CN108614804B - Regularization Kalman filtering method based on signal-to-noise ratio test - Google Patents
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Abstract
The invention provides a regularization Kalman filtering method based on signal-to-noise ratio test. The method comprises the following steps: step 1, obtaining t k+1 State parameter X to be estimated at time k+1 Initial state estimation of
And a typical parameter theta k+1 Least squares estimation of
Step 2, according to the state parameter X to be estimated k+1 The ith parameter of
State estimation of
Determining
The signal-to-noise ratio statistic of (1), 2 …, t-1, t, t is the number of state parameters to be estimated; step 3, according to the above
The signal-to-noise ratio statistic of (2), the state parameter X to be estimated k+1 Dividing into interference-related state parameters and non-interference-related state parameters; step 4, estimating according to least square
Determining ridge parameters of disturbance-related state parameters
Ridge parameters with non-interference state parameters
Step 5, according to the ridge parameters
And
determining a state parameter X to be estimated k+1 To estimate said initial state
And (6) correcting. The invention effectively reduces the introduction of deviation while reducing the estimation variance of the state parameters, so that the estimation result of the state parameters is better in the mean square error sense.
Description
Technical Field
The invention relates to the technical field of dynamic data processing, in particular to a regularization Kalman filtering method based on signal-to-noise ratio test.
Background
Kalman filtering is one of the most common methods for dynamic data processing, and has been extensively studied and widely applied in geodetic surveying, satellite navigation, and satellite orbit determination. The ill-conditioned nature of the observation matrix in the discrete dynamic system can produce very big influence to Kalman filtering state estimation, in order to overcome the influence of the ill-conditioned nature of the observation matrix, improve the precision of parameter estimation, many scholars have given improved algorithm.
Currently, there are some methods proposed by scholars to solve the ill-posed problem in discrete dynamic systems from the perspective of biased estimation: (1) the Biased estimation is combined with Kalman filtering to provide a Biased Kalman Filter. (2) The ridge regression and Kalman filtering are combined, and the gain array is corrected, so that the adverse effect of observation matrix ill-conditioned on a filtering value is overcome. (3) Combining Stein compression estimation and ridge regression with Kalman filtering, providing corresponding compression type Kalman filtering and ridge type Kalman filtering and their algorithms, and providing a method for selecting compression coefficients and ridge parameters.
In fact, the estimated hazard size of each state parameter is different according to the ill-condition of the observation matrix, and the hazard size is related to the magnitude of the state parameter itself and the degree of the complex collinearity of the observation matrix data column corresponding to the parameter.
Disclosure of Invention
The invention provides a two-parameter ridge Kalman filtering method based on signal-to-noise ratio measurement, which measures the influence of the ill-conditioned property of parameter state estimation by using signal-to-noise ratio statistics, takes targeted measures on a Kalman filtering recursion process according to a measurement result, improves a ridge Kalman filtering algorithm, and further reduces the influence of the ill-conditioned property of an observation matrix on the state estimation.
The invention provides a regularization Kalman filtering method based on signal-to-noise ratio test, which comprises the following steps:
step 4, estimating according to least square
Determining ridge parameters of disturbance-related state parameters
Ridge parameters with non-interference state parameters
Step 5, according to the ridge parameters
And
determining a state parameter X to be estimated k+1 To estimate said initial state
And (6) correcting.
Further, the method further comprises:
obtaining t k+1 Normal matrix N of time instants k+1 ；
If the judgment result is positive matrix N k+1 Is greater than the preset condition number threshold, then for t k+1 And carrying out signal-to-noise ratio statistics on each state parameter at each moment.
Further, the step 2 specifically includes:
according to the formula
Determining
Signal to noise ratio statistic of
Wherein the content of the first and second substances,
is composed of
The variance of (c).
Further, the step 3 specifically includes:
if it is
Then the
Is a disturbance related state parameter;
if it is
Then
The non-interference state parameters are obtained;
wherein
Omega is the level of significance and is,
is center x 2 Distribution of
Upper omega quantile.
