阅读说明：本技术 一种基于混合概率选择算子的代价参考粒子滤波方法 (Cost reference particle filtering method based on mixed probability selection operator ) 是由 张新雨 王晓璇 雷碧玉 于 2021-08-30 设计创作，主要内容包括：本发明公开了一种基于混合概率选择算子的CRPF方法,首先随机产生N个实数作为系统状态的样本,所述样本称为粒子,并定义每一粒子对应的代价值均为0,由所述样本与代价值组成的集合称为粒子—代价集合；同时,分别随机产生粒子组成1号子集合与2号子集合,每一个子集合的数量为N；分别计算所有子集合中粒子的代价函数值、风险函数值及权值；对子集合进行重采样、更新、信息交互、粒子选择操作,得到最终的粒子—代价集合并对所得到的集合进行更新；最后系统滤波,最终得到系统状态的最优估计值。本发明解决了现有技术中对噪声统计特性未知系统滤波准确度较低的问题。(The invention discloses a CRPF method based on mixed probability selection operator, firstly, randomly generating N real numbers as samples of system state, wherein the samples are called particles, and defining that the cost value corresponding to each particle is 0, and a set formed by the samples and the cost value is called a particle-cost set; simultaneously, respectively randomly generating particles to form a sub-set 1 and a sub-set 2, wherein the number of each sub-set is N; respectively calculating a cost function value, a risk function value and a weight of the particles in all the subsets; resampling, updating, information interaction and particle selection operations are carried out on the subset to obtain a final particle-cost set, and the obtained set is updated; and finally, filtering the system to finally obtain the optimal estimated value of the system state. The invention solves the problem of lower filtering accuracy of a system with unknown noise statistical characteristics in the prior art.)

1. A CRPF method based on a mixed probability selection operator is characterized by being implemented according to the following steps:

step 1, randomly generating N real numbers as samples of a system state, wherein the samples are called particles, a cost value corresponding to each particle is defined to be 0, and a set formed by the samples and the cost values is called a particle-cost set; simultaneously, respectively randomly generating particles to form a sub-set 1 and a sub-set 2, wherein the number of each sub-set is N;

step 2, respectively calculating cost function values, risk function values and weights of particles in all the subsets;

step 3, resampling the subset, wherein a resampling method selects a polynomial for resampling;

step 4, updating the subset;

step 5, the information interaction of the particles among the subsets, if the set exchange step number is reached, the operation is executed; otherwise, the information interaction operation is not required to be executed;

step 6, generating a particle-cost set finally used for system filtering;

step 7, updating the final particle-cost set;

and 8, filtering the system to finally obtain the optimal estimation value of the system state.

2. The CRPF method based on mixed probability selection operator of claim 1, wherein step 1 generates the initial size N particle-cost set E₀Is shown asWherein the content of the first and second substances,a random estimate of the state of the system is shown,namely, it isObey to uniform distribution U (I)₀) Value of initial cost functionN is the sampling sample number of the system state random estimation value;

at the same time, two subsets of size N are generatedWherein the content of the first and second substances,obey to uniform distribution U (I)₀) J is the number of subsets, j is 1,2, when j is 1,denotes the subset 1, when j is 2,representing the subset number 2.

3. The CRPF method based on hybrid probability selection operator according to claim 2, wherein the step 2 is as follows:

step 2.1, calculating particles in the subset at the time tValue of cost function ofj is 1,2, i is 1,2, …, N, T is 1,2, …, T is the time series length, and the cost function is calculated as follows:

in the formulas (1) and (2),representing the ith particle in the j subset at time tA cost function value of; lambda represents a forgetting factor, 0 < lambda < 1;represents the ith particle in the j subset at the time t-1A cost value function value of;indicating particleThe incremental amount of the cost of (a),indicating particleA cost increment function of (a); y is_tThe measured value of the sensor at the time t is represented, h (-) represents an observation function of the estimated value of the sensor, and the preset parameter q satisfies that q is more than or equal to 1;

step 2.2, from the cost function values of the subset particlesComputingObtaining a value of a risk functionThe formula for the risk function is:

with respect to the formula (3),representing the ith particle in the j subset at time tThe risk function value of (a); λ is more than 0 and less than 1, i is 1,2, …, N, and q is more than or equal to 1; f (-) represents a state transition function of the system state;

step 2.3, from the cost function values of the subset particlesOr value of risk functionCalculating the weight of the particleThe calculation method is as follows:

it can also be calculated by the following method:

in the formulas (4) and (5),i.e. the ith particle in the j subset at the t momentThe particle weight value of (1), and the particle weight value adjusting parameter beta is more than 1; in the formula (4)Are particlesA cost value function value of; in the formula (5)Indicating particleThe value of the risk function of (a),and expressing the minimum value of the risk function in the N random estimation samples of the subset j at the time t, wherein delta is an adjusting parameter, and delta is more than 0 and less than 1.

