A modeling method, system, and readable storage medium for non-cooperative target observation.

Abstract

This invention provides a modeling method, system, and readable storage medium for non-cooperative target observation. The method includes: observing non-cooperative targets using multiple measurement devices in a non-cooperative target observation system; constructing a multi-station nonlinear observation model of the non-cooperative target based on the EMBET estimation method; introducing nonparametric variables to form a multi-station nonlinear semiparametric model based on uncertain error components; constructing a regularization matrix based on factors influencing the error and ensuring its uniqueness; optimizing the multi-station nonlinear semiparametric model based on the optimal regularization matrix set formed by the unique regularization matrix to obtain an optimized semiparametric model; and using the optimized semiparametric model to solve for the position of the non-cooperative target observed by the multiple measurement devices to obtain the target's position. This improves positioning accuracy, accurately identifies observation system errors, and provides a theoretical basis for complex error modeling and identification of non-cooperative targets.

Description

Technical Field

[0001] This invention relates to the field of non-cooperative target observation, and specifically to a modeling method, system, and readable storage medium for non-cooperative target observation. Background Technology

[0002] Target tracking and localization refers to the theory, technology, and methods for extracting the motion state of a target from observational information under a unified spatiotemporal reference. High-precision target localization is a prerequisite and key to missions in strategic deployment, military strikes, and space resource exploration and development.

[0003] With the expansion of applications such as weapon offense and defense and space exploration, the high mobility, cross-regional, multi-mission characteristics of non-cooperative targets and the lack of prior information make it difficult to determine the motion state of the targets. The observation system has the characteristics of high dynamism, nonlinearity, complexity and variability and uncertainty, which leads to unclear error mechanisms and complex and uncertain forms in the observation of non-cooperative targets. It is difficult to accurately construct error models and effectively estimate error parameters, which has become a bottleneck restricting the high-precision positioning of non-cooperative targets.

[0004] Currently, the main methods for modeling and estimating errors in observation systems include parametric methods, nonparametric methods, and semiparametric methods.

[0005] Non-parametric methods eliminate the assumptions about the functional form of the model and transform prior information such as continuity into constraints, thereby controlling the characteristics of the state solution and avoiding the risks caused by representation errors. However, they fail to effectively utilize the prior information that can be effectively modeled in the target motion.

[0006] Semi-parametric methods combine parametric models with non-parametric components, exhibiting greater adaptability and interpretability. They have been widely studied and applied in measurement data processing, leading to the development of estimation methods such as parametric approximation, two-step estimation, and robust estimation. The selection of parameters such as the regularization matrix in the model is a crucial factor affecting the accuracy of semi-parametric estimation, typically relying on practical experience or post-hoc data analysis. Regularization matrices are constructed primarily through spline function fitting, time-series characteristics, prior variance, and distance optimization. However, in multi-station observation systems with non-cooperative targets, observation errors are complex and variable. The aforementioned construction methods do not consider error influencing factors, and a fixed regularization matrix cannot accurately describe the error characteristics of a multi-station system, resulting in decreased positioning accuracy. Therefore, it is necessary to study online adaptive design methods for regularization matrices that consider the influencing factors of multi-station observation system errors. Summary of the Invention

[0007] This invention provides a modeling method, system, and readable storage medium for non-cooperative target observation, which can solve the technical problem in the prior art that "error factors are not considered and the fixed regularization matrix is ​​difficult to accurately describe the error characteristics of multi-station systems, resulting in a decrease in positioning accuracy".

[0008] To achieve the above objectives, in a first aspect, embodiments of the present invention provide a modeling method for non-cooperative target observation, comprising:

[0009] A multi-measurement device observation model of a non-cooperative target observation system is used to observe non-cooperative targets. A multi-station nonlinear observation model of the non-cooperative targets is constructed based on the EMBET estimation method. The multi-station nonlinear observation model includes modelable systematic errors. Nonparametric variables are introduced into the multi-station nonlinear observation model to describe unmodelable observation systematic errors and / or model errors, forming a multi-station nonlinear semiparametric model based on uncertain error components.

[0010] The factors affecting the estimation accuracy of the multi-station nonlinear semiparametric model based on uncertain error components are identified: the regularization matrix. A regularization matrix is ​​constructed based on the factors affecting the error, ensuring that the regularization matrix is ​​unique. The unique regularization matrix has l-point smoothness invariance and elementary transformation invariance. The optimal regularization matrix set constructed by the unique regularization matrix is ​​used to optimize the multi-station nonlinear semiparametric model based on uncertain error components, resulting in an optimized semiparametric model.

[0011] An optimized multi-station nonlinear semi-parametric model was used to solve for the location of non-cooperative targets observed by multiple measurement devices, thus obtaining the location of the non-cooperative targets.

[0012] Secondly, embodiments of the present invention provide a modeling system for non-cooperative target observation, comprising:

[0013] The observation model construction unit is used to observe non-cooperative targets using multiple measurement devices of a non-cooperative target observation system, and to construct a multi-station nonlinear observation model of the non-cooperative targets based on the EMBET estimation method. The multi-station nonlinear observation model includes modelable systematic errors. Nonparametric variables are introduced into the multi-station nonlinear observation model to describe unmodelable observation systematic errors and / or model errors, forming a multi-station nonlinear semiparametric model based on uncertain error components.

[0014] The model optimization unit is used to determine the factors affecting the estimation accuracy of the multi-station nonlinear semiparametric model based on uncertain error components: regularization matrix. A regularization matrix is ​​constructed based on the factors affecting the error, ensuring that the regularization matrix is ​​unique. The unique regularization matrix has l-point smoothness invariance and elementary transformation invariance. The optimal regularization matrix set constructed by the unique regularization matrix is ​​used to optimize the multi-station nonlinear semiparametric model based on uncertain error components, thereby obtaining the optimized semiparametric model.

[0015] The solver unit is used to solve for the position of non-cooperative targets observed by multiple measurement devices using an optimized semi-parametric model, thereby obtaining the position of the non-cooperative targets.

[0016] Thirdly, embodiments of the present invention provide a computer-readable storage medium storing one or more programs, which, when executed by a computer device, cause the computer device to execute the modeling method for non-cooperative target observation described prior to the invention.

[0017] The above technical solution has beneficial effects: it can improve positioning accuracy, accurately identify observation system errors, and provide a theoretical basis for complex error modeling and identification of non-cooperative targets. Attached Figure Description

[0018] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art 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.

[0019] Figure 1 This is a flowchart of a modeling method for non-cooperative target observation according to an embodiment of the present invention;

[0020] Figure 2 This is a structural diagram of a modeling system for non-cooperative target observation according to an embodiment of the present invention;

[0021] Figure 3 This is a schematic diagram of a space target observation system according to an embodiment of the present invention;

[0022] Figure 4 This is a fusion estimation method based on a hybrid semiparametric model, as described in this embodiment of the invention.

[0023] Figure 6 This is a hybrid semiparametric model based on model likelihood values, as described in this embodiment of the invention.

[0024] Figure 5 This is an example of a multi-station ranging and positioning system according to an embodiment of the present invention;

[0025] Figure 7 This is complex error sampling data based on the combined effects of multiple influencing factors in an embodiment of the present invention;

[0026] Figure 8 This is a comparison of the results of various methods under complex periodic errors in the embodiments of the present invention;

[0027] Figure 9 This is a comparison of the results of various methods under the complex FBM error of the present invention embodiments;

[0028] Figure 10 This refers to the estimation accuracy of each algorithm during the optimization process of this invention embodiment;

[0029] Figure 11 This is a comparison of the estimation results of each algorithm after weight optimization in the embodiments of the present invention. Detailed Implementation

[0030] 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.

[0031] like Figure 1 As shown, in conjunction with embodiments of the present invention, a modeling method for non-cooperative target observation is provided, including:

[0032] S101: The non-cooperative target observation system uses multiple measurement devices to observe the non-cooperative target. Based on the EMBET estimation method, a multi-station nonlinear observation model of the non-cooperative target is constructed. The multi-station nonlinear observation model includes modelable systematic errors. Nonparametric variables are introduced into the multi-station nonlinear observation model. The nonparametric variables describe the unmodelable observation systematic errors and / or model errors, forming a multi-station nonlinear semiparametric model based on uncertain error components.

[0033] S102: Determine the factors affecting the estimation accuracy of the multi-station nonlinear semiparametric model based on uncertain error components: the regularization matrix, and construct a regularization matrix based on the factors affecting the error, ensuring that the regularization matrix is ​​unique. The unique regularization matrix has l-point smoothness invariance and elementary transformation invariance. Optimize the multi-station nonlinear semiparametric model based on uncertain error components using the optimal regularization matrix set constructed by the unique regularization matrix to obtain the optimized semiparametric model.

[0034] S103: The optimized semi-parametric model is used to solve for the position of the non-cooperative target observed by multiple measurement devices, and the position of the non-cooperative target is obtained.

