CN110135085B - Fatigue crack evaluation method based on adaptive kernel density estimation auxiliary particle filter - Google Patents

Fatigue crack evaluation method based on adaptive kernel density estimation auxiliary particle filter Download PDF

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CN110135085B
CN110135085B CN201910419770.2A CN201910419770A CN110135085B CN 110135085 B CN110135085 B CN 110135085B CN 201910419770 A CN201910419770 A CN 201910419770A CN 110135085 B CN110135085 B CN 110135085B
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fatigue crack
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CN110135085A (en
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林焱辉
焦鑫磊
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Beihang University
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Abstract

The invention provides a fatigue crack evaluation method based on adaptive kernel density estimation auxiliary particle filtering, which comprises the following steps: determining a key evaluation component, constructing a state equation and an observation equation of a fatigue crack random expansion model, constructing a nonlinear state space model to describe the evaluation process of fatigue crack expansion, and performing crack expansion state simulation calculation on the nonlinear state space model of the fatigue crack expansion of the gyroplane structure by using a self-adaptive kernel density estimation-assisted particle filtering method. Aiming at the condition that a state model is uncertain and large, the resampling step is corrected by using a nuclear density estimation method, so that the particles after resampling keep diversity, the accuracy of a filter estimation result is improved, a self-adaptive nuclear bandwidth selection scheme is provided by using the weight of the particles, a better nuclear density estimation result is obtained and is applied to a particle filter frame, the estimation accuracy of fatigue crack expansion is improved, and the safety of a rotorcraft structure in the using process is improved.

Description

Fatigue crack evaluation method based on adaptive kernel density estimation auxiliary particle filter
Technical Field
The invention belongs to the technical field of computer simulation, and particularly relates to a fatigue crack evaluation method based on adaptive kernel density estimation auxiliary particle filtering.
Background
The reliability of the performance of a system, such as a rotorcraft structure, which is a riveted joint for a helicopter cockpit canopy, is often affected by potential degradation mechanisms inside the system, such as fatigue, wear, corrosion, etc., which eventually lead to system failure over time. Based on historical degradation data or degradation experiment measured data of the system, the degradation process of the system can be dynamically described through various models containing random factors. As sensor technology advances, the state of degradation of a system in operation can be estimated by real-time monitoring of the sensors. Such degradation state evaluation methods based on degradation models and combined with monitoring data have received much attention in recent years.
The damage caused by the degradation process of the system cannot be directly monitored, but can be inferred by combining the measured data and the degradation model by using a state space model. The state space model comprises two parts, namely a state transition equation for describing the randomness of the hidden variables of the system degradation state and a noisy observation equation for describing the relation between the observation value and the hidden variables. Based on a Bayes filtering method and an observed value sequence, the posterior probability distribution of the system degradation state can be updated in a recursion mode, and therefore state estimation of the hidden variables is obtained. Among the Bayes filtering methods, the particle filtering method is widely applied due to the characteristic of being suitable for the nonlinear non-Gaussian state space model.
The particle filtering method based on Monte Carlo simulation is to approximate the probability density function of the system implicit state by a series of weighted particles. Gordon et al propose a significant resampling particle filter (SIR) method, which generally comprises 3 steps: (1) and (3) prediction: sampling predicted particles by using an importance density function based on the particles at the previous moment; (2) updating: calculating a normalized particle weight according to the current observation value; (3) resampling: a new set of equally weighted particles is resampled from the discrete set of weighted particles. In the prediction step, the prior probability density function is usually chosen to facilitate the calculation of the importance density function, ignoring the effect of the latest observations. Doing so easily leads to that only a few particles have significant weights over several iterations, while most particles hardly contribute to approximating the posterior probability density function of the system state, a phenomenon also known as the degradation problem of particle filtering methods. The introduction of the resampling step may alleviate the problem of particle degradation to some extent by duplicating a large number of high-weight particles and rejecting low-weight particles in a randomly selected manner, but results in a large number of low-weight particles being discarded with few "offspring" being generated. This phenomenon, also known as starvation or loss of diversity, also results in increased estimation errors. Therefore, the improved particle filtering method can consider selecting a better importance density function or optimizing a resampling strategy.
