A Strongly Nonlinear Non-Gaussian State Estimation Method Based on Affine Multi-Kernel Maximum Correlation Entropy

Abstract

This invention relates to a strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy, comprising: acquiring process noise and measurement noise to construct a composite non-Gaussian noise model; in the outer iteration of the dynamic framework, linearizing the model based on iterative affine approximation to obtain an optimal local linearization model for constructing a robust cost function; in the inner iteration of the dynamic framework, performing derivative calculation with the objective of maximizing the robust cost function, and using a Kalman gain analytical model containing robust weighting factors for posterior state update, and further recursively correcting the posterior error covariance to obtain the optimal posterior estimate and covariance at the target time, then constructing an inverse weighted multi-kernel cost function, performing derivative calculation with the objective of maximizing the inverse weighted multi-kernel cost function, and using the inverse gain matrix to propagate the optimal posterior estimate and covariance at the target time to the previous target time for inverse correction, in order to solve for the optimal linearization parameters.

Description

Technical Field

[0001] This invention relates to the field of dynamic system state estimation and information fusion technology, and in particular to a strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy. Background Technology

[0002] State estimation is fundamental to dynamic system analysis and control, and its task is to infer the internal state of a system in real time under noisy observations. Since the introduction of Kalman filtering (KF), state estimation has been extended from linear Gaussian scenarios to nonlinear and non-Gaussian scenarios, but existing methods still have significant shortcomings in complex engineering.

[0003] In linear systems where both process noise and measurement noise are zero-mean Gaussian distributed, KF can achieve unbiased optimal estimation due to its advantages such as recursive prediction and update, and optimal mean square error. However, real-world systems generally have strong nonlinearity and complex noise, such as angular velocity coupling in target tracking, attitude updates in inertial navigation, coordinate transformations in radar measurements, heavy tail noise caused by electromagnetic interference, multipath effects, and device aging, multimodal noise formed by the superposition of errors from multiple sensors, and bias noise caused by sensor misalignment, calibration errors, and incomplete models.

[0004] For nonlinear problems, the Extended Kalman Filter (EKF) relies on Taylor expansion for local linearization. Because it ignores higher-order terms, it is prone to cumulative truncation errors under strong nonlinearity or high uncertainty conditions, leading to estimation bias or even divergence. While the Unscented Kalman Filter (UKF) and Capacitive Kalman Filter (CKF) improve nonlinear approximation through deterministic sampling, they are still limited by the point set coverage in strongly nonlinear regions and are essentially still based on Gaussian noise. For non-Gaussian noise, methods such as Maximum Correlation Entropy Kalman Filter (MCKF) and Maximum Correlation Entropy Unscented Kalman Filter (MCUKF) based on the Maximum Correlation Entropy Criterion (MCC) suppress large error samples, improving outlier resistance. However, single-kernel methods are sensitive to kernel bandwidth and struggle to characterize multimodal noise. While hybrid kernels such as Gaussian-Gaussian and Gaussian-Cauchy enhance adaptability to complex noise, the sub-kernel centers are usually fixed at zero, implicitly containing zero mean. In the presence of bias noise, they cannot accurately describe the true error distribution, easily leading to weight mismatch and performance degradation. More importantly, most existing methods separate nonlinear modeling from non-Gaussian noise suppression, causing model error and noise error to couple and amplify each other during the filtering process.

[0005] Therefore, existing technologies have at least three shortcomings: a lack of high-precision approximation methods for strongly nonlinear systems; a lack of effective modeling mechanisms for bias noise; and a lack of a unified framework for coordinating nonlinear modeling errors and complex non-Gaussian noise. New state estimation methods are urgently needed to address these issues. Summary of the Invention

[0006] This invention aims to address the problem of severe performance degradation or even failure of existing state estimation techniques in environments with strong nonlinear systems and biased multimodal non-Gaussian noise. It provides a strong nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy. By constructing an iterative affine approximation, a center-offset multi-kernel mixing function, and a dynamic two-layer iterative optimization framework, it achieves deep integration and synergistic optimization of nonlinear modeling accuracy and noise suppression capability, thereby achieving high-precision and robust state estimation in complex and harsh environments.

[0007] To achieve the above objectives, the present invention provides the following solution: Strongly nonlinear non-Gaussian state estimation methods based on affine multi-kernel maximum correlation entropy include: The process noise and measurement noise are acquired, and the process noise and measurement noise are modeled as a composite non-Gaussian noise model. A dynamic framework for collaborative optimization of inner and outer iterations is constructed. In the outer iteration of the dynamic framework, the composite non-Gaussian noise model is linearized based on the iterative affine approximation to obtain the optimal local linearization model, which is used to construct the robust cost function of the multi-kernel hybrid function and establish the adaptive parameter mechanism of the robust cost function. In the inner iteration of the dynamic framework, the robust cost function is maximized. The robust cost function is differentiated, and a Kalman gain analytical model incorporating robust weighting factors is used to recursively correct the posterior error covariance, obtaining the optimal posterior estimate and covariance at the target time. Further, an inversely weighted multi-kernel cost function is constructed. The inversely weighted multi-kernel cost function is differentiated, and the optimal posterior estimate and covariance at the target time are propagated to the previous target time using the inverse gain matrix. The smoothing estimate and covariance at the previous time are then corrected in reverse to obtain the optimal linearization parameters.

[0008] Optionally, acquiring the process noise and the measurement noise includes: ; ; in, Indicates the system at time 10:00 The true state vector, Represents the corresponding observation vector, and Let represent the nonlinear state transition function and measurement function of the system, respectively, and at each time step... All of the above are considered to be known mappings. This represents the dimension of the state vector. This indicates the dimension of the measurement vector. Indicates process noise. Indicates measurement noise. It is the set of real numbers.

[0009] Optionally, obtaining the optimal local linearization model includes: ; ; in, These are the optimal linear affine regression matrices for the state transition equation and the measurement equation, respectively. These are the bias terms after linearization of the state transition equation and the measurement equation, respectively. For the linearized process noise and measurement noise, These are the linearized residual covariances of the state transition equation and the measurement equation, respectively. For the system at time The true state vector, For the system at time The true state vector.

