CN113325406B - A Passive Localization Method Based on Regularized Constrained Weighted Least Squares - Google Patents

Abstract

The invention provides a regularized constraint weighted least square-based passive positioning method. The method comprises two steps, wherein a positioning model based on an RCTLS (global positioning system) idea is established for a TDOA/FDOA positioning problem in a first step, regularization parameters are solved based on a criterion of minimizing a mean square error, and then a closed-type analytic solution of the model is given through mathematical deduction; and the second step is to establish an equation about the estimation error of the first step by using constraint conditions, then solve the equation, and finally correct the estimation result of the first step by using the solved equation. The method can improve the positioning precision of the positioning method based on the CTLS model, and the performance is more stable under the condition that the coefficient matrix is in a pathological state.

Description

A Passive Location Method Based on Regularized Constrained Weighted Least Squares

Technical Field

The present invention relates to the field of passive radar positioning and can be used to estimate the position and speed of a moving target in real time in military and civilian fields, and in particular to a TDOA/FDOA passive positioning method based on regularized constrained weighted least squares.

Background Art

At present, multi-station passive positioning technology has been widely used in various military and civilian fields, such as sensor networks, radar, navigation, etc. The observations that can be used in multi-station passive positioning technology mainly include angle of arrival (AOA), time difference of arrival (TDOA), frequency difference of arrival (FDOA), etc.

The positioning system combining different measurement parameters can integrate the advantages of different parameters and improve the positioning accuracy to a certain extent. The positioning system combining TDOA and FDOA has been widely studied because it can estimate the position and velocity of the target at the same time. In recent years, algorithms for positioning problems based on TDOA/FDOA have been proposed one after another, including Two-Stage Weighted Least Squares (TSWLS) and Constrained Total Least Squares (CTLS). However, due to the interference of noise, the distribution of receiving stations and target positions, and the influence of the number of receiving stations on the coefficient matrix, the coefficient matrices in the above methods based on CWLS and CTLS models may have ill-conditioned problems in practical applications.

Summary of the invention

The present invention aims at the ill-posed problem of coefficient matrix in TDOA/FDOA passive positioning problem, and proposes a two-stage regularized constrained least squares method (Two-Stage Regularized Constrained TotalLeast Squares, TRCTLS) based on the idea of RCTLS. This method introduces a regularization parameter on the basis of the CTLS model, and improves the root mean square error of positioning and suppresses the ill-posed nature of the coefficient matrix by reasonably selecting the regularization parameter.

The object of the present invention is achieved in that the steps of the present invention are as follows:

Step 1: Establish the positioning equation based on the measured TDOA and FDOA information;

Step 2: Solve the positioning equation based on the idea of RCTLS to obtain a closed-form solution containing regularization parameters;

Step 3: Solve the regularization parameter in the second step based on the criterion of minimizing the mean square error;

Step 4: Use the constraints to establish an equation for the error in the first step and then find a solution that includes the regularization parameter;

Step 5: Also based on the criterion of minimizing the mean square error, the regularization parameter contained in the solution obtained in step 4 is solved, and then the result obtained in step 2 is corrected using the solution obtained in step 4 to obtain the final solution.

The present invention also includes such structural features:

1. The positioning equation based on TDOA/FDOA obtained in step 1 is:

为接收站的位置和速度坐标，u^o为目标的位置和速度坐标，ε_t＝BΔr，为TDOA方程的误差项，

为FDOA方程的误差项，其中

Where r _i1 is the distance difference between the primary station and the secondary station from the target, s _i and

are the position and velocity coordinates of the receiving station, u ^o are the position and velocity coordinates of the target, ε _t = BΔr, is the error term of the TDOA equation,

is the error term of the FDOA equation, where

2. Step 2 is as follows:

根据CTLS方法将基于TDOA/FDOA的定位模型表示为如下形式：Constructing auxiliary variables

According to the CTLS method, the positioning model based on TDOA/FDOA is expressed as follows:

(A+ΔA)θ ₁ =(b+Δb)

Where:

Among them, O _(M-1)×2N is a (M-1)×2N zero matrix, and 0 is a 3×1 zero vector. Solving this equation using the idea of RCTLS can obtain a closed-form solution containing the regularization parameter:

