Integrated System Reliability Evaluation Method and System Based on Nested Moment Integration

Abstract

The present invention belongs to the technical field of integrated system reliability evaluation, and relates to a method and system for integrated system reliability evaluation based on nested moment integration, which includes: S1, generating an initial model parameter set of integrated system reliability parameters; S2, establishing an updated model of integrated system reliability parameters based on nested moment integration, and outputting an integrated system reliability parameter set and weights; S3, establishing a response estimation model of integrated system nested moment integration, and outputting an integrated system reliability evaluation result. The present invention uses orthogonal polynomials to construct the objective function, enhances the numerical stability of the solution process, and improves the algebraic accuracy and the convergence of evaluating the current result by increasing the 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 result by gradually increasing the number of samples; the model evaluation results can all be reused.

Description

Technical Field

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

[0002] The reliability assessment of an integrated system depends on the quantification of the uncertainties of various factors of its components. By effectively quantifying the uncertainties, the reliability of the system can be more accurately evaluated and improved. The uncertainties in an integrated system may stem from various factors, including uncertainties in material properties, environmental uncertainties, uncertainties brought about by model simplification, and uncertainties caused by human factors, etc. By appropriately considering and quantifying these uncertainties, the behavior of the system can be accurately predicted, thereby effectively monitoring the system; as modern integrated systems become increasingly complex, the demand for computing resources in reliability assessment gradually increases. Therefore, in order to quickly obtain accurate reliability assessments with fewer model evaluation times in practical applications, this study proposes a method for reliability assessment of integrated systems based on nested moment integration. This method uses the nested moment integration rule to quantify the uncertainties in the integrated system, and through appropriate variable substitution, transforms the points of the nested moment integration rule into the model input parameter set of the integrated system, weights the corresponding results according to the weights of the nested moment integration rule, thereby obtaining the estimated moments of each order of the system response, and calculating the corresponding reliability index therefrom.

[0003] Methods for quantifying uncertainty in integrated systems can generally be divided into three categories. The first category is the uncertainty quantification method based on asymptotic estimation, including the second-order perturbation method and the Laplace approximation method. These methods fit the original distribution to a Gaussian distribution for subsequent analysis by performing a Taylor expansion at the peak of the distribution and ignoring higher-order terms. To ensure the effectiveness of the fitting, the original distribution is usually required to have asymptotic symmetry and unimodal characteristics, so there are certain limitations in dealing with distributions with large skewness or monotonic distributions. Since the peak is often unknown, it is necessary to calculate the derivative of the model response with respect to the uncertain variables to identify the peak. In addition, this method also needs to calculate the Hessian matrix of the original distribution at the identified peak. For a system with 3 to 5 uncertain variables, the entire process may require dozens to hundreds of model evaluations. The second category is the uncertainty quantification method based on Monte Carlo, which generates a large number of random samples and uses each set of samples to perform statistical analysis on the model. The accuracy of this method is proportional to the square root of the sample size. Therefore, 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 improvement 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 through a trial-and-error method, which still requires a large number of model runs, thus limiting their application in complex models. The third category is the uncertainty quantification method based on surrogate models, including the response surface method, the Kriging method, the polynomial chaos method, and the metamodel method. The core idea of this method is to establish a simplified model based on the relationship between the uncertain input parameters and the 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 needs to 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, and as the polynomial order increases, the number of model evaluations required may increase significantly, resulting in an increase in computational overhead. To date, a large number of studies have focused on these three types of problems, and the aforementioned methods are only a part of them.

