A target state estimation method with colored measurement noise under generalized unknown disturbance

Abstract

The application discloses a target state estimation method with colored measurement noise under generalized unknown disturbance, and one-step prediction of a state at a current moment is obtained based on target state estimation at a previous moment; prediction of reconstructed measurement at the current moment is generated according to the one-step prediction of the state; target state estimation at the current moment is calculated according to the one-step prediction of the state, the prediction of the reconstructed measurement, filter gain and the reconstructed measurement at the current moment; wherein the filter gain is calculated through determined parameters of a minimum upper limit; the application considers coexistence of the generalized unknown disturbance and the colored measurement noise, needs more relaxed conditions for constructing an upper limit of estimation error covariance than directly calculating a theoretical value of the estimation error covariance, designs a general framework of a recursive upper limit state estimator, so as to obtain filter estimation and smoothing estimation simultaneously; the measurement of the target is reconstructed, a minimum upper limit of the estimation error covariance is pursued through parameter optimization, so as to suppress an estimation error peak value and improve estimation precision.

Description

Technical Field

[0001] This invention belongs to the field of target tracking technology, and in particular relates to a method for estimating the state of a target with colored measurement noise under generalized unknown perturbation. Background Technology

[0002] Recursive state estimation methods for discrete-time stochastic systems, such as Kalman filtering / smoothing, are widely used in model-based control, statistical signal processing, target tracking, and fault diagnosis. For linear systems perturbed by Gaussian white noise, classical Kalman filtering / smoothing can obtain the optimal estimation result under the minimum mean square error criterion.

[0003] However, in practical applications, the Gaussian white noise assumption is often an ideal situation. In radar target tracking, due to high-speed radar sampling or continuous external interference, the measurement noise is always colored, which has prompted research on a series of recursive estimation methods under colored noise. Colored noise, especially colored measurement noise, is usually modeled as an autoregressive model perturbed by Gaussian white noise, and a first-order autoregressive process is a common case. Generally speaking, state estimation methods for stochastic systems under colored measurement noise (following a first-order autoregressive process) are usually divided into two categories: state dimension extension methods and measurement difference methods.

[0004] State expansion methods construct new state vectors using system states and colored measurement noise, thus avoiding the presence of colored noise, and then perform standard Kalman filtering / smoothing. However, since the reconstructed measurement equations do not contain measurement noise, the covariance matrix may become singular during Kalman filtering or smoothing, increasing the risk of numerical instability. To overcome this problem, numerical stability of Kalman filtering / smoothing is often achieved by regularizing the gain matrix or adding small positive numbers to the diagonal elements of the estimation error covariance.

[0005] The measurement difference method constructs a new measurement model by using the weighted difference of the original measurements at adjacent sampling times, thereby achieving whitening of the reconstructed measurement noise. However, this method results in the coexistence of system states at multiple sampling times in the reconstructed measurement model, making it impossible to directly use Kalman filtering / smoothing.

[0006] Based on the order in which state vectors are estimated at different time points, the measurement difference method can be further divided into three subcategories. The first subcategory uses the state of the previous time point to represent the state of the current time point in the reconstructed measurement model, directly obtaining a smooth estimate of the state with one-step lag, and recursively deriving the state estimate of the current time point based on the dynamic evolution model. The second subcategory uses inverse operations in the measurement model to transform the state of the previous time point into the state of the current time point, directly obtaining an online estimate of the state of the current time point. However, this inverse operation requires that the transition matrix in the dynamic evolution model must be invertible. The third subcategory is based on the reconstructed measurement model, directly estimating both the state of the current time point and the state of the previous time point simultaneously.

[0007] It has been proven that the three methods described above are theoretically equivalent for linear systems with colored measurement noise (following a first-order autoregressive process). Furthermore, compared to the first two methods, the third method is more suitable for recursive state estimation of nonlinear systems or systems with uncertain parameters under colored measurement noise because it avoids state transitions between different time points in the reconstructed measurement model, thus preventing information loss or model approximation. This results in a wider range of applications and more robust estimation accuracy.

[0008] In fact, in many practical applications, continuous external interference not only causes colored noise in measurements but also introduces unknown disturbances that are difficult to model in measurement models. Taking non-cooperative radar target tracking in a game-theoretic environment as an example, continuous electromagnetic countermeasures inevitably contaminate measurement data with unknown disturbances, making the measurement noise colored rather than white. On the one hand, although a series of recursive filtering and smoothing algorithms exist to handle colored noise, their estimation accuracy is severely degraded by the presence of unknown disturbances. On the other hand, although minimum upper bound filtering exists to achieve recursive estimation of nonlinear systems and jumping Markov systems under generalized unknown disturbances and can effectively suppress estimation peak errors, it requires the system noise to be white noise and is difficult to adapt to colored noise.

[0009] Therefore, it is important and an open problem to propose a recursive estimation method for colored measurement noise under generalized unknown perturbations, based on the practical application of non-cooperative target tracking. Summary of the Invention

[0010] The purpose of this invention is to provide a target state estimation method with colored measurement noise under generalized unknown disturbances, which simultaneously obtains the state filtering estimate at the current time and the state smoothing estimate before the current time, so as to improve the robustness of the state estimation.

