CN114325581A - Elliptical target positioning method with clock synchronization error - Google Patents

Abstract

The invention discloses an elliptical target positioning method with clock synchronization errors, which is characterized in that under the condition that all sensors have clock synchronization errors, a single target is positioned by using an elliptical target positioning system, and all clock synchronization errors are treated as noise; then, the reflection path distance measurement model can be expressed as a standard generalized confidence domain subproblem through processing; finally, the problem can be solved accurately by using an efficient GTRS algorithm, so that the high-precision positioning of the target is realized; the invention has the advantages that: on one hand, the target positioning is realized, and simultaneously, all clock synchronization errors do not need to be estimated, so that the number of variables to be estimated is greatly reduced, and the operation efficiency of the algorithm is improved; on the other hand, the problem is solved by using a GTRS algorithm with extremely low complexity, and efficient and accurate positioning of the target is realized.

Description

Elliptical target positioning method with clock synchronization error

Technical Field

The invention relates to a target positioning technology, in particular to an elliptical target positioning method with clock synchronization errors.

Background

With the rapid development of wireless communication technology, target location technology has also been rapidly developed, and the technology is also applied to various fields in real life, such as: the intelligent driving, the marine navigation, the accurate guidance of missiles and the like, so that the target positioning technology is closely related to all walks of life, and the research on the target positioning technology becomes very significant.

Generally, the target positioning is to collect some measurement information by direct or indirect communication between some sensors with known positions and the target, then collect the measurement information to a data processing center uniformly, and the data processing center runs a corresponding positioning algorithm to complete the positioning of the target. These measurement information typically includes: time of arrival (TOA), time difference of arrival (TDOA), angle of arrival (AOA), Received Signal Strength (RSS), combinations thereof, and the like. Among all the measurement information, the positioning method based on the time measurement information has higher positioning accuracy, and thus it is also widely studied by scholars. Compared with other positioning methods based on time measurement information, the elliptical target positioning method has the following advantages, such as: compared with time of arrival (TOA) based, the elliptical target location method does not require cooperation between the target and the sensors, nor does it require time synchronization between the sensors; the elliptical object location method has a higher location accuracy with all sensors (including the transmitter and receiver) time synchronized than with time difference of arrival (TDOA) based. Meanwhile, the method for positioning the elliptical target can be suitable for all scenes with time asynchronization of the sensors and can also be suitable for scenes with time synchronization of part or all of the sensors, so that the method for positioning the elliptical target has better flexibility. When the ellipse positioning is implemented, firstly, a transmitter transmits a signal, the signal is reflected by a target and then received by a receiver, and the position of the positioned target is exactly positioned at the intersection point of a plurality of ellipses taking the positions of the transmitter and the receiver as focuses.

Ideally, when collecting data using sensors (including transmitters and receivers), only the measurement error of the measured value (delay or distance) needs to be considered, and the clock synchronization error caused by the desynchronization of the sensor clocks does not need to be considered, because the sensors are assumed to be clock-synchronized. However, in a real situation, since the sensors (including the transmitter and the receiver) are affected by the production process, temperature variation, environmental variation, aging degree, and the like, clocks between them may not be synchronized, which may cause the measured value to become too large in error due to the clock synchronization error, thereby affecting the positioning accuracy of the target. Therefore, in order to improve the positioning accuracy of the target, it is necessary to adopt an appropriate method to deal with the clock synchronization error problem. At present, for the situation that clock synchronization errors exist in positioning of an elliptical target, the clock synchronization errors are generally used as variables to be estimated in the existing literature, and this way causes many variables in the elliptical target positioning method, which results in that on one hand, the problem solving in the elliptical target positioning method becomes complicated and may be difficult to solve, and even if the problem can be solved, the positioning accuracy of the target is not high, and the target is susceptible to noise and the performance of the positioning algorithm is unstable; on the other hand, the computation complexity of the elliptical target positioning method is high.

Disclosure of Invention

The invention aims to solve the technical problem of providing an elliptical target positioning method with clock synchronization errors, wherein the clock synchronization errors are treated as noise, and the position of a target can be accurately estimated by using a GTRS algorithm with low complexity after a reflection path distance measurement model is converted.

