CN116629010B - Degradation model confirmation and test design method based on random process - Google Patents
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Abstract
The invention discloses a degradation model confirmation and test design method based on a random process, which adopts complete degradation data in the whole time domain to establish degradation model confirmation measurement indexes, thereby improving the reliability of degradation model confirmation; the invention establishes a degradation model confirmation measurement method based on a random process, introduces nuclear density estimation, and reduces the requirement on the number of test samples; the invention fully considers the design variables of the number and the distribution of the observation time and the number of the test samples, thereby improving the accuracy of the confirmation test; the invention provides a degradation model confirmation test optimization method based on a random process, which is used for carrying out confirmation test design by considering a degradation model with random uncertainty, and simultaneously provides a collaborative optimization algorithm.
Description
Technical Field
The invention relates to a degradation model confirmation technology, in particular to a degradation model confirmation based on a random process and a test design method thereof.
Background
The product life prediction method based on performance degradation is an effective supplement to the traditional product life prediction method, and is an important way for solving the problem of high-reliability and long-service-life product life prediction. The mathematical model describing the degradation process of the product performance is called degradation model, and its reliability directly affects the accuracy of product life prediction. How to quantitatively describe the accuracy and reliability of a degradation model is an open subject. Model validation is the determination of how well a model can accurately describe the real physical world process from the perspective of model use. The existing model confirmation method mainly comprises four types: classical hypothesis testing, bayesian factors, frequency indices, and area metrics. These methods are only applicable to normal models.
Product performance exhibits uncertainty and time-variability in the degradation process due to various uncertainty factors from the manufacturing, processing, and external environments, and the degradation process is typically described using a stochastic process. In recent years, a small number of researchers have studied dynamic model validation metrics. These studies have generally represented the entire stochastic process by constructing different decomposition formulas (e.g., principal component analysis) and the model by reducing dimensions in the time domain to obtain a series of coefficients. And then comparing the simulation model coefficient with the test observation coefficient to obtain the confirmed model measurement. This necessarily loses data information, in particular time information, whereas complete degradation data throughout the time domain is crucial for evaluating degradation model accuracy.
The degradation model confirms that the test design has two conflicting goals of reliability and cost, and a method capable of minimizing the cost of the confirmation test under the condition that the reliability of the confirmation test is satisfied is needed.
Disclosure of Invention
The invention aims to: the invention aims to provide a degradation model confirmation and test design method based on a random process, so that a basis is provided for the confirmation measurement of the degradation model, and the cost of the degradation model confirmation test is reduced under the condition that the reliability of the confirmation test is met.
The technical scheme is as follows: the invention relates to a degradation model confirmation method based on a random process, which comprises the following steps:
s1: m groups of random samples are extracted, and a simulation model T is evaluated S Time of day system response Y s ；
S2: taking N test products, respectively testing T E Obtaining the performance parameters of the product at the moment and obtaining T times of tests E Observation data Y at observation time e And fitting to obtain N times of tests T S Time interpolation test data
S3: obtaining T S Probability density function K of time simulation response s Probability density function K of test data e ；
S4: calculate T S Time simulation response interval I s Evaluation of T S Time trial data interval I e ；
S5: and obtaining a degradation model confirmation measurement index by adopting the superposition rate of the time domain simulation response and the test data probability density function, and judging whether the prediction model is a real degradation model.
The probability density function K of the test data described in the step S3 e Obtained using a nuclear density estimate.
The degradation model in step S5 confirms that the measurement index isWhere A (-) represents the area between the probability density function and the x-axis and U represent the intersection and union, respectively.
A degradation model confirmation test design method based on a random process comprises the following steps:
s1: taking an optimization target as test cost, confirming a measurement index as constraint, and designing variables including the number N of test samples and the number N of test observation moments E And experimental observation time T E The experimental design optimization model is as follows:
min C E (N,N E ,T E )
s.t.E N [W(K s ,K e (N,N E ,T E ))]≥η
N min ≤N≤N max
0≤T i ≤T max (i＝1,…,N E )
wherein E is N [·]Is the expected value of the confirmation index, N min And N max Respectively are provided withFor the minimum and maximum number of samples tested,and->Respectively represent the minimum and maximum observation time quantity, T max Is the maximum experimental observation time; η is a predetermined validation index; total cost C E (N,N E ,T E )＝C u ·N+C m ·N·N E ，C u And C m The cost of a test sample and a test observation are respectively;
s2: according to the total cost of the test C E (N,N E ,T E ) By minimizing the number of test samples N and the number of test observation times N E Number, lowest test cost can be obtained; the experimental design optimization model can be converted into a multi-objective optimization model:
min N,N E
s.t.E N [W(K s ,K e (N,N E ,T E ))]≥η
N min ≤N≤N max
0≤T i ≤T max (i＝1,…,N E )
in the multi-objective optimization model, N and N E Is a discrete variable, T E Is a variable-dimension continuous variable, which leads to a variable-dimension mixed variable optimization problem, and a collaborative optimization method is selected for solving.
