Maximum cross-correlation entropy Kalman filtering method based on rational kernel function

Abstract

The invention discloses a maximum correlation entropy Kalman filtering method based on a rational kernel function, and belongs to the technical field of signal processing. The method specifically comprises the following steps: 1) constructing a linear system equation and a measurement equation; step 2) selecting a kernel width of a rational quadratic kernel function, and initializing a system state and a covariance; 3) according to a system equation, updating one-step prediction of a state and a covariance; (4) the state value is initialized again at the fixed point iteration starting moment; 5) performing system model deformation according to the initial system and the measurement equation to obtain an error vector after deformation; 6) according to a concept based on a weighting criterion and entropy, defining a cost function by using an error vector; 7) for the cost function, solving an optimal solution of a state estimation value according to a maximum correlation entropy principle; and step 8) estimating the variance of a posterior estimation value. Compared with the existing GSKF, HF and MCKF algorithms, the method provided by the invention has the advantage that the accuracy of state estimation and the robustness of estimation are greatly improved.

Description

Technical Field

[0001] This invention belongs to the field of signal processing technology, specifically relating to a maximum correlation entropy Kalman filtering method based on rational kernel functions. Background Technology

[0002] State estimation is a crucial problem in signal processing. The Kalman filter method is one of the important approaches for solving the state estimation problem of linear systems under Gaussian noise. It fully utilizes the system's state model and observation data, solving an optimization problem to minimize the error in state estimation, thereby obtaining the optimal estimate of the system. However, noise contamination due to system model or measurement factors often results in heavy-tailed non-Gaussian noise, which can lead to a decrease in the accuracy of the Kalman filter algorithm or even divergence. Therefore, filtering methods specifically for heavy-tailed non-Gaussian noise are needed.

[0003] For non-Gaussian measurement noise, existing filtering algorithms include Gaussian sum filtering, Huber filtering based on M-estimation, and t-filtering based on Student's method. However, Gaussian sum filtering requires known probability density distribution of the noise, which is difficult to achieve in practical engineering. Huber filtering's influence function does not decrease after the influence parameter γ exceeds 1.345, leading to a decline in estimation performance. Student's t-filtering is only suitable for cases where the system noise covariance and measurement noise covariance are small. Therefore, it is crucial to propose a new Kalman filtering method that improves robustness and noise resistance for linear systems. To this end, researchers have proposed a new method to address heavy-tailed non-Gaussian noise: filtering methods based on maximum correlation entropy, such as maximum correlation entropy Kalman filtering. Unlike traditional filtering methods, correlation entropy includes not only second-order statistical information but also higher-order statistical information, thus achieving better estimation results. Generally, entropy is approximated using a Gaussian kernel function; however, in applications, the Gaussian kernel function can produce singular matrices, limiting its application scope. Summary of the Invention

[0004] This invention addresses the problem of singular matrices arising from Gaussian kernel functions in existing maximum correlation entropy Kalman filtering methods. It proposes a maximum correlation entropy Kalman filtering method based on rational kernel functions. This method performs state estimation for linear systems under non-Gaussian heavy-tailed noise conditions. The application of weighting criteria enhances the robustness and accuracy of the estimation, while the application of rational kernel functions reduces the occurrence of singular matrices.

[0005] To achieve the above objectives, this invention provides the following technical solution: a maximum correlation entropy Kalman filtering method based on a rational kernel function, comprising the following steps:

[0006] Step 1) Construct the linear system equations and measurement equations as follows:

[0007] x k =F k-1 x k-1 +q k-1 ,

[0008] z k =H k x k +r k ,

[0009] Where x k ∈R n Let z be the n-dimensional system state vector at time k. k ∈R m F is the m-dimensional measurement vector at time k; k-1 and H k Let q be the known transition matrix at time k-1 and the measurement matrix at time k, respectively. k-1 ∈R n Let r be the noise of the n-dimensional system at time k-1. k ∈R m Let m be the measurement noise at time k; the system noise follows a Gaussian distribution q. k-1 ~N(0,Q) k-1 The measurement noise is non-Gaussian noise, and q k and r k For unrelated processes, satisfy

[0010]

[0011] Where E[*] represents the mathematical expectation; Q k Let R be the variance matrix of the system noise sequence, assumed to be a non-negative definite matrix; k The variance matrix of the measurement noise sequence is assumed to be a positive definite matrix; δ kj The Kronecker delta function satisfies

[0012]

[0013] Step 2) Select the kernel width c of the rational quadratic kernel function and initialize the system state. Covariance P(0|0);

[0014] Step 3) Update the one-step prediction of the state and covariance based on the system equations:

[0015]

[0016]

[0017] Step 4) Reinitialize the state values ​​at the start of the fixed-point iteration.