[0035] like Figure 2 As shown, in conjunction with embodiments of the present invention, a modeling system for non-cooperative target observation is provided, comprising:

[0036] The observation model construction unit 21 is used to observe non-cooperative targets using multiple measurement devices of a non-cooperative target observation system, and to construct a multi-station nonlinear observation model of the non-cooperative target based on the EMBET estimation method. The multi-station nonlinear observation model includes modelable systematic errors. Nonparametric variables are introduced into the multi-station nonlinear observation model to describe unmodelable observation systematic errors and / or model errors, forming a multi-station nonlinear semiparametric model based on uncertain error components.

[0037] Model optimization unit 22 is used to determine the factors affecting the estimation accuracy of the multi-station nonlinear semiparametric model based on uncertain error components: the regularization matrix, and construct a regularization matrix based on the factors affecting the error, ensuring that the regularization matrix is ​​unique. The unique regularization matrix has l-point smoothness invariance and elementary transformation invariance. The optimal regularization matrix set constructed by the unique regularization matrix is ​​used to optimize the multi-station nonlinear semiparametric model based on uncertain error components, thereby obtaining the optimized semiparametric model.

[0038] Solver 23 is used to solve for the position of non-cooperative targets observed by multiple measurement devices using an optimized semi-parametric model, thereby obtaining the position of the non-cooperative targets.

[0039] Preferably, in S101, a multi-measurement device of a non-cooperative target observation system is used to observe the non-cooperative target. A multi-station nonlinear observation model of the non-cooperative target is constructed based on the EMBET estimation method. The multi-station nonlinear observation model includes modelable systematic errors, including the following:

[0040] A multi-station nonlinear observation model of non-cooperative targets is constructed based on the EMBET estimation method, observing non-cooperative targets using multiple measurement devices in a non-cooperative target observation system.

[0041]

[0042] In the formula, System state, which includes modelable systematic error, e is random error, y is the observations of m measuring devices, and f(·) is the system state in which the system state is modelable systematic error. A function whose form and parameters are known.

[0043] The above-mentioned technical means are also specific embodiments of the observation model construction unit 21.

[0044] Preferably, the multi-station nonlinear semi-parametric model based on uncertain error components includes a pointwise semi-parametric model. The pointwise semi-parametric model does not make time series assumptions about the system state and non-parametric components. The system state and non-parametric components at different times are uncorrelated, and the system state and non-parametric components at each time are estimated independently.

[0045] In S101, nonparametric variables are introduced into the multi-station nonlinear observation model to describe the unmodelable observation system errors and / or model errors, forming a multi-station nonlinear semiparametric model based on uncertain error components, including:

[0046] Nonparametric variables are introduced into the multi-station nonlinear observation model. These nonparametric components are used to describe unmodelable systematic errors or model errors in the multi-station nonlinear observation model corresponding to m measuring devices at a certain time, forming a point-by-point semiparametric model. The point-by-point semiparametric model is expressed as follows:

[0047]

[0048] In the formula, s represents the nonparametric component. Let be the covariance matrix of the random error e, and P be the weight matrix of the observation y; calculate the residuals of the pointwise semiparametric model, and express the residuals of the pointwise semiparametric model in equation (2) as:

[0049]

[0050] The normal equation of equation (3) is:

[0051]

[0052] Where J represents the Jacobian matrix;

[0053] The introduction of nonparametric components leads to the positive semidefinite non-invertible coefficient matrix of normal equation (4). The compensated least squares criterion is used to correct normal equation (4), thereby correcting equation (3):

[0054]

[0055] In equation (4), the factors affecting the estimation accuracy of the multi-station nonlinear semi-parameter model based on uncertain error components are: smoothing factor α and regularization matrix R; a Lagrangian function is constructed to solve equation (5), and the constructed Lagrangian function is expressed as:

[0056]

[0057] make get:

[0058]

[0059] From equation (7), we can obtain:

[0060]

[0061] in, Let J be the Jacobian matrix; the system state estimate and nonparametric estimate obtained by solving equation (5) are:

[0062]

[0063] Where S is an intermediate quantity, the superscript k indicates the kth iteration, and I represents the large identity matrix.

[0064] The above-mentioned technical means are also specific embodiments of the observation model construction unit 21.

[0065] Preferably, the multi-station nonlinear semiparametric model based on uncertain error components further includes a spline-based semiparametric model, which makes time-series assumptions about the system state and nonparametric components, assuming that the system state and semiparametric components satisfy spline functions over a period of time, and jointly estimates the system state and nonparametric components over a period of time.

[0066] In S101, nonparametric variables are introduced into the multi-station nonlinear observation model to describe the unmodelable observation system errors and model errors, forming a multi-station nonlinear semiparametric model based on uncertain error components, including:

[0067] When using a multi-measurement device system to observe a non-cooperative target, it is assumed that the system state and nonparametric components over a period of time conform to a spline function. This is then incorporated into the multi-station nonlinear observation model of the non-cooperative target, forming a semiparametric model based on spline representation. This semiparametric model based on spline representation is expressed as:

[0068]

[0069] Among them, the observed value y i =[y i1 ,y i2 ,…,y im ] T Let represent the observations of the 1st, 2nd, ..., mth measuring devices at time i, e i For m-dimensional random error, the nonparametric component s(t) i )=[s1(t i ),s2(t i ),…,s m (t i )] T s l (t i ), l=1,2,…,m describes the observation y il Unmodelable systematic errors and / or model errors, observation y il It is y i The l components; P iFor t i Time observation value - y i The weight matrix, where the system state β is the polynomial coefficient or spline polynomial coefficient of the non-cooperative objective during this time period;

[0070] The spline-based semiparametric model is applied to t. i Time residual V i =[V i1 V i2 ,…,V im ] T Represented as:

[0071]

[0072] Due to the introduction of nonparametric components, the coefficient matrix of normal equation (4) becomes positive semidefinite and non-invertible. The compensating least squares criterion is used to correct normal equation (4), thus modifying equation (12) as follows:

[0073]

[0074] Based on the properties of the spline-based semiparametric model, such as continuity and differentiability at the observation nodes, we can obtain equation (13) as follows:

[0075]

[0076] In the formula, F and G are sparse matrices determined by the observation sampling interval, constructed to solve equation (14), and the constructed Lagrangian function is:

[0077]

[0078] In the formula, the factors affecting the estimation accuracy of the multi-station nonlinear semiparametric model based on uncertain error components are: smoothing factor α and regularization matrix R, R = FG -1 F T , P = diag(P1, P2, ..., P n ), s=(s T (t1),s T (t2),…,s T (t n )) T ,

[0079] Finding the minimum value of equation (15) yields:

[0080]

[0081] From equation (16), we can obtain:

[0082]

[0083] The system state estimate and nonparametric estimate obtained by solving equation (14) are:

[0084]

[0085] Where S is an intermediate quantity, the superscript k indicates the kth iteration, and I represents the large identity matrix.

[0086] The above-mentioned technical means are also specific embodiments of the observation model construction unit 21.

[0087] Preferably, in S102, a regularization matrix is ​​constructed based on the influencing factors affecting the error, ensuring the regularization matrix is ​​unique. This unique regularization matrix possesses l-point smoothness invariance and elementary transformation invariance, including:

[0088] For the pointwise semi-parametric model, an optimal set of regularization matrices is constructed based on the factors affecting the error, with each error having its own set of regularization matrices; the set of factors affecting the measurement error is denoted as A = {a1, a2, ...}, where a i , i = 1, 2, ... represent the factors influencing the measurement element error, and a represents the factors influencing the measurement element error at the measuring station where the measuring equipment is located. i The set of values ​​below is M. i ={X 01 (a i ),X 02 (a i ),…,X 0m (a i )}, then the influencing factors a of the measurement error i The determined measurement error of station j is X. 0j (a i For functions j = 1, 2, ..., m, let this mapping relationship be:

[0089] X 0j (a i )→g i (X 0j (a i (40)

[0090] Where g i (·) represents the influencing factor a of the observation error on the measurement error. i The function form of M; i Given an ordered set, there exists a set M. i The binary relation H(a) on i ), i = 1, 2, ..., satisfying reflexivity, antisymmetry, and transitivity, using ordered pairs <M i ,H(a iThe influence factor a of the measurement error is expressed as follows: i As an ordered factor;

[0091] Let a be the influencing factor of the measurement error. i The optimal set of regularized matrices M to be determined i is O(a i ),make:

[0092]

[0093] Let g i (X 00 (a i If ))=0, then the symmetric matrix R i1 for

[0094]

[0095] Where p and q represent the p-th row and q-th column of the matrix, respectively;

[0096] According to equation (41), R can be obtained. i2 and R i3 Specific form;

[0097] R ik The function type of s is described by the functional relationship between adjacent points (k = 1, 2, 3). The value of each regularization matrix in the regularization matrix set corresponds to the weight of the measurement error. The functional relationship and weight between the components of the nonparametric components are described by the regularization matrix set. The compensated least squares criterion is used to make R... ik ∈O(a i ), (k=1,2,3); and the optimal regularization matrix set O(a i It has the following two characteristics:

[0098] (1) l-point smoothness invariance: This means that no matter how many adjacent nonparametric components are involved, the regularized matrix obtained by smoothing is in the set of the optimal regularized matrix.

