CN107679279A - A kind of life-span prediction method of model subjects difference parameter Adaptive matching - Google Patents

Abstract

A kind of life-span prediction method of model subjects difference parameter Adaptive matching, equipment degenerated mode is built first, and individual difference parameter is introduced in a model, the distribution that it is best suited according to the maximum likelihood estimation Adaptive matching of the parameter, then use the real-time evaluation status of particle filter method and parameter, on this basis by obtaining residual life probability density to the stochastic simulation of degenerative process, present invention, avoiding the blindness of predesignated individual difference parameter distribution, realizes effective prediction of equipment residual life.

Description

Service life prediction method for model individual difference parameter adaptive matching

Technical Field

The invention belongs to the technical field of residual life prediction of mechanical equipment, and particularly relates to a life prediction method for model individual difference parameter adaptive matching.

Background

The safe and stable operation of the equipment is a prerequisite condition for ensuring personal safety and improving production efficiency, so that the residual service life of the equipment needs to be accurately predicted, and an effective equipment maintenance strategy needs to be formulated according to the residual service life. The method constructs a decay model through theoretical analysis or empirical summary of the decay process of equipment, and then carries out real-time evaluation on model parameters and states according to monitoring data, thereby realizing accurate prediction of the residual life on the basis. To reflect the randomness of the equipment degradation, a stochastic process model is usually used to describe the degradation trend. Because the initial states, the operating conditions and other conditions of different devices are different, the decline often has individual differences, and the individual differences must be taken into consideration when a random process model is established. However, there is currently no general method for dealing with individual differences. Existing methods generally assume that individual difference parameters obey a certain distribution and perform parameter estimation on this basis. This assumption has no corresponding theoretical basis and cannot achieve better practical effects.

Disclosure of Invention

In order to overcome the defects of the prior art, the invention aims to provide a life prediction method for model individual difference parameter adaptive matching, which fully considers the individual difference of equipment degradation, thereby improving the accuracy of the residual life prediction of the equipment.

In order to achieve the purpose, the invention adopts the technical scheme that:

a life prediction method for model individual difference parameter adaptive matching comprises the following steps:

1) establishing a regression model:

wherein s is_kRepresents t_kThe state value observed at the moment, s (t) representing t_kThe state value observed at any later moment t, mu (s (tau), tau and b) is a nonlinear decay rate function, and b in the function is an unknown constant; a is a random variable reflecting individual differences of equipment deterioration, sigma_BThe diffusion coefficient of the regression model is an unknown constant; b (tau) is standard Brownian motion, reflecting degenerated time-varying;

2) discretizing the regression model, and accordingly obtaining a difference sequence of state values obtained by two times of observation:

s_i,j＝s_i,j-1+a_iμ(s_i,j-1,t_j-1,b)Δt_j-1+σ_BdB(Δt_j-1) (2)

wherein s is_i,jRepresenting the ith sample of the existing N fading samples at time t_jObserving the obtained state value; delta S_iA difference sequence of state values obtained by observing the ith sample twice before and after; Δ t_j-1＝t_j-t_j-1Representing an observation time interval; k_iRepresenting the time sequence number corresponding to the last observation of the ith sample;

3) for unknown parametersPerforming maximum likelihood estimation, wherein a_iAs individual difference parameters for each sample, i ═ 1,2, …, N; the method comprises the following specific steps:

3.1) according to the discretization decay model of the step 2), for each sample, the difference sequence of the state values obtained by two times of observation is subjected to normal distributionWhereinFrom t for the ith sample₀ToThe vector formed by multiplying the fading rate of each moment and the observation time interval;from t for the ith sample₀Is timed toObserving a diagonal matrix formed by time intervals in time;

the log-likelihood function is thus listed as follows:

in the formulaA state value sequence formed by state values obtained by observing all samples;

3.2) for a respectively_iAndthe partial derivative is calculated and is made 0, and the estimated values of the parameters are obtained as follows:

3.3) the above equation is back substituted into the log-likelihood function equation (4), and the log-likelihood function is transformed into:

3.4) using one-dimensional optimization method to obtain the corresponding parameter b when the maximum value of the above formula is taken as the estimated valueWill be provided withSolving for equations (5) and (6)Andup to this point, the parameters are unknownThe estimated values of (a) are all obtained;

