Regularization time difference positioning method based on singular value decomposition

Abstract

The invention relates to a regularization time difference positioning method based on singular value decomposition, and the method comprises the steps: introducing a small singular value constraint mechanism based on singular value decomposition in a TDOA pseudo-linear positioning equation solving process, and carrying out the effective inhibition of an ill-conditioned coefficient matrix caused by an unfavorable geometric layout through a regularization matrix. The sensitivity of the least square estimation to a tiny singular value is remarkably reduced, so that the problems of error amplification and solution instability of a traditional algorithm under an ill-conditioned condition are avoided. By analyzing the initial estimation variance, constructing a regularization item related to a small singular value, and correcting the initial solution in combination with a geometrical relationship, the overall mean square error can be remarkably reduced on the premise of keeping relatively low calculation complexity, and the numerical stability and robustness of a solution result are improved.

Description

Technical Field

[0001] This invention relates to a regularized time difference positioning method, and more particularly to a regularized time difference positioning method based on singular value decomposition. Background Technology

[0002] Multi-station passive positioning technology, as a fundamental and key technology in many military and civilian applications such as sensor networks, radar reconnaissance, wireless communication, and navigation, aims to estimate the position of a target by relying solely on observational information of external radiation sources from multiple spatially distributed receiving nodes, even when the target does not actively transmit signals. Existing passive positioning methods commonly used include those based on direction of arrival (DOA), time difference of arrival (TDOA), frequency difference of arrival (FDOA), time of arrival (TOA), and received signal strength (RSS), or positioning systems that jointly process multiple observations. Among these, TDOA-based positioning methods have become an important technical approach in related research and applications due to their relatively simple model construction, low hardware cost, and ability to maintain high positioning accuracy even in multipath or partially obstructed environments.

[0003] In TDOA positioning research, existing techniques generally employ least-squares methods to solve the nonlinear hyperboloid equations formed by TDOA measurements. For example, the typical two-step weighted least-squares method, by pseudo-linearizing the TDOA measurements and solving them step-by-step with weights, can obtain an approximate closed-form solution with low computational complexity, and is therefore widely used in engineering practice. However, this type of least-squares framework is highly dependent on the coefficient matrix structure determined by the geometric relationship between the sensor and the target. When the geometric layout is unfavorable (such as base station distribution biased to one side or insufficient spacing between ranging reference nodes), the coefficient matrix is ​​prone to ill-conditioned characteristics. In this case, even with low observation noise, the least-squares solution process may lead to unstable positioning results and a sharp increase in error due to the significant amplification of errors by the ill-conditioned matrix. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to overcome the shortcomings of the prior art and provide a regularized time difference positioning method based on singular value decomposition.

[0005] This invention is achieved through the following technical solution:

[0006] A regularized time difference localization method based on singular value decomposition, comprising the following steps:

[0007] Step 1: Deploy multiple signal receiving stations, designate one of them as the primary receiving station and the others as secondary receiving stations, and construct an equation for measuring the true value of the distance difference between the signal source and the primary and secondary receiving stations;

[0008] Step 2: Introduce the positional distance relationship between the signal source and the receiving station into the equation for measuring the true value of the distance difference between the signal source and the main receiving station and the auxiliary receiving station, and establish an algebraic relationship between the position of the signal source and the true position of the receiving station;

[0009] Step 3: Rewrite the algebraic relationship between the signal source location and the actual location of the receiving station into vector form, and introduce the relationship between the actual distance difference and the measured value into the algebraic relationship between the signal source location and the actual location of the receiving station to obtain a matrix-form positioning equation set;

[0010] Step 4: Construct the vector X to be determined 0 The cost function is used to obtain the explicit weighted least squares solution of the localization equations, and the estimated vector is obtained through the covariance propagation rate. The covariance matrix;

[0011] Step 5: Find the trace of the covariance matrix of the estimated vector to obtain the global variance of the least squares estimate;

[0012] Step 6: Select the eigenvectors corresponding to the top k small singular values ​​in the trace of the estimated vector that have a significant impact on the estimation result, and construct a regularization matrix;

[0013] Step 7: Construct the constrained regularization cost function of the vector to be regularized from the eigenvectors used in Step 6 to construct the regularization matrix;

[0014] Step 8: Construct auxiliary variables using the actual distance between the signal source and the main receiving station, as well as the geometric relationship between the target location, to correct the errors in the first estimate;

[0015] Step 9: Construct the cost function for the vector Δu to be determined. Obtain the least squares estimate of the estimator by solving the cost function. The explicit estimated vector can be obtained through the covariance propagation rate. The covariance matrix;

[0016] Step 10: Select the eigenvectors corresponding to the small singular values ​​to construct the regularization matrix R2, and obtain the expression for the least squares estimation problem with regularization constraints;

[0017] Step 11: Calculate the estimated value of the error vector Δu based on the expression of the least squares estimation problem with regularization constraints in Step 10;

[0018] Step 12: Subtract the estimated value of the error vector Δu from the first estimated value to obtain the estimated target position of the final signal source.

