CN113452349A - Kalman filtering method based on Bayes sequential importance integral - Google Patents
Kalman filtering method based on Bayes sequential importance integral Download PDFInfo
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Abstract
The invention discloses a Bayesian sequential importance integral-based Kalman filtering method which comprises the steps of establishing a discrete multi-model parameter nonlinear Gaussian system model, constructing truncation priors by soft constraint, constructing mixed Gaussian importance distribution by fusing the truncation priors and state posterior feedback, correcting comprehensive integration points for prediction updating, and fusing target posterior distribution under multimode parameters. By fusing the cut-off prior, the posterior feedback and other constraint information, the importance function covering the multimodal distribution is constructed, the matching degree of the target importance function and the actual target real distribution is improved, and the diversity and the accuracy of the sampling sample are improved. And introducing sequential importance sampling correction integral points, and introducing related information entropy measure in a time updating stage to comprehensively improve the diversity of the integral points and the fault tolerance of prediction covariance. The method can greatly reduce the average error without sacrificing the calculation complexity, and can improve the real-time tracking performance by one order of magnitude when being applied to tracking the maneuvering target in the airspace.
Description
Technical Field
The invention relates to the technical field of target state estimation, in particular to a Kalman filtering method based on Bayesian sequential importance integral.
Background
The filtering refers to the estimation of unknown states or parameters from sensing measurement containing noise, and the nonlinear filtering is one of core technologies for solving the uncertain signal processing problem and is widely applied to the fields of engineering, statistics and mathematics. Bayesian reasoning provides a mathematical mechanism for the filtering problem, however, in practical application, due to the complexity of nonlinear dynamic system functions, closed-form analysis of integral operation of an equation is difficult, and numerical methods are generally used for approximate solution, which mainly include numerical integration and Monte Carlo.
The first category of numerical integration methods is based on a deterministic rule. Among them, gaussian approximation is a common method, which approximates the first second moment of target distribution by gaussian moment matching, and the method is simple to calculate, but the numerical accuracy is general. The unscented transformation technology directly approaches the mean and covariance of the original distribution of the target by selecting a fixed number of sigma points to carry out state transfer. Under the precondition that the covariance matrix is positive, the Unscented Kalman Filter (UKF) algorithm can reach third-order numerical precision. When the integral dimensionality of the state space model is moderate, the numerical integration points can be determined according to Gauss-Hermite rules to carry out multidimensional quadratic integration or volume integration, and if the specific integral conditions under the assumed probability model condition are met, the algorithm can reduce the approximation error to the maximum extent and becomes a high-performance reference technology in the relevant application field. However, the application of the sigma-point Kalman filter is generally limited to Gaussian noise disturbance, and the limitation restricts the application of the numerical integration method in a strong nonlinear scene.
The second class of Monte Carlo methods is based on the idea of random sampling, so that the nonlinear function does not need to be linearized, the nonlinear and non-Gaussian problem of dynamic state estimation can be effectively processed, and the central limit theorem ensures the consistent convergence of the methods. The Markov chain Monte Carlo Markov Chain (MCMC) algorithm generates valid Monte Carlo samples by simulating Markov chains consistent with a target distribution, however, it is difficult to maintain Markov steady-state distributions in a state evolution process with less sample data, and the computational complexity and time cost of improving the algorithm increase with the increase of data volume and acquisition volume. In contrast, the Importance Sampling (IS) algorithm approximates a target importance function by decomposing the mathematical expectation of the target posterior probability density function, generates samples having respective weights from the target importance function, and approximates the target posterior distribution by dirac function weighting. According to the method, the deviation between the target importance function and the real target distribution can be corrected by adjusting the sample weight, so that the establishment of the importance function capable of covering the effective likelihood area is critical to the filtering performance of the Monte Carlo method. In the prior art, sequential monte carlo sampling is also called as a Particle Filter (PF) method, and the importance sampling rule of discrete time dynamic models in several different scientific disciplines is unified under the bayesian filtering meaning. Theoretically, the sampling error of the monte carlo method has dimension invariance, but the rapid attenuation of the sample and the weight degradation caused by dimension disaster still remain a technical problem to be solved.
From the mathematical model analysis of Bayesian probability theory, the estimation of unknown parameters in the nonlinear high-dimensional state space is fuzzy rather than accurately analyzable, and the root cause is not the extrinsic statistical variation, but the ambiguity inherent in the complex nonlinear dynamic system model, mainly including the unpredictability of the target state and the ambiguity of the unstable data. This ambiguity restricts the application of many single non-linear filtering algorithms, and according to the advantages and disadvantages of the above two types of numerical methods, there are many improved algorithms that combine the two types of numerical methods, for example:
and (3) approximating the first second moment of the observation model of the dynamic nonlinear system by adopting an iteration posterior linearization method, and proving the convergence of the algorithm from the angle of statistical analysis. However, linear iterative optimization is easy to fall into local optimization, so that the algorithm lacks strong generalization capability;
and introducing the target importance function into a frame of a Gauss-Hermite numerical integration algorithm, and measuring the convergence error of the importance integration method by adopting a statistical linearization method. Within a certain error range, the algorithm can be expanded to the numerical integration of a non-Gaussian state space model, and the adaptivity of a target importance function still needs to be adjusted through a specific heuristic mechanism;
and dividing a dynamic multi-model state space according to a Rao-Blackwell theory, and estimating model parameters of the system state and state evolution under a condition model by adopting linear Gaussian approximation and a particle filtering method respectively. However, for the state space model calculation of a complex nonlinear dynamic system, the problem of large sample sampling time is still obvious. In fact, consistent or compatible geometric mapping or statistical constraint relationships are prevalent between nonlinear dynamic system models, state variables, and observations.