4) selecting several distributions as alternative distributions, based on several estimated values of the parameter a obtained in step 3)Performing Kolmogorov-Smirnov goodness-of-fit test (K-S test), and selecting the alternative distribution corresponding to the maximum significance level value (p value) in the K-S test as the individual difference parameterMatched estimated distribution

5) Using particle filtering to parametrize according to the observation information of the sample to be predictedNumber a and state s_kThe real-time evaluation is carried out, and the specific steps are as follows:

5.1) from step 4)Distribution of (2)Extracting N from_pParticles of individual parameterSetting an initial weight ofGenerating N_pParticles in individual state

5.2) establishing the following state one-step transfer equation according to the discretization regression model obtained in the step 2):

wherein,andis composed ofThe ith state particle and parameter particle at time instant,the state value of the ith state particle after one-step transfer;

5.3) at t_kObserving the time to obtain a state value s_kThen, the weight values are updated and normalized according to equations (9) to (10):

wherein,is composed ofThe weight of the ith particle at the moment,is t_kThe ith particle normalized front weight value after the moment is updated,is t_kThe weight value of the ith particle after the time updating is normalized;

5.4) resampling according to the weight of the particles obtained in the step 5.3), copying high-weight particles, and removing low-weight particles, namely the resampled particles are required to meet the requirement Wherein P (-) is a probability operator; thereby obtaining a new particle setAndthe new particle set has a weight of each particle ofComputing median of state particles from new set of particlesAnd median of parameter particlesLet it be t_kState and parameter estimates at time;

6) the random simulation conditions for the fading process are set as follows:

6.1) initial conditions for random simulation of the given decay process:

initial time step-Deltal₀；

Initial value of declineWhere Ns represents the total number of simulated traces;

6.2) establishing a state one-step transfer equation of the decline process:

wherein s is_n(l_i+t_k) For the nth simulation track at l_i+t_kThe state value of the time, i ∈ N⁺＝{1,2,3,...}；Δl_i-1The time step length when the step i-1 is transferred is taken as the time step length; l_iThe time duration for the state transition step i,V_i-1to shift the noise in one step, uniform distribution of U (-v) is obeyed_i-1,v_i-1)，

6.3) setting the time step adaptive adjustment logic as follows:

wherein M is_iNumber of simulated tracks that meet or exceed the failure threshold at the i-th transition, M_amThe maximum number of simulation traces allowed to reach or exceed the failure threshold in the state one-step transition;

7) starting random simulation and continuously carrying out state recursion according to the step 6), stopping simulation until all simulation tracks reach or exceed a decline threshold value, and then calculating the remaining life and probability density of each simulation track according to the formulas (13) to (14):

where λ is the failure threshold, L_nThe number of transfer steps required for the nth track to reach the failure threshold,n is 1,2, N, which is the remaining life value of the battery_s；

8) Calculating to have the same remaining lifeIs taken as the remaining life of the mean of the probability density functions of the different trajectories ofProbability density function estimate of (1):

wherein,

the invention has the beneficial effects that: introducing individual difference parameters into the regression model, and adaptively matching the most consistent distribution according to the maximum likelihood estimation value of the parameters; the state and the parameters are evaluated by utilizing the particle filtering, and then the residual life is predicted by randomly simulating the decay process, so that the blindness of pre-designated individual difference parameter distribution is avoided, and the effective prediction of the residual life of the equipment is realized.

Drawings

FIG. 1 is a flow chart of the present invention.

FIG. 2 is fatigue crack data for a metal specimen.

FIG. 3 is a graph showing the predicted life of a sample, and (a) shows the predicted life of a sample 1 to which the method of the present invention is applied; FIG. (b) shows the predicted result when the method of the present invention is applied to sample 11.

Detailed Description

The invention is described in further detail below with reference to the figures and examples.

As shown in fig. 1, a life prediction method for model individual difference parameter adaptive matching includes the following steps:

1) establishing a regression model:

wherein s is_kRepresents t_kThe state value observed at the moment, s (t) representing t_kThe state value observed at any later moment t, mu (s (tau), tau and b) is a nonlinear decay rate function, and b in the function is an unknown constant; a is a random variable reflecting individual differences of equipment deterioration, sigma_BThe diffusion coefficient of the regression model is an unknown constant; b (tau) is standard Brownian motion, reflecting degenerated time-varying;