[0018] Step 5) Deform the system model based on the initial system and measurement equations:

[0019]

[0020] Where I is an n×n identity matrix, and the noise v k for

[0021]

[0022] And the noise variance matrix is:

[0023]

[0024] Among them B k yes The decomposition of Jolisky.

[0025] Multiply both sides of the transformed equations of the system and measurement by the left side. Then there is

[0026]

[0027] Where e x ∈R n e z ∈R m They are respectively:

[0028]

[0029]

[0030] And there are Remember e x,i For vector e x The i-th element, 1≤i≤n, e z,j For vector e z The j-th element, 1≤j≤m;

[0031] Step 6) Based on the concepts of weighting criteria and entropy, the cost function is defined as follows:

[0032]

[0033] Where k RQ (·) is a rational quadratic kernel function:

[0034]

[0035] e is the error, c is the kernel width of the rational quadratic kernel function; and the n+m dimensional vector (l1,l2,…,l)n ,l1',l2',…,l' m The following conditions must be met:

[0036]

[0037]

[0038] Where |e x,i | is the error e x,i The absolute value of |e z,j |Error e z,j The absolute value, and

[0039] Step 7) For this cost function, according to the principle of maximum correlation entropy, find the state x. k The estimated value

[0040] Step 8) Estimate the variance of the final posterior estimate as follows:

[0041]

[0042] If k+1 = N, where N is the preset number of algorithm iterations, then stop the calculation; otherwise, continue executing the above steps.

[0043] Preferably, the kernel width c of a rational quadratic kernel function satisfies the following equation:

[0044]

[0045] Where e z,j R represents the j-th measurement error after model deformation. k,j The ideal covariance R of the measurement noise k The j-th main diagonal element, Represents the true measurement noise covariance The j-th main diagonal element.

[0046] Preferably, according to the principle of maximum correlation entropy, in step 7), state x k The estimated value The following equation must be satisfied:

[0047]

[0048] Solving for the results

[0049]

[0050] The matrix form is as follows:

[0051]

[0052] Then the state estimate x k The matrix form is as follows:

[0053]

[0054] in

[0055]

[0056]

[0057]

[0058] Here:

[0059]

[0060]

[0061] And there are

[0062] Compared with the prior art, the beneficial effects of the present invention are as follows: The present invention proposes a new weighted kernel function maximum cross-correlation entropy Kalman filter (RKMCKF) method by introducing weighting criteria and rational kernel function into MCKF. This method is used for state estimation of linear systems under non-Gaussian heavy-tailed noise. The application of weighting criteria enhances the robustness and accuracy of estimation, and the application of rational kernel function reduces the occurrence of singular matrices.

[0063] This invention utilizes MATLAB software for simulation experiments, comparing the RKMCKF algorithm with existing GSKF, HF, and MCKF algorithms, and concludes that the accuracy and effectiveness of state estimation are greatly improved. Attached Figure Description

[0064] Figure 1 This is a flowchart of an RKMCKF-based method described in this invention.

[0065] Figure 2 The MSE plots of state x1 under Gaussian noise are KF and RKMCKF.

[0066] Figure 3 The MSE plots of state x2 under Gaussian noise are KF and RKMCKF.

[0067] Figure 4 It is the MSE plot of state x1 under mixed Gaussian noise, one of the heavy-tailed non-Gaussian noises, RKMCKF (c=1,2,5,10,15,30).

[0068] Figure 5The MSE plots of state x1 under mixed Gaussian noise, one of the heavy-tailed non-Gaussian noises, are KF, GSKF, HF, MCKF (σ=2) and RKMCKF (c=15).