Further, the step 4 specifically includes:
the ridge parameters
And
respectively according to the following formula
Determining; wherein the content of the first and second substances,
to represent
The maximum value of (a) is,
s is the number of interference-related state parameters.
Further, the state parameter X to be estimated k+1 The correction matrix of (a) is:
the invention has the beneficial effects that:
the regularization Kalman filtering method based on signal-to-noise ratio test provided by the invention divides the state parameters to be estimated into interference parameters and non-interference parameters according to the magnitude of the signal-to-noise ratio statistic of each state parameter; and then estimate based on least squares
Determining two ridge parameters
And
so as to perform different intensity ridge-type corrections on the state estimates of the two part parameters. For interference parameters, the interference parameters are corrected
Relatively large, for non-involved parameters, make it correct ridge parameters
Is relatively small. The refined processing effectively reduces the introduction of deviation in ridge type Kalman filtering while reducing the state parameter estimation variance, so that the state parameter estimation result is better in the mean square error sense.
Drawings
FIG. 1 is a schematic flow chart of a regularization Kalman filtering method based on signal-to-noise ratio test according to an embodiment of the present invention;
FIG. 2 is a diagram illustrating a variation of condition numbers of a normal matrix in each filtering method according to an embodiment of the present invention;
fig. 3 is a schematic diagram illustrating changes of mean square errors in filtering methods according to an embodiment of the present invention;
fig. 4 is a schematic diagram illustrating changes in state estimation errors in each filtering 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 clearer, the technical solutions in the embodiments of the present invention will be clearly described below with reference to the accompanying drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. 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.
Before describing the regularized Kalman filtering method based on signal-to-noise ratio test provided by the invention, a discrete dynamic system, a Kalman filtering elementary equation, a ridge-type Kalman filtering state estimation and the influence of the ill-conditioned observation matrix in the discrete dynamic system on the Kalman filtering state estimation are introduced in the following.
The embodiment of the invention adopts the following state space model to describe a discrete dynamic system:
X k+1 ＝Φ k+1,k X k +W k (1)
Y k ＝H k X k +V k (2)
in the formula, X k ∈R t For discrete dynamic systems at time t k The state of (1); y is k ∈R n Is the corresponding observed signal; phi k+1,k Is a t × t nonsingular matrix, is t k Time to t k+1 A one-step state transition matrix of the moment; h k Is an nxt observation matrix; w k ∈R t Input noise that is normally distributed; v k ∈R n Observed noise that follows a normal distribution. Equation (1) is called a state equation, and equation (2) is called an observation equation.
At the same time, assume W k And V k Satisfy the requirement of
E(W k )＝0,Cov(W k ,W j )＝E(W k W j T )＝Q k δ kj
E(V k )＝0,Cov(V k ,V j )＝E(V k V j T )＝R k δ kj
Cov(W k ,V j )＝E(W k V j T )＝0 (3)
In the formula, Q k For a covariance matrix of the input noise, assume Q k Is a non-negative array, R k To observe the covariance matrix of the noise, let R be assumed k Is a positive definite matrix; delta kj Is a Kronecher-delta function, which is defined as
As can be seen from formula (3), W k And V k Mean value of zero and covariance matrix of Q k And R k Is white uncorrelated noise.
