A Maximum Cross-Correlation Entropy Kalman Filtering Method Based on Random Weighted Criteria

Abstract

The invention provides a maximum cross-correlation entropy Kalman filtering method based on random weighting criteria, comprising: constructing a linear system equation and a measurement equation, selecting a suitable kernel width and initializing a system state and covariance, updating a one-step prediction of the state and covariance according to the system equation, re-initializing the state value at the initial iteration time of a fixed point, deforming the system model according to the initial system and measurement equation and calculating the error after deformation and a kernel function of the error calculated thereby, obtaining two diagonal matrices by the random weighting criteria and the kernel function, and correcting the one-step prediction covariance and the measurement error by the two diagonal matrices to correct the gain matrix, and estimating the system posterior state and covariance. The invention can obtain better performance than Kalman filtering and maximum cross-correlation entropy Kalman filtering for the problem of non-Gaussian heavy-tailed impact noise of a linear model, and can be widely applied to the case where the noise of a linear system is non-Gaussian, so as to improve the filtering estimation accuracy under the non-Gaussian noise situation.

Description

Technical Field

[0001] The invention belongs to the technical field of signal processing, and in particular relates to a maximum cross-correlation entropy Kalman filtering method based on a random weighted criterion. Background Art

[0002] State estimation is an important issue in signal processing. The Kalman filter method is one of the important methods to solve the state estimation problem of linear systems under Gaussian noise. It makes full use of the state model and observation data of the system, and minimizes the error of state estimation by solving the optimization problem, so as to obtain the optimal estimate of the system. However, due to the system model or measurement, the noise is polluted, and heavy-tailed non-Gaussian noise often occurs, which will cause the accuracy of the Kalman filter algorithm to decrease or even diverge. Therefore, a filtering algorithm for non-Gaussian noise is needed.

[0003] For the case where the measurement noise is non-Gaussian, the filtering algorithms that have emerged include: Gaussian sum filtering, M estimation filtering based on Huber technology, and t filtering based on Student's method. However, the use of Gaussian sum filtering requires that the probability density distribution of the noise is known, which is difficult to achieve in engineering practice: Huber technology will not fall back after the influence parameter γ exceeds 1.345, which will lead to a decrease in estimation performance; and Student's t filtering can only be used when the system noise covariance and measurement noise covariance are small. Therefore, it is particularly important to propose a new Kalman filtering method that is suitable for linear systems with improved robustness and noise resistance. To this end, researchers proposed a new method to solve heavy-tailed non-Gaussian noise, namely a filtering method based on maximum cross-correlation entropy, such as maximum cross-correlation entropy Kalman filtering. Different from traditional filtering methods, cross-correlation entropy includes not only second-order statistical information, but also higher-order statistical information, so it can obtain better estimation results. Summary of the invention

[0004] In view of the defects of the prior art, the present invention provides a maximum cross-correlation entropy Kalman filtering method based on a random weighted criterion, which is applicable to linear systems in the case of non-Gaussian noise and improves the robustness and noise resistance of the system.

[0005] In order to achieve the above object, the present invention is implemented through the following technical solutions.

[0006] A maximum cross-correlation entropy Kalman filtering method based on a random weighted criterion comprises the following steps:

[0007] Step 1: Construct the linear system equation and measurement equation as follows:

[0008]

[0009] Where k-1 represents the k-1th moment, x k ∈R n is the n-dimensional system state vector at the kth moment, z k ∈R m is the m-dimensional measurement vector at the kth moment; F k-1 and H k are the known transfer matrix and measurement matrix, q k-1 ∈R n is the n-dimensional system noise at the k-1th moment, r k ∈R m is the m-dimensional measurement noise at the kth moment; the system noise follows the Gaussian distribution q k-1 ~N(0,Q k-1 ), the measurement noise is non-Gaussian and follows a mixed Gaussian distribution r k ~λN(0,R k,1 )+(1-λ)N(0,R k,2 ), q k-1 and r k is uncorrelated process and measurement Gaussian noise, satisfying

[0010]

[0011] where E[·] represents the mathematical expectation, δ kj is the Kronecker sign function, Represents the mixed noise vector r j The transposed vector of ;

[0012] Step 2: Initialization, select a kernel width σ and initialize the system state and covariance P(0|0), let k=1;

[0013] Step 3: Update the prior state according to the system's one-step prediction equation and covariance P k|k-1 ;

[0014]

[0015]

[0016] Step 4: Initialize the state value again at the fixed point iteration time: let t = 1 and