[0010] Optionally, constructing a robust cost function for a multi-core hybrid function and establishing an adaptive parameter mechanism for the robust cost function includes: ; in, The kernel mixing coefficient, for The actual observed value at time [time]. The mean center bias is set for the j-th measurement noise. Indicates the first j The optimal linear affine regression matrix corresponding to each measurement The linearized residual covariance of the state transition equation. Represents the Gaussian kernel function. Represents the Cauchy kernel function. Indicates to One-step prediction update of the state at any given moment. For the first j The linearized residual covariance corresponding to each measurement component For the first j Linearized bias term for each measurement, For the first j The responsibility of each measurement component.

[0011] Optionally, the method further includes: The derivative calculation of the robust cost function includes: ; in, For robust cost function, For the system at time The true state vector, The mixing coefficients of the kernel function, For the first j The adaptive kernel bandwidth corresponding to each measurement. For Gaussian kernel function, For the first j Residual of each measurement channel For Cauchy kernel function, The adaptive kernel bandwidth corresponding to the prediction error. To predict residuals, is the scaling parameter of the Cauchy kernel function. The inverse of the linearized residual covariance of the state transition equation. For the first j The transpose of the optimal linear affine regression matrix corresponding to each measurement. It is the inverse of the linearized residual covariance of the j-th measurement.

[0012] Optionally, the recursive correction of the posterior error covariance using the Kalman gain analytical model containing robust weighting factors includes: ; in, for t The posterior state estimate at time 10:00 To t One-step prediction update of the state at any given moment. To obtain the combined information by vertically stacking the residuals of each measurement channel in channel order, It is the inverse of the covariance of the linearized residuals. For the combined measurement matrix, It is the inverse of the covariance of the linearized residuals of the combined weighted measurement equations.

[0013] Optionally, constructing the inverse weighted multi-core cost function includes: ; in, For Gaussian kernel function, The covariance of the linearized residuals. for t The posterior state estimation at time 10:00 Let be the optimal linear regression matrix for the state transition equation. The bias term is the linearized form of the state transition equation. This is the mean center bias term for process noise. The mixing coefficients of the kernel function, This refers to the process noise after linearization. fort Posterior state estimate at time -1 for t Posterior estimate of covariance at time -1 for t Posterior state estimation at time 10:00 for t The true state at time -1 for t The estimation error of the state at time -1.

[0014] Optionally, calculating the derivative of the inverse weighted multi-core cost function includes: ; in, For inverse weighted multi-core cost function, Let be the optimal linear regression matrix for the state transition equation. for t The true state at time -1 The bias term is the linearized form of the state transition equation. This is the mean center bias term for process noise. The kernel mixing coefficient, For the Gaussian kernel function adaptive bandwidth during reverse update, This is the transpose of the state-optimal linear regression matrix. It is the inverse of the covariance of the linearized residuals. for t Posterior state estimation at time 10:00 The Gaussian kernel function used for measurement. for t Posterior estimate of covariance at time -1 for t The true state at time -1 for t Posterior state estimate at time -1 For the Gaussian kernel function used for the state, is the scaling parameter of the Cauchy kernel function. For the adaptive bandwidth of the Cauchy kernel function, The Cauchy kernel function used for measurement. This is the transpose of the optimal linear regression matrix. For the Cauchy kernel function used for the state, for t Posterior estimate of covariance at time -1.

[0015] Optionally, the inverse gain matrix is ​​defined as follows: ; in, Let be the optimal linear regression matrix for the state transition equation. The Kalman gain during reverse update. for t Posterior estimate of covariance at time -1 Forward update-driven during reverse update t Adaptive weighted intensity at time step This is the transpose of the optimal linear regression moments of the state transition equation. It is the inverse of the covariance of the linearized residuals. For reverse updates t Adaptive weighting strength dominated by the posterior state estimation at time -1.

[0016] The beneficial effects of this invention are as follows: This invention constructs a unified filtering framework that deeply integrates iterative affine approximation and center-offset multi-kernel maximum correlation entropy criterion, realizing the synergistic optimization of nonlinear model linearization and non-Gaussian noise suppression, fundamentally solving the structural error coupling problem caused by the isolated treatment of the two in the prior art.

[0017] This invention proposes a multi-core mixing function with adjustable center parameters, which incorporates the sub-core centers into the optimization process and directly models the statistical characteristics of non-zero mean, skewed, and multimodal noise, thus overcoming the fundamental limitation of the traditional correlation entropy method that implicitly assumes zero mean.

[0018] Furthermore, this invention employs an iterative affine approximation based on statistical linearization to replace the traditional single-point Taylor expansion, and combines a two-layer iterative optimization framework to achieve information backpropagation and dynamic relinearization of linearized points, which significantly reduces the high-order truncation error of strongly nonlinear systems and ensures the consistency of estimation. Attached Figure Description

[0019] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0020] Figure 1 This is a flowchart of the strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy according to an embodiment of the present invention. Figure 2 This is a schematic diagram of the target trajectory simulation results under air traffic control using the strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy, according to an embodiment of the present invention. Figure 3This is a schematic diagram of the target trajectory error results under air traffic control using the strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy according to an embodiment of the present invention. Figure 4 This is a target trajectory error distribution diagram under air traffic control for the strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy, according to an embodiment of the present invention. Figure 5 This diagram illustrates the verification results of the strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy in this embodiment of the invention on a measured dataset of lithium ions during the charging process. Detailed Implementation

[0021] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0022] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0023] like Figure 1 As shown, this embodiment discloses a strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy, including: acquiring process noise and measurement noise, modeling the process noise and measurement noise as a composite non-Gaussian noise model, and constructing a dynamic framework for coordinated optimization of inner and outer iterations; in the outer iteration of the dynamic framework, linearizing the composite non-Gaussian noise model based on iterative affine approximation to obtain an optimal local linearization model, which is used to construct a robust cost function for the multi-kernel mixture function, and establishing an adaptive parameter mechanism for the robust cost function; in the inner iteration of the dynamic framework, ... To maximize the robust cost function, the derivative of the robust cost function is calculated, and a Kalman gain analytical model with robust weighting factors is used to recursively correct the posterior error covariance, obtaining the optimal posterior estimate and covariance at the target time. Further, an inverse weighted multi-kernel cost function is constructed, and to maximize this function, the derivative of the multi-kernel cost function is calculated. The optimal posterior estimate and covariance at the target time are propagated to the previous target time using the inverse gain matrix, and the smoothing estimate and covariance at the previous time are corrected in reverse to obtain the optimal linearization parameters.