The following relevant introduction is made to the basic equation of the kalman filter algorithm:
and (3) state one-step prediction:
one-step prediction covariance matrix:
a filter gain matrix:
and (3) updating the state:
estimating an error variance matrix:
equations (4) to (8) are basic equations of recursive kalman filtering. Given an initial value
And P 0 According to t k+1 Observed quantity Y of time k+1 Then t can be calculated by recursion k+1 State estimation of time of day
From the basic Kalman filtering equation, t k+1 The state estimate for a time instant can in turn be expressed as:
namely, it is
Is a solution of formula (10):
is provided with
N k+1 A normal matrix called kalman filtering. N is a radical of k+1 Is non-negative array, for N k+1 The decomposition of the characteristic value is carried out,
Then
wherein, the first and the second end of the pipe are connected with each other,
if the observation matrix H k+1 Presence of pathological conditions, then H k+1 And
is likely to be such that N k+1 The existence of the ill-conditioned character, which is shown by the practical work,
the control effect on the ill-conditioned nature of the observation matrix is weak and cannot eliminate the adverse effect of the ill-conditioned nature of the observation matrix on the state estimation.
From the formula (12), N can be seen k+1 Presence of one or more small eigenvalues
At this time, if Y k+1 And
there is a small observed error or bias to which the inverse of the small eigenvalue in the above equation amplifies, thereby causing the estimate to deviate far from the true value.
In order to reduce the influence of small characteristic values on the observation result, a ridge type Kalman filtering method is provided on the basis of the algorithm, t k+1 The ridge-type kalman filter state estimation at time is:
ridge-type Kalman filtering utilizes a ridge parameter α k+1 For small eigenvalue
And (4) suppressing is carried out, and the variance of state estimation is reduced, so that the amplification effect of small characteristic values on observation errors is weakened.
However, the ridge-type kalman filter has two drawbacks, one is that no ill-conditioned information is utilized, so that correction of ridge parameters is blindness, and all parameters are corrected completely and consistently; secondly, deviation is introduced by the ridge-type Kalman filtering, and the deviation introduced by the ridge-type Kalman filtering is possibly amplified in the continuous recursion process, so that the estimation precision is influenced.
From the above analysis, t k+1 State estimation of time of day
Associated with the normal matrix (see equation (10)), if normal matrix N k+1 If a pathological condition exists, t is k+1 The state estimate at the time will become very unstable, which is where the ill-conditioned nature of the observation matrix exerts an influence on the state estimate. Normal matrix N k+1 The reason for the morbidity is that the complex collinearity relationship exists between the data columns, thereby causing t k+1 The state estimation at the moment is not good, but not all the state estimation is not good, the ill-conditioned property of the normal matrix only has larger influence on the state parameter estimation corresponding to the data columns participating in the complex collinearity, and has smaller influence on the state parameter estimation corresponding to the data columns not participating in the complex collinearity.
The invention provides a regularization Kalman filtering method (DPRTKF for short) based on signal-to-noise ratio test on the basis of a ridge Kalman filtering method, which corrects the state parameters of two parts with smaller and larger ill-conditioned observed matrixes with different strengths. Fig. 1 is a schematic flow chart of a regularization kalman filtering method based on signal-to-noise ratio test according to an embodiment of the present invention. As shown in fig. 1, the method comprises the steps of:
s101, acquiring t k+1 State parameter X to be estimated at time k+1 Initial state estimation of
And a typical parameter theta k+1 Least squares estimation of
Specifically, the method provided by the invention needs to acquire t in advance k+1 State parameter X to be estimated at time k+1 Initial state estimation of
And a typical parameter theta k+1 Least squares estimation of
Since the invention aims at estimating the state obtained by the existing Kalman filtering method
The correction is performed, so that the initial state estimation obtained by using the existing Kalman filtering method needs to be obtained in advance
And a typical rule parameter theta k+1 Least squares estimation of
For example, given initial values of state parameters
And mean square error thereof
Using the kalman filter primitive equation:
one-step prediction equation of state:
one-step prediction covariance matrix equation:
filter gain matrix equation:
the following can be obtained:
typical parameters
Its least square estimation is
S102, according to the state parameter X to be estimated k+1 The ith parameter of
State estimation of
Determining
The signal-to-noise ratio statistic of (1), 2 …, t-1, t, t is the number of state parameters to be estimated;
s103, according to the
The signal-to-noise ratio statistic of the state parameter X to be estimated k+1 Dividing into interference-related state parameters and non-interference-related state parameters;
s104, estimating according to least squares
Determining ridge parameters of disturbance-related state parameters
Ridge parameters with non-interference state parameters
S105, according to the ridge parameters
And
determining a state parameter X to be estimated k+1 To estimate said initial state
And (6) correcting.