Pitt and shepard propose an Auxiliary Particle Filter (APF) method, an importance probability density function is reshaped by using Auxiliary variables in combination with the latest observed value, and the obtained particles are more likely to approach the true probability density of the state by adjusting the weight of the particles at the previous moment and resampling from the new importance probability density function. Under the condition that the uncertainty of the degradation model is small, the auxiliary variable can well represent prior probability distribution, and the problem of particle degradation can be effectively relieved. However, in the case that the uncertainty of the degradation model is large, it is difficult to ensure that the APF method can obtain an accurate estimation, because the auxiliary variables in the APF method describe the prior distribution by using the point estimation, the information loss is easy to occur. Musso, N, et al consider optimizing the resampling strategy, propose Regularized Particle Filter (RPF) method, based on the discrete distribution of approximate characterization of the particle with weight, utilize kernel function smoothing strategy to reconstruct the continuous approximate distribution of posterior probability density function and resample from Regularized probability density function, thus greatly increased the diversity of the particle, however RPF method considers kernel function bandwidth to select the fixed constant, neglected the over-smoothing or under-smoothing problem that may appear in kernel density estimation, influenced the precision of kernel density estimation.
However, the particle filter method has many problems to be improved in practical application due to the problems of degradation and particle shortage, so that the evaluation accuracy of fatigue crack propagation of the structure of the rotorcraft is low, and the problem of system safety is caused.
Disclosure of Invention
Aiming at the defects in the prior art, the invention aims to provide an improved auxiliary particle filtering method integrating adaptive kernel density estimation resampling to evaluate the fatigue crack state of a rotorcraft structure based on the problem of particle shortage of an auxiliary particle filtering method under the condition of large state noise influence.
A fatigue crack evaluation method based on adaptive kernel density estimation auxiliary particle filtering comprises the following steps:
s1, determining the key evaluation component: for structural components influencing the flight safety of the aircraft in the service process of the rotorcraft, selecting key components with high fault risks to evaluate the crack propagation state;
s2, carrying out physical mechanism analysis on the crack propagation process of the key component, constructing a random propagation model of the fatigue crack as a state equation, determining random parameters describing uncertainty according to the inherent uncertainty existing in the crack propagation process, and determining the distribution of the random parameters;
s3, determining an observation model for describing fatigue cracks and sensor observation data according to the test experimental device, using the observation model as an observation equation, and determining the distribution of model parameters and observation noise;
s4, combining a state equation and an observation equation, and constructing a nonlinear state space model to describe the fatigue crack propagation evaluation process;
s5, carrying out simulation calculation on crack propagation of the nonlinear state space model of fatigue crack propagation of the gyroplane structure based on a self-adaptive kernel density estimation auxiliary particle filtering method; and
and S6, obtaining a simulation estimation result of the crack and carrying out error analysis.
Further, the step S2 specifically includes the following steps:
s21, obtaining a fatigue crack random expansion model of the rotorcraft structure according to the corrected Paris fatigue crack expansion model:
Figure BDA0002065629840000031
wherein: x is the number ofkDenotes the length, x, of the crack at the kth monitoring momentk-1Denotes the length of the crack at the k-1 th monitoring time, N denotes the fatigue load cycle, C1.071 10-10Where m is 2.092, C and m are both material parameters, Δ σk313.7Mpa represents the cyclic stress amplitude at the k-th monitoring instant ukIs process noise, representing the inherent uncertainty of the degradation process, ukObey 0 mean and variance of
Figure BDA0002065629840000032
K is an integer greater than 0, e and pi are natural logarithm and circumferential rate constants, respectively;
s22, for random parameters u existing in the fatigue crack propagation modelkBy adjusting ukVariance (variance)
Figure BDA0002065629840000033
To change the inherent in the modelThe magnitude of the uncertainty.