[0019] Preferably, the equation for measuring the true value of the distance difference between the signal source and the main receiving station and the auxiliary receiving station is:

[0020]

[0021] Where i = 2, 3, ..., N, This represents the true distance difference between the signal source and receiving station 1 and receiving station i. Let i be the actual distance between the signal source and the receiving station i. This represents the actual distance between the signal source and receiving station 1.

[0022] Preferably, the distance relationship between the signal source and the receiving station is as follows:

[0023]

[0024] The algebraic relationship between the location of the signal source and the actual location of the receiving station is as follows:

[0025]

[0026] Preferably, the vector form of the relationship between the signal source location and the actual location of the receiving station is:

[0027] The relationship between the true value and the measured value of the distance difference is r = r 0 +Δr=r 0 +cn;

[0028] The matrix-form positioning equations are as follows: In the formula, n = [n 21 ,n 31 ,…,n N1 ] T , x i1 =x i -x1,y i1 =y i -y1.

[0029] Preferably, the overall variance of the least squares estimate in step 5 is:

[0030]

[0031] in, The covariance matrix of the error vector is obtained by Cholesky decomposition of ψ1. make The trace of the estimated vector is then expressed as:

[0032]

[0033] Where, μ i These are the singular values ​​of H1;

[0034] When the sum of the standard deviation components corresponding to the smallest singular values ​​accounts for 95% or more of the total standard deviation, and a minimum index k is set such that the sum of the variances contributed by the first k smallest singular values ​​accounts for more than 95% of the total variance, that is:

[0035] Preferably, the regularized least squares formula for constructing the regularization matrix from the eigenvectors in step 6 is:

[0036]

[0037] in, V is the regularization matrix. i Let λ1 be the eigenvector corresponding to each singular value in the right singular vector matrix V1 of matrix H1, and let λ1 be the regularization parameter. Let l = 3 be the error vector, and let x be the vector X. o The number of unknowns; σ = L -1 n is whitened noise, and L is obtained by Cholesky decomposition of Q, Q = LL. T .

[0038] Preferably, the least squares solution of the constrained regularized cost function in step 7 is:

[0039]

[0040] in, W1 is the correction matrix, and F is the weighting matrix. i =O (N-1)×(N-1) ,i∈[1,l-1],F l =I (N-1)×(N-1) ,F l+1 =diag{r 21 ,r 31 ,…,r N1}

[0041] Preferably, the actual distance between the signal source and the main receiving station in step 8 is... The target location is (x, y). T , and (x,y) T Geometric relationship is The auxiliary variable is

[0042] The true value exist Performing a first-order Taylor expansion at point U yields the following relationship between the first estimate u1 and the error vector Δu:

[0043]

[0044] in, The position coordinates of the signal source obtained from the first estimate in equation (7);

[0045] Substituting equation (8) into equation (3), we can obtain the matrix equation for the second estimate as follows:

[0046]

[0047] in, This is the error vector for the second estimate.

[0048] Preferably, the expression for the least squares estimation problem with regularization constraints in step 10 is:

[0049]

[0050] Where R2 is the regularization matrix of the second estimate, derived from the vector of the second estimate. The eigenvectors of the small singular values ​​in the covariance matrix are constructed, with λ² being the regularization parameter, calculated based on the minimum mean square error. It is the error vector of the second estimate.

[0051] Preferably, the estimated value of the error vector Δu in step 11 is:

[0052]

[0053] in, R2 is the regularization matrix of the second estimate, and λ2 is the regularization parameter of the second estimate. By adding a regularization term, the influence of each singular value on the variance of the estimate is constrained, so as to improve the stability of the estimate.

[0054] The estimated target location of the final signal source in step 12 is:

[0055]

[0056] The beneficial effects of this invention are:

[0057] This invention introduces a small singular value constraint mechanism based on singular value decomposition (SVD) during the solution of the pseudolinear TDOA positioning equation. It also utilizes a regularization matrix to effectively suppress ill-conditioned coefficient matrices caused by unfavorable geometric layouts, significantly reducing the sensitivity of least squares estimation to small singular values. This avoids the error amplification and solution instability problems that occur in traditional algorithms under ill-conditioned conditions. By analyzing the initial estimate variance, constructing a regularization term related to small singular values, and correcting the preliminary solution based on geometric relationships, this invention can significantly reduce the overall mean square error while maintaining low computational complexity, thus improving the numerical stability and robustness of the solution results. Finally, the proposed two-stage regularization solution framework can achieve higher TDOA positioning accuracy and reliability under complex conditions such as uneven base station distribution, multipath environments, or high measurement noise. Attached Figure Description

[0058] Figure 1 This is a detailed flowchart of a regularized time difference localization method based on singular value decomposition according to the present invention.

[0059] Figure 2 This is a line graph showing the positioning error of the signal source at (4500m, 7000m).

[0060] Figure 3 This is a line graph showing the positioning error at the signal source location (5100m, 8700m). Detailed Implementation

[0061] To enable those skilled in the art to better understand the technical solutions of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and preferred embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0062] This invention provides a regularized time difference positioning method based on singular value decomposition to effectively improve positioning accuracy and stability.

