanying drawings, the drawings are not necessarily drawn to scale unless otherwise specified.

[0088] The embodiment of the present invention uses a four-branch series integrated circuit system as an example to provide an integrated system reliability assessment method based on nested moment integration, such as Figure 1 As shown, an initial model parameter set of integrated system reliability parameters is generated; an integrated system reliability parameter update model based on nested moment integral is established, and an integrated system reliability parameter set and weight are output; a response estimation model of nested moment integral of the integrated system is established, and an integrated system reliability evaluation result is output; it includes:

[0089] Step S1: Generate an initial model parameter set of integrated circuit system reliability parameters.

[0090] According to the distribution type of the random variable of the reliability parameter of the integrated circuit system, the initial integration rule of n points under the corresponding probability density function is generated to obtain the initial model parameter set of the reliability parameter of the integrated circuit system. The initial model parameter set includes the input parameter x j and the corresponding weight ω j , j is a constant, specifically:

[0091] Step S11: When the probability density function of the random variable X of the integrated circuit system reliability parameter is p(x), the k-order origin moment μ of the integrated circuit system reliability parameter k It is expressed as:

[0092] μ k =E[X k ]=∫ x∈Ω x k p(x)dx (5)

[0093] Among them, μ k represents the k-th order origin moment of the integrated circuit system reliability parameter; E represents the mathematical expectation of the random variable X of the integrated circuit system reliability parameter; p(x) represents the distribution function of the integrated circuit system reliability parameter; Ω represents the set of integrated circuit system reliability parameters; X, x, k represent the random variable, random variable value and origin moment order of the integrated circuit system reliability parameter, respectively.

[0094] When the probability density function of the random variable of the integrated circuit system reliability parameter is unknown, the k-order origin moment μ of the integrated circuit system reliability parameter is k Equal to the sample moment of the reliability parameter of the integrated circuit system

[0095]

[0096] in, p n represents the sample moment and sample number of the reliability parameter of the integrated circuit system; x i Represents the sample value of the reliability parameter of the i-th integrated circuit system.

[0097] Step S12: Based on the k-order origin moment μ of the integrated circuit system reliability parameter k , construct the Hankel matrix H of the reliability parameters of the integrated circuit system k :

[0098]

[0099] Among them, H k Hankel matrix representing the reliability parameters of an integrated circuit system.

[0100] Step S13: Hankel matrix H of the reliability parameters of the integrated circuit system k Perform Cholesky decomposition to obtain the upper triangular matrix R k :

[0101]

[0102] Where r represents the upper triangular matrix R k elements.

[0103] Step S14: construct the moment characteristic matrix J of the integrated circuit system reliability parameters:

[0104]

[0105] Where J represents the moment characteristic matrix of the reliability parameters of the integrated circuit system; α j represents the jth first characteristic element of the moment characteristic matrix, and β j represents the jth second characteristic element of the moment characteristic matrix, and

[0106] Step S15: Calculate the eigenvalue λ of the moment characteristic matrix J of the integrated circuit system reliability parameter j With the feature vector v j =(v j1 ,…,v jn ) T , j = 1, ..., n; get the initial model parameter set and corresponding weights, and the model input parameter x j and its corresponding weight ω j It consists of two parts, where x j =λ j ,

[0107] Step S2: Establish an integrated circuit system reliability parameter update model based on nested moment integration, and output the integrated circuit system reliability parameter set and weights.

[0108] Step S21: Determine n input parameters x1,…,x n , n is the number of input parameters, the initialization moment integral algebraic precision m = 3n + 1, the moment integral nesting points are z1, ..., z n+1 and weight

[0109] Step S22: Calculate the orthogonal polynomial φ of the reliability parameters of the first m-order integrated circuit system according to the distribution type of the random variable i (x):

[0110] φ i (X z )=s 1,i+1 X z 0 +s 2,i+1 X z 1 +…+s i+1,i+1 X z i ,i=0,…,m (1)

[0111] Among them, φ i (X z ) represents the orthogonal polynomial of the reliability parameter of the integrated circuit system; s represents the input parameter coefficient of the orthogonal polynomial; X z Represents the monomial basis of the orthogonal polynomial, including the input parameter x and the moment integral nesting point z; i is the input number of the orthogonal polynomial.

