CN113452349B - Kalman filtering method based on Bayes sequential importance integral - Google Patents

Kalman filtering method based on Bayes sequential importance integral Download PDF

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CN113452349B
CN113452349B CN202110720593.9A CN202110720593A CN113452349B CN 113452349 B CN113452349 B CN 113452349B CN 202110720593 A CN202110720593 A CN 202110720593A CN 113452349 B CN113452349 B CN 113452349B
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CN113452349A (en
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张宏伟
张小虎
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Sun Yat Sen University
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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 prior by soft constraint, constructing mixed Gaussian importance distribution by fusing the truncation prior and state posterior feedback, correcting comprehensive integral 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 improve the real-time tracking performance by one order of magnitude when being applied to tracking airspace maneuvering targets.

Description

Kalman filtering method based on Bayes sequential importance integral
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 are based on the idea of random sampling, so that the nonlinear function does not need to be linearized, the nonlinear non-Gaussian problem of dynamic state estimation can be effectively processed, and the central limit theorem ensures the consistent convergence of the method. 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, there are generally consistent or compatible geometric mapping or statistical constraint relationships 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 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 modified 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.
In one embodiment, in step 1, the discrete-time nonlinear gaussian system specifically includes:
Figure BDA0003136380030000031
Figure BDA0003136380030000032
in the formula (I), the compound is shown in the specification,
Figure BDA0003136380030000033
representing a non-linear Gaussian system at time k
Figure BDA0003136380030000034
The state equation of the space,
Figure BDA0003136380030000035
Observation equation of space, n X 、n Z Representing a state variable dimension, a measured sequence dimension, upsilon k 、e k Represents n υ Dimensional process noise, n e Dimensional observation noise, respectively mean value of
Figure BDA0003136380030000036
Standard covariance of ∑ υ 、Σ e The noise of the gaussian noise of (a),
Figure BDA0003136380030000037
denotes the K-th model parameter at the K-th 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, the space-time distribution of the observation noise is truncated, and the truncation prior distribution of the system is constructed as follows:
Figure BDA0003136380030000038
Figure BDA0003136380030000039
in the formula (I), the compound is shown in the specification,
Figure BDA00031363800300000310
truncated prior distribution, X, representing the observed noise soft constraint construction k-1 Representing the state quantities of the non-linear gaussian system at time k-1,
Figure BDA00031363800300000311
denotes the k model parameter sequence from time 1 to time k-1, r 1:k Representing a sequence of potential characteristic variables, Z, in a nonlinear Gaussian system from time 1 to time k 1:k An observation sequence representing a nonlinear Gaussian system from a 1 st time to a k-th time;
Figure BDA0003136380030000041
to assist in indicating the function, the value of which depends on the probability density of the observed noise in the observed model
Figure BDA0003136380030000042
Whether in feasible field I X (Z k ) 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
Figure BDA0003136380030000043
Figure BDA0003136380030000044
Thus, it is possible to prevent the occurrence of,
Figure BDA0003136380030000045
in the formula, p t (X) represents the truncated prior distribution after the nonlinear Gaussian system is corrected, X represents the state variable of a random system,
Figure BDA0003136380030000046
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;
Figure BDA0003136380030000047
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; x 0 Is an initial value of state, α * 、d i Respectively, 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:
Figure BDA0003136380030000048
in the formula (I), the compound is shown in the specification,
Figure BDA0003136380030000049
representing the posterior probability distribution of the target state in a nonlinear gaussian system,
Figure BDA00031363800300000410
representing the likelihood of observation in a nonlinear gaussian system,
Figure BDA00031363800300000411
representing the original prior distribution, X, in a non-linear Gaussian system 0:k Represents the state quantity, X, of the nonlinear Gaussian system from time 0 to time k 1:k-1 Represents the state quantity of the nonlinear Gaussian system from the 1 st moment to the k-1 st moment, Z 1:k-1 Represents the observed quantity of the nonlinear Gaussian system from the 1 st time to the k-1 st time, r 1:k-1 Representing 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:
Figure BDA0003136380030000051
the sub-optimal target importance function of the nonlinear Gaussian system is obtained by decomposing the mathematical expectation integral, i.e.