[0004] Recently, in order to efficiently solve the uncertain quantification problem in complex computational models, a new moment integration quantization method has been proposed. The core idea of this method is to design Gauss integration rules for any input distribution and calculate the origin moments of all orders of the model response. For low-dimensional complex problems, this method can obtain results equivalent 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, by combining sparse tensor product rules, the computational cost can be reduced from exponential to polynomial. However, whether it is the moment integration method or its sparse variant, when increasing the integration nodes to improve the algebraic accuracy and the convergence of the evaluation results, since the integration nodes of two consecutive Gauss integration rules are staggered, the running results of the current model cannot be reused, resulting in a waste of computational resources. The nested moment integration quantization method proposed in this study allows model evaluation to start from a small sample and improve the accuracy of 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 through weighting to obtain a more accurate model estimate. In addition, for one-dimensional problems, this method has the least computational cost for judging the convergence of the current model estimation results, which is particularly useful when evaluating the results of a certain one-dimensional variable of the model. Summary of the Invention

[0005] Aiming at the deficiencies of the prior art, the present invention provides an integrated system reliability evaluation method and system based on nested moment integration, constructs an objective function using orthogonal polynomials to enhance the numerical stability of the solution process, and improves the algebraic accuracy and the convergence of the evaluation results by increasing the integration nodes. The proposed nested moment integration quantization method allows model evaluation to start from a small sample and improve the accuracy of 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 solutions:

[0007] An integrated system reliability evaluation method based on nested moment integration, which includes:

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

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

[0010] S2: Establish an updated model of the integrated system reliability parameters based on nested moment integration, and output the integrated system reliability parameter set and weights;

[0011] S21: Determine n input parameters x1, …, x of the initial model parameter set, n , where n is the number of input parameters, initialize the moment integration algebraic precision m = 3n + 1, and the moment integration nested points are z1, …, z n+1 and weights

[0012] S22: Calculate the orthogonal polynomial φ i (x) of the first m - order integrated system reliability parameters according to the distribution type of the random variable:

[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] where 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 integration nested point z; i is the input number of the orthogonal polynomial;

[0015] S23: Use the orthogonal polynomial of the integrated system reliability parameter 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 integration nested point weight, and e1 represents the identity matrix, and

[0018] S24: Based on the penalty function method, transform the constrained optimization problem of the integrated system reliability parameter into an unconstrained problem to obtain the second objective function of the integrated system reliability parameter update model as:

[0019]

[0020] where 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 for the nested moment integration of the integrated system, and output the integrated system reliability evaluation result;

[0023] Obtain the integrated system reliability parameter set obtained in step S2, construct a response estimation model for the nested moment integration of the integrated system, and the output integrated system reliability evaluation result is:

[0024]

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

[0026] Preferably, in step S1, according to the distribution type of the integrated system reliability parameter random variable, generate n initial integration rules under the corresponding probability density function to obtain the 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 kth moment about the origin μ k of the integrated system reliability parameter is expressed as:

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

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

[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 integrated system reliability parameter; 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 parameters j With the eigenvector v j =(v j1 ,…,v jn ) T, j = 1, …, n; obtain the initial model parameter set and corresponding weights, from the model input parameter x j and its corresponding weight ω j consisting of two parts, where x j = λ j ,

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

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

[0045]

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

[0047]

[0048] wherein, represents the inverse matrix of the upper triangular matrix R m .

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

[0050] Substitute the n input parameters x1, …, x n of the initial model parameter set in step S21 and the moment integral nested points z1, …, z n+1 into the orthogonal polynomial of the integrated system reliability parameters to obtain the Vandermonde coefficient matrix V:

[0051]

[0052] wherein, V represents the Vandermonde coefficient matrix.

[0053] Preferably, in step S24, the constrained optimization problem of the integrated system reliability parameters 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 constraints of the penalty function method that the integration nodes need to be in the set domain and the weights need to be greater than 0, and the conditions are:

[0055]

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

[0057] Obtain the penalty term P of the second objective function:

[0058] P = [P1 … P 3n+2 (14)

[0059] Among them, 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 iteration termination condition of the penalty function method in step S24 is:

[0065]

[0066] Among them, represents the iteration termination condition of the penalty function method; P j represents the j-th penalty term of 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 is less than the preset value ∈, and at the same time all integration nodes fall within the set domain, the negative weight ratio of the model input parameter weights is less than 50%, and the minimum value is greater than -10 -4 , then the condition for upward continuous iteration is satisfied. After adding 1 to the moment integration algebraic accuracy m, perform the iterative optimization of the next round of penalty function method; if the above requirements are still not met at the maximum number of iteration steps, then the condition for downward continuous iteration is satisfied. After subtracting 1 from the moment integration algebraic accuracy m, perform the iterative optimization of the next round of penalty function method;