[0099] (2) Invariance of elementary transformations: Any element in the optimal regularized matrix set will still belong to the optimal regularized matrix set after elementary transformations of the matrix.

[0100] The above-mentioned technical means are also specific embodiments of the model optimization unit 22.

[0101] Preferably, the modeling method for non-cooperative target observation further includes:

[0102] The semi-parametric model based on spline representation has a uniquely determined regularization matrix. Combined with the set of error influencing factors A={a1,a2,…}, the corresponding optimal regularization matrix set can be constructed.

[0103] The above-mentioned technical means are also specific embodiments of the model optimization unit 22.

[0104] Preferably, in S102, the multi-station nonlinear semi-parametric model based on uncertain error components is optimized using an optimal set of regularized matrices constructed from unique regularized matrices to obtain an optimized semi-parametric model, including:

[0105] The geometric precision factor (GDOP) is the magnification factor of the observation error, and equation (46) is used as the objective function for weight optimization:

[0106]

[0107] in, For the accuracy estimate, m p The number of parameters to be estimated refers to the components within the system state.

[0108] The likelihood value of the semiparametric model at each time step is introduced. This semiparametric model includes a pointwise semiparametric model and a spline-based semiparametric model, forming a hybrid semiparametric model based on the model likelihood value. M represents the number of hybrid semiparametric models, and generally M ≥ r; Λ j,k Let j = 1, 2, ..., M be the likelihood values ​​of each model at time k. The likelihood value at time k is calculated as follows:

[0109]

[0110]

[0111] Where r is an intermediate variable;

[0112] By introducing the likelihood value at the current time, the weights of each regularization matrix in the optimal regularization matrix set at the next time are adaptively adjusted, so that the weights of each semiparametric model move closer to the mixed semiparametric model with the largest likelihood value, thereby converging to the optimal weights of the model at the current stage. This adapts to the different requirements of the mixed weights of errors at different stages, with one stage including multiple time points.

[0113] Suppose the mixture semiparametric model with the highest likelihood value at the current time is:

[0114] Λ q,k =max{Λ 1,k ,Λ 2,k ,…,Λ M,k} (49)

[0115] Then, at the next moment, the adjustment method for the mixed weights of the error is as follows:

[0116]

[0117] Where μ is a random number in the interval [0,1].

[0118] The above-mentioned technical means are also specific embodiments of the model optimization unit 22.

[0119] Preferably, in S102, the multi-station nonlinear semi-parametric model based on uncertain error components is optimized using an optimal set of regularized matrices constructed from unique regularized matrices to obtain an optimized semi-parametric model, including:

[0120] (1) Construct the optimal regularization matrix

[0121] Based on equation (42), the optimal regularization matrix set is constructed. The number of hybrid semiparametric models is set to M. Let k = 1, and initialize the proportion of each factor affecting the error in the observation error. The proportion represents the weight of each regularization matrix in the optimal regularization matrix set, so that the weight of each regularization matrix satisfies:

[0122]

[0123] (2) Regularization matrix is ​​used for interactive mixing

[0124] Calculate the mixture value of factors based on weights. By constructing the hybrid regularization matrices using equation (42), a hybrid semiparametric model 1,2,…,M is formed;

[0125] (3) Hybrid semiparametric estimation

[0126] The components α within the smoothness factor α are determined using the generalized cross-validation method or the L-curve method. j,k The system state estimate and nonparametric estimate at the current time can be obtained by equation (10) or (19), respectively.

[0127] (4) Multi-model information fusion

[0128] The estimation accuracy of the M hybrid semiparametric models is calculated according to equation (46). The system state estimate at the current time is obtained by optimal weighted fusion of the hybrid semiparametric models. The optimal weighted fusion method of the hybrid semiparametric models is as follows:

[0129]

[0130] (5) Weight adaptive optimization

[0131] Let k = k + 1, calculate the likelihood value of each hybrid semiparametric model according to equation (48), and then adaptively optimize the weights according to equations (49) and (51); proceed to step (2) until k = N; the likelihood value of the hybrid semiparametric model contains the current information of the model and the model jump information. The current information refers to the likelihood value of each hybrid semiparametric model at the current time, and the model jump information refers to the change of the likelihood value of each hybrid semiparametric model. If the true weights of the factors affecting the error change, the model jumps; the change of the model's likelihood value is fed back to the weights of each regularization matrix at the next time, and the weights are adaptively optimized according to the proportion of factors affecting the error at each stage;

[0132] An optimized semi-parametric model is obtained.

[0133] The above-mentioned technical means are also specific embodiments of the model optimization unit 22.

[0134] In conjunction with embodiments of the present invention, a computer-readable storage medium is provided, the computer-readable storage medium storing one or more programs, which, when executed by a computer device, cause the computer device to perform any of the aforementioned non-cooperative target observation modeling methods.

[0135] The beneficial technical effects achieved by the embodiments of the present invention are as follows:

[0136] A method for modeling and estimating the uncertainty of complex multi-station errors based on a hybrid semi-parametric model was developed, achieving high-precision target estimation and accurate error identification. In non-cooperative target observation systems, errors are complex and variable, error models are uncertain, and parameterized models suffer from representation errors. Therefore, point-by-point and spline multi-station semi-parametric models for non-cooperative targets were constructed.

[0137] Traditional regularization matrices do not consider error factors and cannot accurately describe the error characteristics of multi-station systems. Therefore, a method for constructing regularization matrices based on multiple factors is proposed, and its l-point smoothness invariance and elementary transformation invariance are proved.

[0138] Based on this, we introduced model likelihood information and designed a hybrid semiparametric model based on online adaptive optimization of weights. Compared with the traditional semiparametric model, this model enhances the optimality of the regularization matrix and the optimization efficiency.

[0139] It can improve positioning accuracy and accurately identify observation system errors, providing a theoretical basis for complex error modeling and identification of non-cooperative targets.

[0140] The technical solutions of the present invention will be described in detail below with reference to specific application examples. For technical details not described in the implementation process, please refer to the relevant descriptions above.

[0141] In existing technologies, the error mechanisms of non-cooperative target observation systems are unclear and their forms are complex and uncertain, making it difficult to parameterize and accurately model them, thus restricting high-precision target positioning. This invention proposes a target uncertainty error modeling and estimation method based on hybrid semi-parametric statistics, and constructs a regularization matrix (RM) method. Experiments and numerical simulations demonstrate that this method can accurately identify the sources, forms, and variations of uncertainty errors, achieving high-precision target positioning.

[0142] 1. Regarding the problem of non-cooperative target localization and tracking based on multi-station observations, the steps of this invention are summarized as follows:

[0143] To address the issue of reduced positioning accuracy caused by model representation errors in parametric models, a multi-station nonlinear observation model with uncertainty error components was constructed. The nonlinear compensated least squares method for point-by-point semiparametric models and spline-based semiparametric models was studied, and the theoretical properties of its solutions were proved.

[0144] To address the problem that traditional regularization matrices cannot accurately estimate the errors of multi-station systems, a method for constructing regularization matrices based on multiple factors is proposed. The l-point smoothness invariance and elementary transformation invariance of the optimal regularization matrix set are proven. Furthermore, a state fusion estimation method based on a hybrid semi-parametric model is designed.

[0145] To address the issues of strong randomness and unstable optimization results in the optimization of factor weights in hybrid semiparametric models using traditional optimization algorithms, a weight adaptive optimization method based on model likelihood values ​​is proposed. This method achieves online adaptive optimization of weights by introducing current model information and jump information, thereby improving the global optimality and optimization efficiency of the regularization matrix.

[0146] 2. Multi-station nonlinear observation model based on uncertainty error components (i.e., multi-station nonlinear observation model containing error terms, where the error terms are part of the multi-station nonlinear observation model, and their form is the error model).

[0147] In non-cooperative target observation systems, such as Figure 3 As shown, equipment observation error and observation geometry are important factors affecting target positioning accuracy. When the configuration of the measuring equipment and the target satisfies the extreme value condition of the geometric accuracy factor, the inaccuracy of observation error modeling and estimation becomes an important reason restricting the high-precision positioning of non-cooperative targets.

[0148] EMBET is the most commonly used unified estimation method, which typically models multi-station nonlinear observation models without error terms as defined functions such as linear, periodic, quadratic, and exponential functions.

[0149]

[0150] In the formula, Let be the system state (which includes modelable systematic errors), e be the random error, y be the observations from m measuring devices, and f(·) be the system state. The form of f(·) (form refers to the functional form of f(·), such as linear, quadratic, cubic, etc.) and its parameters are both known. Stable optimal estimates in the least squares sense can be obtained using nonlinear parameter estimation methods such as Gauss-Newton and Levenberg-Marquardt. The optimal estimate is the optimal estimate of the parameter β to be estimated, representing the solution of the non-cooperative target observation system, i.e., the parameter estimate of β.