Figure BDA0003136380030000052
Comprises the following steps:
Figure BDA0003136380030000053
in the formula (I), the compound is shown in the specification,
Figure BDA0003136380030000054
representing a set of model parameters, K representing the number of model parameters,
Figure BDA0003136380030000055
representing model parameters M at time k κ The following observations were:
Figure BDA0003136380030000056
in the formula, H Ja A Jacobian matrix representing the nonlinear observation function, T representing the transpose of the matrix;
system-based suboptimal target importance function sampling N s Importance sample, i-th importance sample under k-th model parameter
Figure BDA0003136380030000057
And its weight
Figure BDA0003136380030000058
Respectively as follows:
Figure BDA0003136380030000059
Figure BDA00031363800300000510
the weights of the importance sample of the nonlinear gaussian system are normalized, namely:
Figure BDA00031363800300000511
in the formula (I), the compound is shown in the specification,
Figure BDA00031363800300000512
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:
Figure BDA0003136380030000061
in the formula (I), the compound is shown in the specification,
Figure BDA0003136380030000062
representing the nth integration point of the system at the kth model parameter at the kth instant,
Figure BDA0003136380030000063
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 order
Figure BDA0003136380030000064
Performing dot product to obtain Bayes sequential importance integral point weight as follows:
Figure BDA0003136380030000065
finally, the Bayesian sequential importance integral point is obtained as
Figure BDA0003136380030000066
In one embodiment, step 6 specifically includes:
variance of likelihood distribution of decomposition subregion:
Figure BDA0003136380030000067
in the formula (I), the compound is shown in the specification,
Figure BDA0003136380030000068
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:
Figure BDA0003136380030000069
in the formula (I), the compound is shown in the specification,
Figure BDA00031363800300000610
represents the integrated integration points of the multiple model sets in the nonlinear Gaussian system,
Figure BDA00031363800300000611
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:
Figure BDA00031363800300000612
Figure BDA00031363800300000613
in the formula (I), the compound is shown in the specification,
Figure BDA00031363800300000614
represents the nth integrated integration point of the nonlinear Gaussian system under the kth model parameter at the kth-1 moment,
Figure BDA00031363800300000615
representing the nth Bayes sequential importance integral point of the nonlinear Gaussian system at the k-1 moment and under the kth model parameter;
calculating the measurement prediction mean of the nonlinear Gaussian system:
Figure BDA00031363800300000616
the covariance of the measurement predictions is calculated:
Figure BDA0003136380030000071
calculating the system's covariance:
Figure BDA0003136380030000072
in the formula (I), the compound is shown in the specification,
Figure BDA0003136380030000073
which represents the mean of the prediction of the measurement,
Figure BDA0003136380030000074
which represents the covariance of the measurement prediction,
Figure BDA0003136380030000075
which represents the sequence of observations of the system,
Figure BDA0003136380030000076
representing the cross covariance of the system.
In one embodiment, step 7 specifically includes:
computing kalman filter gain
Figure BDA0003136380030000077
Figure BDA0003136380030000078
Calculating a filtered mean
Figure BDA0003136380030000079
Figure BDA00031363800300000710
Calculating filter covariance
Figure BDA00031363800300000711
Figure BDA00031363800300000712
Fusing the posterior distribution of the multi-modal filtering output approximate nonlinear Gaussian system state:
Figure BDA00031363800300000713
Figure BDA00031363800300000714
in the formula (I), the compound is shown in the specification,
Figure BDA00031363800300000715
is the mean of the approximate posterior distribution of the nonlinear gaussian system states,
Figure BDA00031363800300000716
is the covariance of the approximate a posteriori distribution of the nonlinear gaussian system states,
Figure BDA00031363800300000717
for model parameters in nonlinear dynamic systems
Figure BDA00031363800300000718
The similarity measure factor of (1).