Further, the step S3 specifically includes the following steps:
s31, determining an observation model based on the characteristics of the experimental sensor:
Figure BDA0002065629840000034
wherein: y iskDenotes the monitored data of the sensor at the time k, and the parameter α is 1.22e in the observation model-8,β＝6.489,∈＝0.0866，xkIndicating the length of the crack at the kth monitoring moment, vkRepresentative of observation noise, vkObedience mean of 0 and variance of
Figure BDA0002065629840000035
(ii) a gaussian distribution of;
s32, collecting monitoring data y from 1 to k time according to the real-time monitoring of the sensor1:k。
Further, the fatigue crack propagation state evaluation process of step S4 specifically includes the steps of:
s41, when k is 1, distributing μ (x) from the initial distribution0) Middle sampling NsParticles of
Figure BDA0002065629840000036
And assign their same weight
Figure BDA0002065629840000037
S42, starting from a time when k is 2, executing the following steps at each time:
s421, giving a particle set
Figure BDA0002065629840000038
According to
Figure BDA0002065629840000039
Calculating an auxiliary variable for each particle;
s422, calculating the weight of the particles in the first stage according to
Figure BDA00020656298400000310
Calculating a particle stage weight based on
Figure BDA00020656298400000311
Normalizing the particle weight;
s423, giving a weighted particle set
Figure BDA00020656298400000312
Resampling N according to a resampling method based on adaptive kernel density estimationsA new particle
Figure BDA00020656298400000313
S424, according to
Figure BDA00020656298400000314
Sampling out NsA predicted particle
Figure BDA00020656298400000315
S425, according to
Figure BDA00020656298400000316
Assigning two-stage particle weights to predicted particles
Figure BDA00020656298400000317
And calculating to obtain the estimation of the k time state of the system
Figure BDA00020656298400000318
y1:k＝{y1,y2,…,ykTherein of
Figure BDA00020656298400000319
Represents the weight of the ith particle at time k, (-) represents a Dirac delta function;
s426, calculating the number of effective samples
Figure BDA0002065629840000041
According to a preset resampling threshold value NthJudging if N iseff<NthResampling NsParticles and assigning their same weight
Figure BDA0002065629840000042
Let k be k +1, the process returns to step S421 to estimate the crack propagation state at the next time.
Further, the resampling method for adaptive kernel density estimation specifically includes the following steps:
s51, giving weighted particle set
Figure BDA0002065629840000043
According to
Figure BDA0002065629840000044
Figure BDA0002065629840000045
Calculating an empirical covariance matrix Sk-1According to
Figure BDA0002065629840000046
The whitening matrix D is obtained by calculationk-1According to
Figure BDA0002065629840000047
Calculating to obtain the optimal fixed core bandwidth h0Where d denotes the dimension of the state vector, cdRepresenting the volume of a d-dimensional sphere, NsRepresenting the number of particles;
s52, from
Figure BDA0002065629840000048
Obtained by resampling method
Figure BDA0002065629840000049
Omitting the particle number ijThen N is obtainedsParticles of equal weight
Figure BDA00020656298400000410
S53, according to the adaptive kernel bandwidth hadIs calculated by
Figure BDA00020656298400000411
Calculating an adaptive kernel bandwidth for each particle
Figure BDA00020656298400000412
And sampling N from kernel functions K (x)sA sample
Figure BDA00020656298400000413
According to
Figure BDA00020656298400000414
Calculating to obtain NsA new particle
Figure BDA00020656298400000415
Still further, the resampling method comprises:
setting j to 1 and m to 1;
if j is<NsFrom a uniform distribution of [0,1 ]]Randomly sampling a number a; otherwise, resampling is finished;
if m is<NsCalculating the sum of the particle weights c if c>a, copying the particles and storing the particles into a new particle set, wherein j is j + 1; otherwise, m is m + 1; if m is equal to NsCopying the NthsAnd (4) changing the particles to a new particle set, and enabling m to be 1 and j to be j + 1. Further, the number of particles NsSet to 400, resample threshold NthIs also set to NsLoad cycle period N1000, initial crack length x0＝0.76mm。
Compared with the prior art, the invention has the technical effects that:
aiming at the problem of particle shortage of the auxiliary particle filter method under the condition that a state model is uncertain and large, a re-sampling step is corrected by using a kernel density estimation method, so that the particles after re-sampling keep diversity, and the accuracy of a filter estimation result is improved; aiming at the problem of kernel bandwidth selection in kernel density estimation, a self-adaptive kernel bandwidth selection scheme is provided by using particle weights, and a better kernel density estimation result is obtained and applied to a particle filter framework; therefore, the method for estimating the auxiliary particle filter based on the self-adaptive nuclear density can greatly improve the accuracy of fatigue crack propagation estimation of the gyroplane structure and improve the safety of the gyroplane structure in the using process.