[0151] To characterize the system errors and model representation errors that cannot be accurately modeled, nonparametric components are introduced into the aforementioned non-cooperative target observation system. A multi-station nonlinear semiparametric model based on uncertain error components is constructed. Depending on whether time-series assumptions are made regarding the system state and nonparametric components, it is divided into pointwise semiparametric models and spline-based semiparametric models. Pointwise semiparametric models do not make time-series assumptions about the system state and nonparametric components; that is, the system state and nonparametric components at different times are uncorrelated, and the system state and nonparametric components at each time point are estimated independently. Spline-based semiparametric models make time-series assumptions about the system state and nonparametric components, assuming that the system state and semiparametric components satisfy spline functions over a period of time and are correlated, allowing for joint estimation of the system state and nonparametric components over a period of time. Making time-series assumptions is suitable when there is some prior information about the target; using prior information to make time-series assumptions about the system state and nonparametric components can improve estimation accuracy.

[0152] 2.1 Pointwise Semiparametric Modeling and Estimation

[0153] In the point-by-point semi-parametric model, the nonparametric components describe the unmodelable systematic errors or model errors in the multi-station nonlinear observation model corresponding to m measurements at a certain time. The observation equation (point-by-point semi-parametric model) is as follows:

[0154]

[0155] In the formula, s represents a nonparametric component, which represents the unmodelable system error or model error, i.e., the error model. Let R be the covariance matrix of the random error e. m Let P represent the m-dimensional real number set, and P be the weight matrix of the observation y. The residual of the pointwise semiparametric model in equation (2) above is expressed as:

[0156]

[0157] The normal equation of equation (3) is:

[0158]

[0159] Where J represents the Jacobian matrix.

[0160] The introduction of nonparametric components results in the positive semidefinite non-invertible coefficient matrix of the normal equation (4). Therefore, the compensated least squares criterion is used to modify the normal equation (4), thereby modifying equation (3):

[0161]

[0162] In the formula, α and R are the given smoothing factor and regularization matrix. A Lagrangian function (6) is constructed to solve equation (5), but it cannot replace equation (5). The constructed Lagrangian function is:

[0163]

[0164] make get:

[0165]

[0166] From equation (7) above, we can obtain:

[0167]

[0168] in, Let J be the Jacobian matrix. Then, the system state estimate and nonparametric estimate obtained by solving equation (5) are:

[0169]

[0170] Where S is an intermediate quantity, the superscript k indicates the kth iteration, and I represents the large identity matrix.

[0171] In a multi-station nonlinear observation model, one part is about the system state and the other part is about the error (nonparametric component). Because it is necessary to estimate the error (nonparametric estimate), it is also necessary to estimate the system state (system state estimate) at the same time.

[0172] 2.2 Semiparametric Modeling and Estimation Based on Spline Representation

[0173] For non-cooperative target observation systems, certain assumptions are made about the system state and nonparametric components over a period of time. These assumptions include that the system state and nonparametric components conform to spline functions over a period of time. A semi-parametric model based on spline representation can then be established as a multi-station nonlinear observation model.

[0174]

[0175] In the formula, the observed value y i =[y i1 ,yi2 ,…,y im ] T Let represent the observations of the 1st, 2nd, ..., mth measuring devices at time i. For the system state and the modelable system error, e i For m-dimensional random error, the nonparametric component s(t) i )=[s1(t i ),s2(t i ),…,s m (t i )] T s l (t i ), l=1,2,…,m describes the observation y il Unmodelable systematic errors and model errors, observation y il It is y i P has l components. i For t i Time observation value y i The weight matrix is ​​given by β, where the system state β is the polynomial coefficient or spline polynomial coefficient of the non-cooperative objective during this time period.

[0176] The semi-parametric model based on spline representation in t i Time residual V i =[V i1 V i2 ,…,V im ] T Represented as:

[0177]

[0178] The introduction of nonparametric components results in the positive semidefinite non-invertible coefficient matrix of normal equation (4). Therefore, the compensating least squares criterion modifies normal equation (4), thereby modifying equation (12) as follows:

[0179]

[0180] Based on the properties of the spline-based semiparametric model, such as continuity and differentiability at the observation nodes, we can obtain equation (13) as follows:

[0181]

[0182] In the formula, F and G are sparse matrices determined by the observation sampling interval, the proof of which is discussed in Section 3. The constructed Lagrangian function is obtained by solving equation (14):

[0183]

[0184] In the formula, the factors affecting the estimation accuracy of the multi-station nonlinear semiparametric model based on uncertain error components are: smoothing factor α and regularization matrix R, R = FG -1 F T , P = diag(P1, P2, ..., P n ), s=(s T (t1),s T (t2),…,s T (t n )) T ,

[0185] Finding the minimum value of equation (15) yields:

[0186]

[0187] From equation (16), we can obtain:

[0188]

[0189] The system state estimate and nonparametric estimate obtained by solving equation (14) are as follows:

[0190]

[0191] Where S is an intermediate quantity, the superscript k indicates the kth iteration, and I represents the large identity matrix.

[0192] 2.3 Accuracy Analysis of Semi-parametric Model Estimation

[0193] Semi-parametric models (including pointwise semi-parametric models and spline-based semi-parametric models) eliminate assumptions about the non-parametric functional forms. Since any model form has representational errors, adopting the non-assumption-free form of semi-parametric models suppresses these errors, making the semi-parametric model closer to reality and thus mitigating the impact of these errors on target positioning accuracy. The properties of its solutions are discussed below:

[0194] In the above semi-parametric model, when rank(RJ) = t, the system state estimate and non-parametric estimate can be uniquely obtained by equation (10) or (19), and when That is, when H is positive semi-definite, under the meaning of mean square error, the estimate obtained by the semi-parametric model is better than the estimate obtained by the parametric model.

[0195]

[0196] M is an intermediate quantity, and the subscript pe indicates the parameter estimate. It is a parametric model The mean square error. Proof: By the extremum condition of the Lagrange function, the normal equation of the above semi-parametric model (after considering the compensated least squares principle, the normal equation (4) becomes a normal equation) is:

[0197]

[0198] Considering If the above formula is true, then and Established simultaneously, that is When rank(RJ) = t, if and only if and Equation (21) holds true. Therefore, when rank(RJ) = t, the system state estimate and nonparametric estimate can be uniquely obtained by equation (10) or (19). The system state is the parameter to be estimated, which is an unknown parameter. The parameter to be estimated is the position and velocity of the target in a non-cooperative target observation system.

[0199] The mean square error is defined as:

[0200]

[0201] Where E represents expectation.

[0202] The mean square error measures the accuracy of the system state estimate, and can be expressed as follows: Decomposed into:

[0203]

[0204] Where D represents variance.

[0205] tr(·) denotes the trace of the matrix within (). From equation (7), we can obtain:

[0206]

[0207] Let G = (J) T PJ) -1 J T P, then therefore

[0208]

[0209] in, Let e ​​represent the covariance of the random error e.

[0210] From equation (24), we can obtain

[0211]

[0212] Similarly, the mean square error of the system state estimate of the semi-parametric model can be obtained as follows:

[0213]

[0214] Therefore, subtracting and rearranging the previous two equations (27) and (28) yields:

[0215]

[0216] Right now The proposition is true.

[0217] 3. A Regularization Matrix Construction Method Based on Factors Affecting Error

[0218] In the analysis of the accuracy of semi-parametric model estimation, as shown in equation (27), the smoothing factor α and the regularization matrix R are important factors affecting the accuracy of semi-parametric model estimation, because only these two parameters need to be determined in advance and can be adjusted; different values ​​affect the final solution of the semi-parametric model. The smoothing factor α can be obtained by constructing a generalized cross-validation function from the residuals and by using the signal norm. and noise norm vn(α)=V T The L-curve constructed by PV (a current technology) is used to determine the model. The regularization matrix R describes the functional relationship and weights between the nonparametric components. Traditional semiparametric estimation determines R based on experience or prior information, without considering the influence of observation errors (i.e., systematic errors) during target motion. It cannot accurately reflect the complex observation errors in the actual measurement process and does not effectively reduce model representation errors.

[0219] In the semiparametric models (2) and (11), the regularization matrix is ​​an important factor affecting the estimation accuracy of the semiparametric model.

[0220] To address the aforementioned issues, a regularization matrix R needs to be constructed. First, the uniqueness of the regularization matrix R for the spline-based semiparametric model is proven, since the spline-based semiparametric model can uniquely determine the regularization matrix R, eliminating the need for its construction. A single pointwise semiparametric model cannot uniquely determine R, so a regularization matrix R for the pointwise semiparametric model needs to be constructed based on the factors influencing the error. The optimality of this construction method is then proven. Finally, to address the complex and variable nature of the factors influencing the error, a hybrid semiparametric model is proposed, and a weight optimization method is designed.

[0221] The properties and construction methods of its regularization matrix are discussed in two cases below.

[0222] 3.1 Uniqueness of the Regularization Matrix R of a Semi-parametric Model Based on Spline Representation

[0223] Spline functions are used to constrain the motion state and nonparametric parameters of a non-cooperative target at a certain stage. The uniqueness of its regularization matrix can be proven using the properties of spline functions. The time interval [a, b] is divided as follows: Δ: a < t1 < t2 < ... < t n <b, without loss of generality, assume that some component s(t) of the nonparametric component is in t∈[t i ,t i+1 The inner part can be represented as:

[0224]

[0225] Let s(t) i ) = s i Based on the continuity of s″(t) at the node, we have:

[0226]

[0227] Where h represents the sampling time interval.