Disclosure of Invention
Aiming at the problems that the likelihood function, the target importance function and the actual target distribution are not matched under the conditions of model inaccuracy and data instability in nonlinear filtering in the prior art, the invention provides the Kalman filtering method based on the Bayesian sequential importance integral, which can greatly reduce the average error without sacrificing the calculation complexity and improve the real-time tracking performance by one order of magnitude when applied to tracking the maneuvering target in the airspace.
In order to achieve the above object, the present invention provides a kalman filtering method based on bayesian sequential importance integral, which comprises the following steps:
step 7, calculating Kalman filtering gain, filtering mean value and filtering covariance, and finally fusing posterior distribution of the multi-modal filtering approximation system;
and 8, taking the approximate posterior distribution as the original prior distribution of the next moment, and carrying out the next round of system state filtering.
In one embodiment, in step 1, the discrete-time nonlinear gaussian system specifically includes:
in the formula (I), the compound is shown in the specification,representing a non-linear Gaussian system at time kThe state equation of the space,Observation equation of space, nX、nZRepresenting a state variable dimension, a measured sequence dimension, upsilonk、ekRepresents nυDimensional process noise, neDimensional observation noise, respectively mean value ofStandard covariance of ∑υ、ΣeThe noise of the gaussian noise of (a),denotes the kth model parameter at the kth time, and K denotes the total number of model parameters.
In one embodiment, step 2 specifically includes:
step 2.1, according to the soft constraint theory, truncating the space-time distribution of the observation noise, and constructing the truncated prior distribution of the system, wherein the truncated prior distribution comprises the following steps:
in the formula (I), the compound is shown in the specification,truncated prior distribution, X, representing the observed noise soft constraint constructionk-1Representing the state quantities of the non-linear gaussian system at time k-1,denotes the k model parameter sequence from time 1 to time k-1, r1:kRepresenting a sequence of potential characteristic variables, Z, in a nonlinear Gaussian system from time 1 to time k1:kRepresenting an observation sequence of the nonlinear Gaussian system from the 1 st moment to the k-th moment;to assist in indicating the function, the value of which depends on the probability density of the observed noise in the observation modelWhether in feasible field IX(Zk) Internal;
step 2.2, assuming the modified truncation prior distribution in the nonlinear Gaussian system as Gaussian distribution, the mean value is approximated by the Newton interior point method
Thus, it is possible to prevent the occurrence of,
in the formula, pt(X) represents the truncated prior distribution after the nonlinear Gaussian system is corrected, X represents the state variable of a random system,representing the feasible region center of the state distribution of the real target of the kth model parameter at the k moment, namely, the mean value of the truncated prior distribution;representing a second moment of a feasible region of a k-th model parameter statistical center at the k moment, namely, a covariance of truncated prior distribution; x0Is an initial value of state, α*、diRespectively, the iteration step length and the search direction of the backtracking method.
In one embodiment, in step 3, the posterior probability distribution of the target state in the nonlinear gaussian system can be expressed as:
in the formula (I), the compound is shown in the specification,representing the posterior probability distribution of the target state in a nonlinear gaussian system,representing the likelihood of observation in a nonlinear gaussian system,representing the original prior distribution, X, in a non-linear Gaussian system0:kRepresents the state quantity, X, of the nonlinear Gaussian system from time 0 to time k1:k-1Represents the state quantity of the nonlinear Gaussian system from the 1 st moment to the k-1 st moment, Z1:k-1Represents the observed quantity, r, of the nonlinear Gaussian system from the 1 st time to the k-1 st time1:k-1Representing potential characteristic variables in the nonlinear gaussian system from time 1 to time k-1.
In one embodiment, step 4 specifically includes:
the mathematical expectation integral of the posterior probability density of the state and the model parameter in the nonlinear Gaussian system is obtained based on the posterior probability distribution of the target state in the nonlinear Gaussian system, and is as follows:
the sub-optimal target importance function of the nonlinear Gaussian system is obtained by decomposing the mathematical expectation integral, i.e.Comprises the following steps:
in the formula (I), the compound is shown in the specification,representing a set of model parameters, K representing the number of model parameters,representing model parameters M at time kκThe following observations were:
in the formula, HJaA Jacobian matrix representing the nonlinear observation function, T representing the transpose of the matrix;
system-based suboptimal target importance function sampling NsImportance sample, i-th importance sample under the k model parameterAnd its weightRespectively as follows:
the weights of the importance sample of the nonlinear gaussian system are normalized, namely:
in the formula (I), the compound is shown in the specification,the weights of the importance sample of the normalized nonlinear gaussian system are represented.
In one embodiment, step 5 specifically includes:
since the characteristic root of the constructed Hermite polynomial of the order l is determined, a Hermite integral point under the nonlinear Gaussian system model parameters is selected according to Gauss-Hermite rule and is determined, and the weight of the Hermite integral point is constant and is expressed as follows:
in the formula (I), the compound is shown in the specification,representing the nth integration point of the system at the kth model parameter at the kth instant,representing the weight of the nth integration point of the system under the kth model parameter at the kth moment;
integrating point weight determination of importance sampling sample weight of nonlinear Gaussian system and Hermite polynomial of l orderPerforming dot product to obtain Bayes sequential importance integral point weight as follows:
In one embodiment, step 6 specifically includes:
variance of likelihood distribution of decomposition subregion:
in the formula (I), the compound is shown in the specification,the mean square error of the system truncation prior distribution of the variance of the likelihood distribution of the sub-region at the k-1 moment and the k model parameter is represented;
constructing a comprehensive integral point of a multi-model set in a nonlinear Gaussian system, wherein the comprehensive integral point is as follows:
in the formula (I), the compound is shown in the specification,represents the integrated integration points of the multiple model sets in the nonlinear Gaussian system,representing the nth Bayes sequential importance integral point of the nonlinear Gaussian system at the k-1 moment;
substituting the comprehensive integral point into the state equation of the nonlinear Gaussian system to calculate the mean value and the covariance of state prediction:
in the formula (I), the compound is shown in the specification,represents the nth integrated integration point of the nonlinear Gaussian system under the kth model parameter at the kth-1 moment,representing the nth Bayes sequential importance integral point of the nonlinear Gaussian system at the k-1 moment under the kth model parameter;
calculating the measured prediction mean of the nonlinear Gaussian system:
in the formula (I), the compound is shown in the specification,which represents the mean of the prediction of the measurement,which represents the covariance of the measurement prediction,which represents the sequence of observations of the system,representing the cross-covariance of the system.
