CN114221638B - A Maximum Cross-Correlation Entropy Kalman Filtering Method Based on Random Weighted Criteria - Google Patents
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Abstract
本发明提供一种基于随机加权准则的最大互相关熵卡尔曼滤波方法，包括：构建线性系统方程和量测方程、选择合适的核宽且初始化系统状态和协方差、根据系统方程更新状态和协方差的一步预测、在固定点初始迭代时刻再次初始化状态值、根据初始系统和量测方程进行系统模型变形且计算出变形后的误差和由此计算出误差的核函数、由随机加权准则和核函数得出两个对角阵、两个对角阵来修正一步预测协方差和量测误差从而修正增益矩阵、估计出系统后验状态和协方差，本发明对于线性模型的非高斯重尾冲击噪声问题，可以获得比卡尔曼滤波和最大互相关熵卡尔曼滤波更好的性能，可广泛应用于线性系统的噪声为非高斯情况，提高非高斯噪声情形下的滤波估计精度。
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
The invention belongs to the technical field of signal processing, and particularly relates to a maximum cross-correlation entropy Kalman filtering method based on a random weighting criterion.
Background
State estimation is an important issue in signal processing. The Kalman filtering method is one of important methods for solving the problem of state estimation under Gaussian noise of a linear system, fully utilizes a state model and observation data of the system, and minimizes the error of the state estimation by solving the problem of optimization, so as to obtain the optimal estimation of the system. However, the noise is polluted due to the system model or measurement, and the heavy tail non-gaussian noise is often generated, which causes the accuracy of the kalman filtering algorithm to be reduced or even spread, so that the filtering algorithm for the non-gaussian noise is required.
For the case that the measured noise is non-Gaussian, filtering algorithms which have appeared at present are Gaussian sum filtering, M estimation filtering based on Huber technology and t filtering based on Student's method. However, the probability density distribution of noise is known by using Gaussian and filtering, which is difficult to realize in engineering practice, because the Huber technology cannot drop after the influence parameter gamma exceeds 1.345, so that the estimation performance is reduced, and student t filtering can only be used in the case that the covariance of system noise and the covariance of measurement noise are small. Therefore, it is important to propose a new kalman filtering method suitable for the linear system, which improves both the robustness and the noise immunity. For this reason, researchers have proposed a new method for solving heavy-tail non-gaussian noise, i.e., a filtering method based on maximum cross-correlation entropy, such as maximum cross-correlation entropy kalman filtering, etc. Unlike conventional filtering methods, the cross-correlation entropy includes not only second-order statistical information but also higher-order statistical information, so that a better estimation effect can be obtained.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a maximum cross-correlation entropy Kalman filtering method based on a random weighting criterion, which is suitable for a linear system under the condition that noise is non-Gaussian, and improves the robustness and noise immunity of the system.
In order to achieve the above object, the present invention is achieved by the following technical means.
A maximum cross-correlation entropy Kalman filtering method based on a random weighting criterion comprises the following steps:
step one, constructing a linear system equation and a measurement equation as follows:
Wherein k-1 represents k-1 time, x k∈Rn is n-dimensional system state vector at k time, z k∈Rm is m-dimensional measurement vector at k time, F k-1 and H k are known transfer matrix and measurement matrix respectively, q k-1∈Rn is n-dimensional system noise at k-1 time, r k∈Rm is m-dimensional measurement noise at k time, system noise obeys Gaussian distribution q k-1～N(0,Qk-1), measurement noise is non-Gaussian obeys mixed Gaussian distribution r k～λN(0,Rk,1)+(1-λ)N(0,Rk,2),qk-1 and r k and is uncorrelated process and measurement Gaussian noise, and the requirements are met
Where E [. Cndot. ] represents a mathematical expectation, delta kj is a Croneck sign function,A transpose vector representing the hybrid noise vector r j;
initializing, selecting a kernel width sigma, and initializing the system state And covariance P (0|0), let k=1;
Step three, updating the prior state according to a one-step prediction equation of the system And covariance P k|k-1;
step four, initializing the state value again at the fixed point iteration moment, wherein t=1 and
Fifthly, performing system model deformation according to the initial system and the measurement equation, and calculating an error of the new model, thereby calculating a kernel function of the error;
firstly, reconstructing a state equation and a measurement equation:
Wherein E [. Cndot. ] represents mathematical expectation, P k|k-1 is a state one-step prediction error covariance matrix at the kth time, R k is a measurement noise covariance matrix at the kth time, B P (k|k-1) is a matrix obtained by performing Cholesky decomposition on P k|k-1, Is the transposed matrix of B P (k|k-1), and similarly, B R (k) is a matrix obtained by performing Cholesky decomposition on R k,Is the transpose of B R (k), B k is a new diagonal matrix of B P (k|k-1) and B R (k);
In equation (d) Is multiplied by simultaneously two sides ofObtaining:
Dk＝Wkxk+ek
Wherein the method comprises the steps of The i-th column element of the error vector e k is:
ek(i)＝di(k)-wi(k)xk(i)
where D i (k) is the i-th element of D k, W i (k) is the i-th row element of matrix W k, x k (i) here represents the i-th state quantity of x k, and D k is an l=n+m-dimensional vector;
Step six, obtaining two diagonal arrays by a random weighting criterion and a kernel function;
due to the random weighting criteria, a new cost function is defined:
Wherein the method comprises the steps of G σ (·) gaussian kernel:
Here, we take:
then x k (i) optimal solution:
the matrixing form is as follows:
Wherein the method comprises the steps of And is also provided with Obtaining two diagonal matrixes
Step seven, two diagonal arraysTo correct one-step prediction covarianceAnd measurement error covariance
Thereby correcting the gain matrix;
estimating the posterior state of the system filtering
Sum covariance
If k+1=n, where N is the preset algorithm iteration number, stopping calculation, otherwise continuing to execute the above steps.