[0069] Step S242: If the iteration termination condition and m < 3n + 1, or at the maximum number of iteration steps, the iteration termination condition If m > 3n + 1, the iteration termination condition is satisfied. In the first case, m remains unchanged. In the second case, the algebraic accuracy m of the moment integral is decreased by 1, and the iterative optimization process of the penalty function method is ended;

[0070] Step S243: If m < 3n + 1 and the iteration termination condition is met at the maximum number of iteration steps Then the condition for continuing the downward iteration is satisfied. After decreasing the algebraic accuracy m of the moment integral by 1, the iterative optimization of the next round of the penalty function method is carried out; If the iteration termination condition And m > 3n + 1, then the condition for continuing the upward iteration is satisfied. After increasing the algebraic accuracy m of the moment integral by 1, the iterative optimization of the next round of the penalty function method is carried out.

[0071] The second aspect of the present invention proposes an integrated system reliability evaluation system for an integrated system reliability evaluation method based on nested moment integral. Based on the nested moment integral method, it accurately evaluates the reliability of the integrated system using small samples, and it includes: a model startup unit, a multi-order orthogonal construction unit, a target optimization generation unit, a comprehensive regularization solution unit, an adaptive accuracy 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, obtain a moment feature matrix based on the moment integral method, and the eigenvalues of the moment feature matrix are the initial model parameter set;

[0073] 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 Sarkovskii's theorem, and obtain the three-term recurrence formula of the corresponding orthogonal polynomials;

[0074] The target optimization generation unit is used to generate an unconstrained target optimization function. The difference operation between the product of the Vandermonde matrix constructed based on orthogonal polynomials and the weight vector and the unit vector is the initial target optimization function. Based on the penalty function method, a cost is imposed for violating the constraints to transform it into an unconstrained final target optimization function;

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

[0076] The adaptive accuracy adjustment unit is used to generate a nested integration rule with the highest algebraic accuracy of the moment integral. According to the solution situation of the nested integration rule under the algebraic accuracy of the moment integral, the value of the algebraic accuracy of the moment integral is dynamically adjusted, so as to obtain a nested integration rule with the highest algebraic accuracy of the moment integral;

[0077] The global optimization and adjustment unit is used to output an accurate integrated system reliability estimate. Based on the model response estimate results obtained from the initial model parameter set and the updated model parameter set, it calculates the relative change rate of the estimate results, thereby dynamically adjusting the number of nesting times, and finally outputs an estimate 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 general nested integration rule for any input distribution, aiming to calculate the origin moments of each order of the model response, so as to conduct the reliability assessment of the integrated system. This nested integration rule is based on the Gaussian integration rule and is a nested integration rule with the smallest calculation increment for improving the algebraic accuracy. This innovation enables efficient and accurate random analysis when dealing with complex distributions, significantly enhancing the ability to evaluate the reliability of the integrated system.

[0080] (2) The present invention uses orthogonal polynomials to construct the objective function, greatly enhancing the numerical stability in the solution process. At the same time, it provides a simple method for obtaining the orthogonal polynomial expressions corresponding to any probability density function. This method not only improves the calculation accuracy and stability but also provides an effective solution for dealing with different probability density functions, expanding the applicable range of this technology in practical applications.

[0081] (3) The nested moment integration quantization method proposed by the present invention 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 at each step can be reused and combined with the new model evaluation quantities by weighting to obtain a more accurate model estimate. BRIEF DESCRIPTION OF THE DRAWINGS

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

[0083] Figure 2 is a schematic diagram of the integrated system for controlling the water temperature in a boiler, which is a specific embodiment of the present invention;

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

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

[0086] Figure 5 is a schematic diagram of the redundant flight control integrated system, which is another specific embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