[0017] Step 5: Deform the system model according to the initial system and measurement equations, calculate the error of the new model, and thus calculate the kernel function of the error;

[0018] First, the state equation and measurement equation are reconstructed:

[0019]

[0020]

[0021]

[0022] Where: E[·] represents the mathematical expectation, P k|k-1 is the one-step forecast error covariance matrix of the state at the kth moment, R k is the measurement noise covariance matrix at the kth moment, B P (k|k-1) is the pair of P k|k-1 The matrix obtained after Cholesky decomposition is, It is B P The transposed matrix of (k|k-1), similarly B R (k) is R k The matrix obtained after Cholesky decomposition is, It is B R The transposed matrix of (k), B k By B P (k|k-1) and B R (k) The new diagonal matrix formed;

[0023] In the equation Multiply both sides of have to:

[0024] D k =W k x k +e k

[0025] in Error vector e k The elements in column i are:

[0026] e k (i) = d i (k)-w i (k)x k (i)

[0027] Where: d i (k) is D k The i-th element of i (k) is the matrix W k The i-th row element, x k (i) represents x here k The i-th state quantity, and D k is a L = n + m dimensional vector;

[0028] Step 6: Obtain two diagonal matrices by using the random weighted criterion and kernel function;

[0029] Due to the random weighting criterion, a new cost function is defined:

[0030]

[0031] in G σ (·) Gaussian kernel function:

[0032] Take it here:

[0033] Then x k The optimal solution for (i) is:

[0034]

[0035] The matrix form is:

[0036]

[0037] in and Get two diagonal matrices

[0038] Step 7: Two diagonal matrices To correct the one-step forecast covariance and the measurement error covariance

[0039]

[0040] Thus the gain matrix is ​​corrected;

[0041]

[0042] Step 8: Estimate the posterior state of the system filter

[0043]

[0044] and covariance

[0045]

[0046] If k+1=N, where N is the preset number of algorithm iterations, then stop the calculation; otherwise, continue to execute the above steps.

[0047] Compared with the prior art, the present invention has the following beneficial effects:

[0048] The present invention improves the accuracy of MCKF by introducing random weighting criterion into MCKF; the present invention adopts random weighting theory in the cost function, increases the correlation entropy, complies with the maximum correlation entropy criterion (MCC), and improves the accuracy of estimation; experiments show that compared with traditional KF algorithm and MCKF algorithm, the accuracy of state estimation and the effectiveness of estimation are improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 is a flow chart based on the RWMCKF method described in the present invention;

[0050] Figure 2 is the probability density function of state x1 under KF, MCKF (σ=2) and RWCKF;

[0051] Figure 3 is the probability density function of state x2 under KF, MCKF (σ=2) and RWCKF;

[0052] Figure 4 is the probability density function of state x1 under MCKF (σ=2) and RWCKF;

[0053] Figure 5 is the probability density function of state x2 under MCKF (σ=2) and RWCKF. DETAILED DESCRIPTION

[0054] The present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. In order to better understand the method of the present invention, the network structure of the present invention is first described in detail.

[0055] The present invention is a method for maximum cross-correlation entropy Kalman filtering based on a random weighted criterion, comprising the following steps:

[0056] Step 1: Construct the linear system equation and measurement equation as follows:

[0057]

[0058] Where k-1 represents the k-1th moment, x k ∈R n is the n-dimensional system state vector at the kth moment, z k ∈R m is the m-dimensional measurement vector at the kth moment; F k-1 and H k are the known transfer matrix and measurement matrix, q k-1 ∈R n is the n-dimensional system noise at the k-1th moment, r k ∈R mis the m-dimensional measurement noise at the kth moment, assuming that the system noise follows a Gaussian distribution q k-1 ~N(0,Q k-1 ), the measurement noise is non-Gaussian and follows a mixed Gaussian distribution r k ~λN(0,R k,1 )+(1-λ)N(0,R k,2 ), q k-1 and r k is uncorrelated process and measurement Gaussian noise, satisfying

[0059]

[0060] where E[·] represents the mathematical expectation, δ kj is the Kronecker sign function, Represents the mixed noise vector r j The transposed vector of ;

[0061] Step 2: Initialization, select a suitable kernel width σ through experience, and initialize the system state and covariance P(0|0), let k=1;

[0062] Step 3: Update the prior state according to the system's one-step prediction equation and covariance P k|k-1 ;

[0063]

[0064]

[0065] Step 4: Initialize the state value again at the fixed point iteration time: let t = 1 and

[0066] Step 5: Deform the system model according to the initial system and measurement equations, calculate the error of the new model, and thus calculate the kernel function of the error;