[0024] Specifically, this embodiment discloses a strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy. Its key feature is that, for complex systems simultaneously exhibiting strong nonlinearity, multimodal distribution, heavy-tailed characteristics, and non-zero mean bias noise, a state estimation framework integrating affine linearization, multi-kernel hybrid cost function, and two-layer iterative optimization is constructed, including the following steps: Step S1: First, model the target state transition and measurement generation, then model the complex non-Gaussian noise with multimodal, heavy-tailed characteristics and non-zero mean bias, and finally initialize the motion state parameters. Step S2: Linearize the nonlinear model based on iterative affine approximation, and calculate the optimal approximation affine parameters of the nonlinear state function and measurement function through weighted statistics to construct the optimal local linearization model; Step S3: Design a multi-kernel hybrid function with adjustable center parameters to construct a robust cost function that integrates prediction error terms and multimodal measurement residual terms, and establish an adaptive parameter mechanism for the cost function so that the cost function can adapt to time-varying noise environments to match real-time noise statistical characteristics. Step S4: Maximize the robust cost function, derive the analytical expression of the Kalman gain including the robust weighting factor, combine the combined innovation and the combined measurement matrix to achieve the closed-loop update of the state estimate, and then perform the recursive correction of the a priori error covariance. Step S5: Construct a dynamic framework for collaborative optimization of inner and outer iterations. Through multiple backward propagation of information over time and dynamic correction of linearization points, achieve a synergistic improvement in nonlinear modeling accuracy and noise suppression capability, and complete real-time high-precision tracking of the target state.

[0025] In step 1, the target transition state and measurement generation modeling, non-Gaussian distribution modeling, and state parameter initialization are as follows: (1); (2); in, Indicates the system at time 10:00 The true state vector, Represents the corresponding observation vector. Function and Let represent the nonlinear state transition function and measurement function of the system, respectively, and at each time step... All of the above are considered to be known mappings. This represents the dimension of the state vector. This represents the dimension of the measurement vector. To characterize the inherent uncertainties in the system, process noise is introduced. and measurement noise , for 3D real vector space, for A real vector space. These two represent the random disturbances in the system's dynamics process and the uncertainties in the observation process, respectively, assuming they are independent. Correspondingly, the covariance matrices of the process noise and the measurement noise are denoted as... and In complex dynamic environments, when process noise and measurement noise exhibit statistical bias and heavy-tailed characteristics, the zero-mean Gaussian mixture model of traditional filters, if it has significant limitations, will directly lead to linearization distortion in state prediction and underestimation of error covariance, ultimately resulting in a decline in filtering performance. To address this issue, this invention employs a hybrid multi-component student model. Modeling noise using distributions offers the advantage of flexibly representing the bias and heavy-tailed characteristics of noise. If process noise... and measurement noise They respectively satisfy the following formulas: (3); in and These represent the mean center bias terms for process noise and measurement noise, respectively. and They are students t The scale matrix of the distribution and Students Distribute the corresponding degrees of freedom parameters. Represents the noise distribution coefficient. Students t distributed.

[0026] Step 2 linearizes the nonlinear model based on iterative affine approximation to construct an optimal locally linearized model: (4); (5); in, These are the optimal linear affine regression matrices for the state transition equation and the measurement equation, respectively. These are the bias terms for the state transition equation and the measurement equation, respectively. For the linearized process noise and measurement noise, These are the linearized residual covariances of the state transition equation and the measurement equation, respectively.

[0027] Design a multi-kernel hybrid function with adjustable center parameters to construct a robust cost function that integrates prediction error terms and multimodal measurement residual terms, and establish an adaptive parameter mechanism for the cost function so that the cost function can adapt to time-varying noise environments to match real-time noise statistical characteristics.

[0028] (6); in, N To measure the number of components, The kernel mixing coefficient, for The actual observed value at time [time]. For the first j Mean center bias of measurement noise Indicates the first j The optimal linear affine regression matrix corresponding to each measurement The linearized residual covariance of the state transition equation. Represents the Gaussian kernel function. Represents the Cauchy kernel function. Indicates to t One-step prediction update of the state at any given moment. For the first j The linearized residual covariance corresponding to each measurement component For the first j Linearized bias term for each measurement, For the first j The responsibility of each measurement component.

[0029] By maximizing the robust cost function, an analytical expression for the Kalman gain including a robust weighting factor is derived. This expression is then combined with the combined innovation and combined measurement matrix to achieve a closed-loop update of the state estimate, followed by recursive correction of the a posteriori error covariance. Indicates the first Residual of each measurement channel Indicates the prediction error. Indicates new information from the combination. For the first j The adaptive kernel bandwidth corresponding to each measurement. The adaptive kernel bandwidth corresponding to the prediction error. for t The posterior state estimation at time 10:00 for t The state at time -1 for t The state posterior at time -1 This represents the degrees of freedom of the Cauchy kernel function.

[0030] (7); in, For the first j The inverse of the linearized residual covariance of a measurement Indicates the first j The inverse of the optimal linear affine regression matrix corresponding to each measurement.

[0031] but tThe posterior state at time 1 can be estimated by the following formula: (8) in, It is the inverse of the covariance of the linearized residuals of the combined weighted measurement equations.

[0032] To further refine the estimation, an inverse weighted multi-kernel cost function is constructed based on the above and its derivative is calculated: (9); in, For inverse weighted multi-core cost function, Let be the optimal linear regression matrix for the state transition equation. for t The true state at time -1 The bias term is the linearized form of the state transition equation. This is the mean center bias term for process noise. The kernel mixing coefficient, For the Gaussian kernel function adaptive bandwidth during reverse update, This is the transpose of the state-optimal linear regression matrix. It is the inverse of the covariance of the linearized residuals. for t Posterior state estimation at time 10:00 For use t The Gaussian kernel function at time posterior time. for t Posterior estimate of covariance at time -1 for t The true state at time -1 for t Posterior state estimate at time -1 For use t The Gaussian kernel function of the state at time -1 is the scaling parameter of the Cauchy kernel function. For the adaptive bandwidth of the Cauchy kernel function, For use t The Cauchy kernel function at time posterior. This is the transpose of the optimal linear regression matrix. For use t The Cauchy kernel function for the state at time -1. Then... t Re-estimation of the state at time -1 Represented as: (10); in The Kalman gain during reverse update. for t Posterior state estimate at time -1 for t Posterior state estimation at time 10:00 Indicates to t One-step prediction update of the state at any given time.

