CN107421543B - Implicit function measurement model filtering method based on state dimension expansion - Google Patents

Implicit function measurement model filtering method based on state dimension expansion Download PDF

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CN107421543B
CN107421543B CN201710478805.0A CN201710478805A CN107421543B CN 107421543 B CN107421543 B CN 107421543B CN 201710478805 A CN201710478805 A CN 201710478805A CN 107421543 B CN107421543 B CN 107421543B
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state quantity
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宁晓琳
孙晓函
吴伟仁
房建成
刘刚
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Beihang University
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    • G01—MEASURING; TESTING
    • G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C21/00—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00
    • G01C21/20—Instruments for performing navigational calculations

Abstract

The invention relates to a hidden function measurement model filtering method based on state dimension expansion, namely IAUKF. In this method, the quantity measurement is extended into the state quantity, while the zero vector is treated as an equivalent quantity measurement for filter updating. The IAUKF achieves better estimation performance than the IAEKF and IEKF. Especially when the measurement noise increases, the performance can be greatly improved compared with the implicit UKF.

Description

Implicit function measurement model filtering method based on state dimension expansion
Technical Field
The invention belongs to the field of autonomous navigation of spacecrafts, and relates to a filtering method of an implicit function measurement model of state dimension expansion.
Background
Kalman in 1960 proposed a linear optimal recursive filtering method, Kalman Filter (KF). Initially, KF is only applicable to linear systems, and as the demand for filtering nonlinear systems expands, filtering methods such as Extended Kalman Filter (EKF), Unscented Kalman Filter (uvf), Particle Filter (PF), etc. are gradually proposed and continuously developed. The measurement models in the classical kalman filtering algorithm all have explicit expressions, however, in many practical problems, constraints of state quantity and measurement quantity are often implicit, and explicit measurement models are not easy or cannot be obtained, which is an implicit measurement model filtering problem.
There are two main types of methods for solving the state estimation problem including the implicit measurement model at home and abroad. The first type is the IEKF proposed by sotto et al, obtained by linearizing an Implicit measurement equation at a reference point and taking a second order form, in combination with the conventional EKF algorithm with an explicit measurement equation, through an Implicit Extended Kalman Filter (IEKF). The second type is a filtering method of an implicit measurement model containing Iterative measurement updating, namely Iterative IEKF, which is provided by Steffen on the basis of analyzing the IEKF method. Both the methods are established on the basis of EKF, a Jacobian matrix needs to be calculated during application, linearization errors can reduce the precision of a filtering algorithm, divergence of filtering results can be caused, and the calculation of the Jacobian matrix is usually complex.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the method overcomes the defects of the prior art, provides a hidden function measurement model filtering method based on state dimension expansion, and obtains better estimation performance compared with IAEKF and IEKF. Especially when the measurement noise increases, the estimation performance can be greatly improved compared with the implicit UKF.
The invention provides a hidden function measurement model filtering method based on state dimension expansion, namely IAUKF. In the method, the quantity measurement is extended into the state quantity, and meanwhile, the zero vector is regarded as the equivalent quantity measurement to perform filtering updating, and the method specifically comprises the following steps:
first, the true quantity of k time is measured
Figure BDA0001328745700000021
Extending to state quantity XkIn the structure of the expanded state quantity
Figure BDA0001328745700000022
And
Figure BDA0001328745700000023
establishing a system model meeting the dimension expansion state quantity;
and secondly, initializing, solving the state quantity of the initial moment according to the system model, and substituting the state quantity of the initial moment and the covariance matrix thereof into the expanded state quantity constructed in the first step
Figure BDA0001328745700000024
And
Figure BDA0001328745700000025
in each case are denoted by
Figure BDA0001328745700000026
And
Figure BDA0001328745700000027
thirdly, time updating, namely calculating the predicted state according to the expanded state quantity of the initial time and the covariance matrix obtained in the second step
Figure BDA0001328745700000028
And error covariance matrix thereof
Figure BDA0001328745700000029
At the rest of the time, in the state of calculating the prediction
Figure BDA00013287457000000210
And error covariance matrix thereof
Figure BDA00013287457000000211
Then, the state quantity obtained in the fourth step is updated according to the measurement
Figure BDA00013287457000000212
And corresponding error covariance matrix
Figure BDA00013287457000000213
To obtain;
fourthly, measuring and updating, and solving the updated state estimation according to the state prediction value obtained by time updating and solving
Figure BDA00013287457000000214
And corresponding error covariance matrix
Figure BDA00013287457000000215
And then returning to the third step, and realizing loop solution until the filtering is finished.