[0067] First, the state equation and measurement equation are reconstructed:

[0068]

[0069]

[0070]

[0071] Where: E[·] represents the mathematical expectation, P k|k-1 is the one-step forecast error covariance matrix of the state at the kth moment, R k is the measurement noise covariance matrix at the kth moment, B P (k|k-1) is the pair of Pk|k-1 The matrix obtained after Cholesky decomposition is, It is B P The transposed matrix of (k|k-1). Similarly, B R (k) is R k The matrix obtained after Cholesky decomposition is, It is B R The transposed matrix of (k), B k By B P (k|k-1) and B R (k) The new diagonal matrix formed;

[0072] In the equation Multiply both sides of have to:

[0073] D k =W k x k +e k

[0074] in Error vector e k The elements in the i-th column are:

[0075] e k (i) = d i (k)-w i (k)x k (i)

[0076] where d i (k) is D k The i-th element of i (k) is the matrix W k The i-th row element, x k (i) represents x here k The i-th state quantity, and D k It is a L=n+m dimensional vector.

[0077] Step 6: Obtain two diagonal matrices by using the random weighted criterion and kernel function;

[0078] Due to the random weighting criterion, a new cost function is defined:

[0079]

[0080] in G σ (·) Gaussian kernel function:

[0081] Take it here:

[0082] Then xk The optimal solution for (i) is:

[0083]

[0084] The matrix form is:

[0085]

[0086] in and Get two diagonal matrices

[0087] Step 7: Two diagonal matrices To correct the one-step forecast covariance and the measurement error covariance

[0088]

[0089]

[0090] Thus the gain matrix is ​​corrected;

[0091]

[0092] Step 8: Estimate the posterior state and covariance of the system filter, specifically,

[0093]

[0094]

[0095] At this point, the estimation of the target state parameters can be completed. and the state estimation error covariance matrix It will be used to estimate the state parameters at the next moment.

[0096] The random weighted maximum cross-correlation entropy Kalman filter (RWMCKF) algorithm adopted by the present invention performs state estimation for linear systems under non-Gaussian heavy-tailed noise conditions, and the application of random weighted criteria enhances the robustness of the system and improves the accuracy of filtering. The present invention uses MATLAB simulation software to perform simulation experiments, and compares the RWMCKF algorithm with existing filtering algorithms KF and MCKF, and it is found that the accuracy of state estimation and the effectiveness of estimation are greatly improved.

[0097] The present invention is demonstrated below by means of specific embodiments.

[0098] Embodiment 1

[0099] Consider the general linear system model:

[0100]

[0101]

[0102] Where θ = π / 18, and the system noise is Gaussian noise q i (k-1)~N(0,2)(i=1,2), the measurement noise is non-Gaussian mixed Gaussian noise r(k)~0.9N(0,1)+0.1N(0,100).

[0103] According to the above-mentioned maximum relevant entropy Kalman filtering method based on random weighted criteria, the initial value of the state is taken as The error covariance matrix is ​​P(0|0)=diag(100,100). The kernel width σ of the Gaussian kernel function is set to 0.1, 0.5, 1, 2, 3, 5, 8, and 10, respectively, and compared with the KF algorithm and MCKF (σ=2), and the obtained Figure 2 , 3 The probability density function of the two states x1 and x2 under different filtering algorithms. The simulation results of the maximum correlation entropy Kalman filtering method based on the random weighted criterion provided by the present invention are shown in the curve RWMCKF. It can be seen that compared with the traditional KF and MCKF, the performance advantage is obvious: the KF algorithm is biased in the case of heavy-tailed measurement noise, MCKF and RWMCKF are unbiased, but the unbiasedness of the RWCKF algorithm is more advantageous than that of the MCKF algorithm.

[0104] The influence of the kernel width σ of the Gaussian kernel function on the maximum correlation entropy Kalman filter of the random weighted criterion provided by the present invention can be obtained by Figure 3 , 4 It is concluded that the similarity between RWCKF and MCKF is that the effectiveness is best when σ=2.

[0105] Embodiment 2:

[0106] Change the simulation model to one-dimensional linear uniform acceleration motion. The system model and measurement model are as follows:

[0107]

[0108]

[0109] Where Δt = 0.1s, and both the system noise and the measurement noise are nonlinear mixed Gaussian noise:

[0110] q1(k-1)~0.9N(0,0.01)+0.1N(0,1)

[0111] q2(k-1)~0.9N(0,0.01)+0.1N(0,1)

[0112] q3(k-1)~0.9N(0,0.01)+0.1N(0,1)

[0113] r(k)~0.8N(0,0.01)+0.2N(0,100).

