Particle filter method based on state trajectory clustering similarity and application

Abstract

The application provides a particle filtering method based on state trajectory clustering similarity and application, and comprises the following steps: S1, initialization; S2, assuming a prior probability density distribution as an importance probability density function, that is, a proposal distribution guides particles to perform sequential importance sampling, and then the importance weight of the sampling particles is recursively calculated; S3, combining the consistency principle of the system true state and the sampling particle state trajectory, similarity measurement is performed on the current filtering and the future multi-stage Gaussian filtering predicted measurement information by using a data mining thought, that is, a clustering analysis method; S4, the proposal distribution in S2 is modified by using the similarity measurement in S3; S5, the importance weight calculation in the sequential importance sampling process in S2 is updated by using the proposal distribution modified in S4, the particle weight is normalized, and then the estimation value of the system state at the current moment is obtained; S6, steps S2-S5 are repeated, and the estimation values of the system state at different moments are sequentially output. The application can improve the particle degradation phenomenon.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of nonlinear filtering, in particular to a particle filtering method based on state trajectory clustering similarity and application. BACKGROUND

[0002] Nonlinear phenomena are ubiquitous in natural engineering technical fields, and state estimation as a hot research topic with important theoretical and practical value has been applied in many aspects, such as target tracking and navigation, fault diagnosis and detection, process feedback and control, biochemical reaction and extraction, economic prediction and regulation, etc. Many solutions have been proposed by domestic and foreign scholars, and the more classic ones are Luenberger observer, Kalman filter, unknown input observer, high gain observer, proportional integral observer, non-smooth observer and robust observer, etc. For example, Kalman filtering obtains the linear minimum variance estimation of the updating process of the conditional probability density of random variables through recursive formula under the premise of satisfying Gaussian noise distribution, and then develops multiple improved methods such as extended Kalman, unscented Kalman, high-order extended Kalman and constant gain extended Kalman filter. The above methods have great advantages and relatively wide application range for the state estimation problem of nonlinear systems, but there is still a lot of optimization and improvement space for the complexity of different actual engineering nonlinearities, the uncertainty of environmental noise, etc.

[0003] With the rapid development of digital computer computing power, as the particle filtering method based on Bayesian theory and Monte Carlo idea shows strong advantages and application potential in solving nonlinear, non-Gaussian system evaluation, its prototype-sequential importance sampling (SIS) was formed in the middle of last century, which was mainly applied in physics and automatic control disciplines. Subsequently, due to the inherent sample degeneration phenomenon of the method itself and the restriction of computer hardware conditions, the research of particle filtering method slowed down. Until 1993, Gordon and Salmond introduced the resampling step into the SIS algorithm to form the sequential important resampling method (SIR), which improved and laid the foundation of the related theory of particle filtering, and opened a new round of development boom, but the resampling method improved the particle degeneration phenomenon while leading to the particle impoverishment problem. At the same time, around the improvement and development of stochastic probability theory and Monte Carlo method, auxiliary particle filtering and Gaussian (and) particle filtering methods were proposed. Auxiliary particle filtering (APF) proposed by Guarniero approximates the optimal proposal distribution according to the latest observation data, and represents the prior probability density of the current state by introducing an auxiliary variable, but when the system state process noise is strong, the filtering accuracy is difficult to guarantee due to insufficient information. Gaussian particle filter (GPF) designed by Kotecha uses Gaussian distribution to approximate the posterior probability density distribution of the system state under the basic framework of SIS, and only gets the mean and variance of the probability density each time. Gaussian Sum Particle filter (GSPF) proposed by Djuric uses a weighted sum of a finite number of Gaussian densities to approximate the posterior non-Gaussian probability density, which has the same principle as GPF. The filtering effect of the two Gaussian algorithms depends heavily on the degree of nonlinearity of the problem, and is limited to the dimension of system variables.

[0004] The standard particle filter method is simple in structure and easy to operate. The algorithm approximates the estimation value to the true state value in the optimal estimation. However, there are still some defects and deficiencies in the practical engineering application of this method, mainly in the following two aspects: (1) The standard particle filter method introduces an importance probability density distribution in the sequential importance sampling process, which causes the particle weight variance to accumulate with the increase of the number of iterations, and the importance weight of most particles tends to zero, i.e. the particle degeneracy phenomenon, resulting in a serious waste of computing resources, and the approximate estimation cannot accurately describe the posterior distribution of the true state. More seriously, this degeneration phenomenon cannot be avoided due to the defects of the method itself. (2) Resampling strategy is an effective means to improve the particle degeneration phenomenon. By resampling the discrete approximate posterior probability distribution obtained by importance sampling, the samples with larger weights are copied multiple times under the guidance of particle motion and the distribution of the previous state, so that the number of effective particles increases to suppress the degeneration purpose. At the same time, the resampling process may cause the abandonment or loss of some particles with low weights, so that the particles after resampling deviate from the true state posterior region too early, resulting in sample impoverishment problem, and finally leading to the increase of state estimation variance and the significant decline of filtering performance. The main measures to improve the algorithm are to select a reasonable proposal distribution and to improve the resampling.

[0005] In the selection of reasonable proposal distribution, Xu Bo et al. (Application No. 202011072157.7) used multi-rate consistent fusion technology to obtain the influence degree of local observation nodes on global state quantity; Xia Wei et al. (Application No. 202010042906.5) introduced the Kullback-Leibler (K-L) divergence defined by the true posterior distribution of the target state and the particle posterior distribution to define the cost function; Li Liangqun et al. (Application No. 201910650468.8) introduced TSK fuzzy model to model the dynamic system of the target; Zhou Zhaihe et al. (Application No. 201910291650.9) used a new quaternion distribution as the standard distribution of quaternions on the unit hypersphere; Liu Yu et al. (Application No. 201410783563.2) introduced network credibility to perform interaction and fusion of local credibility through consistency iteration; Wang Hongjian et al. (Application No. 201310296086.2) selected fading factor and weakening factor to optimize the design of STSRCKF; Xia Yuanqing et al. (Application No. 201310645786.8) obtained the measurement value at time k and calculated the mean and variance of N filter processes in parallel; Zhu Zhiyu et al. (Application No. 201110308010.8) used artificial fish swarm algorithm to guide prior particles to move to the high likelihood area; Yang Meng et al. (Application No. 201010121599.6) adopted SSUKF algorithm based on SSUT transformation of hyperspherical single-line sampling; Yang Meng et al. (Application No. 201010121571.2) used particle swarm optimization algorithm to optimize the estimated state to tend to the area with larger posterior probability density; Ji Hongbing et al. (Application No. 200810232762.9) used jump quasi-Monte Carlo random sample sequence to transform into quasi-Gaussian sample sequence obeying specified distribution; Zhao Qingjie et al. (Application No. 200710099440.7) used prior probability distribution, extended Kalman filter, unscented Kalman filter, and other divide-and-conquer sampling strategies to process sample particles; The above methods can all achieve the purpose of optimizing or updating the proposal distribution, thereby improving the particle degradation phenomenon.

[0006] In the aspect of improving the resampling method, Liu Haitao et al. (Application No. 202110195743.9) cross-process the preset weight threshold according to the adaptive mutation strategy; Lu Zhan et al. (Application No. 201611165889.4) use the variational Bayesian method to obtain the distribution of unknown parameters in the Gaussian mixture model through a method of cyclic iteration; Huang Zhenjin et al. (Application No. 201810980396.9) generate a new population by fission, mutation, crossover, and selection of the particle population through the method of classification evolution; Pan Jiahui et al. (Application No. 201510531989.3) retain part of the weight information of the particle by drawing on the idea of guided filtering; Wang Jun et al. (Application No. 201410241879.9) accept the probability model based on the GPU architecture maximum likelihood sampling; He Kang et al. (Application No. 201410052491.4) output and replace small weight particles by comparing and exchanging the FIFO device; Yu Xuelian et al. (Application No. 201510493464.5) set a reasonable weight threshold and record the number of random numbers falling within the threshold as the number of subsequent particle replication; Zhou Yun et al. (Application No. 201510494727.4) design an importance sampling weight adaptive optimization method; Wei Guohua et al. (Application No. 201410397456.6) group particles by importance weight and calculate and compare the effective number for linear processing; Li Hongwei (Application No. 201310695501.1) optimizes the sampling particles by difference iteration to obtain the optimal particle set; Cong Li et al. (Application No. 201010121623.6) design a parallel genetic resampling method to establish an information exchange model; Qin Honglei et al. (Application No. 200910238800.6) calculate the fitness of the offspring population and arrange them in descending order, and use the small environment to limit competition and selection to make the particles move towards high fitness; Peng Xiyuan et al. (Application No. 201810136919.8) adjust the particle weight by using the similarity between the particle observation vector and the system state. The above methods can improve or correct the resampling process, thereby solving the problem of particle impoverishment.