[0112] The input parameter coefficients s of the orthogonal polynomial are the m-order upper triangular matrix R m The elements of the inverse matrix are:

[0113] Calculate the first m-order origin moments and use the Hankel matrix H of the integrated circuit system reliability parameters k Get the m-order upper triangular matrix R m :

[0114]

[0115] Calculate the upper triangular matrix R m The inverse matrix of :

[0116]

[0117] in, Represents the upper triangular matrix R mThe inverse matrix of .

[0118] Step S23: construct the first objective function r of the integrated circuit system reliability parameter update model using the orthogonal polynomial of the integrated circuit system reliability parameter:

[0119] r=V·w-e1 (2)

[0120] Where r represents the first objective function; V represents the Vandermonde coefficient matrix; w represents the moment integral nesting point weight, and e1 represents the identity matrix, and

[0121] The Vandermonde coefficient matrix V is obtained according to the orthogonal polynomial of the integrated circuit system reliability parameter in step S22, specifically:

[0122] The n input parameters x1,…,x n and moment integral nested points z1,…,z n+1 Substituting the orthogonal polynomial of the integrated circuit system reliability parameter, the Vandermonde coefficient matrix V is obtained as follows:

[0123]

[0124] Where V represents the Vandermonde coefficient matrix.

[0125] Step S24: Based on the penalty function method, the constrained optimization problem of the reliability parameters of the integrated circuit system is converted into an unconstrained problem to obtain the second objective function Specifically:

[0126] Calculate the penalty term P of the second objective function, and set the constraint of the penalty function method as follows: the integral node needs to be in the set domain, the weight needs to be greater than 0, and the conditions are:

[0127]

[0128] Among them, P j represents the jth penalty term of the iteration termination condition; a and b represent the upper and lower bounds of the set domain respectively.

[0129] Get the second objective function penalty term P:

[0130] P=[P1 … P 3n+2 ] (14)

[0131] Wherein, P represents the penalty term of the second objective function.

[0132] Calculate the penalty factor c of the penalty term:

[0133] c=max{10 3,1 / ||r|| 2} (15)

[0134] Among them, max represents the maximum value function.

[0135] Construct the second objective function

[0136]

[0137] in, represents the second objective function; P represents the penalty term of the second objective function; c represents the penalty factor.

[0138] Step S241: When m=3n+1, if the iteration termination condition Less than the preset value∈, all integration nodes fall within the set domain, the negative weight ratio of the model input parameter weight is less than 50%, and the minimum value is greater than -10 -4 , then the upward iteration condition is satisfied, and the algebraic precision m of the moment integral is increased by 1, and the next round of iterative optimization of the penalty function method is carried out; if the above requirements are still not met at the maximum number of iterations, the downward iteration condition is satisfied, and the algebraic precision m of the moment integral is reduced by 1, and the next round of iterative optimization of the penalty function method is carried out; the iterative termination condition of the penalty function method for:

[0139]

[0140] in, represents the iterative termination condition of the penalty function method; P j The j-th penalty term representing the iteration termination condition.

[0141] Step S242: If the iteration termination condition And m<3n+1, or the iteration termination condition is reached at the maximum number of iterations And if m>3n+1, the iteration termination condition is met. In the first case, m remains unchanged. In the second case, the moment integral algebraic precision m is reduced by 1, and the iterative optimization process of the penalty function method is terminated.

[0142] Step S243: If m<3n+1 and the iteration is terminated at the maximum number of iterations If the condition for continuing iteration is met, the algebraic precision m of the moment integral is reduced by 1, and the next round of iterative optimization of the penalty function method is performed; if the iteration termination condition If m>3n+1, the condition for continuing to iterate upward is met. After the moment integral algebraic precision m is increased by 1, the next round of iterative optimization of the penalty function method is carried out.

