CN107886058B - Noise-related two-stage volume Kalman filtering estimation method and system - Google Patents

Noise-related two-stage volume Kalman filtering estimation method and system Download PDF

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CN107886058B
CN107886058B CN201711045230.XA CN201711045230A CN107886058B CN 107886058 B CN107886058 B CN 107886058B CN 201711045230 A CN201711045230 A CN 201711045230A CN 107886058 B CN107886058 B CN 107886058B
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张露
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Quzhou University
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Abstract

The invention discloses a noise-related two-stage volume Kalman filtering estimation method and a system, wherein a noise-related system is transformed by adopting an identity deformation method to establish a system model; converting the system model from a noise-related system to a noise-unrelated system by adding a coefficient, and establishing a new system model; and then, carrying out recursive calculation on the noise parameters in the new system model and the two-stage filtering to obtain a noise-related two-stage volume Kalman filtering estimator. The two-stage volume information filtering algorithm related to noise provided by the invention eliminates a Jacobian matrix by utilizing the approximate relation between the product of cross covariance and error covariance and the Jacobian matrix, and ensures the application of the algorithm in a high-dimensional nonlinear system.

Description

Noise-related two-stage volume Kalman filtering estimation method and system
Technical Field
The present invention relates to the field, and in particular, to a noise-dependent two-stage volumetric Kalman filter estimation method and system.
Background
The precondition of various two-stage filtering algorithms derived at present is that a nonlinear Gaussian system is assumed to be noise-independent, that is, state equation noise and measurement equation noise are not related and are Gaussian white noise, which is the noise condition under an ideal state. In practical application, however, the noise correlation of the system generally exists, for example, the noise correlation occurs due to the dual influence of the internal components of the system and the external environment change; the system for measuring the colored noise is subjected to noise dimension expansion, and the original system can be converted into a noise related system after the system is expanded into a state; in systems requiring multi-sensor information fusion, such as moving target tracking, there are also a lot of noise-related situations. For noise correlation systems, the conventional solution is to ignore the correlated noise and use conventional two-stage volumetric Kalman filtering to estimate, which inevitably reduces the estimation accuracy. According to the scheme, the conversion coefficient matrix is introduced, the noise-related system is converted into the uncorrelated system, the conversion relation between the uncorrelated system and the uncorrelated system is obtained, estimation is carried out, the related noise is fully considered, and accurate tracking of the noise-related system is achieved.
The pure azimuth tracking system tracks the state of a moving target through two sensors to obtain nonlinear measured values, each sensor can only obtain an angle observation value of the target state, and the two angle observation values are marked as alphai，kAnd betai，kThe observations of the two angles form the position of the intersection in the plane coordinates. For two sensors S of a rectangular coordinate systemi1And Si2(i-1, 2, …, N) are fixed to the stage P, respectively1And P2And the distance between them is d. With a plurality of sensors fixed to the platform Pj(j { (S) } at (1, 2)1，j，Pj)，(S2，j，Pj)，…，(SN，j，Pj) Corresponding to a non-linear measurement value of { (α)1，k，β1，k)，(α2，k，β2，k)，…，(αN，k，βN，k)}。
The kinetic model is a four-dimensional nonlinear system, xk＝[x1，k x2，k y1，k y2，k]T, wherein x1，kAnd x2，kIs the displacement component in the east and north directions, y1，kAnd y2，kIs the velocity component relative to the displacement component, taking the movement of the target as the CV model, the equation of state and the variance of the deviation are as follows:
Figure BDA0001452110810000021
Figure BDA0001452110810000022
wherein
Figure BDA0001452110810000023
Process noise variance
Figure BDA0001452110810000024
The tracking period T is 1 s.