[0069] Figure 6 It is the MSE plot of state x2 under mixed Gaussian noise, one of the heavy-tailed non-Gaussian noises, RKMCKF (c=1,2,5,10,15,30).

[0070] Figure 7 The MSE plots of state x2 under mixed Gaussian noise, one of the heavy-tailed non-Gaussian noises, are KF, GSKF, HF, MCKF (σ=2) and RKMCKF (c=15).

[0071] Figure 8 It is the MSE plot of state x1 under RKMCKF (c=1,2,5,10,15,30) conforming to t-distribution non-Gaussian noise.

[0072] Figure 9 The MSE plots of state x1 under non-Gaussian noise with a t-distribution are KF, GSKF, HF, MCKF (σ=2) and RKMCKF (c=15).

[0073] Figure 10 It is the MSE plot of state x2 under RKMCKF (c=1,2,5,10,15,30) conforming to t-distribution non-Gaussian noise.

[0074] Figure 11 The MSE plots of state x2 under non-Gaussian noise with a t-distribution are KF, GSKF, HF, MCKF (σ=2) and RKMCKF (c=15). Detailed Implementation

[0075] The embodiments of the present invention will be described in detail below with reference to the accompanying drawings and examples, so as to fully understand how the present invention uses technical means to solve technical problems and achieve technical effects and to implement it accordingly.

[0076] like Figure 1 As shown, a maximum cross-correlation entropy Kalman filter method based on rational kernel functions is performed according to the following steps:

[0077] Step 1) Construct the linear system equations and measurement equations as follows:

[0078] x k =F k-1 x k-1 +q k-1 ,

[0079] z k =H k xk +r k ,

[0080] Where x k ∈R n Let z be the n-dimensional system state vector at time k. k ∈R m F is the m-dimensional measurement vector at time k; k-1 and H k Let q be the known transition matrix at time k-1 and the measurement matrix at time k, respectively. k-1 ∈R n Let r be the noise of the n-dimensional system at time k-1. k ∈R m Let m be the measurement noise at time k; the system noise follows a Gaussian distribution q. k-1 ~N(0,Q) k-1 The measurement noise is non-Gaussian noise, and q k and r k For unrelated processes, satisfy

[0081]

[0082] Where E[*] represents the mathematical expectation;

[0083] Step 2) Select the kernel width c of the rational quadratic kernel function and initialize the system state. Let the covariance be P(0|0), and let k = 1;

[0084] The kernel width c of a rational quadratic kernel function satisfies the following equation:

[0085]

[0086] Where e z,j R represents the j-th measurement error after model deformation. k,j The ideal covariance R of the measurement noise k The j-th main diagonal element, Represents the true measurement noise covariance The j-th main diagonal element.

[0087] Step 3) Update the one-step prediction of the state and covariance based on the system equations:

[0088]

[0089]

[0090] Step 4) Reinitialize the state values ​​at the start of the fixed-point iteration, letting t = 1 and

[0091] Step 5) Deform the system model based on the initial system and measurement equations:

[0092]

[0093] Where I is an n×n identity matrix, and the noise v k for

[0094]

[0095] And the noise variance matrix is:

[0096]

[0097] Among them B k yes The decomposition of Jolisky;

[0098] Multiply both sides of the transformed equations of the system and measurement by the left side. Then there is

[0099]

[0100] Where e x ∈R n e z ∈R m They are respectively:

[0101]

[0102]

[0103] And there are Remember e x,i For vector e x The i-th element, 1≤i≤n, e z,j For vector e z The j-th element, 1≤j≤m;

[0104] Step 6) Based on the concepts of weighting criteria and entropy, the cost function is defined as follows:

[0105]

[0106] Where k RQ (·) is a rational quadratic kernel function:

[0107]

[0108] e is the error, c is the kernel width of the rational quadratic kernel function; and the n+m dimensional vector (l1,l2,…,l) n ,l1',l2',…,l' m The following conditions must be met:

[0109]

[0110]

[0111] Where |e x,i | is the error e x,i The absolute value of |e z,j |Error e z,j The absolute value, and

[0112] Step 7) For this cost function, find the state variable x. k The estimated value The following equation must be satisfied:

[0113]

[0114] Solving for the results

[0115]

[0116] Matrix form

[0117]

[0118] Then the state estimate The matrix form is

[0119]

[0120] in

[0121]

[0122]

[0123]

[0124] Here,

[0125]

[0126]

[0127] And there are

[0128] Step 8) Estimate the variance of the final posterior estimate as follows:

[0129]

[0130] If k+1 = N, where N is the preset number of algorithm iterations, then stop the calculation; otherwise, continue executing the above steps.