4. The CRPF method based on hybrid probability selection operator according to claim 3, wherein the step 3 is as follows:

step 3.1, in the interval [0,1 ]]In the above, a random number set { u } that is subject to uniform distribution is randomly generated_i}_i＝1:NAnd each element in the set satisfies independent same distribution;

step 3.2, cumulative value of weight Iⁱ＝cdf{u_iThe rule cdf denotes the cumulative distribution function of the particle weights, with respect to the random number u_iExistence of

Step 3.3, let the weight w of the particle_k1/N, after a polynomial resampling operation, a set of numbers of replicated particles { v } is obtainedⁱ}_i＝1:N，0≤vⁱ≤N，vⁱThe number of times of copying the ith particle after resampling;

step 3.4, as known from the formula (4), the smaller the cost function value of the particle is, the larger the weight is, so that the more times the particle with the large weight is copied in the resampling step is, and meanwhile, the cost function value corresponding to the particle with the large weight is stored; after the polynomial resampling operation, a set of the t-time particle-cost function values is finally obtained

5. The CRPF method based on mixed probability selection operator according to claim 4, wherein the step 4 is as follows:

step 4.1, subset No. 1The updating of the cost function value and the weight value corresponding to the particle is as follows:

when t is 1, the particles in the subset 1 are updated toWhen t is greater than or equal to 2, the particles are updated based on Gaussian distribution, i.e. Representing the state value of the particle at time t-1 after being subjected to polynomial resampling,representing covariance matrix, identity functionAnd variance σ_tThe calculation method of (c) is as follows:

in equations (6) and (7), cov (·) represents the operation of covariance;

the updating of the cost function value is realized by formulas (1) and (2); updating the weight according to a formula (4) or a formula (5); after updating, a new No. 1 particle-cost set at the time t is obtained

Step 4.2, subset No. 2The updating of the particles and the cost function values and weights corresponding to the particles is as follows:

when t is 1, the particles in the subset 2 are updated toWhen t is more than or equal to 2, the updating of the particles in the subset No. 2 is realized based on Cauchy distribution, i.e. the updating is realized by using Cauchy distributionGamma (gamma > 0) represents a scale parameter, l₀Representing the position parameters, gamma and l₀The value of (a) is system dependent; the updating method of the cost function value and the weight is the same as that of the subset 1, and finally, a new particle-cost set 2 at the moment t is obtained

6. The CRPF method based on hybrid probability selection operator of claim 5, wherein the two subsets perform information interaction operation between subsets every G generation in step 5, if T ≠ m, then it is not necessary to perform operation on subsets, m is integer multiple of G, m < T, so thatIf t is equal to m, performing information interaction operation, specifically as follows:

step 5.1, get the particle-cost set of the subset No. 1And sorted subset No. 2First M particles and set of corresponding cost values of the particlesM ∈ {1,2, …, N }, and is setMiddle particleCorresponding cost valueSatisfies the conditionsMerging the two sets into a new particle-cost set

Step 5.2, assembleSorting the cost values of the particles in ascending order to obtain a setNamely, it isSimultaneous aggregationMiddle particleCost value ofMust satisfy the conditionThen, get the collectionThe first N particles and their corresponding cost values are grouped togetherA particle set as the m-time of the No. 1 subset;

step 5.3, for subset No. 2Similar operations are carried out in step 5.1 and step 5.2 to obtain the particle-cost set of the m generation 2 subsetAnd are assembledEach particle ofCost value ofSatisfy the requirement of

Finally obtaining the particle-cost set of the No. 1 subset through the step 5Particle-cost set of subset 2