[0007] From the research status in recent years, the improvement methods for particle filtering are focused on the improvement or solution of particle degradation and impoverishment. The solution of the above two problems mostly depends on or relies on swarm intelligence, bionics, genetic mutation, and other iterative optimization algorithms, which are complex in algorithm programming, time-consuming, and prone to local optimization, and even iterative divergence, which seriously affects the timeliness and calculation accuracy of the state estimation of the particle filtering method, and hinders its application and development. In addition, the system nonlinearity represented by the high complexity of the current natural engineering structure is getting higher and higher, and the bimodal complex characteristics of the observation likelihood function make the filtering state estimation problem more difficult to solve. Although the conventional improved filtering methods mentioned above can alleviate the particle degradation and impoverishment phenomenon to some extent, they still have problems such as low filtering estimation accuracy and poor method stability, which cannot meet the modern engineering requirements. SUMMARY

[0008] To solve the technical problems in the background art, the present application provides a particle filtering method based on state trajectory clustering similarity, which can improve the particle degradation phenomenon.

[0009] To achieve the above objective, the present application adopts the following technical solution:

[0010] A particle filtering method based on state trajectory clustering similarity, comprising the following steps:

[0011] S1, initializing sampling particles and corresponding weights: since the posterior probability density function is unknown, an importance probability density function is selected to meet the Monte Carlo sampling, so as to approximately obtain a particle set;

[0012] S2, sequential importance sampling: assuming that the prior probability density distribution is an importance probability density function, i.e., the proposal distribution guides the particle to perform sequential importance sampling, and then the importance weight of the sampling particle is recursively calculated;

[0013] S3, measuring the state trajectory by clustering similarity: taking the mathematical model of the well-posed space state trajectory problem as the starting point, combining the consistency principle of the real state of the system and the state trajectory of the sampling particle, and using the clustering analysis method of the data mining idea to measure the similarity of the current filtering and the future multi-stage Gaussian filtering prediction measurement information;

[0014] S4, modifying the importance probability density function: using the similarity measurement of step S3 to modify the proposal distribution in step S2;

[0015] S5, updating the importance weight of the sampling particle: updating the importance weight calculation in the sequential importance sampling process in step S2 by the proposal distribution modified in step S4, normalizing the particle weight, and then obtaining the estimation value of the system state at the current time;

[0016] S6, steps S1-S5 constitute an iteration process of the state estimation of the particle filtering method; repeating steps S2-S5, and sequentially outputting the estimation value of the system state at different times.

[0017] Further, the step S1 comprises: initializing the sampling particle and the particle weight is where X0 is the particle set at the initial time t0, is the state vector of the i-th sampling particle at the initial time t0, N is the number of generated particles, and p(X0) is the initial probability density function.

[0018] Further, the step S2 comprises: since the marginal posterior filtering distribution p(x k|y 1:k ) is unknown, so the importance probability density function q(x k |x k-1 ,y 1:k ) is constructed to meet the requirements of the Monte Carlo sampling method, so as to make up for the defect that the sampling from the target distribution is impossible, and the prior probability density distribution in the following formula is usually selected as the importance probability density function in the sequential importance resampling process:

[0019] q(x k |x k-1 ,y 1:k ) = K(x k |x k-1 ).

[0020] Further, the step S3 is based on the principle of consistency of the spatial vector trajectory, that is, the closer the state trajectory of a particle to the true state trajectory, the higher the degree to which the state of the particle represents the true state, and the step S3 comprises:

[0021] S31, selecting a particle set at k to k+L+l time points, wherein L and l are pre-set constants, the original state trajectory x of the particle is subjected to the SIS filtering process, and the modified state trajectory x' j (i) is subjected to the GPF prediction process; since the true state information is unknown in the particle filtering method, the consistency of the state trajectory is characterized by the measurement likelihood function, and the measurement likelihood trajectory corresponding to the state of the sampling particle is solved according to the state trajectory of the particle;

[0022] S32, using the distance clustering similarity to analyze the consistency of the trajectory of the sampling particle, and calculating the distance similarity measure of the reference trajectory and the measurement likelihood trajectory corresponding to the state of the sampling particle, wherein the reference trajectory is the measurement likelihood trajectory corresponding to the true state of the system.

[0023] Further, the measurement likelihood trajectory corresponding to the state of the sampling particle in the step S31 is solved according to the following formula:

[0024]

[0025] wherein the observation noise υ k = 0, the observation equation H(*) is a known function determined by a specific research object in the absence of noise interference, and the measurement likelihood trajectory corresponding to the true state of the system, that is, the reference trajectory is {Y k} = {y j : j = k, …, k+L+l}.

[0026] ​Further, the step S32 comprises: selecting distance clustering similarity to analyze the consistency of the trajectory, and the distance similarity between the reference trajectory and the measurement likelihood trajectory corresponding to the sampling particle state is calculated as follows:

[0027]

[0028] Wherein dis(*) represents the distance similarity function, and dis(*)≥0, S is a metric type parameter;

[0029] In order to increase the reliability of the method, the above formula is exponentially transformed:

[0030]

[0031] Wherein λ is a gradient factor.

[0032] Further, the step S4 comprises: according to the distance similarity between the reference trajectory and the measurement likelihood trajectory corresponding to the sampling particle state obtained in the step S32, the importance probability density function replaced by the prior density function q(x k |x k-1 ,y k ) is corrected by similarity according to the following formula:

[0033] q(x k |x k-1 ,y k )=d k *p(x k |x k-1 )。

[0034] Further, the step S5 comprises: updating the importance weight of the first-order Markov model by using the importance probability density function corrected in the step S4, and the importance weight corresponding to the sampling particle at the time k and k+L+l can be calculated respectively as follows:

[0035]

[0036]

[0037] Wherein p υ (*) is the probability density function of the measurement likelihood noise;

[0038] The importance weight is normalized according to the following formula:

[0039]

[0040]

[0041] ​​Set the SIS particle distribution at time k And the corresponding importance weight The posterior probability density distribution π that can approximate the system state k|k The GPF particle distribution at time k+L+1 And the corresponding importance weight The predictive probability density function π that can be approximately expressed k+L+l|k Therefore, the state estimate value x of the system k Can be obtained by filtering operation, and the state estimate value x k+L+l Can be obtained by a prediction step:

[0042] Filtering process: consistent with the SIS (GPF) filtering process, extracting a particle set

[0043] Updating process: solving the corresponding weight And And normalizing it to obtain And Estimate the system state x according to the following formula k And x k+L+l :

[0044]

[0045]

[0046] The present application also provides a social and economic development evaluation method based on state trajectory clustering similarity particle filtering, comprising the following steps:

[0047] S1, using state trajectory clustering similarity particle filtering to establish a one-dimensional nonlinear system-single variable non-stationary social and economic development growth model, the growth model has the characteristics of high nonlinearity and bimodal state quantity distribution;

[0048] S2, initialize the sampling particles and the corresponding weight: since the posterior probability density function of the growth model is unknown, the importance probability density function is usually selected to meet the Monte Carlo sampling, so as to approximately obtain the particle set;

[0049] S3, sequential importance sampling: assuming that the prior probability density distribution of the growth model is used as the importance probability density function, that is, the proposed distribution guides the particle to perform sequential importance sampling, and then the importance weight of the sampling particle is recursively calculated;

[0050] S4, measure the state trajectory clustering similarity: taking the growth model as the starting point, combining the consistency principle of the real state of the system and the state trajectory of the sampling particle, and using the data mining idea-clustering analysis method to measure the similarity of the current filtering and the future multi-stage Gaussian filtering prediction measurement information;

[0051] S5, revising the importance probability density function: revising the proposal distribution in step S3 by using the similarity measurement in step S4;

[0052] S6, updating the importance weight of the sampling particle: updating the importance weight calculation in the sequential importance sampling process in step S3 by step S5, normalizing the particle weight, and then obtaining the state estimation value at the current time;

[0053] S7, steps S2-S6 constitute an iteration process of the growth model; repeating steps S3-S6 outputs the state estimation value of the growth model at different times in turn.

[0054] The application also provides a lithium ion battery health state evaluation method based on a state trajectory clustering similarity particle filtering method, comprising the following steps:

[0055] S1, combining the equivalent circuit model and relying on the electrochemical characteristic attenuation mechanism, using the state trajectory clustering similarity particle filtering to establish a lithium ion battery state tracking and remaining useful life prediction model;

[0056] S2, initializing the sampling particle and the corresponding weight: since the posterior probability density function of the prediction model is unknown, the importance probability density function is usually selected to meet the Monte Carlo sampling, so as to approximately obtain the particle set;

[0057] S3, sequential importance sampling: assuming that the prior probability density distribution of the prediction model is the importance probability density function, that is, the proposal distribution guides the particle to perform sequential importance sampling, and then recursively calculates the importance weight of the sampling particle;

[0058] S4, clustering similarity measurement of state trajectory: taking the lithium ion battery state tracking and remaining useful life prediction model as the starting point, combining the consistency principle of system true state and sampling particle state trajectory, and using the data mining idea--clustering analysis method to measure the similarity of the current filtering and future multi-stage Gaussian filtering prediction measurement information;

[0059] S5, revising the importance probability density function: revising the proposal distribution in step S3 by using the similarity measurement in step S4;

[0060] S6, updating the importance weight of the sampling particle: updating the importance weight calculation in the sequential importance sampling process in step S3 by step S5, normalizing the particle weight, and then obtaining the state tracking and remaining useful life prediction value at the current time;

[0061] S7, steps S2-S6 constitute an iteration process of the lithium-ion battery state tracking and RUL prediction model; steps S3-S6 are repeated to sequentially output the predicted state tracking and remaining useful life estimation values of the prediction model at different times.