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

[0088] Taking a four-branch series integrated circuit system as an example, an integrated system reliability evaluation method based on nested moment integration is provided in an embodiment of the present invention. As Figure 1 shown, an initial model parameter set of integrated system reliability parameters is generated; an integrated system reliability parameter update model based on nested moment integration is established to output an integrated system reliability parameter set and weights; an integrated system nested moment integration response estimation model is established to output an integrated system reliability evaluation result, which 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 integrated circuit system reliability parameter, n-point initial integration rules under the corresponding probability density function are generated to obtain an initial model parameter set of the integrated circuit system reliability parameter. The initial model parameter set includes input parameter x j and corresponding weight ω j , where 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-th order origin moment μ k of the integrated circuit system reliability parameter is expressed as:

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

[0093] where 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, and k respectively represent the random variable, random variable value, and origin moment order of the integrated circuit system reliability parameter.

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

[0095]

[0096] Among them, p n represents the sample moment and the number of samples 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: According to the k-th order origin moment μ k of the reliability parameter of the integrated circuit system, construct the Hankel matrix H k of the reliability parameter of the integrated circuit system:

[0098]

[0099] Among them, H k represents the Hankel matrix of the reliability parameter of the integrated circuit system.

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

[0101]

[0102] Among them, r represents the element of the upper triangular matrix R k .

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

[0104]

[0105] Among them, J represents the moment characteristic matrix of the reliability parameter of the integrated circuit system; α j represents the j-th first characteristic element of the moment characteristic matrix, and β j represents the j-th second characteristic element of the moment characteristic matrix, and

[0106] Step S15: Calculate the eigenvalues λ j and eigenvectors v j =(v j1 ,…,v jn ) T of the moment characteristic matrix J of the reliability parameter of the integrated circuit system, j = 1,…,n; obtain the initial model parameter set and the corresponding weights, which are composed of the model input parameter x j and its corresponding weight ω j , where x j =λ j ,

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

[0108] Step S21: Determine n input parameters x1, …, x of the initial model parameter set, n , where n is the number of input parameters, initialize the algebraic precision of moment integration m = 3n + 1, and the nested points of moment integration are z1, …, z n+1 and weights

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

[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] where 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 nested points of moment integration z; i is the input number of the orthogonal polynomial.

[0112] The input parameter coefficient s of the orthogonal polynomial is the element of the inverse matrix of the m - order upper triangular matrix R m , specifically:

[0113] Calculate the first m - order origin moments, and obtain the m - order upper triangular matrix R k according to the Hankel matrix H m of the reliability parameter of the integrated circuit system:

[0114]

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

[0116]

[0117] where represents the upper triangular matrix R mThe inverse matrix of .

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

[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 integrated circuit system reliability parameters is converted into an unconstrained problem to obtain the second objective function Specifically:

[0126] Calculate the penalty term P of the second objective function. Set the constraint of the penalty function method as follows: the integration node must be in the set domain, the weight must be greater than 0, and the conditions must be met:

[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] Among them, 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 is less than the preset value ∈, and at the same time all integration nodes fall within the set domain, the negative weight ratio of the model input parameter weights is less than 50%, and the minimum value is greater than -10 -4 , then the condition for continuing the upward iteration is satisfied. After adding 1 to the moment integration algebraic accuracy m, the iterative optimization of the penalty function method is carried out in the next round; if the above requirements are still not met at the maximum number of iteration steps, then the condition for continuing the downward iteration is satisfied. After subtracting 1 from the moment integration algebraic accuracy m, the iterative optimization of the penalty function method is carried out in the next round; the iteration termination condition of the penalty function method is:

[0139]

[0140] Among them, represents the iteration termination condition of the penalty function method; P j represents the j-th penalty term of the iteration termination condition.

[0141] Step S242: If the iteration termination condition and m < 3n + 1, or at the maximum number of iteration steps the iteration termination condition and m > 3n + 1, then the iteration termination condition is satisfied. In the first case, m remains unchanged, and in the second case, the moment integration algebraic accuracy m is subtracted by 1, and the iterative optimization process of the penalty function method ends.