The first step is to construct an expanded state quantity, and establish a system model meeting the dimension expansion state quantity as follows:
Figure BDA00013287457000000216
wherein, XkAnd ZkRespectively the state quantity and the quantity measurement at the time k,
Figure BDA00013287457000000217
the state quantity after the expansion is represented,
Figure BDA00013287457000000218
indicating measurement of Z from actual quantitykAnd the measurement noise vkMeasuring the true quantity obtained by mixing;
Figure BDA00013287457000000219
in the formula, Fa(. a) and
Figure BDA00013287457000000220
a state model representing the extended state quantities and their state errors, respectively, h (-) is a non-linear explicit function.
In the second step, the state quantity after the dimension expansion
Figure BDA00013287457000000221
And covariance matrix thereof
Figure BDA00013287457000000222
The initialization is as follows:
Figure BDA0001328745700000031
Figure BDA0001328745700000032
wherein the content of the first and second substances,
Figure BDA0001328745700000033
E[x]denotes the expected value of x, Z1And N1And n and m respectively represent the dimensionality of the state vector and the measurement vector.
In the third step, the time is updated as follows:
Figure BDA0001328745700000034
Figure BDA0001328745700000035
in the formula (I), the compound is shown in the specification,
Figure BDA0001328745700000036
is satisfied with a mean value of
Figure BDA0001328745700000037
Covariance of
Figure BDA0001328745700000038
Point of (a), wiIs the weight of the ith Sigma point,
Figure BDA0001328745700000039
is composed of
Figure BDA00013287457000000310
The covariance matrix of (a), wherein,
Figure BDA00013287457000000311
Figure BDA00013287457000000312
Figure BDA00013287457000000313
in the formula, naA dimension representing the state quantity after expansion, tau is a scaling parameter,
Figure BDA00013287457000000314
represents the square root of the matrix
Figure BDA00013287457000000315
The ith dimensional column vector of (1).
In the fourth step, the measurement is updated as follows:
Figure BDA00013287457000000316
Figure BDA00013287457000000317
Figure BDA00013287457000000318
wherein the content of the first and second substances,
Figure BDA00013287457000000319
for filtering the gain matrix, Ykk-1In order to measure the predicted value of the measurement,
Figure BDA00013287457000000320
Figure BDA00013287457000000321
respectively corresponding error covariance matrices.
Compared with the prior art, the invention has the advantages that:
(1) according to the invention, the state quantity and the real quantity measurement are expanded into a new state quantity, and meanwhile, the zero vector is regarded as an equivalent quantity measurement to carry out filtering updating, so that better estimation performance is obtained compared with IAEKF and IEKF, and the estimation precision is improved; especially when the measurement noise increases, the performance can be greatly improved compared with the implicit UKF.
(2) The method omits the process of solving the Jacobian matrix in the existing method, and reduces the calculated amount.