Compared with the prior art, the invention has the beneficial effects that:
The invention improves the accuracy of MCKF by introducing MCKF a random weighting criterion, increases the related entropy by adopting the random weighting theory in the cost function, accords with the maximum related entropy criterion (MCC), improves the estimation accuracy, and improves the state estimation accuracy and the estimation effectiveness compared with the traditional KF algorithm and MCKF algorithm through experiments.
Drawings
FIG. 1 is a flow chart of a RWMCKF-based method according to the present invention;
Fig. 2 is a probability density function for state x 1 at KF, MCKF (σ=2), and RWCKF;
Fig. 3 is a probability density function for state x 2 at KF, MCKF (σ=2), and RWCKF;
Fig. 4 is a probability density function for state x 1 at MCKF (σ=2) and RWCKF;
fig. 5 is a probability density function for state x 2 at MCKF (σ=2) and RWCKF.
Detailed Description
The invention is described in further detail below with reference to the drawings and examples. For a better understanding of the method of the present invention, the network structure of the present invention will be described in detail.
The invention relates to a maximum cross-correlation entropy Kalman filtering method based on a random weighting criterion, which comprises the following steps:
step one, constructing a linear system equation and a measurement equation as follows:
Wherein k-1 represents the k-1 moment, x k∈Rn is the n-dimensional system state vector at the k moment, z k∈Rm is the m-dimensional measurement vector at the k moment, F k-1 and H k are respectively known transfer matrix and measurement matrix, q k-1∈Rn is the n-dimensional system noise at the k-1 moment, r k∈Rm is the m-dimensional measurement noise at the k moment, the system noise is assumed to obey Gaussian distribution q k-1～N(0,Qk-1), the measurement noise is non-Gaussian mixture Gaussian distributions r k～λN(0,Rk,1)+(1-λ)N(0,Rk,2),qk-1 and r k are uncorrelated process and measurement Gaussian noise, and the requirements are satisfied
Where E [. Cndot. ] represents a mathematical expectation, delta kj is a Croneck sign function,A transpose vector representing the hybrid noise vector r j;
initializing, selecting a proper kernel width sigma through experience, and initializing system state And covariance P (0|0), let k=1;
Step three, updating the prior state according to a one-step prediction equation of the system And covariance P k|k-1;
step four, initializing the state value again at the fixed point iteration moment, wherein t=1 and
Fifthly, performing system model deformation according to the initial system and the measurement equation, and calculating an error of the new model, thereby calculating a kernel function of the error;
firstly, reconstructing a state equation and a measurement equation:
Wherein E [. Cndot. ] represents mathematical expectation, P k|k-1 is a state one-step prediction error covariance matrix at the kth time, R k is a measurement noise covariance matrix at the kth time, B P (k|k-1) is a matrix obtained by performing Cholesky decomposition on P k|k-1, Is the transposed matrix of B P (k|k-1). Similarly, B R (k) is a matrix obtained by performing Cholesky decomposition on R k,Is the transpose of B R (k), B k is a new diagonal matrix of B P (k|k-1) and B R (k);
In equation (d) Is multiplied by simultaneously two sides ofObtaining:
Dk＝Wkxk+ek
Wherein the method comprises the steps of The i-th column element of the error vector e k is:
ek(i)＝di(k)-wi(k)xk(i)
Where D i (k) is the i-th element of D k, W i (k) is the i-th row element of matrix W k, x k (i) here represents the i-th state quantity of x k, and D k is an l=n+m-dimensional vector.
