CN113503879B - Dynamic adaptive Kalman filter method based on ensemble empirical mode decomposition - Google Patents

Dynamic adaptive Kalman filter method based on ensemble empirical mode decomposition Download PDF

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CN113503879B
CN113503879B CN202110779301.9A CN202110779301A CN113503879B CN 113503879 B CN113503879 B CN 113503879B CN 202110779301 A CN202110779301 A CN 202110779301A CN 113503879 B CN113503879 B CN 113503879B
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CN113503879A (en
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张京娟
周文翔
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Beihang University
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    • G—PHYSICS
    • 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
    • G—PHYSICS
    • 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
    • G01C21/10—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00 by using measurements of speed or acceleration
    • G01C21/12—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00 by using measurements of speed or acceleration executed aboard the object being navigated; Dead reckoning
    • G01C21/16—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00 by using measurements of speed or acceleration executed aboard the object being navigated; Dead reckoning by integrating acceleration or speed, i.e. inertial navigation
    • G01C21/165—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00 by using measurements of speed or acceleration executed aboard the object being navigated; Dead reckoning by integrating acceleration or speed, i.e. inertial navigation combined with non-inertial navigation instruments

Abstract

The invention discloses a dynamic adaptive Kalman filter method based on ensemble empirical mode decomposition, which is characterized in that satellite signals during integrated navigation are preprocessed by using an Ensemble Empirical Mode Decomposition (EEMD) method to obtain dynamic real-time changing noise; performing variance calculation on the obtained satellite signal (GNSS) noise, and forming an observation noise covariance matrix of a Kalman filter by using the variance; and taking the ratio of the actual residual covariance matrix to the theoretical residual covariance matrix as a self-adaptive factor, and self-adaptively adjusting the process noise covariance matrix of the Kalman filter. The dynamic adaptive Kalman filtering algorithm is suitable for the combined navigation process of satellite signals and Inertial Navigation Signals (INS), and is beneficial to improving the filtering precision of a Kalman filter.

Description

Dynamic adaptive Kalman filter method based on ensemble empirical mode decomposition
Technical Field
The invention relates to a method for improving positioning precision of a Kalman filter in the field of navigation, which is suitable for the field of integrated navigation of satellite signals and inertial navigation signals, in particular to a dynamic adaptive Kalman filter method based on Ensemble Empirical Mode Decomposition (EEMD).
Background
Currently, the most common navigation systems for objects such as drones, missiles, and automobiles generally include a Global Navigation Satellite System (GNSS) and an Inertial Navigation System (INS). The navigation positioning algorithm mainly adopts a Kalman filter to perform combined navigation on the satellite signal and the inertial navigation signal. The Kalman filter can obtain a filtering signal with positioning accuracy superior to that of a satellite signal and an inertial navigation signal, so that the Kalman filter algorithm is widely concerned since the Kalman filter algorithm is provided.
The main problem faced by conventional kalman filters is the uncertainty of the filter parameters. In the field of integrated navigation, uncertainty parameters of a filter mainly comprise an observation noise covariance matrix and a process noise covariance matrix. In actual use, the observed noise and process noise of the system will typically vary over time. Conventional filter parameters are typically based on empirically set constant values, which can lead to model parameter mismatch problems with actual parameters, which can greatly affect the performance of the kalman filter.
Disclosure of Invention
The invention solves the problems: the dynamic adaptive Kalman filter method based on Ensemble Empirical Mode Decomposition (EEMD) is provided, and the filtering precision of the traditional Kalman filter can be effectively improved. The method comprises the steps of adopting a set empirical mode decomposition method to extract noise from satellite signals, dynamically calculating the variance of the noise in real time to obtain an observation noise covariance matrix of a filter, and carrying out self-adaptive adjustment on a process noise covariance matrix in the filter according to the ratio of an actual residual covariance matrix to a theoretical residual covariance matrix.