[0033] A dynamic framework for collaborative optimization of inner and outer iterations is constructed. Through multiple backpropagations of information over time and dynamic corrections of linearization points, a synergistic improvement in nonlinear modeling accuracy and noise suppression capability is achieved, resulting in collaborative optimization of the model and estimate. The inner iteration focuses on solving the problem under fixed linearization parameters. t The optimal robust state estimate at time -1 is obtained; the outer iteration dynamically reconstructs the linearization point and corrects the covariance using the inner results, ensuring the consistency of statistical linearization. Both iteratively progress, working together to achieve the optimal balance between estimation accuracy and numerical stability. In the current... k The linearization parameters of the outermost iteration Given the given conditions, the inner iterations apply a cost function based on mixed correlation entropy. Optimization is performed to solve for the optimal state estimate. The maximum correlation entropy criterion, by introducing a Gaussian kernel function to weight the residuals, effectively reduces the weight of outlier observations and model bias, thus endowing the algorithm with strong robustness. Specifically, in each iteration... The weight matrix and covariance scaling term will be recalculated based on the current residual, and the gradient will be adjusted accordingly. Set to zero to update state estimate This iterative process continues until the state update amount tends to stabilize, that is, it satisfies... End at time, To solve for the threshold when estimating the optimal state, In the first k The inner layer iteration under the next outer layer iteration l The next iteration t State estimation at time 10:00 In the first k The inner layer iteration under the next outer layer iteration l +1 iterations t State estimation at time 1. At this point, the output is... That is, under the current linearization parameters t Robust optimal estimate at time step; then based on The latest posterior information at time step is propagated backward through the inverse gain matrix to At any given moment, an information consistency correction mechanism is formed. This mechanism utilizes the current more accurate posterior estimate. Inversely correct the smoothing estimate from the previous time step. and its corresponding covariance .

[0034] To further overcome the inherent approximation error of single-step linearization, this invention constructs an outer iterative structure in addition to the core filtering step. Its core idea is: based on the correction value from the previous time step... and Regenerate Sigma points and execute the first step. k +1 statistical linearization yields a set of higher-precision results. k Affine parameters corresponding to +1 outer iteration ,in For the first k The optimal linear regression matrix for the nonlinear state equation corresponding to the +1 outer iteration. No. k +1 outer iteration of the nonlinear state equation bias term after affine approximation For the first k The covariance of the linearized residuals of the nonlinear state equation corresponding to +1 outer iteration. For the first k The optimal linear regression matrix of the measurement equation corresponding to the +1 outer iteration. For the first k The bias term of the measurement equation after the +1 outer iteration is obtained by affine approximation. For the first k The covariance of the linearized residuals of the measurement equation corresponding to +1 outer iteration. The correction amount for the outer iterations up to the linearization point is sufficiently small, i.e. Stop when the linearization parameters are obtained, and the linearization parameters obtained at this point are considered to be optimal. To solve for the threshold of the optimal affine approximation parameters, In the first k +1 outer iterations for the first... t Backward estimation at time -1.

[0035] This dynamic bi-level iterative optimization framework organically integrates the robust non-Gaussian estimation of the inner layer with the statistical consistency correction of the outer layer. The inner layer ensures the estimation accuracy under anomalous disturbances, while the outer layer improves the ability to approximate the dynamics of the real system from the root of linearization.

[0036] More specifically, this embodiment provides a strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy, including the following steps: Step S1: To characterize the unavoidable uncertainties in the actual system, process noise and measurement noise are modeled as a composite non-Gaussian noise model with multimodal distribution, heavy-tailed characteristics and non-zero mean bias, providing a modeling object for non-Gaussian noise processing. Step S2: Linearize the nonlinear model based on iterative affine approximation, calculate the optimal approximation affine parameters of the nonlinear state function and measurement function through weighted statistics, and construct the optimal local linearization model so that the subsequent multi-kernel cost function can differentiate the weights of the residuals of each mode. Step S3: Design a multi-kernel hybrid function with adjustable center parameters to construct a robust cost function that integrates prediction error terms and multimodal measurement residual terms, and establish an adaptive parameter mechanism for the cost function so that the cost function can adapt to time-varying noise environments to match real-time noise statistical characteristics. Step S4: Using the maximum correlation entropy as the optimization criterion, by maximizing the robust cost function, the analytical expression of the Kalman gain containing the robust weighting factor is derived. The closed-loop update of the state estimate is achieved by combining the combined innovation and the combined measurement matrix, and the recursive correction of the a priori error covariance is then performed. Step S5: Construct a dynamic framework for the coordinated optimization of inner and outer iterations. Through multiple backward propagation of information over time and dynamic correction of linearization points, the accuracy of nonlinear modeling and noise suppression capabilities are improved in a coordinated manner, thereby achieving coordinated optimization of the model and the estimate.

[0037] In many practical engineering systems, the dynamic evolution of states and its observation mechanisms often exhibit significant nonlinear characteristics, making system modeling and state estimation more complex. To characterize the dynamic behavior of such nonlinear systems, this invention considers a discrete-time nonlinear state-space model. Without loss of generality, if the system is affected by additive noise, its state transition equation and measurement equation are expressed as follows: (11); (12); in, Indicates the system at time 10:00 The true state vector, Represents the corresponding observation vector. Function and Let represent the nonlinear state transition function and measurement function of the system, respectively, and at each time step... All of the above are considered to be known mappings. This represents the dimension of the state vector. This represents the dimension of the measurement vector. To characterize the inherent uncertainties in the system, process noise is introduced. and measurement noise These two variables respectively reflect the random disturbances in the system's dynamics process and the uncertainties in the observation process, assuming they are independent of each other. Accordingly, the covariance matrices of the process noise and the measurement noise are denoted as follows: and In complex dynamic environments, when process noise and measurement noise exhibit statistical bias and heavy-tailed characteristics, the zero-mean Gaussian mixture model of traditional filters, if it has significant limitations, will directly lead to linearization distortion in state prediction and underestimation of error covariance, ultimately resulting in a decline in filtering performance. To address this issue, this invention employs a hybrid multi-component student model. Modeling noise using distributions offers the advantage of flexibly representing the bias and heavy-tailed characteristics of noise. If process noise... and measurement noise They respectively satisfy the following non-Gaussian noise models: (13); in, and These represent the mean center bias terms for process noise and measurement noise, respectively. and They are students t The scale matrix of the distribution and For the corresponding degrees of freedom parameters, This represents the noise distribution coefficient.