3. Step 3 is as follows:

This step is mainly to solve the regularization parameter in the solution obtained in step 2. Through mathematical derivation, the mean square error of the estimation error in the first step of estimation can be expressed as:

Then, based on the criterion of minimizing the mean square error, the regularization parameter can be solved as follows:

Step 4 is as follows:

令

将步骤2中辅助变量的真实值

和

在u₁和

处进行一阶泰勒展开可得：Let the estimated errors in step 2 be Δu＝u ₁ -u ^o ,

make

The true value of the auxiliary variable in step 2

and

In u ₁ and

Performing a first-order Taylor expansion at gives:

代入步骤2中的定位模型可得：Substituting the above formula and the formula Δu＝u ₁ -u ^o ,

Substituting into the positioning model in step 2, we get:

(M+ΔM)θ ₂ =N+ΔN

Where:

Based on the idea of RCTLS, we can solve the equation:

Similar to the idea of step 3, step 5 also estimates the regularization parameter of the solution obtained in step 4 based on the criterion of minimizing the mean square error.

Compared with the prior art, the beneficial effects of the present invention are as follows: theoretical derivation and simulation experiments show that the method of the present invention can further reduce the root mean square error (RMSE) of the algorithm on the basis of the CTLS method by reasonably selecting the regularization parameter, and the positioning performance is also more stable when the coefficient matrix is ill-conditioned.

BRIEF DESCRIPTION OF THE DRAWINGS

Figure 1 is a model of a receiving station positioning a moving target;

Fig. 2 is an implementation flow chart of the present invention;

Figure 3 shows the distribution of receiving stations and target locations in three positioning scenarios;

Figure 4 shows the root mean square error of position estimation of the three methods in scenario 1;

Figure 5 shows the root mean square error of velocity estimation of the three methods in scenario 1;

Figure 6 shows the root mean square error of position estimation for the three methods in scenario 2;

Figure 7 shows the root mean square error of speed estimation of the three methods in scenario 2;

Figure 8 shows the root mean square error of position estimation of the three methods in scenario 3;

Figure 9 shows the root mean square error of speed estimation of the three methods in scenario three.

DETAILED DESCRIPTION

The present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments.

A positioning method based on regularized constrained total least squares, specifically comprising:

第i个接收站的位置和速度分别为s_i＝[x_i,y_i,z_i]^T、

则接收站与目标辐射源之间的真实距离可以表示为：(1) As shown in Figure 1, M receiving stations are used to locate the moving target radiation source in three-dimensional space. The position and velocity of the target radiation source are u ^o = [x, y, z] ^T ,

The position and velocity of the i-th receiving station are s _i = [x _i , y _i , z _i ] ^T ,

Then the actual distance between the receiving station and the target radiation source can be expressed as:

Select any one of the receiving stations as the main station, numbered 1, and the rest of the receiving stations as auxiliary stations. The actual distance difference between the main station and each auxiliary station is:

由真实的TDOA数据计算得到，其向量形式可记为

由上述可得基于TDOA的定位方程：in

Calculated from real TDOA data, its vector form can be expressed as

From the above, the positioning equation based on TDOA can be obtained:

的定义式进行求导可得接收站与目标辐射源之间真实的距离变化率的定义式：right

By taking the derivative of the definition, we can get the definition of the actual rate of change of distance between the receiving station and the target radiation source:

Taking the time derivative of the TDOA equation gives the positioning equation based on FDOA:

是由真实的FDOA信息推导出的真实距离差变化率，其向量形式可记为

假设利用实际测量得到的TDOA和FDOA数据得到的距离差向量和距离差变化率向量分别为r＝[r₂₁,r₃₁,…,r_M1]^T和

可将其描述为真实值与噪声值相加的形式：in

is the rate of change of the actual distance difference derived from the actual FDOA information, and its vector form can be expressed as

Assume that the distance difference vector and the distance difference change rate vector obtained by using the TDOA and FDOA data obtained by actual measurement are r = [r ₂₁ , r ₃₁ , …, r _M1 ] ^T and