The technical scheme adopted by the invention for solving the technical problems is as follows: an ellipse target positioning method with clock synchronization error is characterized by comprising the following steps:

step 1: in a multi-input multi-output radar system, establishing a K-dimensional coordinate system as a reference coordinate system, setting a sensor with M + N known real positions and a target with 1 unknown real position, and setting that each sensor has a clock synchronization error; then taking M sensors as transmitters for transmitting signals, taking the rest N sensors as receivers for receiving signals, and recording the real position of the ith transmitter in a reference coordinate system as t_iThe real position of the jth receiver in the reference coordinate system is denoted as s_jThe real position of the target in the reference coordinate system is recorded as u^oTransmitting the ithThe clock synchronization error existing in the machine is recorded as

The clock synchronization error existing in the jth receiver is recorded as

The signal propagation distance error caused by the clock synchronization error existing in the ith transmitter is recalculated and recorded as

And calculating the signal propagation distance error caused by the clock synchronization error existing in the jth receiver, and recording as

Wherein, the value of K is 2 or 3, M is more than or equal to 2, N is more than or equal to 2, i is more than or equal to 1 and less than or equal to M, j is more than or equal to 1 and less than or equal to N, c represents the speed of signal propagation and is the speed of light;

step 2: calculating the sum of the measured distances from the ith transmitter to the target and from the target to the jth receiver, namely the distance measurement value of the reflection path from the ith transmitter to the jth receiver after the signal transmitted by the ith transmitter is reflected by the target and received by the jth receiver, and recording as r_i,j，r_i,j＝τ_i,jX c; wherein, tau_i,jRepresenting the flight time of the signal transmitted by the ith transmitter after being reflected by the target and received by the jth receiver;

and step 3: according to r_i,jAnd constructing a reflection path distance measurement model, which is described as follows:

then, all distance measurement values in the reflection path distance measurement model, signal propagation distance errors caused by all clock synchronization errors and all measurement noises are expressed in a vector form respectively, and the correspondence is recorded as r, delta r and n, wherein r is [ r ═ r_1,1,…,r_1,N,…,r_M,1,…,r_M,N]^T，

n＝[n_1,1,...,n_1,N,...,n_M,1,...,n_M,N]^T(ii) a Wherein the symbol "| | |" is a two-norm symbol, | | | | u | ", which is a binary norm symbol^o-t_i| represents the real distance from the ith transmitter to the target, and | u^o-s_jI represents the real distance from the target to the jth receiver, n_i,jRepresenting the measurement noise on the reflected path of the signal transmitted by the ith transmitter reflected by the target and received by the jth receiver]"is a vector or matrix representing a symbol, the superscript" T "representing a transpose, r_1,1Distance measurement, r, representing the reflection path of a signal transmitted by a 1 st transmitter reflected by a target and received by a 1 st receiver_1,NDistance measurement, r, representing the reflection path of a signal transmitted by a 1 st transmitter reflected by a target and received by an Nth receiver_M,1Distance measurement, r, representing the reflection path of the signal transmitted by the Mth transmitter reflected by the target and received by the 1 st receiver_M,NA distance measurement, Δ r, representing a reflection path of a signal transmitted by an Mth transmitter reflected by a target and received by an Nth receiver₁ ^tIndicating the signal propagation distance error caused by the clock synchronization error present at the 1 st transmitter,

representing the error in the propagation distance of the signal, Δ r, caused by the clock synchronization error present in the Mth transmitter₁ ^sIndicating the signal propagation distance error caused by the clock synchronization error present at the 1 st receiver,

representing the error in the propagation distance of the signal caused by the clock synchronization error present in the Nth receiver, N_1,1Representing the measurement noise on the reflected path of the signal transmitted by the 1 st transmitter after reflection by the target and received by the 1 st receiver, n_1,NRepresenting the measurement noise, N, on the reflected path of the signal transmitted by the 1 st transmitter after reflection by the target and received by the Nth receiver_M,1Representing the measurement noise on the reflected path of the signal transmitted by the Mth transmitter after reflection by the target and received by the 1 st receiver, n_M,NRepresenting the measurement noise on the reflection path of the Mth transmitter reflected by the target and received by the Nth receiver, the mean value of the Deltar obeys 0, and the covariance matrix is Q_△rWith n obeying a mean of 0 and a covariance matrix of Q_nIs in a Gaussian distribution of (a) and (b) are independent of each other, Q_△rCovariance matrix, Q, representing Deltar_nA covariance matrix representing n;

and 4, step 4: model the distance measurement of the reflection path

Split into a first sub-model

And a second submodel r_i,j-(r_1,j-||u^o-t₁||)＝||u^o-t_i||+△r_i ^t+n_i,j-△r₁ ^t-n_1,jI 2, …, M, j 1,2, …, N; then squaring both sides of the equation for the first submodel and ignoring its second order noise term