[0038] like The posterior distribution at time t follows a Gaussian distribution. Furthermore, since process noise is independent of the state, and its expectation is... ,but State prior estimation at time 1 It can be represented as (14); in, To t All measurement sequences up to time -1 To change the state from t -1 time is mapped to t The nonlinear state transition function at time t.

[0039] This shows that the bias of process noise only manifests as a shift in the state transition mean and does not affect the nonlinear propagation of higher-order covariances. Therefore, the covariance of the prior error... Represented as: (15); in, For students The covariance of the distribution. Further, the cross-covariance between the state and the nonlinear propagation can be defined. : (16); in, for State prior estimation at time step.

[0040] This invention is for dimensional state vector Still select symmetrically Sigma point and their corresponding weights These points are propagated through a nonlinear state transition model to obtain the transformed point set. Then formulas (14)-(16) can be further approximated as weighted summation: (17); (18); (19); Based on the above statistics, this invention seeks the optimal affine approximation of the nonlinear state equation in the sense of minimum mean square error. The optimal linear regression matrix for the nonlinear state equation... and bias terms It can be calculated using the following formula: (20); (twenty one); Meanwhile, the covariance of the linearized residuals of the nonlinear state equations Defined as: (twenty two); Therefore, we can obtain One-step prediction of state at time step : (twenty three); and The true state of a moment Represented as: (twenty four); in, This represents the linearized process noise. Therefore, the one-step prediction error... Represented as: (25); but t Covariance of prediction error at time It can be calculated using the following formula: (26); As can be seen from the above derivation, the affine prediction model, while preserving the statistical characteristics of the nonlinear system, incorporates the bias term. The state transition mean term is naturally incorporated, thus enabling explicit modeling of the system bias during the prediction phase; the corresponding prediction covariance... It not only reflects the second-order statistical characteristics of nonlinear propagation error, but also... The tail weights of the distribution characterize the potential heavy-tailed uncertainty, providing statistically robust prior information for subsequent measurement updates. and During the measurement phase, the following is introduced: Actual observation value at time This is used to correct the prediction, thereby obtaining a more accurate posterior state estimate. Similar to the prediction step, measurement updates also involve updating the nonlinear measurement function. Statistical linearization. It should be noted that measurement noise is independent, therefore... , This is the mean of the measurement noise. Measurement one-step prediction. Measurement of new information covariance and the cross-covariance between state and measurement Defined by the following integral: (27); (28); (29); in for The effective covariance of the measurement noise distribution, and only if the noise distribution degrees of freedom The second moment exists. To achieve an efficient approximation, the Sigma point sampling method is used to calculate the above integral. A one-step prediction is then made based on the state. Covariance of one-step prediction error The following state prediction Sigma points can be obtained. and The measurement Sigma point is obtained through propagation via the measurement model. The measurement one-step prediction is obtained by weighted summation of the above integrals. Measurement of new information covariance and the cross-covariance between state and measurement : (30); (31); (32); in, For the measurement prediction point set, This is the set of state prediction points.

[0041] Based on the above statistics, the key parameters for the optimal affine approximation required for the measurement equation can be calculated: the optimal linear regression matrix of the measurement equation. Bias terms and linearized residual covariance The calculation is as follows: (33); (34); (35); at this time, The measurement state at time t is represented as follows: (36); At this point, the system's state transition and measurement model have both been transformed into a linear form in a statistical sense.

[0042] However, when the noise mean is not zero, a fixed zero-center kernel will lead to a mismatch in the similarity metric center in scenarios with non-zero mean residuals, thus causing a bias in weight allocation. Therefore, to characterize the similarity metric with biased non-Gaussian noise, this invention constructs the following robust cost function: (38); at this time, N To measure the number of components, The kernel mixing coefficient is used to control the proportion and heavy-tailedness of the Gaussian kernel function and Cauchy kernel function in the measurement residuals. Indicates the first j The optimal linear affine regression matrix corresponding to each measurement The linearized residual covariance of the state transition equation. Represents the Gaussian kernel function. This represents the Cauchy kernel function. Indicates to t One-step prediction update of the state at any given moment. For the first j The linearized residual covariance corresponding to each measurement component For the first j Linearized bias term for each measurement, For the first j The degree of responsibility for each measurement component Indicates the first j The mean center bias of the measurement noise. For the first j The adaptive kernel bandwidth corresponding to each measurement. The adaptive kernel bandwidth corresponding to the prediction error. for t The posterior state estimation at time 10:00 This represents the degrees of freedom of the Cauchy kernel function. Indicates the first Residual of each measurement channel Indicates the prediction error, if Under the condition of local consistency, The optimal state estimate at time step 1 can be obtained by maximizing the cost function. Obtain. For the cost function about Differentiation yields: (39); For the posterior responsibility of the noise component. For the first j The scaling matrix corresponding to each measurement noise component at the current time; , The adaptive weighting strengths of the measurement and prediction terms were characterized separately, assuming... ,but Let the gradient of the cost function be... We can obtain: (40); At this time, the j The measurement residuals can be simplified to: (41); Define the combined information obtained by vertically stacking the residuals of each measurement channel in channel order. Combined measurement matrix Inverse of the covariance of the linearized residuals of the combined weighted measurement equation : (42); (43); achievable t The posterior update equation for the temporary state at time t: (44); The robust form of the Kalman gain is: (45); Based on the true state and the estimated posterior state, at this time t The estimation error of the state at time step 1 can be expressed as: (46); make ,at this time t A posteriori and t The difference between the actual state at any given moment It can be represented as follows: (47); The error covariance of the posterior estimate with biased non-Gaussian noise can then be expressed as: (48); intermediate variables It can be calculated as follows: (49); in, For the first N The autocovariance matrix of the noise residuals of each measurement channel.

