High-dimensional nonlinear system state estimation method based on tensor unscented kalman filter

Abstract

This invention relates to the field of nonlinear system state estimation technology, specifically to a method for estimating the state of high-dimensional nonlinear systems based on tensor unscented Kalman filtering. This invention is applicable to system state estimation methods based on tensor unscented Kalman filtering in high-dimensional nonlinear scenarios such as multi-sensor fusion, autonomous driving, and aerospace. For example, it estimates the state of a target node, such as its position and velocity, to obtain estimated values ​​of the state tensor. This invention optimizes the accuracy, efficiency, and robustness of real-time state estimation in high-dimensional nonlinear environments, filling the performance gap of traditional estimation methods in high-dimensional dynamic data processing, which suffer from "insufficient accuracy, structural destruction, and poor real-time performance." Its application scope widely covers multiple industries with stringent requirements for high-dimensional nonlinear dynamic data processing, possessing strong scenario adaptability and engineering practical value.

Description

Technical Field

[0001] This invention belongs to the field of nonlinear system state estimation technology, specifically relating to a method for estimating the state of high-dimensional nonlinear systems based on tensor unscented Kalman filtering. Background Technology

[0002] In real-time data processing applications (such as autonomous driving and aerospace), filtering algorithms are the core technology for extracting system state information from observation data. For linear systems, Kalman filtering (KF) achieves state prediction and updating through a recursive framework, but it cannot handle nonlinear systems. To address the nonlinearity problem, existing technologies have proposed extended Kalman filtering (EKF) and unscented Kalman filtering (UKF): EKF locally linearizes the nonlinear function through a first-order Taylor expansion, resulting in significant linearization errors and a sharp drop in estimation accuracy under strongly nonlinear scenarios; UKF generates sigma points to approximate the nonlinear distribution through unscented transformation (UT), avoiding linearization errors, but its computational complexity increases dramatically when processing high-dimensional data, making it difficult to meet real-time requirements.

[0003] Tensors, as an efficient representation tool for high-dimensional data, can preserve the inherent structural information of multi-dimensional data and have been used to improve traditional filtering algorithms, forming methods such as Tensor Extended Kalman Filter (TEKF). However, TEKF still relies on linearization approximation, and its performance is limited in strongly nonlinear systems. At the same time, traditional UKF does not take advantage of the structural advantages of tensors, and requires vectorization before processing high-dimensional tensor data, which destroys the data structure and increases the computational cost.

[0004] Therefore, existing technologies suffer from three major problems: "low estimation accuracy in strongly nonlinear scenarios", "high computational complexity in processing high-dimensional data", and "performance loss due to data structure destruction". There is an urgent need for a state estimation method that can balance nonlinear approximation accuracy, high-dimensional data processing efficiency, and preservation of structural information. Summary of the Invention

[0005] To address the shortcomings of existing methods such as large linearization errors in EKF, complex high-dimensional computations in traditional UKF, and poor adaptability to strong nonlinearity in TEKF, this invention provides a state estimation method for high-dimensional nonlinear systems that integrates tensor operations and unscented Kalman filtering, achieving state estimation with "no linearization errors, preservation of high-dimensional structure, and controllable computational efficiency." This invention is applicable to system state estimation methods based on tensor unscented Kalman filtering in high-dimensional nonlinear scenarios such as multi-sensor fusion, autonomous driving, and aerospace, for example, estimating the state of target nodes, such as position and velocity, to obtain estimated values ​​of the state tensor.

[0006] The technical solution adopted in this invention is:

[0007] A state estimation method for high-dimensional nonlinear systems based on tensor unscented Kalman filtering is used to estimate the real-time state of a system in high-dimensional data processing scenarios, obtaining an estimate of the state tensor; it includes the following steps:

[0008] S1. Initialization, specifically including:

[0009] S11, Defining the system in The initial state estimation tensor at time t, containing zero-mean Gaussian noise perturbation, is: The initial covariance tensor is ,in, and The size of each dimension of the high-dimensional state tensor. , ;

[0010] S12, Sigma sampling point parameter configuration, setting the Sigma point adjustment parameter to the scaling parameter. This is used to control the dispersion of sigma points; secondary scaling parameters. , The dimension of the state tensor, used to adapt to high-dimensional tensor distributions; shape parameter. This is used to adapt to the Gaussian distribution;

[0011] S13. Calculate the first-order and second-order weights of the Sigma sampling points:

[0012] ,

[0013] ,

[0014] in, , For first-order weights, , , For second-order weights, the two must satisfy the weight normalization condition. , ;

[0015] S2, Predicted State, specifically including:

[0016] S21. If this is the first iteration, then the current covariance tensor... For the initial covariance tensor Conversely, it is the covariance tensor estimated during the iteration. Then, perform High-Order Singular Value Decomposition (HOSVD) on the current covariance tensor to obtain the square root tensor. and satisfy ,in For Einstein;

[0017] S22. Generate 2D+1 state prediction Sigma sampling points and define the square root tensor. The column vector is , And generate accordingly sigma points in tensor form The central sigma point is The positive direction sigma point is The negative sigma point is ;

[0018] S23. One-step state prediction for Sigma sampling points: Perform state transition for each Sigma point to obtain the predicted state Sigma point. , ;in, For the system state transition model ;

[0019] S24. Calculate the predicted state and the predicted covariance, and calculate the predicted state tensor based on the first-order weights. ; Calculate the predicted covariance tensor based on second-order weights ,in, For the tensor Kronecker product; for Noise of zero-mean Gaussian distribution process at time step The corresponding covariance tensor ;

[0020] S3, Update, specifically including:

[0021] S31, Predicting covariance tensor Perform Higher-Order Singular Value Decomposition (HOSVD) to obtain the predicted square root tensor. and satisfy ;

[0022] S32. Generate 2D+1 observation-prediction Sigma sampling points and predict the square root tensor. The column vector is , And based on the predicted state tensor obtained in S24 generate sigma points in tensor form ; where the central sigma point is The positive direction sigma point is The negative sigma point is ;

[0023] S33, Sigma sampling point one-step observation prediction: Perform observation mapping for each Sigma point to obtain the observation prediction Sigma point. , ,in, For observation model ;in, and To measure the magnitude of each dimension of the high-dimensional observation tensor, , ;

[0024] S34. Calculate the observation covariance and the state observation cross-covariance, and predict the sigma point based on the observations. Obtain the observation prediction tensor Based on this, the observation-prediction covariance tensor can be obtained. and the state-observation cross-covariance tensor ,in, For the tensor Kronecker product; for Time-mean Gaussian distribution observation noise The corresponding covariance tensor ;

[0025] S35. Calculate the Kalman gain tensor. Based on the cross-covariance tensor and the observation covariance tensor, calculate the Kalman gain in tensor form. ;

[0026] S36. Update the state estimate and covariance estimate. At this stage, Observation tensor at time Input the system and use the Einstein product to achieve state estimation updates. Covariance Estimation Update ;

[0027] S4, Order Check if there is a next observation. If so, return to S2; otherwise, end the state estimation and output the target's state estimate at the current time. With state covariance estimation .

[0028] The beneficial effects of this invention are as follows: compared with the UKF and TEKF algorithms, it has a significantly lower root mean square error (RMSE) at the same time scale and better filtering accuracy; although the average execution time of a single recursion is slightly higher than the latter two, the benefit of improved accuracy far outweighs the slight increase in time cost, achieving a good balance between filtering accuracy and running efficiency. Attached Figure Description

[0029] Figure 1 Technical solution flowchart;

[0030] Figure 2 Comparison of RMSE of estimation results of various algorithms under small initial perturbation

[0031] Figure 3 Comparison of RMSE of estimation results of various algorithms under large initial state perturbation

[0032] Figure 4 Comparison chart of average time per step for each algorithm Detailed Implementation

[0033] The technical solution of the present invention will be further described below with reference to the accompanying drawings and simulations.

[0034] This embodiment uses The system state is carried out using a third-order complex tensor to simulate a high-dimensional, highly nonlinear scenario. The state transition function employs a third-order CP decomposition. The observation function is designed with cross-coupling mappings and introduces inter-dimensional interference. The covariance of both process and observation noise is set to... and set low ( ),high( Two initial covariance perturbation scenarios were tested. TEKF and traditional UKF were used as comparison algorithms, and 100 Monte Carlo simulations were conducted under standard hardware and software conditions.

[0035] The implementation process follows the TUKF core workflow S1~S4: first, parameter initialization and weight calculation are completed, and then the covariance square root tensor is solved based on HOSVD decomposition and generated. The system uses tensor form sigma points; state transitions and predictions are achieved through CP decomposition. Simultaneously, the tensor Kronecker product and Einstein product are used to calculate covariance and cross-covariance. A pseudo-inverse is employed to solve for the Kalman gain. After updating the state and covariance, the system recursively executes 200 steps, simultaneously recording key data such as error and time consumption.