In one embodiment, step 7 specifically includes:
Fusing the posterior distribution of the multi-modal filtering output approximate nonlinear Gaussian system state:
in the formula (I), the compound is shown in the specification,is the mean of the approximate posterior distribution of the nonlinear gaussian system states,is the covariance of the approximate a posteriori distribution of the nonlinear gaussian system states,for model parameters in nonlinear dynamic systemsThe similarity measure factor of (1).
In one of the embodiments, the first and second electrodes are,the acquisition process comprises the following steps:
from the viewpoint of information theory learning, the related information entropy can measure the generalized similarity between any two random variables, and in order to measure the influence of parameters such as truncation prior and posterior feedback in a nonlinear dynamic system on the system state and parameter estimation, the kernel function is defined as follows:
the objective function is constructed according to the maximum relevant information entropy criterion as follows:
according to the Bayesian sequential importance integral-based Kalman filtering method provided by the invention, the importance function covering multimodal distribution is constructed by fusing the constraint information such as truncation prior and posterior feedback, so that the matching degree of the target importance function and the actual target real distribution is improved, and the diversity and the accuracy of a sampling sample are improved. And the sequential importance sampling correction Gauss-Hermite integral point is introduced, so that the adaptivity and the accuracy of the sampling integral point are enhanced, and meanwhile, the diversity of the integral point and the fault tolerance of the prediction covariance are comprehensively improved by introducing the relevant information entropy measure in the time updating stage. The method can greatly reduce the average error without sacrificing the calculation complexity, and can improve the real-time tracking performance by one order of magnitude when being applied to tracking the maneuvering target in the airspace.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the structures shown in the drawings without creative efforts.
FIG. 1 is a schematic flow chart of a Kalman filtering method based on Bayesian sequential importance integral in an embodiment of the present invention;
FIG. 2 is a diagram illustrating a filtering error of sampling 10 samples in example 1 according to an embodiment of the present invention;
FIG. 3 is a diagram illustrating a filtering error of sampling 200 samples in example 1 according to an embodiment of the present invention;
FIG. 4 is a diagram illustrating a filtering error of sampling 300 samples in example 1 according to an embodiment of the present invention;
FIG. 5 is a graph illustrating a filtering error of 500 samples in example 1 according to an embodiment of the present invention;
FIG. 6 is a diagram of civil aviation tracks and radar observations collected by ADS-B in example 2 of an embodiment of the present invention;
FIG. 7 is a schematic diagram of the position root mean square error of IMMEKF, IMMUK, MMRBPF and the invention in tracking a maneuvering target in two-dimensional space in example 2 of an embodiment of the invention;
FIG. 8 is a schematic diagram of the position root mean square error of IMMEKF, IMMUK, MMRBPF and the present invention tracking a maneuvering target in the X-axis direction in example 2 of an embodiment of the present invention;
FIG. 9 is a schematic diagram of the position root mean square error of IMMEKF, IMMUK, MMRBPF and the present invention tracking a maneuvering target in the Y-axis direction in example 2 of an embodiment of the present invention.
The implementation, functional features and advantages of the objects of the present invention will be further explained with reference to the accompanying drawings.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
It should be noted that all the directional indicators (such as up, down, left, right, front, and rear … …) in the embodiment of the present invention are only used to explain the relative position relationship between the components, the movement situation, etc. in a specific posture (as shown in the drawing), and if the specific posture is changed, the directional indicator is changed accordingly.
In addition, the descriptions related to "first", "second", etc. in the present invention are only for descriptive purposes and are not to be construed as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one such feature. In the description of the present invention, "a plurality" means at least two, e.g., two, three, etc., unless specifically limited otherwise.
In the present invention, unless otherwise expressly stated or limited, the terms "connected," "secured," and the like are to be construed broadly, and for example, "secured" may be a fixed connection, a removable connection, or an integral part; the connection can be mechanical connection, electrical connection, physical connection or wireless communication connection; they may be directly connected or indirectly connected through intervening media, or they may be connected internally or in any other suitable relationship, unless expressly stated otherwise. The specific meanings of the above terms in the present invention can be understood by those skilled in the art according to specific situations.
In addition, the technical solutions in the embodiments of the present invention may be combined with each other, but it must be based on the realization of those skilled in the art, and when the technical solutions are contradictory or cannot be realized, such a combination of technical solutions should not be considered to exist, and is not within the protection scope of the present invention.