The regularization Kalman filtering method based on signal-to-noise ratio test provided by the invention divides the state parameters to be estimated into interference parameters and non-interference parameters according to the magnitude of the signal-to-noise ratio statistic of each state parameter; and then estimate based on least squares
Determining two ridge parameters
And
so as to perform different intensity ridge-type corrections on the state estimates of the two part parameters. For interference parameters, the parameters are corrected
Relatively large, for non-involved parameters, make it correct ridge parameters
Is relatively small. The refined processing effectively reduces the introduction of deviation in ridge Kalman filtering while reducing the state parameter estimation variance, so that the state parameter estimation result is better in mean square error sense.
On the basis of the above embodiment, the method further comprises the steps of:
obtaining t k+1 Normal matrix N of time instants k+1 ；
If judging the learning method matrix N k+1 Is greater than the preset condition number threshold, then for t k+1 Each state parameter at a momentThe numbers are counted for signal to noise ratio.
In particular, the Fa matrix N k+1 If the condition number of the observation matrix is greater than the preset condition number threshold, the observation matrix is considered to be a sick matrix, and the ill-condition of the observation matrix can adversely affect the state estimation result of the state parameter to be estimated, so that the t needs to be estimated k+1 And carrying out signal-to-noise ratio statistics on each state parameter at each moment. Therefore, the state parameters to be estimated can be grouped for the following signal-to-noise ratio statistics based on each state parameter, and the state parameters to be estimated are divided into non-interference state parameters with small influence and interference state parameters with large influence.
On the basis of the foregoing embodiments, the step S102 in the method specifically includes: according to the formula
Determining
Signal to noise ratio statistic of
Wherein the content of the first and second substances,
is composed of
The variance of (c).
On the basis of the foregoing embodiments, the step S103 in the method specifically includes:
if it is
Then
The state parameter is a non-interference state parameter;
if it is
Then
Is a disturbance related state parameter;
wherein
Omega is the level of significance and is,
is center x 2 Distribution of
Upper omega quantile.
In particular, when
Considering the complex collinearity to estimate the corresponding state
The damage of the (A) is serious, and the estimation effect is not good; when in use
Considering the complex collinearity to estimate the corresponding state
The damage of (2) is small, and the estimation effect is good. By calculating the SNR statistic, t can be calculated k+1 The state parameter of the moment is divided into two parts
The s parameters for which the SNR statistic is smaller are
The complex collinearity is greatly damaged and is called as interference-related state parameter; the t-s parameters with larger signal-to-noise ratio statistics are
They are less compromised by complex collinearity and are referred to as non-interfering state parameters. For the
Because the estimation is poor, the parameters of the part are greatly corrected; for the
With minor modifications thereto. In practice, the threshold value
Can be determined flexibly according to the actual situation.
On the basis of the foregoing embodiments, step S104 in the method specifically includes:
the ridge parameters
And
respectively according to the following formula
Determining; wherein the content of the first and second substances,
to represent
The maximum value of (a) is,
s is interferenceThe number of state parameters.
Specifically, all signal-to-noise ratio statistics are combined
Rearranged in descending order and numbered as
Order to
On the basis of the above embodiments, the state parameter X to be estimated in the method k+1 The correction matrix of (a) is:
in particular, the correction matrix Z k+1 For t rows and t columns matrix, the first s rows of the matrix are all main diagonals
For correcting disturbance-related state parameters
The front and back t-s rows of the matrix are main diagonals
For correcting non-interference state parameters
The invention is further illustrated by the following further specific examples.
and 3, updating the state:
step 4. judging method matrix
If the condition number is greater than 500, if so, go to step 5, otherwise return to step 2.