[0011] This invention adopts the following technical solution: a method for estimating the state of a target with colored measurement noise under generalized unknown perturbation, comprising the following steps:

[0012] Based on the target state estimation of the previous time step, the current state is predicted in one step.

[0013] Based on the state, a prediction of the reconstructed measurement at the current moment is generated in one step;

[0014] The target state estimate at the current time is calculated based on the state one-step prediction, the prediction of the reconstructed measurement, the filter gain, and the reconstructed measurement at the current time.

[0015] The filter gain is calculated using a parameter that determines the minimum upper limit.

[0016] Furthermore, the calculation of the target's state estimation information at the current moment based on the state one-step prediction information, measurement one-step prediction information, filter gain, and measurement information at the current moment includes:

[0017]

[0018] in, This represents the target state estimate at time k obtained based on the measurement sequence from time 1 to time k. This represents a one-step prediction of the state at the current time based on the measurement sequence from time 1 to k-1, y k This represents the reconstructed measurement of the target at time k. This represents the prediction of the reconstructed measurement at time k based on the measurement sequence from time 1 to k-1. This represents the filter gain at time k-0.

[0019] Furthermore, the prediction of generating the reconstructed measurement at the current moment based on the state in one step includes:

[0020]

[0021] Among them, H k Let v be the measurement matrix at time k, and n be the colored noise v at time k. k The order of the autoregressive process, C i,k H represents the coefficient of the i-th matrix in the autoregressive model at time k. k-i Let be the measurement matrix at time ki. The result is a smoothed estimate of the target state at time ki, obtained from the measurement sequence from time 1 to time k-1. When k < n, let n = k.

[0022] Furthermore, the calculation method for the reconstructed measurement at the current moment is as follows:

[0023]

[0024] Among them, z k Here is the measurement at time k, and n is the colored noise v at time k. k The order of the autoregressive process, Ci,k Let z be the coefficient of the i-th matrix in the autoregressive model at time k. k-i The measurement is taken at time ki.

[0025] Furthermore, the filter gain is calculated using parameters determined by the minimum upper limit, including:

[0026] The upper limit of the information covariance from the previous time step to the current time step is determined based on the minimum upper limit.

[0027] Calculate the filter gain at the current moment based on the upper limit of the new information covariance.

[0028] Furthermore, the upper bound of the information covariance from the previous time step to the current time step, based on the minimum upper bound, includes:

[0029]

[0030] in, R is the upper bound of the innovation covariance at time k, obtained from the measurement sequence at times 1 to k-1. k Let τ be the covariance of the Gaussian white noise at time k. k A is the parameter for determining the minimum upper limit. k Let be the error gain matrix at time k, and Ω be an adjustable prior positive definite matrix.

[0031]

[0032] This is the upper bound covariance matrix of the prediction at time k, obtained based on the measurement sequence from time 1 to k-1. for The 1st to nth x row, nth x +1 to 2n x Sub-block matrix of columns, n x Let the dimension of the target state be . Estimation of the two target states at time k-1 The upper limit of the error covariance, For Z-based 1:k-1 State estimation at time k For Z-based 1:k-1 The smooth estimation results of the target state at time ki. for The 1st to nth x row, nth x +1 to 2n x Sub-block matrix of columns, n x Let the dimension of the target state be . Estimation of the two target states at time k-1 The upper limit of the error covariance, For Z-based 1:k-1 The smoothed upper bound covariance matrix obtained by smoothing estimation at time ki. for The 1st to nth x row, nth x +1 to 2n x Sub-block matrix of columns, n x Let the dimension of the target state be . Estimation of the two target states at time k-1 The upper limit of the error covariance, For Z-based 1:k-1 For a smooth estimation of the target state at time kj, C i,k Z represents the coefficient of the i-th matrix in the autoregressive model at time k. 1:k-1 This represents the sequence of quantities from time 1 to time k-1.

[0033] Furthermore, the method for calculating the parameters for determining the minimum upper limit is as follows:

[0034]

[0035] Where, σ max (·) represents the largest eigenvalue of the corresponding matrix, σ min (·) represents the largest eigenvalue of the corresponding matrix. := means to define ... as ...,

[0036] Furthermore, it also includes a smooth update of all target state estimates prior to the current time step, specifically using the following method:

[0037] according to Calculate the smooth gain from time k to time ki.

[0038] according to The target state at time ki is estimated using a smoothing method.

[0039] Another technical solution of the present invention: a target state estimation device with colored measurement noise under generalized unknown disturbance, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the above-described method.

[0040] The beneficial effects of this invention are as follows: By considering the coexistence of generalized unknown disturbances and colored measurement noise, and taking into account that constructing an upper limit of the estimation error covariance requires more relaxed conditions than directly calculating the theoretical value of the estimation error covariance, this invention designs a general framework for a recursive upper limit state estimator to simultaneously obtain filtered and smoothed estimates; the measurement of the target is reconstructed, and the minimum upper limit of the estimation error covariance is pursued through parameter optimization to suppress the peak value of the estimation error and improve the estimation accuracy. Attached Figure Description

[0041] Figure 1 This is a schematic diagram of time-varying unknown perturbation in an embodiment of the present invention;

[0042] Figure 2 This is a graph showing the estimation error curves of different methods in the embodiments of the present invention;

[0043] Figure 3 The graph shows the root mean square error curves of different methods in the embodiments of the present invention.