[0114] According to the above-mentioned maximum correlation entropy Kalman filtering method based on random weighted criteria, the filtering algorithm is first initialized. The initial state value and the initial covariance value are set to x(0)=

[001] T ,P(0|0)=diag(0.01,0.01,0.01). The kernel width σ of the Gaussian kernel function is set to 2 (that is, the relatively optimal kernel width in Example 1), and the mean square error table under the KF, MCKF (σ=2) and RWMCKF (σ=2) algorithms can be obtained:

[0115] Table I Mean square error of KF, MCKF (σ=2) and RWMCKF (σ=2) algorithms

[0116]

[0117] Table I gives the mean square errors of the three algorithms, KF, MCKF (σ=2) and RWMCKF (σ=2). It can be concluded from Table I that under the influence of the same mixed Gaussian system and measurement noise, the mean square error of the maximum correlation entropy Kalman filter variance based on the random weighted criterion is the smallest, indicating that the performance of this algorithm is greatly improved compared with KF and MCKF.

[0118] The above is only a specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by a person skilled in the art within the technical scope disclosed by the present invention should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention shall be based on the protection scope of the claims.

Claims

1. A maximum cross-correlation entropy Kalman filtering method based on a random weighted criterion, characterized in that: The following steps are involved: Step 1: Construct the linear system equation and measurement equation as follows: Where k-1 represents the k-1th moment, x k ∈R n is the n-dimensional system state vector at the kth moment, z k ∈R m is the m-dimensional measurement vector at the kth moment; F k-1 and H k are the known transfer matrix and measurement matrix, q k-1 ∈R n is the n-dimensional system noise at the k-1th moment, r k ∈R m is the m-dimensional measurement noise at the kth moment; the system noise follows the Gaussian distribution q k-1 ~N(0,Q k-1 ), where N(0,Q k-1 ) means the mean is 0 and the variance is Q k-1 The measurement noise is non-Gaussian and follows a mixed Gaussian distribution r k ~λN(0,R k,1 )+(1-λ)N(0,R k,2 ), where N(0,R k,1 ) means the mean is 0 and the variance is R k,1 Normal distribution; N(0,R k,2 ) has a mean of 0 and a variance of R k,2 Normal distribution of q k-1 and r k is uncorrelated process and measurement Gaussian noise, satisfying where E[·] represents the mathematical expectation, δ kj is the Kronecker sign function, Represents the mixed noise vector r j The transposed vector, R k The measurement noise covariance matrix at the kth moment is: and Represent the above vector q respectively j and r j The transposed vector of ; Step 2: Initialization, select a kernel width σ and initialize the system state and covariance P(0|0), let k=1; Step 3: Update the prior state according to the system's one-step prediction equation and covariance P k|k-1 ; Step 4: Initialize the state value again at the fixed point iteration time: let t = 1 and Step 5: Deform the system model according to the initial system and measurement equations, calculate the error of the new model, and thus calculate the kernel function of the error; First, the state equation and measurement equation are reconstructed: Where: E[·] represents the mathematical expectation, P k|k-1 is the one-step forecast error covariance matrix of the state at the kth moment, R k is the measurement noise covariance matrix at the kth moment, B P (k|k-1) is the pair of P k|k-1 The matrix obtained after Cholesky decomposition is, It is B P The transposed matrix of (k|k-1), similarly B R (k) is R k The matrix obtained after Cholesky decomposition is, It is B R The transposed matrix of (k), B k By B P (k|k-1) and B R (k) The new diagonal matrix formed; In the equation Multiply both sides of have to: D k =W k x k +e k in Error vector e k The elements in the i-th column are: e k (i)=d i (k)-w i (k)x k (i) Where: d i (k) is D k The i-th element of i (k) is the matrix W k The i-th row element, x k (i) represents x here k The i-th state quantity, and D k is a L = n + m dimensional vector; Step 6: Obtain two diagonal matrices by using the random weighted criterion and kernel function; Due to the random weighting criterion, a new cost function is defined: in G σ (·) Gaussian kernel function: Take it here: Then x k The optimal solution for (i) is: The matrix form is: in and Get two diagonal matrices Step 7: Two diagonal matrices To correct the one-step forecast covariance and measurement error covariance in and C x (k) and C z The inverse matrix of (k); Thus the gain matrix is ​​corrected; Step 8: Estimate the posterior state of the system filter and covariance If k+1=N, where N is the preset number of algorithm iterations, then stop the calculation; otherwise, continue to execute the above steps.

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