And 5, measuring the signal-to-noise ratio, and determining interference-related state parameters and non-interference-related state parameters.
Step 6, determining two ridge parameters
And parameter c k+1 。
And 7, correcting the state estimation by utilizing the two-parameter ridge Kalman filtering, and calculating a mean square error matrix
Wherein the content of the first and second substances,
step 8. order
And returning to the step 2, and aligning the next time by using Kalman filtering
And estimating the state parameters.
From the above, it can be seen that:
therefore, it is not only easy to use
Is that
A linear transformation of (a);
and because
Therefore, it is not only easy to use
Is a state parameter X k A biased estimate of (d);
at the same time
Therefore, it is not only easy to use
Is a state parameter X k A compressed biased estimation of (2).
In addition to this, the present invention is,
due to the fact that
Therefore, it is not only easy to use
And because, when selecting the appropriate initial value
And mean square error thereof
Time, Kalman filtering
Can be an unbiased estimation, while the ordinary ridge-type Kalman filtering
And regularized filtering provided by the present invention
Are all Kalman filtering
Is estimated by compression of
Filtering Kalman
Has a compressive strength greater than
Filtering Kalman
The compressive strength of (a), so from a deflection point of view,
is greater than
To be smaller than
Due to the fact that
So in the condition number sense
Algorithm condition number less than Kalman filter estimation
Larger than ridge Kalman filtering
The algorithm condition number.
In conclusion, the regularization Kalman filtering method based on signal-to-noise ratio test provided by the invention not only has the anti-attitude property which is not possessed by the common Kalman filtering, but also overcomes the blind and excessive anti-attitude property of the common ridge Kalman filtering, thereby reducing the damage of excessive deviation caused by excessive anti-attitude property and achieving more accurate and proper effect.
The invention is further illustrated by the following further specific examples.
To verify the correctness of the pathological analysis herein and the validity of the new algorithm, the following algorithm was used to perform the simulation test. Setting a one-step State transition TorqueMatrix phi k+1,k Is composed of
Design observation matrix H k Of 5 column vectors of
a k1 ＝[15.57 44.02 20.42 18.74 49.20 44.92 55.48 59.28 94.39 128.02 96.00 131.42 127.21 252.90 409.20 463.70 510.22] T
a k2 ＝[2643 2048 3940 6505 5723 11520 5779 5969 8461 20106 13313 10771 45543 36194 34703 39204 86533] T
a k4 ＝[18.0 9.5 12.8 36.7 35.7 24.0 43.3 46.7 76.7 180.5 60.9 103.7 126.8 157.7 169.4 331.4 371.6] T
a k5 ＝[4.45 6.92 4.28 3.90 5.50 4.60 5.62 5.15 6.18 6.15 5.88 4.68 4.88 5.57 10.78 7.05 6.35] T
a k3 ＝2a k1 +0.5a k4 +e k ，e k ～N 17 (0,0.05 2 I)
H k ＝[a k1 a k2 a k3 a k4 a k5 ]
Q k ＝I 5 ，R k ＝0.5 2 ×I 17 Covariance matrices, initial values, of state noise sequence and observation noise sequence, respectively
Is composed of
Wherein X 0 ＝[200,15,35,16,-2.8,6] Τ In the true value, the value of,
is the initial error matrix.
In the embodiment of the invention, the data columns of the observation matrix are set to have approximate linear relation, so that the normal matrix has serious ill-conditioned property. Three filtering algorithms are used for calculation and their condition numbers, mean square errors and deviations are compared, and the effect pair is shown in fig. 2 to 4.