The embodiment discloses a Kalman filtering method based on Bayesian sequential importance integration, which mainly comprises two aspects. The first aspect is to construct a soft constraint parameter model for truncation prior and state posterior feedback, and to modulate the inconsistency of multi-likelihood distribution by adopting a heuristic algorithm, thereby reducing the deviation of importance distribution and real target distribution caused by uncertainty of artificial subjective experience modeling. The second aspect modifies the system integral point distribution and its weight, reducing the deviation between the likelihood distribution and the target distribution caused by the unpredictability of the nonlinear gaussian system. Referring to fig. 1, the kalman filtering method based on the bayesian sequential importance integral in this embodiment specifically includes the following steps:
step 7, calculating Kalman filtering gain, filtering mean value and filtering covariance, and finally fusing posterior distribution of the multi-modal filtering approximation system;
and 8, taking the approximate posterior distribution as the original prior distribution at the next moment, and carrying out the next round of system state filtering.
In this embodiment, constraint information such as truncation prior and posterior feedback is incorporated into the system state evolution process, and a model parameter set is defined asWherein K is more than or equal to 2, and the model parameter set comprises K mutually independent model parameters. Under the Bayes condition, each model parameter is used as a random variable added by the system to be estimated together with a target state, namely the state estimation under the multi-mode parameter. Therefore, in this embodiment, the state model and the observation model for establishing the k-time nonlinear gaussian system are respectively:
in the formula (I), the compound is shown in the specification,respectively representing time-of-k systemsThe state model equation of the space,Equation of the observation model of space, nX、nZRespectively representing a state variable dimension, a measured sequence dimension, upsilonk、ekRespectively represent nυDimensional process noise, neDimensional observation noise, respectively mean value ofStandard covariance of ∑υ、ΣeThe noise of the gaussian noise of (a),representing the kth model parameter at time kth.
According to a Bayes inference integral model, defining one-step prediction distribution of state variables in the discrete time nonlinear Gaussian system as follows:
in the formula (I), the compound is shown in the specification,one-step prediction of state variables in a representation system, Xk-1Representing the state quantity of the system at time k-1,denotes the k model parameter, Z, at time k-11:k-1Represents observation from time 1 to time kThe sequence of the amounts is such that,a gaussian distribution representing a one-step prediction of state variables in the system,the mean value of the states is represented,representing the state variance, defined as:
wherein the superscript T represents the transpose operation of the matrix,jacobian matrix of nonlinear observation functions, denoted
Similarly, the probability density function of the observation likelihood in the discrete time system can be deducedComprises the following steps:
according to the energy conservation theorem, in the actual physical dynamic system model, the energy distribution of random observation noise is bounded. Therefore, in this embodiment, a truncation constraint is applied to the bayesian state space model of the nonlinear dynamic system according to the truncation theory, and preferably, the truncation prior distribution of the nonlinear gaussian system is constructed by observing the soft constraint of the noise space-time distribution, which is:
in the formula (I), the compound is shown in the specification,representing a truncated prior distribution constructed based on soft constraints of observed noise,sequence representing the k-th model parameter from time 1 to time k-1, r1:kRepresenting a sequence of potential characteristic variables in the system from the 1 st moment to the k-th moment;to assist in indicating the function, the value of which depends on the probability density of the observed noise in the observation modelWhether in feasible field IX(Zk) Internal;
subsequently, the mean and covariance of the modified truncated prior distribution in the system are approximated, and the observation model of the system is set to satisfy the locally differentiable, Jacobian matrix of the nonlinear observation functionAre present. Thus, the initial value X of the state of the system0Distributed in feasible region IX(Zk) Thereby approximating the center of the real target state distribution by a heuristic optimization methodIn this embodiment, the panorama newton method is used to iteratively solve the center of the feasible domain of the truncated space, and the back tracing backtracking algorithm is used to obtain:
wherein alpha is*、diRespectively the iteration step length and the search direction of the backtracking method; calculating the second moment, i.e. variance, of the feasible region according to the statistical center
Thus, the modified truncated prior distribution in the system is obtained as follows:
in the formula, pt(X) represents the truncated prior distribution of the nonlinear Gaussian system after correction, as in the previous paragraphX represents a state variable of a stochastic system,representing the feasible region center of the state distribution of the real target of the kth model parameter at the k moment, namely, the mean value of the truncated prior distribution;the second moment of the feasible region of the statistical center of the k-th model parameter at time k, i.e., the covariance of the truncated prior distribution, is represented.
Obtaining a posterior probability distribution of a target state in the nonlinear Gaussian system based on the corrected truncated prior probability density and the original prior probability density according to a total probability formula of Bayes inference, wherein the posterior probability distribution comprises:
in the formula (I), the compound is shown in the specification,posterior probability distribution, p, representing target states in a non-linear Gaussian systemlik(. represents the observed likelihood of the system, ptDenotes the truncated prior distribution of the system, po() represents the original a priori distribution of the system,probability density function of observed noise, X, representing a model of the system0:kRepresenting the sequence of state quantities of the system from time 0 to time k, Z1:k-1Represents the observation sequence of the system from time 1 to time k-1, X1:k-1Representing the sequence of state quantities of the nonlinear Gaussian system from the 1 st to the k-1 st instants, r1:k-1Representing potential characteristic variable sequences in the nonlinear gaussian system from the 1 st moment to the k-1 st moment.
