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

A Lifespan Prediction Method Based on Adaptive Matching of Model Individual Difference Parameters

technical field

The invention belongs to the technical field of remaining life prediction of mechanical equipment, and in particular relates to a life prediction method for adaptive matching of model individual difference parameters.

Background technique

The safe and stable operation of equipment is a prerequisite for ensuring personal safety and improving production efficiency. Therefore, it is necessary to accurately predict the remaining life of equipment and formulate effective equipment maintenance strategies accordingly. The model-based life prediction method is one of the methods to achieve accurate prediction of the remaining life. This method builds a decline model through theoretical analysis or experience summary of the equipment decline process, and then evaluates the model parameters and status in real time according to the monitoring data. Based on this Accurate prediction of remaining life is achieved on the basis of In order to reflect the randomness of equipment decline, a stochastic process model is usually used to describe its decline trend. Due to the different initial states and operating conditions of different equipment, there are often individual differences in the decline, and individual differences must be taken into account when establishing stochastic process models. But there is currently no general approach to deal with individual differences. Existing methods usually assume that the individual difference parameters obey a certain distribution, and estimate the parameters on this basis. This assumption has neither a corresponding theoretical basis nor a good practical effect.

Contents of the invention

In order to overcome the above-mentioned shortcomings of the prior art, the object of the present invention is to provide a life prediction method for adaptive matching of model individual difference parameters, which fully considers individual differences in equipment degradation, thereby improving the accuracy of equipment remaining life prediction.

In order to achieve the above object, the technical scheme that the present invention takes is:

A life prediction method for adaptive matching of model individual difference parameters, comprising the following steps:

1) Establish a recession model:

Among them, s _k represents the state value observed at time t _k , s(t) represents the state value observed at any time t after t _k , μ(s(τ),τ,b) is the nonlinear decay rate function, and the function Among them, b is an unknown constant; a is a random variable, reflecting the individual differences of equipment decline, σ _B is the diffusion coefficient of the decline model, which is an unknown constant; B(τ) is the standard Brownian motion, reflecting the time-varying nature of the decline;

2) Discretize the recession model, and obtain the difference sequence of the state values obtained by the two observations:

s _i,j ＝s _i,j-1 +a _i μ(s _i,j-1 ,t _j-1 ,b)Δt _j-1 +σ _B dB(Δt _j-1 ) (2)

Among them, s _{i, j} represent the state value observed by the i-th sample in the existing N decaying samples at time t _j ; ΔS _i is the difference sequence of the state value observed twice before and after the i-th sample; Δt _j-1 = t _j -t _j-1 , which represents the observation time interval; K _i represents the time sequence number corresponding to the last observation of the i-th sample;

3) For unknown parameters Perform maximum likelihood estimation, where a _i is the individual difference parameter of each sample, i=1,2,...,N; the specific steps are as follows:

3.1) According to the discretized decay model in step 2), for each sample, the difference sequence of the state values obtained by the two observations before and after it obeys the normal distribution in for the ith sample from t ₀ to The vector formed by multiplying the decay rate at each moment and the observation time interval; For the i-th sample from time t ₀ to Diagonal matrix composed of observation time intervals within a moment;

The log-likelihood function is thus listed as follows:

In the formula The state value sequence composed of state values observed for all samples;

3.2) For a _i and Find the partial derivative, and set the partial derivative to 0, the estimated value of the parameter is as follows:

3.3) Substitute the above formula back into the logarithmic likelihood function formula (4), and transform the logarithmic likelihood function into:

3.4) Use the one-dimensional optimization method to obtain the parameter b corresponding to the maximum value of the above formula, and use it as an estimated value Will Bring in formulas (5) and (6) to solve with So far, unknown parameters The estimated value of has been obtained;

4) Select several distributions as alternative distributions, according to several estimated values of parameter a obtained in step 3) Carry out the Kolmogorov-Smirnov goodness of fit test, that is, the KS test, and select the significance level value in the KS test, that is, the alternative distribution corresponding to the maximum p value as the individual difference parameter Match the estimated distribution

5) According to the observation information of the sample to be predicted, the particle filter is used to evaluate the parameter a and the state _sk in real time, and the specific steps are as follows:

5.1) from step 4) Distribution Extract N _p parameter particles from Set the initial weight to Generate N _p state particles

5.2) According to the discretized decay model obtained in step 2), the following state one-step transition equation is established:

in, with for The i-th state particle and parameter particle at time, is the state value of the i-th state particle after one-step transfer;

5.3) After observing the state value s _k at time t _k , the weights are updated and normalized according to equations (9)-(10):

in, for The weight value of the i-th particle at time, is the normalized weight of the _i -th particle after updating at time tk, is the normalized weight of the i-th particle after updating at time t _k ;

5.4) Perform resampling according to the weights of particles obtained in step 5.3), copy high-weight particles, and remove low-weight particles, that is, the particles after resampling are required to satisfy where P( ) is a probability operator; thus a new particle set is obtained with The weight of each particle in the new particle set is Calculate the state particle median value from the new particle set and the median value of the parameter particle Take it as the state and parameter estimates at time t _k ;