7. The CRPF method based on hybrid probability selection operator according to claim 6, wherein the step 6 is as follows:

in the particle-cost set E^j(j is 1,2), the i-th particle at time t and the value of the cost function corresponding to the particle are takenIf number 1 subset E¹Middle particleValue of cost function ofGreater than subset 2E²Middle particleValue of cost function ofNamely, it isThen selectEnter the final set of particle-costOtherwise, selectingEntry CollectionAfter the operation, a particle-cost set required by system filtering is obtained

8. The CRPF method based on hybrid probability selection operator according to claim 7, wherein the step 7 is as follows:

referring to the updating method of the No. 1 subset in the step 4, the particle-cost set is updatedPerforming update operation to obtain updated particle-cost set

9. The CRPF method based on the hybrid probability selection operator according to claim 8, wherein the step 8 is as follows:

calculating the weight corresponding to the particles in the particle-cost set E obtained in the step 7 according to the method in the step 2.3, and performing weighted average processing to finally obtain the optimal estimation of the system state at the moment t as follows

In the formula (8), the first and second groups,namely the optimal estimated value of the system state at the moment t of the system, namely the final result required to be solved in the whole filtering process,is a random estimate of the state of the system at time t,is a random estimated valueThe corresponding weight, T, is 1,2, …, T.

Technical Field

The invention belongs to the technical field of nonlinear filtering, and particularly relates to a cost reference particle filtering method based on a mixed probability selection operator.

Background

In many industrial control processes, all data measured by sensors contain noise and cannot be completely eliminated, while many internal system states are not directly measurable. Therefore, in the system state analysis process, it is necessary to filter data including a large amount of noise measured by the sensor, so as to obtain the measured optimal value or the system state optimal estimation value that cannot be measured as much as possible. For example, in the non-linear and non-gaussian lithium ion battery life prediction, the remaining life of the lithium ion battery cannot be directly measured on line, and can only be estimated through the related parameters of charging voltage, current and the like measured on line. Modern filtering techniques have great advantages in this area because of their predictive and estimation roles. Cost Reference Particle Filter (CRPF) can handle the filtering problem of non-linearity and unknown noise statistical characteristics, but the standard CRPF resamples using traditional resampling methods such as polynomial resampling, random resampling, and system resampling. Although these resampling methods can increase the number of effective particles, some particles with large weights are copied in large quantities, so that the particles are too concentrated in the same area, and the resampling result is poor in particle diversity. Therefore, the resampling process of the standard CRPF has the problem of poor particle diversity, and the precision and accuracy of filtering are further influenced. Therefore, the standard CRPF needs to be further optimized, so as to improve the filtering precision and accuracy of the CRPF, and further improve the precision of estimating the remaining life of the lithium ions.

Disclosure of Invention

The invention aims to provide a CRPF method based on a mixed probability selection operator, which solves the problem of low filtering accuracy of a system with unknown noise statistical characteristics in the prior art.

The technical scheme adopted by the invention is that the CRPF method based on the mixed probability selection operator is implemented according to the following steps:

step 1, randomly generating N real numbers as samples of a system state, wherein the samples are called particles, a cost value corresponding to each particle is defined to be 0, and a set formed by the samples and the cost values is called a particle-cost set; simultaneously, respectively randomly generating particles to form a sub-set 1 and a sub-set 2, wherein the number of each sub-set is N;

step 2, respectively calculating cost function values, risk function values and weights of particles in all the subsets;

step 3, resampling the subset, wherein a resampling method selects a polynomial for resampling;

step 4, updating the subset;

step 5, the information interaction of the particles among the subsets, if the set exchange step number is reached, the operation is executed; otherwise, the information interaction operation is not required to be executed;

step 6, generating a particle-cost set finally used for system filtering;

step 7, updating the final particle-cost set;

and 8, filtering the system to finally obtain the optimal estimation value of the system state.

The present invention is also characterized in that,

step 1 generating particle-cost set E with initial size N₀Is shown asWherein the content of the first and second substances,a random estimate of the state of the system is shown,namely, it isObey to uniform distribution U (I)₀) Value of initial cost functioni is 1,2, …, N, N is the sampling sample number of the system state random estimation value;

at the same time, two subsets of size N are generatedWherein the content of the first and second substances,obey to uniform distribution U (I)₀) J is the number of subsets, j is 1,2, when j is 1,denotes the subset 1, when j is 2,representing the subset number 2.