The technical solution of the invention is as follows: a dynamic adaptive Kalman filter method based on Ensemble Empirical Mode Decomposition (EEMD) comprises the following steps:
step 1: decomposing satellite signals during the combined navigation into sub-quantities with different layers by using an ensemble empirical mode decomposition method (EEMD), classifying the sub-quantities into noise items and signal items by using energy spectrum density (PSD) indexes of the sub-quantities, and extracting satellite signal noise with real-time change and fixed window length;
step 2: calculating the variance of the satellite signal noise with the fixed length extracted in the step 1 to obtain an observation noise covariance matrix of a Kalman filter;
and step 3: and (3) obtaining an observation noise covariance matrix based on the step (2), and adjusting the process noise covariance matrix of the Kalman filter in a real-time self-adaptive manner by taking the ratio of the actual residual covariance matrix to the theoretical residual covariance matrix as a self-adaptive factor, wherein the adjustment of the process noise covariance matrix cannot be influenced by the uncertainty of the observation noise covariance matrix, so that the influence of the inaccuracy of the observation and process noise covariance matrices on the normal work of the Kalman filter is solved.
In the step 1, the EEMD is a self-adaptive signal decomposition algorithm, which can decompose an original signal into a plurality of subcomponents whose frequencies are arranged from low to high, and because the difference of the index values of the Power Spectral Density (PSD) between a low-frequency signal component and a high-frequency noise component is large, the noise component with a small PSD in the subcomponents can be identified by using the index, and the noise self-components are added and reconstructed to obtain satellite signal noise which changes in real time;
the step 1 is realized by the following steps:
the process of calculating the covariance matrix of the observation noise at the time k by using the EEMD is as follows, wherein a satellite signal comprises three channels which are vertically and horizontally higher, variance values corresponding to the satellite signal noise of the three channels are respectively calculated, the calculation of one channel is as follows, and the calculation processes of the other two channels are the same:
(1.1) adding a decomposition window length of n to the satellite signal, and taking out [ k-n: k ] satellite signals for a time period;
(1.2) a pair [ k-n: the satellite signal in the k ] time period is decomposed by using an EEMD algorithm, and according to the decomposition principle, the original signal is decomposed into the following form:
Figure BDA0003155587860000031
Figure BDA0003155587860000032
where o (t) is the original satellite signal containing noise, B f (t) is the F component of the decomposed real satellite signal, F is the number of real satellite signal components in the component, A j (t) J is the jth component of the decomposed noise signal, J is the number of noise signal components in the component, and A (t) is the reconstructed noise signal; t is the t-th point of the satellite signal in the window, and t belongs to [ k-n, k ∈]The reconstructed noise signal a (t) is a superposition of noise components.
The step 2 is specifically realized as follows: the satellite signal contains three channels which are higher than the longitude and the latitude, and the variance values of the noise of the three satellite signal channels which are respectively calculated are used as an observation noise covariance matrix R of the Kalman filter k The value of the diagonal element of (1), i.e. the observation noise covariance matrix R of the Kalman filter at the moment k k (ii) a Wherein the variance calculation process is as follows:
Figure BDA0003155587860000033
wherein A is satellite signal noise, n is window length for calculating noise, D is variance corresponding to noise in the window length, i represents ith point of data in the window length, A is the noise of satellite signal, n is the window length for calculating noise, D is the variance corresponding to noise in the window length, i represents the ith point of data in the window length, and A is the variance of noise in the window length i Indicates the magnitude of the noise at the ith point,
Figure BDA0003155587860000034
representing the mean of the noise over the window length.