In the specific implementation process, for a complex nonlinear function, the integral of the normalization constant of the denominator in the Bayesian model is difficult to perform closed analytic solution, so that the p (X | Z) can not be directly distributed from the target posterior part1:k) The samples are sampled. To solve this problem, in the importance sampling method, a target importance function pi (X | Z) is obtained by decomposing the mathematical expectation of the posterior probability density of the target state1:k) Sampling N according to the target importance functionsThe individual weight samples approximate the posterior distribution of the target state. Assuming the system state equation is f (-), the mathematical expectation of the posterior distribution of the target states can be computed by decomposition by:
from the above equation, p (X | Z) is a posterior probability density function regardless of the target state1:k) Whether it is non-zero, the target importance function is non-zero, and pi (X | Z)1:k)≥p(X|Z1:k). Assuming a known observation likelihood plik(Z1:k|Xk) And state prior p (X)0:k) The expression for sampling the weight samples according to the target importance function is:
Xi~π(X|Z1:k),i=1,…,Ns
state prediction f (X)i) Is the importance sample XiThe normalized weight of (a) is expressed as:
the goal is to sample weight samples from the importance function to approximate the true target state distribution. According to the monte carlo law, the posterior probability density function of the approximate target state weighted by the dirac function δ (·) is:
it can be known from the derivation process of the importance sampling algorithm that the selection of the target importance function is a key factor influencing the filtering performance of the importance sampling method. Due to the signal truncation problem in the process from continuous integration of the target posterior probability density to numerical approximation, the target importance function and the real distribution of the actual target have deviation. In this case, the weight samples randomly extracted from the target importance function may not completely describe the posterior distribution of the actual target, and for this problem, in this embodiment, the truncation prior, the state posterior feedback, and the potential characteristic variables in the system are used as auxiliary variables to be merged into the construction of the target importance function, so as to obtain the mathematical expected product of the posterior probability density of the state and the model parameters in the system as follows:
in the formula (I), the compound is shown in the specification,a suboptimal target importance function for a nonlinear gaussian system is:
in the formula (I), the compound is shown in the specification,a set of model parameters is represented which,representing model parameters M at time kκThe following observations are:
in the formula, HJaJacobian matrices representing non-linear observation functions, as in the preceding
Sampling samples based on suboptimal target importance function, sampling sample of ith importance under kappa model parameterAnd its weightComprises the following steps:
in the formula, NsFor the number of importance samples, the weights of the importance samples of the non-linear gaussian system are normalized, i.e.:
in the formula (I), the compound is shown in the specification,the weights of the importance sample of the normalized nonlinear gaussian system are represented.
In the specific implementation process, for each one-dimensional gaussian distribution integral N (X; m, Σ), we can use Gauss-Hermite integral approximation, and in this embodiment, we define a one-dimensional l-order Hermite polynomial as:
according to Gauss-Hermite rule, the I characteristic roots of Hermite polynomial are selected as integration points, and accurate approximation can be carried out on 2l-1 order (maximum order) polynomial. It should be noted that, in order to be numerically more stable, the eigenvalue of the three-diagonal matrix is selected as the Hermite polynomial H in this embodimentlRoot of (X), i.e. xiiI is 1, …, l. Furthermore, a Hermite polynomial does not need to be constructed and the characteristic root of the Hermite polynomial is calculated, and the weight of the corresponding integral point can be solved in a closed mode, namely:
since the characteristic root of the Hermite polynomial of order l is determined, the integral point under the system model parameter in the embodiment is determined according to Gauss-Hermite rule, and the weight is constant and expressed asWherein the content of the first and second substances,representing the nth integration point of the system at the kth model parameter at the kth instant,representing the weight of the nth integration point of the system at the kth model parameter at the kth time instant. Generally, a nonlinear function model of a dynamic system can be constructed according to engineering practice and experimental reasoning, but due to the limitation of human knowledge and modeling errors caused by uncertainty of human subjective experience, the state of the dynamic system is unpredictable, so that the observation likelihood of the system is inconsistent with the posterior distribution of an actual target. Therefore, the definite integration point generated according to the Gauss-Hermite rule in this uncertain case is not accurate, and it is difficult to characterize the true distribution of the actual target state. Based on this, in the embodiment, by constructing the suboptimal target importance function under the multi-model parameter of the system, the sample distribution can cover the effective observation likelihood region of the system target state mapping as much as possible, so that the deviation between the target importance function and the actual target distribution is corrected.
According to the consistent convergence theorem of the Monte Carlo numerical method, the weight of an importance sampling sample of a nonlinear Gaussian system and the determined integral point weight of an l-order Hermite polynomialPerforming dot product to obtain Bayes sequential importance integral point weight as follows:
finally, the Bayesian sequential importance integral point is obtained asNamely, the initial parameter setting of the Bayes sequential importance integral filtering method in the embodiment is completed, and then the filtering method is used for carrying out subsequent target prediction, updating and fusion under the multi-mode parametersAnd (6) testing distribution. The method specifically comprises the following steps:
first, the variance of the subregion likelihood distributions is decomposed:
in the formula (I), the compound is shown in the specification,the mean square error of the system truncation prior distribution of the variance of the likelihood distribution of the sub-region at the k-1 moment and the k model parameter is represented;
and then constructing a comprehensive integral point of a multi-model set in the nonlinear Gaussian system, wherein the comprehensive integral point is as follows:
in the formula (I), the compound is shown in the specification,represents the nth integrated integration point of the nonlinear Gaussian system under the kth model parameter at the kth-1 moment,representing the nth Bayes sequential importance integral point of the nonlinear Gaussian system at the k-1 moment under the kth model parameter;
substituting the comprehensive integral point into a state equation of the nonlinear Gaussian system to calculate the mean value and the covariance of state prediction:
in the formula (I), the compound is shown in the specification,is the average of the prediction of the states,covariance predicted for the state;
calculating the measurement prediction mean, covariance and cross covariance of the nonlinear Gaussian system:
in the formula (I), the compound is shown in the specification,which represents the mean of the prediction of the measurement,which represents the covariance of the measurement prediction,which represents the sequence of observations of the system,representing the cross-covariance of the system;
And (3) fusing the multi-mode filtering output approximation to obtain the approximate posterior distribution of the nonlinear Gaussian system state:
in the formula (I), the compound is shown in the specification,is the mean of the approximate posterior distribution of the nonlinear gaussian system states,is the covariance of the approximate a posteriori distribution of the nonlinear gaussian system states,for model parameters in nonlinear dynamic systemsThe obtaining process of the similarity measure factor is as follows:
from the information theory learning point of view, the related information entropy can measure the generalized similarity between any two random variables. Therefore, in order to measure the influence of parameters such as truncation prior and posterior feedback in a nonlinear dynamic system on the state and parameter estimation of the system, the kernel function is defined as follows:
the objective function is constructed according to the maximum relevant information entropy criterion as follows:
for the convergence of the method in this embodiment, in the discrete nonlinear dynamic system state and the observation model, we respectively use process noise and observation noise to represent the uncertainty of the system signal caused by inaccurate model and random noise. In addition to this uncertainty, the true state parameterAnd measuringThere often exists a causal relationship between them. In the embodiment, truncation prior and state posterior feedback are introduced into the state evolution process, and the model parameters and the system state are estimated together, namelyThe central limit theorem of the majority ensures that the importance sampling under the multi-model parameters meets the requirement of consistent convergence, and the integral point under the modified multi-model parameters is obtained by the weight dot product of the importance sample and the Gauss-Hermite integral pointIs still in a certain Euclidean real space, so the method in this embodiment is in a Bayesian state modelThe space converges.