2) discretizing the regression model, and accordingly obtaining a difference sequence of state values obtained by two times of observation:

s_i,j＝s_i,j-1+a_iμ(s_i,j-1,t_j-1,b)Δt_j-1+σ_BdB(Δt_j-1) (2)

wherein s is_i,jRepresenting the ith sample of the existing N fading samples at time t_jObserving the obtained state value; delta S_iA difference sequence of state values obtained by observing the ith sample twice before and after; Δ t_j-1＝t_j-t_j-1Representing an observation time interval; k_iRepresenting the time sequence number corresponding to the last observation of the ith sample;

3) for unknown parametersPerforming maximum likelihood estimation, wherein a_iAs individual difference parameters for each sample, i ═ 1,2, …, N; the method comprises the following specific steps:

3.1) according to the discretization decay model of the step 2), for each sample, the difference sequence of the state values obtained by two times of observation is subjected to normal distributionWhereinFrom t for the ith sample₀ToThe vector formed by multiplying the fading rate of each moment and the observation time interval;from t for the ith sample₀Is timed toObserving a diagonal matrix formed by time intervals in time;

the log-likelihood function is thus listed as follows:

in the formulaA state value sequence formed by state values obtained by observing all samples;

3.2) for a respectively_iAndthe partial derivative is calculated and is made 0, and the estimated values of the parameters are obtained as follows:

3.3) the above equation is back substituted into the log-likelihood function equation (4), and the log-likelihood function is transformed into:

3.4) using one-dimensional optimization method to obtain the corresponding parameter b when the maximum value of the above formula is taken as the estimated valueWill be provided withSolving for equations (5) and (6)Andup to this point, the parameters are unknownThe estimated values of (a) are all obtained;

4) selecting several distributions as alternative distributions, based on several estimated values of the parameter a obtained in step 3)Performing Kolmogorov-Smirnov goodness-of-fit test (K-S test), and selecting the alternative distribution corresponding to the maximum significance level value (p value) in the K-S test as the individual difference parameterMatched estimated distribution

5) Using particle filtering to pair parameter a and state s according to observation information of sample to be predicted_kThe real-time evaluation is carried out, and the specific steps are as follows:

5.1) from step 4)Distribution of (2)Extracting N from_pParticles of individual parameterSetting an initial weight ofGenerating N_pParticles in individual state

5.2) establishing the following state one-step transfer equation according to the discretization regression model obtained in the step 2):

wherein,andis composed ofThe ith state particle and parameter particle at time instant,the state value of the ith state particle after one-step transfer;

5.3) at t_kObserving the time to obtain a state value s_kThen, the weight values are updated and normalized according to equations (9) to (10):

wherein,is composed ofThe weight of the ith particle at the moment,is t_kThe ith particle normalized front weight value after the moment is updated,is t_kThe weight value of the ith particle after the time updating is normalized;

5.4) resampling according to the weight of the particles obtained in the step 5.3), copying high-weight particles, and removing low-weight particles, namely the resampled particles are required to meet the requirement Wherein P (-) is a probability operator; thereby obtaining a new particle setAndthe new particle set has a weight of each particle ofComputing median of state particles from new set of particlesAnd median of parameter particlesLet it be t_kState and parameter estimates at time;

6) the random simulation conditions for the fading process are set as follows:

6.1) initial conditions for random simulation of the given decay process:

initial time step-Deltal₀；

Initial value of declineWhere Ns represents the total number of simulated traces;

6.2) establishing a state one-step transfer equation of the decline process:

wherein s is_n(l_i+t_k) For the nth simulation track at l_i+t_kThe state value of the time, i ∈ N⁺＝{1,2,3,...}；Δl_i-1The time step length when the step i-1 is transferred is taken as the time step length; l_iThe time duration for the state transition step i,V_i-1to shift the noise in one step, uniform distribution of U (-v) is obeyed_i-1,v_i-1)，

6.3) setting the time step adaptive adjustment logic as follows:

wherein M is_iNumber of simulated tracks that meet or exceed the failure threshold at the i-th transition, M_amThe maximum number of simulation traces allowed to reach or exceed the failure threshold in the state one-step transition;

7) starting random simulation and continuously carrying out state recursion according to the step 6), stopping simulation until all simulation tracks reach or exceed a decline threshold value, and then calculating the remaining life and probability density of each simulation track according to the formulas (13) to (14):

where λ is the failure threshold, L_nThe number of transfer steps required for the nth track to reach the failure threshold,n is 1,2, N, which is the remaining life value of the battery_s；