[0131] The effectiveness of the present invention will be verified through simulation using specific embodiments below.

[0132] The verification system is as follows:

[0133]

[0134]

[0135] Where θ = π / 18, and the system noise is Gaussian noise w i (k-1)~N(0,1)(i=1,2), the measurement noise is non-Gaussian mixed Gaussian noise.

[0136] Based on the weighted criterion-based rational kernel function maximum correlation entropy Kalman filtering method described above, the initial state value is taken as... The error covariance matrix is ​​P(0|0) = diag(1,1).

[0137] Non-Gaussian measurement noise will be discussed in the following cases:

[0138] Case 1: v(k) ~ N(0,0.1), which is Gaussian noise;

[0139] Case 2: v(k) ~ 0.9N(0,1) + 0.1N(0,100), which is mixed Gaussian noise;

[0140] Case 3: v(k) ~ t(1), which is the student t-distribution noise with 2 degrees of freedom;

[0141] For these three types of noise, the kernel parameter c of the rational kernel function is set to 1, 2, 5, 10, 15, and 30, respectively. It can be seen from the MSE table of each state variable that for different types of non-Gaussian noise, the Kalman filter method with maximum correlation entropy of rational kernel function can achieve its optimal result when the kernel parameter c is set to a suitable value.

[0142] Figures 2 to 11 The figure shows the simulation results of an estimation method based on the maximum correlation entropy Kalman filter with a rational kernel function, as described in the above embodiments.

[0143] Figures 2 to 11 The graph shows the mean square error of the state variables under different noise interference conditions using different filtering algorithms. The smaller the mean square error of the state variables under different filtering estimates, the better.

[0144] From the simulation results diagram Figure 2 and Figure 3It can be seen that when the measurement noise is Gaussian noise, the filtering effect of the rational kernel-based maximum correlation entropy Kalman filter (RKMCKF) method of this invention is related to the size of the kernel parameter c. When the kernel parameter c (c<2) is particularly small, the estimation effect is not ideal. As the kernel parameter c increases, especially when the kernel parameter c≥15, the filtering effect of the traditional Kalman filter (KF) algorithm is approximately the same as that of the RKMCKF method of this invention. This shows that under Gaussian noise, when the kernel parameter c is sufficiently large, this invention can achieve the effect of Kalman filtering, achieving good estimation performance and strong robustness.

[0145] When the measurement is interfered with by Gaussian mixed noise, which is one of the non-Gaussian heavy-tailed noises, by Figure 4 and Figure 6 As can be seen, the RKMCKF method of this invention is affected by the kernel parameter c. When the kernel parameter c is very small (c≤2), the estimation effect is not ideal. As the kernel parameter c increases, the estimation effect improves, reaching its best when c increases to 15. However, when c further increases to 30, the estimation effect decreases. This indicates that for the RKMCKF method proposed in this invention, finding a suitable kernel parameter c is crucial to achieving a good estimation effect. After selecting a suitable kernel parameter (c=15), the measurement was tested under mixed Gaussian noise interference. Figure 5 and Figure 7 The mean square error plots of the RKMCKF (c=15) method of this invention are presented compared with those of KF, GSKF, HF, and MCKF (σ=2). These plots show that, with a suitable kernel parameter (c=15), the estimation performance of this invention is superior to the other three non-Gaussian noise filtering estimation methods compared.