To obtain

And squares the equation of the second submodel on both sides and ignores its second order noise term (Δ r)₁ ^t-△r_i ^t+n_1,j-n_i,j)²To obtain

Wherein r is_1,jIndicating that the signal transmitted by the 1 st transmitter is reflected by the target and then is received by the jth receiverDistance measurement of the received reflection path, t₁Representing the true position of the 1 st transmitter in the reference frame, n_1,jRepresenting the measurement noise on the reflected path of the signal transmitted by the 1 st transmitter reflected by the target and received by the jth receiver,

representing the true distance of the reflected path of the signal transmitted by the 1 st transmitter after reflection by the target and received by the jth receiver,

the real distance of a reflection path which represents that a signal transmitted by the ith transmitter is reflected by a target and then is received by the jth receiver is represented;

and 5: according to

And

constructing a constrained weighted least squares problem, described as:

where min () is the minimization function, s.t. denotes "constrained to … …", g is the optimization variable, g ═ u^T,||u-t₁||]^TU denotes a position variable of the object, g_(K+1)Denotes the K +1 th element in g, (b-Ag)^TW^-1(b-Ag) is an objective function,

b、b₁and b₂Are all introduced coefficient vectors, s₁Representing the true position, s, of the 1 st receiver in the reference frame_NRepresenting the true position, t, of the Nth receiver in the reference frame₂Representing the true position, t, of the 2 nd transmitter in the reference frame_MRepresenting the true position of the Mth transmitter in the reference coordinate system, r_2,1Indicates the 2 nd hairDistance measurement, r, of reflected path of signal transmitted by transmitter after reflection by target and received by 1 st receiver_2,NA distance measurement representing a reflected path of a signal transmitted by the 2 nd transmitter reflected by the target and received by the nth receiver,

A、A₁and A₂Are all an introduced coefficient matrix, W^-1Denotes the inverse of W, W being the introduced weight matrix,

F₁and F₂Are all an introduced intermediate coefficient matrix, F₁＝2HP₁，F₂＝2HP₂，H、P₁、P₂Are all introduced intermediate coefficient matrices, H ═ blkdiag (H)₁,H₂)，H₁、H₂Are all the introduced intermediate coefficient matrixes,

representing the true distance of the reflected path of the signal transmitted by the 1 st transmitter reflected by the target and received by the 1 st receiver,

representing the true distance of the reflected path of the signal transmitted by the 2 nd transmitter reflected by the target and received by the 1 st receiver,

representing the true reflection path of the signal transmitted by the 1 st transmitter reflected by the target and received by the Nth receiverThe distance between the first and second electrodes,

representing the true distance of the reflected path that the mth transmitter transmits after reflection by the target and is received by the nth receiver, blkdiag () is a block diagonal operation function, diag () is an element diagonal operation function,

I_Mrepresenting an identity matrix of dimension M x M,

a column vector of dimension M × 1, representing the 1 st element as 1 and the remaining elements as 0_MRepresenting all 1 vectors, symbols, of dimension Mx 1

Is a symbol of a kronecker product operation, I_NIdentity matrix with dimension NxN, 1_NRepresenting a full 1 vector of dimension N × 1;

step 6: the constrained weighted least squares problem is transformed into a standard generalized confidence domain subproblem, described as:

wherein, a is the introduced coefficient vector,

d is an introduced coefficient matrix, and D is blkdiag (I)_K,-1)，I_KAn identity matrix having a dimension of K × K;

and 7: adopting GTRS algorithm to solve the standard generalized confidence domain subproblem, the concrete process is as follows: according to the generalized confidence domain subproblem of the standard, a Lagrangian function is obtained, and is marked as L (g, lambda), and L (g, lambda) is (b-Ag)^TW^-1(b-Ag)+λ(g-a)^TD (g-a); then, for L (g, lambda) ═ b-Ag^TW^-1(b-Ag)+λ(g-a)^TD (g-a) obtains a partial derivative related to g, and the partial derivative related to g is 0, so that g (lambda) is equal to (A)^TW^-1A+λD)^-1(A^TW^-1b + λ Da); then, g (lambda) is changed to (A)^TW^-1A+λD)^-1(A^TW^-1b + λ Da) into the constraint (g-a) in the standard generalized confidence domain sub-problem^TD (g-a) ═ 0, and defines f (λ) ═ g (λ) -a)^TD (g (. lamda) -a); then searching out an optimal value lambda of lambda by utilizing dichotomy^*So that f (λ)^*) 0; then will lambda^*Substituting g (λ) ═ a^TW^-1A+λD)^-1(A^TW^-1b + λ Da), mixing g (λ)^*) As the optimal value g of g^*(ii) a Finally according to g^*An estimate of u is obtained and is denoted as u^*，u^*＝g^* _(1:K)(ii) a Wherein L () is a Lagrangian function representation form, g in L (g, λ), λ is a parameter of L (), λ represents a Lagrangian multiplier, g (λ) represents a function of g with respect to λ, f (λ) is an introduced intermediate function, u (λ) is a function of L (g, λ), u (u) is a function of L (g, λ) and u (g, λ) is a function of L (g, λ) and u (u) is a function of L (g, λ) and u (a) is a function of L (i) and a parameter of L^*＝g^* _(1:K)Wherein ═ is an assignment symbol, g^* _(1:K)Denotes g ^*1 st to kth element in (1).