8) Calculating to have the same remaining lifeIs taken as the remaining life of the mean of the probability density functions of the different trajectories ofProbability density function estimate of (1):

wherein,

to verify the effectiveness of the present invention, the metal specimen fatigue crack data provided in the literature statistical methods for reliability data (New York: John Wiley & Sons,1998, pp 639) was used for verification. The crack data is shown in fig. 2, where the data set contains crack data for a total of 21 metal samples, each sample being pretreated before the experiment with 0.9 inch long added crack as the initial defect; the crack length was recorded every 10000 times the number of stress cycles increased during the experiment. When the crack length reached or exceeded 1.6 inches, the specimen was considered to have failed and the test was terminated; further, when the number of revolutions reached 120000, the experiment was terminated regardless of whether the crack length reached 1.6 inches. After the experiment was completed, a total of 12 specimens out of the 21 specimens failed.

The decay process is modeled by selecting three random process models (I: polynomial model; II: exponential model; III: state transfer model), and the service life of 12 failure samples in the data set is predicted successively by using the method. The remaining 20 samples, except itself, are taken at each prediction as a training set to estimate the parameters b andand the distribution of the individual difference parameter a. On the basis, the state and parameter evaluation is carried out by using particle filtering, and the service life is predicted by random simulation of the fading process. After the prediction is finished, a prediction result graph is prepared by taking sample numbers 1 and 11 as examples, as shown in fig. 3.

As can be seen from the figure, as the predicted time approaches the final service life of the bearing, the predicted results of the three models gradually converge to the real service life. However, the predicted value of the remaining life of the sample 1 corresponding to fig. 3- (a) converges to the real life faster under the state transfer model, and the convergence rate is slower under the exponential model and the polynomial model; the sample 11 corresponding to fig. 3- (b) performs better under the exponential model and the polynomial model, and performs poorly under the state transfer model, that is, the prediction effect of none of the three models is completely better than that of the other two models.

The life prediction method of model individual difference parameter adaptive matching provided by the invention can predict the residual life of the metal material in failure modes such as fatigue and the like, and can also be used for the life prediction problem of other electromechanical equipment. It should be noted that the regression model used in the method of the present invention is a general formula, and when an implementer applies the method of the present invention to other products, the implementer can select a corresponding random process model according to the different regression trends of the object to be predicted, so that the method can adapt to the application requirements of different devices. In addition, the invention provides a concept of 'individual difference parameter adaptive matching + state and parameter effective evaluation + residual life random simulation', and the invention is subjected to replacement and modification of indexes, parameters or models and the like on the premise of not departing from the concept of the invention and is also considered as the protection scope of the invention.

Claims (1)

1. A life prediction method for model individual difference parameter adaptive matching is characterized by comprising the following steps:

1) establishing a regression model:

<mrow> <mi>s</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>s</mi> <mi>k</mi> </msub> <mo>+</mo> <mi>a</mi> <munderover> <mo>&amp;Integral;</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mi>t</mi> </munderover> <mi>&amp;mu;</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>(</mo> <mi>&amp;tau;</mi> <mo>)</mo> <mo>,</mo> <mi>&amp;tau;</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>&amp;tau;</mi> <mo>+</mo> <munderover> <mo>&amp;Integral;</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mi>t</mi> </munderover> <msub> <mi>&amp;sigma;</mi> <mi>B</mi> </msub> <mi>d</mi> <mi>B</mi> <mrow> <mo>(</mo> <mi>&amp;tau;</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>

wherein s is_kRepresents t_kThe state value observed at the moment, s (t) representing t_kThe state value observed at any later moment t, mu (s (tau), tau and b) is a nonlinear decay rate function, and b in the function is an unknown constant; a is a random variable reflecting individual differences of equipment deterioration, sigma_BThe diffusion coefficient of the regression model is an unknown constant; b (tau) is standard Brownian motion, reflecting degenerated time-varying;

2) discretizing the regression model, and accordingly obtaining a difference sequence of state values obtained by two times of observation:

s_i,j＝s_i,j-1+a_iμ(s_i,j-1,t_j-1,b)Δt_j-1+σ_BdB(Δt_j-1) (2)