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Other features, objects and advantages of the present application will become more apparent upon reading of the following detailed description of non-limiting embodiments thereof, made with reference to the accompanying drawings.
FIG. 1 is a flow chart of a fatigue crack assessment method based on adaptive kernel density estimation assisted particle filtering in accordance with the present invention;
FIG. 2 is a schematic flow chart of fatigue crack growth state evaluation;
FIG. 3 is a diagram showing simulation results under the condition that the noise variance is set to 0.1; and
fig. 4 is a diagram showing the simulation result under the condition that the noise variance is set to 0.6.
Detailed Description
The present application will be described in further detail with reference to the following drawings and examples. It is to be understood that the specific embodiments described herein are merely illustrative of the relevant invention and not restrictive of the invention. It should be noted that, for convenience of description, only the portions related to the related invention are shown in the drawings.
It should be noted that the embodiments and features of the embodiments in the present application may be combined with each other without conflict. The present application will be described in detail below with reference to the embodiments with reference to the attached drawings.
FIG. 1 shows an adaptive kernel density estimation-based particle filter-aided fatigue crack assessment method according to the invention, which comprises the following steps:
s1, determining the key evaluation component: and for structural components influencing the flight safety of the aircraft in the service process of the rotorcraft, selecting key components with high fault risks to evaluate the fatigue crack state.
S2, performing physical mechanism analysis on the crack propagation process of the key component, constructing a random propagation model of the fatigue crack as a state equation, determining a random parameter describing the uncertainty according to the inherent uncertainty existing in the crack propagation process, and determining the distribution of the parameter.
And S3, determining an observation model for describing fatigue cracks and sensor observation data according to the test experimental device, using the observation model as an observation equation, and determining model parameters and observation noise distribution.
S4, combining a state equation and an observation equation, and constructing a nonlinear state space model to describe the fatigue crack propagation evaluation process;
and S5, performing simulation calculation on the crack propagation state of the nonlinear state space model of fatigue crack propagation of the rotorcraft structure based on the adaptive kernel density estimation auxiliary particle filtering method.
And S6, obtaining a simulation estimation result of the crack and carrying out error analysis.
In one embodiment, step S2 specifically includes the following steps:
s21, obtaining a fatigue crack random expansion model of the rotorcraft structure according to the corrected Paris fatigue crack expansion model:
Figure BDA0002065629840000061
wherein: x is the number ofkDenotes the length, x, of the crack at the kth monitoring momentk-1Denotes the length of the crack at the k-1 th monitoring time, N denotes the fatigue load cycle, C1.071 10-10Where m is 2.092, C and m are both material parameters, Δ σk313.7Mpa represents the cyclic stress amplitude at the k-th monitoring instant ukIs process noise, representing the inherent uncertainty of the degradation process, ukObey 0 mean and variance of
Figure BDA0002065629840000062
K is an integer greater than 0, e and piNatural logarithmic and circumferential rate constants, respectively.