[0143] Step S25: Solve the second objective function in step S24 to obtain the integrated circuit system reliability parameter set x1,…,xn ,z1,…,z n+1 and weights ω1,…,ω 2n+1 .

[0144] Step S3: Establish a response estimation model of the nested moment integral of the integrated circuit system and output the reliability assessment result of the integrated circuit system.

[0145] The integrated circuit system reliability parameter set obtained in step S2 is obtained, a response estimation model of the nested moment integral of the integrated circuit system is constructed, and the integrated circuit system reliability assessment result is output as:

[0146]

[0147] Among them, I n ψ represents the reliability evaluation result of the integrated circuit system; g(x) represents the integrated circuit system model response expression; p(x) represents the integrated circuit system reliability parameter distribution function; Ξ j represents the j-th response matrix of the response estimation model, and ω j represents the jth weight matrix of the response estimation model response quantity, and θ j represents the reliability assessment correction parameter; Ω represents the reliability parameter set of the integrated circuit system.

[0148] In a specific embodiment, in order to solve the reliability problem of the integrated system for controlling the water temperature in the boiler, the schematic diagram of the integrated system for controlling the water temperature is as follows: Figure 2 As shown, the water temperature of the boiler tank is required to be stable at a given value. The voltage signal is transmitted to the monitoring device in real time through the temperature sensor, converted into a digital signal through the EM235 analog input module and sent to the PLC for PID adjustment. Finally, the heating of the heating wire is adjusted by controlling the voltage of the heating device. The reliability of the water temperature control integrated system is jointly determined by the reliability of the four components of the power supply, temperature sensor, monitoring device and heating device. Assume that the performance of each component can be represented by the first variable x1 and the second variable x2 according to its respective material properties. The input variables affecting the four components show two types of correlation, nonlinear or linear, and have four limit states. The mathematical expression of the structural function of the system reliability response is:

[0149]

[0150] The input variables are modeled by two independent uniform random variables X i ~U(0,1). The failure threshold is defined as g(x)≤0, and the reliability of the system is P r =1-P f=1-P(g(x)≤0); start the evaluation from the reliability parameters of a single water temperature control integrated circuit system, nesting step by step, and replace the variables according to the following formula Figure 3 The results of the integration rules at all levels in are transformed into the model input parameter set of this problem:

[0151] ξ=(τ+1) / 2 (18)

[0152] Among them, τ represents Figure 3 The set of integration nodes of each level of integration rules in ; ξ represents the set of model input parameters of the problem.

[0153] Substitute it into the integrated circuit system reliability evaluation formula in S3 to calculate the relative change rate of two adjacent results, as shown in Table 1. r1 represents the result of a single model evaluation using the 1-point moment integration method, M r2 Indicates that in M r1 The result obtained by nesting it on the basis of M r3 Indicates that in M r2 The results are obtained by nesting them on the basis of . At this time, the relative change rates of the mean and variance of the continuous nesting results are less than 0.1%, and the results are considered to have converged. r3 This is the final result. 6 The Monte Carlo method is used for comparison, and the probability density image is obtained as follows Figure 4 As shown, MC represents the result of the Monte Carlo method. At this time, the response of the system is close to the normal distribution. Therefore, the reliability calculated by the Monte Carlo method is 99.2907%. r3 The reliability of the results is 99.2902%, which shows that the accuracy of the proposed method reaches one hundred thousandth, and its calculation amount is only one ten-thousandth of the Monte Carlo method.

[0154]

[0155] Table 1

[0156] In another specific embodiment, the reliability of a redundant flight control integrated system is analyzed as a case study. Since the UAV has special requirements for the reliability of the flight control computer system, a redundant architecture design is adopted to ensure the flight safety and mission capability of the UAV. Redundant architecture design refers to the deliberate addition of additional components or functions during design and construction to improve the reliability and availability of the integrated system. Its basic concept is to reduce or prevent the impact of system failures through backup and substitution.