(1) As can be seen from fig. 2, the normal Kalman filter algorithm has a condition number of 10 for the normal Kalman filter algorithm 4 The order of magnitude of (2) exceeds a threshold value of 500, which shows that complex collinearity of an observation matrix can really cause ill-conditioned law matrix, and the condition numbers of law matrices of an RTKF algorithm (namely, ordinary ridge-type Kalman filtering in the figure) and a DPRTKF algorithm (namely, two-parameter ridge-type Kalman filtering in the figure) are effectively controlled.
(2) It can be seen from fig. 3 that the RTKF algorithm and the DPRTKF algorithm effectively weaken the ill-posed nature in the Kalman filtering, and the DPRTKF algorithm is superior to the general Kalman filtering algorithm and the RTKF algorithm in the mean square error sense.
(3) As can be seen from fig. 4, compared with the Kalman filter algorithm, the RTKF algorithm properly sacrifices the unbiased property of the state estimation in exchange for the great reduction of the variance, so that the estimation result is relatively stable, while the DPRTKF algorithm reduces the introduction of the deviation while reducing the state estimation variance, so that the euclidean distance between the solution and the true value is smaller than the Kalman filter algorithm and the RTKF algorithm.
Finally, it should be noted that: the above examples are only intended to illustrate the technical solution of the present invention, but not to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.
Claims (4)
1. A regularization Kalman filtering method based on signal-to-noise ratio test is applied to a state parameter estimation process of a pathologically observed matrix in a discrete dynamic system, and a state space model is adopted to describe the discrete dynamic system:
X k+1 ＝Φ k+1,k X k +W k (1)
Y k ＝H k X k +V k (2)
in the formula, X k ∈R t For discrete dynamic systems at time t k The state of (1); y is k ∈R n Is the corresponding observed signal; phi k+1,k Is a t × t nonsingular matrix, is t k Time to t k+1 A one-step state transition matrix of the moment; h k Is an nxt observation matrix; w k ∈R t Input noise that is normally distributed; v k ∈R n Observed noise that is normally distributed; wherein the discrete dynamic system is any one of a geodetic surveying system, a satellite navigation system and a satellite orbit determination system, and the method comprises:
step 1, obtaining t k+1 State parameter X to be estimated at time k+1 Initial state estimation of
And a typical parameter theta k+1 Least squares estimation of
Step 2, according to the state parameter X to be estimated k+1 The ith parameter of
State estimation of
Determining
The signal-to-noise ratio statistic of (1), 2 …, t-1, t, t is the number of state parameters to be estimated; the step 2 specifically comprises the following steps:
according to the formula
Determining
Signal to noise ratio statistic of
Wherein the content of the first and second substances,
is composed of
The variance of (a);
step 3, according to the above
The signal-to-noise ratio statistic of (2), the state parameter X to be estimated k+1 Dividing into interference-related state parameters and non-interference-related state parameters; the method specifically comprises the following steps:
if it is
Then
Is a disturbance related state parameter;
if it is
Then
The non-interference state parameters are obtained;
wherein
Omega is the level of significance and is,
is center x 2 Distribution of
Upper omega quantile of (a);
step 4, estimating according to least square
Determining ridge parameters of disturbance-related state parameters
Ridge parameters with non-interference state parameters
Step 5, according to the ridge parameters
And
determining a state parameter X to be estimated k+1 To estimate said initial state
And (6) correcting.
2. The method of claim 1, further comprising, prior to step 1:
obtaining t k+1 Normal matrix N of time instants k+1 ；
If judging the learning method matrix N k+1 Is greater than the preset condition number threshold, then for t k+1 And carrying out signal-to-noise ratio statistics on each state parameter at each moment.
3. The method according to claim 1, wherein step 4 is specifically:
the ridge parameters
And
respectively according to the following formula
Determining; wherein the content of the first and second substances,
to represent
The maximum value of (a) is,
s is the number of interference state parameters, and all signal-to-noise ratio statistics are calculated
Rearranged in descending order and numbered as
4. Method according to claim 3, characterized in that said state parameter X to be estimated is a parameter X k+1 The correction matrix of (a) is:
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