The step 2 is as follows:

step 2.1, calculating particles in the subset at the time tValue of cost function ofj is 1,2, i is 1,2, …, N, T is 1,2, …, T is the time series length, and the cost function is calculated as follows:

in the formulas (1) and (2),representing the ith particle in the j subset at time tA cost function value of; lambda represents a forgetting factor, 0 < lambda < 1;represents the ith particle in the j subset at the time t-1A cost value function value of;indicating particleThe incremental amount of the cost of (a),indicating particleA cost increment function of (a); y is_tThe measured value of the sensor at the time t is represented, h (-) represents an observation function of the estimated value of the sensor, and the preset parameter q satisfies that q is more than or equal to 1;

step 2.2, from the cost function values of the subset particlesCalculating to obtain the risk function valueThe formula for the risk function is:

with respect to the formula (3),representing the ith particle in the j subset at time tThe risk function value of (a); λ is more than 0 and less than 1, i is 1,2, …, N, and q is more than or equal to 1; f (-) represents a state transition function of the system state;

step 2.3, from the cost function values of the subset particlesOr value of risk functionCalculating the weight of the particleThe calculation method is as follows:

it can also be calculated by the following method:

in the formulas (4) and (5),i.e. the ith particle in the j subset at the t momentThe particle weight value of (1), and the particle weight value adjusting parameter beta is more than 1; in the formula (4)Are particlesA cost value function value of; in the formula (5)Indicating particleThe value of the risk function of (a),and expressing the minimum value of the risk function in the N random estimation samples of the subset j at the time t, wherein delta is an adjusting parameter, and delta is more than 0 and less than 1.

The step 3 is as follows:

step 3.1 inInterval [0,1]In the above, a random number set { u } that is subject to uniform distribution is randomly generated_i}_i＝1：NAnd each element in the set satisfies independent same distribution;

step 3.2, cumulative value of weight Iⁱ＝cdf{u_iThe rule cdf denotes the cumulative distribution function of the particle weights, with respect to the random number u_iExistence of

Step 3.3, let the weight w of the particle_k1/N, after a polynomial resampling operation, a set of numbers of replicated particles { v } is obtainedⁱ}_i＝1：N，0≤vⁱ≤N，vⁱThe number of times of copying the ith particle after resampling;

step 3.4, as known from the formula (4), the smaller the cost function value of the particle is, the larger the weight is, so that the more times the particle with the large weight is copied in the resampling step is, and meanwhile, the cost function value corresponding to the particle with the large weight is stored; after the polynomial resampling operation, a set of the particle-cost function values at the time t is finally obtained

The step 4 is as follows:

step 4.1, subset No. 1The updating of the cost function value and the weight value corresponding to the particle is as follows:

when t is 1, the particles in the subset 1 are updated toWhen t is greater than or equal to 2, the particles are updated based on Gaussian distribution, i.e.Representing the state value of the particle at time t-1 after being subjected to polynomial resampling,representing covariance matrix, identity functionAnd variance σ_tThe calculation method of (c) is as follows:

in equations (6) and (7), cov (·) represents the operation of covariance.

The updating of the cost function value is realized by formulas (1) and (2); updating the weight according to a formula (4) or a formula (5); after updating, a new No. 1 particle-cost set at the time t is obtained

Step 4.2, subset No. 2The updating of the particles and the cost function values and weights corresponding to the particles is as follows:

when t is 1, the particles in the subset 2 are updated toWhen t is more than or equal to 2, the updating of the particles in the subset No. 2 is realized based on Cauchy distribution, i.e. the updating is realized by using Cauchy distributionGamma (gamma > 0) represents a scale parameter, l₀Representing the position parameters, gamma and l₀The value of (a) is system dependent; the updating method of the cost function value and the weight is the same as that of the subset 1, and finally, a new particle-cost set 2 at the moment t is obtained

The step 5 is as follows:

in step 5, performing information interaction operation between the two subsets every G generations, if T is not equal to m, then not needing to operate the subsets, m is integral multiple of G, m is less than T, and orderIf t is equal to m, performing information interaction operation, specifically as follows:

step 5.1, get the particle-cost set of subset No. 1And sorted subset No. 2First M particles and set of corresponding cost values of the particlesAnd are assembledMiddle particleCorresponding cost valueSatisfies the conditionsMerging the two sets into a new particle-cost set