[0142] Step S243: If m < 3n + 1 and at the maximum number of iteration steps the iteration termination condition then the condition for continuing the downward iteration is satisfied. After subtracting 1 from the moment integration algebraic accuracy m, the iterative optimization of the penalty function method is carried out in the next round; if the iteration termination condition and m > 3n + 1, then the condition for continuing the upward iteration is satisfied. After adding 1 to the moment integration algebraic accuracy m, the iterative optimization of the penalty function method is carried out in the next round.

[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 for the nested moment integral of the integrated circuit system and output the reliability evaluation result of the integrated circuit system.

[0145] Obtain the reliability parameter set of the integrated circuit system obtained in Step S2, construct a response estimation model for the nested moment integral of the integrated circuit system, and output the reliability evaluation result of the integrated circuit system as:

[0146]

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

[0148] In a specific embodiment, for the reliability problem of the water temperature control integrated system in a boiler, the schematic diagram of the water temperature control integrated system is as Figure 2 shown. It is required that the water temperature in the boiler inner tank be stabilized 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, and finally the heating of the heating wire is adjusted by controlling the voltage of the heat adjustment device. The reliability of the water temperature control integrated system is jointly determined by the reliabilities of four components: the power supply, the temperature sensor, the monitoring device, and the heating device. Assume that the performance of each component can be represented by the first variable x1 and the second variable x2 according to their respective material characteristics. Among them, the input variables affecting the four components show two types of correlation relationships: non-linear or linear, and they have four limit states. The mathematical expression of the structural function for constructing 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] Where τ represents Figure 3 The set of integration nodes of the integration rules at all levels 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 and 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 based on M r3 Indicates that in M r2 The results are obtained by nesting them based on the continuous nesting results. 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 The final result is obtained. 6 For comparison, the Monte Carlo method of Figure 4 As shown, MC represents the result of Monte Carlo method. At this time, the response of the system is close to the normal distribution. Therefore, the reliability calculated by 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. Because drones place special demands on the reliability of their flight control computer systems, a redundant architecture design is employed to ensure flight safety and mission capability. Redundant architecture design involves intentionally adding additional components or functions during design and construction to improve the reliability and availability of the integrated system. The fundamental concept is to reduce or prevent the impact of system failures through backup and replacement.

[0157] A schematic diagram of a redundant flight control integrated system is shown in the following figure: 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 5 modules in the main module group. When a certain module in the main module group fails, the redundant management module accesses a certain module in the redundant module group to replace it and participate in the voting. When the number of modules participating in the voting 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 independently and identically distributed, the reliability of the system is:

[0158]

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

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

[0161] ξ = 0.04τ + 0.86(20)

[0162] Among them, τ represents Figure 3 the integration node set of the integration rules at all levels in; ξ represents the model input parameter set of this problem.

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

[0164]

[0165] Table 2

[0166] The second aspect of the embodiments of the present invention provides an integrated system reliability evaluation system for an integrated system reliability evaluation method based on nested moment integration. Based on the nested moment integration method, it can accurately evaluate the reliability of an integrated circuit system using small samples. It 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 reliability parameters of the integrated circuit system, and obtain a moment feature matrix based on the moment integration method. The eigenvalues of the moment feature matrix are 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 Sharkovsky's theorem, and obtain the three-term recurrence formula of the corresponding orthogonal polynomials.

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

[0170] The comprehensive regularization solution 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 regularization techniques to obtain a solution that meets the accuracy requirements.

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

[0172] The global optimization adjustment unit is used to output an accurate reliability estimate of the integrated circuit system. Based on the model response estimate results obtained from the initial model parameter set and the updated model parameter set, calculate the relative change rate of the estimate results, and thus dynamically adjust the number of nesting times, and finally output an estimate 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 evaluation method based on nested moment integration, which constructs an objective function using orthogonal polynomials to enhance the numerical stability of the solution process; the nested moment integration quantization method proposed in this study allows model evaluation to start from small samples, improves the accuracy of prediction results by gradually increasing the number of samples, enables the reuse of the model evaluation results at each step, and obtains a more accurate model estimate by weighted combination with new model evaluation quantities; for low-dimensional complex problems, it can obtain results equivalent 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, by combining sparse tensor product rules, the computational cost is reduced from exponential to polynomial level; through the computational analysis of integrated series circuits and redundant systems, it is proved that this method has good practical use effects and meets the actual engineering requirements.