[0063] To solve the above technical problems, the present invention adopts the following technical solution:

[0064] The regularized time difference localization method based on singular value decomposition includes the following steps:

[0065] Step 1: Deploy multiple signal receiving stations, designating one as the primary receiving station and the others as secondary receiving stations. Specifically, the primary receiving station is designated as receiving station 1, and the secondary receiving stations include receiving station 2, ..., receiving station N. Next, construct an equation for the actual distance difference between the signal source and the primary and secondary receiving stations, providing a foundational distance difference relationship for subsequent TDOA positioning. The actual distance difference measurement equation is as follows:

[0066]

[0067] Where i = 2, 3, ..., N, This represents the true distance difference between the signal source and receiving station 1 and receiving station i. Let i be the actual distance between the signal source and the receiving station i. This represents the actual distance between the signal source and receiving station 1.

[0068] Step 2: Determine the distance between the signal source and the receiving station. In equation (1), u 0 =(x 0 ,y 0 ) T For the location of the signal source, s i =(x i ,y i ) T Let i∈{1,N} be the location of the receiving station. Through squaring and rearranging, a set of TDOA positioning equations is constructed to establish an algebraic relationship between the signal source location and the actual location of the receiving station:

[0069]

[0070] Step 3: Write equation (2) in vector form. Considering the inherent errors in actual TDOA measurements, the relationship between the true distance difference and the measured value is r = r 0 +Δr=r 0 +cn, where Δr=cn is the distance difference error, c is the speed of light, and n=[n 21 ,n 31 ,…,n N1 ] T Let be the TDOA error. Introducing equation (2) and neglecting the second-order error term, we can obtain the matrix-form positioning equations:

[0071]

[0072] In the formula, n = [n 21 ,n 31 ,…,n N1 ] T , x i1 =x i -x1,y i1 =y i -y1.

[0073] Step 4: Construct the vector X to be determined 0The cost function is used to obtain the explicit weighted least squares solution of the localization equations, and the estimated vector is obtained through the covariance propagation rate. The covariance matrix.

[0074] Step 5: The trace of the covariance matrix is ​​the sum of the variances of the parameter estimates, reflecting the magnitude of the parameter estimate variances. Taking the trace of the covariance matrix of the estimated vectors yields the overall variance of the least squares estimate:

[0075]

[0076] in, The covariance matrix of the error vector can be obtained by performing Cholesky decomposition on ψ1. make The trace of the estimated vector can then be expressed as:

[0077]

[0078] Where, μ i Let H1 be the singular values. Considering the contribution of each singular value of matrix H1 to the estimated variance, we define small singular values ​​as those whose sum of standard deviation components corresponding to the smallest singular values ​​accounts for 95% or more of the total standard deviation, thus indicating that these small singular values ​​have a significant impact on the estimation results. We then search for the smallest index k such that the sum of the variance contributed by the first k smallest singular values ​​accounts for more than 95% of the total variance, i.e.:

[0079] Step 6: Select the eigenvectors corresponding to the first k small singular values ​​in equation (5) that have a significant impact on the estimation results to construct a regularization matrix. The regularized least squares can then be expressed as:

[0080]

[0081] in, V is the regularization matrix. i Let λ1 be the eigenvector corresponding to each singular value in the right singular vector matrix V1 of matrix H1, and let λ1 be the regularization parameter. Let l = 3 be the error vector, and let x be the vector X. o The number of unknowns; σ = L -1 n is whitened noise, and L is obtained by Cholesky decomposition of Q, Q = LL. T .

[0082] Step 7: Construct the constrained regularization cost function of the vector to be solved in equation (6). By solving the constrained regularization cost function, the least squares solution under the regularization constraint can be obtained.

[0083]

[0084] in, W1 is the correction matrix, and F is the weighting matrix. i =O (N-1)×(N-1) ,i∈[1,l-1],F l =I (N-1)×(N-1) ,F l+1 =diag{r 21 ,r 31 ,…,r N1 Since equation (7) contains information about the vector to be determined in the matrix, the estimated value must be obtained through step 4 first, and then the first estimated value can be obtained using equation (7). By using the regularized least squares method, constructing the regularization matrix through the eigenvalues ​​of small singular values, and obtaining the regularization parameter based on the minimum mean square error, the amplification effect on the estimation error caused by the ill-conditioned coefficient matrix can be effectively reduced.

[0085] Step 8: Since equation (7) is based on the assumption The values ​​are obtained under the assumption that the parameters are independent of each other, so the estimates under this assumption contain errors. Therefore, the estimation method used here is... Relative to the target position (x, y) T geometric relationship Constructing auxiliary variables To correct the error in the first estimate, and to bring the true value to the correct value. exist Performing a first-order Taylor expansion at point U yields the following relationship between the first estimate u1 and the error vector Δu:

[0086]

[0087] in, Let be the location coordinates of the signal source obtained from the first estimate in equation (7). Substituting equation (8) into equation (3), we can obtain the matrix equation for the second estimate as follows:

[0088]

[0089] Where A2 is the coefficient matrix in the second estimate, and B2 is the measurement matrix. This is the corresponding error vector.