[0044] Figure 4 This is a graph showing the root mean square error curves of CMUBE filtering estimation and smoothing estimation in an embodiment of the present invention;

[0045] Figure 5 These are position estimation ARMSEs under different measurement noise levels in embodiments of the present invention;

[0046] Figure 6 These are position estimation ARMSEs under different degrees of unknown interference in the embodiments of the present invention. Detailed Implementation

[0047] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0048] In numerous practical applications, such as non-cooperative target tracking in adversarial game scenarios, continuous external electromagnetic interference not only colors the system noise but also introduces unknown disturbances in the measurement model that are difficult to model. Existing Kalman filtering / smoothing methods for colored noise, whether based on state dimension extension or measurement difference, are only applicable to cases where the measurement noise is colored; their estimation accuracy deteriorates significantly when faced with unknown disturbances. On the other hand, existing minimum upper bound filtering methods are often designed for nonlinear systems or jumping Markov systems under unknown disturbances. While they can effectively suppress the peak value of the estimation error, they require the system noise to be white noise, making them unsuitable for cases with colored noise. Furthermore, they can only obtain online filtered estimation results of the system state at the current moment and cannot obtain online smoothed estimation results of the system state before the current moment.

[0049] This invention designs a minimum upper bound state estimation method for colored measurement noise under generalized unknown perturbations, belonging to the field of information fusion and target tracking technology. In many practical applications based on state estimation, measurement data are often contaminated by a series of uncertain factors, especially colored noise and unknown interference. Take radar target tracking in a game-theoretic environment as an example. Tracking radar always samples rapidly to accumulate as many target measurements as possible to accurately track the observed target; conversely, the tracked target usually uses electromagnetic interference, such as range gate dragging and angle interference, to avoid being continuously tracked. In this case, due to rapid sampling and continuous electromagnetic interference, colored noise and time-varying unknown interference always coexist in radar measurements.

[0050] This invention designs a minimum upper bound state estimation method for colored measurement noise under generalized unknown perturbations, belonging to the field of information fusion and target tracking technology. In many practical applications based on state estimation, measurement data are often contaminated by a series of uncertain factors, especially colored noise and unknown interference. Take radar target tracking in a game-theoretic environment as an example. Tracking radar always samples rapidly to accumulate as many target measurements as possible to accurately track the observed target; conversely, the tracked target usually uses electromagnetic interference, such as range gate dragging and angle interference, to avoid being continuously tracked. In this case, due to rapid sampling and continuous electromagnetic interference, colored noise and time-varying unknown interference always coexist in radar measurements.

[0051] Consider the following discrete-time stochastic system with colored measurement noise under generalized unknown disturbances:

[0052] x k =F k-1 x k-1 +w k-1 (1)

[0053] z k =H k x k +A k δ k +v k (2)

[0054] Where, x k and z k Indicates system status and measurements, w k Errors in modeling target motion, such as unknown maneuvers, can be described as having a mean of zero and a covariance of Q. k Gaussian white noise, F k H is the state transition matrix. k For the measurement matrix, A k This is the error gain matrix, where the subscript k represents time k. Additionally, there is colored noise v. k It follows an nth-order autoregressive process:

[0055] v k =C 1,k v k-1 +…+C n,k v k-n +e k (3)

[0056] Here, C i,k Let e ​​be the coefficient of the i-th matrix in the autoregressive model at time k. k With zero mean and covariance R k Gaussian white noise. Here, for any two sampling times k and t, Moreover, the initial states x0 and w k and e t None of them are relevant. Unknown perturbation δ k satisfy:

[0057]

[0058] t' and t" represent different sampling times, where δ k The sampling is uncorrelated with process noise and measurement noise at future time points, but its correlation with process noise and measurement noise at previous time points is not restricted. The former implies that the system under consideration is a practically realizable causal system, while the latter indicates that δ... k It can represent a series of uncertainties, and is therefore called generalized unknown disturbance.

[0059] This invention discloses a target state estimation method with colored measurement noise under generalized unknown disturbance, comprising the following steps: obtaining a one-step prediction of the current state based on the target state estimation of the previous time step; generating a prediction of the reconstructed measurement of the current time step based on the one-step prediction of the state step; calculating the target state estimation of the current time step based on the one-step prediction of the state step, the prediction of the reconstructed measurement, the filter gain, and the reconstructed measurement of the current time step; wherein, the filter gain is calculated by determining the minimum upper limit parameter.

[0060] This invention addresses the coexistence of generalized unknown disturbances and colored measurement noise. Considering that constructing an upper limit for the estimation error covariance requires more relaxed conditions than directly calculating the theoretical value of the estimation error covariance, a general framework for a recursive upper limit state estimator is designed to simultaneously obtain filtered and smoothed estimates. The target measurement is reconstructed, and parameter optimization is used to pursue the minimum upper limit of the estimation error covariance in order to suppress the peak value of the estimation error and improve the estimation accuracy.

[0061] The purpose of this invention is to measure sequence Z 1:k :={z1,…,z k Under the condition of}, online recursive estimation of x t Here, t≤k. In fact, if δ does not exist...k Then x t The classic minimum mean square error estimate, considering only colored measurement noise, can be obtained through a Kalman-like filtering and smoothing process.

[0062] However, due to δ k The existence of δ necessitates either identifying δ in the design of the recursive state estimator or... k To correct the corresponding estimate, either δ needs to be compensated. k The impact on the estimation error covariance. However, neither of these methods is feasible for the system under consideration.