In said step 2, τ_i,jThe acquisition mode is as follows: the signal transmitted by the ith transmitter is provided with a time stamp, the signal transmitted by the ith transmitter is reflected by a target and then received by the jth receiver, and tau is calculated according to the time stamp of the received signal recorded by the jth receiver_i,j。

In step 7, an optimal value λ of λ is searched by using a dichotomy^*So that f (λ)^*) In 0, the dichotomy search interval is

Gamma is

The maximum eigenvalue of (d); wherein γ is an introduced intermediate symbol.

Compared with the prior art, the invention has the advantages that:

1) the method of the invention treats the clock synchronization error existing in the sensor as noise, does not need to estimate all clock synchronization errors, greatly reduces the number of variables to be estimated, reduces the calculation complexity and improves the operation efficiency.

2) The method of the invention adopts a mode similar to TDOA difference processing, and selects the first transmitter as a reference, so that after ingenious transformation, the processed model only contains | | | u^o-t₁And | | is a two-norm variable, the number of the variables is reduced again, and the complexity of the algorithm is further reduced.

3) The method converts the weighted least square problem with the constraint into a standard generalized confidence domain subproblem, and the complexity of solving the standard generalized confidence domain subproblem is extremely low, so that the calculation complexity is reduced.

4) The method can achieve the target positioning precision of the Cramer-Rao lower bound (CRLB) under the condition of not very large noise.

5) Compared with the existing classical two-step weighted least square (TSWLS) method, the method has more stable performance, and the advantage is particularly obvious in the case of large noise.

Drawings

FIG. 1 is a block diagram of an overall implementation of the method of the present invention;

FIG. 2 is a schematic diagram of a system model of 2 transmitters and 3 receivers;

FIG. 3 shows the method of the present invention (GTRS) and the existing two-step weighted least squares method (TSWLS), the existing maximum likelihood estimation Method (ML) based on the original problem and the Clarmet-Lour boundary (CRLB) at σ_△rA comparison plot of the log Mean Square Error (MSE) of the location of the target location as a function of the measured noise power for the case of 15 (m);

FIG. 4 shows the relationship between the present invention method (GTRS) and the existing two-step weighted least squares method (TSWLS), the existing maximum likelihood estimation Method (ML) based on the original problem and the Cramer-Row lower bound (CRLB)

A comparison graph of the logarithmic Mean Square Error (MSE) for the location of the target position as a function of the clock-synchronous error noise power change.

Detailed Description

The invention is described in further detail below with reference to the accompanying examples.

The invention provides an ellipse target positioning method with clock synchronization error, the general implementation block diagram of which is shown in figure 1, and the method comprises the following steps:

step 1: in a multiple-input multiple-output (MIMO) radar system, establishing a K-dimensional coordinate system as a reference coordinate system, setting a sensor with M + N known real positions and a target with 1 unknown real position, and setting that each sensor has a clock synchronization error; then taking M sensors as transmitters for transmitting signals, taking the rest N sensors as receivers for receiving signals, and recording the real position of the ith transmitter in a reference coordinate system as t_iThe real position of the jth receiver in the reference coordinate system is denoted as s_jThe real position of the target in the reference coordinate system is recorded as u^oThe clock synchronization error of the ith transmitter is recorded as

The clock synchronization error existing in the jth receiver is recorded as

The signal propagation distance error caused by the clock synchronization error existing in the ith transmitter is recalculated and recorded as

And calculating the signal propagation distance error caused by the clock synchronization error existing in the jth receiver, and recording as

Wherein, the value of K is 2 or 3, M is more than or equal to 2, N is more than or equal to 2, i is more than or equal to 1 and less than or equal to M, j is more than or equal to 1 and less than or equal to N, c represents the speed of signal propagation and is the speed of light,a random error exists between the actual clock of each sensor and the standard clock of the sensor in the multi-input multi-output radar system, and the error is the clock synchronization error of each sensor.