[0090] Step 9: Construct the cost function for the vector Δu to be determined. Solve the cost function to obtain the least-squares estimate of the estimator. Similarly, the explicit estimated vector can be obtained through the covariance propagation rate. The covariance matrix.

[0091] Step 10: Calculating the trace of the estimated vector reveals the factors causing its variance. Selecting the eigenvectors corresponding to the small singular values ​​allows us to construct a regularization matrix R². At this point, the least squares estimation problem with regularization constraints can be expressed as...

[0092]

[0093] Where R2 is the regularization matrix of the second estimate, derived from the second estimate vector. The eigenvectors of the small singular values ​​in the covariance matrix are constructed, with λ² being the regularization parameter, calculated based on the minimum mean square error. It is the error vector of the second estimate.

[0094] Step 11: Solve the least squares problem with regularization constraints in equation (10), and the estimated value of the error vector Δu can be obtained as follows:

[0095]

[0096] in, Due to the error vector It contains information about the value to be estimated, Δu, so let This facilitates calculations in matrix differentiation. R² is the regularization matrix for the second estimate, and λ² is the regularization parameter for the second estimate. By adding a regularization term, the influence of singular values ​​on the variance of the estimate is constrained, thereby improving the stability of the estimate.

[0097] Step 12: Finally, subtract the error estimate obtained from equation (11) from the first estimate to obtain the final estimated target position of the signal source:

[0098]

[0099] Furthermore, step 1 includes the following steps:

[0100] Let the position coordinates of the signal source be u. O =(x,y) T The coordinates of the receiving station are s i =(x i ,y i ) T , i∈{1,N},[·] 0 This represents the true value of [·].

[0101] Taking the first receiving station as the master receiving station, the actual distance from the signal source to receiving station i is:

[0102]

[0103] The relationship between the true and measured values ​​of the distance difference between the signal source and receiving stations 1 and i is as follows:

[0104]

[0105] Where, Δr i1 For distance difference measurement error, n i1 This represents the TDOA measurement error, and n = [n 21 ,n 31 ,…,n N1 ] T .

[0106] Furthermore, step 3 includes the following steps:

[0107] Error vector The error of the i-th row can be expressed as:

[0108]

[0109] Where, x i1 =x i -x1,y i1 =y i -y1, and satisfies in actual measurement Therefore, the assumption here is reasonable. Represented in vector form as: in It is a diagonal matrix consisting of the distances from the signal source to each auxiliary station.

[0110] Furthermore, step 4 includes the following steps:

[0111] The least squares estimation problem of equation (3) can be expressed as follows:

[0112]

[0113] The least squares problem above can be expressed as a minimization problem satisfying the following objective function.

[0114]

[0115] In the formula, let equation (17) be applied to... The partial derivative is equal to zero.

[0116]

[0117] The least squares solution of equation (3) is:

[0118]

[0119] in, Error vector The covariance matrix.

[0120] Furthermore, step 5 includes the following steps:

[0121] The covariance matrix of the estimated value can be obtained using the covariance propagation rate.

[0122]

[0123] because And X is a deterministic parameter, therefore Substituting the covariance matrix of B1 into equation (20), we can obtain the covariance matrix of the estimated value as follows:

[0124]

[0125] Cholesky decomposition of ψ1 yields At this point, the covariance matrix of the estimated value can be expressed as:

[0126]

[0127] make Singular value decomposition of H1 yields

[0128]

[0129] The trace of the covariance matrix is ​​the sum of the variances of the parameter estimates, and it can reflect the overall magnitude of the parameter estimate variances. From equations (22) and (23), the overall variance of the least squares estimate is obtained as follows:

[0130]

[0131] Where, μ i Let H1 be the singular values. From the above formula, we can see that the estimated variance of the parameters can be considered as the sum of the variance components caused by each singular value. The smaller the singular value, the greater its impact on the variance, and the larger the variance component it causes. When the sum of the standard deviation components corresponding to small singular values ​​accounts for 95% or more of the total standard deviation, these small singular values ​​are considered to have a significant impact on the result.

[0132] Furthermore, step 8 includes the following steps:

[0133] The distance difference r = r that includes the error vector 0 +Δr=r 0 Substituting +cn into equation (9), the error vector is... The error of the i-th row can be expressed as

[0134]

[0135] Where, ΔB i ,ΔA i Let the i-th row vector of the matrix be represented as follows:

[0136] Furthermore, step 10 includes the following steps:

[0137] Using the error vector obtained in step 8, the covariance matrix of the error term can be calculated as follows:

[0138]

[0139] Cholesky decomposition of ψ2 The covariance matrix of the second estimate can be expressed as:

[0140]

[0141] make And perform singular value decomposition on H2, let V′ i Let V2 be the eigenvectors corresponding to the singular values ​​of the right singular vector matrix V2 of H2.