[0062] By adopting the technical scheme, the present application has the following beneficial effects:

[0063] The particle filtering method based on state trajectory clustering similarity improves the particle filtering method itself without relying on external methods, and re-architects the algorithm logic framework: for the particle degradation phenomenon, the trajectory consistency degree of the particles is measured by using the clustering similarity theory, and the higher the consistent similarity degree is, the closer the particles are to the real state; the importance weight of the updated particles is measured, the particle degradation phenomenon is improved, the resampling process is replaced by the measure of integrating the update compensation scheme (clustering measurement) into the latest measurement information in the importance sampling process, so that the improved filtering method has no resampling process, and the particle depletion problem can be fundamentally eliminated; the particle degradation phenomenon is effectively improved, and the timeliness is also significantly improved. BRIEF DESCRIPTION OF DRAWINGS

[0064] Figure 1 The flowchart of the particle filtering method based on state trajectory clustering similarity for a preferred embodiment of the present application;

[0065] Figure 2 The flowchart of the standard particle filtering method in the prior art;

[0066] Figure 3 Comparison of state estimation results of SIR, APF, GPF and CSPF (Gaussian noise);

[0067] Figure 4 Comparison of state estimation results of SIR, APF, GPF and CSPF (non-Gaussian noise);

[0068] Figure 5 Comparison of RMSE results of SIR, APF, GPF and CSPF (including ECSPF and CCSPF) four types of state estimation methods (15 times of random simulation);

[0069] Figure 6 Battery equivalent circuit model;

[0070] Figure 7 Comparison of lithium-ion battery state tracking and RUL prediction results of the SIR method;

[0071] Figure 8 Comparison of lithium-ion battery state tracking and RUL prediction results of the APF method;

[0072] Figure 9 Comparison of lithium-ion battery state tracking and RUL prediction results for the ECSPF method;

[0073] Figure 10 Comparison of lithium-ion battery state tracking and RUL prediction results for the CCSPF method; DETAILED DESCRIPTION

[0074] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by a person of ordinary skill in the art without creative work fall within the scope of the present application.

[0075] It should be noted that when a component is referred to as being "fixed" to another component, it can be directly on the other component or intervening components can also be present. When a component is referred to as being "connected" to another component, it can be directly connected to the other component or intervening components can also be present. When a component is referred to as being "disposed" on another component, it can be directly on the other component or intervening components can also be present. The terms "vertical", "horizontal", "left", "right", and similar expressions used herein are for illustrative purposes only.

[0076] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs. The terminology used in the description herein is for describing particular embodiments only and is not intended to be limiting of the application. As used herein, the term "and / or" includes any and all combinations of one or more of the associated listed items.

[0077] Please refer to Figure 1 A particle filtering method based on state trajectory clustering similarity is provided in a preferred embodiment of the present application, comprising the following steps:

[0078] S1, initializing sampling particles and corresponding weights: since the posterior probability density function is unknown, an importance probability density function is selected to meet Monte Carlo sampling, so as to approximately obtain a particle set.

[0079] The step S1 comprises: sampling particle initialization And the particle weight is Wherein, X0 is a particle set at an initial time t0, is a state vector of the i th sampling particle at the initial time t0, N is the number of generated particles, and p(X0) is an initial probability density function, which is specifically:

[0080] S11, Optimal Bayesian Recursive Optimal Particle Filter, in particular:

[0081] Set (Ω, P, F) as a random probability space, define two real vector random processes: X = {X k ,k∈N} and Y = {Y k ,k∈N\{0}} respectively, where X is the state process and Y is the measurement information. Assume n x and n y are the dimensions of the state and measurement information corresponding to the state space respectively. Assume The probability density measure space on the set is the set of probability measures on the largest dimensional Euclidean space Set (E, d) as the measure space, and are two continuous function sequences: E→E, where the space E will be the set of all probability measures on the n-dimensional Euclidean space in the random filtering.

[0082] Most nonlinear systems can be represented in the form of a dynamic state space (DSS) model as follows:

[0083] X k = F(X k-1 , ω k ) (State Model) (1)

[0084] Y k = H(X k , υ k ) (Measurement Equation) (2)

[0085] where k represents the discrete time (stage) index, is the set of system states at stage k, represents the measurement information corresponding to the state X k . F(*) and H(*) are known state transition and observation functions, respectively, corresponding to the state transition probability kernel density K(X k |X k-1 ) and the measurement likelihood probability density g(Y k |X k ) in statistical description. System noise and observation noise are independent random variables following arbitrary form probability density function (PDF).

[0086] The spatial state X follows a first-order Markov process, i.e. the state X k at the current time is only related to the state X k-1 at the previous time. The initial distribution X0~μ(dX0), the state transition probability kernel density K(X k |X k-1 ) and the measurement likelihood probability density g(Y k |X k ) are Lebesgue measures, i.e.

[0087] K(X k |X k-1 ) = P ω (X k -F(X k-1 )) (3)

[0088] g(Y k |X k ) = P ω (Y k -H(X k )) (4)

[0089] From the perspective of Bayesian theory, the state estimation problem is to recursively calculate the credibility of the current state X 1：k based on a series of existing data Y k (pre-knowledge), and the credibility measure is the joint posterior filtering distribution P(X k |Y 1:k ). Therefore, the recursive calculation needs to be realized through two steps of prediction and update.

[0090] The step S11 specifically comprises the following steps:

[0091] S111, prediction: the state transition probability kernel density K(X k |X k-1 ) is calculated by using the state equation, i.e. the state transition formula (1) in the dynamic state space (DSS) model, and the joint posterior filtering distribution P(X k-1 |Y 1:k-1 ) at the k-1 time is used to predict the joint posterior filtering distribution P(X k |Y 1:k-1 ) at the k time, i.e.

[0092]

[0093]

[0094] S112, update: the measurement information g(Y k |X k ) at the k time is used to update the joint posterior filtering distribution P(X k |Y 1:k-1 ) at the k time.k |X k Update the joint posterior filter distribution P(X) k |Y 1:k-1 ), and thus obtain state X k The joint posterior filtering distribution P(X) k |Y 1:k ),Right now

[0095]

[0096] S213, System state probability density measure π k:l|m Defined as follows, it represents the probability of the state occurring from time k to l, given that the observation information from time 1 to m is known:

[0097]

[0098] The joint posterior probability density distribution π can be derived using Bayes' theorem. 0:k|k (dX 0:k ),Right now

[0099]

[0100]

[0101] Similarly, the marginal posterior probability density distribution π can be obtained. k|k-1 (dX k ),Right now

[0102]

[0103]

[0104] S12, Markov process probability measure initialization. In step S12, it is assumed that v is a probability measure. Let K represent any function, K represent the transition kernel density satisfying the Markov process, and Θ and Ψ represent arbitrary function variables. The specific steps are as follows:

[0105] S121, the following calculation method is defined:

[0106]

[0107]

[0108]

[0109] S122, according to the above definition of symbols, for any function The prediction and update process of Bayesian theory can be redefined, and can be obtained from the prediction formula, i.e., formula (5):

[0110]

[0111] The update formula, i.e., formula (6) is obtained:

[0112]

[0113]

[0114] S2, Sequential Importance Sampling: Assuming the prior probability density distribution as the importance probability density function, i.e., the proposal distribution guides the particles to perform sequential importance sampling, and then recursively calculates the importance weight of the sampled particles.

[0115] In step S2, since the marginal posterior filtering distribution p(x k |y 1:k ) is unknown, the importance probability density function q(x k |x k-1 ,y 1:k ) needs to be constructed to meet the requirements of the Monte Carlo sampling method, thereby making up for the defect that sampling cannot be performed from the target distribution. Generally, the prior probability density distribution in the following formula is selected as the importance probability density function in the sequential importance resampling process (SIR):

[0116] q(x k |x k-1 ,y 1:k ) = K(x k |x k-1 ).

[0117] Specifically, step S2 includes:

[0118] S21, the Monte Carlo sampling method calculates the integral analytical solution.