[0043] Then its block The elements are: (50); For mixed students Noise, including: (51); therefore: (52); in, For the first j The scaling matrix corresponding to each measurement noise component at the current time.

[0044] In the aforementioned filtering framework, this invention obtains Posterior estimation at time However, since the system dynamic model and noise distribution may have non-Gaussian or time-varying biases, forward filtering alone cannot fully utilize the full potential. Use real-time information to correct Time-time estimation. Therefore, it can be further achieved by constructing an inverse cost function based on the posterior residual. Self-consistent correction of time-state and adaptive optimization of transition model parameters. To this end, this invention constructs the following inverse weighted multi-kernel cost function: (53); State re-estimation at time step by maximizing the cost function Obtain. To Taking the derivative and setting the gradient to 0, we get: (54); set up Then the above equation can be simplified to the following matrix balance condition: (55); Substituting into the linearized approximation: (56); The solution can be obtained t The reverse update equation for the state at time -1: (57); And define the inverse gain matrix. for: (58); at this time The state can be re-estimated as: (59); For recalculation The error covariance at time step can be used to reconstruct the state error as follows: (60); in It can also reflect the statistical coupling of the state in the estimation domain. Taking the covariance of both sides and assuming that the estimation error and the estimated value are approximately uncorrelated, the following covariance relationship can be obtained by rearranging the terms in the above equation: (61); To eliminate the overlap between the estimation error and the state estimate, the following approximation is used: (62); By the variance decomposition theorem, under the condition of unbiased estimation: (63); in and Let the error covariances before and after the update be respectively. Substituting these values ​​into the above equation, we get: (64); (65); By combining the covariance equations, we can obtain: (66); This recursion essentially feeds back the incremental information from the current measurement update to the previous time step through a linear mapping, thereby improving the confidence level of the previous time step in a strictly semi-definite sense, thus achieving time consistency correction and information feedback in the multi-layer filtering system.

[0045] In complex systems characterized by strong nonlinearity and non-Gaussian noise, traditional single-step statistical linearization strategies have significant limitations. Their affine filtering structures often fail to fully capture the system's higher-order nonlinear characteristics due to improper selection of linearization points, leading to accumulated estimation bias and distortion of the covariance matrix—the so-called inconsistency problem. To fundamentally address this challenge, this invention proposes a dynamic two-layer iterative optimization framework. This framework achieves a synergistic improvement in robust estimation and linearization accuracy through the collaboration of inner and outer iterations: the inner iteration focuses on solving the problem under fixed linearization parameters. t The optimal robust state estimate at time -1 is obtained; the outer iteration dynamically reconstructs the linearization point and corrects the covariance using the inner results, ensuring the consistency of statistical linearization. Both iteratively progress, working together to achieve the optimal balance between estimation accuracy and numerical stability. In the current... k The linearization parameters of the outermost iteration Given the given conditions, the inner iterations apply a cost function based on mixed correlation entropy. Optimization is performed to solve for the optimal state estimate. The maximum correlation entropy criterion, by introducing a Gaussian kernel function to weight the residuals, effectively reduces the weight of outlier observations and model bias, thus endowing the algorithm with strong robustness. Specifically, in each iteration... The weight matrix and covariance scaling term will be recalculated based on the current residual, and the gradient will be adjusted accordingly. Set to zero to update state estimate This iterative process continues until the state update amount tends to stabilize, that is, it satisfies... End at time, To solve for the threshold when estimating the optimal state, In the first k The inner layer iteration under the next outer layer iteration l The next iteration t State estimation at time 10:00 In the first k The inner layer iteration under the next outer layer iteration l +1 iterations t State estimation at time 1. At this point, the output is... That is, under the current linearization parameters t Robust optimal estimate at time step; then based on The latest posterior information at time step is propagated backward through the inverse gain matrix to At any given moment, an information consistency correction mechanism is formed. This mechanism utilizes the current more accurate posterior estimate. Inversely correct the smoothing estimate from the previous time step. and its corresponding covariance .

[0046] To further overcome the inherent approximation error of single-step linearization, this invention constructs an outer iterative structure in addition to the core filtering step. Its core idea is: based on the correction value from the previous time step... and Regenerate Sigma points and execute the first step. k +1 statistical linearization yields a set of higher-precision results. k Affine parameters corresponding to +1 outer iteration ,in For the first k The optimal linear regression matrix for the nonlinear state equation corresponding to the +1 outer iteration. No. k +1 outer iteration of the nonlinear state equation bias term after affine approximation For the first k The covariance of the linearized residuals of the nonlinear state equation corresponding to +1 outer iteration. For the first k The optimal linear regression matrix of the measurement equation corresponding to the +1 outer iteration. For the first k The bias term of the measurement equation after the +1 outer iteration is obtained by affine approximation. For the first k The covariance of the linearized residuals of the measurement equation corresponding to +1 outer iteration. The correction amount for the outer iterations up to the linearization point is sufficiently small, i.e. Stop when the linearization parameters are obtained, and the linearization parameters obtained at this point are considered to be optimal. To solve for the threshold of the optimal affine approximation parameters, In the first k +1 outer iterations for the first... t Backward estimation at time -1.

[0047] This dynamic bi-level iterative optimization framework organically integrates the robust non-Gaussian estimation of the inner layer with the statistical consistency correction of the outer layer. The inner layer ensures the estimation accuracy under anomalous disturbances, while the outer layer improves the ability to approximate the dynamics of the real system from the root of linearization.