[0174] The embodiments described above are only descriptions of the preferred embodiments of the present invention and do not limit the scope of the present invention. Without departing from the design spirit of the present invention, various deformations and improvements made by those of ordinary skill in the art to the technical solutions of the present invention shall fall within the protection scope determined by the claims of the present invention.

Claims

1. An integrated system reliability evaluation method based on nested moment integration, characterized in that It includes: S1: Generate an initial model parameter set for the reliability parameters of the integrated system; According to the distribution type of the random variables of the reliability parameters of the integrated system, generate the initial integration rules for n points under the corresponding probability density function, and obtain the initial model parameter set of the reliability parameters of the integrated system. The initial model parameter set includes input parameter x j and the corresponding weight ω j , where j is a constant; S2: Establish an updated model for the reliability parameters of the integrated system based on nested moment integration, and output the reliability parameter set and weights of the integrated system; S21: Determine n input parameters x1, …, x of the initial model parameter set, where n is the number of input parameters, initialize the moment integral algebraic precision m = 3n + 1, and the moment integral nested points are z1, …, z n , and the weight w = ω1, …, ω n+1 ; 2n+1 T ; S22: Calculate the orthogonal polynomial φ of the first m - order integrated system reliability parameters according to the distribution type of the random variable 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 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 nested point z; i is the input number of the orthogonal polynomial; S23: Use the orthogonal polynomials of the reliability parameters of the integrated system to construct the first objective function r of the updated model for the reliability parameters of the integrated system: r = V·w - e1(2) where r represents the first objective function; V represents the Vandermonde coefficient matrix; w represents the moment integral nested point weights, and w = ω1, …, ω 2n+1 T ; e1 represents the identity matrix, and e1 = 1, 0, …, 0 T ; S24: Convert the constrained optimization problem of the integrated system reliability parameters into an unconstrained problem based on the penalty function method, and obtain the second objective function of the integrated system reliability parameter update model. Among them, 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 for the nested moment integration of the integrated system, and output the reliability evaluation result of the integrated system; Obtain the reliability parameter set of the integrated system obtained in step S2, construct a response estimation model for the nested moment integration of the integrated system, and the output reliability evaluation result of the integrated system is: 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; Ξ j represents the j-th response matrix of the response estimation model, and ω j represents the j-th weight matrix of the response quantity of the response estimation model, and θ j represents the correction parameter for reliability evaluation; Ω represents the set of reliability parameters of the integrated system.

2. The reliability evaluation method of the integrated system based on nested moment integration according to claim 1, characterized in that: In step S1, according to the distribution type of the random variables of the reliability parameters of the integrated system, generate n-point initial integration rules under the corresponding probability density function to obtain the initial model parameter set, specifically: S11: When the probability density function of the random variable X of the integrated system reliability parameter is p(x), the k-th order origin moment μ of the integrated system reliability parameter k is expressed as: μ k = EX k = ∫ x∈Ω x k pxdx(5) Among them, μ k represents the k-th 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, and k respectively represent the random variable, the numerical value of the random variable, and the order of the origin moment of the reliability parameter of the integrated system; When the probability density function of the random variable of the reliability parameter of the integrated system is unknown, the k-th order origin moment μ of the reliability parameter of the integrated system k is equal to the sample moment of the reliability parameter of the integrated system Among them, p n represents the sample moment and the number of samples 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: Construct the Hankel matrix H of the reliability parameters of the integrated system according to the k-th origin moment μ of the reliability parameters of the integrated system k : k ​ Among them, H k represents the Hankel matrix of the integrated system reliability parameters; S13: Perform Cholesky decomposition on the Hankel matrix H of the integrated system reliability parameters k to obtain an upper triangular matrix R k : where r represents the elements of the upper triangular matrix R k ; S14: Construct the moment characteristic matrix J of the reliability parameters of the integrated system; Among them, J represents the moment characteristic matrix of the reliability parameters of the integrated system; α j represents the j-th first characteristic element of the moment characteristic matrix, and β j represents the j-th second characteristic element of the moment characteristic matrix, and S15: Calculate the eigenvalues λ of the moment characteristic matrix J of the integrated system reliability parameters j and the eigenvectors v j = v j1 , …, v jn T , j = 1, …, n; obtain the initial model parameter set and the corresponding weights. The model input parameters consist of x j and its corresponding weights ω j . Among them, x j = λ j , 3. The integrated system reliability evaluation 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 Elements of the inverse matrix, specifically: Calculate the first m origin moments and obtain the upper triangular matrix R of order m based on the Hankel matrix H of the integrated system reliability parameters k m :​ Calculate the inverse matrix of the upper triangular matrix R m : Among them, represents the inverse matrix of the upper triangular matrix R m .