[0063] Therefore, this invention develops a novel recursive state estimator in which the estimation error covariance is replaced by its upper limit, and the final state estimate is obtained by minimizing this upper limit of the estimation error covariance in the case of colored measurement noise. Here, when unknown disturbances and colored measurement noise coexist, constructing the upper limit of the estimation error covariance and performing recursion requires more relaxed conditions and is easier to implement than directly calculating the theoretical value of the estimation error covariance. Furthermore, in practical applications such as radar target tracking, it is often necessary to suppress the peak value of the estimation error to avoid track loss, and minimizing the upper limit of the estimation error covariance ensures this better than minimizing the mean square error covariance.

[0064] This invention proposes a recursive minimum upper bound state estimation design process for colored measurement noise under generalized unknown perturbations, as follows:

[0065] First, a measurement model perturbed by zero-mean Gaussian white noise is reconstructed using the measurement difference method. This model includes state vectors at a series of different sampling times and generalized unknown disturbances (GUDs). Then, the definition of a recursive upper bound state estimator is given, and the recursive upper bound state estimate (CUBE) under colored measurement noise (CMN) is derived through inequality operations on positive semi-definite matrices. Finally, by numerically approximating and optimizing the minimum upper bound of the estimation error covariance, the minimum upper bound state estimate under CMN, i.e., CMUBE, is obtained.

[0066] First, according to equations (1), (2), and (3), the reconstructed measurement is defined, that is, the calculation method of the reconstructed measurement at the current moment is as follows: Among them, z k Here is the measurement at time k, and n is the colored noise v at time k. k The order of the autoregressive process, C i,k Let z be the coefficient of the i-th matrix in the autoregressive model at time k. k-i The measurement is taken at time ki.

[0067] Measurements z at different times k-i Substitute y k From the definition we get:

[0068]

[0069] Clearly, in the above equation, for any t ≥ 0, and The design of the subsequent recursive state estimator will be based on the reconstructed measurement sequence Y. 1:k ={y1,…,y k Expanding it, it contains the same valid information as Z1:k.

[0070] remember For x k In Y 1:l Estimation under the given conditions For x k In Y 1:l The error under the given conditions, if l = k, represents the filtered estimate, and l > k, represents the smoothed estimate. The superscript "~" indicates the error, and "^" indicates the estimate. Since y k Also depends on x k ,x k-1 ,…,x k-n Therefore, it is necessary to obtain the filtering result from the previous time step. Smooth with previous moments Only then can we obtain an estimate of the current moment. Here, t = kn,…,k-2. Therefore, a generalized recursive upper bound state estimator (UBE) is defined as follows, which includes both filtered and smoothed estimations.

[0071] Definition: If there exists a series of positive definite matrices and The following conditions must be met:

[0072]

[0073]

[0074]

[0075]

[0076] An estimator having the following form is called a recursive upper bound state estimator (UBE):

[0077]

[0078] Where t < k; “:=" means “to define ... as ..."; Indicates based on measurement sequence Y 1:k The obtained y j The estimate, These are the upper covariance matrices for prediction, filtering, smoothing, and innovation, respectively.

[0079] Based on the above definition of recursive UBE, assume and If the condition k≥1 holds true, then the recursive structure of UBE under colored measurement noise is given as follows.

[0080] State prediction:

[0081]

[0082]

[0083]

[0084]

[0085]

[0086]

[0087]

[0088] In other words, the prediction of the reconstructed measurement at the current moment is generated by one step prediction based on the state and is obtained by formula (15). For the prediction of the reconstructed measurement at time k based on the measurement sequence from time 1 to k-1, H k Let v be the measurement matrix at time k, and n be the colored noise v at time k. k The order of the autoregressive process, C i,k H represents the coefficient of the i-th matrix in the autoregressive model at time k. k-i Let be the measurement matrix at time ki. The result is a smoothed estimate of the target state at time ki, obtained from the measurement sequence from time 1 to time k-1. When k < n, let n = k.

[0089] The upper limit of the information covariance from the previous time step to the current time step, based on the minimum upper limit, is obtained by formula (16), where, For Z-based 1:k-1 The upper bound of the new information covariance at time k is obtained, R k Let τ be the covariance of the Gaussian white noise at time k. k A is the parameter for determining the minimum upper limit. k Let be the error gain matrix at time k, and Ω be an adjustable prior positive definite matrix. For Z-based 1:k-1The upper prediction covariance matrix at time k obtained from the measurement sequence at times 1 to k-1. for The 1st to nth x row, nth x +1 to 2n x Sub-block matrix of columns, n x Let the dimension of the target state be . Estimation of the two target states at time k-1 The upper limit of the error covariance, For Z-based 1:k-1 State estimation at time k For Z-based 1:k-1 The smooth estimation results of the target state at time ki. for The 1st to nth x row, nth x +1 to 2n x Sub-block matrix of columns, n x Let the dimension of the target state be . Estimation of the two target states at time k-1 The upper limit of the error covariance, For Z-based 1:k-1 The smoothed upper bound covariance matrix obtained by smoothing estimation at time ki. for The 1st to nth x row, nth x +1 to 2n x Sub-block matrix of columns, n x Let the dimension of the target state be . Estimation of the two target states at time k-1 The upper limit of the error covariance, For Z-based 1:k-1 The smooth estimation result of the target state at time kj, C i,k Z represents the coefficients of the i-th matrix in the autoregressive model at time k. 1:k-1 This represents all measurements from time 1 to time k-1, i.e., the measurement sequence.