[0157] A schematic diagram of a redundant flight control integrated system is shown in Figure 5As shown in the figure, the flight control integrated system consists of four parts: the main module group, the redundant module group, the redundant management module and the voter. The flight control of the UAV is jointly determined by the five modules in the main module group. When a module in the main module group fails, the redundant management module accesses a module in the redundant module group to participate in the vote instead. When the number of modules participating in the vote in the voter is less than 5, the redundant flight control integrated system fails. Considering that the reliability of each module in the main module group and the redundant module group is independent and identically distributed, the reliability of the system is:

[0158]

[0159] Among them, x follows the uniform distribution U(0.82,0.9).

[0160] According to the following variable substitution formula, Figure 3 The results of the integration rules at all levels in are transformed into the model input parameter set of this problem:

[0161] ξ=0.04τ+0.86(20)

[0162] Among them, τ represents Figure 3 The set of integration nodes of each level of integration rules in ; ξ represents the set of model input parameters of the problem.

[0163] The mean and variance of system reliability are calculated using the nested moment integration method, and the results are shown in Table 2. r1 represents the result of a single model evaluation using the 1-point moment integration method, M r2 Indicates that in M r1 The result obtained by nesting it on the basis of M r3 Indicates that in M r2 The results are obtained by nesting them on the basis of . At this time, the relative change rates of the mean and variance of the continuous nesting results are less than 0.1%, and the results are considered to have converged. r3 This is the final result, M r3 The variance is 10 -4 , indicating that the dispersion of the results is extremely low, so the reliability of the system is the mean result, which is 98.1341%. The sample size is 10 6 The Monte Carlo method is used as a comparison, and the corresponding reliability is 98.1353%. It can be seen that the accuracy of the proposed method reaches the ten thousand level, and its calculation amount is only one ten thousandth of the Monte Carlo method.

[0164]

[0165] Table 2

[0166] The second aspect of an embodiment of the present invention proposes an integrated system reliability assessment system based on a nested moment integration method. Based on the nested moment integration method, a small sample is used to accurately assess the reliability of the integrated circuit system. The system includes: a model startup unit, a multi-order orthogonal construction unit, a target optimization generation unit, a comprehensive regularization solution unit, an adaptive precision adjustment unit, and a global optimization adjustment unit.

[0167] The model startup unit is used to generate an initial model parameter set of the integrated circuit system reliability parameters, and obtains a moment characteristic matrix based on a moment integral method, and the eigenvalue of the moment characteristic matrix is ​​the initial model parameter set.

[0168] The multi-order orthogonal construction unit is used to generate arbitrary order orthogonal polynomials corresponding to any probability density function, generate a set of orthogonal polynomial systems based on the Schakowsky theorem, and obtain the three-term recursive formula of the corresponding orthogonal polynomials.

[0169] The target optimization generation unit is used to generate an unconstrained target optimization function. The initial target optimization function is obtained by multiplying the Vandermonde matrix constructed based on orthogonal polynomials with the weight vector and performing a difference operation with the unit vector. Based on the penalty function method, a cost is imposed on violating the constraints to convert it into an unconstrained final target optimization function.

[0170] The comprehensive regularization solving unit is used to solve the target optimization function. The norm of the target optimization function is minimized based on the Gauss-Newton method and regularization techniques to obtain a solution that satisfies the accuracy.

[0171] The adaptive precision adjustment unit is used to generate a nested integration rule with the highest moment integral algebraic precision, and dynamically adjust the value of the moment integral algebraic precision according to the solution of the nested integration rule under the moment integral algebraic precision, so as to obtain the nested integration rule with the highest moment integral algebraic precision.

[0172] The global optimization adjustment unit is used to output accurate integrated circuit system reliability estimation. Based on the model response estimation results obtained based on the initial model parameter set and the updated model parameter set, the relative change rate of the estimation results is calculated, thereby dynamically adjusting the number of nesting times and finally outputting an estimation result that meets the accuracy requirements.