Step six, obtaining two diagonal arrays by a random weighting criterion and a kernel function;
due to the random weighting criteria, a new cost function is defined:
Wherein the method comprises the steps of G σ (·) gaussian kernel:
Here, we take:
then x k (i) optimal solution:
the matrixing form is as follows:
Wherein the method comprises the steps of And is also provided with Obtaining two diagonal matrixes
Step seven, two diagonal arraysTo correct one-step prediction covarianceAnd measurement error covariance
Thereby correcting the gain matrix;
Estimating the posterior state and covariance of the system filtering, specifically,
At this time, the estimation of the target state parameter can be completedSum-state estimation error covariance matrixWill be used for the estimation of the state parameter at the next moment.
The invention adopts the random weighted maximum cross-correlation entropy Kalman filtering (RWMCKF) algorithm to perform state estimation on the linear system under the condition of non-Gaussian heavy tail noise, and the application of the random weighting criterion enhances the robustness of the system and improves the filtering precision. According to the invention, MATLAB simulation software is used for carrying out simulation experiments, RWMCKF algorithm is compared with the existing filtering algorithms KF and MCKF to obtain the state estimation accuracy and estimation effectiveness are greatly improved.
The invention is illustrated by the following specific examples.
Example 1
Consider a general linear system model:
Wherein θ=pi/18, and the system noise is Gaussian noise q i (k-1) to N (0, 2) (i=1, 2), and the measurement noise is non-Gaussian mixed Gaussian noise r (k) to 0.9N (0, 1) +0.1N (0, 100).
According to the maximum correlation entropy Kalman filtering method based on the random weighting criterion, the state initial value is takenThe error covariance matrix takes P (0|0) =diag (100 ). The kernel width σ of the gaussian kernel function is set to 0.1, 0.5, 1,2, 3,5, 8, 10, and compared with KF algorithm and MCKF (σ=2), respectively, to obtain probability density functions of the two states x 1、x2 of fig. 2 and 3 under different filtering algorithms. The simulation result of the maximum correlation entropy Kalman filtering method based on the random weighting criterion is shown as a curve RWMCKF, compared with the traditional KF and MCKF, the performance advantage is obvious, the KF algorithm is biased under the condition of heavy tail noise measurement, MCKF and RWMCKF are unbiased, but the unbiasedness of the RWCKF algorithm is more advantageous than that of the MCKF algorithm.
The effect of the kernel width σ of the gaussian kernel on the maximum correlation entropy kalman filter of the provided random weighting criteria of the present invention can be derived from fig. 3, 4 that RWCKF and MCKF have the similarity that σ=2 is most effective.
Embodiment two:
the simulation model is replaced by one-dimensional linear uniform acceleration motion, and the system model and the measurement model are as follows:
wherein Δt=0.1 s, and both the system noise and the measurement noise are nonlinear mixed gaussian noise:
q1(k-1)~0.9N(0,0.01)+0.1N(0,1)
q2(k-1)~0.9N(0,0.01)+0.1N(0,1)
q3(k-1)~0.9N(0,0.01)+0.1N(0,1)
r(k)~0.8N(0,0.01)+0.2N(0,100)。
According to the maximum correlation entropy Kalman filtering method based on the random weighting criterion, a filtering algorithm is initialized first. The state initial value and the covariance initial value are set to x (0) = [001] T, P (0|0) =diag (0.01,0.01,0.01), respectively. The kernel width σ of the gaussian kernel function is set to 2 (i.e., the relatively optimal kernel width in example 1), then a mean square error table under the algorithms KF, MCKF (σ=2) and RWMCKF (σ=2) can be obtained:
mean square error under the algorithms of tables IKF, MCKF (σ=2) and RWMCKF (σ=2)
Table I shows the mean square error of KF, MCKF (sigma=2) and RWMCKF (sigma=2), respectively, and it can be derived from Table I that the mean square error of the maximum correlation entropy Kalman filtering variance based on the random weighting criterion is minimum under the same Gaussian mixture system and measurement noise influence, which shows that the performance of the algorithm is greatly improved compared with KF and MCKF.
The foregoing is merely illustrative of the present invention, and the present invention is not limited thereto, and any changes or substitutions easily contemplated by those skilled in the art within the scope of the present invention should be included in the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.
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