S22 random parameter u existing in fatigue crack propagation modelkBy adjusting ukVariance (variance)
Figure BDA0002065629840000063
To change the magnitude of the inherent uncertainty in the model.
In one embodiment, step S3 specifically includes the following steps:
s31, determining an observation model based on the characteristics of the experimental sensor:
Figure BDA0002065629840000064
wherein y iskDenotes the monitored data of the sensor at the time k, and the parameter α is 1.22e in the observation model-8,β＝6.489,∈＝0.0866，xkIndicating the length of the crack at the kth monitoring moment, vkRepresentative of observation noise, vkObedience mean of 0 and variance of
Figure BDA0002065629840000065
A gaussian distribution of (a).
S32, collecting monitoring data y from 1 to k time according to the real-time monitoring of the sensor1:k。
The fatigue crack random propagation model and the observation model of the gyroplane structure are constructed through the steps, and a foundation is laid for simulation calculation of crack propagation in the next step, which is one of important invention points of the invention.
In one embodiment, and with reference to FIG. 2, the fatigue crack growth state assessment process includes the following steps:
s41, when k is 1, distributing μ (x) from the initial distribution0) Middle sampling NsParticles of
Figure BDA0002065629840000066
And assign their same weight
Figure BDA0002065629840000067
S42, starting from a time when k is 2, executing the following steps at each time:
s421, giving a particle set
Figure BDA0002065629840000068
According to
Figure BDA0002065629840000069
Calculating an auxiliary variable for each particle;
s422, calculating the weight of the particles in the first stage according to
Figure BDA00020656298400000610
Calculating a particle stage weight based on
Figure BDA00020656298400000611
Normalizing the particle weight;
s423, giving a weighted particle set
Figure BDA00020656298400000612
Resampling N according to a resampling method based on adaptive kernel density estimationsA new particle
Figure BDA00020656298400000613
S424, according to
Figure BDA00020656298400000614
Sampling out NsA predicted particle
Figure BDA00020656298400000615
S425, according to
Figure BDA00020656298400000616
Assigning two-stage particle weights to predicted particles
Figure BDA00020656298400000617
And calculate to obtainEstimation of state at time k to system
Figure BDA0002065629840000071
y1:k＝{y1,y2,…,ykTherein of
Figure BDA0002065629840000072
Represents the weight of the ith particle at time k, (-) represents a Dirac delta function;
s426, calculating the number of effective samples
Figure BDA0002065629840000073
According to a preset resampling threshold value NthJudging if N iseff<NthResampling NsParticles and assigning their same weight
Figure BDA0002065629840000074
Let k be k +1, the process returns to step S421 to estimate the crack propagation state at the next time.
In one embodiment, a resampling method for adaptive kernel density estimation comprises the steps of:
s51, giving weighted particle set
Figure BDA0002065629840000075
According to
Figure BDA0002065629840000076
Figure BDA0002065629840000077
Calculating an empirical covariance matrix Sk-1According to
Figure BDA0002065629840000078
The whitening matrix D is obtained by calculationk-1According to
Figure BDA0002065629840000079
Calculating to obtain the optimal fixed core bandwidth h0D representsDimension of the state vector, cdRepresenting the volume of a d-dimensional sphere, NsThe number of particles is expressed.