[0173] The beneficial effects of the present invention are as follows: the present invention provides an integrated system reliability assessment method based on nested moment integrals, uses orthogonal polynomials to construct an objective function, and enhances the numerical stability of the solution process; the nested moment integral quantization method proposed in this study allows model evaluation to start from a small sample, and improves the accuracy of the prediction result by gradually increasing the number of samples, so that the model evaluation result of each step can be reused, and a more accurate model estimate can be obtained by weighted combination with a new model evaluation quantity; for low-dimensional complex problems, a small amount of model evaluation can be used to obtain results equivalent to thousands or even millions of model runs using the Monte Carlo method; for high-dimensional complex problems, the computational overhead is reduced from the exponential level to the polynomial level by combining the sparse tensor product rule; the computational analysis of integrated series circuits and redundant systems proves that the method has a good practical effect and meets actual engineering needs.

[0174] The embodiments described above are only descriptions of the preferred implementation modes of the present invention, and are not intended to limit the scope of the present invention. Without departing from the design spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should all fall within the protection scope determined by the claims of the present invention.

Claims

1. An integrated system reliability assessment method based on nested moment integration, characterized in that: It includes: S1: Generate the initial model parameter set of integrated system reliability parameters; According to the distribution type of the random variable of the reliability parameter of the integrated system, the initial integration rule of n points under the corresponding probability density function is generated to obtain the initial model parameter set of the reliability parameter of the integrated system. The initial model parameter set includes the input parameter x j and the corresponding weight ω j , j is a constant; S2: Establish an integrated system reliability parameter update model based on nested moment integration, and output the integrated system reliability parameter set and weight; S21: Determine the n input parameters x1,…,x of the initial model parameter set n , n is the number of input parameters, the initialization moment integral algebraic precision m = 3n + 1, the moment integral nesting points are z1, ..., z n+1 and weights w=ω1,…,ω 2n+1 T ; S22: According to the distribution type of random variables, calculate the orthogonal polynomial φ of the reliability parameters of the first m-order integrated system i x: φ i X z =s 1,i+1 X z 0 +s 2,i+1 X z 1 +…+s i+1,i+1 X z i ,i=0,…,m(1) Among them, φ i X z represents the orthogonal polynomial of the reliability parameter of the integrated system; s represents the input parameter coefficient of the orthogonal polynomial; X z Represents the monomial basis of the orthogonal polynomial, including the input parameter x and the moment integral nesting point z; i is the input number of the orthogonal polynomial; S23: Use the orthogonal polynomial of the integrated system reliability parameters to construct the first objective function r of the integrated system reliability parameter update model: r=V·w-e1(2) Where r represents the first objective function; V represents the Vandermonde coefficient matrix; w represents the moment integral nesting point weight, and w=ω1,…,ω 2n+1 T ; e1 represents the identity matrix, and e1=1,0,…,0 T ; S24: Based on the penalty function method, the constrained optimization problem of the integrated system reliability parameters is transformed into an unconstrained problem, and the second objective function of the integrated system reliability parameter update model is obtained. in, represents the second objective function; P represents the penalty term of the second objective function; c represents the penalty factor; S25: Solve the second objective function in step S24 to obtain the integrated system reliability parameter set x1,…,x n ,z1,…,z n+1 and weights ω1,…,ω 2n+1 ; S3: Establish a response estimation model of nested moment integration of the integrated system and output the reliability assessment results of the integrated system; Obtain the integrated system reliability parameter set obtained in step S2, build the response estimation model of the nested moment integral of the integrated system, and output the integrated system reliability assessment result as follows: Among them, I n ψ represents the reliability evaluation result of the integrated system; gx represents the response expression of the integrated system model; px represents the reliability parameter distribution function of the integrated system; j represents the j-th response matrix of the response estimation model, and ω j represents the jth weight matrix of the response estimation model response quantity, and θ j represents the reliability assessment correction parameter; Ω represents the reliability parameter set of the integrated system.