The collaborative optimization method in the step S2 comprises the following steps:
s1: setting initial parameters of test design including N min ，N max ，And eta;
s2: selecting the number of test samples n=n max Number of test observation times
S3: optimization test observation time T E The step takes the model confirmation measurement index as an optimization target, and tests the observation time T E For optimizing variables, the optimization model is:
max E N [W(K s ,K e (N,N E ,T E ))]
s.t.0≤T i ≤T max (i＝1,…,N E )
wherein E is N [·]Is the expected value of the degradation model validation metric;
s4: confirming the measurement index E if the degradation model of the current optimal test scheme N [W(K s ,K e (N,N E T, E ))]Recording the optimal test observation time T under the number of test samples and the number of test observation time when the number of test samples is larger than or equal to eta E Step S5 is entered; otherwise, increasing the number N of test observation moments E ＝N E +1, returning to step S3;
s5: optimizing the number N of test samples, wherein the step takes the number N of test samples as an optimization target, confirms the measurement index as a constraint, and the optimization model is as follows:
min N
s.t.E N [W(K s ,K e (N,N E ,T E ))]≥η；
N min ≤N≤N max
s6: recording the current optimal test scheme including the number N of test samples and the number N of test observation moments E And experimental observation time T E ；
S7: if the number of test observation times N E The number of test observation moments is increased by less than the number of maximum test observation moments E ＝N E +1, returning to step S3; otherwise, enter step S8;
s8: and obtaining a Pa Lei Tuojie set meeting the reliability of the confirmation test, calculating the test cost of each test scheme, and selecting the optimal test scheme.
The beneficial effects are that: compared with the prior art, the invention has the following advantages:
1. the invention adopts the complete degradation data in the whole time domain to establish the degradation model confirmation measurement index, thereby improving the reliability of the degradation model confirmation.
2. The invention establishes a degradation model confirmation measurement method based on a random process, introduces nuclear density estimation, and reduces the requirement on the number of test samples.
3. The invention fully considers the design variables of the number and the distribution of the observation time and the number of the test samples, and improves the accuracy of the confirmation test.
4. The invention provides a degradation model confirmation test optimization method based on a random process, which is used for carrying out confirmation test design by considering a degradation model with random uncertainty, and simultaneously provides a collaborative optimization algorithm.
Drawings
FIG. 1 is a schematic representation of the coincidence rate of time t-test and simulated probability density functions;
FIG. 2 is a flowchart of a degradation model validation experiment design based on a stochastic process;
FIG. 3 is a schematic view of a composite laminate;
fig. 4 is a validation test optimal solution set.
Detailed Description
The technical scheme of the invention is further described below with reference to the accompanying drawings.
A degradation model confirmation method based on a random process comprises the following steps:
extracting M groups of simulation parameter samples X S ＝[X 1 ,X 2 ,…,X M ]，X i (i=1, 2, …, M) can contain random variables, random fields and random processes. Substituting M groups of parameter samples into degradation model to calculate response In order to eliminate numerical errors of continuous data simulated by discrete data, the number of simulation model observation moments is increased appropriately.
Calculate T S Time simulation response intervalAnd probability density function
I s (t i )＝[μ s (t i )-3σ s (t i )μ s (t i )+3σ s (t i )](i＝1,2,…,N S )
Wherein mu s (t i ) Sum sigma s (t i ) Respectively t i Mean and standard deviation of time-of-day simulation response.
Simulation response probability density function K s (t i )(i＝1,2,…,N S ) Obtained using a histogram evaluation.
Taking N test products to obtain N tests T E Test data at the time of observationFitting to obtain N times of tests T S Time interpolation test data
Calculate T S Time trial data intervalAnd probability density function
Wherein x is min (t i ) And x max (t i ) Respectively t i Minimum and maximum values of time trial data.