[0090] Filter update, that is, calculating the target's state estimate at the current time based on the state one-step prediction information, the measurement one-step prediction information, the filter gain, and the measurement information at the current time:

[0091]

[0092] in, This represents the target state estimate at time k obtained based on the measurement sequence from time 1 to time k. This represents a one-step prediction of the state at the current time based on the measurement sequence from time 1 to k-1, y k This represents the reconstructed measurement of the target at time k. This represents the prediction of the reconstructed measurement at time k based on the measurement sequence from time 1 to k-1. This represents the filter gain at time k-0.

[0093]

[0094] Smooth Update:

[0095]

[0096]

[0097]

[0098] Here, nonnegative τ k Here are the parameters to be optimized: Ω is a manually set adjustable prior positive definite matrix, and the filter gain is... yes and The function, smoothing gain yes and The function, yes off-diagonal sub-block matrix, It is a state estimation The upper limit of the error covariance.

[0099] The recursive process of UBE under colored measurement noise needs to be executed step by step from the case where CMN follows a first-order autoregressive process to the case where CMN follows an nth-order autoregressive process, until k≥n. Initialization in

[0100] UBE provides a recursive estimate of the upper bound of the estimation error covariance under colored measurement noise. However, in practical applications such as non-cooperative target tracking, the focus is on minimizing the peak value of the tracking error, i.e., minimizing the upper bound of the estimation error covariance. Therefore, the minimum upper bound recursive state estimate under CMN for the considered system, i.e., CMUBE, is presented below.

[0101] Assumption and For any τ k If true, then CMUBE has the following recursive structure:

[0102]

[0103]

[0104]

[0105]

[0106]

[0107]

[0108] in,

[0109] It should be noted that, in the embodiments of the present invention, let τ k Achieve its optimal result This is then applied to the above calculations to improve the accuracy of target estimation.

[0110] In fact, due to the theoretical new information covariance matrix S k|k-1 It is difficult to compute, and comparing positive definite matrices requires mapping them to scalars, such as norms, determinants, or traces. Finding the optimal solution is difficult. Therefore, in In the actual solution process, unbiased estimation is used. To replace S k|k-1 ,here, This represents the prediction of the reconstructed measurement. Based on this, The approximate solution, i.e., the method for calculating the parameters determining the minimum upper limit, is as follows:

[0111]

[0112] Where, σ max (·) represents the largest eigenvalue of the corresponding matrix, σ min (·) represents the largest eigenvalue of the corresponding matrix, and := means defining ... as ....

[0113] here, The filter gain at time k-0 corresponding to the minimum upper bound recursive state estimator. This represents the smoothing gain of the minimum upper bound recursive state estimator at time k to time ki. and This refers to the minimum upper limit of the final output innovation covariance, filter estimation error covariance, and smoothing estimation error covariance. In this embodiment of the invention, the filter gain is calculated using the parameters for determining the minimum upper limit, including: the upper limit of the innovation covariance from the previous time to the current time based on the parameters for determining the minimum upper limit; and the filter gain at the current time calculated based on the upper limit of the innovation covariance.

[0114] In fact, in the reconstructed measurement model, there are unknown perturbations at multiple times, and the rigorous formula for the upper bound of the innovation covariance should include multiple τ. k-i Let i = 0, 1, ..., n. Each τ k-i Corresponding to the corresponding δ k-i At the same time, for each τ k-i If we strictly consider their calculation formulas, it involves... A series of calculations are performed, where q = 0, 1, ..., n. However, there is only one constraint to achieve parameter optimization, namely... This constraint alone is insufficient to optimize all the aforementioned free parameters. Therefore, considering feasibility, the recursive upper bound state estimator under colored measurement noise is designed using... To represent y k The uncertainty arising from the coexistence of all unknown disturbances, and only for τ k Optimization is performed. It should be noted that by optimizing τ... k This can guarantee the calculated satisfy And the conclusions drawn from this and Of course, this simplification comes at the cost of increasing the minimum upper limit of the new information covariance.

[0115] Finally, the pseudocode flow of CMUBE is shown in Table 1. When the system running time k is less than n, the colored measurement noise follows a k-th order autoregressive process. The proposed CMUBE degenerates into a recursive minimum upper bound state estimate of the colored measurement noise following a k-th order autoregressive process under generalized unknown perturbations. The obtained state estimate is as follows: Here, i = 0,…,k. As the system runtime gradually increases to greater than n, CMUBE achieves a recursive minimum upper bound state estimate of colored measurement noise following an nth-order autoregressive process under generalized unknown disturbances, and outputs… i = 0, ..., k. At this point, to achieve smooth estimation... The output (l>n) can be set as follows: And in Season C k-i =O, which will not affect the modeling of colored measurement noise following an nth-order autoregressive process.

[0116] Table 1

[0117]

[0118]

[0119] In addition, to better illustrate the effectiveness of the technical solution of the present invention, the embodiments of the present invention also verify the effectiveness of the proposed CMUBE through a typical example of non-cooperative target tracking under continuous external interference.