<mrow> <msub> <mi>&amp;Delta;S</mi> <mi>i</mi> </msub> <mo>=</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>s</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>s</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msub> <mi>s</mi> <mrow> <mi>i</mi> <mo>,</mo> <msub> <mi>K</mi> <mi>i</mi> </msub> </mrow> </msub> <mo>-</mo> <msub> <mi>s</mi> <mrow> <mi>i</mi> <mo>,</mo> <msub> <mi>K</mi> <mi>i</mi> </msub> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>

wherein s is_i,jRepresenting the ith sample of the existing N fading samples at time t_jObserving the obtained state value; delta S_iA difference sequence of state values obtained by observing the ith sample twice before and after; Δ t_j-1＝t_j-t_j-1Representing an observation time interval; k_iRepresenting the time sequence number corresponding to the last observation of the ith sample;

3) for unknown parametersPerforming maximum likelihood estimation, wherein a_iAs individual difference parameters for each sample, i ═ 1,2, …, N; the method comprises the following specific steps:

3.1) according to the discretization decay model of the step 2), for each sample, the difference sequence of the state values obtained by two times of observation is subjected to normal distributionWhereinFrom t for the ith sample₀ToThe vector formed by multiplying the fading rate of each moment and the observation time interval;from t for the ith sample₀Is timed toObserving a diagonal matrix formed by time intervals in time;

the log-likelihood function is thus listed as follows:

in the formulaA state value sequence formed by state values obtained by observing all samples;

3.2) for a respectively_iAndthe partial derivative is calculated and is made 0, and the estimated values of the parameters are obtained as follows:

<mrow> <msub> <mover> <mi>a</mi> <mo>^</mo> </mover> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msubsup> <mi>&amp;omega;</mi> <mi>i</mi> <mi>T</mi> </msubsup> <msubsup> <mi>&amp;Sigma;</mi> <mi>i</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>&amp;Delta;S</mi> <mi>i</mi> </msub> </mrow> <mrow> <msubsup> <mi>&amp;omega;</mi> <mi>i</mi> <mi>T</mi> </msubsup> <msubsup> <mi>&amp;Sigma;</mi> <mi>i</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>&amp;omega;</mi> <mi>i</mi> </msub> </mrow> </mfrac> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>...</mo> <mi>N</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msubsup> <mover> <mi>&amp;sigma;</mi> <mo>^</mo> </mover> <mi>B</mi> <mn>2</mn> </msubsup> <mo>=</mo> <mfrac> <mrow> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </msubsup> <msup> <mrow> <mo>(</mo> <msub> <mi>&amp;Delta;S</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;omega;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msubsup> <mi>&amp;Sigma;</mi> <mi>i</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>&amp;Delta;S</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <msub> <mi>&amp;omega;</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </msubsup> <msub> <mi>K</mi> <mi>i</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>

3.3) the above equation is back substituted into the log-likelihood function equation (4), and the log-likelihood function is transformed into:

3.4) using one-dimensional optimization method to obtain the corresponding parameter b when the maximum value of the above formula is taken as the estimated valueWill be provided withSolving for equations (5) and (6)Andup to this point, the parameters are unknownThe estimated values of (a) are all obtained;

4) selecting several distributions as alternative distributions, based on several estimated values of the parameter a obtained in step 3)Selecting by performing Kolmogorov-Smirnov goodness-of-fit test, i.e., K-S testAlternative distributions corresponding to the level of significance, i.e. when the p-value is maximal, in the K-S test as parameters of individual differencesMatched estimated distribution

5) Using particle filtering to pair parameter a and state s according to observation information of sample to be predicted_kThe real-time evaluation is carried out, and the specific steps are as follows:

5.1) from step 4)Distribution of (2)Extracting N from_pParticles of individual parameterSetting an initial weight ofGenerating N_pParticles in individual state

5.2) establishing the following state one-step transfer equation according to the discretization regression model obtained in the step 2):

<mrow> <msubsup> <mover> <mi>s</mi> <mo>~</mo> </mover> <mi>k</mi> <mi>i</mi> </msubsup> <mo>=</mo> <msubsup> <mi>s</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>i</mi> </msubsup> <mo>+</mo> <msubsup> <mi>a</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>i</mi> </msubsup> <mi>&amp;mu;</mi> <mrow> <mo>(</mo> <msubsup> <mi>s</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>i</mi> </msubsup> <mo>,</mo> <msub> <mi>t</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mover> <mi>b</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <msub> <mi>&amp;Delta;t</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mover> <mi>&amp;sigma;</mi> <mo>^</mo> </mover> <mi>B</mi> </msub> <mi>d</mi> <mi>B</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;Delta;t</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>

wherein,andis t_k-1The ith state particle and parameter particle at time instant,the state value of the ith state particle after one-step transfer;