[0119] As can be seen from the Markov process, for a general nonlinear, non-Gaussian system, it is difficult to calculate the analytical solution of the Bayesian theory, and thus the exact solution of the posterior probability cannot be obtained. The particle filter fully utilizes the optimal Bayesian recursive theory and uses the Monte Carlo sampling method to replace the integral operation method to solve the complex problem. Step 21 includes:

[0120] S211, using a series of sampled particle sets to infinitely approximate the posterior probability density distribution π k|k , and then estimating the current state X k of the system by calculating the sample expectation value;

[0121] S212, using the importance probability density function q(x k |y1:k ) to approximate the sampling and obtain the particle set Further calculate the posterior measure distribution (sampling particles)

[0122]

[0123] S213, as the number of sampling particles N→∞, the posterior measure distribution (sampling particles) Close to the true posterior probability density distribution π k|k Where δ(*) represents the Dirac Delta function.

[0124] S22, select the prior probability density update sequential importance sampling process, step 22 includes:

[0125] S221, since the marginal posterior filtering distribution p(x k |y 1:k ) is unknown, it is necessary to construct an importance probability density function q(x k |x k-1 ,y 1:k ) to meet the requirements of the Monte Carlo sampling method, and to make up for the defect that it cannot be sampled from the target distribution. Usually in the sequential importance resampling process (Sequential Importance Resampling, SIR) process, the importance probability density function satisfies the following conditions:

[0126] q(x k |x k-1 ,y 1:k )=K(x k |x k-1 )

[0127] Therefore, the weight of the sequential importance sampling (Sampling Importance Sampling) process can be obtained according to the Bayesian filtering:

[0128]

[0129] Substitute the prior probability density into the importance weight calculation to obtain:

[0130]

[0131] As can be seen, the importance weight at time k is proportional to the weight at the previous time and the measurement likelihood probability.

[0132] S222, set the posterior empirical measure distribution (sampling particles) at time k-1 Determine that the known, then the particle set at time k is The prediction measure distribution (sample particles) of the prediction phase is thus obtained is:

[0133]

[0134] S223, when the sample particle N is large enough, the prediction measure distribution (sample particles) The prior probability density distribution π close to the true state k|k-1 , the prediction measure distribution (sample particles) is substituted into the update formula, i.e. formula (6), to obtain the Monte Carlo approximation of the posterior measure distribution (sample particles)

[0135]

[0136] That is

[0137]

[0138] wherein, represents all sample particles The importance weight of the normalized importance probability density function q(x k |y 1:k ) is the importance weight of the posterior measure distribution (sample particles) is the weighted sum of Dirac Delta functions. Steps S1-S2 are called SIS filtering. During the iteration of the above algorithm, the variance of the sample particle importance weight will increase, which will cause the effective number of particles in the state space to decrease, and thus cause the filtering performance to decrease. To overcome this drawback, the standard particle filtering method usually uses a resampling method to solve the degeneracy problem, by copying the particles with high weights and discarding the particles with low weights, so that the particle set is aggregated in the high-probability posterior region, so as to obtain an "unweighted" empirical distribution approximation

[0139]

[0140] It is inferred that the essence / aim of the resampling process is achieved by resampling N times from the posterior measure distribution (sample particles) , so that the new particle set obtained thereby is close to the posterior probability density distribution π of the system true state k|k . Common resampling methods include random, systematic, polynomial and residual resampling. The specific process of the standard particle filtering method is as follows Figure 2 .

[0141] The standard particle filter method is simple in method structure and easy to perform operation, and the algorithm approximate estimation value converges to the true state value under optimal estimation. However, the method still has certain defects and deficiencies in actual engineering application, mainly including the following two aspects: (1) the standard particle filter method introduces the importance probability density distribution in the sequential importance sampling process, so that the particle weight variance will continuously accumulate with the increase of the iteration number of the algorithm, and the importance weight corresponding to most particles also tends to zero, that is, the particle degradation phenomenon. Thus, the calculation resource is seriously wasted, and the approximate estimation cannot accurately describe the posterior distribution of the true state, and more seriously, the degradation phenomenon cannot be avoided due to the defects of the method itself. (2) The resampling strategy is an effective means to improve the particle degradation phenomenon. The discrete approximate posterior probability distribution obtained by importance sampling is resampled, so that the samples with large weights are copied multiple times under the guidance of the particle motion and the distribution of the previous state, so that the effective particle number is increased to suppress the degradation purpose. At the same time, the resampling process may cause the abandonment or loss of some particles with low weights, so that the particles after resampling deviate from the true state posterior region too early, the sample impoverishment problem is caused, and finally the state estimation variance is large and the filtering performance is greatly reduced.

[0142] The embodiment of the application proposes an improved particle filter method, which uses a clustering similarity method to measure the distance discrimination of the system true and sampled particle state information set, including the measurement information of the current (stage) state filtering and the future multi (stage) state prediction, to guide the generation and improvement of the new proposal distribution, and then update the importance sampling process weight calculation, make up for the defects of the prior probability density instead of the importance function in the traditional PF method, effectively improve the particle degradation phenomenon, significantly improve the estimation method accuracy and robustness, and at the same time, discard the resampling strategy in the traditional PF method for the purpose of improving the particle degradation phenomenon in the algorithm process, fundamentally solve the particle degradation problem, and effectively improve the efficiency of the method. Specifically, the step S3 is used to measure the clustering similarity of the system true and sampled particle state information set, to guide the generation and improvement of the new proposal distribution, and then update the importance sampling process weight calculation.

[0143] S3, clustering similarity measurement state trajectory: taking the mathematical model of the well-posed space state trajectory problem as the starting point, combining the consistency principle of the system true state and the sampled particle state trajectory, and using the data mining idea--clustering analysis method to measure the similarity of the current filtering (normal trajectory) and the future multi-stage Gaussian filtering (modified trajectory) predicted measurement information.

[0144] The step S3 is based on the principle of spatial vector trajectory consistency, i.e. the closer the state trajectory of a particle to the real state trajectory, the higher the degree of representing the real state. In the embodiment, the step S3 comprises:

[0145] S31, selecting a particle set at time k to k+L+l, wherein the particle set at time k to k+L+l in the step S31 is wherein L and l are preset constants, the original state trajectory of the particle obeys the SIS filtering process, and the modified state trajectory of the particle x' obeys the GPF prediction process. j (i) obeys the GPF prediction process; since the real state information is unknown in the particle filtering method, the consistency of the state trajectory is characterized by the measurement likelihood function, and the measurement likelihood trajectory corresponding to the state of the sampling particle is solved according to the state trajectory of the particle;

[0146] Definition and are respectively the real state trajectory of the system at time k to l and the measurement likelihood trajectory corresponding to the real state of the system, and and are respectively the state trajectory of the sampling particle at time k to l and the measurement likelihood trajectory corresponding to the state of the sampling particle.

[0147] In the embodiment, the measurement likelihood trajectory corresponding to the state of the sampling particle is solved by the following formula (7) in the step S32:

[0148]

[0149] i.e.

[0150]

[0151] wherein the observation noise υ k = 0, and the observation equation H(*) is a known function determined by the specific research object without noise interference.

[0152] S32, using distance clustering similarity to analyze the consistency of the state trajectory of the sampling particle, calculating the distance similarity measure of the reference trajectory and the measurement likelihood trajectory corresponding to the state of the sampling particle, wherein the reference trajectory is the measurement likelihood trajectory corresponding to the real state of the system.

[0153] In the embodiment, the reference trajectory in the step S32 is k} = {y j : j = k, …, k+L+l}; the reference trajectory is the measurement likelihood information of the real state of the system, which is a known condition.

[0154] The distance similarity measure of the reference trajectory and the measurement likelihood trajectory corresponding to the sampling particle state is calculated as follows:

[0155]

[0156] where dis(*) represents a distance similarity function, dis(*) ≥ 0, and S is a measure type parameter;

[0157] An exponential transformation is performed on the formula (8) to increase the reliability of the method:

[0158] where λ is a gradient factor.

[0159] S4, correcting the importance probability density function: the proposed distribution in step S2 is corrected by using the similarity measure of step S3, so that the update compensation scheme-state trajectory clustering similarity is effectively improved by the particle degradation phenomenon by means of targeted integration of the latest measurement information in the importance sampling process.

[0160] In the embodiment, according to the distance similarity measure of the reference trajectory and the measurement likelihood trajectory of the sampling particle obtained in step S32, the importance probability density function q(x k |x k-1 ,y k ) replaced by the prior density function is corrected according to the following formula:

[0161] q(x k |x k-1 ,y k ) = d k *p(x k |x k-1 ) (9)

[0162] S5, updating the importance weight of the sampling particle: the importance weight calculation in the sequential importance sampling process in step S2 is updated by the proposed distribution corrected in step S4, and the particle weight is normalized, and then the estimation value of the system state at the current time is obtained.

[0163] The method for updating the importance weight in step S5 is that the importance weight calculation of the first-order Markov model is updated by using the importance probability density function corrected in step S4, and the importance weight corresponding to the sampling particle at the time k and k+L+l can be calculated respectively as and The calculation is as follows:

[0164]

[0165]

[0166] where pυ (*) is the probability density function of the measurement noise.