Observing the function according to the cross principle
Figure BDA0001452110810000025
In a multi-sensor system, the measurement equation is:
Figure BDA0001452110810000026
wherein h is1，k(xk)＝h2，k(xk)＝…＝hN，k(xk)＝hk(xk)。
Let ui，k＝ciωk，k-1Then there is
Figure BDA0001452110810000027
Figure BDA0001452110810000031
The initial state estimate and covariance matrix are:
Figure BDA0001452110810000032
Figure BDA0001452110810000033
Figure BDA0001452110810000034
the simulation time was 200 seconds, and 1000 Monte Carlo simulations were performed for both algorithms. The algorithm error is calculated using the root mean square error (RMSE error) as follows:
Figure BDA0001452110810000035
wherein M is the Monte Carlo number of times,
Figure BDA0001452110810000036
and
Figure BDA0001452110810000037
respectively represent x under the nth Monte Carlo simulation*State values and estimated values of.
In summary, the problems of the prior art are as follows: in the prior art, correlated noise is not analyzed, so that the estimation precision is reduced; the tracking result is poor; correlated noise is not considered in the estimation process, the correlated noise is considered according to the condition of the irrelevant noise, and the difficulty of the solution is that the correlated noise cannot be considered while the position of the aircraft is estimated; the filtering divergence phenomenon occurs, and further tracking estimation can not be carried out at all.
Disclosure of Invention
In order to solve the problems of the prior art, the embodiment of the invention provides a method. The invention can be used in the field of target tracking of single or multiple aircrafts, and is particularly explained by taking a pure azimuth tracking system as an example.
The invention is realized in such a way that a noise-related two-stage volume Kalman filtering estimation method comprises the following steps:
transforming the noise-related system by adopting an identity deformation method, and establishing a system model;
converting the system model from a noise-related system to a noise-unrelated system by adding a coefficient, and establishing a new system model;
and then, carrying out recursive calculation on the noise parameters in the new system model and the two-stage filtering to obtain a noise-related two-stage volume Kalman filtering estimator.
Further, the step of transforming the noise-related system by using the method of constant deformation and establishing the system model is specifically as follows:
the noise correlation system is a nonlinear gaussian system:
xk+1＝fk(xk)+ωk+1，k； (1)
zk＝hk(xk)+υk； (2)
where k is a discrete time series, xk∈Rn×1Is the state vector of the system, zk∈Rm×1Is a measurement vector, f (-) and h (-) are known nonlinear state transfer functions and measurement functions and are in xkProcess noise sequence omega of continuous micro processk+1，kAnd measuring the noise sequence upsilonkAre Gaussian white noise sequences with the mean value E (omega)k+1，k)＝qk，E(υk)＝rkVariance Qk+1，kAnd RkThe following conditions are satisfied:
Figure BDA0001452110810000041
initial statex0And omegak+1，k、υkIrrelevant, and satisfy:
Figure BDA0001452110810000042
further, the step of converting the system model from a noise-related system to a noise-unrelated system by adding a coefficient to establish a new system model specifically comprises:
converting the noise-related system into an uncorrelated system through constant deformation, and then performing filtering estimation;
from the model equation (2):
zk-hk(xk)-υk＝0；
let ΔkThe undetermined coefficients are as follows:
Δk(zk-hk(xk)-υk)＝0 (3)；
substituting the formula (1) and finishing to obtain:
Figure BDA0001452110810000043
wherein
Fk(xk)＝fk(xk)+Δk(zk-hk(xk)) (5)；
Figure BDA0001452110810000044
The models shown in equations (1) and (2) are converted into:
Figure BDA0001452110810000051
zk＝hk(xk)+υk (8)；
wherein
Figure BDA0001452110810000052
Converting a noise-related system to a noise-independent system, there are:
Figure BDA0001452110810000053
unfolding to obtain:
Figure BDA0001452110810000054
when equation (9) is satisfied, the noise-independent system process noise and the metrology noise are uncorrelated;
using a conversion model method to obtain a noise-related nonlinear Gaussian filter formula, and performing equal angular notation t expression;
Figure BDA0001452110810000055
Figure BDA0001452110810000056
Figure BDA0001452110810000057
Figure BDA0001452110810000058