Step 5.2, assembleSorting the cost values of the particles in ascending order to obtain a setNamely, it isSimultaneous aggregationMiddle particleCost value ofMust satisfy the conditionThen, get the collectionThe first N particles and their corresponding cost values are grouped togetherA particle set as the m-time of the No. 1 subset;

step 5.3, for subset No. 2Similar operations of step 5.1 and step 5.2 are carried out to obtain the particle-cost set of the m generation 2 subsetAnd are assembledEach particle ofCost value ofSatisfy the requirement of

Finally obtaining the particle-cost set of the No. 1 subset through the step 5Particle-cost set of subset number 2

The step 6 is as follows:

in the particle-cost set E^j(j is 1,2), the i-th particle at time t and the value of the cost function corresponding to the particle are takenIf number 1 subset E¹Middle particleValue of cost function ofGreater than subset 2E²Middle particleValue of cost function ofNamely, it isThen selectEntering a final particle-cost set tableOtherwise, selectingEntry CollectionAfter the operation, a particle-cost set required by system filtering is obtained

Step 7 is specifically as follows:

referring to the updating method of the No. 1 subset in the step 4, the particle-cost set is updatedPerforming update operation to obtain updated particle-cost set

The step 8 is as follows:

calculating the weight corresponding to the particles in the particle-cost set E obtained in the step 7 according to the method in the step 2.3, and performing weighted average processing to finally obtain the optimal estimation of the system state at the moment t as follows

In the formula (8), the first and second groups,namely the optimal estimated value of the system state at the moment t of the system, namely the final result required to be solved in the whole filtering process,is a random estimate of the state of the system at time t,is a random estimated valueThe corresponding weight, T, is 1,2, …, T.

The CRPF method based on the mixed probability selection operator has the beneficial effects that the system with unknown noise statistical characteristics is processed by the CRPF method based on the mixed probability selection operator. And performing standard CRPF operation on two subsets with the same size, updating the particle sets of the subsets respectively based on Gaussian distribution and Cauchy distribution, and enabling the two subsets to perform information interaction operation between the subsets at a specified step number. And then, comparing the corresponding cost function values of the particles in the two sub-group sets, and selecting the particles with small cost values to enter a particle-cost set for finally performing system filtering so as to replace a resampling link of a standard CRPF method. And then, updating the particle-cost set obtained through selection, and finally, carrying out filtering processing on the system. According to the invention, through information interaction and particle selection operation of the subset, the particle diversity is improved, the precision and accuracy of filtering are improved, and the effect of system filtering is better.

Drawings

FIG. 1 is a schematic view of the information interaction operation of the present invention;

FIG. 2 is a schematic view of the particle selection operation of the present invention;

FIG. 3 is a graph of a battery capacity degradation data distribution according to an embodiment of the present invention;

FIG. 4 is a graph comparing the filtering results of the present invention with a standard CRPF;

FIG. 5 is a graph comparing the absolute deviation of the present invention from a standard CRPF;

FIG. 6 is a MAPE comparison of the invention with a standard CRPF.