[0228] Therefore, equation (31) can be expressed as:

[0229]

[0230] Since s′(t) is continuous at the nodes, i.e., s′(t) i +0)=s′(t i -0), therefore, from equation (32) we can obtain:

[0231]

[0232] Therefore, we can conclude that: F T s=Gc(34)

[0233] In the formula: s=(s1,s2,…,s n ) T c = (c2, c3, ..., c n-1 ) T F and G are shown below:

[0234]

[0235] Note that s″′(t) in each subinterval [t i ,t i+1 ] is a constant, that is Then the corresponding equation (14) for the compensated least squares criterion is:

[0236]

[0237] From equation (32), we can see that

[0238]

[0239] From equation (35), we can see that

[0240]

[0241] Therefore, this estimation method based on the spline representation of the semiparametric model can uniquely determine the regularization matrix R;

[0242] Common factors influencing measurement errors include observation line distance, timing error, station location error, environmental factors, and angle difference. Based on the unique regularization matrix R, and combined with the set of error influencing factors A = {a1, a2, ...}, a corresponding optimal regularization matrix set is constructed.

[0243] 3.2 Construction of Regularization Matrix Based on Factors Affecting Error

[0244] The regularization matrix R reflects the functional type of the nonparametric components, i.e., the model error. Therefore, the selection of the regularization matrix is ​​largely determined by the influencing factors of the non-cooperative target observation system error. Common influencing factors of measurement errors include observation line distance, timing error, station location error, environmental factors, and angle difference. Below, for a point-by-point semi-parametric model, we construct the optimal set of regularization matrices based on the factors affecting the error; each error has its own set of regularization matrices.

[0245] The set of factors influencing the measurement error is A = {a1, a2, ...}, where a i , i = 1, 2, ... represent the factors influencing the measurement element error, and a represents the factors influencing the measurement element error at the measuring station where the measuring equipment is located. i The set of values ​​below is M. i ={X 01 (a i ),X 02 (a i ),…,X 0m (a i )}, then the influencing factors a of the measurement error i The determined measurement error of station j is X. 0j (a i Let the function be a function of j = 1, 2, ..., m, and denote this mapping relationship as .

[0246] X 0j (a i )→g i (X 0j (a i (40)

[0247] Where g i (·) represents the influencing factor a of the observation error on the measurement error. i The function form of generally includes linear, quadratic, exponential, etc. There exists a set M.i The binary relation H(a) on i ), i = 1, 2, ..., satisfying reflexivity, antisymmetry, and transitivity, i.e., M i For ordered sets, use ordered pairs. <M i ,H(a i The factor representing the influence of measurement error is called a. i These are ordered factors, specifically factors a influencing the measurement error. i The observation errors generated by different devices can be ordered. In non-cooperative target observation systems, the factors influencing measurement errors are generally ordered. Traditional regularization matrices do not utilize this information and cannot effectively reduce model representation errors.

[0248] Let a be the influencing factor of the measurement error. i The optimal set of regularized matrices M to be determined i is O(a i ),make

[0249]

[0250] Let g i (X 00 (a i If ))=0, then the symmetric matrix R i1 for

[0251]

[0252] Where p and q represent the p-th row and q-th column of the matrix, respectively.

[0253] Similarly, R can be obtained from equation (41). i2 and R i3 The specific form of R. ik The function type of s is described by the functional relationship between adjacent points (k = 1, 2, 3). The value of each regularization matrix in the regularization matrix set corresponds to the weight of the nonparametric component, i.e., the measurement error. The regularization matrix set describes the functional relationship and weight between the components of the nonparametric component. Due to the use of the compensated least squares criterion, R... ik ∈O(a i ), (k=1,2,3). And the optimal regularization matrix set O(a i It has the following two characteristics:

[0254] 1) Point-level smoothness invariance. The above R... ik (k = 1, 2, 3) corresponds to the sum of squares of the differences between two adjacent nonparametric components. Similarly, this can be extended to three-point smoothing:

[0255]

[0256] Then R i ∈O(a i This can then be extended to smoothing at points l ≤ m.

[0257] l refers to the number of points used, and equation (41) utilizes s. j-1 s j These two points represent two-point smoothing; Equation (43) utilizes s j-1 s j s j+1 These three points represent three-point smoothing. The meaning of l-point smoothing invariance is that the regularization matrix obtained by smoothing several points (several adjacent nonparametric components) is within the set of optimal regularization matrices.

[0258] 2) Invariance of elementary transformations. in, Let be the elementary transformation matrix for arbitrary row and column swaps. The meaning of elementary transformation invariance is that any element in the optimal regularized matrix set (i.e., any regularized matrix) still belongs to the optimal regularized matrix set after elementary transformation.

[0259] The advantage of each of these two properties is that elements in the optimal regularized matrix set, after undergoing l-point smoothing or elementary transformations, still belong to this optimal regularized matrix set; and all elements in the optimal regularized matrix set possess optimality and are consistent. Proof:

[0260] 1) Based on the compensated least squares criterion, the regularization matrix is ​​based on l-point smoothing. The functional relationship between adjacent l nonparametric components accurately reflects the weights of s components, therefore we have

[0261] 2) Depend on

[0262]

[0263] It can be known The mean square error of the system state estimate is used to obtain:

[0264]

[0265] Let α′=α / λ, then Equivalent to the aforementioned minimization problem, therefore

[0266] In this section, for non-cooperative target tracking and localization scenarios, due to the random and variable motion state of non-cooperative targets and the diversity of measurement models, the factors influencing errors are numerous and the error forms are uncertain. Regularization matrices constructed based on single factors cannot accurately estimate observation errors. Therefore, a set of error influencing factors is constructed, and a hybrid semi-parametric model is formed through the interaction of corresponding regularization matrix sets.

[0267] 4. State estimation algorithm based on hybrid semiparametric model

[0268] Based on the set of error influencing factors A = {a1, a2, ...}, a corresponding set of regularization matrices is constructed. Combined with the set of regularization matrices constructed in section 3.2, a hybrid semi-parametric model is formed through the interaction of models (each regularization matrix corresponds to a pointwise semi-parametric model or a spline-based semi-parametric model), thereby obtaining the system state estimate and non-parametric estimates. For example... Figure 4 As shown.

[0269] Figure 4 In this process, the optimal regularization matrix is ​​constructed using equation (42), and the system state and nonparametric estimates are obtained using equations (10) or (19). The smoothing factor α is determined by the generalized cross-validation method or the L-curve method. The mixed regularization matrix is ​​not a simple weighted sum of the individual regularization matrices, but rather a mixture of factors calculated using weights. Then, it is constructed using equation (42). The weight p i The estimation accuracy can be iteratively optimized using an optimization algorithm. Since the true state of non-cooperative targets is difficult to obtain, and considering that the geometrical precision factor (GDOP) is a magnification factor of the observation error, the following formula is used as the objective function of the weight optimization problem:

[0270]

[0271] In the formula For the accuracy estimate, m p The number of parameters to be estimated (the number of components in the system state).

[0272] If gradient descent is used to optimize the weights, since

[0273]

[0274] As can be seen from equations (10) and (42), the gradient calculation is large and the weight optimization efficiency is low. When the heuristic algorithm is applied to the weight optimization problem of the interactively formed hybrid semiparametric model, due to the randomness of the search, there are problems such as unstable optimization results, easy to get trapped in local optima, and low convergence efficiency. On the other hand, in addition to the diversity of measurement systems, the motion of non-cooperative targets has the characteristics of multi-stage, large span and random variation, which leads to complex and varied error sources. The weights of each factor change at all times. The weights obtained based on global accuracy optimization remain unchanged and cannot meet the requirements of high-precision positioning. Therefore, it is necessary to find an adaptive online adjustment method for the weights. Introducing the likelihood value information of the semiparametric model at each time step, the algorithm is adjusted as follows:

[0275] Figure 6 This is a mixture of semiparametric models based on model likelihood values, where M is the number of mixture of semiparametric models, and generally M ≥ r. Λ j,k Let j = 1, 2, ..., M be the likelihood values ​​of each model at time k, calculated as follows:

[0276]

[0277] Here, r is an intermediate variable.

[0278] By introducing the likelihood value of the hybrid semiparametric model at the current time step, the weights of each regularization matrix at the next time step are adaptively adjusted so that the weights of each semiparametric model move closer to the hybrid semiparametric model with the largest likelihood value, thereby converging to the optimal weights of the model at the current stage. This adapts to the different requirements of error mixing weights at different stages, with each stage including multiple time steps.

[0279] Suppose the mixed semiparametric model with the highest likelihood value at the current time is:

[0280] Λ q,k =max{Λ 1,k ,Λ 2,k ,…,Λ M,k} (49)

[0281] Then, in the next moment, the mixed weights will be adjusted as follows:

[0282]

[0283] In the formula, μ is a random number in the interval [0,1]. That is, the weights of each hybrid semiparametric model converge towards the weights of the model with the highest likelihood value. The step size depends on the likelihood value of the semiparametric model and the likelihood value of the optimal hybrid semiparametric model; the larger the difference, the larger the step size, thus improving the convergence efficiency of the algorithm. μ helps each hybrid semiparametric model escape local optima, thereby strengthening the global optimality of the algorithm.