6) Set the stochastic simulation conditions of the recession process as follows:

6.1) Given the initial conditions of the stochastic simulation of the recession process:

Initial time step - Δl ₀ ;

Recession initial value where Ns represents the total number of simulated trajectories;

6.2) Establish the one-step transition equation of the recession process state:

Among them, s _n (l _i +t _k ) is the state value of the nth simulated trajectory at the moment l _i +t _k , i∈N ⁺ ={1,2,3,...}; Δl _i-1 is The time step of the i-1th step transition; l _i is the duration of the i-step state transition, V _i-1 is one-step transfer noise, which obeys the uniform distribution U(-v _i-1 ,v _i-1 ),

6.3) Set the time step adaptive adjustment logic as:

Among them, M _i is the number of simulated trajectories that reach or exceed the failure threshold when the i-th step is transferred, and M _am is the maximum number of simulated trajectories that are allowed to reach or exceed the failure threshold in the state one-step transfer;

7) Start stochastic simulation according to step 6) and continue state recursion until all simulated trajectories reach or exceed the decay threshold, then the simulation stops, and then calculate the remaining life of each simulated trajectory and its probability density according to formulas (13)-(14) :

Among them, λ is the failure threshold, L _n is the number of transfer steps required for the nth trajectory to reach the failure threshold, Its remaining life value, n=1,2,...,N _s ;

8) Calculation has the same remaining life The mean value of the probability density function of different trajectories, which is regarded as the remaining life is Estimated probability density function for :

in,

Beneficial effects of the present invention: Introduce the individual difference parameter in the recession model, and adaptively match its most consistent distribution according to the maximum likelihood estimation value of the parameter; use the particle filter to evaluate the state and parameters, and then pass the decay process Stochastic simulation is used to predict the remaining life, which avoids the blindness of pre-specifying the distribution of individual difference parameters, and realizes the effective prediction of the remaining life of equipment.

Description of drawings

Fig. 1 is the flow chart of the present invention.

Figure 2 shows the fatigue crack data of metal samples.

Fig. 3 is the prediction result figure of sample life, and figure (a) is the prediction result when the method of the present invention is applied to sample 1; Fig. (b) is the prediction result when the method of the present invention is applied to sample 11.

detailed description

The present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments.

As shown in Figure 1, a life prediction method for adaptive matching of model individual difference parameters includes the following steps:

1) Establish a recession model:

Among them, s _k represents the state value observed at time t _k , s(t) represents the state value observed at any time t after t _k , μ(s(τ),τ,b) is the nonlinear decay rate function, and the function Among them, b is an unknown constant; a is a random variable, reflecting the individual differences of equipment decline, σ _B is the diffusion coefficient of the decline model, which is an unknown constant; B(τ) is the standard Brownian motion, reflecting the time-varying nature of the decline;

2) Discretize the recession model, and obtain the difference sequence of the state values obtained by the two observations:

s _i,j ＝s _i,j-1 +a _i μ(s _i,j-1 ,t _j-1 ,b)Δt _j-1 +σ _B dB(Δt _j-1 ) (2)

Among them, s _{i, j} represent the state value observed by the i-th sample in the existing N decaying samples at time t _j ; ΔS _i is the difference sequence of the state value observed twice before and after the i-th sample; Δt _j-1 = t _j -t _j-1 , which represents the observation time interval; K _i represents the time sequence number corresponding to the last observation of the i-th sample;

3) For unknown parameters Perform maximum likelihood estimation, where a _i is the individual difference parameter of each sample, i=1,2,...,N; the specific steps are as follows:

3.1) According to the discretized decay model in step 2), for each sample, the difference sequence of the state values obtained by the two observations before and after it obeys the normal distribution in for the ith sample from t ₀ to The vector formed by multiplying the decay rate at each moment and the observation time interval; For the i-th sample from time t ₀ to Diagonal matrix composed of observation time intervals within a moment;

The log-likelihood function is thus listed as follows:

In the formula The state value sequence composed of state values observed for all samples;

3.2) For a _i and Find the partial derivative, and set the partial derivative to 0, the estimated value of the parameter is as follows:

3.3) Substitute the above formula back into the logarithmic likelihood function formula (4), and transform the logarithmic likelihood function into:

3.4) Use the one-dimensional optimization method to obtain the parameter b corresponding to the maximum value of the above formula, and use it as an estimated value Will Bring in formulas (5) and (6) to solve with So far, unknown parameters The estimated value of has been obtained;

4) Select several distributions as alternative distributions, according to several estimated values of parameter a obtained in step 3) Carry out the Kolmogorov-Smirnov goodness of fit test, that is, the KS test, and select the significance level value in the KS test, that is, the alternative distribution corresponding to the maximum p value as the individual difference parameter Match the estimated distribution