[0167] The importance weights calculated are normalized according to the following formula:

[0168]

[0169]

[0170] Set the SIS particle distribution at time k and the corresponding importance weights The posterior probability density distribution π that can approximately represent the system state k|k The GPF particle distribution at time k+L+1 and the corresponding importance weights The predicted probability density function π that can be approximately expressed k+L+l|k Therefore, the system state estimate x k can be obtained by filtering operation, and the system state estimate x k+L+l can be obtained by the prediction step, specifically:

[0171] Filtering process: the filtering step is consistent with the SIS (GPF) filtering process of the prior art, and a particle set

[0172] Updating process: the corresponding weights and weights are solved and normalized to obtain and

[0173] Estimate the system state x k and x k+L+l :

[0174]

[0175]

[0176] S6, steps S1-S5 constitute an iteration process of the particle filtering method for state estimation; repeat steps S2-S5 to sequentially output the system state estimates at different times.

[0177] Unlike the standard particle filtering method, the filtering method of the present embodiment consists of three parts of prediction, update and filtering, without resampling strategy and related steps. The particle filtering method (CSPF) process of the present embodiment is as follows Figure 1 .

[0178] This invention also proposes and proves the particle filtering method based on state trajectory clustering similarity and its convergence theorem. The rationality of the particle filtering method proposed in this invention is mainly based on the Bootstrap Filtering theory, and Bayesian state estimation can be implemented using the weighted bootstrap method: assuming particle samples... It follows a continuous probability density function R(x), while the desired posterior probability density distribution π k|k Given that R(x) and S(x) have a constant proportional relationship, and S(x) is a known function, then as the sample size N→∞, from and their corresponding weights The discrete distribution of the particles approximates the true posterior probability density. Referring to Bayesian theory in this example algorithm, the posterior probability density distribution π... k|k And the measured likelihood function and the predicted probability density distribution π k|k-1 The product of π and π is proportional to a constant, and the predicted probability density distribution π is... k|k-1 This can be equivalent to R(x), where the weights are used in the importance sampling process. If we consider the observation likelihood function as equivalent to S(x), then the method is clearly in line with the theory of Booster Filtering and is reasonable and effective.

[0179] (1) The specific steps for proposing and proving the convergence of the CSPF method are as follows:

[0180] a. Assumption The probability density measure space over a set is a maximally dimensional Euclidean space with a convergent topological structure. Let (E,d) be the probability measure set on the given surface, and let (E,d) be the measure space. and Given two continuous function sequences: E→E, where in random filtering the space E will be an n-dimensional Euclidean space. All probability measure spaces on.

[0181] b. Set a k Representing the measure π k-1|k-1 →π k|k-1 The mapping relationship, b k Represents the measure π k|k-1 →π k|k The mapping relationship. Then a k for The mapping (prediction) on the measure set satisfies

[0182]

[0183] This holds true for any measure ν. Therefore, it holds true for continuous functions. Substituting into the prediction formula, we get:

[0184]

[0185] Therefore, the expression for the predicted probability density distribution (true state) is:

[0186] π k|k-1 =a k (π k-1|k-1 )

[0187] Similarly, b k for The mapping relationship (update) on the measure set, combined with the update formula, yields...

[0188]

[0189] That is, the posterior probability density distribution of the Bayesian filter can be expressed as:

[0190]

[0191] The operator “⊙” represents a composite mapping function.

[0192] c. Set h k Represents the measure π k-1|k-1 →π k|k The transformation function, h 1:k Represents the measure π 0|0 →π k|k If the transformation function is given, then the Bayesian filtering process can be expressed as:

[0193]

[0194] In the abstract environment setting, the particle filtering method uses the Monte Carlo method to solve the difficulty of finding the integral analytical solution in Bayesian theory. Its main principle is to generate a set of sampled samples {x1, x2, ..., x...} from the target distribution π(x). N This can be used to approximate a partial feature of the target distribution. This feature is only the expectation of the "good performance" function and can be approximated as the average value, i.e.:

[0195]

[0196] Where, when N→∞, the estimated value is Converging to the expected value definition

[0197]

[0198]

[0199] Using the Law of Large Numbers and the Central Limit Theorem, we can deduce that as N→∞:

[0200]

[0201] That is,

[0202]

[0203] It is obvious that the disturbance caused by Monte Carlo sampling method to the integral operation analytical solution is inevitable, mainly due to the approximate estimate value is based on random and limited sample set. But under the guarantee of law of large numbers and central limit theorem, when the number of particles tends to infinity, the disturbance is very small, and satisfies the Gaussian distribution, that is

[0204]

[0205] It can be seen that when N→∞, The estimated value converges to the true expected value, and the estimated variance decreases with the increase of the number of particles N.

[0206] d. Through the above analysis, particle filtering is based on Bayesian filtering combined with Monte Carlo sampling method, that is, sampling disturbance function c N Its disturbance formula can be expressed as:

[0207]

[0208] The process formula of particle filtering method can be expressed as:

[0209]

[0210] Where κ is the initial value π 0|0 And the improved method in this paper uses clustering analysis method to measure the similarity of multi-stage measurement information, instead of the marginal posterior filtering distribution p(x k |x k-1 ) in the sequential importance sampling process:

[0211] q(x k |x k-1 ,y k ) = d k (i) K(dx k |x k-1 )

[0212] The updated importance weight calculation w k (i) is:

[0213]

[0214] The updated formula of the improved method is:

[0215]

[0216] Where, Representative improvement method measurement The mapping relationship, Monte Carlo measure Referring to the above definitions and formulas, the improved particle filter can be expressed as:

[0217]

[0218] e. Convergence Theorem of the CSPF Method: Assume the state transition kernel function K satisfies a first-order Markov process, and measure the likelihood function g in... It is continuous in range, bounded, and strictly greater than 0. In Monte Carlo sampling, the perturbation c... N Below is the posterior measure distribution (sampled particles) of the improved particle filtering method in this embodiment. Converging to the posterior probability density distribution π of the Bayesian optimal filter theoretical value (true state value) k|k ,Right now

[0219]

[0220] f. Convergence Theorem Proof: In the particle filtering method, the Monte Carlo sampling perturbation is random and uncertain. (Set c) N ，θ For random perturbations, the number of sampled particles N>0 and the independent variable θ∈Ω, for all measures have

[0221]

[0222] in, It is a random variable with independent and identically distributed measure v. According to the simplified mechanism, we can obtain...

[0223] π k|k =b k (π k|k-1 )=b k ⊙a k (π k-1|k-1 )

[0224]

[0225] The measure of the Bayesian prediction phase at time k is a. k The sampling perturbation measure in the particle filter prediction stage is c. N (a k Using independent and identically distributed variables V j , and have to

[0226]

[0227] Where ||*|| represents the value at the threshold. the supremum norm of,

[0228] Thus the sum expectation of the sampling particle number N from 1 to infinity is calculated

[0229]

[0230] Thus it is deduced that

[0231]

[0232] Accordingly, the measure at a certain time in the prediction stage can be expressed as

[0233]

[0234] The probability measure of the update stage is updated simultaneously For and any get

[0235]

[0236] Thus it can be concluded that the improved method is the same as the Bayesian filter update stage measure, that is

[0237]

[0238] The above formula is solved simultaneously, and

[0239]

[0240] Therefore, the particle filter method based on trajectory consistent clustering similarity proposed in the application still converges to the true value under the disturbance of Monte Carlo sampling, and the convergence theorem of the CSPF method is proved.

[0241] (2) The specific steps of proposing and proving the convergence of the CSPF method results are as follows:

[0242] a. On the basis of the convergence proof of the improved method, the convergence of the mean square error of the results is proved. It is assumed that in the field range, denotes a sequence of random probability measures, and satisfies For any function From the formula of the convergence theorem of the method, it can be obtained that

[0243]

[0244] b. Convergence theorem of the results of CSPF: it is assumed that the state transition kernel function K satisfies the first-order Markov process, and the measurement likelihood function g is continuous, bounded and strictly greater than 0 in the range. For any function there exists a real constant C k|k satisfies

[0245]

[0246] where

[0247] The result convergence theorem is proved: the steps of the improved particle filtering method of the embodiment are divided into two parts, and the prediction and update steps are proved respectively.

[0248] c. In the prediction stage of the optimal Bayesian recursive optimization particle filtering step, the set condition of the result convergence theorem is satisfied, and there exists a real constant C when k≥0 k|k-1 , the predicted measure distribution (sampling particle) satisfies

[0249]

[0250] where The proof is completed by using the inductive framework method.

[0251] (1) When k=0, for any function there exists a real constant C 0|0 satisfies

[0252]

[0253] When k=0 in the improved particle filtering method of the embodiment that is, N particles are obtained from the prior probability density measure π 0|0 , and each is independently and identically distributed. By using the Marcinkiewicz-Zygmund inequality, it can be obtained that

[0254]

[0255]

[0256] Therefore, when k=0, the result convergence theorem is proved.