[0142] The specific steps of this method are illustrated below through a concrete example:

[0143] A pseudo-linear system of equations is obtained using TDOA. Let the position coordinates of the signal source be u. O =(x,y) T The coordinates of the receiving station are s i =(x i ,y i ) T , i∈{1,N},[·] 0 This represents the true value of [·]. Taking the first receiving station as the master receiving station, the true distance from the signal source to receiving station i is:

[0144]

[0145] The relationship between the true and measured values ​​of the distance difference between the signal source and receiving stations 1 and i is as follows:

[0146]

[0147] in, This represents the true distance difference between the signal source and receiving station 1 and receiving station i. Let i be the actual distance between the signal source and the receiving station i. Δr represents the actual distance between the signal source and receiving station 1. i1 For distance difference measurement error, n i1Let n be the measurement error of TDOA and n = [n 21 ,n 31 ,…,n N1 ] T .

[0148] The positional distance relationship between the signal source and the receiving station Introducing equation (1.2), and squaring both sides, we can simplify to obtain:

[0149]

[0150] Write the N-1 equations in vector form Considering the inherent errors in actual TDOA measurements, the relationship between the true distance difference and the measured value is r = r 0 +Δr=r 0 By introducing equation (1.3) and ignoring the second-order error term, we can obtain the matrix-form positioning equations.

[0151]

[0152] In the formula, n = [n 21 ,n 31 ,…,n N1 ] T , x i1 =x i -x1,y i1 =y i -y1, Error vector The error of the i-th row can be expressed as:

[0153]

[0154] Where, x i1 =x i -x1,y i1 =y i -y1, and satisfies in actual measurement Therefore, the assumption here is reasonable. Represented in vector form as:

[0155] The least squares estimation method is used to estimate the target information. First, the noise vector is whitened. Let the covariance matrix of the TDOA error be Q = cov(n). Then, a Cholesky decomposition of Q yields Q = LL. T Let σ = L -1 n is whitened noise. The least squares estimation problem in this case can be formulated as:

[0156]

[0157] The least squares estimate at this point is:

[0158]

[0159] in, Error vector The covariance matrix. Since matrix P in the formula contains the information to be estimated, we first let ψ1 = Q to obtain the first estimate by least squares estimation, that is:

[0160]

[0161] A regularized estimation model is constructed by analyzing the estimated covariance matrix in equation (1.7). The covariance matrix of the estimated value can be obtained using the covariance propagation law.

[0162]

[0163] because And X is a deterministic parameter, therefore Substituting the covariance matrix of B1 into equation (1.9), we obtain the covariance matrix of the estimated value as follows:

[0164]

[0165] Cholesky decomposition of ψ1 yields At this point, the covariance matrix of the estimated value can be expressed as:

[0166]

[0167] make Singular value decomposition of H1 yields:

[0168]

[0169] The trace of the covariance matrix is ​​the sum of the variances of the parameter estimates, and it can reflect the overall magnitude of the parameter estimate variances. From equations (1.11) and (1.12), the overall variance of the least squares estimate is:

[0170]

[0171] Where, μ i Let H1 be the singular values. From the above equation, we can see that the estimated variance of the parameters can be regarded as the sum of the variance components caused by each singular value. The smaller the singular value, the greater its influence on the variance and the larger the variance component it causes.

[0172] When a matrix is ​​ill-conditioned, its condition number becomes very high, and the condition number is expressed as the ratio of the largest singular value to the smallest singular value. Therefore, it is reasonable to assume that the effect of ill-conditioning is mainly manifested in the amplification of variance by small singular values. Small singular values ​​have a significant impact on the estimated variance, which is extremely amplified by small singular values. This situation reduces the reliability of least squares estimation, making it difficult to obtain accurate parameter estimates.

[0173] Equation (1.13) can also be understood as each singular value contributing to the standard deviation of the parameter estimate. To identify the small singular values ​​that have the greatest impact on the standard deviation of the parameter estimate and to suppress them through regularization, this paper distinguishes small singular values ​​by quantifying the relative contribution of each singular value to the total standard deviation. All singular values ​​are sorted in ascending order, and their cumulative standard deviation contribution is calculated. When the sum of the standard deviation components corresponding to the small singular values ​​accounts for 95% or more of the total standard deviation, these small singular values ​​are considered to be components that have a significant impact on the result. A minimum index k is found such that the sum of the variances contributed by the first k smallest singular values ​​accounts for more than 95% of the total variance, i.e.:

[0174]

[0175] In the singular value matrix ∑, let μ k To determine the boundary values ​​of small singular values, a regularization matrix is ​​constructed by selecting the eigenvectors corresponding to the small singular values. The regularized least squares can then be expressed as:

[0176] min(‖σ‖ 2 +λ1X T R1X)

[0177] stA1X-B1=cPLσ(1.15)

[0178] in, V i Let σ be the eigenvector corresponding to each singular value in the right singular vector matrix V1 of matrix H1; σ = L -1 Let n be the whitening noise of n. The least squares optimization problem in the above equation can be expressed as a minimization problem satisfying the following objective function:

[0179]

[0180] The above formula is correct. Taking the partial derivative, we get:

[0181]

[0182] in:

[0183]

[0184] F i =O (N-1)×(N-1) i = 1, ..., l-1

[0185] F l =I (N-1)×(N-1)

[0186] F l+1 =diag{r 21 ,r 31 ,…,r N1} (1.19)

[0187] Matrix Λ1 also contains information to be estimated. The information of Λ1 is obtained by using the estimation result of equation (1.18). At this time, the solution of the equation can be obtained as follows:

[0188]

[0189] The regularization parameter is obtained below using the minimum mean square error, assuming the following relationship exists between the first estimate and the true value:

[0190]

[0191] Substituting equation (1.21) into equation (1.17) yields:

[0192]

[0193] make From equation (1.22), we can obtain:

[0194] ΔX=-(SA1+λ1R1) -1 (SW1σ+λ1R1X) (1.23)

[0195] Let SW1σ = J, then we can obtain:

[0196]

[0197] Typically, matrix SA1 is a full-rank matrix. There exists an orthonormal matrix that diagonalizes SA1:

[0198] SA1 = W T diag{u1,u2,u3,u4}W (1.25)

[0199] Suppose that the eigenvector W of SA1 and the eigenvector V of R1 i If the features are orthogonal (or partially overlapping), then SA1+λ1R1 can be viewed as adding regularization to the small singular value portion of SA1. For small singular values ​​i≥k, the regularization term λ1R1 is equivalent to adding a "perturbation" of λ in the direction of the corresponding eigenvector, thus we can obtain:

[0200]

[0201] Where, δ ik =1 (when i≥k1), otherwise 0. For [(SA1+λ1R1)] -1 ] T (SA1+λ1R1) -1 Eigenvalue decomposition yields:

[0202] [(SA1+λ1R1) -1 ] T (SA1+λ1R1) -1 =WDW T (1.27)

[0203] Where D is a diagonal matrix whose elements are the reciprocal of the square of the sum of singular values ​​and regularization parameters, and the diagonal elements can be represented as:

[0204]

[0205] Taking the expectation of equation (1.28), the mean square error of the estimated solution is:

[0206]

[0207] Let C1 = W T J, C2 = W T If RX, then C1 and C2 are two column vectors. Let C1 = [c 11 ,c 12 ,…,c 1l ] T C2 = [c 21 ,c 22 ,…,c 2l ] T , then E[ΔX T ΔX] can be expressed as:

[0208]

[0209] From the above equation, it can be seen that the mean square error of the estimate is a function of λ1. To obtain the minimum mean square error estimate, taking the partial derivative of equation (1.30) yields:

[0210]

[0211] The Newton-Raphson method is used to analyze equation (1.31). First, the objective function is defined as:

[0212]

[0213] Differentiating equation (1.32) yields:

[0214]

[0215] After obtaining the functions f(λ1) and f′(λ1), the equation f(λ1) = 0 is solved iteratively. Specifically, given an initial guess value... Using iterative formulas Perform iterations. In each iteration, calculate the function value. and its derivative The estimated value of λ1 is then updated according to the iterative formula.

[0216] The iterative process continues until the pre-defined convergence condition is met. Where ε is a sufficiently small positive number, representing the acceptable error range. When the convergence condition is met, then... This is the approximate zero of the equation f(λ1)=0, which is also the key value for equation (1.31) to turn from negative to positive.

[0217] By leveraging the inherent physical constraints of the auxiliary variable r1 and the target position (x, y), a Taylor expansion is used to explicitly establish the propagation relationship between the position error (Δx, Δy) and the distance error, thereby enabling a more accurate construction of the error covariance matrix. Assuming the target position estimated by equation (1.20) is... The estimation error can then be expressed as: The true value exist A first-order Taylor expansion at this point yields:

[0218]

[0219] In the above formula, r1 is the target position of the first estimated value. The calculation yields the following result. Substituting equation (1.34) into equation (1.4), we get:

[0220]

[0221] in:

[0222]

[0223] The distance difference r = r that includes the error vector 0 +Δr=r 0 Substituting +cn into equation (1.35), the error vector... The error of the i-th row can be expressed as:

[0224]

[0225] Where, ΔBi ,ΔA i Let the i-th row vector of the matrix be represented as follows:

[0226] At this time, the error vector The covariance matrix is Cholesky decomposition of ψ2 yields Cholesky decomposition of ψ2 The covariance matrix of the solution to the matrix equation of the second estimate can be expressed as:

[0227]

[0228] make And perform singular value decomposition on H2, let V′ i Let V2 be the eigenvectors corresponding to the singular values ​​of the right singular vector matrix V2 of H2.