[0120] The target moves in approximately uniform linear motion on the o-ξη plane, and its dynamic evolution model is shown in equation (1). The target state... and Where T = 0.5, q = 0.005. Initial state x0 = (10, 0.25, 12, 0.2) T The system runs for 120 sampling times.

[0121] The radar located at the origin of the o-ξη plane detects the target and obtains the target's position observation at each sampling time, i.e. Due to the high-speed sampling of the radar and its continuous external interference, the measurement noise is colored noise, and its modeling is shown in equation (3). Here, the colored measurement noise follows a second-order autoregressive process. C i,k =O, i>2. Meanwhile, R k =σ 2 I², σ = 0.3. Furthermore, unknown disturbances... like Figure 1 As shown, and the corresponding coefficient matrix is ​​A. k =ρI2, where ρ = 5.

[0122] The filter output of the proposed CMUBE The outputs are compared with those of the Kalman filter (KF), the Kalman filter (CKF) output under a second-order autoregressive process in CMN, and the minimum upper bound filter (MUBF) output for linear stochastic systems. The initial estimates of all methods are consistent, i.e. here, It is a random selection.

[0123] In a single Monte Carlo simulation, the estimation errors of the filtered outputs of all comparison methods are as follows: Figure 2 As shown; in 1000 Monte Carlo simulations, the root mean square errors (RMSEs) of the filter outputs of all contrasting methods are as follows: Figure 3 As shown in Table 2, the average RMSEs (ARMSEs) of the filtered outputs from these methods are also shown in Table 2.

[0124] Table 2

[0125]

[0126] Here are:

[0127]

[0128]

[0129] in, Represents state x in the i-th Monte Carlo simulation kThe j-th element, x represents the corresponding estimator. j represents the j-th dimension element without considering the time factor, N represents the total number of Monte Carlo simulations, and K represents the total sampling step size of the system operation.

[0130] like Figure 2 As shown, the top left figure shows the position error curves of each method on the x-axis, the bottom left figure shows the position error curves of each method on the y-axis, the top right figure shows the velocity error curves of each method on the x-axis, and the bottom right figure shows the position error curves of each method on the y-axis. The estimation errors of all comparison methods (i.e.,...) The values ​​fluctuate around zero, reflecting that the estimation results of each method are essentially unbiased. Furthermore, compared to other methods, Figure 2 The estimation error of CMUBE is close to 0 at most sampling times. More importantly, the peak value of the estimation error of CMUBE is always minimal.

[0131] In addition, such as Figure 3 As shown, the top left figure shows the root mean square error (RMSE) curves of each method's position on the x-axis, the bottom left figure shows the RMSE curves of each method's position on the y-axis, the top right figure shows the RMSE curves of each method's velocity on the x-axis, and the bottom right figure shows the RMSE curves of each method's velocity on the y-axis. At most sampling times, CMUBE's RMSEs are consistently much lower than those of the other methods, which is also verified by the RMSEs in Table 2. Furthermore, among all the compared methods, CMUBE's RMSEs change the slowest. It should be noted that although MUBF also performs state estimation recursion in the form of upper bound filtering, the presence of colored measurement noise significantly degrades MUBF's filtering accuracy. In conclusion, because both colored measurement noise and unknown perturbations are considered simultaneously, CMUBE's filtering accuracy is the best among all the compared algorithms.

[0132] In addition, this invention also compares the estimation accuracy of CUMBE filtered output and smoothed output.

[0133] Due to its unique structure, the proposed CMUBE can not only output filter estimation Moreover, it can output a smooth estimate. Here, i > 0. To compare the estimation accuracy of filtering and smoothing, based on 1000 Monte Carlo implementations, the CMUBE output... and The RMSE curve is as follows Figure 4 As shown, and the corresponding ARMSEs are shown in Table 3. Although the colored measurement noise follows a second-order autoregressive process, by letting C 3,k =O, thus obtaining the three-step lag smoothing estimate. (equivalent to) This demonstrates that CMUBE can obtain a smooth estimate with any lag step size.

[0134] Table 3

[0135]

[0136] like Figure 4 As shown, the top left graph is the root mean square error (RMSE) curve of the asynchronous lag smoothing estimate on the x-axis, the bottom left graph is the RMSE curve of the asynchronous lag smoothing estimate on the y-axis, the top right graph is the RMSE curve of the asynchronous lag smoothing estimate on the x-axis, and the bottom right graph is the RMSE curve of the asynchronous lag smoothing estimate on the y-axis. As l = 0, 1, 2, 3 increases, more and more "future" measurements are used to update the current state estimate. The RMSEs continuously decrease at most sampling times. Furthermore, as shown in Table 3, for both position and velocity estimation, the RMSEs also decrease along the ξ and η axes according to... and The order gradually decreases. In summary, in CMUBE, the estimation accuracy of smoothing is better than that of filtering, and the estimation accuracy of l-th hysteresis smoothing is always better than that of l-1-th hysteresis smoothing, verifying the effectiveness of the proposed method.

[0137] Finally, this invention also compares the estimation accuracy under different measurement noise levels.