[0146] When the measurement is interfered with by another type of non-Gaussian noise conforming to a t-distribution, the mean square error plot is as follows: Figures 8 to 11 The estimation results obtained from these figures are similar to those under the second non-Gaussian noise condition: from Figure 8 and Figure 10 It can be seen that when the kernel parameter c is relatively small (c≤2), the performance of the RKMCKF method of this invention is not ideal. As the kernel parameter c increases, the estimation effect improves, and the RKMCKF method of this invention achieves the best estimation effect when the kernel parameter c = 15. However, when the kernel parameter is further increased to c = 30, the mean square error increases. This indicates that for the RKMCKF method proposed in this invention, a larger kernel parameter c is not always better; there is an upper limit to the kernel parameter c, and exceeding this limit actually worsens the estimation effect. Selecting a suitable kernel parameter (c = 15) is recommended. Figure 9 and Figure 11The mean square error (MSE) plots of the RKMCKF (c=15) method of this invention, compared with KF, GSKF, HF, and MCKF (σ=2), are presented respectively. These plots show that the MSE of the proposed method is smaller than that of the other methods. Therefore, it can be concluded that the RKMCKF method of this invention is superior to other methods when measurements are interfered with by non-Gaussian noise.

[0147] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A maximum correlation entropy Kalman filter method based on rational kernel functions, characterized in that: Follow these steps: Step 1) Construct the linear system equations and measurement equations as follows: x k =F k-1 x k-1 +q k-1 , z k =H k x k +r k , Where x k ∈R n Let z be the n-dimensional system state vector at time k. k ∈R m F is the m-dimensional measurement vector at time k; k-1 and H k Let q be the known transition matrix at time k-1 and the measurement matrix at time k, respectively. k-1 ∈R n Let r be the noise of the n-dimensional system at time k-1. k ∈R m Let m be the measurement noise at time k; the system noise follows a Gaussian distribution q. k-1 ~N(0,Q) k-1 The measurement noise is non-Gaussian noise, and q k and r k For unrelated processes, satisfy Where E[*] represents the mathematical expectation, Q k Let R be the variance matrix of the system noise sequence, assumed to be a non-negative definite matrix; k The variance matrix of the measurement noise sequence is assumed to be a positive definite matrix; δ kj Let the Kronecker Delta function satisfy... Step 2) Select the kernel width c of the rational quadratic kernel function and initialize the system state. Covariance P(0|0); Step 3) Update the one-step prediction of the state and covariance based on the system equations: Step 4) Reinitialize the state values ​​at the start of the fixed-point iteration. Step 5) Deform the system model based on the initial system and measurement equations: Where I is an n×n identity matrix, and the noise v k for And the noise variance matrix is: Among them B k yes The decomposition of Jolisky; Multiply both sides of the transformed equations of the system and measurement by the left side. Then there is Where e x ∈R n e z ∈R m They are respectively: And there are Remember e x,i For vector e x The i-th element, 1≤i≤n, e z,j For vector e z The j-th element, 1≤j≤m; Step 6) Based on the concepts of weighting criteria and entropy, the cost function is defined using the error vector as follows: Where k RQ (·) is a rational quadratic kernel function: e is the error, c is the kernel width of the rational quadratic kernel function; and the n+m dimensional vector (l1,l2,…,l) n ,l'1,l'2,…,l' m The following conditions must be met: Where |e x,i | is the error e x,i The absolute value of |e z,j |Error e z,j The absolute value, and Step 7) For this cost function, according to the principle of maximum correlation entropy, find the state x. k The estimated value Step 8) Estimate the variance of the final posterior estimate as follows: If k+1 = N, where N is the preset number of algorithm iterations, then stop the calculation; otherwise, continue executing the above steps.

2. The maximum cross-correlation entropy Kalman filter method based on a rational kernel function according to claim 1, characterized in that: In step 7), state x k The estimated value The following equation must be satisfied: Solving for the results The matrix form is as follows: Then the state estimate x k The matrix form is as follows: in Here: And there are 3. The maximum cross-correlation entropy Kalman filter method based on a rational kernel function according to claim 1, characterized in that: The kernel width c of a rational quadratic kernel function satisfies the following equation: Where e z,j R represents the j-th measurement error after model deformation. k,j The ideal covariance R of the measurement noise k The j-th main diagonal element, Represents the true measurement noise covariance The j-th main diagonal element.

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