In one of the embodiments, the first and second parts of the device,
Figure BDA00031363800300000719
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:
Figure BDA00031363800300000720
the objective function is constructed according to the maximum relevant information entropy criterion as follows:
Figure BDA0003136380030000081
according to the Bayesian sequential importance integral-based Kalman filtering method, the importance function covering multimodal distribution is constructed by fusing constraint information such as truncation prior and 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 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 of a filtering error for sampling 10 samples in example 1 of 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 present invention will be further described with reference to the accompanying drawings.
Detailed Description
The technical solutions in the embodiments of the present invention will be described clearly and completely 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 explicitly specified 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 1, establishing a discrete-time nonlinear Gaussian system (hereinafter referred to as a 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 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 as
Figure BDA0003136380030000101
Wherein 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:
Figure BDA0003136380030000102
Figure BDA0003136380030000103
in the formula (I), the compound is shown in the specification,
Figure BDA0003136380030000104
respectively representing time-of-k systems
Figure BDA0003136380030000105
The state model equation of the space,
Figure BDA0003136380030000106
Observation model equation of space, n X 、n Z Respectively representing a state variable dimension, a measured sequence dimension, upsilon k 、e k Respectively represent n υ Dimensional process noise, n e Dimensional observation noise, respectively mean value of
Figure BDA0003136380030000107
Standard covariance of ∑ υ 、Σ e The noise of the gaussian noise of (a),
Figure BDA0003136380030000108
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:
Figure BDA0003136380030000109
in the formula (I), the compound is shown in the specification,
Figure BDA00031363800300001010
one-step prediction of state variables in a representation system, X k-1 Representing the state quantity of the system at time k-1,
Figure BDA00031363800300001011
denotes the k model parameter, Z, at time k-1 1:k-1 Representing the observation sequence from time 1 to time k,
Figure BDA0003136380030000111
a one-step predicted gaussian distribution representing state variables in the system,
Figure BDA0003136380030000112
the mean value of the states is represented,
Figure BDA0003136380030000113
representing the state variance, defined as:
Figure BDA0003136380030000114
Figure BDA0003136380030000115
wherein the superscript T represents the transpose operation of the matrix,
Figure BDA0003136380030000116
jacobian matrix of nonlinear observation functions, denoted as
Figure BDA0003136380030000117
Similarly, the probability density function of the observation likelihood in the discrete time system can be deduced
Figure BDA0003136380030000118
Comprises the following steps:
Figure BDA0003136380030000119
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:
Figure BDA00031363800300001110
Figure BDA00031363800300001111
in the formula (I), the compound is shown in the specification,
Figure BDA00031363800300001112
representing a truncated prior distribution constructed based on soft constraints of observed noise,
Figure BDA00031363800300001113
sequence representing the k-th model parameter from time 1 to time k-1, r 1:k Representing a sequence of potential characteristic variables in the system from the 1 st moment to the k-th moment;
Figure BDA0003136380030000121
to assist in indicating the function, the value of which depends on the probability density of the observed noise in the observation model
Figure BDA0003136380030000122
Whether in feasible field I X (Z k ) 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 function
Figure BDA0003136380030000123
Are present. Thus, the initial value X of the state of the system 0 Distributed in feasible region I X (Z k ) Thereby approximating the center of the real target state distribution by a heuristic optimization method
Figure BDA0003136380030000124
In 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:
Figure BDA0003136380030000125
wherein alpha is * 、d i Respectively 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
Figure BDA0003136380030000126
Thus, the modified truncated prior distribution in the system is obtained as follows:
Figure BDA0003136380030000127
in the formula, p t (X) represents the truncated prior distribution of the non-linear Gaussian system after correction, as in the previous paragraph
Figure BDA0003136380030000128