[0257] (2) When k-1, for any function suppose there exists a real constant C k-1|k-1 satisfies

[0258]

[0259] Then at k, according to the initialization process of the Markov process probability measure in the improved method step, it can be concluded that

[0260]

[0261] Substitute the prediction stage formula into the above equation, we have

[0262]

[0263] At the same time, set ζ k-1 For the particle set The generated σ-field space domain, combined with the Monte Carlo idea, we have

[0264]

[0265] Since And substitute the above equation

[0266]

[0267] At the same time, refer to the assumption formula, there must be a real constant, we have

[0268]

[0269] Using Minkowski inequality theorem, we have

[0270]

[0271] Where, Thus, the convergence of the prediction stage result is proved.

[0272] d. In the update stage, meet the method convergence theorem setting conditions and the prediction stage result convergence conclusion. For any function When k≥0, there must be a real constant C k|k , the posterior measure distribution (sampling particles) Meet

[0273]

[0274] Where

[0275] Through the update formula, we have

[0276]

[0277]

[0278] Combined with the update stage in the improved method steps, substitute the improved method update formula, we have

[0279]

[0280] Similarly, using Minkowski inequality inequality, we have

[0281]

[0282] Therefore, the result convergence theorem is proved.

[0283] The application also applies the particle filtering method based on state trajectory clustering similarity to the evaluation of social and economic development. The superiority of the improved method of the application is not only reflected in the application in nonlinear Gaussian systems, but also applicable to nonlinear non-Gaussian systems. The application selects a typical one-dimensional nonlinear system, a single variable non-stationary growth model, to verify the effectiveness and timeliness of the performance of the CSPF method. The specific steps are as follows:

[0284] a. A one-dimensional nonlinear system, a single variable non-stationary growth model, is established by using a nonlinear algorithm, as shown in the following formula (11). The selected mathematical model has the characteristics of high nonlinearity and bimodal state quantity distribution, and is widely used in the fields of social and economic development evaluation, short-term prediction of insurance stocks and bank interest rates, etc.:

[0285]

[0286]

[0287] In the formula: υ k is the observation noise; λ is the gradient factor of the filtering method of the embodiment; k is the time; x k is the state level at time k; y k is the observation information at time k; δ, α, β, ε are parameters of the non-stationary growth model respectively; ω k is the process noise.

[0288] b. Initialization of sampling particles and corresponding weights: since the posterior probability density function of the growth model is unknown, an importance probability density function is usually selected to meet the Monte Carlo sampling, so as to approximately obtain a particle set;

[0289] c. Sequential importance sampling: assuming that the prior probability density distribution of the growth model is used as the importance probability density function, that is, the proposal distribution guides the particle to perform sequential importance sampling, and then the importance weight of the sampling particle is recursively calculated;

[0290] d. Clustering similarity measurement of state trajectory: taking the growth model as the starting point, combining the consistency principle of the real state of the system and the state trajectory of the sampling particle, and using the clustering analysis method of the data mining idea to measure the similarity of the current filtering and the future multi-stage Gaussian filtering prediction measurement information;

[0291] e. Correction of importance probability density function: the proposal distribution in step c is corrected by using the similarity measurement in step d;

[0292] f. Update the importance weight of the sampling particle: the importance weight calculation in the sequential importance sampling process in step c is updated by the proposal distribution revised in step e, and the particle weight is normalized to obtain the state estimation value at the current time;

[0293] g. Steps b-f constitute an iteration process of the growth model; repeat steps c-f to sequentially output the state estimation value of the growth model at different times.

[0294] In this embodiment, the observation noise is set as υ k ~N(0, R k ), and R k =1; the initial state x0=0.1, α=0.5, β=25, γ=8, δ=1.2, ε=20. The number of particles N=100 is selected, the simulation step number k=1, 2,…, T, T=5000. In the CSPF algorithm, the correlation coefficient L=2, l=1, the gradient factor λ=1.3, and the running simulation number is 100.

[0295] The computer processor speed used in the above model is selected as 2.7GHz, and the memory capacity is 8Gbyte. The root mean squared error (RMSE) is used as the performance evaluation method of the five algorithms, and represent the system true value and the state estimation value, respectively, that is,

[0296]

[0297] In this example, two types of process noise are selected, Gaussian noise ω k_G ~N(ω k ;0,1) and non-Gaussian (heavy-tailed) noise where ξ=0.8, The non-Gaussian noise ω k_NG also has obvious heavy-tailed distribution characteristics, and the Monte Carlo (acceptance-rejection sampling) method is used to realize noise sampling, and the confidence level is 97.5%.

[0298] In this embodiment, different particle filtering algorithms are used to evaluate the state of social and economic development, and the simulation results are shown in Figures 3-4 .

[0299] The clustering similarity metric used in this embodiment is distance similarity and other methods. The distance similarity method has many types, such as Euclidean distance, Mahalanobis distance, Chebyshev distance, and Manhattan distance. This embodiment selects Euclidean distance and Chebyshev distance as examples for description. The particle filter method based on state trajectory clustering similarity using Euclidean distance (Euclidean Distance Clustering Similarity Particle Filter, hereinafter referred to as “ECSPF”) represents the Euclidean space distance similarity algorithm (metric type S = 2). The particle filter method based on state trajectory clustering similarity using Chebyshev distance (Chebyshev Distance Clustering Similarity Particle Filter, hereinafter referred to as “CCSPF”) represents the Chebyshev space distance similarity algorithm (metric type S→∞).

[0300] Figure 3 and Figure 4 The figure legend shows the comparison of the state estimation results of different particle filter algorithms in a single run of a nonlinear system under Gaussian and non-Gaussian process noise environments. The state estimation results of different algorithms can be directly compared. Figures 3-4 In the figure, “CSPF” is the simulation result curve of the particle filter method based on state trajectory clustering similarity of this embodiment, Figures 3-4 In the figure, “SIR” is the simulation result curve of the traditional standard particle filter algorithm; “APF” is the simulation result curve of the auxiliary particle filter algorithm; “GPF” is the simulation result curve of the Gaussian particle filter algorithm; and “true” represents the true value curve of the social and economic development state.

[0301] In order to comprehensively show the comparison of the state estimation results of various algorithms, Figure 5 The comparison of the RMSE results of 15 runs is given. The solid line and the algorithm with the G subscript represent the selection of Gaussian noise ω k_G for the system noise. The dashed line and the algorithm with the NG subscript represent the selection of non-Gaussian noise ω k_NG . Table 1 and Table 2 respectively show the accuracy improvement of the ECSPF algorithm and the CCSPF algorithm of this embodiment compared with other algorithms after 100 simulation runs, so as to quantitatively compare the mean and variance of the estimation results of different algorithms.

[0302] Table 1: RMSE expectation and variance comparison of state estimation results (SIR, APF, GPF, ECSPF and CCSPF five methods are randomly simulated 100 times)

[0303]

[0304] Table 2: Accuracy improvement rate comparison of RMSE expectation and variance (SIR, APF, GPF, ECSPF and CCSPF five methods are randomly simulated 100 times)

[0305]

[0306] From Figure 3 and Figure 4 , it can be seen that the state estimation result accuracy of the CSPF algorithm is obviously better than that of other algorithms, and the estimation result trend is relatively more complete and close to the true state change. From Figure 5 and Table 1 and Table 2, it can be seen that the RMSE values (averages) and variances of the ECSPF and CCSPF algorithm results are lower than those of the SIR, APF and GPF algorithms. The prediction accuracy (variance) of the ECSPF in the Gaussian noise environment is improved by 65-69% and 67-89% compared with the SIR and APF, respectively. The prediction accuracy (variance) of the ECSPF in the non-Gaussian noise environment is improved by 52-57% and 14-54% compared with the SIR and APF, respectively. Therefore, the filtering method of the embodiment has higher result accuracy and stronger stability, and the performance of the Euclidean space distance similarity algorithm is particularly significant.

[0307] Under the condition of determining the same known time setting, the state estimation result accuracy level of each algorithm in a single operation is obtained by adjusting the sampling particle number N and the simulation times T. Therefore, the time efficiency of the filtering method of the embodiment is verified by comparing the corresponding operation cost, as shown in Table 3. Among them, the filtering method of the embodiment represented by ECSPF and CCSPF in the Guass and Non-Guass noise environment is compared with the accuracy of SIR, APF and GPF algorithm. From Table 3, it can be seen that under the same operation cost, that is, when the operation time and simulation step number in the Guass (Non-Guass) environment are the same, the sampling particle number N used by the filtering method of the embodiment is the least, and the accuracy (RMSE) is obviously higher than that of the other three algorithms, which is about 3 times (2 times) relationship, thereby verifying that the filtering method of the embodiment has high operation efficiency and outstanding time efficiency advantage.