2. The integrated system reliability assessment method based on nested moment integration according to claim 1, characterized in that: In step S1, according to the distribution type of the random variable of the reliability parameter of the integrated system, the n-point initial integration rules under the corresponding probability density function are generated to obtain the initial model parameter set, which is specifically: S11: When the probability density function of the random variable X of the integrated system reliability parameter is p(x), the k-order origin moment μ of the integrated system reliability parameter k It is expressed as: μ k =EX k =∫ x∈Ω x k pxdx(5) Among them, μ k represents the k-order origin moment of the reliability parameter of the integrated system; E represents the mathematical expectation of the random variable X of the reliability parameter of the integrated system; p(x) represents the distribution function of the reliability parameter of the integrated system; Ω represents the set of reliability parameters of the integrated system; X, x, k represent the random variable, random variable value and origin moment order of the reliability parameter of the integrated system respectively; When the probability density function of the random variable of the integrated system reliability parameter is unknown, the k-order origin moment μ of the integrated system reliability parameter is k Equal to the sample moment of the integrated system reliability parameter in, p n represents the sample moment and sample number of the reliability parameter of the integrated system; x i represents the sample value of the reliability parameter of the i-th integrated system; S12: According to the k-order origin moment μ of the integrated system reliability parameter k , construct the Hankel matrix H of the integrated system reliability parameters k : Among them, H k Hankel matrix representing the reliability parameters of the integrated system; S13: Hankel matrix H for integrated system reliability parameters k Perform Cholesky decomposition to obtain the upper triangular matrix R k : Where r represents the upper triangular matrix R k Elements of S14: Construct the moment characteristic matrix J of the integrated system reliability parameters: Where J represents the moment characteristic matrix of the integrated system reliability parameters; α j represents the jth first characteristic element of the moment characteristic matrix, and β j represents the jth second characteristic element of the moment characteristic matrix, and S15: Calculate the eigenvalue λ of the moment characteristic matrix J of the integrated system reliability parameter j With the feature vector v j =v j1 ,…,v jn T , j = 1, ..., n; get the initial model parameter set and corresponding weights, and the model input parameter x j and its corresponding weight ω j It consists of two parts, where x j =λ j , 3. The integrated system reliability assessment method based on nested moment integration according to claim 1, characterized in that: The input parameter coefficient s of the orthogonal polynomial in step S22 is an m-order upper triangular matrix R m The elements of the inverse matrix are: Calculate the first m-order origin moments and use the Hankel matrix H of the integrated system reliability parameters k Get the m-order upper triangular matrix R m : Calculate the upper triangular matrix R m The inverse matrix of : in, Represents the upper triangular matrix R m The inverse matrix of .

4. The integrated system reliability assessment method based on nested moment integration according to claim 1 is characterized in that: The Vandermonde coefficient matrix V in step S23 is obtained according to the orthogonal polynomial of the integrated system reliability parameter in step S22, specifically: The n input parameters x1,…,x n and moment integral nested points z1,…,z n+1 Substituting the orthogonal polynomial of the integrated system reliability parameter, we get the Vandermonde coefficient matrix V: Where V represents the Vandermonde coefficient matrix.

5. The integrated system reliability assessment method based on nested moment integration according to claim 1 is characterized in that: In step S24, the constraint optimization problem of the reliability parameters of the integrated system is converted into an unconstrained problem based on the penalty function method, and the second objective function is obtained. Specifically: Calculate the penalty term P of the second objective function, and set the constraint of the penalty function method as follows: the integral node needs to be in the set domain, the weight needs to be greater than 0, and the conditions are: Among them, P j represents the jth penalty term of the iteration termination condition; a and b represent the upper and lower bounds of the set domain respectively; Get the second objective function penalty term P: P=[P1…P 3n+2 ](14) Where P represents the penalty term of the second objective function; Calculate the penalty factor c of the penalty term: c=max10 3 ,1 / r 2 (15) Among them, max represents the maximum value function; Construct the second objective function 6. The integrated system reliability assessment method based on nested moment integration according to claim 1 is characterized in that: Iteration termination condition of penalty function method in step S24 for: in, is the iterative termination condition of the penalty function method; P j is the jth penalty term for the iteration termination condition.