Figure BDA0001452110810000059
further, the step of obtaining a noise-related two-stage volume Kalman filter estimator by using the recursive calculation of the noise parameter and the two-stage filtering in the new system model specifically comprises:
non-linear gaussian system with random bias:
Figure BDA0001452110810000061
where k is a discrete time series, xk∈Rn×1Is the state vector of the system, bk∈Rp×1Is the systematic deviation vector, zk∈Rm×1Is a measurement vector, fk(. and h)k(. is) a known nonlinear state transfer function and metrology function at xkProcess noise sequence
Figure BDA0001452110810000062
Offset noise sequence
Figure BDA0001452110810000063
And measuring the noise sequence upsilonkAre Gaussian white noise sequences, the deviation noise is uncorrelated with the process noise and the measurement noise, wherein the average value is
Figure BDA0001452110810000064
E(υk)＝rkVariance of
Figure BDA0001452110810000065
Figure BDA0001452110810000066
And RkThe following conditions are satisfied:
Figure BDA0001452110810000067
initial state x0、b0And omegak+1，k、υkIrrelevant, and satisfy:
Figure BDA0001452110810000068
Figure BDA0001452110810000069
Figure BDA00014521108100000610
order to
Figure BDA00014521108100000611
Hk(Xk)＝hk(xk)+Fkbk；
The system model given by equation (15) is rewritten as follows:
Xk+1＝Γk(Xk)+ωk
Zk＝Hk(Xk)+υk
wherein
Figure BDA00014521108100000612
Based on the identity transform, the model (15) becomes a noise-independent system as shown in equations (7) and (8), as shown in equation (17):
Figure BDA00014521108100000613
wherein Fk(Xk)＝Γk(Xk)+Δk(Zk-Hk(Xk))，
Figure BDA0001452110810000071
Initializing state conditions:
Figure BDA0001452110810000072
Figure BDA0001452110810000073
for k＝1，2，…，N do；
step one, time updating:
1) assuming a posteriori density function at known k-1 time
Figure BDA0001452110810000074
To Pk-1|k-1Cholesky decomposition is carried out to obtain
Figure BDA0001452110810000075
2) Calculating volume points
Figure BDA0001452110810000076
And propagation volume point
Figure BDA0001452110810000077
Wherein, i is 1, 2, m is 2nx；
3) Let mk-1＝qk-1-ΔkrkM is scaled according to the state vector dimension and the deviation vector dimensionkPartitioning is carried out, and then:
Figure BDA0001452110810000078
same pair rkThe partitioning is carried out as follows:
Figure BDA0001452110810000079
by mk-1Estimating noise dependent unbiased filter state prediction values
Figure BDA00014521108100000710
Sum biased filter state prediction
Figure BDA00014521108100000711
4) Order to
Figure BDA00014521108100000712
According to the dimension of the block matrix in the two-stage transformation formula, the method comprises the following steps of
Figure BDA0001452110810000081
Partitioning:
Figure BDA0001452110810000082
in the same way, order
Figure BDA0001452110810000083
Partitioning the state noise variance matrix:
Figure BDA0001452110810000084
by means of coupling relationships
Figure BDA0001452110810000085
Estimating noise-dependent unbiased filter state error covariance
Figure BDA0001452110810000086
Sum-biased filter state error covariance
Figure BDA0001452110810000087
Step two, measurement updating:
A) decomposition of
Figure BDA00014521108100000819
To obtain
Figure BDA0001452110810000088
B) Calculating volume points
Figure BDA0001452110810000089
And propagation volume points propagated through the measurement equation
Figure BDA00014521108100000810
Wherein, i is 1, 2.. times, m;
C) estimating noise-related metrology predictions
Figure BDA00014521108100000811
D) Estimating noise-related metrology error covariance
Figure BDA00014521108100000812
Cross covariance correlated with noise
Figure BDA00014521108100000813
E) Will be a formula
Figure BDA00014521108100000814
Partitioning according to corresponding dimensionality to obtain a partitioning gain matrix:
Figure BDA00014521108100000815
estimating noise dependent unbiased filter kalman gain
Figure BDA00014521108100000816
Noise dependent biased filter Kalman gain
Figure BDA00014521108100000817
F) By mk-1And rkComputing noise-dependent unbiased filter state estimate
Figure BDA00014521108100000818
State estimation of a biased filter in relation to noise
Figure BDA0001452110810000091
G) By means of
Figure BDA0001452110810000092
Calculating noise-correlated estimation error covariance of an unbiased filter
Figure BDA0001452110810000093
Noise-dependent biased filter estimation error covariance
Figure BDA0001452110810000094
And (6) ending.