X represents a state variable of a stochastic system,
Figure BDA0003136380030000129
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;
Figure BDA00031363800300001210
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:
Figure BDA0003136380030000131
in the formula (I), the compound is shown in the specification,
Figure BDA0003136380030000132
posterior probability distribution, p, representing target states in a non-linear Gaussian system lik (. represents the observed likelihood of the system, p t Denotes the truncated prior distribution of the system, p o () represents the original a priori distribution of the system,
Figure BDA0003136380030000133
probability density function of observed noise, X, representing a model of the system 0:k Representing the sequence of state quantities of the system from time 0 to time k, Z 1:k-1 Represents the observation sequence of the system from time 1 to time k-1, X 1:k-1 Representing the sequence of state quantities of the nonlinear Gaussian system from the 1 st to the k-1 st instants, r 1:k-1 Representing 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 solve by closed analysis, so that the p (X | Z) distribution can not be directly obtained from the target posterior distribution in general 1:k ) The sample is 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 state 1:k ) Sampling N according to the target importance function s The 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:
Figure BDA0003136380030000141
from the above equation, p (X | Z) is a posterior probability density function regardless of the target state 1:k ) Whether it is non-zero, the target importance function is non-zero, and pi (X | Z) 1:k )≥p(X|Z 1:k ). Assuming a known observation likelihood p lik (Z 1:k |X k ) And state prior p (X) 0:k ) The expression for sampling the weight samples according to the target importance function is:
X i ～π(X|Z 1:k ),i＝1,…,N s
state prediction f (X) i ) Is the importance sample X i The normalized weight of (a) is expressed as:
Figure BDA0003136380030000142
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:
Figure BDA0003136380030000143
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:
Figure BDA0003136380030000151
in the formula (I), the compound is shown in the specification,
Figure BDA0003136380030000152
a suboptimal target importance function for a nonlinear gaussian system is:
Figure BDA0003136380030000153
in the formula (I), the compound is shown in the specification,
Figure BDA0003136380030000154
a set of model parameters is represented which,
Figure BDA0003136380030000155
representing model parameters M at the k-th time κ The following observations are:
Figure BDA0003136380030000156
in the formula, H Ja Jacobian matrices representing non-linear observation functions, as in the preceding
Figure BDA0003136380030000157
Sampling samples based on suboptimal target importance function, sampling sample of ith importance under kappa model parameter
Figure BDA0003136380030000158
And its weight
Figure BDA0003136380030000159
Comprises the following steps:
Figure BDA00031363800300001510
Figure BDA00031363800300001511
in the formula, N s For the number of importance samples, the weights of the importance samples of the non-linear gaussian system are normalized, i.e.:
Figure BDA00031363800300001512
in the formula (I), the compound is shown in the specification,
Figure BDA00031363800300001513
representing samples of importance of a normalized nonlinear gaussian systemAnd (4) weighting.
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:
Figure BDA0003136380030000161
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 embodiment l Root of (X), i.e. xi i I 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:
Figure BDA0003136380030000162
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 as
Figure BDA0003136380030000163
Wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003136380030000164
representing the nth integration point of the system at the kth model parameter at the kth instant,
Figure BDA0003136380030000165
the weight of the system at the nth integration point at the kth model parameter at the kth time is shown. In general, a nonlinear function model of a dynamic system can be constructed according to engineering practice and experimental reasoning, but modeling errors due to limitations of human knowledge and uncertainty of human subjective experience can causeThe dynamic system state is unpredictable, so that the system observation likelihood is inconsistent with the posterior distribution of the 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 polynomial
Figure BDA0003136380030000166
Performing dot product to obtain Bayes sequential importance integral point weight as follows:
Figure BDA0003136380030000167
finally obtaining Bayes sequential importance integral points as
Figure BDA0003136380030000168
