CN113325406B - Regularized constraint weighted least square-based passive positioning method - 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

Regularized constraint weighted least square-based passive positioning method

Technical Field

The invention relates to the field of passive radar positioning, which can estimate the position and the speed of a moving target in real time in the field of army and civilian, in particular to a TDOA/FDOA passive positioning method based on regularization constraint weighted least square.

Background

Currently, the multi-station passive positioning technology is widely applied to various military and civil fields, such as sensor networks, radars, navigation and the like. Observables that may be utilized by multi-station passive positioning techniques mainly include angle of Arrival (AOA), time difference of Arrival (Time Difference of Arrival, TDOA), frequency difference of Arrival (Frequency Difference of Arrival, FDOA), and the like.

The positioning system combining different measurement parameters can combine the advantages of different parameters, so that the positioning accuracy is improved to a certain extent, and the positioning system combining TDOA and FDOA can estimate the position and speed of a target at the same time, so that extensive research is obtained. In recent years, algorithms for TDOA/FDOA-based positioning problems have been proposed successively, including the Two-step weighted least squares method (Two-Stage Weighted Least Squares, TSWLS), constrained overall least squares (Constrained Total Least Squares, CTLS), and the like. However, due to the interference of noise, the distribution of the receiving stations and the target positions, and the influence of the number of the receiving stations on the coefficient matrix, the coefficient matrix in the methods based on the CWLS and CTLS models may have a pathological problem in practical application.

Disclosure of Invention

Aiming at the coefficient matrix pathological problem in the TDOA/FDOA passive positioning problem, the invention provides a Two-step regularization constraint least squares method (Two-Stage Regularized Constrained Total Least Squares, TRCTLS) based on the concept of RCTLS, and the method introduces regularization parameters on the basis of a CTLS model, and improves the root mean square error of positioning and suppresses the pathological condition of the coefficient matrix by reasonably selecting the regularization parameters.

The purpose of the invention is realized in the following way: the method comprises the following steps:

step 1: establishing a positioning equation according to the measured TDOA and FDOA information;

step 2: solving a positioning equation based on the concept of RCTLS to obtain a closed solution containing regularization parameters;

step 3: solving the regularization parameters in the second step based on a criterion of minimizing the mean square error;

step 4: establishing an equation about the estimation error of the first step by using constraint conditions, and then solving a solution containing regularization parameters;

step 5: 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.

The invention also includes such structural features:

1. the positioning equation based on TDOA/FDOA obtained in the step 1 is as follows:

wherein r is _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 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 +.>

2. The step 2 is specifically as follows:

construction auxiliary variable

The TDOA/FDOA-based positioning model is expressed according to the CTLS method as follows:

(A+ΔA)θ ₁ ＝(b+Δb)

wherein:

wherein O is _(M-1)×2N A zero matrix of (M-1) x 2N, and 0 is a zero vector of 3 x 1. Solving the equation by using the concept of RCTLS to obtain a closed solution containing regularization parameters:

3. the step 3 is specifically as follows:

the regularization parameters in the solution obtained in the step 2 are solved, and the mean square error of the estimation error in the first step of estimation can be expressed as by mathematical derivation:

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

the step 4 is specifically as follows:

let the estimation errors in step 2 be Δu=u, respectively ₁ -u ^o 、

Let->

The true value of the auxiliary variable in step 2 +.>

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 this equation, again based on the concept of RCTLS, can result in:

as with the idea of step 3, step 5 is also to estimate the regularization parameters of the solution obtained in step 4 based on a criterion that minimizes the mean square error.

Compared with the prior art, the invention has the beneficial effects that: theoretical deduction and simulation experiments show that the method can further reduce root mean square error (Root Mean Square Error, RMSE) of an algorithm on the basis of a CTLS method by reasonably selecting regularization parameters, and the positioning performance is more stable when a coefficient matrix is in a pathological state.

Drawings

FIG. 1 is a model of a receiving station locating a moving object;

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

FIG. 3 is a distribution of receiving stations and target locations in three positioning scenarios;

FIG. 4 is a position estimation root mean square error for three methods in scenario one;

FIG. 5 is a root mean square error of velocity estimates for three methods in scenario one;

FIG. 6 is a root mean square error of position estimates for three methods in scenario two;

FIG. 7 is a root mean square error of velocity estimation for three methods in scenario two;

FIG. 8 is a position estimation root mean square error for three methods in scenario three;

fig. 9 is a root mean square error of velocity estimation for three methods in scenario three.

Detailed Description

The invention is described in further detail below with reference to the drawings and the detailed description.