S52, from
Figure BDA00020656298400000710
Obtained by resampling method
Figure BDA00020656298400000711
Omitting the particle number ijThen N is obtainedsParticles of equal weight
Figure BDA00020656298400000712
S53, according to the adaptive kernel bandwidth hadIs calculated by
Figure BDA00020656298400000713
Calculating an adaptive kernel bandwidth for each particle
Figure BDA00020656298400000714
And sampling N from kernel functions K (x)sA sample
Figure BDA00020656298400000715
According to
Figure BDA00020656298400000716
Calculating to obtain NsA new particle
Figure BDA00020656298400000717
The fatigue crack propagation state evaluation step and the resampling method for adaptive kernel density estimation are important points of the invention, and are mainly reflected in that: aiming at the problem of particle shortage of the auxiliary particle filter method under the condition of large state uncertainty, the resampling step is corrected by using a kernel density estimation method, so that the particles after resampling keep diversity, and the accuracy of the filter estimation result is improved; aiming at the problem of kernel bandwidth selection in kernel density estimation, a self-adaptive kernel bandwidth selection scheme is provided by using particle weights, and a better kernel density estimation result is obtained and applied to a particle filter framework; the method is also called a method for estimating auxiliary particle filtering based on self-adaptive nuclear density, can greatly improve the accuracy of evaluating the fatigue crack propagation of the structure of the rotorcraft, and improves the safety of the structure of the rotorcraft in the using process.
The resampling method used in the invention comprises the following steps:
setting j to 1 and m to 1;
if j is<NsFrom a uniform distribution of [0,1 ]]Randomly sampling a number a; otherwise, resampling is finished;
if m is<NsCalculating the sum of the particle weights c if c>a, copying the particles and storing the particles into a new particle set, wherein j is j + 1; otherwise, m is m + 1; if m is equal to NsCopying the NthsAnd (4) changing the particles to a new particle set, and enabling m to be 1 and j to be j + 1.
In the simulation calculation, the number of particles NsSet to 400, resample threshold NthIs also set to NsNumber of load cycles N1000, initial crack length x00.76mm, observation noise vkIs set to 0.01, the process noise ukThe variance of (c) is set to 0.1 and 0.6, respectively, to describe the different magnitudes of uncertainty inherent in the degradation process.
The nonlinear state space model of the fatigue crack propagation of the rotorcraft structure established above is utilized, and the beneficial effects of the fatigue crack propagation simulation method are verified according to the simulation result. The simulation experiment compares the evaluation results of the importance resampling particle filter method (SIR), the auxiliary particle filter method (APF) and the adaptive kernel density estimation auxiliary particle filter method (AKAPF) proposed in the invention. Wherein the criterion for numerically comparing the estimated errors of the three methods is the root mean square error, which is defined as follows:
Figure BDA0002065629840000081
wherein: t is the number of the iteration steps,
Figure BDA0002065629840000082
is an estimate of the state, xtrueIs the true value of the state. The mean (mean) and variance (var) of the root mean square error of the three methods after 100 experiments were calculated in this experiment.
Fig. 3 and 4 are graphs showing comparison of evaluation results of a random experiment of the importance resampling particle filter method (SIR), the auxiliary particle filter method (APF), and the adaptive kernel density estimation auxiliary particle filter based method (AKAPF) proposed in the present invention, fig. 3 is a result with a process noise variance set to 0.1, and fig. 4 is a result with a process noise variance set to 0.6. In the figure, the horizontal axis represents the cycle period of the load, and the vertical axis represents the propagation state of the fatigue crack. The open origin represents the true value of crack propagation, the dotted line represents the Auxiliary Particle Filter (APF) estimation, the dotted line represents the estimation of the importance resampling method (SIR), and the solid line represents the estimation of the proposed method (AKAPF). As can be seen from the figure, in the state space model of fatigue crack propagation, when the process noise variance becomes larger from 0.1 to 0.6, the method for estimating auxiliary particle filter (AKAPF) based on adaptive kernel density proposed in the present invention has higher estimation accuracy than the other two methods.
Table 1 shows the calculated root mean square error mean and variance for 100 experiments with process noise variance of 0.1 and 0.6, respectively, for the three methods. As can be seen from the figure, the adaptive kernel density estimation assisted particle filtering (AKAPF) -based method proposed in the present invention results in a smaller and more stable root mean square error after multiple experiments than the other two methods, indicating that the proposed method has higher estimation accuracy.