5.3) at t_kObserving the time to obtain a state value s_kThen, the weight values are updated and normalized according to equations (9) to (10):

<mrow> <msubsup> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>k</mi> <mi>i</mi> </msubsup> <mo>=</mo> <msubsup> <mi>w</mi> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>i</mi> </msubsup> <mfrac> <mn>1</mn> <msqrt> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> <msubsup> <mover> <mi>&amp;sigma;</mi> <mo>^</mo> </mover> <mi>B</mi> <mn>2</mn> </msubsup> <msub> <mi>t</mi> <mi>k</mi> </msub> </mrow> </msqrt> </mfrac> <mi>exp</mi> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <msup> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mi>k</mi> </msub> <mo>-</mo> <msubsup> <mover> <mi>s</mi> <mo>~</mo> </mover> <mi>k</mi> <mi>i</mi> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msubsup> <mover> <mi>&amp;sigma;</mi> <mo>^</mo> </mover> <mi>B</mi> <mn>2</mn> </msubsup> <msub> <mi>t</mi> <mi>k</mi> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msubsup> <mi>w</mi> <mi>k</mi> <mrow> <mi>i</mi> <mo>*</mo> </mrow> </msubsup> <mo>=</mo> <mfrac> <msubsup> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>k</mi> <mi>i</mi> </msubsup> <mrow> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>N</mi> <mi>s</mi> </mrow> </msubsup> <msubsup> <mover> <mi>w</mi> <mo>~</mo> </mover> <mi>k</mi> <mi>i</mi> </msubsup> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>

wherein,is t_k-1The weight of the ith particle at the moment,is t_kThe ith particle normalized front weight value after the moment is updated,is t_kThe weight value of the ith particle after the time updating is normalized;

5.4) resampling according to the weight of the particles obtained in the step 5.3), copying high-weight particles, and removing low-weight particles, namely the resampled particles are required to meet the requirement Wherein P (-) is a probability operator; thereby obtaining a new particle setAndthe new particle set has a weight of each particle ofComputing median of state particles from new set of particlesAnd median of parameter particlesLet it be t_kState and parameter estimates at time;

6) the random simulation conditions for the fading process are set as follows:

6.1) initial conditions for random simulation of the given decay process:

initial time step-Deltal₀；

Decline initial value-Where Ns represents the total number of simulated traces;

6.2) establishing a state one-step transfer equation of the decline process:

<mrow> <msub> <mi>s</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>s</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>l</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mover> <mi>a</mi> <mo>&amp;OverBar;</mo> </mover> <mi>k</mi> </msub> <mi>&amp;mu;</mi> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mi>n</mi> </msub> <mo>(</mo> <mrow> <msub> <mi>l</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo> <mo>,</mo> <msub> <mi>l</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>&amp;Delta;l</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>

wherein s is_n(l_i+t_k) For the nth simulation track at l_i+t_kThe state value of the time, i ∈ N⁺＝{1,2,3,...}；Δl_i-1The time step length when the step i-1 is transferred is taken as the time step length; l_iThe time duration for the state transition step i,V_i-1to shift the noise in one step, uniform distribution of U (-v) is obeyed_i-1,v_i-1)，

6.3) setting the time step adaptive adjustment logic as follows:

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;l</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>&amp;Delta;l</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;l</mi> <mi>i</mi> </msub> <mo>=</mo> <mi>m</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>M</mi> <mrow> <mi>a</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>&amp;Delta;l</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>/</mo> <msub> <mi>M</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;Delta;l</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mn>0</mn> <mo>&lt;</mo> <msub> <mi>M</mi> <mi>i</mi> </msub> <mo>&amp;le;</mo> <msub> <mi>M</mi> <mrow> <mi>a</mi> <mi>m</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;l</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mi>a</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>&amp;Delta;l</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>/</mo> <msub> <mi>M</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mi>i</mi> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <msub> <mi>M</mi> <mi>i</mi> </msub> <mo>&gt;</mo> <msub> <mi>M</mi> <mrow> <mi>a</mi> <mi>m</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>

wherein M is_iNumber of simulated tracks that meet or exceed the failure threshold at the i-th transition, M_amThe maximum number of simulation traces allowed to reach or exceed the failure threshold in the state one-step transition;

7) starting random simulation and continuously carrying out state recursion according to the step 6), stopping simulation until all simulation tracks reach or exceed a decline threshold value, and then calculating the remaining life and probability density of each simulation track according to the formulas (13) to (14):