Detailed Description

The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

When the CRPF method based on the mixed probability selection operator is used for processing a system with unknown noise statistical characteristics, in the mixed probability selection operator, the selection operator obeying Gaussian distribution can improve the local search capability, the operator obeying Cauchy distribution can improve the global search capability, and the mixed probability selection operator based on the two distributions can improve the convergence rate and the filtering precision. Firstly, initializing a particle-cost set (particles are system state values generated randomly) with the size of N and used for final system filtering and two subsets with the size of N according to the operation of a standard CRPF, wherein the two subsets are respectively a subset 1 and a subset 2, the subset 1 is updated based on Gaussian distribution in the updating stage of the particle set, and the subset 2 is similarly operated based on Cauchy distribution. And then, respectively outputting the particle-cost subsets obtained by updating the two subsets, wherein the operation inside the subsets is the same as the standard CRPF. And when the specified step number is reached, performing information interaction operation on the two subsets. Taking the subset combination No. 1 as an example, the first M particles corresponding to the subset No. 2 after the whole subset No. 1 and the cost value (i.e., the cost function value) are sorted in ascending order are combined into a new particle-cost set. And sorting the cost values of the new set in ascending order, taking the set formed by the first N corresponding particles with the cost values of the new set as a new set of the No. 1 subset, and then carrying out similar operation on the No. 2 subset. And then, performing particle selection operation on the two subsets, namely comparing cost function values corresponding to the particles in the two subsets in each step, and selecting the optimal particles and the cost values corresponding to the optimal particles for realizing the particle-cost set of the final system filtering. The specific selection standard is as follows: in each particle selection process, if the cost value of the particles in the subset 1 is greater than that of the subset 2, selecting the particles in the subset 2 and the corresponding cost values of the particles into a particle-cost set finally used for system filtering; otherwise, selecting the particles in the No. 1 subset and the corresponding cost values of the particles. The particle-cost set finally used for filtering is generated through the processes of information interaction and particle selection, namely, particles in two generated subsets are respectively updated based on Gaussian distribution and Cauchy distribution according to the size of a cost function value, the formed particle set comprises both particles obeying the Gaussian distribution and particles obeying the Cauchy distribution, and the operation of a mixed probability selection operator is realized. And after the operation of particle selection is completed, updating and weighted average processing of the weight value are carried out on the obtained particle-cost set, and finally the optimal estimated value of the system state at each moment is obtained, so that the filtering process of the system is completed.

The invention relates to a CRPF method based on a mixed probability selection operator, which is implemented according to the following steps:

step 1, as shown in fig. 1, randomly generating N real numbers as samples of a system state, wherein the samples are called particles, a cost value corresponding to each particle is defined to be 0, and a set composed of the samples and a cost value is called a particle-cost set; simultaneously, respectively randomly generating particles to form a sub-set 1 and a sub-set 2, wherein the number of each sub-set is N;

step 1 generating particle-cost set E with initial size N₀Is shown asWherein the content of the first and second substances,a random estimate of the state of the system is shown,(I₀is system dependent), i.e.Obey to uniform distribution U (I)₀) Value of initial cost functioni is 1,2, …, N, N is the sampling sample number of the system state random estimation value;

at the same time, two subsets of size N are generatedWherein the content of the first and second substances,obey to uniform distribution U (I)₀) J is the number of subsets, j is 1,2, when j is 1,denotes the subset 1, when j is 2,denotes the subset No. 2, the other parameters have the same meaning.

Step 2, respectively calculating cost function values, risk function values and weights of particles in all the subsets;

the step 2 is as follows:

step 2.1, calculating particles in the subset at the time tValue of cost function ofj is 1,2, i is 1,2, …, N, T is 1,2, …, T is the time series length, and the cost function is calculated as follows:

in the formulas (1) and (2),representing the ith particle in the j subset at time tA cost function value of; lambda represents a forgetting factor, 0 < lambda < 1;when represents t-1Carve the ith granule in the j subsetA cost value function value of;show particlesThe incremental amount of the cost of (a),indicating particleA cost increment function of (a); y is_tThe measured value of the sensor at the time t is represented, h (-) represents an observation function of the estimated value of the sensor, and the preset parameter q satisfies that q is more than or equal to 1;

step 2.2, from the cost function values of the subset particlesCalculating to obtain the risk function valueThe formula for the risk function is:

with respect to the formula (3),representing the ith particle in the j subset at time tThe risk function value of (a); λ is more than 0 and less than 1, i is 1,2, …, N, and q is more than or equal to 1; f (-) represents a state transition function of the system state;

step 2.3 Generation from subset of particlesValue of valence functionOr value of risk functionCalculating the weight of the particleThe calculation method is as follows:

it can also be calculated by the following method:

in the formulas (4) and (5),i.e. the ith particle in the j subset at the t momentThe particle weight value of (1), and the particle weight value adjusting parameter beta is more than 1; in the formula (4)Are particlesA cost value function value of; in the formula (5)Indicating particleThe value of the risk function of (a),and expressing the minimum value of the risk function in the N random estimation samples of the subset j at the time t, wherein delta is an adjusting parameter, and delta is more than 0 and less than 1.