It is another object of the present invention to provide a noise-dependent two-stage volumetric Kalman filter estimation system.
The technical scheme provided by the embodiment of the invention has the following beneficial effects: the invention provides a noise-related Two-stage volume Kalman filtering algorithm (Two-stage volume tube Filter with corrected noise, TSC KF-CN), which provides a Two-stage volume Kalman filtering algorithm of a transformation model based on a minimum variance estimation criterion, and converts a noise-related system into an uncorrelated system and obtains a conversion relation between the Two by introducing a conversion coefficient matrix.
In practical applications, noise-dependent nonlinear systems are very common, and if the correlated noise is not considered, the accuracy of the estimation is necessarily reduced by still applying the conventional two-stage filtering algorithm. The invention provides a two-stage volume Kalman filtering algorithm of a transformation model based on a minimum variance estimation criterion, a conversion coefficient matrix is introduced, a noise correlation system is converted into an uncorrelated system and a conversion relation between the uncorrelated system and the uncorrelated system is obtained, in practical application, the noise correlation is taken into consideration as a condition, the method is used in the field of target tracking of single or multiple aircrafts, the tracking precision is superior to the condition of neglecting noise correlation and counting, and a better tracking result is obtained.
In a noise-related nonlinear system, the estimated value of each time position of the TSCKF-CN algorithm is superior to a two-stage volume Kalman algorithm without considering the related noise, the estimation precision advantage is obvious, and in some times, the two-stage volume Kalman filtering does not consider the related noise, so that the filtering divergence phenomenon occurs, and further tracking estimation cannot be carried out at all.
Drawings
Fig. 1 is a flowchart of a noise-dependent two-stage volumetric Kalman filter estimation method according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, embodiments of the present invention will be described in detail with reference to the accompanying drawings.
In the prior art, correlated noise is not analyzed, so that the estimation precision is reduced; the tracking result is poor.
The present invention will be described in detail below with reference to the accompanying drawings.
As shown in fig. 1, the method for estimating a noise-related two-stage volume Kalman filter according to an embodiment of the present invention includes:
s101: transforming the noise-related system by adopting an identity deformation method, and establishing a system model;
s102: converting the system model from a noise-related system to a noise-unrelated system by adding a coefficient, and establishing a new system model;
s103: and then, carrying out recursive calculation on the noise parameters in the new system model and the two-stage filtering to obtain a noise-related two-stage volume Kalman filtering estimator.
The invention is further described with reference to specific examples.
Optionally, the step of transforming the noise-related system by using the method of identity deformation and establishing a system model specifically includes:
consider the following nonlinear gaussian system:
xk+1＝fk(xk)+ωk+1，k (4-1)
zk＝hk(xk)+υk (4-2)
where k is a discrete time series and where,
Figure BDA0001452110810000101
is the state vector of the system and,
Figure BDA0001452110810000102
is a measurement vector, f (-) and h (-) are known nonlinear state transfer functions and measurement functions and are in xkProcess noise sequence omega of continuous micro processk+1，kAnd measuring the noise sequence upsilonkAre Gaussian white noise sequences with the mean value E (omega)k+1，k)＝qk，E(υk)＝rkVariance Qk+1，kAnd RkThe following conditions are satisfied:
Figure BDA0001452110810000111
initial state x0And omegak+1，k、υkIrrelevant, and satisfy:
Figure BDA0001452110810000112
optionally, the step of adding a coefficient to convert the system model from a noise-related system to a noise-unrelated system, and the step of establishing a new system model specifically includes:
and transforming the model, converting the noise-related system into an uncorrelated system through constant deformation, and then performing filtering estimation.