Theoretically, the numerical approximation error and the state dimension n of the Monte Carlo importance sampling methodXIndependently, the error term is represented as O (n)x -1/2). In the method for constructing the importance function covering multi-likelihood distribution, the matching degree between the multi-model parameter likelihood and the actual distribution of the actual target is measured through the maximum correlation information entropy criterion, and the dimensionality reduction of the multi-model state space is realized through the characteristic manifold structure of the kernel function of the model parameters. Therefore, when the state dimension of the discrete nonlinear dynamic system is determined, the mixed Gaussian importance function under multiple model parametersIs calculated byDown to
The performance of the kalman filtering method based on the bayesian sequential importance integral in this embodiment is evaluated in the following with specific examples.
Example 1: for the one-dimensional univariate non-stationary growth model, the Bayesian sequential importance integral filtering method (SIQF) in the embodiment is adopted for distribution and compared with Extended Kalman Filtering (EKF), Unscented Kalman Filtering (UKF), traditional particle filtering (GPF), extended Kalman particle filtering (EPF) and unscented Kalman filtering (UPF) algorithms.
The state equation and the observation equation model of the one-dimensional univariate non-stationary growth model are respectively
Where the coefficients are known constants, a is 0.5, b is 2.5, c is 8,initial value of state X0In [0,1 ]]Average value within the range. Upsilon iskIs subject to a parameter of [2,3]Is detected by the gamma distribution of (1). e.g. of the typekIs the observed noise with a mean of 0 and a variance of 0.5.
Fig. 2-5 are schematic diagrams of filter errors for sampling 10, 200, 300, 500 samples, respectively. From the qualitative comparative analysis of the filter error trend, it can be known that: compared with the extended Kalman filtering algorithm and the unscented Kalman filtering algorithm, the traditional particle filtering algorithm has the advantage of processing non-Gaussian noise. The extended Kalman particle filter and the unscented Kalman particle filter are generated by nonlinear state estimation generated by the extended Kalman filter and the unscented Kalman filter algorithm, and the distribution of the estimation of the extended Kalman filter and the unscented Kalman filter is deviated from the actual target, so that the estimation errors of the extended Kalman particle filter and the unscented Kalman particle filter are larger. As the number of samples increases, the filtering error decreases accordingly, however, the amount of calculation increases sharply. Fig. 5 is a graph comparing the filtering performance of the numerical method based on monte carlo sampling alone when the number of samples increases to 500. Along with the advance of time, in the state evolution process, the estimation errors of the extended Kalman particle filter, the unscented Kalman particle filter and the traditional particle filter algorithm are obviously increased after 30s, and the main reason of the phenomenon is that the mismatch of an importance function and the real distribution of an actual target causes sample diversity attenuation and weight degradation. Compared with the unscented Kalman particle filter algorithm, the method has the advantage that the average error is reduced by 63% under the condition of not sacrificing the calculation complexity. The method is mainly characterized in that the algorithm is provided to fuse truncation priors and posterior feedback to construct the target importance function covering multiple likelihoods, the problem of the deviation between the target importance function and the actual target real distribution is effectively solved, and the diversity and the accuracy of a sampling sample are improved.
Example 2: the method is compared with an interactive multi-model extended Kalman filter (IMMEKF), an interactive multi-model unscented Kalman filter (IMMUKF) and a Rao-Blackwell-based multi-model particle filter (MMRBPF) algorithm by taking actual measurement airspace civil aviation tracks of an ADS-B system as target data.