A positioning method based on regularization constraint total least square specifically comprises the following steps:

(1) As shown in FIG. 1, the moving target radiation source is positioned in three-dimensional space by M receiving stations, and the position and the speed of the target radiation source are u respectively ^o ＝[x,y,z] ^T 、

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

The true distance between the receiving station and the target radiation source can be expressed as:

any one of the receiving stations is selected as a master station, the number is 1, and the rest of the receiving stations are auxiliary stations, so that the true distance difference between the master station and each auxiliary station is as follows:

wherein the method comprises the steps of

Calculated from the real TDOA data, the vector form thereof can be expressed as +.>

From the above-described TDOA-based positioning equations:

for a pair of

Deriving the definition of the rate of change of the actual distance between the receiving station and the target radiation source:

taking the time derivative of the TDOA equation yields the FDOA-based positioning equation:

wherein the method comprises the steps of

Is the true range difference rate derived from the true FDOA information, and its vector form can be described 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 ], respectively ₂₁ ,r ₃₁ ,…,r _M1 ] ^T And->

It can be described as the form of the addition of a true value to a noise value:

r＝r ^o +Δr＝r ^o +cΔt

wherein the measurement noise contained in the TDOA and FDOA measurement data is Δr= [ Δr ], respectively ₂₁ ,Δr ₃₁ ,…，Δr _M1 ] ^T and

Assuming they obey a gaussian distribution with zero mean, their covariance matrix is:

TDOA measurement r=r to contain noise ^o +Δr and FDOA measurements

Substituting into the TDOA equation and the FDOA equation and ignoring the second order error term can obtain a TDOA/FDOA equation containing the error term:

wherein ε is _t And B deltar, is the error term of the TDOA equation,

as an error term of the FDOA equation,

(2) Construction auxiliary variable

First, a positioning equation containing error terms is expressed as a matrix form according to a CTLS model:

(A+ΔA)θ ₁ ＝(b+Δb)

wherein:

wherein O is _(M-1)×2N A zero matrix of (M-1) x 2N, and 0 is a zero vector of 3 x 1. As can be seen from the expression of matrix A, the unreasonable distribution of the receiving station and the target position and the noise interference can lead to the occurrence of the disease state of matrix A, for example, when the coordinates of the receiving station are relatively close to each other on the x-axis, the condition number of matrix A can be obviously increased, and the matrix can be obtainedThe condition is now in progress, resulting in an increased sensitivity of the localization result to noise. And introducing regularization parameters is a means for effectively suppressing matrix morbidity. 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 column number of matrix Δa (l=8), F ₁ To F ₉ The value of (2) 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} )

wherein: sigma (sigma) ₁₁ ＝∑ ₂₂ ＝diag(r ₂₁ ,r ₃₁ ,…,r _M1 )、

The whitening process is performed on n considering that the errors in n have correlation and have different variances. Let q=e [ nn ] ^T ]Cholesky decomposition of Q yields q=pp ^T The whitening vector of n is σ=p ^-1 n, at the same time, 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 concept, the positioning equation can be converted into the following form:

min(‖σ‖ ² +λ ₁ ‖θ ₁ ‖ ² )

through simple mathematical operation, the above formula can be converted into the problem of minimum value of the following functions:

the method can be obtained after the above derivation:

in the middle of

Further solving the equation to obtain the RCTLS solution of the equation:

(3) The solution obtained in the previous step does not take into account constraints, and therefore in this step the constraint will be used to reconstruct a set of equations to complete the correction of the first step estimate. Assuming the true position and velocity of the target to be u ^o And

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

The estimation error in the first step is Δu=u ₁ -u ^o 、

Let->

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

And->

In u ₁ And->

The first-order Taylor expansion is carried out:

recording the auxiliary variable estimated value obtained in the first step as r ₁ And

the expressions α and β are:

α＝(u ₁ -s ₁ )/r ₁

let Δu=u ₁ -u ^o 、

And +.>

And->

The taylor expansion of (2) is substituted into the positioning equation:

(M+ΔM)θ ₂ ＝N+ΔN

wherein:

wherein Δm and Δn may be represented as:

ΔM＝[U ₁ n，U ₂ n，…，U _k n]

ΔN＝U _k+1 n

where k is the column number of the matrix (k=6), U ₁ ,U ₂ …U _k The specific expression of (2) is obtained after whitening n:

ΔM＝[V ₁ n,V ₂ n，…,V _k n]

ΔN＝V _k+1 n

wherein V is _i ＝U _i P is (M+ΔM) θ ₂ =n+Δn can be converted into the following form:

wherein the method comprises the steps of

The above equation is solved by using the RCTLS algorithm in the first step, and the method can be obtained:

wherein the method comprises the steps of

The final estimated values of the target position and velocity are +.>

(4) The regularization parameters in steps (2) and (3) need to be solved from the principle of minimizing the mean square error. Firstly, solving regularization parameters in the solving step (2), and supposing the estimated value in the step (2)