Drawings
FIG. 1 is a flow chart of a filtering method of a hidden function measurement model based on state dimension expansion according to the present invention;
Detailed Description
Fig. 1 shows a flow chart of a filtering method of an implicit function measurement model based on state dimension expansion. The following detailed description of the embodiments of the present invention:
common nonlinear systems all contain explicit metrology models, and such systems can be described as:
Figure BDA0001328745700000041
in the formula, the state equation f (-) and the measurement equation h (-) are both non-linear explicit functions. XkAnd wkRespectively representing the state vector at time k and its noise, Xk+1Representing the state vector at time k + 1. ZkAnd vkRespectively representing the measurement vector at time k and its noise. In practical application, it can be considered that both the state vector and the measurement vector are affected by zero-mean and uncorrelated white gaussian noise, that is, the state quantity noise and the measurement noise respectively obey:
Figure BDA0001328745700000042
wherein Q iskAnd NkRespectively representing the covariance matrix corresponding to the state noise and the measurement noise, and the specific value is determined by engineering experience or system parameters.
However, in many practical problems, the state quantity and the actual quantity measurement are constrained in a measurement model in an implicit function form, and an explicit measurement equation is not easy or available, and such problems are filtering problems of the implicit measurement model. Such implicit metrology models can be described as a system as follows:
Figure BDA0001328745700000043
wherein the state quantity XkSum quantity measurement ZkThe constituent implicit function h (·) ═ 0 constraint.
First, to deal with the nonlinear system problem with implicit metrology models, the state quantities and true quantity measurements can be expanded into a new state quantity:
Figure BDA0001328745700000051
in the formula, the superscript a marks the state extension,
Figure BDA0001328745700000052
indicating measurement of Z from actual quantitykAnd the measurement noise vkThe actual amount of measurements made by the mixing is measured,
Figure BDA0001328745700000053
indicating the expanded state quantities.
Second, note h (X) in System (3)k,Zk+vk) Zero vector equal to m dimensions, so the zero vector can be regarded as an equivalent quantity measurement YkNamely:
Figure BDA0001328745700000054
due to expanded state quantity
Figure BDA0001328745700000055
Is derived from the previous state quantity XkAnd true quantity measurement
Figure BDA0001328745700000056
The structure is that the measurement model (5) can be rewritten as:
Figure BDA0001328745700000057
therefore, the system model for establishing the dimension expansion state quantity satisfaction is as follows:
Figure BDA0001328745700000058
in the formula, Fa(. a) and
Figure BDA0001328745700000059
state model representing the extended state quantities and its state errors, Fa(. a) and
Figure BDA00013287457000000510
the calculation formula of (a) is as follows:
Figure BDA00013287457000000511
wherein the content of the first and second substances,
Figure BDA00013287457000000512
the covariance matrix of (A) is defined as
Figure BDA00013287457000000513
The IAUKF method comprises the following concrete implementation steps:
1. initialization
Initial state estimation and corresponding error covariance matrix
Figure BDA00013287457000000514
And P0Respectively setting as follows:
Figure BDA0001328745700000061
in the formula, E [ x ]]Expected value, state error covariance matrix Q representing xkCovariance matrix N with measurement errorskThe corresponding values at the initial time are selected respectively.
Initial extended state quantity
Figure BDA0001328745700000062
Should be constructed according to equation (4), however, in practical applications, the true quantity is measured
Figure BDA0001328745700000063
Not available, the present invention measures Z with an actual quantity1Measurement in place of a true quantity
Figure BDA0001328745700000064
At the moment, a noise covariance matrix is introduced as N1Thereby, the state quantity after dimension expansion
Figure BDA0001328745700000065
And covariance matrix thereof
Figure BDA0001328745700000066
May be respectively configured as:
Figure BDA0001328745700000067
Figure BDA0001328745700000068
in the formula, Z1And N1And n and m respectively represent the dimensionality of the state vector and the measurement vector.