[0048] To comprehensively evaluate the estimation accuracy and robustness of the proposed algorithm in strongly nonlinear dynamic systems and non-Gaussian noise environments, this invention constructs a typical air traffic control (Air) scenario. T A simulated Raffic Control (ATC) scenario. The target aircraft performs continuous turns and reverse maneuvers relative to a ground-based radar in the horizontal plane. This process simultaneously involves highly nonlinear state evolution and strongly geometrically constrained radar measurements, effectively testing the anti-jamming capability of the filter under complex conditions. The system state vector is defined as: ,in and They respectively represent the target at shaft and The position and velocity components of the axis, The target turning rate. The initial state is set to... , , The initial covariance is set to The values ​​of each component are as follows: , , , , The target motion follows a discretized constant turning model, and its state transition matrix can be expressed as: (66); in The sampling period is 0.5 seconds. This represents the turn rate. To enhance the dynamic complexity of the trajectory and simulate continuous maneuvers and reverse turns in real flight, the angular velocity is inverted after every 100 steps. The process noise covariance is: (67); Among them dynamic parameters , Ground-based radar sensors provide four types of measurements: range, azimuth, elevation, and radial rate. The observation equation can be expressed as: (68); in Installation height of the m radar antenna The measurement noise is represented by a non-Gaussian distribution, and its statistical properties are selected according to the experimental setup to verify the robustness of the proposed algorithm under abnormal measurements. The measurement noise is modeled using a mixture distribution. First, a uniform random variable is selected from the noise components corresponding to different mixing coefficients; then, each noise component is represented by a variable with [degrees of freedom]. students Distributed generation, mixing coefficient set to Noise mixing component number The mean values ​​of the two noise components are set as follows: , This is used to describe the mean behavior of a sensor under electromagnetic interference or system drift. The noise covariance matrix is ​​uniformly set to... , , , ,as well as The initial state of the target is: The experiment consisted of 200 Monte Carlo simulations, each generating independent noise samples randomly. Each trial was run on the same real-world trajectory and passed the root mean square error (RMSE). The filtering performance was evaluated to verify the stability and accuracy advantages of the proposed method under strongly nonlinear and non-Gaussian noise environments. Indicates the number of Monte Carlos. Indicates the estimated state. Represents the actual trajectory.

[0049] Depend on Figure 2 As shown in the target trajectory estimation diagram, both methods can achieve basic tracking of the real motion trajectory, indicating that each filter has a certain degree of adaptability in extended target tracking. Further observation reveals that the method proposed in this invention, compared with Extended-MCKF (Extended Maximum Correlation Entropy Kalman Filter), can more accurately approximate the real trajectory when the target is maneuvering and subject to biased noise interference, especially in the turning and angular velocity inverse regions. The comparative methods, on the other hand, exhibit significant lag and jitter, fully demonstrating the accurate capture capability of this invention for strongly nonlinear dynamic features through linearization and a two-layer iterative mechanism. Figure 3 The root mean square error (RMSE) significantly reflects this phenomenon: the RMSE of the method in this invention is significantly lower than that of Extend-MCKF throughout the entire process. Furthermore, as time increases, Extend-MCKF exhibits obvious divergence characteristics, with its error continuously rising and fluctuating significantly, reflecting its sensitivity to heavy-tailed outliers in the measurement. In contrast, the method in this invention maintains a consistently low error level with small fluctuations, verifying the natural suppression capability of the multi-kernel maximum correlation entropy criterion for anomalous observations. Figure 4 The error probability density distribution shows that the error of the proposed method is more concentrated than that of Extend-MCKF, with smaller overall error values ​​and higher peak values, exhibiting a clear "high and narrow" error distribution. In contrast, the error distribution of the comparison method is relatively dispersed, indicating a higher probability of larger deviations. These results fully demonstrate the significant advantages of the proposed method in terms of estimation accuracy and stability.

[0050] To accurately characterize the nonlinear dynamic behavior of lithium-ion batteries, this invention employs an Enhanced Self-Correcting (ESC) model as the physical basis for the state estimation algorithm. This model, by organically integrating linear dynamic components with nonlinear correction elements, can characterize the complex physicochemical processes within the battery in multiple dimensions, exhibiting significant advantages, particularly in describing voltage hysteresis characteristics with a substantial memory effect. The system abstracts the battery's dynamic behavior into a three-dimensional state-space model, defining... The state vector at time t is .in, In a charged state, For the diffusion current of the equivalent RC network branch, To characterize the hysteresis characteristics, the discretized state transition equations of the system are as follows: (69); The model parameters and their physical meanings are defined as follows: SOC evolution mechanism: based on the ampere-hour integral method. The sampling period is Rated capacity of the battery (unit: Ah). For Coulomb efficiency, for Load current at time t. Polarization kinetic response: is the polarization attenuation coefficient, where The polarization time constant; For the corresponding input gain. Hysteresis dynamic evolution: and This is used to describe the dynamic transition of hysteresis states as current accumulates. Among them... This is a dimensionless hysteresis rate constant. (Sign function) Defined as follows: 1 when the current is positive (discharging), -1 when the current is negative (charging), and 0 when at rest. System noise: The process noise is represented by its covariance matrix. Battery terminal voltage As the system's observed output, its nonlinear measurement equation integrates thermodynamic potential energy, polarization loss, and hysteresis effect: (70); The terms used in the measurement process are defined as follows: 1. 1. The open-circuit voltage in equilibrium is a strongly nonlinear function of the state of charge (SOC), typically obtained through a lookup table generated from prior data. 2. Hysteresis voltage component ( ): The maximum offset constant for dynamic hysteresis. The amplitude of the instantaneous hysteresis change; As the current sign memory factor, when hour Otherwise, maintain the state from the previous moment. 3. Impedance voltage drop: Corresponding to instantaneous ohmic loss, This reflects the diffusion pressure drop caused by ion migration. 4. Measurement noise: This provides reliable data support for evaluating the robustness of the estimation algorithm under complex noise. To evaluate the robustness of the proposed algorithm under complex disturbances, this invention uses a dataset of a lithium battery's 40°C constant current charging process for verification: Depend on Figure 5 The SOC estimation results for lithium-ion batteries show that the proposed method, compared to Extend-MCKF, can more accurately estimate the SOC change trend throughout the charging process. Especially in the high SOC region where the OCV curve is steepest in the later stages of charging, the estimated trajectory of the proposed method is highly consistent with the actual SOC curve, while the comparison method shows significant cumulative deviation. The RMSE change trend shows that the SOC estimation error of the proposed method is significantly lower than that of the comparison method throughout the entire process. Furthermore, as the charging progresses, the error of the comparison method continuously increases, showing an accelerated growth trend, while the error growth of the proposed method is more gradual, maintaining a consistently low error level. These results fully demonstrate the accuracy advantage and robust performance of the proposed method in state estimation of strongly nonlinear systems.