Namely, the initialization parameter setting of the bayesian sequential importance integral filtering method in the embodiment is completed, and then the subsequent target posterior distribution under the multi-mode parameters is predicted, updated and fused by the filtering method. The method specifically comprises the following steps:
first, the variance of the subregion likelihood distributions is decomposed:
Figure BDA0003136380030000171
in the formula (I), the compound is shown in the specification,
Figure BDA0003136380030000172
the variance of the likelihood distribution of the sub-region is expressed in the system under the k-1 th model parameterTruncating the mean square error of the prior distribution;
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:
Figure BDA0003136380030000173
in the formula (I), the compound is shown in the specification,
Figure BDA0003136380030000174
represents the nth integrated integration point of the nonlinear Gaussian system under the kth model parameter at the kth-1 moment,
Figure BDA0003136380030000175
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:
Figure BDA0003136380030000176
Figure BDA0003136380030000177
in the formula (I), the compound is shown in the specification,
Figure BDA0003136380030000178
is the average of the prediction of the states,
Figure BDA0003136380030000179
covariance predicted for the state;
calculating the measurement prediction mean, covariance and cross covariance of the nonlinear Gaussian system:
Figure BDA00031363800300001710
Figure BDA00031363800300001711
Figure BDA00031363800300001712
in the formula (I), the compound is shown in the specification,
Figure BDA00031363800300001713
represents the mean of the predictions of the measurements,
Figure BDA00031363800300001714
which represents the covariance of the measurement prediction,
Figure BDA00031363800300001715
which represents the sequence of observations of the system,
Figure BDA00031363800300001716
representing the cross-covariance of the system;
computing kalman filter gain
Figure BDA00031363800300001717
Figure BDA00031363800300001718
Calculating a filtered mean value
Figure BDA00031363800300001719
Figure BDA00031363800300001720
Calculating the covariance of the filter
Figure BDA00031363800300001721
Figure BDA0003136380030000181
And (3) fusing the multi-mode filtering output approximation to obtain the approximate posterior distribution of the nonlinear Gaussian system state:
Figure BDA0003136380030000182
Figure BDA0003136380030000183
in the formula (I), the compound is shown in the specification,
Figure BDA0003136380030000184
is the mean of the approximate posterior distribution of the nonlinear gaussian system states,
Figure BDA0003136380030000185
is the covariance of the approximate a posteriori distribution of the nonlinear gaussian system states,
Figure BDA0003136380030000186
for model parameters in nonlinear dynamic systems
Figure BDA0003136380030000187
The 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:
Figure BDA0003136380030000188
the objective function is constructed according to the maximum relevant information entropy criterion as follows:
Figure BDA0003136380030000189
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 parameter
Figure BDA00031363800300001810
And measuring
Figure BDA00031363800300001811
There 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, namely
Figure BDA00031363800300001812
The 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 point
Figure BDA00031363800300001813
Still in a certain euclidean real space, the method in this embodiment therefore converges in the bayesian state model space.
Theoretically, the numerical approximation error and the state dimension n of the Monte Carlo importance sampling method X Independently, 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. Thus, when the state dimension of the discrete nonlinear dynamical system is determined, the multiple modelsGaussian mixture importance function under parameters
Figure BDA0003136380030000191
Is calculated by
Figure BDA0003136380030000192
Fall to
Figure BDA0003136380030000193
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 the algorithms of Extended Kalman Filtering (EKF), Unscented Kalman Filtering (UKF), traditional particle filtering (GPF), extended Kalman particle filtering (EPF) and unscented Kalman filtering (UPF) are compared.
The state equation and the observation equation model of the one-dimensional univariate non-stationary growth model are respectively
Figure BDA0003136380030000194
Figure BDA0003136380030000195
Where the coefficients are known constants, a is 0.5, b is 2.5, c is 8,
Figure BDA0003136380030000196
initial value of state X 0 In [0,1 ]]The average value in the range. Upsilon is k Is subject to a parameter of [2,3]Is detected by the gamma distribution of (1). e.g. of the type k Is the observed noise with a mean of 0 and a variance of 0.5.