And the true value theta ₁ The relation between the two is:

substituting the above into the function F (θ) ₁ ) Can be found in the derivatives of:

order the

Then it is obtainable by the formula:

order the

The method can obtain:

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 )

typically the matrix SA is a full order matrix, then there is a orthonormal matrix such that SA diagonalizes:

SA＝P ^T diag{u ₁ ，u ₂ ，…，u ₈ }P

for [ (SA+lambda ] ₁ I _8×8 ) ^-1 ] ^T (SA+λ ₁ I _8×8 ) ^-1 The eigenvalue decomposition can be carried out to obtain:

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

where P is an 8 x 8 orthonormal matrix, d=diag { (i) ₁ +λ ₁ ) ^-2 ，(u ₂ +λ ₁ ) ^-2 ，…，(u ₈ +λ ₁ ) ^-2 }. Considering E (J) =0, for Δθ ₁ ^T Δθ ₁ The mean square error of the estimated value obtained in the first step is expected to be obtained by the expression of (a):

let C ₁ ＝PJ、C ₂ ＝Pθ ₁ C is then ₁ 、C ₂ Two column vectors. Set C ₁ ＝[c ₁₁ ，c ₁₂ ，…，c ₁₈ ] ^T 、C ₂ ＝[c ₂₁ ，c ₂₂ ，…,c ₂₈ ] ^T E [ delta ] theta ₁ ^T Δθ ₁ ]Can be expressed as:

it can be seen that the mean square error of the result obtained in the first step using the RCTLS algorithm is related to lambda ₁ To find the lambda corresponding to the minimum mean square error ₁ Value of delta theta ₁ ^T Δθ ₁ The derivation can be obtained:

the analysis can be carried out when

When (I)>

At this time, the mean square error is estimated with lambda ₁ Monotonically decreasing; when->

When (I)>

At this time, the mean square error is estimated with lambda ₁ Monotonically increasing. Thus, lambda can be obtained ₁ The minimum value is +.>

The minimum value is taken here as: />

Similarly, lambda in step (3) can be obtained ₂ Is a value of (a). Note that in the method of step (2), Λ sums

All contain an unknown number theta ₁ Therefore, in the actual calculation, Λ=0, +.>

After obtaining the initial estimated value, substituting it into matrix Λ and +.>

Pairs of expressions of (a)Λ and->

Update is performed, after which new Λ and +.>

Recalculating theta ₁ The final theta can be obtained by circulating for 3 to 5 times ₁ Similarly, in the second step, the second step is performed on ₂ Similar processing is required when solving.

To further illustrate this embodiment, a simulation analysis was performed on the algorithm, with simulation experiments performed in 3 scenarios, where the position and velocity of the target radiation source in scenario 1 and scenario 2 are u= [285,325,275, respectively] ^T m and

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

The distribution of the positions of the receiving stations in 3 scenes and the target in the three-dimensional space is shown in fig. 2, and it can be seen that the overall distribution of the receiving stations in 3 scenes is basically similar, and only the coordinates of 2 receiving stations in scene 2 on the z axis are slightly different from those in scene 1 and scene 3. Scene 1 is a positioning scene when the coefficient matrix does not appear in a pathological state, but in scene 2, the coefficient matrix appears in a pathological state because the coordinates of the receiving stations are close on the z-axis, and in scene 3, the coefficient matrix appears in a pathological state because the distance between the target radiation source and 5 receiving stations is close. Performance analysis using root mean square error in all 3 scenarios, using CRLB is used as a standard for measuring estimation accuracy, and a TSWLS method and a CTLS method are simultaneously used as a comparison method.

Fig. 4 and 5, fig. 6 and fig. 7, and fig. 8 and fig. 9 respectively reflect the positioning root mean square errors of the three methods in the first scene, the second scene and the third scene, so that it can be seen that the algorithm adopted by the 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 the coefficient matrix has a pathological problem.

It will be appreciated by those skilled in the art that, in the foregoing method according to the embodiments of the present application, the sequence number of each step does not mean that the execution sequence of each step should be determined by the function and the internal logic, and should not limit the implementation process of the embodiments of the present application in any way.

Finally, it should be noted that the above embodiments are only intended to describe the technical solution of the present invention and not to limit the technical method, the present invention extends to other modifications, variations, applications and embodiments in application, and therefore all such modifications, variations, applications, embodiments are considered to be within the spirit and scope of the teachings of the present invention.

In summary, the invention provides a Regularization Constraint Total Least Squares (RCTLS) based closed-form analysis 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.

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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