It can be described as the sum of the true value and the noise value:

r＝ ^ro +Δr＝ ^ro +cΔt

假设它们服从均值为零的高斯分布，则其协方差矩阵为：The measurement noises included in the TDOA and FDOA measurement data are Δr = [Δr ₂₁ , Δr ₃₁ , …, Δr _M1 ] ^T and

Assuming they follow a Gaussian distribution with mean zero, their covariance matrix is:

代入TDOA方程和FDOA方程中并忽略二阶误差项可以得到包含误差项的TDOA/FDOA方程：The TDOA measurement value r = r ^o + Δr and the FDOA measurement value containing noise are

Substituting into the TDOA equation and FDOA equation and ignoring the second-order error term, we can obtain the TDOA/FDOA equation including the error term:

为FDOA方程的误差项，

Where ε _t = BΔr, is the error term of the TDOA equation,

is the error term of the FDOA equation,

首先根据CTLS模型将包含误差项的定位方程表示为矩阵形式：(2) Constructing auxiliary variables

First, according to the CTLS model, the positioning equation containing the error term is expressed in matrix form:

(A+ΔA)θ ₁ =(b+Δb)

Where:

可以看出ΔA和Δb中的噪声是具有相关性的，可以将矩阵ΔA和Δb表示为：Among them, O _(M-1)×2N is a zero matrix of (M-1)×2N, and 0 is a 3×1 zero vector. From the expression of matrix A, it can be seen that the unreasonable distribution of receiving stations and target positions and the interference of noise may cause matrix A to become ill-conditioned. For example, when the coordinates of the receiving stations are close on the x-axis, the condition number of matrix A will increase significantly, and the matrix will become ill-conditioned, resulting in increased sensitivity of positioning results to noise. Introducing regularization parameters is an effective means to suppress the ill-conditioned problem of matrices. Let the noise matrix

It can be seen that the noise in ΔA and Δb is correlated, and the matrices ΔA and Δb can be expressed as:

ΔA=[F ₁ n, F ₂ n,..., F _l n]

Δb＝F _l+1 n

Where l is the number of columns of the matrix ΔA (l=8), the values of _F1 to _F9 can be expressed as:

F _i =O _{2(M-1)×2(M-1)} (i＝1,2,…,6)

F ₇ = -2 × I _{2 (M-1) × 2 (M-1} )

考虑到n中各项误差具有相关性且有着不同的方差，对其进行白化处理。令Q＝E[nn^T]，对Q做Cholesky分解可得Q＝PP^T，则n的白化向量为σ＝P^-1n，同时可以得到ΔA＝[G₁σ,G₂σ,…，G_lσ]、Δb＝F_l+1n，其中G_i＝F_iP，基于RCTLS思想可将定位方程转换为如下形式：Where: ∑ ₁₁ =∑ ₂₂ =diag(r ₂₁ ,r ₃₁ ,…,r _M1 ),

Considering that the errors in n are correlated and have different variances, they are whitened. Let Q = E[nn ^T ], and perform Cholesky decomposition on Q to obtain Q = PP ^T , then the whitened vector of n is σ = P ^-1 n, and at the same time, we can obtain ΔA = [G ₁ σ, G ₂ σ, ..., G _l σ], Δb = F _l+1 n, where G _i = F _i P. Based on the RCTLS idea, the positioning equation can be converted into the following form:

min(‖σ‖ ² +λ ₁ ‖θ ₁ ‖ ² )

After simple mathematical operations, the above formula can be converted into the problem of finding the minimum value of the following function:

After taking the derivative of the above formula, we can get:

进而对该式进行求解可得方程的RCTLS解：In the formula

Then, the RCTLS solution of the equation can be obtained by solving the equation:

从第一步得到的目标估计位置和速度分别为u₁和

则第一步的估计误差为Δu＝u₁-u^o、

令

将第一步中辅助变量的真实值

和

在u₁和

处进行一阶泰勒展开可得：(3) The solution obtained in the previous step does not take into account the constraints, so in this step, we will use the constraints to reconstruct a set of equations to correct the estimated values in the first step. Assume that the actual position and velocity of the target are u ^o and