Fig. 2-5 are schematic diagrams of filtering errors for samples of 10, 200, 300, and 500, 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 shows the filtering performance of the numerical method based on monte carlo sampling compared alone when the number of samples is increased 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 for 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 experiment data of the airspace target are as follows: one is located at [0m,0m ]] T Is observed in a two-dimensional space [0km,30km ]]×[0km,12km]In-flight airspace targets. The target state variable consists of a position,Speed and turn rate composition, expressed as
Figure BDA0003136380030000201
Wherein omega k,t The 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:
Figure BDA0003136380030000202
Figure BDA0003136380030000203
wherein T is sampling time interval of radar, is taken as 1s, and standard deviation of process noise is sigma v ＝0.1km/s 2 ，σ ω ＝0.1rad/s 2 . The observation model of the radar is the same as that in the present embodiment, and the nonlinear measurement function is:
Figure BDA0003136380030000204
Figure BDA0003136380030000211
wherein (X) s ,Y s ) Is the position coordinate of radar, r and theta are radar distance measurement and angle measurement, respectively, and standard deviation of observation noise is sigma r 0.1km and σ θ ＝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
Figure BDA0003136380030000212
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 (7)

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 integral points according to a Gauss-Hermite rule, and correcting the Hermite integral points based on the importance sampling samples to obtain Bayes sequential importance integral 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;
step 8, taking the approximate posterior distribution as the original prior distribution of the next moment, and carrying out the system state filtering of the next round;
in step 1, the discrete-time nonlinear gaussian system specifically includes:
Figure FDA0003679360850000011
Figure FDA0003679360850000012
in the formula (I), the compound is shown in the specification,
Figure FDA0003679360850000013
representing a non-linear Gaussian system at time k
Figure FDA0003679360850000014
The state equation of the space,
Figure FDA0003679360850000015
Observation equation of space, n X 、n Z Representing a state variable dimension, a measured sequence dimension, upsilon k 、e k Represents n υ Dimensional process noise, n e Dimensional observation noise, respectively mean value
Figure FDA0003679360850000016
Standard covariance of ∑ υ 、Σ e The noise of the gaussian noise of (a) is,
Figure FDA0003679360850000017
represents the kth model parameter at the kth time, and K represents the total number of model parameters;
in step 3, the posterior probability distribution of the target state in the nonlinear gaussian system can be expressed as:
Figure FDA0003679360850000018
in the formula (I), the compound is shown in the specification,
Figure FDA0003679360850000019
representing the posterior probability distribution of the target state in a nonlinear gaussian system,
Figure FDA0003679360850000021
representing the likelihood of observation in a nonlinear gaussian system,
Figure FDA0003679360850000022
representing the original prior distribution, X, in a non-linear Gaussian system 0:k Represents the state quantity, X, of the nonlinear Gaussian system from time 0 to time k 1:k-1 Represents the state quantity of the nonlinear Gaussian system from the 1 st moment to the k-1 st moment, Z 1:k-1 Represents the observed quantity, r, of the nonlinear Gaussian system from the 1 st time to the k-1 st time 1:k-1 Representing potential characteristic variables in the nonlinear gaussian system from time 1 to time k-1,
Figure FDA0003679360850000023
denotes the k model parameter sequence from time 1 to time k-1, Z 1:k Representing a sequence of observations of a nonlinear Gaussian system from time 1 to time k, r 1:k Representing a sequence of potential characteristic variables, X, in a nonlinear Gaussian system from time 1 to time k k-1 Representing the state quantities of the nonlinear gaussian system at time k-1.
2. The bayesian sequential importance integral-based kalman filtering method according to claim 1, 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:
Figure FDA0003679360850000024
Figure FDA0003679360850000025
in the formula (I), the compound is shown in the specification,
Figure FDA0003679360850000026
representing a truncated prior distribution constructed by observing noise soft constraints;
Figure FDA0003679360850000027
to assist in indicating the function, the value of which depends on the probability density of the observed noise in the observation model
Figure FDA0003679360850000028
Whether in feasible field I X (Z k ) 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
Figure FDA0003679360850000029
Figure FDA00036793608500000210
Thus, it is possible to prevent the occurrence of,
Figure FDA00036793608500000211
in the formula, p t (X) represents the truncated prior distribution after the nonlinear Gaussian system is corrected, X represents the state variable of a random system,
Figure FDA00036793608500000212
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;
Figure FDA00036793608500000213
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; x 0 Is an initial value of state, α * 、d i Respectively, the iteration step length and the search direction of the backtracking method.
3. The Bayesian sequential importance integral-based Kalman filtering method according to claim 2, 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:
Figure FDA0003679360850000031
the sub-optimal target importance function of the nonlinear Gaussian system is obtained by decomposing the mathematical expectation integral, i.e.