In step 3, the expression of the adaptive factor is as follows:
Figure BDA0003155587860000035
in the formula, s k Denotes the adaptation factor, k denotes the k-th time instant, k-1 denotes the k-1-th time instant, m denotes the window length used for calculating the adaptation factor, e denotes the e-th point within the window length in which the adaptation factor is calculated, d k-e Is the residual at time k-e, R k Is the observed noise covariance matrix, Q, at time k of the Kalman filter k-1 Is the process noise covariance matrix, H, at time k-1 of the Kalman filter k Is an observation matrix of a Kalman filter, phi k/k-1 Is a one-step state transition matrix of the kalman filter,
Figure BDA0003155587860000036
is an estimated state covariance matrix of the Kalman filter, trace represents the trace-finding operation of the matrix in brackets, where d k-e The expression of (A) is as follows:
Figure BDA0003155587860000041
in the formula (I), the compound is shown in the specification,
Figure BDA0003155587860000042
is a predicted state vector, Z k Is an observation vector, d k Is the residue of time kA difference;
at this time, the equation of the kalman filter becomes the following form:
Figure BDA0003155587860000043
Figure BDA0003155587860000044
Figure BDA0003155587860000045
Figure BDA0003155587860000046
Figure BDA0003155587860000047
wherein k represents the kth time, k-1 represents the kth time,
Figure BDA0003155587860000048
the state vector is estimated and the estimated state vector,
Figure BDA0003155587860000049
is a predicted state covariance matrix, K k Is the gain of the Kalman filter, I is the unit matrix, Q k Is the process noise covariance matrix at time k of the kalman filter.
Compared with the prior art, the invention has the advantages that:
(1) According to the method, the original satellite signals are subjected to noise extraction processing by using a set empirical mode decomposition method, and the variance values of the noise extracted by using three channels of the satellite signals form the observation noise covariance matrix of the Kalman filter, so that the observation noise covariance matrix of the Kalman filter closer to an actual value can be obtained, the traditional method for determining parameters according to experience is eliminated, and the filtering precision of the Kalman filter can be improved.
(2) The method is self-adaptive and can improve algorithm efficiency to a certain extent.
(3) The method utilizes the ratio of the actual value and the theoretical value of the residual covariance matrix as the adaptive factor to adjust the process noise covariance matrix of the filter, so that the process noise covariance matrix is closer to the actual value, and the filtering precision of the Kalman filter can be improved.
(4) On the basis, the invention simultaneously adjusts two main parameters of the observation noise covariance matrix and the process noise covariance matrix which influence the performance of the Kalman filter in real time, and solves the influence of the uncertainty of the parameters of the filter on the performance of the filter.
In a word, the key point of the method is to dynamically obtain the observation noise covariance matrix of the Kalman filter in real time by using a method of ensemble empirical mode decomposition. And the process noise covariance matrix is adjusted in real time by adding the adaptive factor, so that the influence of the uncertainty of the parameters in the using process of the Kalman filter algorithm on the filtering performance is solved.
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FIG. 1 is a flow chart of the operation of the present invention;
FIG. 2 is a flow chart of a specific algorithm of the dynamic adaptive Kalman filter in the present invention;
FIG. 3 is a graph of satellite signal noise in accordance with the present invention;
FIG. 4 is a diagram of the effect of extracting noise by empirical mode decomposition according to the present invention;
FIG. 5 is a comparison graph of the experimental effect of the dynamic adaptive Kalman filter of the present invention;
FIG. 6 is a diagram illustrating the variation of the adaptive factors in the dynamic adaptive Kalman filter according to the present invention.
Detailed Description
The process of the present invention is described in detail below with reference to specific examples.
As shown in fig. 1, the present invention is a dynamic adaptive kalman filter method based on Ensemble Empirical Mode Decomposition (EEMD), comprising the following steps:
(1) Assuming that the noise distribution of the satellite signal is as shown in fig. 3, to verify the effect of the proposed method, the noise signal is set to a step change with a standard deviation from 10-30 m.
A sliding window is added to the satellite signals containing noise, and the signals in the window are decomposed by using a set empirical mode decomposition method. And screening out noise signals according to the energy spectral density characteristics of the decomposed sub-signals, and calculating the variance of the noise signals.