Step 3, resampling the subset, wherein a resampling method selects a polynomial for resampling;

the step 3 is as follows:

step 3.1, in the interval [0,1 ]]In the above, a random number set { u } that is subject to uniform distribution is randomly generated_i}_i＝1：NAnd each element in the set satisfies independent same distribution;

step 3.2, cumulative value of weight Iⁱ＝cdf{u_iThe rule cdf denotes the cumulative distribution function of the particle weights, with respect to the random number u_iExistence of

Step 3.3, let the weight w of the particle_k1/N, after a polynomial resampling operation, a set of numbers of replicated particles { v } is obtainedⁱ}_i＝1：N，0≤vⁱ≤N，vⁱThe number of times of copying the ith particle after resampling;

step 3.4, as known from the formula (4), the smaller the cost function value of the particle is, the larger the weight is, so that the more times the particle with the large weight is copied in the resampling step is, and meanwhile, the cost function value corresponding to the particle with the large weight is stored; after the polynomial resampling operation, a set of the particle-cost function values at the time t is finally obtainedj＝1，2。

Step 4, updating the subset;

the step 4 is as follows:

step 4.1, subset No. 1The updating of the cost function value and the weight value corresponding to the particle is as follows:

when t is 1, number 1Particle update of subset toWhen t is greater than or equal to 2, the particles are updated based on Gaussian distribution, i.e.Representing the state value of the particle at time t-1 after being subjected to polynomial resampling,representing covariance matrix, identity functionAnd variance σ_tThe calculation method of (c) is as follows:

in equations (6) and (7), cov (·) represents the operation of covariance.

The updating of the cost function value is realized by formulas (1) and (2); updating the weight according to a formula (4) or a formula (5); after updating, a new No. 1 particle-cost set at the time t is obtained

Step 4.2, subset No. 2The updating of the particles and the cost function values and weights corresponding to the particles is as follows:

when t is 1, the particles in the subset 2 are updated toWhen t is greater than or equal to 2, in subset No. 2The updating of the particles being effected on the basis of the Cauchy distribution, i.e.Gamma (gamma > 0) represents a scale parameter, l₀Representing the position parameters, gamma and l₀The value of (a) is system dependent; the updating method of the cost function value and the weight is the same as that of the subset 1, and finally, a new particle-cost set 2 at the moment t is obtained

Step 5, the information interaction of the particles among the subsets, if the set exchange step number is reached, the operation is executed; otherwise, the information interaction operation is not required to be executed;

in step 5, performing information interaction operation between the two subsets every G generations, if T is not equal to m, then not needing to operate the subsets, m is integral multiple of G, m is less than T, and orderIf t is equal to m, performing information interaction operation, specifically as follows:

step 5.1, get the particle-cost set of subset No. 1And sorted subset No. 2First M particles and set of corresponding cost values of the particlesAnd are assembledTherein is sizedCorresponding cost valueSatisfies the conditionsMerging the two sets into a new particle-cost set

Step 5.2, assembleSorting the cost values of the particles in ascending order to obtain a setNamely, it isSimultaneous aggregationMiddle particleCost value ofMust satisfy the conditionThen, get the collectionThe first N particles and their corresponding cost values are grouped togetherA particle set as the m-time of the No. 1 subset;

step 5.3, for subset No. 2Similar operations of step 5.1 and step 5.2 are carried out to obtain the particle-cost set of the m generation 2 subsetAnd are assembledEach particle ofCost value ofSatisfy the requirement of

Finally obtaining the particle-cost set of the No. 1 subset through the step 5Particle-cost set of subset number 2

Step 6, generating a particle-cost set finally used for system filtering;

the step 6 is as follows:

as shown in fig. 2, in the particle-cost set E^j(j is 1,2), the i-th particle at time t and the value of the cost function corresponding to the particle are takenIf number 1 subset E¹Middle particleValue of cost function ofGreater than subset 2E²Middle particleValue of cost function ofNamely, it isThen selectEnter the final set of particle-costsOtherwise, selectingEntry CollectionAfter the operation, a particle-cost set required by system filtering is obtained

Step 7, updating the final particle-cost set;

step 7 is specifically as follows:

referring to the updating method of the No. 1 subset in the step 4, the particle-cost set is updatedPerforming update operation to obtain updated particle-cost set

And 8, filtering the system to finally obtain the optimal estimation value of the system state.