4. The reliability evaluation method of the integrated system based on nested moment integration according to claim 1, wherein: The Vandermonde coefficient matrix V in step S23 is obtained according to the orthogonal polynomials of the reliability parameters of the integrated system in step S22, specifically: Substitute the n input parameters x1, …, x of the initial model parameter set in step S21 n and the moment integral nested points z1, …, z n+1 into the orthogonal polynomial of the integrated system reliability parameters to obtain the Vandermonde coefficient matrix V: Where, V represents the Vandermonde coefficient matrix.

5. The reliability evaluation method of the integrated system based on nested moment integration according to claim 1, wherein: In step S24, the constrained optimization problem of the integrated system reliability parameters is converted into an unconstrained problem based on the penalty function method to obtain the second objective function Specifically: Calculate the penalty term P of the second objective function, and set the constraints of the penalty function method that the integration nodes need to be in the set domain and the weights need to be greater than 0. The conditions are satisfied as: where P j represents the j-th penalty term for the iteration termination condition; a and b respectively represent the upper and lower bounds of the setting domain; Obtain the penalty term P of the second objective function: 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) Where, max represents the maximum value function; Construct the second objective function 6. The integrated system reliability evaluation method based on nested moment integration according to claim 1, characterized in that: Iterative termination condition of the penalty function method in step S24 is as follows: Among them, is the iteration termination condition of the penalty function method; P j is the j-th penalty term of the iteration termination condition.

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

8. An integrated system reliability evaluation system for the integrated system reliability evaluation 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, accurately evaluate the reliability of the integrated system using small samples, which includes: a model startup unit, a multi-order orthogonal construction unit, a target optimization generation unit, a comprehensive regularization solution unit, an adaptive accuracy adjustment unit, and a global optimization adjustment unit; The model startup unit is used to generate an initial model parameter set for the reliability parameters of the integrated system, obtain the moment characteristic matrix based on the moment integration method, and the eigenvalues of the moment characteristic matrix are the initial model parameter set; 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 Sharkovsky's theorem, and obtain the three-term recurrence formula of the corresponding orthogonal polynomials; The target optimization generation unit is used to generate an unconstrained target optimization function. The difference operation between the product of the Vandermonde matrix constructed based on orthogonal polynomials and the weight vector and the unit vector is the initial target optimization function. Based on the penalty function method, a cost is imposed for violating the constraints to convert it into an unconstrained final target optimization function; The comprehensive regularization solution 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 regularization techniques to obtain a solution that meets the accuracy; The adaptive accuracy adjustment unit is used to generate a nested integration rule with the highest moment integration algebraic accuracy, and dynamically adjust the value of the moment integration algebraic accuracy according to the solution of the nested integration rule under the moment integration algebraic accuracy, so as to obtain a nested integration rule with the highest moment integration algebraic accuracy; The global optimization and adjustment unit is used to output an accurate integrated system reliability estimate. Based on the model response estimation results obtained from the initial model parameter set and the updated model parameter set, it calculates the relative change rate of the estimation results, thereby dynamically adjusting the number of nesting times, and finally outputs an estimation result that meets the accuracy requirements.

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