The process of calculating the covariance matrix of the observed noise at time k by using the EEMD is as follows (the satellite signal usually contains three channels, i.e. one of the channels is taken as an example, and the other two channels are the same):
(1) Adding a decomposition window length with the length of n to the satellite signal, and taking out [ k-n: k ] satellite signals of a time period.
(2) For [ k-n: k ] the satellite signal of the time period is decomposed by using an EEMD algorithm, and the original signal is decomposed into the following form:
Figure BDA0003155587860000061
Figure BDA0003155587860000062
where o (t) is the original satellite signal containing noise, B f (t) is the F component of the real satellite signal after decomposition, and F is the number of the real satellite signal components in the components. A. The j (t) is the jth component of the decomposed noise signal, J is the number of noise signal components in the component, A (t) is the reconstructed noise signal, t is the tth point of the satellite signal in the window, and t belongs to [ k-n, k ∈]。
Wherein the real satellite signal component and the noise signal component are distinguished by the difference between the PSD values of the components. Defining if the PSD difference between two successive components isIf the ratio is more than 100 times, the larger of the two components is B F (t), the last of the signal components; the smaller of the two components is A 1 (t), the first component of the noise component.
Figure BDA0003155587860000063
Finally, the reconstructed noise signal a (t) is defined as the sum of the superposition of the noise components.
Fig. 4 shows the ratio of the variance estimated by the EEMD algorithm to the variance of the actual noise, where the straight line represents the standard deviation estimated by the EEMD algorithm and the star line represents the standard deviation of the actual noise. As can be seen from fig. 4, the estimated variance of the EEMD algorithm is substantially consistent with the actual variance, and can well follow the variation of the actual noise variance, and the average difference between the two is 1.45m.
By calculating the variance of the reconstructed noise signal a (t), the observation noise covariance matrix of the kalman filter can be obtained. The variance is calculated as follows:
Figure BDA0003155587860000064
in the formula, A is a satellite signal noise sequence, n is a window length used for calculating noise, D is a variance corresponding to the noise in the window, and i represents the ith point of data in the window length. A. The i Represents the magnitude of the noise at the ith point,
Figure BDA0003155587860000065
representing the mean of the noise over the window length.
And D, the satellite signal obtained by calculation contains three channels which are higher in longitude and latitude, the solving process is only the solving of one channel, and the solving processes of the other two channels are consistent with the channel. Taking the variance values obtained by the calculation of the three channels as diagonal elements of the observation noise covariance matrix R, namely obtaining the observation noise covariance matrix R of the Kalman filter at the moment k k 。
Figure BDA0003155587860000071
In the formula, D1, D2, and D3 are variances of three channels in the longitude and latitude of the satellite signal obtained in the steps (1) and (2), respectively.
(3) On the premise of ensuring the accuracy of an observed noise covariance matrix of the Kalman filter through the steps, calculating an adaptive factor according to the ratio of the actual residual covariance matrix to the theoretical residual covariance matrix, and if the process noise covariance matrix is inaccurate, adjusting the adaptive factor by using the adaptive factor; the expression form of the adaptive factor is as follows:
Figure BDA0003155587860000072
in the formula, s k Denotes an adaptation factor, k denotes a k-th time instant, k-1 denotes a k-1-th time instant, m denotes a window length for calculating the adaptation factor, e denotes an e-th point within the window length for calculating the adaptation factor, d k-e Is the residual error at time k-e, R k Is the observed noise covariance matrix, Q, of the filter at time k k-1 Is the process noise covariance matrix of the filter at time k-1, H k Is the observation matrix of the filter, phi k/k-1 Is a one-step state transition matrix, P, of the filter k-1 Is the estimated state covariance matrix of the filter, trace represents the tracing operation of the matrix in brackets. Wherein d is k-e The expression of (A) is as follows:
Figure BDA0003155587860000073
in the formula (I), the compound is shown in the specification,
Figure BDA0003155587860000074
is a predicted state vector, Z k Is an observation vector, d k The residual error at time k.