2. Time updating
At time k, the estimation value of the expansion state quantity obtained at the last time is needed first
Figure BDA0001328745700000069
The correction is performed according to equation (4). However, in practical applications, the true quantity is measured
Figure BDA00013287457000000610
Not available, we measured Z with the actual quantitykMeasurement in place of a true quantity
Figure BDA00013287457000000611
At this time, a measurement noise v is introducedkAnd its noise covariance matrix Nk. Therefore, the dimension expansion state quantity
Figure BDA00013287457000000612
And covariance matrix thereof
Figure BDA00013287457000000613
The correction can be made as follows:
Figure BDA00013287457000000614
Figure BDA00013287457000000615
in the formula (I), the compound is shown in the specification,
Figure BDA00013287457000000616
and Pk-1Are respectively shown to be included in
Figure BDA00013287457000000617
And
Figure BDA00013287457000000618
the unexpanded state quantity estimate in (1) and its error covariance matrix.
Likewise, the IAUKF method performs probability deduction based on UT transforms. Satisfy the mean value of
Figure BDA00013287457000000619
Covariance of
Figure BDA0001328745700000071
2n ofa+1 Sigma points are equivalent to
Figure BDA0001328745700000072
The Sigma points are propagated through the system model (7) to obtain corresponding propagated Sigma points, and the propagated Sigma points can be used for calculating the predicted state
Figure BDA0001328745700000073
And error covariance matrix thereof
Figure BDA0001328745700000074
This particular set of Sigma is determined according to the following formula:
Figure BDA0001328745700000075
in the formula, naA dimension representing the state quantity after expansion, numerically equal to n + m, τ being a scaling parameter,
Figure BDA0001328745700000076
represents the square root of the matrix
Figure BDA0001328745700000077
Of the ith-dimensional column vector, wiIs the weight of the ith Sigma point.
The Sigma point-by-state model passes as:
Figure BDA0001328745700000078
the state prediction value and its error covariance matrix can be calculated as follows:
Figure BDA0001328745700000079
Figure BDA00013287457000000710
3. measurement update
According to equation (6), the Sigma point of the estimator measurement can be calculated according to the following equation:
Figure BDA00013287457000000711
due to the prediction of the Sigma points obtained according to equation (16)Value of
Figure BDA00013287457000000712
Not the true state at time k, contains errors. Therefore, equation (18) is also not equal to its true value of 0, which provides information about the predicted state
Figure BDA00013287457000000713
Can be used for state correction.
Thus, the predicted value of the measurement can be calculated as:
Figure BDA00013287457000000714
the corresponding error covariance matrix can be obtained by:
Figure BDA0001328745700000081
then, the gain matrix is filtered
Figure BDA0001328745700000083
Updated state estimation
Figure BDA0001328745700000084
And corresponding error covariance matrix
Figure BDA0001328745700000085
Can be obtained by calculation according to a UKF method respectively:
Figure BDA0001328745700000086
Figure BDA0001328745700000087
Figure BDA0001328745700000088
table 1 and Table 2 show the comparison of the navigation results of the IAUKF, the IEKF, the IAEKF, and the implicit UKF.
TABLE 1 navigation results of four filtering methods
Figure BDA0001328745700000089
TABLE 2 navigation results of four filtering methods under different measurement noises
Figure BDA00013287457000000810
Figure BDA0001328745700000091
Table 1 compares the filtering results of the four filtering methods when the equivalent measurement noise is relatively small and is 1'; table 2 compares the filtering results of the four filtering methods with different measured noise. It can be seen that: when the measurement noise is small, compared with the IAEKF and the IEKF, the estimation precision of the position of the IAUKF is greatly improved; when the measurement noise is gradually increased, the estimation performance of the IAUKF can be greatly improved compared with the implicit UKF.
Those skilled in the art will appreciate that the invention may be practiced without these specific details.