[0284] The specific implementation process of the algorithm is as follows:

[0285] (1) Construct the optimal regularization matrix

[0286] First, based on the observation model and target motion characteristics, a set of factors influencing the error is constructed, and the corresponding factor values ​​for each station are calculated. Then, g is determined. i Construct the optimal regularization matrix according to equation (42) in the form of (·). Set the number of models M, let k=1, and initialize the proportion of each error's influencing factors in the observation error, i.e., the weights of each regularization matrix, to satisfy:

[0287]

[0288] (2) Regularization matrix is ​​used for interactive mixing

[0289] Calculate the mixture value of factors based on weights. Then, by using equation (42), a hybrid semiparametric model 1,2,…,M is formed by constructing the hybrid regularization matrices.

[0290] (3) Hybrid semiparametric estimation

[0291] The components α within the smoothness factor α are determined using the generalized cross-validation method or the L-curve method. j,k Then, the state estimate and nonparametric estimate at the current time can be obtained through equation (10) or (19). Nonparametric variables represent the unmodelable errors of the observation systems at each station at the current moment.

[0292] (4) Multi-model information fusion

[0293] The estimation accuracy of the M hybrid semiparametric models is calculated according to equation (46). The system state estimate at the current time with higher accuracy is obtained by optimal weighted fusion of the hybrid semiparametric models. The optimal weighted fusion method is as follows:

[0294]

[0295] (5) Weight adaptive optimization

[0296] Let k = k + 1, calculate the likelihood value of each hybrid semiparametric model according to equation (48), and then adaptively optimize the weights according to equations (49) and (51). Proceed to step (2) until k = N.

[0297] During the movement of a non-cooperative target, the observation system error arises from the interaction and mixing of various factors, and the proportion of each factor changes constantly. Therefore, the aforementioned model-based likelihood value-based adaptive weight optimization method is adopted. The model's likelihood value includes current model information (the likelihood value of each mixed semi-parametric model at the current time) and model jump information (the change in the likelihood value of each mixed semi-parametric model; a change indicates a jump). If the true weights of the factors affecting the error change, i.e., the model jumps, the change in the model's likelihood value is fed back to the weights of the regularization matrices at the next time step. The weights are adaptively optimized based on the proportion of influencing factors of the error at each stage, accurately identifying the sources of error from multiple stations, the proportion of each error factor, and the error form, thus improving estimation accuracy. Therefore, the weights are adaptively optimized based on the observation information at each time step, without the need to set iteration stopping conditions, improving the optimization efficiency and robustness of the method.

[0298] In summary, regarding the completeness of the factor set, due to the lack of prior information on non-cooperative targets and the potential coupling between different factors, a model set with too many factors increases computational complexity. Typically, based on the measurement system's methodology, the actual measurement scenario, and the production background of the observation equipment, factors that significantly influence station errors are selected to form the factor set, thereby generating an optimal regularized matrix set that accurately describes the types and forms of errors across multiple stations.

[0299] 5. Simulation Experiments and Analysis

[0300] To verify the effectiveness of the proposed multi-station hybrid semi-parametric model and weighted adaptive optimization method, a non-cooperative target positioning experiment was conducted using the target GPS trajectory, station relative position information, and observation information in a real multi-station ranging and positioning system, combined with simulation data of complex observation errors.

[0301] To test the system error identification capability of the multi-station hybrid semi-parametric model, simulations were conducted based on observational data with complex error forms considering multiple error factors. Constant, periodic, semi-parametric, polynomial periodic system error models and a hybrid semi-parametric model were constructed respectively, and their system errors and state estimation accuracy were compared.

[0302] The target motion state is divided into multiple stages, with each error influencing factor contributing a different proportion to the system error at each stage. In the hybrid semi-parametric model, particle swarm optimization, genetic algorithm, simulated annealing, and adaptive weight optimization based on model likelihood are used to optimize the weights of each error factor. The optimization efficiency, optimization results, error identification results, and state estimation accuracy of each algorithm are compared. Finally, the impact of the number of models M on the algorithm's accuracy and efficiency is considered.

[0303] 5.1 Simulation Test Scenario

[0304] Convert the GPS target trajectory and station position into relative coordinates, such as Figure 5 The figure shows a ranging and positioning system with n=6 stations. After parameter significance testing, the target trajectory approximately satisfies a cubic polynomial, and the system state, i.e., the target position, is X=(x,y,z). T satisfy

[0305]

[0306] In the formula, a i ,b i (i = 1, 2, 3) are the polynomial coefficients, and t is the sampling time. The experiment selects n = 80 data points uniformly sampled from t ∈ [0.7s, 1.5s].

[0307] Considering the complex errors at each station caused by the combined effects of various factors during target movement, including the observation line distance (distance between the station and the target), the rate of change of distance, and the station's timing error (which can be obtained from prior information through static experiments), the error amplitude is determined based on the factor values ​​at each station. On the other hand, the form of error is also an important factor affecting positioning accuracy. The error identification capability of the model is verified using periodic error and fractal Brownian motion (FBM) error, respectively. Figure 7 As shown, S represents the measuring station. Brownian motion {B(t), t≥0} satisfies B(0)=0, and B(t) has stationary independent increments. Fractal Brownian Motion B H (t), 0 < H < 1, and more generalized fractal geometry is a mathematical model proposed by Mandelbrot and Ness:

[0308]

[0309] In the formula, H is the Hurst exponent, which characterizes the long memory property of time series. H B(t) is a generalization of B(t). The increment has stability, self-similarity and autocorrelation within any time window, and can be used to simulate noisy time trajectories with fractal characteristics.

[0310] 5.2 Experimental Validation of Multi-Station Hybrid Semi-parametric Model

[0311] The observation information is generated using the GPS target trajectory and the relative position of the station as shown in Equation (53), and periodic error and FBM error based on multiple factors are added to it respectively. The random error is set to e ~ N(0, 0.01·I). Constant error model (CVEM), periodic error model (PEM), semi-parametric error model (SPEM), polynomial periodic error model (PPEM) and hybrid semi-parametric error model (HSPEM) are constructed respectively. In SPEM and HSPEM, the smoothing parameter α is determined according to the generalized cross-validation method (GCV). In SPEM, the regularization matrix is ​​determined by the traditional time series characteristic method, that is, it satisfies HSPEM, on the other hand, is determined using a multi-factor-based construction method. The state estimation error, observation estimation (OC) residuals, and systematic error identification for each model are as follows: Figure 8 and Figure 9 As shown.

[0312] from Figure 8 and Figure 9 The following results and analysis can be obtained from this:

[0313] (1) Under the two complex error conditions, the trend of GCV(α) is consistent. This is because when α increases, αs T With the increased weight of Rs, the nonparametric components of each station are smoother, thus better reflecting the actual error and reducing the representation error.

[0314] (2) The HSPEM positioning error, considering multiple factors, exhibits zero mean, while the positioning errors of CVEM, PEM, SPEM, and PPEM are significantly periodic and non-stationary. The positioning error of the proposed method is significantly smaller than that of traditional parameter estimation methods and semi-parametric methods.

[0315] (3) In terms of overall position accuracy and measurement residuals, HSPEM is also significantly better than traditional methods. As can be seen from Table 1, the proposed method significantly improves positioning accuracy, while the traditional semi-parametric estimation method has limited accuracy improvement compared to the parametric method. Therefore, for non-cooperative targets with complex errors, multiple factors should be considered to construct a regularization matrix.

[0316] (4) CVEM, PEM, SPEM and PPEM methods cannot accurately identify station systematic errors, while HSPEM method can accurately identify the systematic errors of each station under complex conditions.

[0317] Table 1. Estimation accuracy of each method under two complex error conditions.

[0318]

[0319] 5.2 Experimental Verification of the Adaptive Weight Optimization Method

[0320] Error sources during target movement are complex and varied. A regularization matrix should be set up considering multiple influencing factors. Determining the weights of each factor for non-cooperative targets is a key issue in HSPEM, directly determining the system's estimation accuracy. Within the HSPEM framework, the target movement process is divided into two stages, with the proportions of observation line distance, distance change rate, and station timing error in each stage being p... (1) =(0.2,0.1,0.7) and p (2) = (0.2, 0.6, 0.2). The weights of each factor were optimized using Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Simulated Annealing (SA), and Model Likelihood Adaptive Optimization (MLAO) algorithms, respectively. The parameter settings are shown in Table 2.

[0321] Table 2 Algorithm Parameter Settings

[0322]

[0323]

[0324] During the optimization process, the objective functions of PSO, GA, and SA are shown in equation (44), with the maximum number of iterations set to 200. The optimization method of MLAO is shown in equations (46)-(48). Therefore, the number of iterations of MLAO is the number of sampling points for the target motion. The estimation accuracy of each algorithm during the optimization process is as follows: Figure 10 As shown in (a). The proportions of each error influencing factor and the optimization time obtained from the optimization of each algorithm are shown in Table 3. The results are as follows: [Table data would be inserted here]. Figure 10 (b)

[0325] Table 3. Optimization weights and optimization times for each algorithm.