Thus, at this point the process noise covariance matrix of the kalman filter becomes:
Figure BDA0003155587860000075
in the formula, Q k Is the original process noise covariance matrix at time k, s k For the adaptive factor at time k, the adaptive factor,
Figure BDA0003155587860000076
for the process noise covariance matrix after the time k is adjusted by the adaptive factor,
at this time, the kalman filter equation becomes the following form:
Figure BDA0003155587860000081
Figure BDA0003155587860000082
Figure BDA0003155587860000083
Figure BDA0003155587860000084
Figure BDA0003155587860000085
wherein k represents the kth time, k-1 represents the kth time,
Figure BDA0003155587860000086
the state vector is estimated and the estimated state vector,
Figure BDA0003155587860000087
is a predicted state covariance matrix, K k Is the Kalman Filter gain, Q k Is the process noise covariance matrix at time k of the filter.
The method can be obtained through the process, the Kalman filter observes real-time acquisition of the noise covariance matrix and adjusts the process noise covariance matrix in real time, and after the observation noise covariance matrix is determined through an EEMD algorithm, the adjustment of the process noise covariance matrix cannot be influenced by the uncertainty of the observation noise covariance matrix; by the two parts, the influence of the inaccuracy of the covariance matrix of observation and process noise on the normal operation of the Kalman filter can be solved.
A detailed flow chart of the algorithm execution process described above is shown in fig. 2.
Setting the actual noise of the satellite signal as dynamic noise (within 0-200 s, noise standard deviation is within 2m and within 400-600 s, noise standard deviation is within 100m and 30 m) for verifying the validity of the proposed algorithm. And compared with a conventional kalman filter (the standard deviation of the observation noise of the conventional kalman filter is set to a constant value of 30 m).
In fig. 5, the straight line represents the filtering result of the kalman filter, and the asterisk represents the filtering result of the dynamic adaptive kalman filter proposed by the present invention. The simulation result in fig. 5 shows that, when the model noise is not matched with the actual noise, the dynamic adaptive kalman filter of the present invention has a better performance effect than the conventional kalman filter; when the model noise is matched with the actual noise, the performance effect of the dynamic adaptive Kalman filter is consistent with that of the traditional Kalman filter; the dynamic adaptive Kalman filtering method provided by the invention is very effective.
FIG. 6 is a diagram illustrating the variation of the adaptive factors during the Kalman filtering operation. The fluctuation of the adaptive factor at the initial moment is caused by inaccurate parameter initial value setting of the Kalman filter; at 200s and 400s, the adaptive factor changes with the change of the satellite signal noise and quickly converges to the vibration around 1 again, and the result shows the effectiveness of the adaptive factor under the condition of the change of the signal noise.
Although illustrative embodiments of the present invention have been described above to facilitate the understanding of the present invention by those skilled in the art, it should be understood that the present invention is not limited to the scope of the embodiments, and various changes may be made apparent to those skilled in the art as long as they are within the spirit and scope of the present invention as defined and defined by the appended claims, and all matters of the invention which utilize the inventive concepts are protected.