From the model equation (4-2), it can be derived:
zk-hk(xk)-υk＝0
let ΔkThe undetermined coefficients are as follows:
Δk(zk-hk(xk)-υk)＝0 (4-3)
substituting into equation (4-1) and working up gives:
Figure BDA0001452110810000113
wherein
Fk(xk)＝fk(xk)+Δk(zk-hk(xk)) (4-5)；
Figure BDA0001452110810000114
The models shown in equations (4-1) and (4-2) are converted into:
Figure BDA0001452110810000115
zk＝hk(xk)+υk (4-8)；
wherein
Figure BDA0001452110810000116
Converting the model from a noise-related system to a noise-independent system, i.e. making the system process noise uncorrelated with the measurement noise, then:
Figure BDA0001452110810000117
unfolding to obtain:
Figure BDA0001452110810000121
namely, when the formula (4-9) is satisfied, the process noise and the measurement noise in the system model are not related any more, a nonlinear Gaussian filter algorithm can be used for calculation, and in order to distinguish the nonlinear Gaussian filter formula, the noise-related nonlinear Gaussian filter formula obtained by using a conversion model method is represented by a corner mark t.
Figure BDA0001452110810000122
Figure BDA0001452110810000123
Figure BDA0001452110810000124
Figure BDA0001452110810000125
Figure BDA0001452110810000126
The invention provides a noise-related Two-stage volume Information filtering estimation algorithm (TSCIF-CN), which eliminates a Jacobian matrix by utilizing the product of cross covariance and error covariance and the approximate relation of the Jacobian matrix, and ensures the application of the algorithm in a high-dimensional nonlinear system.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention are intended to be included within the scope of the present invention.

Claims (2)

1. A noise-correlated two-stage volumetric Kalman filter estimation method, operating in a pure azimuth tracking system onboard in target tracking of a single or multiple aircraft, the noise-correlated two-stage volumetric Kalman filter estimation method comprising:
transforming the noise-related system by adopting an identity deformation method, and establishing a system model;
converting the system model from a noise-related system to a noise-unrelated system by adding a coefficient, and establishing a new system model;
then, the recursive calculation of the noise parameters in the new system model and the two-stage filtering is carried out to obtain a two-stage volume Kalman filtering estimator related to the noise;
the steps of adopting the method of constant deformation to transform the noise-related system and establishing the system model are as follows:
the noise correlation system is a nonlinear gaussian system:
xk+1＝fk(xk)+ωk+1，k； (1)
zk＝hk(xk)+υk； (2)
where k is a discrete time series, xk∈Rn×1Is the state vector of the system, zk∈Rm×1Is a measurement vector, f (-) and h (-) are known nonlinear state transfer functions and measurement functions and are in xkProcess noise sequence omega of continuous micro processk+1，kAnd measuring the noise sequence upsilonkAre Gaussian white noise sequences with the mean value E (omega)k+1，k)＝qk，E(vk)＝rkVariance Qk+1，kAnd RkThe following conditions are satisfied:
Figure FDA0002863915340000011
initial state x0And omegak+1，k、υkIrrelevant, and satisfy:
Figure FDA0002863915340000012
the step of converting the system model from a noise-related system to a noise-unrelated system by adding the coefficient to establish a new system model specifically comprises the following steps:
converting the noise-related system into an uncorrelated system through constant deformation, and then performing filtering estimation;
from the model equation (2):
zk-hk(xk)-υk＝0；
let ΔkThe undetermined coefficients are as follows:
Δk(zk-hk(xk)-vk)＝0 (3)；
substituting the formula (1) and finishing to obtain:
Figure FDA0002863915340000021
wherein
Fk(xk)＝fk(xk)+Δk(zk-hk(xk)) (5)；
Figure FDA0002863915340000022