[0229] make The estimation result of the same equation (1.20) can be obtained as follows:

[0230]

[0231] in

[0232]

[0233] Matrices W2 and Λ2 contain the quantities to be estimated, so we need to assume W2 = Q and Λ2 = ON. -1×1 Substituting λ2 = 0 into equation (1.39) to obtain an initial estimation result, we then use the updated matrix information to estimate the final result. The estimation steps for λ2 at this point are the same as those for the first estimation. The final estimated target position of the signal source is:

[0234]

[0235] The effects of this invention can be further illustrated by the following simulation experiments:

[0236] A. Simulation conditions

[0237] To verify the positioning performance of the proposed method, simulation experiments were designed for two target positioning scenarios. The scenarios were constructed as follows: four fixed receiving stations (rectangular arrangement) were deployed within a two-dimensional plane (10000m, 10000m), and benign and ill-conditioned locations were selected within this area as the signal source positions for the two different scenarios. The positioning performance of two different Chan methods, the TRCTLS method, and the proposed method were analyzed in both the benign and ill-conditioned scenarios.

[0238] B. Simulation Content

[0239] This section evaluates the localization performance of the proposed algorithm through simulation experiments. The accuracy of target position estimation is measured by the root mean square error (RMSE), which is expressed as follows:

[0240]

[0241] in, Let L represent the estimated value from the i-th Monte Carlo simulation, and L represent the Monte Carlo iteration number. In this experiment, the number of Monte Carlo iterations is set to 10000. The diagonal elements of the covariance matrix Q of the TDOA measurement error are... All other elements are in, The variance of noise is measured for TDOA. In the simulations below, the distance noise δ is used as the standard. d =cδ t As an independent variable.

[0242] To distinguish ill-conditioned phenomena at different locations, the condition number of the coefficient matrix A1 in the matrix equation is defined as follows:

[0243]

[0244] in, These are the maximum and minimum singular values ​​of A1, respectively. In the simulation, it is set that if cond(A1) > 1000 at a point, the coefficient matrix at that location is considered ill-conditioned; otherwise, it is considered well-conditioned. In the simulation, the location (4500m, 7000m) is selected as the well-conditioned signal source location, and the location (5100m, 8700m) is selected as the ill-conditioned signal source location.

[0245] C. Simulation Results

[0246] Figure 2 This is a line graph showing the positioning error at the signal source location (4500m, 7000m). Figure 3 This is a line graph showing the positioning error at the signal source location (5100m, 8700m). The horizontal axis represents the distance difference measurement error, and the vertical axis represents the RMSE of the positioning method.

[0247] like Figure 2 As shown, in non-illness scenarios, the RMSE curve of the proposed SVD-based localization method almost coincides with CRLB, and its accuracy is close to the theoretical optimum, significantly outperforming the TRCTLS method and the classic Chan method; Figure 3As shown, in ill-conditioned scenarios, the proposed method still maintains a relatively close approximation to CRLB, demonstrating a particularly strong robustness advantage. In contrast, the TRCTLS method shows a significant performance decline, while the Chan method deteriorates sharply and deviates rapidly from CRLB. This fully demonstrates that the proposed algorithm has significant advantages in suppressing matrix ill-conditioning and resisting noise amplification effects, maintaining stable and excellent positioning accuracy even in ill-conditioned scenarios.

[0248] 10,000 points are uniformly selected in the interval (10,000m, 10,000m). When 10lg(σ) d When RMSE = 5, simulation experiments were conducted on each positioning method, and RMSE was used as the evaluation index. The positioning RMSE of each method was determined by performing 10,000 Monte Carlo simulations, and the results are shown in Table 1.

[0249] Positioning method Mean RMSE / m Maximum RMSE Chan method 51.127 205749.329 TRCTLS method 3.585 216.935 Proposed method 2.279 23.571

[0250] Table 1 shows the mean RMSE of the three positioning methods. The maximum RMSE in Table 1 represents the maximum positioning error of the three methods within the range of (10000m, 10000m). By comparing the simulation results of the Chan method, the TRCTLS method, and the proposed method, it can be concluded that in scenarios with ill-conditioned problems, the positioning performance of the TRCTLS method is better than that of the Chan method. The mean positioning error of the proposed method is much smaller than that of the other two methods, and the positioning accuracy is improved by approximately 36.43% compared to the TRCTLS method.