<mrow> <msub> <mi>l</mi> <msub> <mi>L</mi> <mi>n</mi> </msub> </msub> <mo>=</mo> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <msub> <mi>L</mi> <mi>n</mi> </msub> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>&amp;Delta;l</mi> <mi>i</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>l</mi> <mo>=</mo> <msub> <mi>l</mi> <msub> <mi>L</mi> <mi>n</mi> </msub> </msub> <mo>)</mo> </mrow> <mo>&amp;cong;</mo> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>&amp;lambda;</mi> <mo>-</mo> <msub> <mover> <mi>s</mi> <mo>&amp;OverBar;</mo> </mover> <mi>k</mi> </msub> <mo>-</mo> <msub> <mover> <mi>a</mi> <mo>&amp;OverBar;</mo> </mover> <mi>k</mi> </msub> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <msub> <mi>L</mi> <mi>n</mi> </msub> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <mi>&amp;mu;</mi> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mi>n</mi> </msub> <mo>(</mo> <msub> <mi>l</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>l</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> <msub> <mi>&amp;Delta;l</mi> <mi>i</mi> </msub> </mrow> <mrow> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>l</mi> <msub> <mi>L</mi> <mi>n</mi> </msub> </msub> </msubsup> <msubsup> <mover> <mi>&amp;sigma;</mi> <mo>^</mo> </mover> <mi>B</mi> <mn>2</mn> </msubsup> <mi>d</mi> <mi>&amp;tau;</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msub> <mover> <mi>a</mi> <mo>&amp;OverBar;</mo> </mover> <mi>k</mi> </msub> <mi>&amp;mu;</mi> <mrow> <mo>(</mo> <msub> <mi>s</mi> <mi>n</mi> </msub> <mo>(</mo> <msub> <mi>l</mi> <msub> <mi>L</mi> <mi>n</mi> </msub> </msub> <mo>+</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>l</mi> <msub> <mi>L</mi> <mi>n</mi> </msub> </msub> <mo>+</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <msubsup> <mover> <mi>&amp;sigma;</mi> <mo>^</mo> </mover> <mi>B</mi> <mn>2</mn> </msubsup> </mfrac> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;times;</mo> <mfrac> <msubsup> <mover> <mi>&amp;sigma;</mi> <mo>^</mo> </mover> <mi>B</mi> <mn>2</mn> </msubsup> <msqrt> <mrow> <mn>2</mn> <mi>&amp;pi;</mi> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>l</mi> <msub> <mi>L</mi> <mi>n</mi> </msub> </msub> </msubsup> <msubsup> <mover> <mi>&amp;sigma;</mi> <mo>^</mo> </mover> <mi>B</mi> <mn>2</mn> </msubsup> <mi>d</mi> <mi>&amp;tau;</mi> </mrow> </msqrt> </mfrac> <mi>exp</mi> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <msup> <mrow> <mo>(</mo> <mi>&amp;lambda;</mi> <mo>-</mo> <msub> <mover> <mi>s</mi> <mo>&amp;OverBar;</mo> </mover> <mi>k</mi> </msub> <mo>-</mo> <msub> <mover> <mi>a</mi> <mo>&amp;OverBar;</mo> </mover> <mi>k</mi> </msub> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <msub> <mi>L</mi> <mi>n</mi> </msub> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <mi>&amp;mu;</mi> <mo>(</mo> <msub> <mi>s</mi> <mi>n</mi> </msub> <mo>(</mo> <msub> <mi>l</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>,</mo> <msub> <mi>l</mi> <mi>i</mi> </msub> <mo>+</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo>)</mo> <msub> <mi>&amp;Delta;l</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mn>2</mn> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>l</mi> <msub> <mi>L</mi> <mi>n</mi> </msub> </msub> </msubsup> <msubsup> <mover> <mi>&amp;sigma;</mi> <mo>^</mo> </mover> <mi>B</mi> <mn>2</mn> </msubsup> <mi>d</mi> <mi>&amp;tau;</mi> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>

where λ is the failure threshold, L_nThe number of transfer steps required for the nth track to reach the failure threshold,n is 1,2, N, which is the remaining life value of the battery_s；

8) Calculating to have the same remaining lifeIs taken as the remaining life of the mean of the probability density functions of the different trajectories ofProbability density function estimate of (1):

wherein,