Figure BDA0002065629840000091
TABLE 1
Compared with simulation experiments, the method has the advantages that aiming at the problem of particle shortage existing in the auxiliary particle filtering method, the resampling step is corrected by utilizing a kernel density estimation method, and a new particle set is resampled from a continuous approximate probability density function of a particle set at the previous moment, so that the obtained particles are more dispersed and distributed at different positions of a state space, and only state information at certain points is contained, the diversity of the particle set after resampling is maintained, meanwhile, the risk that estimation errors are generated due to the fact that certain particles are accidentally endowed with great stage weight is reduced, and the accuracy of a filtering estimation result is improved; aiming at the problem of kernel bandwidth selection existing in kernel density estimation, one-stage weight of a particle auxiliary variable is fused into a selection scheme of self-adaptive kernel bandwidth, so that the problem of over-smoothing or under-smoothing possibly occurring in the kernel density estimation is further relieved, and a better kernel density estimation result is obtained and applied to a particle filter frame; therefore, the method for estimating the auxiliary particle filter based on the self-adaptive nuclear density can greatly improve the accuracy of the fatigue crack propagation estimation of the rotorcraft structure, and improves the safety of the rotorcraft structure in the using process.
Finally, it should be noted that: although the present invention has been described in detail with reference to the above embodiments, it should be understood by those skilled in the art that: modifications and equivalents may be made thereto without departing from the spirit and scope of the invention and it is intended to cover in the claims the invention as defined in the appended claims.

Claims (4)

1. A fatigue crack evaluation method based on adaptive kernel density estimation auxiliary particle filtering is characterized by comprising the following steps:
s1, determining the key evaluation component: for structural components influencing the flight safety of the aircraft in the service process of the rotorcraft, selecting key components with high fault risks to evaluate the crack propagation state;
s2, carrying out physical mechanism analysis on the crack propagation process of the key component, constructing a random propagation model of the fatigue crack as a state equation, determining random parameters describing uncertainty according to the inherent uncertainty existing in the crack propagation process, and determining the distribution of the random parameters;
s3, determining an observation model for describing fatigue cracks and sensor observation data according to the test experimental device, using the observation model as an observation equation, and determining the distribution of model parameters and observation noise;
s4, combining a state equation and an observation equation, and constructing a nonlinear state space model to describe the fatigue crack propagation evaluation process;
s5, carrying out simulation calculation on the crack propagation state of the nonlinear state space model of fatigue crack propagation of the rotorcraft structure based on the self-adaptive kernel density estimation auxiliary particle filtering method;
s6, obtaining a simulation estimation result of the crack, and carrying out error analysis;
the fatigue crack propagation state evaluation process of step S4 specifically includes the steps of:
s41, when k is 1, distributing μ (x) from the initial distribution0) Middle sampling NsParticles of
Figure FDA0002560351120000011
And assign their same weight
Figure FDA0002560351120000012
S42, starting from a time when k is 2, executing the following steps at each time:
s421, giving a particle set
Figure FDA0002560351120000013
According to
Figure FDA0002560351120000014
Calculating an auxiliary variable for each particle;
s422, calculating the weight of the particles in the first stage according to
Figure FDA0002560351120000015
Calculating a particle stage weight based on
Figure FDA0002560351120000016
Normalizing the particle weight;
s423, giving a weighted particle set
Figure FDA0002560351120000017
Resampling N according to a resampling method based on adaptive kernel density estimationsA new particle
Figure FDA0002560351120000018
S424, according to
Figure FDA0002560351120000019
Sampling out NsA predicted particle
Figure FDA00025603511200000110
S425, according to
Figure FDA00025603511200000111