Fig. 2 shows a schematic diagram of a system model of 2 transmitters and 3 receivers.

Step 2: calculating the sum of the measured distances from the ith transmitter to the target and from the target to the jth receiver, namely the distance measurement value of the reflection path from the ith transmitter to the jth receiver after the signal transmitted by the ith transmitter is reflected by the target and received by the jth receiver, and recording as r_i,j，r_i,j＝τ_i,jX c; wherein, tau_i,jRepresenting the time of flight that the signal transmitted by the ith transmitter is reflected off of the target and received by the jth receiver.

In this example, in step 2,. tau._i,jThe acquisition mode is as follows: the signal transmitted by the ith transmitter is provided with a time stamp, the signal transmitted by the ith transmitter is reflected by a target and then received by the jth receiver, and tau is calculated according to the time stamp of the received signal recorded by the jth receiver_i,j。

And step 3: according to r_i,jAnd constructing a reflection path distance measurement model, which is described as follows:

then, all distance measurement values in the reflection path distance measurement model, signal propagation distance errors caused by all clock synchronization errors and all measurement noises are expressed in a vector form respectively, and the correspondence is recorded as r, delta r and n, wherein r is [ r ═ r_1,1,…,r_1,N,…,r_M,1,…,r_M,N]^T，

n＝[n_1,1,...,n_1,N,...,n_M,1,…,n_M,N]^T(ii) a Wherein the symbol "| | |" is twoNorm symbol, | | u^o-t_i| represents the real distance from the ith transmitter to the target, and | u^o-s_jI represents the real distance from the target to the jth receiver, n_i,jRepresenting the measurement noise on the reflected path of the signal transmitted by the ith transmitter reflected by the target and received by the jth receiver]"is a vector or matrix representing a symbol, the superscript" T "representing a transpose, r_1,1Distance measurement, r, representing the reflection path of a signal transmitted by a 1 st transmitter reflected by a target and received by a 1 st receiver_1,NDistance measurement, r, representing the reflection path of a signal transmitted by a 1 st transmitter reflected by a target and received by an Nth receiver_M,1Distance measurement, r, representing the reflection path of the signal transmitted by the Mth transmitter reflected by the target and received by the 1 st receiver_M,NDistance measurement value, Deltar, representing the reflection path of the signal transmitted by the Mth transmitter reflected by the target and received by the Nth receiver₁ ^tIndicating the signal propagation distance error caused by the clock synchronization error present at the 1 st transmitter,

indicating the signal propagation distance error, Δ r, caused by the clock synchronization error present in the Mth transmitter₁ ^sIndicating the signal propagation distance error caused by the clock synchronization error present at the 1 st receiver,

representing the error in the propagation distance of the signal caused by the clock synchronization error present in the Nth receiver, N_1,1Representing the measurement noise on the reflected path of the signal transmitted by the 1 st transmitter after reflection by the target and received by the 1 st receiver, n_1,NRepresenting the measurement noise, N, on the reflected path of the signal transmitted by the 1 st transmitter after reflection by the target and received by the Nth receiver_M,1Representing the measurement noise on the reflected path of the signal transmitted by the Mth transmitter after reflection by the target and received by the 1 st receiver, n_M,NIndicating that the signal transmitted by the Mth transmitter has been reflected by the targetThe measured noise on the reflected path received by the Nth receiver, Δ r obeys a mean of 0 and a covariance matrix of Q_△rWith n obeying a mean of 0 and a covariance matrix of Q_nIs in a Gaussian distribution of (a) and (b) are independent of each other, Q_△rCovariance matrix, Q, representing Deltar_nRepresenting the covariance matrix of n.