Probability density function K of test sample e (t i )(i＝1,2,…,N S ) The method is obtained by adopting nuclear density estimation and comprises the following steps:
set Y 1 ,...,Y N For N independent random variables distributed in the same way, K e As a probability density function thereof, K e Nuclear density estimation of (2)Can be expressed as
Wherein, K (·) is a kernel function, h is a bandwidth, and is used to control the smoothness of the kernel function. Currently, the commonly used kernel functions are trigonometric kernel functions, uniform kernel functions, epanechikov kernel functions, triweight kernel functions, gaussian kernel functions, etc., wherein the Gaussian kernel functions are expressed as
Obtaining a degradation model confirmation measurement index based on a random process by using the coincidence rate between probability density functions of the random process of the simulation response and experimental observation data in a time domain, wherein the degradation model confirmation measurement index is expressed as
Where A (-) represents the area between the probability density function and the x-axis. The coincidence rate of the test and simulation probability density functions at the moment t is shown in figure 1. The confirmation measure index is a dimensionless model confirmation index, and canThe method can be used for simultaneously evaluating degradation indexes of a plurality of different dimensions, and the values of the degradation indexes are divided into two cases: (1) The simulated response does not overlap the interval of the test data, where W (K s ,K e ) =0, the simulation model is completely untrusted; (2) The simulated response coincides with the interval of the test data, where 0 < W (K s ,K e )≤1，W(K s ,K e ) The larger represents the closer the simulation model is to the actual physical model, for an ideal simulation model, W (K s ,K e )＝1。
The model confirmation index is the core of a confirmation test, and the degradation model confirmation test design method based on the random process establishes a confirmation test design optimization model by using prior information obtained from a prediction model. According to the above procedure for establishing the validation index, in the validation test, the design variables include the number of test samples N, the number of test observation times N E And experimental observation time T E 。
The experimental design optimization model is as follows:
min C E (N,N E ,T E )
s.t.E N [W(K s ,K e (N,N E ,T E ))]≥η
N min ≤N≤N max
0≤T i ≤T max (i＝1,…,N E )
wherein E is N [·]Is the expected value of the confirmation index, N min And N max Respectively the minimum value and the maximum value of the number of test samples,and->Respectively represent the minimum and maximum observation time quantity, T max Is the largest testObserving time; η is a predetermined confirmation index, the larger the value of η is, meaning that the more stringent the requirements for model confirmation are; total cost C E (N,N E ,T E )＝C u ·N+C m ·N·N E ，C u And C m The cost of one test sample and one test observation, respectively.
According to the total cost of the test C E (N,N E ,T E ) By minimizing the number of test samples N and the number of test observation times N E The number, the lowest test cost is obtained. The experimental design optimization model is converted into a multi-objective optimization model:
min N,N E
s.t.E N [W(K s ,K e (N,N E ,T E ))]≥η
N min ≤N≤N max
0≤T i ≤T max (i＝1,…,N E )
in the multi-objective optimization model, N and N E Is a discrete variable, T E Is a continuous variable of varying dimensions, which results in the difficulty of solving the optimization model. For the problem of optimizing the variable-dimension mixed variables, the invention relates to a collaborative optimization method, which optimizes the test observation time and the test sample number respectively, and comprises the following steps as shown in fig. 2, wherein the steps are as follows:
step 1: setting initial parameters of test design including N min ，N max ，And eta.
Step 2: selecting the number of test samples n=n max Number of test observation times
Step 3: optimization testTime T of observation E The step takes the model confirmation measurement index as an optimization target, and tests the observation time T E For optimizing variables, the optimization model is:
max E N [W(K s ,K e (N,N E ,T E ))]
s.t.0≤T i ≤T max (i＝1,…,N E )
the optimization model is a continuous optimization problem, and the invention adopts a particle swarm algorithm to solve.
Step 4: metrics E of the current optimal test scheme N [W(K s ,K e (N,N E ,T E ))]Recording the optimal test observation time T under the number of test samples and the number of test observation time when the number of test samples is larger than or equal to eta E Step 5 is entered; otherwise, increasing the number N of test observation moments E ＝N E +1, go back to step 3.
Step 5: optimizing the number N of test samples, wherein the step takes the number N of test samples as an optimization target, confirms the measurement index as a constraint, and the optimization model is as follows:
min N
s.t.E N [W(K s ,K e (N,N E ,T E ))]≥η
N min ≤N≤N max
the optimization model is a discrete optimization problem, and the invention adopts an enumeration method to solve.
Step 6: recording the current optimal test scheme including the number N of test samples and the number N of test observation moments E And experimental observation time T E 。
Step 7: if the number of test observation times N E The number of test observation moments is increased by less than the number of maximum test observation moments E ＝N E +1, returning to step 3; otherwise, step 8 is entered.
And 8, obtaining a Pareto (Pareto) solution set meeting the confirmation test reliability, calculating the test cost of each test scheme, and selecting an optimal test scheme.
Confirmation test design method example:
taking the fatigue test of the woven scrim/epoxy composite laminate at 60% stress level as an example, a schematic of the test specimen is shown in fig. 3. The stiffness degradation of the composite material under fatigue load is described by the wiener process, expressed as
E(c)＝E 0 (1-(μ(c)+σB(c)))
Wherein E is 0 For the material stiffness corresponding to the initial cycle, c represents the normalized fatigue cycle parameter, c=n/N f Wherein N and N f The number of application cycles and the final settling period, respectively. Sigma is the diffusion parameter. Mu (c) is the drift parameter, expressed as
Wherein E is d ＝0.3E 0 Is the final degradation of the modulus of elasticity. q and p are model parameters. For the experimental model, the relevant parameters are shown in table 1. Table 2 lists the relevant parameters of the model validation test design and fig. 4 shows the Pareto solution set for the validation test.