Figure FDA0003679360850000032
Comprises the following steps:
Figure FDA0003679360850000033
in the formula，
Figure FDA0003679360850000034
Representing a set of model parameters, K representing the number of model parameters,
Figure FDA0003679360850000035
representing model parameters M at time k κ The following observations were:
Figure FDA0003679360850000036
in the formula, H Ja A Jacobian matrix representing the nonlinear observation function, T representing the transpose of the matrix;
system-based suboptimal target importance function sampling N s Importance sample, i-th importance sample under the k model parameter
Figure FDA0003679360850000037
And their weights
Figure FDA0003679360850000038
Respectively as follows:
Figure FDA0003679360850000039
Figure FDA00036793608500000310
normalizing the weight of the importance sampling sample of the nonlinear Gaussian system, namely:
Figure FDA00036793608500000311
in the formula (I), the compound is shown in the specification,
Figure FDA0003679360850000041
the weights of the importance sample of the normalized nonlinear gaussian system are represented.
4. The Bayesian sequential importance integral-based Kalman filtering method according to claim 3, wherein the step 5 specifically comprises:
since the characteristic root of the constructed Hermite polynomial of the order I is determined, the Hermite integral point under the nonlinear Gaussian system model parameters is selected according to Gauss-Hermite rule and determined, and the weight of the Hermite integral point is constant and is expressed as follows:
Figure FDA0003679360850000042
in the formula (I), the compound is shown in the specification,
Figure FDA0003679360850000043
representing the nth integration point of the system at the kth model parameter at the kth instant,
Figure FDA0003679360850000044
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 order
Figure FDA0003679360850000045
Performing dot product to obtain Bayes sequential importance integral point weight as follows:
Figure FDA0003679360850000046
finally obtaining Bayes sequential importance integral points as
Figure FDA0003679360850000047
5. The Bayesian sequential importance integral-based Kalman filtering method according to claim 4, wherein the step 6 specifically comprises:
variance of likelihood distribution of decomposition subregion:
Figure FDA0003679360850000048
in the formula (I), the compound is shown in the specification,
Figure FDA0003679360850000049
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:
Figure FDA00036793608500000410
in the formula (I), the compound is shown in the specification,
Figure FDA00036793608500000411
represents the integration points of the multiple model sets in the nonlinear Gaussian system,
Figure FDA00036793608500000412
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:
Figure FDA00036793608500000413
Figure FDA00036793608500000414
in the formula (I), the compound is shown in the specification,
Figure FDA0003679360850000051
represents the nth integrated integration point of the nonlinear Gaussian system under the kth model parameter at the kth-1 moment,
Figure FDA0003679360850000052
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:
Figure FDA0003679360850000053
the covariance of the measurement predictions is calculated:
Figure FDA0003679360850000054
computing the cross-covariance of the system:
Figure FDA0003679360850000055
in the formula (I), the compound is shown in the specification,
Figure FDA0003679360850000056
represents the mean of the predictions of the measurements,
Figure FDA0003679360850000057
which represents the covariance of the measurement prediction,
Figure FDA0003679360850000058
which represents the sequence of observations of the system,
Figure FDA0003679360850000059
representing the cross-covariance of the system.
6. The Bayesian sequential importance integral-based Kalman filtering method according to claim 5, wherein the step 7 specifically comprises:
computing Kalman filter gains
Figure FDA00036793608500000510
Figure FDA00036793608500000511
Calculating a filtered mean
Figure FDA00036793608500000512
Figure FDA00036793608500000513
Calculating filter covariance
Figure FDA00036793608500000514
Figure FDA00036793608500000515
Fusing the posterior distribution of the multi-modal filtering output approximate nonlinear Gaussian system state:
Figure FDA00036793608500000516
Figure FDA00036793608500000517
in the formula (I), the compound is shown in the specification,
Figure FDA00036793608500000518
is the mean of the approximate posterior distribution of the nonlinear gaussian system states,
Figure FDA00036793608500000519
is the covariance of the approximate a posteriori distribution of the nonlinear gaussian system states,
Figure FDA00036793608500000520
for model parameters in nonlinear dynamic systems
Figure FDA00036793608500000521
The similarity measure factor of (c).
7. The Bayesian sequential importance integral-based Kalman filtering method of claim 6,
Figure FDA00036793608500000522
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:
Figure FDA0003679360850000061
the objective function is constructed according to the maximum relevant information entropy criterion as follows:
Figure FDA0003679360850000062
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