[0308] Table 3: Time efficiency analysis of state estimation results (SIR, APF, GPF, ECSPF and CCSPF five methods are randomly simulated 100 times)

[0309]

[0310] The application also applies the state trajectory clustering similarity particle filtering method in battery health diagnosis management. Another application of the application is directed to the lithium ion battery health state evaluation problem and takes the lithium ion battery health state evaluation problem as the research object. Different algorithms are used to perform state tracking training on historical capacity samples, to optimize identification of capacity degradation physical experience model parameters and process noise distribution information, to determine and identify the optimal, and to construct a lithium ion battery remaining useful life (RUL) prediction model based on particle filtering (improved method). The specific steps are as follows:

[0311] a. In combination with the equivalent circuit model and relying on the electrochemical characteristic attenuation mechanism, the state trajectory clustering similarity particle filtering is used to establish a lithium ion battery state tracking and remaining useful life (RUL) prediction model.

[0312] In the embodiment, the lithium ion battery state tracking and RUL prediction accuracy depends to a great extent on the lithium ion battery capacity degradation physical model, and the development of the model needs to be obtained through known system physical knowledge. Usually, characteristic information is obtained from a sensor measurement device, which includes battery parameter information such as charge and discharge voltage, current, power, electrochemical impedance spectrum, frequency, and temperature, the internal relationship is captured, and an equivalent circuit model capable of representing battery capacity degradation trend information is built, as shown in Figure 6 . Figure 6 C DL is the double-layer capacitance, V C is the double-layer capacitance voltage, R CT is the charge transfer resistance, R W is the Warburg impedance, I R is the polarization current, R E is the electrolyte resistance, I L is the load current, V L is the terminal voltage, and OCV is the open circuit voltage.

[0313] In combination with the equivalent circuit model and relying on the electrochemical characteristic attenuation mechanism, the relationship between capacity degradation and battery internal impedance is determined through statistical regression theory analysis of experimental data, the degradation characteristic that the impedance increases with time is simulated, and a double exponential degradation experience degradation model equation is obtained. The equation corresponds to the state transition equation:

[0314] C ap = a * exp(b * k) + c * exp(d * k)

[0315] In the equation, C ap is the lithium ion battery capacity (A·h), k is the number of charge and discharge cycle periods, a and c are unknown model parameters related to the battery impedance, and b and d are related to the capacity degradation rate.

[0316] The tracking training model is created by using battery capacity history samples, real-time estimation of experience degradation model parameters and noise multi-feature optimization, and the identified physical model can effectively converge to the gradual trend state of the real degradation of the battery, thereby providing reasonable identification parameters and effective initial values for establishing the RUL prediction model.

[0317] The state space model is updated based on the state transition equation representing the recursive law of the system and the observation equation converting the implicit information of the system into visible output (the observation equation can be composed according to the existing battery capacity and measurement noise, which belongs to the prior art, and is not described here to save space), combined with the actual observation output information, the state estimation at the current time is given according to the state prediction result at the last time and the transition equation, and the error between the actual measurement information (noise interference) at the current time and the estimated observation value is used to generate a correction term with a weight to realize the updating process of the state, thereby obtaining a state estimation with high credibility and a prediction system model, i.e., a lithium ion battery state tracking and remaining useful life prediction model, to realize the processes of lithium ion battery state tracking and remaining useful life prediction evaluation.

[0318] b, initialization of sampling particles and corresponding weights: since the posterior probability density function of the prediction model is unknown, the importance probability density function is usually selected to meet the Monte Carlo sampling, so as to approximately obtain the particle set;

[0319] c, sequential importance sampling: assuming that the prior probability density distribution of the prediction model is used as the importance probability density function, i.e., the proposal distribution guides the particle to perform sequential importance sampling, and the importance weight of the sampling particle is recursively calculated;

[0320] d, clustering similarity measurement of state trajectory: taking the lithium ion battery state tracking and remaining useful life prediction model as the starting point, combining the consistency principle of the real state of the system and the state trajectory of the sampling particle, and using the data mining idea--clustering analysis method, the similarity of the current filtering (normal trajectory) and the future multi-stage Gaussian filtering (modified trajectory) prediction measurement information is measured;

[0321] e, modification of importance probability density function: the proposal distribution in step c is modified by using the similarity measurement in step d, so that the update compensation scheme--state trajectory clustering similarity is effectively integrated into the latest measurement information in the importance sampling process, thereby effectively improving the particle degradation phenomenon;

[0322] f, updating the importance weight of the sampling particle: the importance weight calculation in the sequential importance sampling process in step c is updated by the proposal distribution modified in step d, and the particle weight is normalized, and then the state tracking and RUL prediction value at the current time are obtained;

[0323] g, steps b-f constitute an iteration process of state tracking and RUL prediction model of lithium ion battery. Steps c-f are repeated to sequentially output the predicted state tracking and remaining useful life of the prediction model at different times.

[0324] The embodiment follows the capacity state tracking training parameter identification and battery RUL prediction process, and simulates and compares SIR, APF and CSPF algorithms (Gauss particle filtering is limited to variable dimension and is not suitable for this application). The state tracking parameter identification is performed before S, that is, the optimal model parameters that can minimize the error between the predicted value and the true value of the experiment are estimated, the predicted battery capacity value is executed after S, and it is judged whether the failure threshold is reached.

[0325] The application studies the capacity degradation raw data (Battery_Capacity.dat) of lithium ion battery, which is derived from the open source experimental data of the Center of Advanced Life Cycle Engineering (CALCE) of the University of Maryland. 80% of the rated capacity of the battery is taken as the failure threshold T of the end of life, and the true failure threshold is 133 cycles according to the open source data.

[0326] The state tracking set S is selected as 65, the particle number N is 100, and the simulation results are as follows Figures 7-10 As shown in Table 4 and Table 5, Figures 7-10 The state tracking and prediction accuracy of different algorithms can be intuitively judged and compared, and Table 4 and Table 5 give the quantitative comparison of the state tracking and prediction results obtained by multiple simulation calculations. The state tracking effect indexes include the sum of squares due to error (SSE), mean squared error (MSE), root mean squared error (RMSE), R_square (coefficient of determination), SSE, MSE and RMSE are closer to 0, R_square is closer to 1, and the state tracking effect is better. The prediction accuracy effect indexes include the average value and variance of the predicted failure threshold, the relative error and RMSE.

[0327] Table 4: Comparison of state tracking effect of lithium ion battery (SIR, APF, ECSPF and CCSPF are randomly simulated 1000 times)

[0328]

[0329] Table 5: Comparison of results of RUL prediction of lithium-ion batteries (SIR, APF, ECSPF and CCSPF are simulated 100 times)

[0330]

[0331] From Figures 7-10 As can be seen from Table 4 and Table 5, compared with SIR and APF algorithms, the state tracking index of the ECSPF and CCSPF filtering methods proposed in the embodiment is obviously superior and the effect is the best, among which the Euclidean distance measure is optimal, and at the same time, it provides accuracy guarantee for the same nature of the degradation mechanism embodied in the state tracking part of the early stage of the battery life cycle and the mechanism of the prediction part. From Figures 7-10 As can be seen from Table 4 and Table 5, after 100 times of simulation of SIR, APF, ECSPF and CCSPF, the average failure threshold of RUL prediction is 113, 115, 123 and 122 cycles respectively, and the relative error is maintained within 15.0%. Taking ECSPF as an example, the algorithm prediction threshold variance, relative error and RMSE are all greatly reduced compared with SIR and APF, and the prediction trend is relatively closer to the real capacity degradation curve, among which the relative error is increased by about 50% and 45% respectively, indicating that the prediction accuracy of the filtering algorithm proposed in the embodiment is higher, the prediction threshold variance is reduced by about 25 times and 80 times respectively, the RMSE is increased by about 55% and 60% respectively, which represents that the particle set of the prediction result state of the filtering algorithm proposed in the embodiment has the smallest dispersion degree, the lowest uncertainty expression degree and stronger robust stability.

[0332] The above particle filtering method based on state trajectory clustering similarity improves the particle filtering method itself without relying on external methods, and re-architects the algorithm logic framework: for the particle degradation phenomenon, the trajectory consistency degree of the particles is measured by using clustering similarity theory, and the higher the consistent similarity degree is, the closer to the real state it represents. The importance weight of the updated particle is measured in this way to improve the particle degradation phenomenon. The updated compensation scheme (clustering measurement) is introduced into the latest measurement information in the importance sampling process to replace the resampling process, so that the improved filtering method has no resampling process, and therefore the particle impoverishment problem can be fundamentally eliminated, the particle degradation phenomenon can be effectively improved, and the timeliness of the algorithm can be greatly improved.

[0333] It can be understood that the clustering similarity measurement method is not limited to the distance similarity of the embodiment, and other clustering similarity measurement methods can also be used, such as cosine degree calculation and kernel function calculation, which are not limited in the particle filtering method based on state trajectory clustering similarity of the present application.