[0036] The simulation results are shown in the figure:

[0037] Validation results show that TUKF reduces the overall average RMSE by 25%-31% compared to TEKF and by 15%-18% compared to traditional UKF in both initial scenarios, with a stable RMSE as low as 0.035-0.041. The average execution time per step is 0.365ms. Although the computational complexity is slightly higher than the other two algorithms, it still meets the real-time requirements. Furthermore, it exhibits smaller error fluctuations, more stable gains, and no abnormal situations, demonstrating superior robustness compared to the comparison algorithms.

[0038] In summary, TUKF achieves a balance between accuracy, efficiency, and stability in high-dimensional, strongly nonlinear systems, effectively avoiding linearization errors and data structure corruption. It can be migrated to fields such as multi-sensor fusion and aerospace without complex adjustments, and has strong practical engineering value.

Claims

1. A method for estimating the state of a high-dimensional nonlinear system based on tensor unscented Kalman filtering, used to estimate the system state in high-dimensional data processing scenarios and obtain the estimated value of the state tensor; characterized in that, Includes the following steps: S1. Initialization, specifically including: S11, Defining the system in The initial state estimation tensor at time t, containing zero-mean Gaussian noise perturbation, is: The initial covariance tensor is ,in, and The size of each dimension of the high-dimensional state tensor. , ; S12, Sigma sampling point parameter configuration, setting the Sigma point adjustment parameter to the scaling parameter. This is used to control the dispersion of sigma points; secondary scaling parameters. , The dimension of the state tensor, used to adapt to high-dimensional tensor distributions; shape parameter. This is used to adapt to the Gaussian distribution; S13. Calculate the first-order and second-order weights of the Sigma sampling points: ， ， in, , For first-order weights, , , For second-order weights, the two must satisfy the weight normalization condition. , ; S2, Predicted State, specifically including: S21. If this is the first iteration, then let the current covariance tensor be the initial covariance tensor. Otherwise, let the current covariance tensor be the covariance tensor estimated by the iterative process. Perform higher-order singular value decomposition (HOSVD) on the current covariance tensor to obtain the square root tensor. and satisfy ,in For Einstein; S22. Generate 2D+1 state prediction Sigma sampling points and define the square root tensor. The column vector is , And generate accordingly sigma points in tensor form The central sigma point is The positive direction sigma point is The negative sigma point is ; S23. One-step state prediction for Sigma sampling points: Perform state transition for each Sigma point to obtain the predicted Sigma point. , ;in, For the system state transition model ; S24. Calculate the predicted state and the predicted covariance, and calculate the predicted state tensor based on the first-order weights. ; Calculate the predicted covariance tensor based on second-order weights ,in, For the tensor Kronecker product; for Noise of zero-mean Gaussian distribution process at time step The corresponding covariance tensor ; S3, Update, specifically including: S31, Predicting covariance tensor Perform Higher-Order Singular Value Decomposition (HOSVD) to obtain the predicted square root tensor. and satisfy ; S32. Generate 2D+1 observation-prediction Sigma sampling points and predict the square root tensor. The column vector is , ; and based on the predicted state tensor obtained in S24 generate sigma points in tensor form ; where the central sigma point is The positive direction sigma point is The negative sigma point is ; S33, Sigma sampling point one-step observation prediction: Perform observation mapping for each sigma point to obtain the observation prediction sigma point. , ,in, For observation model ;in, and To measure the magnitude of each dimension of the high-dimensional observation tensor, , ; S34. Calculate the observation covariance and the state observation cross-covariance, and predict the sigma point based on the observations. Obtain the observation prediction tensor Based on this, the observation-prediction covariance tensor is obtained. and the state-observation cross-covariance tensor ,in, For the tensor Kronecker product; for Time-mean Gaussian distribution observation noise The corresponding covariance tensor ; S35. Calculate the Kalman gain tensor. Based on the cross-covariance tensor and the observation covariance tensor, calculate the Kalman gain in tensor form. ; S36. Update the state estimate and covariance estimate. Observation tensor at time Input the system and use the Einstein product to achieve state estimation updates. Covariance Estimation Update ; S4, Order Check if there is a next observation. If so, return to S2; otherwise, end the state estimation and output the target's state estimate at the current time. With state covariance estimation .

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