The estimated position and velocity of the target obtained from the first step are u ₁ and

Then the estimation error of the first step is Δu＝u ₁ -u ^o ,

make

The true value of the auxiliary variable in the first step

and

In u ₁ and

Performing a first-order Taylor expansion at gives:

则α和β的表达式为：Let the estimated values of the auxiliary variables obtained in the first step be r ₁ and

Then the expressions of α and β are:

α＝(u ₁ −s ₁ )/r ₁

以及

和

的泰勒展开式代入定位方程中可得：Δu＝u ₁ -u ^o ,

as well as

and

Substituting the Taylor expansion of into the positioning equation yields:

(M+ΔM)θ ₂ =N+ΔN

Where:

Where ΔM and ΔN can be expressed as:

ΔM=[U ₁ n, U ₂ n,..., U _k n]

ΔN＝U _k+1 n

Where k is the number of columns of the matrix (k=6), the specific expressions of U ₁ ,U ₂ …U _k are omitted, and after whitening n, we can get:

ΔM＝[V ₁ n,V ₂ n,…,V _k n]

ΔN＝V _k+1 n

Where V _i = U _i P, then (M+ΔM)θ ₂ = N+ΔN can be converted into the following form:

利用第一步的RCTLS算法对上式进行求解可得：in

Using the first step of the RCTLS algorithm to solve the above equation, we can get:

则目标位置和速度的最终估计值分别为

in

The final estimated values of the target position and velocity are

与真实值θ₁之间的关系为：(4) It is necessary to solve the regularization parameters in steps (2) and (3) based on the principle of minimizing the mean square error. First, solve the regularization parameter in step (2). Assume that the estimated value in step (2) is

The relationship between the true value θ ₁ is:

Substituting the above formula into the derivative of function F(θ ₁ ) yields:

则由上式可得：make

From the above formula we can get:

可得：make

We can get:

Δθ ₁ ^T Δθ ₁ =(J ^T +λ ₁ θ ₁ ^T )[(SA+λ ₁ I _8×8 ) ^-1 ] ^T (SA+λ ₁ I _8×8 ) ^-1 (J+λ ₁ θ ₁ )

Usually the matrix SA is a full rank matrix, then there exists a standard orthogonal matrix that diagonalizes SA:

SA＝P ^T diag{u ₁ , u ₂ ,..., u ₈ }P

Performing eigenvalue decomposition on [(SA+λ ₁ I _8×8 ) ^-1 ] ^T (SA+λ ₁ I _8×8 ) ^-1 yields:

[(SA+λ ₁ I _8×8 ) ^-1 ] ^T (SA+λ ₁ I _8×8 ) ^-1 =P ^T DP

Where P is a 8×8 standard orthogonal matrix, D=diag{(i ₁ +λ ₁ ) ^-2 ,(u ₂ +λ ₁ ) ^-2 ,…,(u ₈ +λ ₁ ) ^-2 }. Considering E(J)=0, taking the expected value of the expression of Δθ ₁ ^T Δθ _1, the mean square error of the estimated value obtained in the first step is:

Let C ₁ = PJ, C ₂ = Pθ ₁ , then C ₁ and C ₂ are two column vectors. Let C ₁ = [c ₁₁ , c ₁₂ , … , c ₁₈ ] ^T , C ₂ = [c ₂₁ , c ₂₂ , … , c ₂₈ ] ^T , then E[Δθ ₁ ^T Δθ ₁ ] can be expressed as:

It can be seen that the mean square error of the result obtained by the RCTLS algorithm in the first step is a function of λ _1. In order to obtain the λ ₁ value corresponding to the minimum mean square error, the derivative of Δθ ₁ ^T Δθ ₁ is obtained:

时，

此时估计均方误差随λ₁单调递减；当

时，

此时估计均方误差随λ₁单调递增。因此可以得到λ₁最小值的取值区间为

则本文中取极小值为：Analyzing the above formula, we can get that

hour,

At this time, the estimated mean square error decreases monotonically with λ ₁ ;

hour,

At this time, the estimated mean square error increases monotonically with λ _1. Therefore, the range of the minimum value of λ ₁ can be obtained as