7. The integrated system reliability assessment method based on nested moment integration according to claim 1 is characterized in that: The iterative optimization process of the penalty function method in step S24 is specifically as follows: Step S241: When m=3n+1, if the iteration termination condition Less than the preset value∈, all integration nodes fall within the set domain, the negative weight ratio of the model input parameter weight is less than 50%, and the minimum value is greater than -10 -4 , then the condition for continuing the iteration upward is satisfied, and the moment integral algebraic precision m is increased by 1, and the next round of iterative optimization of the penalty function method is carried out. If the above requirements are still not met at the maximum number of iteration steps, the condition for continuing the iteration downward is satisfied, and the moment integral algebraic precision m is reduced by 1, and the next round of iterative optimization of the penalty function method is carried out; Step S242: If the iteration termination condition And m<3n+1, or the iteration termination condition is reached at the maximum number of iterations If m>3n+1, the iteration termination condition is met. In the first case, m remains unchanged. In the second case, the algebraic precision of the moment integral m is reduced by 1, and the iterative optimization process of the penalty function method is terminated. Step S243: If m<3n+1 and the iteration is terminated at the maximum number of iterations If the condition for continuing iteration is met, the algebraic precision m of the moment integral is reduced by 1, and the next round of iterative optimization of the penalty function method is performed; if the iteration termination condition If m>3n+1, the condition for continuing to iterate upward is met. After the moment integral algebraic precision m is increased by 1, the next round of iterative optimization of the penalty function method is carried out.

8. An integrated system reliability assessment system according to the integrated system reliability assessment method based on nested moment integration according to any one of claims 1 to 7, characterized in that: Based on the nested moment integration method, the reliability of the integrated system is accurately evaluated using small samples, which includes: model startup unit, multi-order orthogonal construction unit, target optimization generation unit, comprehensive regularization solution unit, adaptive precision adjustment unit and global optimization adjustment unit; The model startup unit is used to generate an initial model parameter set of the integrated system reliability parameters, and obtain a moment characteristic matrix based on a moment integration method, wherein the eigenvalue of the moment characteristic matrix is ​​the initial model parameter set; The multi-order orthogonal construction unit is used to generate an arbitrary order orthogonal polynomial corresponding to an arbitrary probability density function, generate a set of orthogonal polynomial systems based on the Schakowsky theorem, and obtain three recursive formulas corresponding to the orthogonal polynomials; The target optimization generation unit is used to generate an unconstrained target optimization function. The initial target optimization function is obtained by multiplying the weight vector by the Vandermonde matrix constructed based on the orthogonal polynomial and performing a difference operation with the unit vector. The penalty function method is used to impose a cost for violating the constraint to convert it into an unconstrained final target optimization function. The comprehensive regularization solving unit is used to solve the target optimization function, and minimize the norm of the target optimization function based on the Gauss-Newton method and the regularization technique, so as to obtain a solution that satisfies the accuracy; The adaptive precision adjustment unit is used to generate a nested integration rule with the highest moment integral algebraic precision, and dynamically adjust the value of the moment integral algebraic precision according to the solution of the nested integration rule under the moment integral algebraic precision, so as to obtain the nested integration rule with the highest moment integral algebraic precision; The global optimization adjustment unit is used to output an accurate integrated system reliability estimation, calculate the relative change rate of the estimation result based on the model response estimation result obtained based on the initial model parameter set and the updated model parameter set, thereby dynamically adjusting the number of nesting times, and finally outputting an estimation result that meets the accuracy requirements.

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