[0138] To further verify the effectiveness of the proposed CMUBE in both filtering and smoothing, it was implemented based on 1000 Monte Carlo simulations. Figure 5 The ARMSEs for position estimation using KF, Kalman smoothing (KS), CKF, MUBF, and CMUBE at different sensor accuracy levels are presented. Here, the measurement noise standard deviation σ for each dimension increases from σ = 0.1 to σ = 0.5 in increments of 0.05. Smoothing estimates with one-step, two-step, and three-step lags are obtained for KS and CMUBE, respectively.

[0139] like Figure 5 As shown, the left figure is the average root mean square error (RMSE) curve of each method's position on the x-axis under different measurement noise levels, and the right figure is the average RMSE curve of each method's position on the y-axis under different measurement noise levels. For KS and CMUBE, regardless of the value of σ (i.e., the measurement noise level), the larger the hysteresis step size of the state smoothing, the smaller the corresponding ARMSEs, and the ARMSEs of the smoothing estimate are always smaller than those of the filtering estimate. More importantly, at each measurement noise level, the ARMSEs of CMUBE are significantly smaller than those of the other comparative methods.

[0140] Meanwhile, as σ increases, the ARMSEs of KF, KS, CKF, and CMUBE all show a trend of first decreasing and then increasing. This is because in these recursive estimates, the posterior estimate is a weighted combination of the prior estimate and the measurement. When the measurement noise level is small, such as when σ = 0.1, the posterior estimate mainly depends on the measurement, resulting in a large ARMSE due to the presence of unknown disturbances. As the measurement noise level increases, the posterior estimate's dependence on the measurement decreases, while its dependence on the prior estimate increases. Since the target dynamics model and the initial estimate are relatively accurate, the obtained ARMSEs decrease as σ increases from 0.1 to 0.25. However, after reaching this critical point, the measurement noise level continues to increase, i.e., after σ > 0.25. In this case, although the posterior estimate's dependence on the measurement does not increase, the estimation error is mainly affected by the uncertainty caused by measurement noise and unknown disturbances, resulting in a gradual increase in the corresponding ARMSEs.

[0141] Furthermore, considering different levels of unknown perturbation, i.e., ρ increasing from ρ = 2.5 to ρ = 7.5 in increments of 0.5, based on 1000 Monte Carlo simulations, Figure 6 The ARMSEs for the estimated locations of KF, KS, CKF, MUBF, and CMUBE are given. Here, KS and CMUBE simultaneously output one-step, two-step, and three-step lag smoothing estimates.

[0142] like Figure 6 As shown, the left figure is the average root mean square error (ARMSE) curve of each method's position on the x-axis under different levels of unknown perturbation, and the right figure is the average root mean square error (ARMSE) curve of each method's position on the y-axis under different levels of unknown perturbation. As the value of ρ increases, i.e., the uncertainty caused by the unknown perturbation increases, the ARMSEs of all compared algorithms increase. More importantly, for each ρ value, CMUBE's ARMSEs are always much smaller than those of the other compared algorithms. Furthermore, as the value of ρ increases, the increase in CMUBE's ARMSEs is significantly smaller than that of the other compared algorithms, indicating that the proposed method has a strong ability to suppress unknown perturbations. In addition, in Figure 6 In both KS / KF and CMUBE, for each ρ value, the ARMSEs of the l+1 step lag smoothing estimate are less than those of the l-step lag smoothing estimate, and the ARMSEs of the smoothing estimate are always less than those of the filtered estimate. This is consistent with the fact that using more effective measurements can improve the estimation accuracy.

[0143] In summary, high-precision state estimation is always required in many practical applications, such as non-cooperative target tracking in adversarial game environments. However, the continuous disturbances brought about by adversarial games not only transform system noise into colored noise but also introduce unknown perturbations in the measurement equations that are difficult to model. Addressing the limitations of existing methods, which can only achieve high-precision state estimation under single colored measurement noise or unknown perturbations and are difficult to extend to situations where colored measurement noise and unknown perturbations coexist, this invention proposes a recursive minimum upper bound state estimation method for colored measurement noise under generalized unknown perturbations.

[0144] First, by employing multi-step measurement differencing, an equivalent measurement model containing state vectors from multiple time points is reconstructed, achieving measurement noise whitening. Second, considering that calculating the upper bound of the estimation error covariance recursively often requires more relaxed conditions than calculating the theoretical value of the estimation error covariance, a general framework for recursive upper bound state estimation is defined to simultaneously output filtered and smoothed estimates. Within this framework, a recursive upper bound state estimator under colored measurement noise is derived, and by minimizing the upper bound of the estimation error covariance, a recursive minimum upper bound state estimation method under colored measurement noise is proposed, achieving high-precision filtered and smoothed estimate outputs. Simulations of a typical non-cooperative target tracking scenario in a game-theoretic environment demonstrate that the proposed method can effectively handle generalized unknown disturbances and colored measurement noise. Compared to existing Kalman filtering, Kalman smoothing, Kalman filtering under colored measurement noise, and minimum upper bound filtering, it achieves higher estimation accuracy, and the accuracy of the output smoothed estimate is superior to that of the filtered estimate.

[0145] The present invention also discloses a target state estimation device with colored measurement noise under generalized unknown disturbance, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the above-described method when executing the computer program.

[0146] The present invention also discloses an embodiment that provides a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the steps in the above-described method embodiments.

[0147] The present invention also provides a computer program product that, when run on a data storage device, enables the data storage device to implement the steps in the above-described method embodiments.