[0251] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A singular value decomposition based regularized time difference of arrival positioning method, characterized in that, The method comprises the following steps: Step 1: deploying a plurality of signal receiving stations, setting one of the plurality of signal receiving stations as a main receiving station and the others as auxiliary receiving stations, and constructing a real value measurement equation of a distance difference between a signal source and the main receiving station and the auxiliary receiving stations; Step 2: introducing a position distance relationship between the signal source and the receiving stations into the real value measurement equation of the distance difference between the signal source and the main receiving station and the auxiliary receiving stations, and establishing an algebraic relationship between a signal source position and a real position of the receiving stations; Step 3: rewriting the algebraic relationship between the signal source position and the real position of the receiving stations into a vector form, introducing a relationship between a real value and a measurement value of the distance difference into the algebraic relationship between the signal source position and the real position of the receiving stations to obtain a matrix form of a positioning equation group; Step 4: Constructing the vector X to be solved 0 of the positioning equation set, and obtaining the estimated vector of the covariance matrix of the vector X Step 5: obtaining an overall variance of a least square estimation by taking a trace of a covariance matrix of an estimation vector; Step 6: selecting eigenvectors corresponding to k small singular values which have a significant influence on an estimation result in the trace of the estimation vector to construct a regularization matrix; Step 7: constructing a constrained regularization cost function of a to-be-solved vector of the eigenvector construction regularization matrix in step 6, and obtaining a first estimation value of the signal source position according to the constrained regularization cost function; Step 8: constructing an auxiliary variable by a real distance between the signal source and the main receiving station and a geometric relationship of the target position to correct an error existing in the first estimation value, and obtaining a matrix equation of a second estimation value of the signal source position according to a relationship between the first estimation value and an error vector and the matrix form of the positioning equation group in step 3; Step 9: Construct the cost function of the unknown vector Δu, and obtain the least square estimate of the estimator by solving the cost function, and obtain the explicit estimate vector through the covariance propagation rate covariance matrix of the unknown vector Δu; Step 10: selecting the eigenvectors corresponding to the small singular values to construct a regularization matrix R2 to obtain an expression of a least square estimation problem with a regularization constraint; Step 11: calculating an estimation value of the error vector Δu according to the expression of the least square estimation problem with the regularization constraint in step 10; Step 12: obtaining a final estimation target position of the signal source by taking an error estimation value of the first estimation value minus the estimation value of the error vector Δu.

2. The regularized hyperbolic positioning method of claim 1, wherein, The real value measurement equation of the distance difference between the signal source and the main receiving station and the auxiliary receiving stations is: where i = 2, 3,..., N, is the real value of the distance difference between the signal source and the receiving station 1 and the receiving station i, is the real distance between the signal source and the receiving station i, is the real distance between the signal source and the receiving station 1.

3. The regularized hyperbolic positioning method according to claim 2, characterized in that, The position distance relationship between the signal source and the receiving stations is: The algebraic relationship between the signal source position and the real position of the receiving stations is:

4. The regularized hyperbolic positioning method of claim 3, wherein, The vector form of the relationship between the signal source position and the true position of the receiving station is The relationship between the distance difference true value and the measured value is r = r 0 + Δr = r 0 + cn; The positioning equation set in matrix form is wherein, n = [n 21 ,n 31 ,…,n N1 ] T , x i1 = x i -x1,y i1 = y i -y1.

5. The regularized hyperbolic positioning method according to claim 4, characterized in that, The overall variance of the least square estimation in step 5 is: where is the covariance matrix of the error vector, and Let The trace of the estimated vector is then given by where μ i is the singular value of H1; The proportion of the sum of the standard deviation components corresponding to small singular values in the total standard deviation reaches 95% or more, and a minimum index k is set such that the sum of the variances contributed by the first k smallest singular values occupies 95% or more of the total variance, that is:

6. The regularized hyperbolic positioning method according to claim 5, characterized in that, The regularization least square formula of the regularization matrix constructed by the eigenvectors in step 6 is: wherein, is a regularization matrix, V i is an eigenvector corresponding to each singular value in the right singular vector matrix V1 of the matrix H1, and λ1 is a regularization parameter, is an error vector, and l = 3 is the vector X o is the number of unknowns in the equation; σ = L -1 n is the whitened noise of n, and L is obtained by Cholesky decomposition of Q to obtain Q = LL T .

7. The regularized hyperbolic positioning method according to claim 6, characterized in that, The least square solution of the constrained regularization cost function in step 7 is wherein, is a correction matrix, W1is a weighting matrix, F i = O (N-1)×(N-1) , i ∈ [1, l - 1], F l = I (N-1)×(N-1) , F l+1 = diag{r 21 , r 31 , …, r N1}.

8. The regularized hyperbolic positioning method according to claim 7, characterized in that, The true distance between the signal source and the main receiving station in step 8 is The target position is (x, y) T , and (x, y) T The geometric relationship is The auxiliary variable is The real value At A first order Taylor expansion at u0can be made to obtain the relationship of the first estimate u1and the error vector Δu: wherein the position coordinates of the signal source for the first estimate obtained as formula (7); The matrix equation of the second estimation value can be obtained by introducing formula (8) into formula (3): wherein is the error vector for the second estimate.

9. The regularized hyperbolic positioning method according to claim 8, characterized in that, The expression of the least square estimation problem with the regularization constraint in step 10 is: where R2 is a regularization matrix of the second estimate constructed from the eigenvectors of the small singular values in the covariance matrix of the second estimate vector λ2 is a regularization parameter calculated based on the minimum mean square error, is an error vector of the second estimate.

10. The regularized hyperbolic positioning method of claim 9, wherein, The estimation value of the error vector Δu in step 11 is: wherein, R2 is a regularization matrix of the second estimated value, and λ2 is a regularization parameter of the second estimated value, and by adding a regularization term, the influence of each singular value on the estimated value variance is constrained to improve the stability of the estimated value; The final estimation target position of the signal source in step 12 is:

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