The step 8 is as follows:

calculating the weight corresponding to the particles in the particle-cost set E obtained in the step 7 according to the method in the step 2.3, and performing weighted average processing to finally obtain the optimal estimation of the system state at the moment t as follows

In the formula (8), the first and second groups,namely the optimal estimated value of the system state at the moment t of the system, namely the final result required to be solved in the whole filtering process,is a random estimate of the state of the system at time t,is a random estimated valueThe corresponding weight, T, is 1,2, …, T.The filtering result of the system can provide necessary data support and basic guarantee for the subsequent analysis and control of the system.

The invention discloses a cost reference particle filtering method based on a mixed probability selection operator, which is used for processing a system with unknown noise statistical characteristics, adopting subsets which are updated respectively based on Gaussian distribution and Cauchy distribution to carry out information interaction and particle selection operation on the subsets, increasing the diversity of particles, updating the set after obtaining a particle-cost set required by system filtering, finally calculating the weight corresponding to the particles in the updated particle set, carrying out weighted average processing, and finally obtaining the optimal estimated value of the system state, namely the final system filtering result, thereby realizing the filtering operation on the observation data collected by a sensor. The invention can filter under the condition of unknown process noise and measurement noise without a noise prior distribution function, improves the precision and accuracy of the filtering result, reduces the filtering deviation of the system, improves the filtering effect and is beneficial to further analyzing and controlling the system. The invention solves the problem of lower filtering accuracy of a system with unknown noise statistical characteristics in the prior art.

Examples

In this embodiment, the battery capacity of the lithium battery with model number Li-ion 18650 and rated capacity 2Ah is predicted, and the battery capacity fading data is shown in fig. 3.

In order to better predict the capacity degradation process of the battery, a system state space model is determined, wherein a state equation is shown as an equation (9), and an observation equation is shown as an equation (10). Initial values of model parameters are set to a₁＝1.942，b₁＝-2.052×10^-3，c₁＝1.57×10^-7，d₁＝0.07406。

In formula (9), t is 2, 3_t、c_tIs an estimate of an impedance-related parameter in the battery at time t, b_t、d_tIs an estimate of a battery aging rate related parameter at time t, w_a、w_b、w_c、w_dProcess noise representing a, b, c, d, respectively; q in formula (10)_tMeasuring the noise v for an estimate of the battery capacity at time t_tN (0, 1), N (0, 1) is a standard normal distribution, i.e., a Gaussian distribution with a mean of 0 and a variance of 1.

The simulation step number T is set to 168, the particle numbers of the particle set of the standard CRPF and the CRPF method (the invention) based on the mixed probability selection operator are set to be N-150, and the resampling step uses a polynomial resampling method. In the present invention, the number 1 subset and the number 2 subset exchange information between subsets every G-10 generation, and the number of excellent individuals of the other subset is taken as M-70 when exchanging. In particular, the subset No. 2 of the present invention is updated based on formula (11) following the cauchy distribution when updating the set of particles.

In the formula (11), the reaction mixture,respectively representing the ith particle at the moment t of the No. 2 subset obtained by updating the parameters a, b, c and d.

The parameter q involved in the step of setting the invention is 2, and the forgetting factor lambda is 1 × 10^-6The parameter β is 1.5, and the adjustment parameter δ is 0.001.

Completing the information interaction operation shown in FIG. 1 and the particle selection operation shown in FIG. 2; processing the B0007 battery capacity decline data in the figure 3; the filtering result of the system is obtained as shown in fig. 4.

In order to more accurately analyze the effectiveness of the invention, the absolute deviation, RMSE (mean square error) and MAPE (mean absolute percentage error) are selected as system indexes to be compared and analyzed. FIG. 5 is a graph comparing the deviation of a standard CRPF with the present invention; FIG. 6 is a graph comparing standard CRPF with MAPE per cycle of the present invention; as can be seen from Table 1: after improvement, the RMSE of the complete filtering process is reduced to 51.482% of that of the standard CRPF method, and the overall MAPE of the invention is reduced by 0.50775% compared with the MAPE of the standard CRPF method.

TABLE 1

The experimental results show that compared with the standard CRPF, the method has the advantages of smaller deviation, better filtering effect and higher filtering accuracy and precision (namely the CRPF method based on the mixed probability selection operator).