The civil flight path and radar observation data collected by ADS-B are shown in FIG. 6, where the curve with high smoothness is the GPS located flight path and the curve with low smoothness is the active radar observed data. The sparse observation experimental data of the airspace target are as follows: one is located at [0m,0m ]]TIs observed in a two-dimensional space [0km,30km ]]×[0km,12km]In-flight airspace targets. The target state variables are composed of position, speed and turn rate and are expressed asWherein ω isk,tThe turning rate of the target t at the k-th time. The motion model of the target is the same as the state model in this embodiment, and the state transition matrix and the process noise are respectively:
wherein T is sampling time interval of radar, is taken as 1s, and process noise standard deviation is sigmav=0.1km/s2,σω=0.1rad/s2. The observation model of the radar is the same as that in the present embodiment, and the nonlinear measurement function is:
wherein (X)s,Ys) Is the position coordinate of the radar, r andtheta is radar range finding and angle measuring, and standard deviation of observation noise is sigmar0.1km and σθ=3mrad。
FIGS. 7-9 show the root mean square error of the position of the maneuvering target tracked by the IMMEKF, the IMMUK, the MMRBPF and the four filtering modes of the method in the embodiment in the two-dimensional space and the x-axis and y-axis moving directions respectively. From qualitative comparison of the root mean square error trend of the four filtering modes, it can be known that: compared with the traditional interactive multi-model filtering algorithm and the Rao-Blackwell multi-model particle filtering algorithm, the tracking effect of the method has great advantages in the aspect of filtering precision. The method is mainly characterized in that the maximum correlation information entropy criterion is adopted to measure the truncation prior and the posterior feedback constraint information, the mixed Gaussian target importance function constructed under the multi-model parameters covers the multi-likelihood information, and the deviation between the target importance function and the actual distribution of the actual target state is effectively reduced, so that the anti-interference capability of the system state to random noise in the dynamic evolution process is improved. By modifying the self-adaptability of the Hermite integral point in the Euclidean real number space, the deviation between the observation likelihood and the actual target real distribution is effectively reduced, and therefore the accuracy of state prediction and the fault tolerance of covariance are improved.
In addition, to compare the real-time performance of the present embodiment method tracking a maneuvering target. Table 1 counts error means and average execution time required for performing 100 rounds of monte carlo experiments in four filtering manners including IMMEKF, IMMUK, MMRBPF and the method SIQF of the embodiment. From a quantitative comparison of these two parameters it follows that: compared with the traditional nonlinear filter based on an interactive multi-model algorithm, the implementation time of the Monte Carlo experiment of the method is increased, and the main reason is that the algorithm takes the truncation prior and the state feedback of the system as system constraint information, and integrates the iterative constraint optimization into the importance function construction and the prediction and update calculation process of the integral point. Compared with the RBMMPF filtering algorithm, the filtering precision of the method is equivalent, and the average execution time of the Monte Carlo experiment is reduced by one order of magnitude. The method is mainly characterized in that in the process of constructing the multi-likelihood importance function, effective dimension reduction of a multi-model state space is realized through a characteristic manifold structure of a likelihood kernel function, and meanwhile, the adaptability of the algorithm for learning knowledge in different fields is enhanced.
Mean average execution time of filtering errors of 1100 Monte Carlo wheel experiments in Table
In summary, for the problem that the target importance function and the actual target real distribution and the problem that the likelihood function and the actual target real distribution have deviations, the present embodiment provides a bayesian sequential importance integral filtering method according to the soft constraint theory and the maximum correlation information entropy criterion. And a target importance function covering multi-likelihood information is constructed by fusing truncation prior and state posterior feedback, the matching degree of the target importance function and actual target real distribution is improved, and the diversity and accuracy of a sampling sample are improved. And the sequential importance samples are introduced to correct Gauss-Hermite integration points, so that the adaptivity and the accuracy of sampling integration points are enhanced. Simulation experiments prove that the method improves the capability of the Bayesian filter for resisting multi-peak likelihood and non-Gaussian noise interference.
The above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention, and all modifications and equivalents of the present invention, which are made by the contents of the present specification and the accompanying drawings, or directly/indirectly applied to other related technical fields, are included in the scope of the present invention.
Claims (9)
1. A Kalman filtering method based on Bayes sequential importance integral is characterized by comprising the following steps:
step 1, establishing a discrete-time nonlinear Gaussian system;
step 2, truncating the space-time distribution of the observation noise according to a soft constraint theory, constructing truncation prior distribution of the system, and obtaining the revised truncation prior distribution in the system by adopting a global Newton method based on the current observed quantity and the initial state value;
step 3, according to a total probability formula of Bayesian inference, obtaining posterior probability distribution of a target state in the system based on the corrected truncated prior distribution and the original prior distribution;
step 4, obtaining a suboptimal target importance function of the system based on the posterior probability distribution of the target state, and sampling an importance sampling sample of the system based on the suboptimal target importance function;
step 5, selecting Hermite integration points according to a Gauss-Hermite rule, and correcting the Hermite integration points based on the importance sampling samples to obtain Bayes sequential importance integration points;
step 6, constructing comprehensive integral points of a multi-model set in the system based on Bayes sequential importance integral points, obtaining a mean value and covariance of system state prediction based on the comprehensive integral points, and measuring the predicted mean value, covariance and cross covariance;
step 7, calculating Kalman filtering gain, filtering mean value and filtering covariance, and finally fusing posterior distribution of the multi-modal filtering approximation system;
and 8, taking the approximate posterior distribution as the original prior distribution of the next moment, and carrying out the next round of system state filtering.
2. The bayesian sequential importance integral-based kalman filtering method according to claim 1, wherein in step 1, the discrete-time nonlinear gaussian system specifically comprises:
in the formula (I), the compound is shown in the specification,representing a non-linear Gaussian system at time kThe state equation of the space,Observation equation of space, nX、nZRepresenting a state variable dimension, a measured sequence dimension, upsilonk、ekRepresents nυDimensional process noise, neDimensional observation noise, respectively mean value ofStandard covariance of ∑υ、ΣeThe noise of the gaussian noise of (a),denotes the kth model parameter at the kth time, and K denotes the total number of model parameters.