TABLE 1 parameters of degradation model
Table 2 parameters relating to the test design
The minimum number of experimental observations is 14 and at least 98 experimental samples are required. When the number of test observation times was 20, the number of test samples was reduced to 30. The cost of all test protocols in the Pareto solution set was calculated according to table 2, the lowest test cost being 9280$ (n=31, N E ＝19，)。
Claims (3)
1. A degradation model confirmation method based on a random process, comprising the steps of:
s1: m groups of random samples are extracted, and a simulation model T is evaluated S Time of day system response Y s ；
S2: taking N test products, respectively testing T E Obtaining the performance parameters of the product at the moment and obtaining T times of tests E Observation data Y at observation time e And fitting to obtain N times of tests T S Time interpolation test data
S3: obtaining T S Probability density function K of time simulation response s Probability density function K of test data e ；
S4: calculate T S Time simulation response interval I s Evaluation of T S Time trial data interval I e ；
S5: obtaining a degradation model confirmation measurement index by adopting the coincidence rate of the time domain simulation response and the test data probability density function, and judging whether the prediction model is a real degradation model or not; the degradation model confirms that the measurement index isWhere A (-) represents the area between the probability density function and the x-axis and U represent the intersection and union, respectively.
2. The method for determining a degradation model based on a random process according to claim 1, wherein the probability density function K of the test data in step S3 e Obtained using a nuclear density estimate.
3. The degradation model confirmation test design method based on the random process is characterized by comprising the following steps of:
s1: trial-forming with optimization targetsThe method confirms that the measurement index is constraint, and the design variables comprise the number N of test samples and the number N of test observation moments E And experimental observation time T E The experimental design optimization model is as follows:
minC E (N，N E ，T E )
s.t.E N [W(K s ,K e (N,N E ，T E ))]≥η
N min ≤N≤N max
0≤T i ≤T max
wherein i=1, …, N E ，E N [·]Is the expected value of the confirmation index, N min And N max Respectively the minimum value and the maximum value of the number of test samples,and->Respectively represent the minimum and maximum observation time quantity, T max Is the maximum experimental observation time; η is a predetermined validation index; total cost C E (N，N E ，T E )＝C u ·N+C m ·N·N E ，C u And C m The cost of a test sample and a test observation are respectively;
s2: according to the total cost of the test C E (N,N E ,T E ) By minimizing the number of test samples N and the number of test observation times N E Number, lowest test cost can be obtained; the experimental design optimization model can be converted into a multi-objective optimization model:
minN,N E
s.t.E N [W(K s ,K e (N，N E ,T E ))]≥η
N min ≤N≤N max
0≤T i ≤T max
wherein i=1, …, N E In the multi-objective optimization model, N and N E Is a discrete variable, T E Is a continuous variable with variable dimension, which leads to the problem of optimizing the mixed variable with variable dimension, and a collaborative optimization method is selected for solving;
s2.1: setting initial parameters of test design including N min ，N max ，And eta;
s2.2: selecting the number of test samples n=n max Number of test observation times
S2.3: optimization test observation time T E The step takes the model confirmation measurement index as an optimization target, and tests the observation time T E For optimizing variables, the optimization model is:
max E N [W(K s ,K e (N,N E ,T E ))]
s.t.0≤T i ≤T max
wherein E is N [·]Is the expected value of the degradation model validation metric; i=1, …, N E ；
S2.4: confirming the measurement index E if the degradation model of the current optimal test scheme N [W(K s ,K e (N,N E ,T E ))]Recording the optimal test observation time T under the number of test samples and the number of test observation time when the number of test samples is larger than or equal to eta E Step S2.5 is entered; otherwise, increasing the number N of test observation moments E ＝N E +1, returning to step S2.3;
s2.5: optimizing the number N of test samples, wherein the step takes the number N of test samples as an optimization target, confirms the measurement index as a constraint, and the optimization model is as follows:
s2.6: recording the current optimal test scheme including the number N of test samples and the number N of test observation moments E And experimental observation time T E ；
S2.7: if the number of test observation times N E The number of test observation moments is increased by less than the number of maximum test observation moments E ＝N E +1, returning to step S2.3; otherwise, enter step S2.8;
s2.8: and obtaining a Pa Lei Tuojie set meeting the reliability of the confirmation test, calculating the test cost of each test scheme, and selecting the optimal test scheme.
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