Claims (5)

1. A hidden function measurement model filtering method based on state dimension expansion is used for autonomous navigation of a spacecraft and is characterized in that a state model is constructed by taking the position and the speed of the spacecraft as state quantities, and the method comprises the following steps:
first, the true quantity of k time is measured
Figure FDA0002450419860000011
Extending to state quantity XkIn the structure of the expanded state quantity
Figure FDA0002450419860000012
And
Figure FDA0002450419860000013
establishing a system model meeting the dimension expansion state quantity;
and secondly, initializing, solving the state quantity of the initial moment according to the system model, and substituting the state quantity of the initial moment and the covariance matrix thereof into the expanded state quantity constructed in the first step
Figure FDA0002450419860000014
And
Figure FDA0002450419860000015
in each case are denoted by
Figure FDA0002450419860000016
And
Figure FDA0002450419860000017
thirdly, time updating, namely calculating the predicted state according to the expanded state quantity of the initial time and the covariance matrix obtained in the second step
Figure FDA0002450419860000018
And error covariance matrix thereof
Figure FDA0002450419860000019
At the rest of the time, in the state of calculating the prediction
Figure FDA00024504198600000110
And error covariance matrix thereof
Figure FDA00024504198600000111
Then, the state quantity obtained in the fourth step is updated according to the measurement
Figure FDA00024504198600000112
And corresponding error covariance matrix
Figure FDA00024504198600000113
To obtain;
fourthly, measuring and updating, and solving the updated state estimation according to the state prediction value obtained by time updating and solving
Figure FDA00024504198600000114
And corresponding error covariance matrix
Figure FDA00024504198600000115
And then returning to the third step, and realizing loop solution until the filtering is finished.
2. The method of claim 1, wherein the filtering method comprises: the first step is to construct an expanded state quantity, and establish a system model meeting the dimension expansion state quantity as follows:
Figure FDA00024504198600000116
wherein, XkAnd ZkRespectively the state quantity and the quantity measurement at the time k,
Figure FDA00024504198600000117
the state quantity after the expansion is represented,
Figure FDA00024504198600000118
indicating measurement of Z from actual quantitykAnd the measurement noise vkMeasuring the true quantity obtained by mixing;
Figure FDA00024504198600000119
in the formula, Fa(. a) and
Figure FDA00024504198600000120
a state model representing the extended state quantities and their state errors, respectively, h (-) is a non-linear explicit function.
3. The method of claim 1, wherein the filtering method comprises: in the second step, the state quantity after the dimension expansion
Figure FDA0002450419860000021
And covariance matrix thereof
Figure FDA0002450419860000022
The initialization is as follows:
Figure FDA0002450419860000023
Figure FDA0002450419860000024
wherein the content of the first and second substances,
Figure FDA0002450419860000025
E[x]denotes the expected value of x, Z1And N1And n and m respectively represent the dimensionality of the state vector and the measurement vector.
4. The method of claim 1, wherein the filtering method comprises: in the third step, the time is updated as follows:
Figure FDA0002450419860000026
Figure FDA0002450419860000027
in the formula (I), the compound is shown in the specification,
Figure FDA0002450419860000028
is satisfied with a mean value of
Figure FDA0002450419860000029
Covariance of
Figure FDA00024504198600000210
Point of (a), wiIs the weight of the ith Sigma point,
Figure FDA00024504198600000211
is composed of
Figure FDA00024504198600000212
The covariance matrix of (a), wherein,
Figure FDA00024504198600000213
Figure FDA00024504198600000214
Figure FDA00024504198600000215
in the formula, naA dimension representing the state quantity after expansion, tau is a scaling parameter,
Figure FDA00024504198600000216
represents the square root of the matrix
Figure FDA00024504198600000217
The ith dimensional column vector of (1).
5. The method of claim 1, wherein the filtering method comprises: in the fourth step, the measurement is updated as follows:
Figure FDA00024504198600000218
Figure FDA0002450419860000031
Figure FDA0002450419860000032
wherein the content of the first and second substances,
Figure FDA0002450419860000033
for filtering the gain matrix, Yk|k-1In order to measure the predicted value of the measurement,
Figure FDA0002450419860000034
Figure FDA0002450419860000035
respectively corresponding error covariance matrices.
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