[0326] <![CDATA[p (1) ]]> <![CDATA[p (2) ]]> t(s) Truth value [0.2,0.1,0.7] [0.2,0.6,0.2] - PSO [0.1958,0.7811,0.0231] [0.1958,0.7811,0.0231] 2390.23 GA [0.2018,0.7706,0.0275] [0.2018,0.7706,0.0275] 5952.77 SA [0.2754,0.5666,0.1580] [0.2754,0.5666,0.1580] 19595.98 MLAO (M=100) [0.1969,0.1026,0.7005] [0.2179,0.6313,0.1508] 895.74 MLAO (M=60) [0.1766,0.1098,0.7137] [0.2181,0.6519,0.1300] 212.32 MLAO (M=20) [0.1640,0.1255,0.7105] [0.2214,0.6415,0.1370] 70.87

[0327] The state estimation accuracy and system error identification results of the hybrid semiparametric estimation methods with corresponding weights are as follows: Figure 11 As shown in Table 4. Based on the above results, the following conclusions can be drawn:

[0328] (1) The PSO, GA and SA algorithms optimize the weights based on the global accuracy of the trajectory. However, the weight optimization gets stuck in local minima, resulting in large errors and an inability to identify changes in the factors affecting the error. Constant error weights lead to low positioning accuracy, and the above algorithms have many parameters in the weight optimization problem, resulting in low optimization efficiency.

[0329] (2) The MLAO algorithm can accurately and quickly converge to the optimal error factor weights at each stage, and can accurately identify the model transition time. For example Figure 10As shown in (a), the model jumps at the 41st sampling point, meaning the weight of the error factor changes, resulting in a brief increase in positioning accuracy followed by rapid convergence. The positioning accuracy in the second stage is lower than that in the first stage because the observation error is larger in the second stage.

[0330] (3) In the MLAO algorithm, the larger the number of models M, the better the global optimality of the algorithm, but the optimization efficiency will be reduced. The smaller M is, the larger the "peak" when the model jumps.

[0331] (4) The positioning errors in the second stage of PSO, GA, and SA algorithms are relatively small because the observation errors in the second stage are larger and account for a larger proportion of the optimization objective function. The positioning accuracy of the above algorithms is low and the error identification is inaccurate. In contrast, the MLAO algorithm adaptively adjusts the weights, identifies the source of error, accurately identifies the form of error, and significantly improves the positioning accuracy.

[0332] In summary, in non-cooperative target observation systems with complex errors, the hybrid semi-parametric model based on the MLAO algorithm can accurately identify the sources and forms of errors, adaptively optimize the weights of error factors in stages, improve optimization efficiency and global optimality, and significantly enhance positioning accuracy.

[0333] Table 4 Comparison of accuracy of each algorithm after weight optimization

[0334]

[0335]

[0336] The beneficial technical effects achieved by the embodiments of the present invention are as follows:

[0337] A method for modeling and estimating the uncertainty of complex multi-station errors based on a hybrid semi-parametric model was developed, achieving high-precision target estimation and accurate error identification. In non-cooperative target observation systems, errors are complex and variable, error models are uncertain, and parameterized models suffer from representation errors. Therefore, point-by-point and spline multi-station semi-parametric models for non-cooperative targets were constructed.

[0338] Traditional regularization matrices do not consider error factors and cannot accurately describe the error characteristics of multi-station systems. Therefore, a method for constructing regularization matrices based on multiple factors is proposed, and its l-point smoothness invariance and elementary transformation invariance are proved.

[0339] Based on this, we introduced model likelihood information and designed a hybrid semiparametric model based on online adaptive optimization of weights. Compared with the traditional semiparametric model, this model enhances the optimality of the regularization matrix and the optimization efficiency.

[0340] Simulation experiments demonstrate the effectiveness of the model and method, which can improve positioning accuracy, accurately identify observation system errors, and provide a theoretical basis for complex error modeling and identification of non-cooperative targets.

[0341] It should be understood that the specific order or hierarchy of steps in the disclosed process is an example of an exemplary method. Based on design preferences, it should be understood that the specific order or hierarchy of steps in the process may be rearranged without departing from the scope of this disclosure. The appended method claims provide elements of various steps in an exemplary order and are not intended to limit the scope to the specific order or hierarchy described.

[0342] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A modeling method for non-cooperative target observation, characterized in that, include: A multi-measurement device observation model of a non-cooperative target observation system is used to observe non-cooperative targets. A multi-station nonlinear observation model of the non-cooperative targets is constructed based on the EMBET estimation method. The multi-station nonlinear observation model includes modelable systematic errors. Nonparametric variables are introduced into the multi-station nonlinear observation model to describe unmodelable observation systematic errors and / or model errors, forming a multi-station nonlinear semiparametric model based on uncertain error components. The factors affecting the estimation accuracy of the multi-station nonlinear semiparametric model based on uncertain error components are identified: the regularization matrix. A regularization matrix is ​​constructed based on the factors affecting the error, ensuring that the regularization matrix is ​​unique. The unique regularization matrix has l-point smoothness invariance and elementary transformation invariance. The optimal regularization matrix set constructed by the unique regularization matrix is ​​used to optimize the multi-station nonlinear semiparametric model based on uncertain error components, resulting in an optimized semiparametric model. An optimized semi-parametric model is used to solve for the location of non-cooperative targets observed by multiple measurement devices, and the location of the non-cooperative targets is obtained. A multi-station nonlinear observation model of a non-cooperative target is constructed using a multi-measurement device observation system to observe the non-cooperative target. This model includes modelable systematic errors, including: A multi-station nonlinear observation model of non-cooperative targets is constructed based on the EMBET estimation method, observing non-cooperative targets using multiple measurement devices in a non-cooperative target observation system. In the formula, The system state includes modelable systematic errors, where e is the random error, y is the observations from m measuring devices, and f(·) represents the system state. A function whose formal function and parameters are known.

2. The modeling method for non-cooperative target observation according to claim 1, characterized in that, The multi-station nonlinear semi-parametric model based on uncertain error components includes a pointwise semi-parametric model. The pointwise semi-parametric model does not make time series assumptions about the system state and non-parametric components. The system state and non-parametric components at different times are uncorrelated, and the system state and non-parametric components at each time are estimated independently. Nonparametric variables are introduced into the multi-station nonlinear observation model to describe unmodelable observation system errors and / or model errors, forming a multi-station nonlinear semiparametric model based on uncertain error components, including: Nonparametric variables are introduced into the multi-station nonlinear observation model. These nonparametric components are used to describe unmodelable systematic errors or model errors in the multi-station nonlinear observation model corresponding to m measuring devices at a certain time, forming a point-by-point semiparametric model. The point-by-point semiparametric model is expressed as follows: In equation β, s represents the nonparametric component. Let be the covariance matrix of the random error e, and P be the weight matrix of the observation y; calculate the residuals of the pointwise semiparametric model, and express the residuals of the pointwise semiparametric model in equation (2) as: The normal equation of equation (3) is: Where J represents the Jacobian matrix; The introduction of nonparametric components leads to the positive semidefinite non-invertible coefficient matrix of normal equation (4). The compensated least squares criterion is used to correct normal equation (4), thereby correcting equation (3): In equation (4), the factors affecting the estimation accuracy of the multi-station nonlinear semi-parameter model based on uncertain error components are: smoothing factor α and regularization matrix R; a Lagrangian function is constructed to solve equation (5), and the constructed Lagrangian function is expressed as: make get: From equation (7), we can obtain: in, Let J be the Jacobian matrix; the system state estimate and nonparametric estimate obtained by solving equation (5) are: Where S is an intermediate quantity, the superscript k indicates the kth iteration, and I represents the large identity matrix.