5) According to the observation information of the sample to be predicted, the particle filter is used to evaluate the parameter a and the state _sk in real time, and the specific steps are as follows:

5.1) from step 4) Distribution Extract N _p parameter particles from Set the initial weight to Generate N _p state particles

5.2) According to the discretized decay model obtained in step 2), the following state one-step transition equation is established:

in, with for The i-th state particle and parameter particle at time, is the state value of the i-th state particle after one-step transfer;

5.3) After observing the state value s _k at time t _k , the weights are updated and normalized according to equations (9)-(10):

in, for The weight value of the i-th particle at time, is the normalized weight of the _i -th particle after updating at time tk, is the normalized weight of the i-th particle after updating at time t _k ;

5.4) Perform resampling according to the weights of particles obtained in step 5.3), copy high-weight particles, and remove low-weight particles, that is, the particles after resampling are required to satisfy where P( ) is a probability operator; thus a new particle set is obtained with The weight of each particle in the new particle set is Calculate the state particle median value from the new particle set and the median value of the parameter particle Take it as the state and parameter estimates at time t _k ;

6) Set the stochastic simulation conditions of the recession process as follows:

6.1) Given the initial conditions of the stochastic simulation of the recession process:

Initial time step - Δl ₀ ;

Recession initial value where Ns represents the total number of simulated trajectories;

6.2) Establish the one-step transition equation of the recession process state:

Among them, s _n (l _i +t _k ) is the state value of the nth simulated trajectory at the moment l _i +t _k , i∈N ⁺ ={1,2,3,...}; Δl _i-1 is The time step of the i-1th step transition; l _i is the duration of the i-step state transition, V _i-1 is one-step transfer noise, which obeys the uniform distribution U(-v _i-1 ,v _i-1 ),

6.3) Set the time step adaptive adjustment logic as:

Among them, M _i is the number of simulated trajectories that reach or exceed the failure threshold when the i-th step is transferred, and M _am is the maximum number of simulated trajectories that are allowed to reach or exceed the failure threshold in the state one-step transfer;

7) Start stochastic simulation according to step 6) and continue state recursion until all simulated trajectories reach or exceed the decay threshold, then the simulation stops, and then calculate the remaining life of each simulated trajectory and its probability density according to formulas (13)-(14) :

Among them, λ is the failure threshold, L _n is the number of transfer steps required for the nth trajectory to reach the failure threshold, Its remaining life value, n=1,2,...,N _s ;

8) Calculation has the same remaining life The mean value of the probability density function of different trajectories, which is regarded as the remaining life is Estimated probability density function for :

in,

In order to verify the effectiveness of the present invention, the metal sample fatigue crack data provided in the document Statistical Methods for Reliability Data (New York: John Wiley & Sons, 1998, pp 639) was used for verification. The crack data is shown in Figure 2. There are 21 crack data of metal samples in the data set. Each sample is pretreated to add a 0.9-inch long crack as an initial defect; during the experiment, the number of stress cycles increases by 10,000 records A crack length. When the crack length reaches or exceeds 1.6 inches, the sample is considered to have failed and the test is terminated; in addition, when the number of revolutions reaches 120,000, the test is terminated regardless of whether the crack length reaches 1.6 inches or not. After the experiment, a total of 12 samples out of the above 21 samples failed.

Select following three kinds of stochastic process models (I: polynomial model; II: exponential model; III: state transfer model) to model the decay process respectively, and use the method of the present invention to carry out successively on 12 failure samples in the above-mentioned data set life expectancy. For each prediction, the remaining 20 samples except itself are taken as the training set to estimate the parameters b and The value of , and the distribution of the individual difference parameter a. On this basis, the particle filter is used for state and parameter evaluation, and the life prediction is carried out through stochastic simulation of the decay process. After the prediction is over, take samples No. 1 and No. 11 as examples to draw the prediction results, as shown in Figure 3.

It can be seen from the figure that as the prediction time approaches the final life of the bearing, the prediction results of the above three models gradually converge to the real life. However, the remaining life prediction value of sample 1 corresponding to Figure 3-(a) converges to the real life quickly under the state transfer model, and the convergence rate is slow under the exponential model and polynomial model; while Figure 3-(b) The corresponding sample 11 performed well under the exponential model and the polynomial model, and performed poorly under the state transfer model, that is, none of the three models had a better predictive effect than the other two models.

A life prediction method for self-adaptive matching of model individual difference parameters proposed by the invention can predict the remaining life of metal materials under failure modes such as fatigue, and can also be used for life prediction problems of other electromechanical equipment. It should be noted that the decline model used in the method of the present invention is a general formula. When the implementer applies the method of the present invention to other products, he can select the corresponding stochastic process model according to the difference in the decline trend of the object to be predicted, so that It adapts to the application requirements of different devices. In addition, the present invention provides an idea of "self-adaptive matching of individual difference parameters + effective evaluation of state and parameters + random simulation of remaining life". The proposed replacement and modification should also be regarded as the protection scope of the present 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,