The minimum value in this paper is:

均含有未知数θ₁，因此在实际的计算中先假设Λ＝0、

得到初始的估计值后将其代入矩阵Λ和

的表达式中对Λ和

进行更新，之后利用新的Λ和

重新计算θ₁，如此循环3～5次即可得到最终的θ₁，同样地，在第二步对θ₂进行求解时也需要进行类似的处理。Similarly, the value of λ ₂ in step (3) can be obtained. Note that in the method of step (2), Λ and

Both contain unknown number θ ₁ , so in the actual calculation, we first assume that Λ＝0,

After obtaining the initial estimate, substitute it into the matrix Λ and

In the expression of

Update, and then use the new Λ and

Recalculate θ ₁ , and repeat this process 3 to 5 times to obtain the final θ ₁ . Similarly, similar processing is required when solving θ ₂ in the second step.

场景3中目标辐射源的位置和速度为u＝[50,320,110]^Tm和

每个场景均设置5个接收站，3个场景中的接收站与目标的位置在三维空间中的分布如图2所示，可以看出，3个场景中的接收站的整体分布基本接近，只有场景2中有2个接收站在z轴上的坐标与场景1和场景3中略有不同，这种设置可减少不同场景中接收站的分布方式对定位精度的影响。场景1为系数矩阵并未出现病态时的定位场景，而场景2中则因为接收站在z轴上坐标接近而导致系数矩阵出现病态，场景3则是因为目标辐射源距离5个接收站的距离接近而导致系数矩阵出现病态。在3个场景中均使用均方根误差进行性能分析，使用CRLB作为衡量估计精度的标准，同时使用TSWLS方法和CTLS方法作为对比方法。In order to further illustrate this embodiment, a simulation analysis is performed on the algorithm. The simulation experiment is carried out in three scenarios, where the position and velocity of the target radiation source in scenario 1 and scenario 2 are u = [285, 325, 275] ^T m and

The position and velocity of the target radiation source in scene 3 are u = [50, 320, 110] ^T m and

Five receiving stations are set up in each scene. The distribution of the receiving stations and the positions of the targets in the three scenes in three-dimensional space is shown in Figure 2. It can be seen that the overall distribution of the receiving stations in the three scenes is basically close. Only in scene 2, the coordinates of two receiving stations on the z-axis are slightly different from those in scenes 1 and 3. This setting can reduce the impact of the distribution of receiving stations in different scenes on the positioning accuracy. Scene 1 is a positioning scene when the coefficient matrix is not ill-conditioned, while in scene 2, the coefficient matrix is ill-conditioned because the coordinates of the receiving stations on the z-axis are close. In scene 3, the coefficient matrix is ill-conditioned because the distance between the target radiation source and the five receiving stations is close. In all three scenes, the root mean square error is used for performance analysis, and CRLB is used as the standard for measuring estimation accuracy. The TSWLS method and the CTLS method are used as comparison methods.

Figures 4 and 5, Figures 6 and 7, and Figures 8 and 9 respectively reflect the root mean square errors of positioning of the three methods in scenario 1, scenario 2, and scenario 3. It can be seen that the algorithm adopted in the present invention reduces the root mean square error of the algorithm on the basis of the CTLS method, and improves the stability of the algorithm when pathological problems occur in the coefficient matrix.

Those skilled in the art will understand that in the above method of the specific implementation mode of the present application, the serial number of each step does not mean the order of execution. The execution order of each step should be determined by its function and internal logic, and should not constitute any limitation on the implementation process of the specific implementation mode of the present application.

Finally, it should be noted that the above embodiments are only used to describe the technical solution of the present invention rather than to limit the technical method. The present invention can be extended to other modifications, changes, applications and embodiments in application, and therefore it is believed that all such modifications, changes, applications and embodiments are within the spirit and teaching scope of the present invention.