3. The modeling method for non-cooperative target observation according to claim 2, characterized in that, The multi-station nonlinear semi-parametric model based on uncertain error components also includes a spline-based semi-parametric model. The spline-based semi-parametric model makes time-series assumptions about the system state and non-parametric components, assuming that the system state and semi-parametric components satisfy spline functions over a period of time, and jointly estimates the system state and non-parametric components over a period of time. Nonparametric variables are introduced into the multi-station nonlinear observation model to describe the unmodelable observation system errors and model errors, forming a multi-station nonlinear semiparametric model based on uncertain error components, including: When using a multi-measurement device system to observe a non-cooperative target, it is assumed that the system state and nonparametric components over a period of time conform to a spline function. This is then incorporated into the multi-station nonlinear observation model of the non-cooperative target, forming a semiparametric model based on spline representation. This semiparametric model based on spline representation is expressed as: Among them, the observed value y i =[y i1 ,y i2 ,…,y im ] T Let represent the observations of the 1st, 2nd, ..., mth measuring devices at time i, e i For m-dimensional random error, the nonparametric component s(t) i )=[s1(t i ),s2(t i ),…,s m (t i )] T s l (t i ), l=1,2,…,m describes the observation y il Unmodelable systematic errors and / or model errors, observation y il It is y i The l components; P i For t i Time observation value - y i The weight matrix, where the system state β is the polynomial coefficient or spline polynomial coefficient of the non-cooperative objective during this time period; The spline-based semiparametric model is applied to t. i Time residual V i =[V i1 V i2 ,…,V im ] T Represented as: Due to the introduction of nonparametric components, the coefficient matrix of normal equation (4) becomes positive semidefinite and non-invertible. The compensating least squares criterion is used to correct normal equation (4), thus modifying equation (12) as follows: Based on the properties of the spline-based semiparametric model, such as continuity and differentiability at the observation nodes, we can obtain equation (13) as follows: In the formula, F and G are sparse matrices determined by the observation sampling interval, constructed to solve equation (14), and the constructed Lagrangian function is: In the formula, the factors affecting the estimation accuracy of the multi-station nonlinear semiparametric model based on uncertain error components are: smoothing factor α and regularization matrix R, R = FG -1 F T , P = diag(P1, P2, ..., P n ), s=(s T (t1),s T (t2),…,s T (t n )) T , Finding the minimum value of equation (15) yields: From equation (16), we can obtain: The system state estimate and nonparametric estimate obtained by solving equation (14) are: Where S is an intermediate quantity, the superscript k indicates the kth iteration, and I represents the large identity matrix.

4. The modeling method for non-cooperative target observation according to claim 3, characterized in that, A regularization matrix is ​​constructed based on the factors influencing the error, ensuring its uniqueness. This unique regularization matrix possesses l-point smoothness invariance and elementary transformation invariance, including: For the pointwise semi-parametric model, an optimal set of regularization matrices is constructed based on the factors influencing the error, with each error having its own set of regularization matrices. The set of factors influencing the measurement error is denoted as A = {a1, a2, ...}, where a... i i = 1, 2, representing the factors influencing the measurement element error, and the measurement station where the measuring equipment is located. i The set of values ​​below is M. i ={X 01 (a i ),X 02 (a i ),…,X 0m (a i )}, then the influencing factors a of the measurement error i The determined measurement error of station j is X. 0j (a i For functions j = 1, 2, ..., m, let this mapping relationship be: X 0j (a i )→g i (X 0j (a i )) (40) Where g i (·) represents the influencing factor a of the observation error on the measurement error. i The function form of M; i Given an ordered set, there exists a set M. i The binary relation H(a) on i ), i = 1, 2, ..., satisfying reflexivity, antisymmetry, and transitivity, using ordered pairs <M i ,H(a i The influence factor a of the measurement error is expressed as follows: i As an ordered factor; Let a be the influencing factor of the measurement error. i The optimal set of regularized matrices M to be determined i is O(a i ),make: Let g i (X 00 (a i If ))=0, then the symmetric matrix R i1 for Where p and q represent the p-th row and q-th column of the matrix, respectively; According to equation (41), R can be obtained. i2 and R i3 Specific form; R ik The function type of s is described by the functional relationship between adjacent points (k = 1, 2, 3). The value of each regularization matrix in the regularization matrix set corresponds to the weight of the measurement error. The functional relationship and weight between the components of the nonparametric components are described by the regularization matrix set. The compensated least squares criterion is used to make R... ik ∈O(a i ), (k=1,2,3); and the optimal regularization matrix set O(a i It has the following two characteristics: (1) l-point smoothness invariance: This means that no matter how many adjacent nonparametric components are involved, the regularized matrix obtained by smoothing is in the set of the optimal regularized matrix. (2) Invariance of elementary transformations: Any element in the optimal regularized matrix set will still belong to the optimal regularized matrix set after elementary transformations of the matrix.

5. The modeling method for non-cooperative target observation according to claim 4, characterized in that, Also includes: The semi-parametric model based on spline representation has a uniquely determined regularization matrix. Combined with the set of error influencing factors A={a1,a2,…}, the corresponding optimal regularization matrix set can be constructed.

6. The modeling method for non-cooperative target observation according to claim 5, characterized in that, The multi-station nonlinear semiparametric model based on uncertain error components is optimized using an optimal set of regularized matrices constructed from unique regularized matrices, resulting in an optimized semiparametric model, including: The geometric precision factor (GDOP) is the magnification factor of the observation error, and equation (46) is used as the objective function for weight optimization: in, For the accuracy estimate, m p The number of parameters to be estimated refers to the components within the system state. The likelihood value of the semiparametric model at each time step is introduced. This semiparametric model includes a pointwise semiparametric model and a spline-based semiparametric model, forming a hybrid semiparametric model based on the model likelihood value. M represents the number of hybrid semiparametric models, and generally M ≥ r; Λ j,k Let j = 1, 2, ..., M be the likelihood values ​​of each model at time k. The likelihood value at time k is calculated as follows: Where r is an intermediate variable; By introducing the likelihood value at the current time, the weights of each regularization matrix in the optimal regularization matrix set at the next time are adaptively adjusted, so that the weights of each semiparametric model move closer to the mixed semiparametric model with the largest likelihood value, thereby converging to the optimal weights of the model at the current stage. This adapts to the different requirements of the mixed weights of errors at different stages, with one stage including multiple time points. Suppose the mixture semiparametric model with the highest likelihood value at the current time is: L q,k =max{Λ 1,k ,L 2,k ,…,L M,k } (49) Then, at the next moment, the adjustment method for the mixed weights of the error is as follows: Where μ is a random number in the interval [0,1].

7. The modeling method for non-cooperative target observation according to claim 6, characterized in that, The multi-station nonlinear semiparametric model based on uncertain error components is optimized using an optimal set of regularized matrices constructed from unique regularized matrices, resulting in an optimized semiparametric model, including: (1) Construct the optimal regularization matrix Based on equation (42), the optimal regularization matrix set is constructed. The number of hybrid semiparametric models is set to M. Let k = 1, and initialize the proportion of each factor affecting the error in the observation error. The proportion represents the weight of each regularization matrix in the optimal regularization matrix set, so that the weight of each regularization matrix satisfies: (2) Regularization matrix is ​​used for interactive mixing Calculate the mixture value of factors based on weights. By constructing the hybrid regularization matrices using equation (42), a hybrid semiparametric model 1,2,…,M is formed; (3) Hybrid semiparametric estimation The components α within the smoothness factor α are determined using the generalized cross-validation method or the L-curve method. j,k The system state estimate and nonparametric estimate at the current time can be obtained by equation (10) or (19), respectively. (4) Multi-model information fusion The estimation accuracy of the M hybrid semiparametric models is calculated according to equation (46). The system state estimate at the current time is obtained by optimal weighted fusion of the hybrid semiparametric models. The optimal weighted fusion method of the hybrid semiparametric models is as follows: (5) Weight adaptive optimization Let k = k + 1, calculate the likelihood value of each hybrid semiparametric model according to equation (48), and then adaptively optimize the weights according to equations (49) and (51); proceed to step (2) until k = N; the likelihood value of the hybrid semiparametric model contains the current information of the model and the model jump information. The current information refers to the likelihood value of each hybrid semiparametric model at the current time, and the model jump information refers to the change of the likelihood value of each hybrid semiparametric model. If the true weights of the factors affecting the error change, the model jumps; the change of the model's likelihood value is fed back to the weights of each regularization matrix at the next time, and the weights are adaptively optimized according to the proportion of factors affecting the error at each stage; An optimized semi-parametric model is obtained.

8. A modeling system for non-cooperative target observation, characterized in that, include: The observation model construction unit is used to observe non-cooperative targets using multiple measurement devices of a non-cooperative target observation system, and to construct a multi-station nonlinear observation model of the non-cooperative targets based on the EMBET estimation method. The multi-station nonlinear observation model includes modelable systematic errors. Nonparametric variables are introduced into the multi-station nonlinear observation model to describe unmodelable observation systematic errors and / or model errors, forming a multi-station nonlinear semiparametric model based on uncertain error components. The model optimization unit is used to determine the factors affecting the estimation accuracy of the multi-station nonlinear semiparametric model based on uncertain error components: regularization matrix. A regularization matrix is ​​constructed based on the factors affecting the error and the regularization matrix is ​​made unique. The unique regularization matrix has l-point smoothness invariance and elementary transformation invariance. The optimal regularization matrix set constructed by the unique regularization matrix is ​​used to optimize the multi-station nonlinear semiparametric model based on uncertain error components to obtain the optimized semiparametric model. The solution unit is used to solve for the position of non-cooperative targets observed by multiple measurement devices using an optimized semi-parametric model, thereby obtaining the position of the non-cooperative targets; The observation model building unit is specifically used for: A multi-station nonlinear observation model of non-cooperative targets is constructed based on the EMBET estimation method, observing non-cooperative targets using multiple measurement devices in a non-cooperative target observation system. In the formula, The system state includes modelable systematic errors, where e is the random error, y is the observations from m measuring devices, and f(·) represents the system state. A function whose formal function and parameters are known.

9. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores one or more programs that, when executed by a computer device, cause the computer device to perform the modeling method for non-cooperative target observation as described in any one of claims 1-7.

Images