3. The bayesian sequential importance integral-based kalman filtering method according to claim 2, wherein step 2 specifically includes:
step 2.1, according to the soft constraint theory, truncating the space-time distribution of the observation noise, and constructing the truncated prior distribution of the system, wherein the truncated prior distribution comprises the following steps:
in the formula (I), the compound is shown in the specification,truncated prior distribution, X, representing the observed noise soft constraint constructionk-1Representing the state quantities of the non-linear gaussian system at time k-1,denotes the k model parameter sequence from time 1 to time k-1, r1:kRepresenting a sequence of potential characteristic variables, Z, in a nonlinear Gaussian system from time 1 to time k1:kRepresenting an observation sequence of the nonlinear Gaussian system from the 1 st moment to the k-th moment;to assist in indicating the function, the value of which depends on the probability density of the observed noise in the observation modelWhether in feasible field IX(Zk) Internal;
step 2.2, assuming the modified truncation prior distribution in the nonlinear Gaussian system as Gaussian distribution, the mean value is approximated by the Newton interior point method
Thus, it is possible to prevent the occurrence of,
in the formula, pt(X) represents the truncated prior distribution after the nonlinear Gaussian system is corrected, X represents the state variable of a random system,representing the feasible region center of the state distribution of the real target of the kth model parameter at the k moment, namely, the mean value of the truncated prior distribution;representing a second moment of a feasible region of a k-th model parameter statistical center at the k moment, namely, a covariance of truncated prior distribution; x0Is an initial value of state, α*、diRespectively, the iteration step length and the search direction of the backtracking method.
4. The Bayesian sequential importance integral-based Kalman filtering method according to claim 3, wherein in step 3, the posterior probability distribution of the target state in the nonlinear Gaussian system can be expressed as:
in the formula (I), the compound is shown in the specification,representing the posterior probability distribution of the target state in a nonlinear gaussian system,representing the likelihood of observation in a nonlinear gaussian system,representing the original prior distribution, X, in a non-linear Gaussian system0:kRepresents the state quantity, X, of the nonlinear Gaussian system from time 0 to time k1:k-1Represents the state quantity of the nonlinear Gaussian system from the 1 st moment to the k-1 st moment, Z1:k-1Represents the observed quantity, r, of the nonlinear Gaussian system from the 1 st time to the k-1 st time1:k-1Representing potential characteristic variables in the nonlinear gaussian system from time 1 to time k-1.
5. The Bayesian sequential importance integral-based Kalman filtering method according to claim 4, wherein the step 4 specifically comprises:
the mathematical expectation integral of the posterior probability density of the state and the model parameter in the nonlinear Gaussian system is obtained based on the posterior probability distribution of the target state in the nonlinear Gaussian system, and is as follows:
the sub-optimal target importance function of the nonlinear Gaussian system is obtained by decomposing the mathematical expectation integral, i.e.Comprises the following steps:
in the formula (I), the compound is shown in the specification,representing a set of model parameters, K representing the number of model parameters,representing model parameters M at time kκThe following observations were:
in the formula, HJaA Jacobian matrix representing the nonlinear observation function, T representing the transpose of the matrix;
system-based suboptimal target importance function sampling NsImportance sample, i-th importance sample under the k model parameterAnd its weightRespectively as follows:
the weights of the importance sample of the nonlinear gaussian system are normalized, namely:
6. The Bayesian sequential importance integral-based Kalman filtering method according to claim 5, wherein the step 5 specifically comprises:
since the characteristic root of the constructed Hermite polynomial of the order l is determined, a Hermite integral point under the nonlinear Gaussian system model parameters is selected according to Gauss-Hermite rule and is determined, and the weight of the Hermite integral point is constant and is expressed as follows:
in the formula (I), the compound is shown in the specification,representing the nth integration point of the system at the kth model parameter at the kth instant,representing the weight of the nth integration point of the system under the kth model parameter at the kth moment;
integrating point weight determination of importance sampling sample weight of nonlinear Gaussian system and Hermite polynomial of l orderPerforming dot product to obtain Bayes sequential importance integral point weight as follows:
7. The Bayesian sequential importance integral-based Kalman filtering method according to claim 6, wherein step 6 specifically comprises:
variance of likelihood distribution of decomposition subregion:
in the formula (I), the compound is shown in the specification,the mean square error of the system truncation prior distribution of the variance of the likelihood distribution of the sub-region at the k-1 moment and the k model parameter is represented;
constructing a comprehensive integral point of a multi-model set in a nonlinear Gaussian system, wherein the comprehensive integral point is as follows:
in the formula (I), the compound is shown in the specification,represents the integrated integration points of the multiple model sets in the nonlinear Gaussian system,representing the nth Bayes sequential importance integral point of the nonlinear Gaussian system at the k-1 moment;
substituting the comprehensive integral point into the state equation of the nonlinear Gaussian system to calculate the mean value and the covariance of state prediction:
in the formula (I), the compound is shown in the specification,represents the nth integrated integration point of the nonlinear Gaussian system under the kth model parameter at the kth-1 moment,representing the nth Bayes sequential importance integral point of the nonlinear Gaussian system at the k-1 moment under the kth model parameter;
calculating the measured prediction mean of the nonlinear Gaussian system:
8. The Bayesian sequential importance integral-based Kalman filtering method according to claim 7, wherein the step 7 specifically comprises:
Fusing the posterior distribution of the multi-modal filtering output approximate nonlinear Gaussian system state:
in the formula (I), the compound is shown in the specification,is the mean of the approximate posterior distribution of the nonlinear gaussian system states,is the covariance of the approximate a posteriori distribution of the nonlinear gaussian system states,for model parameters in nonlinear dynamic systemsThe similarity measure factor of (1).
9. The Bayesian sequential importance integral-based Kalman filtering method of claim 8,the acquisition process comprises the following steps:
from the viewpoint of information theory learning, the related information entropy can measure the generalized similarity between any two random variables, and in order to measure the influence of parameters such as truncation prior and posterior feedback in a nonlinear dynamic system on the system state and parameter estimation, the kernel function is defined as follows:
the objective function is constructed according to the maximum relevant information entropy criterion as follows:
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