In summary, the present invention proposes a closed-form analytical method based on regularized constrained total least squares (RCTLS). The method is divided into two steps. In the first step, a positioning model based on the RCTLS concept is established for the TDOA/FDOA positioning problem, and the regularization parameter is solved based on the criterion of minimizing the mean square error. Then, the closed-form analytical solution of the model is given through mathematical derivation; the second step is to use the constraints to establish the equation about the estimation error of the first step and then solve it, and finally use the obtained solution to correct the estimation result of the first step. The method of the present invention can improve the positioning accuracy of the positioning method based on the CTLS model, and the performance is more stable when the coefficient matrix is ill-conditioned.

Claims (3)

1. A passive positioning method based on regularization constraint weighted least square is characterized by comprising the following steps: the method comprises the following steps:

step one: establishing a positioning equation according to the measured TDOA and FDOA information;

step two: solving a positioning equation based on the concept of RCTLS to obtain a closed solution containing regularization parameters;

construction auxiliary variable

u ^o ＝[x,y,z] ^T 、

The position and the velocity vector of the target radiation source to be measured are respectively, wherein [. Cndot.] ^T Representing a transpose operation on a vector or matrix, r ₁ ^o For the true distance of the main receiving station from the target radiation source, < >>

To r is ₁ ^o Taking the distance change rate obtained by the derivative, and expressing a positioning model based on TDOA/FDOA as the following form according to the CTLS method:

(A+ΔA)θ ₁ ＝(b+Δb)

wherein:

wherein: s is(s) _i ＝[x _i ,y _i ,z _i ] ^T 、

The position and velocity of the ith receiving station, respectively; r is (r) _i1 The true distance difference between the main receiving station 1 and each auxiliary receiving station i and the target radiation source is obtained;

The difference value of the change rate of the distance between the main receiving station 1 and each auxiliary receiving station i is obtained; o (O) _(M-1)×2N A zero matrix of (M-1) x 2N, 0 being a zero vector of 3 x 1; r= [ r ] ₂₁ ,r ₃₁ ,...,r _M1 ] ^T And

respectively deriving a distance difference vector and a distance difference change rate vector by using TDOA and FDOA data obtained by actual measurement; Δr= [ Δr ] ₂₁ ,Δr ₃₁ ,...,Δr _M1 ] ^T

Noise contained in TDOA and FDOA data, respectively; solving the equation by using the concept of RCTLS to obtain a closed solution containing regularization parameters:

wherein:

and->

Lambda is defined as an intermediate variable ₁ For regularization parameters, ++>

Step three: solving the regularization parameters in the second step based on a criterion of minimizing the mean square error;

step four: establishing an equation about the estimation error of the first step by using constraint conditions, and then solving a solution containing regularization parameters;

let the estimation errors in the second step be Δu=u, respectively ₁ -u ^o 、

Let->

The true position and velocity of the target is u ^o And->

The estimated position and velocity of the target obtained from the first step are u ₁ And->

The true value of the auxiliary variable in the second step +.>

And->

In u ₁ And->

The first-order Taylor expansion is carried out:

the above formula and formula Δu=u ₁ -u ^o 、

Substituting the positioning model in the step 2 to obtain:

(M+ΔM)θ ₂ ＝N+ΔN

wherein:

solving the equation based on the concept of RCTLS can be achieved:

wherein:

and->

For the purpose of defining the intermediate variables,

step five: and solving regularization parameters contained in the solution obtained in the step 4 based on a criterion of minimizing the mean square error, and correcting the result obtained in the second step by using the solution obtained in the fourth step to obtain a final solution.

2. The regularization constraint weighted least squares based passive positioning method of claim 1, wherein: the positioning equation based on TDOA/FDOA obtained in the first step is as follows:

wherein: r is (r) _i1 S is the measured distance difference between the primary station and the secondary station to the target _i And

for receiving the position and velocity coordinates of the station, u ^o And->

Epsilon for the position and velocity coordinates of the object _t =bΔr, error term of TDOA equation, +.>

Is the error term of the FDOA equation, wherein +.>

3. The regularization constraint weighted least squares based passive positioning method of claim 2, wherein: the third step is as follows: solving regularization parameters in the solution obtained in the second step, and expressing the mean square error of the estimation error in the first step of estimation as:

solving the above equation based on a criterion that minimizes the mean square error may result in regularization parameters of:

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