The models shown in equations (1) and (2) are converted into:
Figure FDA0002863915340000023
zk＝hk(xk)+υk (8)；
wherein
Figure FDA0002863915340000024
Converting a noise-related system to a noise-independent system, there are:
Figure FDA0002863915340000025
unfolding to obtain:
Figure FDA0002863915340000026
when equation (9) is satisfied, the noise-independent system process noise and the metrology noise are uncorrelated;
using a conversion model method to obtain a noise-related nonlinear Gaussian filter formula, and performing equal angular notation t expression;
Figure FDA0002863915340000027
Figure FDA0002863915340000028
Figure FDA0002863915340000031
Figure FDA0002863915340000032
Figure FDA0002863915340000033
Figure FDA0002863915340000034
then, the recursive calculation of the noise parameter and the two-stage filtering in the new system model is used to obtain a noise-related two-stage volume Kalman filtering estimator, which comprises the following steps:
step one, time updating:
1) assuming a posteriori density function at known k-1 time
Figure FDA0002863915340000035
To Pk-1|k-1Cholesky decomposition is carried out to obtain
Figure FDA0002863915340000036
2) Calculating volume points
Figure FDA0002863915340000037
And propagation volume point
Figure FDA0002863915340000038
Wherein, i is 1, 2, m is 2nx；
3) Let mk-1＝qk-1-ΔkrkM is scaled according to the state vector dimension and the deviation vector dimensionkPartitioning is carried out, and then:
Figure FDA0002863915340000039
same pair rkThe partitioning is carried out as follows:
Figure FDA00028639153400000310
by mk-1Estimating noise dependent unbiased filter state prediction values
Figure FDA0002863915340000041
Sum biased filter state prediction
Figure FDA0002863915340000042
4) Order to
Figure FDA0002863915340000043
According to the dimension of the block matrix in the two-stage transformation formula, the method comprises the following steps of
Figure FDA0002863915340000044
Partitioning:
Figure FDA0002863915340000045
in the same way, order
Figure FDA0002863915340000046
Partitioning the state noise variance matrix:
Figure FDA0002863915340000047
by means of coupling relationships
Figure FDA0002863915340000048
Estimating noise-dependent unbiased filter state error covariance
Figure FDA0002863915340000049
Sum-biased filter state error covariance
Figure FDA00028639153400000410
Step two, measurement updating:
A) decomposition of
Figure FDA00028639153400000411
To obtain
Figure FDA00028639153400000412
B) Calculating volume points
Figure FDA00028639153400000413
And propagation volume points propagated through the measurement equation
Figure FDA00028639153400000414
Wherein, i is 1, 2.. times, m;
C) estimating noise-related metrology predictions
Figure FDA00028639153400000415
D) Estimating noise-related metrology error covariance
Figure FDA00028639153400000416
Cross covariance correlated with noise
Figure FDA00028639153400000417
E) Will be a formula
Figure FDA00028639153400000418
Partitioning according to corresponding dimensionality to obtain a partitioning gain matrix:
Figure FDA0002863915340000051
estimating noise dependent unbiased filter kalman gain
Figure FDA0002863915340000052
Noise dependent biased filter Kalman gain
Figure FDA0002863915340000053
F) By mk-1And rkComputing noise-dependent unbiased filter state estimate
Figure FDA0002863915340000054
State estimation of a biased filter in relation to noise
Figure FDA0002863915340000055
G) By means of
Figure FDA0002863915340000056
Calculating noise-correlated estimation error covariance of an unbiased filter
Figure FDA0002863915340000057
Noise-dependent biased filter estimation error covariance
Figure FDA0002863915340000058
2. A noise-correlated two-stage volumetric Kalman filter estimation system of the noise-correlated two-stage volumetric Kalman filter estimation method according to claim 1.
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