Assigning two-stage particle weights to predicted particles
Figure FDA00025603511200000112
And calculating to obtain the estimation of the k time state of the system
Figure FDA00025603511200000113
Wherein
Figure FDA00025603511200000114
Represents the weight of the ith particle at time k, (-) represents a Dirac delta function;
s426, calculating the number of effective samples
Figure FDA00025603511200000115
According to a preset resampling threshold value NthJudging if N iseff＜NthResampling NsParticles and assigning their same weight
Figure FDA00025603511200000116
Returning to step S421 to estimate the crack propagation state at the next time when k is k + 1;
the resampling method for adaptive kernel density estimation specifically comprises the following steps:
s51, giving weighted particle set
Figure FDA0002560351120000021
According to
Figure FDA0002560351120000022
Figure FDA0002560351120000023
Calculating an empirical covariance matrix Sk-1According to
Figure FDA0002560351120000024
The whitening matrix D is obtained by calculationk-1According to
Figure FDA0002560351120000025
Calculating to obtain the optimal fixed core bandwidth h0Where d denotes the dimension of the state vector, cdRepresenting the volume of a d-dimensional sphere, NsRepresenting the number of particles;
s52, from
Figure FDA0002560351120000026
Obtained by resampling method
Figure FDA0002560351120000027
Omitting the particle number ijThen N is obtainedsParticles of equal weight
Figure FDA0002560351120000028
S53, according to the adaptive kernel bandwidth hadIs calculated by
Figure FDA0002560351120000029
Calculating an adaptive kernel bandwidth for each particle
Figure FDA00025603511200000210
And sampling N from kernel functions K (x)sA sample
Figure FDA00025603511200000211
According to
Figure FDA00025603511200000212
Calculating to obtain NsA new particle
Figure FDA00025603511200000213
The resampling method comprises the following steps:
setting j to 1 and m to 1;
if j < NsFrom a uniform distribution of [0,1 ]]Randomly sampling a number a; otherwise, resampling is finished;
if m < NsCalculating the sum of the weight values of the particles, and if c is larger than a, copying the particles and storing the particles into a new particle set, wherein j is j + 1; otherwise, m is m + 1; if m is equal to NsCopying the NthsAnd (4) changing the particles to a new particle set, and enabling m to be 1 and j to be j + 1.
2. The adaptive kernel density estimation-based particle filter-aided fatigue crack assessment method according to claim 1, wherein said step S2 specifically comprises the steps of:
s21, obtaining a fatigue crack random expansion model of the rotorcraft structure according to the corrected Paris fatigue crack expansion model:
Figure FDA00025603511200000214
wherein: x is the number ofkIs shown asLength of crack at k monitoring instants, xk-1Denotes the length of the crack at the k-1 th monitoring time, N denotes the fatigue load cycle, C1.071 10-10Where m is 2.092, C and m are both material parameters, Δ σk313.7Mpa, representing the amplitude of the cyclic stress at the k-th monitoring instant, ukIs process noise, representing the inherent uncertainty of the degradation process, ukObey 0 mean and variance of
Figure FDA00025603511200000215
K is an integer greater than 0, e and pi are natural logarithm and circumferential rate constants, respectively;
s22, for random parameters u existing in the fatigue crack propagation modelkBy adjusting ukVariance (variance)
Figure FDA00025603511200000216
To change the magnitude of the inherent uncertainty in the model.
3. The adaptive kernel density estimation-based particle filter-aided fatigue crack assessment method according to claim 2, wherein said step S3 specifically comprises the steps of:
s31, determining an observation model based on the characteristics of the experimental sensor:
Figure FDA0002560351120000031
wherein: y iskDenotes the monitored data of the sensor at the time k, and the parameter α is 1.22e in the observation model-8，β＝6.489，∈＝0.0866，
Figure FDA0002560351120000032
β -6.489 in (1) gives an index term, xkIndicating the crack length at the k-th monitoring moment, vkRepresentative of observation noise, vkObedience mean of 0 and variance of
Figure FDA0002560351120000033
(ii) a gaussian distribution of;
s32, collecting monitoring data y from 1 to k frog moments according to the real-time monitoring of the sensor1：k。
4. The adaptive kernel density estimation-based particle filter-aided fatigue crack assessment method according to claim 1, wherein the number of particles N issSet to 400, resample threshold NthIs set to NsLoad cycle period N1000, initial crack length x0＝0.76mm。
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