[0051] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy, characterized in that, include: The process noise and measurement noise are acquired, and the process noise and measurement noise are modeled as a composite non-Gaussian noise model. A dynamic framework for collaborative optimization of inner and outer iterations is constructed. In the outer iteration of the dynamic framework, the composite non-Gaussian noise model is linearized based on the iterative affine approximation to obtain the optimal local linearization model, which is used to construct the robust cost function of the multi-kernel hybrid function and establish the adaptive parameter mechanism of the robust cost function. In the inner iteration of the dynamic framework, the robust cost function is maximized. The robust cost function is differentiated, and a Kalman gain analytical model incorporating robust weighting factors is used to recursively correct the posterior error covariance, obtaining the optimal posterior estimate and covariance at the target time. Further, an inversely weighted multi-kernel cost function is constructed. The inversely weighted multi-kernel cost function is differentiated, and the optimal posterior estimate and covariance at the target time are propagated to the previous target time using the inverse gain matrix. The smoothing estimate and covariance at the previous time are then corrected in reverse to obtain the optimal linearization parameters.

2. The strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy according to claim 1, characterized in that, Acquiring the process noise and the measurement noise includes: ； ； in, Indicates the system at time... The true state vector, Represents the corresponding observation vector, and Let represent the nonlinear state transition function and measurement function of the system, respectively, and at each time step... All of the above are considered to be known mappings. This represents the dimension of the state vector. This indicates the dimension of the measurement vector. Indicates process noise. Indicates measurement noise. It is the set of real numbers.

3. The strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy according to claim 1, characterized in that, Obtaining the optimal local linearization model includes: ； ； in, These are the optimal linear affine regression matrices for the state transition equation and the measurement equation, respectively. These are the bias terms after linearization of the state transition equation and the measurement equation, respectively. For the linearized process noise and measurement noise, These are the linearized residual covariances of the state transition equation and the measurement equation, respectively. For the system at time The true state vector, For the system at time The true state vector.

4. The strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy according to claim 1, characterized in that, Constructing a robust cost function for a multi-core hybrid function and establishing an adaptive parameter mechanism for the robust cost function includes: ； in, The kernel mixing coefficient, for The actual observed value at time , The mean center bias is set for the j-th measurement noise. Indicates the first j The optimal linear affine regression matrix corresponding to each measurement The linearized residual covariance of the state transition equation. Represents the Gaussian kernel function. Represents the Cauchy kernel function. Indicates to One-step prediction update of state at time step. For the first j The linearized residual covariance corresponding to each measurement component For the first j Linearized bias term for each measurement, For the first j The responsibility of each measurement component.

5. The strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy according to claim 1, characterized in that, The method also includes: The derivative calculation of the robust cost function includes: ； in, For robust cost function, For the system at time The true state vector, The mixing coefficients of the kernel function, For the first j The adaptive kernel bandwidth corresponding to each measurement. For Gaussian kernel function, For the first j Residual of each measurement channel For Cauchy kernel function, The adaptive kernel bandwidth corresponding to the prediction error. To predict residuals, is the scaling parameter of the Cauchy kernel function. The inverse of the linearized residual covariance of the state transition equation. For the first j The transpose of the optimal linear affine regression matrix corresponding to each measurement. It is the inverse of the linearized residual covariance of the j-th measurement.

6. The strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy according to claim 1, characterized in that, The recursive correction of the posterior error covariance using the Kalman gain analytical model with robust weighting factors includes: ； in, for t The posterior state estimation at time 10:00 To t One-step prediction update of state at time step. To obtain the combined information by vertically stacking the residuals of each measurement channel in channel order, It is the inverse of the covariance of the linearized residuals. For the combined measurement matrix, It is the inverse of the covariance of the linearized residuals of the combined weighted measurement equations.

7. The strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy according to claim 1, characterized in that, Constructing the inverse weighted multi-core cost function includes: ； in, For Gaussian kernel function, The covariance of the linearized residuals. for t The posterior state estimation at time 10:00 Let be the optimal linear regression matrix for the state transition equation. The bias term is the linearized form of the state transition equation. This is the mean center bias term for the process noise. The mixing coefficients of the kernel function, This refers to the process noise after linearization. for t Posterior state estimate at time -1 for t Posterior estimate of covariance at time -1 for t Posterior state estimation at time 10:00 for t The true state at time -1 for t The estimation error of the state at time -1.

8. The strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy according to claim 1, characterized in that, The derivative calculation of the inverse weighted multi-core cost function includes: ； in, For inverse weighted multi-core cost function, Let be the optimal linear regression matrix for the state transition equation. for t The true state at time -1 The bias term is the linearized form of the state transition equation. This is the mean center bias term for the process noise. The kernel mixing coefficient, For the Gaussian kernel function adaptive bandwidth during reverse update, This is the transpose of the state-optimal linear regression matrix. It is the inverse of the covariance of the linearized residuals. for t Posterior state estimation at time 10:00 The Gaussian kernel function used for measurement. for t Posterior estimate of covariance at time -1 for t The true state at time -1 for t Posterior state estimate at time -1 For the Gaussian kernel function used for the state, is the scaling parameter of the Cauchy kernel function. For the adaptive bandwidth of the Cauchy kernel function, The Cauchy kernel function used for measurement. This is the transpose of the optimal linear regression matrix. For the Cauchy kernel function used for the state, for t Posterior estimate of covariance at time -1.

9. The strongly nonlinear non-Gaussian state estimation method based on affine multi-kernel maximum correlation entropy according to claim 1, characterized in that, The inverse gain matrix is ​​defined as follows: ； in, Let be the optimal linear regression matrix for the state transition equation. The Kalman gain during reverse update. for t Posterior estimate of covariance at time -1 Forward update-driven during reverse update t Adaptive weighted intensity at time step This is the transpose of the optimal linear regression moments of the state transition equation. It is the inverse of the covariance of the linearized residuals. For reverse updates t Adaptive weighting strength dominated by the posterior state estimation at time -1.

Images