[0334] The above description is for the preferred embodiment of the present application, but the embodiment is not intended to limit the scope of the patent application of the present application. Any equivalent changes or modifications made under the technical spirit of the present application should be covered by the patent scope of the present application.

Claims

1. A particle filtering method based on state trajectory clustering similarity, characterized in that, Includes the following steps: S1, Initialize the sampled particles and their corresponding weights: Since the posterior probability density function is unknown, we select the importance probability density function to satisfy Monte Carlo sampling, thereby approximating the particle set; S2, Sequential Importance Sampling: Assuming the prior probability density distribution is used as the importance probability density function, that is, the proposal distribution guides the particles to perform sequential importance sampling, and then recursively calculates the importance weight of the sampled particles. S3, Clustering Similarity Measurement of State Trajectory: Taking the mathematical model of the spatial state trajectory problem with well-posedness as the starting point, and combining the consistency principle between the actual state of the system and the state trajectory of sampled particles, the clustering analysis method, a data mining idea, is used to measure the similarity between the current filtering and the future multi-stage Gaussian filtering prediction measurement information. S4, Correct the importance probability density function: Use the similarity metric from step S3 to correct the proposal distribution in step S2; S5, Update the importance weights of sampled particles: Update the importance weights in the sequential importance sampling process in step S2 using the proposed distribution corrected in step S4, and normalize the particle weights to obtain the estimated value of the system state at the current moment. S6, steps S1-S5 constitute one iteration of the state estimation process of the particle filter method; repeat steps S2-S5 to output the estimated values ​​of the system state at different times in sequence.

2. The particle filtering method based on state trajectory clustering similarity as described in claim 1, characterized in that, Step S1 includes: sampling particle initialization And the particle weights are Where X0 is the set of particles at the initial time t0. Let be the state vector of the i-th sampled particle at the initial time t0, N be the number of particles generated, and p(X0) be the initial probability density function.

3. The particle filtering method based on state trajectory clustering similarity as described in claim 1, characterized in that, Step S2 includes: due to the edge posterior filtering distribution p(x) k |y 1:k Since the desired outcome is unknown, it is necessary to construct an importance probability density function q(x). k |x k-1 ,y 1:k To meet the requirements of the Monte Carlo sampling method and thus compensate for the inability to sample from the target distribution, the prior probability density distribution in the following formula is selected as the importance probability density function during the sequential importance resampling process: q(x k |x k-1 ,y 1:k )=K(x k |x k-1 )。 4. The particle filtering method based on state trajectory clustering similarity as described in claim 1, characterized in that, Step S3 is based on the principle of spatial vector trajectory consistency, that is, the closer the state trajectory of a particle is to the true state trajectory, the higher the degree to which the particle's state represents the true state. Step S3 includes: S31, Select the set of particles from time k to k+L+l, where L and l are pre-set constants, and the original state trajectory of the particles. The corrected state trajectory x' of the particle follows the SIS filtering process. j (i) It follows the GPF prediction process. Since the true state information in the particle filtering method is unknown, the consistency of the state trajectory is characterized by the measurement likelihood function. The measurement likelihood trajectory corresponding to the sampled particle state is solved based on the state trajectory of the particle. S32, Distance clustering similarity is selected to analyze the consistency of sampled particle trajectories, and the distance similarity measure between the reference trajectory and the measurement likelihood trajectory corresponding to the sampled particle state is calculated. The reference trajectory is the measurement likelihood trajectory corresponding to the true state of the system.

5. The particle filtering method based on state trajectory clustering similarity as described in claim 4, characterized in that, The measurement likelihood trajectory corresponding to the sampled particle state in step S31 is calculated using the following formula. Right now Among them, the observation noise υ k =0, the observation equation H(*) is a known function determined by a specific research object under noise-free conditions, and the measurement likelihood trajectory corresponding to the true state of the system, i.e., the reference trajectory, is {Y}. k }={y j :j=k,…,k+L+l}.

6. The particle filtering method based on state trajectory clustering similarity as described in claim 4, characterized in that, Step S32 includes: using distance clustering similarity to analyze trajectory consistency, the calculation of the distance similarity metric between the reference trajectory and the measured likelihood trajectory corresponding to the sampled particle state is as follows: Where dis(*) represents the distance similarity function, and dis(*)≥0, and S is the metric type parameter; To increase the reliability of the method, the above equation is transformed exponentially: Where λ is the gradient factor.

7. The particle filtering method based on state trajectory clustering similarity as described in claim 6, characterized in that, Step S4 includes: based on the distance similarity metric between the reference trajectory obtained in step S32 and the measured likelihood trajectory corresponding to the sampled particle state, applying the following formula to correct the importance probability density function q(x) replaced by the prior density function. k |x k-1 ,y k ): q(x k |x k-1 ,y k )=d k *p(x k |x k-1 )。 8. The particle filtering method based on state trajectory clustering similarity as described in claim 7, characterized in that, Step S5 includes: updating the importance weights of the first-order Markov model using the importance probability density function corrected in step S4; calculating the importance weights corresponding to the sampled particles at times k and k+L+l respectively. and The calculation is as follows: Where, p v (*) represents the probability density function of the measurement likelihood noise; The importance weights are normalized according to the following formula: Set the SIS particle distribution at time k and corresponding importance weights The posterior probability density distribution π that can approximately represent the system state k|k GPF particle distribution at time k+L+l and corresponding importance weights It can approximate the prediction probability density function π k+L+l|k Therefore, the system's state estimate x k The state estimate x can be obtained through filtering. k+L+l This can be obtained through the prediction step: Filtering process: The steps are the same as the SIS (GPF) filtering process, extracting the particle set. Update process: Solve for the corresponding weights and And normalize it to obtain and The system state x is estimated according to the following formula. k and x k+L+l : (Filtering) (predict).

9. A socio-economic development evaluation method based on state trajectory clustering similarity particle filtering, characterized in that, Includes the following steps: S1. A one-dimensional nonlinear system-univariate nonstationary socioeconomic development growth model is established using state trajectory clustering similarity particle filtering. This growth model has highly nonlinear characteristics and bimodal characteristics of state quantity distribution. S2, Initialize the sampled particles and their corresponding weights: Since the posterior probability density function of the growth model is unknown, the importance probability density function is selected to satisfy Monte Carlo sampling, thereby approximating the particle set; S3, Sequential Importance Sampling: Assuming the prior probability density distribution of the growth model is used as the importance probability density function, that is, the proposal distribution guides the particles to perform sequential importance sampling, and then recursively calculates the importance weight of the sampled particles. S4, Clustering Similarity Measurement of State Trajectory: Taking this growth model as a starting point, and combining the principle of consistency between the actual state of the system and the state trajectory of sampled particles, the data mining idea - clustering analysis method is used to measure the similarity between the current filtering and the future multi-stage Gaussian filtering prediction measurement information. S5, Correct the importance probability density function: Use the similarity metric from step S4 to correct the proposal distribution in step S3; S6, Update the importance weights of sampled particles: Update the importance weights calculated in the sequential importance sampling process in step S3 through step S5, and normalize the particle weights to obtain the state estimate at the current time. S7, steps S2-S6 constitute one iteration of the growth model; repeat steps S3-S6 to output the state estimates of the growth model at different times.

10. A method for assessing the health status of lithium-ion batteries based on state trajectory clustering similarity particle filtering, characterized in that, Includes the following steps: S1. Combining the equivalent circuit model and relying on the electrochemical characteristic decay mechanism, a lithium-ion battery state tracking and remaining service life prediction model is established by using state trajectory clustering similarity particle filtering. S2, Initialize sampling particles and corresponding weights: Since the posterior probability density function of the prediction model is unknown, an importance probability density function is selected to satisfy Monte Carlo sampling, thereby approximating the particle set; S3, Sequential Importance Sampling: Assuming the prior probability density distribution of the prediction model is used as the importance probability density function, that is, the proposal distribution guides the particles to perform sequential importance sampling, and then recursively calculates the importance weight of the sampled particles. S4, Clustering similarity measurement of state trajectory: Taking the lithium-ion battery state tracking and remaining service life prediction model as the starting point, and combining the consistency principle between the actual system state and the sampled particle state trajectory, the data mining idea - clustering analysis method is used to measure the similarity between the current filtering and the future multi-stage Gaussian filtering prediction measurement information. S5, Correct the importance probability density function: Use the similarity metric from step S4 to correct the proposal distribution in step S3; S6, Update the importance weights of sampled particles: Update the importance weights calculated in the sequential importance sampling process in step S3 through step S5, and normalize the particle weights to obtain the predicted values ​​of the current state tracking and remaining lifetime. S7, steps S2-S6 constitute one iteration of the lithium-ion battery state tracking and RUL prediction model; repeat steps S3-S6 to output the state tracking and remaining lifespan prediction estimates of the prediction model at different times.

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