[0148] If the integrated unit module is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, all or part of the processes in the methods of the above embodiments of the present invention can be implemented by a computer program instructing related hardware. The computer program can be stored in a computer-readable storage medium, and when executed by a processor, it can implement the steps of the various method embodiments described above. The computer program includes computer program code, which can be in the form of source code, object code, executable files, or certain intermediate forms. The computer-readable medium can include at least: any entity or device capable of carrying computer program code to a storage device, a recording medium, a computer memory, a read-only memory (ROM), a random access memory (RAM), an electrical carrier signal, a telecommunication signal, and a software distribution medium. Examples include USB flash drives, portable hard drives, magnetic disks, or optical disks.

[0149] In the above embodiments, the descriptions of each embodiment have different focuses. For parts that are not described in detail or recorded in a certain embodiment, please refer to the relevant descriptions of other embodiments.

[0150] Those skilled in the art will recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed in this invention can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementations should not be considered beyond the scope of this invention.

[0151] In the embodiments provided by this invention, it should be understood that the disclosed apparatus and methods can be implemented in other ways. For example, the apparatus embodiments described above are merely illustrative. For instance, the division of modules or units is merely a logical functional division, and in actual implementation, there may be other division methods. For example, multiple modules or components may be combined or integrated into another system, or some features may be ignored or not executed.

[0152] The modules described as separate components may or may not be physically separate. The components shown as modules may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs.

Claims

1. A method for estimating the state of a target with colored measurement noise under generalized unknown perturbation, characterized in that, Includes the following steps: Based on the target state estimation of the previous time step, the current state is predicted in one step. Based on the state, a prediction of the reconstructed measurement at the current moment is generated in one step; The target state estimate at the current moment is calculated based on the state one-step prediction, the prediction of the reconstructed measurement, the filter gain, and the reconstructed measurement at the current moment. The filter gain is calculated using parameters that determine the minimum upper limit, including: The upper limit of the information covariance from the previous time step to the current time step is determined based on the minimum upper limit. Calculate the filter gain at the current moment based on the upper limit of the new information covariance; The upper limit of the information covariance from the previous time step to the current time step, based on the minimum upper limit, includes: ， in, This is the upper bound of the information covariance at time k, obtained based on the measurement sequence from time 1 to k-1. Let $k$ be the covariance of the Gaussian white noise at time $k$. The parameter for determining the minimum upper limit. Let k be the error gain matrix at time k. It is an adjustable prior positive definite matrix. , Let k be the measurement matrix at time k. Based on The obtained upper bound covariance matrix of the prediction at time k, where n is the colored noise at time k. The order of the autoregressive process is followed. for The 1st to line, number to Sub-block matrix of columns, Estimation of the two target states at time k-1 The upper limit of the error covariance, Based on State estimation at time k Based on The smooth estimation results of the target state at time ki. Let be the measurement matrix at time ki. In the autoregressive model at time k, the first... matrix coefficients, for The 1st to line, number to Sub-block matrix of columns, Let the dimension of the target state be . Estimation of the two target states at time k-1 The upper limit of the error covariance, Based on The smoothed upper bound covariance matrix obtained by smoothing estimation at time ki. for The 1st to line, number to Sub-block matrix of columns, Let the dimension of the target state be . Estimation of the two target states at time k-1 The upper limit of the error covariance, Based on The smoothed estimation results of the target state at time kj. This represents the measurement sequence from time 1 to time k-1; The method for calculating the parameters for determining the minimum upper limit is as follows: ， in, The largest eigenvalue of the corresponding matrix. The largest eigenvalue of the corresponding matrix. , This means to define... as... , This represents the reconstructed measurement of the target at time k. This represents the prediction of the reconstructed measurement at time k based on the measurement sequence from time 1 to k-1.

2. The target state estimation method with colored measurement noise under generalized unknown perturbation as described in claim 1, characterized in that, The calculation of the target state estimate at the current moment based on the state one-step prediction, the reconstruction measurement prediction, the filter gain, and the reconstruction measurement at the current moment includes: ， in, This represents the target state estimate at time k obtained based on the measurement sequence from time 1 to time k. This indicates a one-step prediction of the state at the current time based on the measurement sequence from time 1 to k-1. This represents the filter gain at time k-0.

3. The target state estimation method with colored measurement noise under generalized unknown perturbation as described in claim 2, characterized in that, The prediction for generating the reconstructed measurement at the current moment based on the state includes: ， in, The result is a smoothed estimate of the target state at time ki, obtained from the measurement sequence from time 1 to time k-1. When k < n, let n = k.

4. The target state estimation method with colored measurement noise under generalized unknown perturbation as described in claim 3, characterized in that, The calculation method for the reconstruction measurement at the current moment is as follows: ， in, Here is the measurement at time k, and n is the colored noise at time k. The order of the autoregressive process is followed. In the autoregressive model at time k, the first... matrix coefficients, The measurement is taken at time ki.

5. The target state estimation method with colored measurement noise under generalized unknown perturbation as described in claim 4, characterized in that, It also includes a smooth update of all target state estimates prior to the current moment, specifically using the following method: according to Calculate the smooth gain from time k to time ki. ; according to The target state at time ki is estimated by smoothing.

6. A target state estimation device with colored measurement noise under generalized unknown disturbance, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the method as described in any one of claims 1-5.

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