Claims (3)

1. A dynamic adaptive Kalman filter method based on ensemble empirical mode decomposition is characterized by comprising the following steps:
step 1: decomposing satellite signals during the combined navigation into sub-quantities with different layers by using an ensemble empirical mode decomposition method (EEMD), and classifying the sub-quantities into noise items and signal items by using energy spectrum density (PSD) indexes of the sub-quantities so as to extract satellite signal noise with real-time change and fixed window length;
step 2: calculating the variance of the satellite signal noise with the fixed length extracted in the step 1 to obtain an observation noise covariance matrix of a Kalman filter;
and step 3: based on the observation noise covariance matrix obtained in the step 2, the ratio of the actual residual covariance matrix and the theoretical residual covariance matrix is used as a self-adaptive factor to self-adaptively adjust the process noise covariance matrix of the Kalman filter in real time, and the adjustment of the process noise covariance matrix cannot be influenced by the uncertainty of the observation noise covariance matrix, so that the influence of the inaccuracy of the observation and process noise covariance matrices on the normal work of the Kalman filter is solved;
in step 3, the adaptive factor is expressed as follows:
Figure FDA0003925657700000011
in the formula, s k Denotes the adaptation factor, k denotes the kth time instant, k-1 denotes the kth-1 time instant, m denotes the value for calculating the adaptationWindow length of the factor, e denotes the e-th point within the window length of the calculation of the adaptation factor, d k-e Is the residual at time k-e, R k Is the observed noise covariance matrix, Q, at time k of the Kalman filter k-1 Is the process noise covariance matrix, H, of the Kalman filter at time k-1 k Is an observation matrix of a Kalman filter, phi k/k-1 Is a one-step state transition matrix of the kalman filter,
Figure FDA0003925657700000012
is an estimated state covariance matrix of the Kalman filter, trace represents the trace-finding operation of the matrix in brackets, where d k-e The expression of (A) is as follows:
Figure FDA0003925657700000013
in the formula (I), the compound is shown in the specification,
Figure FDA0003925657700000021
is a predicted state vector, Z k Is an observation vector, d k Residual error at the moment k;
at this time, the equation of the kalman filter becomes the following form:
Figure FDA0003925657700000022
Figure FDA0003925657700000023
Figure FDA0003925657700000024
Figure FDA0003925657700000025
Figure FDA0003925657700000026
wherein k represents the kth time, k-1 represents the kth time,
Figure FDA0003925657700000027
is to estimate the state vector of the state,
Figure FDA0003925657700000028
is a predicted state covariance matrix, K k Is the gain of the Kalman filter, I is the unit matrix, Q k Is the process noise covariance matrix at time k of the kalman filter.
2. The method of claim 1, wherein: the specific implementation process of the step 1 is as follows:
the process of calculating the covariance matrix of the observation noise at the moment k by adopting the EEMD is as follows, wherein the satellite signal comprises three channels which are vertically and horizontally higher, the variance values corresponding to the satellite signal noise of the three channels are respectively calculated, the calculation of one channel is as follows, and the calculation processes of the other two channels are the same:
(1.1) adding a decomposition window length of length n to the satellite signal, taking out [ k-n: k ] satellite signals for a time period;
(1.2) a pair [ k-n: k ] the satellite signal in the time period is decomposed by the EEMD algorithm, and the original signal is decomposed into the following form according to the decomposition principle:
Figure FDA0003925657700000029
Figure FDA00039256577000000210
where o (t) is the original satellite signal containing noise, B f (t) F is the F component of the real satellite signal after decomposition, F is the number of the real satellite signal components in the component, A j (t) J is the jth component of the decomposed noise signal, J is the number of noise signal components in the component, and A (t) is the reconstructed noise signal; t is the t-th point of the satellite signal in the window, and t belongs to [ k-n, k ]]The reconstructed noise signal a (t) is a superposition of noise components.
3. The method of claim 1, wherein: the step 2 is specifically realized as follows: the satellite signal contains three channels which are higher than the longitude and the latitude, and the variance values of the noise of the three satellite signal channels which are respectively calculated are used as an observation noise covariance matrix R of the Kalman filter k The value of the diagonal element of (1), i.e. the observation noise covariance matrix R of the Kalman filter at the moment k is obtained k (ii) a Wherein the variance calculation process is as follows:
Figure FDA0003925657700000031
wherein A is satellite signal noise, n is window length for calculating noise, D is variance corresponding to noise in the window length, i represents ith point of data in the window length, A is the noise of satellite signal, n is the window length for calculating noise, D is the variance corresponding to noise in the window length, i represents the ith point of data in the window length, and A is the variance of noise in the window length i Represents the magnitude of the noise at the ith point,
Figure FDA0003925657700000032
representing the mean of the noise over the window length.
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