And 4, step 4: model the distance measurement of the reflection path

Split into a first sub-model

And a second submodel r_i,j-(r_1,j-||u^o-t₁||)＝||u^o-t_i||+△r_i ^t+n_i,j-△r₁ ^t-n_1,jI 2, …, M, j 1,2, …, N; then squaring both sides of the equation for the first submodel and ignoring its second order noise term

To obtain

And squares the equation of the second submodel on both sides and ignores its second order noise term (Δ r)₁ ^t-△r_i ^t+n_1,j-n_i,j)²To obtain

Wherein r is_1,jDistance measurement, t, representing the reflection path of a signal transmitted by the 1 st transmitter reflected by a target and received by the jth receiver₁Representing the true position of the 1 st transmitter in the reference frame, n_1,jRepresenting the measurement noise on the reflected path of the signal transmitted by the 1 st transmitter reflected by the target and received by the jth receiver,

representing the true distance of the reflected path of the signal transmitted by the 1 st transmitter after reflection by the target and received by the jth receiver,

representing the true distance of the reflected path of the signal transmitted by the ith transmitter after being reflected by the target and received by the jth receiver.

And 5: according to

And

constructing a constrained weighted least squares problem, described as:

where min () is the minimization function, s.t. denotes "constrained to … …", g is the optimization variable, g ═ u^T,||u-t₁||]^TU denotes a position variable of the object, g_(K+1)Denotes the K +1 th element in g, (b-Ag)^TW^-1(b-Ag) is an objective function,

b、b₁and b₂Are all introduced coefficient vectors, s₁Representing the true position, s, of the 1 st receiver in the reference frame_NRepresenting the true position, t, of the Nth receiver in the reference frame₂Representing the true position, t, of the 2 nd transmitter in the reference frame_MRepresenting the true position of the Mth transmitter in the reference coordinate system, r_2,1Distance measurement, r, representing the reflection path of the signal transmitted by the 2 nd transmitter reflected by the target and received by the 1 st receiver_2,NA distance measurement representing a reflected path of a signal transmitted by the 2 nd transmitter reflected by the target and received by the nth receiver,

A、A₁and A₂Are all an introduced coefficient matrix, W^-1Denotes the inverse of W, W being the introduced weight matrix,

F₁and F₂Are all an introduced intermediate coefficient matrix, F₁＝2HP₁，F₂＝2HP₂，H、P₁、P₂Are all introduced intermediate coefficient matrices, H ═ blkdiag (H)₁,H₂)，H₁、H₂Are all the introduced intermediate coefficient matrixes,

representing the true distance of the reflected path of the signal transmitted by the 1 st transmitter reflected by the target and received by the 1 st receiver,

representing the true distance of the reflected path of the signal transmitted by the 2 nd transmitter reflected by the target and received by the 1 st receiver,

representing the true distance of the reflected path of the signal transmitted by the 1 st transmitter reflected by the target and received by the nth receiver,

representing the true distance of the reflected path of the mth receiver after the mth transmitter transmits a signal reflected by the target, blkdiag () is a block diagonal operation function (i.e., the matrix block is taken as the element on the main diagonal, and the other elements are all 0),diag () is an element diagonal operation function (i.e., vector elements are elements on the main diagonal, other elements are all 0),

I_Mrepresenting an identity matrix of dimension M x M,

a column vector of dimension M × 1, representing the 1 st element as 1 and the remaining elements as 0_MRepresenting all 1 vectors, symbols, of dimension Mx 1

Is a symbol of a kronecker product operation, I_NIdentity matrix with dimension NxN, 1_NRepresenting a full 1 vector of dimension N x 1.

The reflection path distance measurement model is divided into two, and divided into a first sub-model (i 1) and a second sub-model (i 2.., M), and then the two are subjected to difference processing (r)_1,j-||u^o-t₁||，r_i,j-(r_1,j-||u^o-t₁| |)) constructs a weighted least squares problem with constraints, which helps to effectively reduce the number of variables, thereby greatly reducing the computational complexity.

The reflection path distance measurement model is subjected to mathematical processing and converted into a weighted least square problem with constraint, and a foundation is laid for the next conversion into a standard generalized confidence domain subproblem.

Step 6: the constrained weighted least squares problem is transformed into a standard Generalized confidence domain sub-problem (GTRS), described as:

wherein, a is the introduced coefficient vector,

d is an introduced coefficient matrix, and D is blkdiag (I)_K,-1)，I_KRepresenting an identity matrix of dimensions K x K.

And 7: adopting GTRS algorithm to solve the standard generalized confidence domain subproblem, the concrete process is as follows: according to the generalized confidence domain subproblem of the standard, a Lagrangian function is obtained, and is marked as L (g, lambda), and L (g, lambda) is (b-Ag)^TW^-1(b-Ag)+λ(g-a)^TD (g-a); then, for L (g, lambda) ═ b-Ag^TW^-1(b-Ag)+λ(g-a)^TD (g-a) obtains a partial derivative related to g, and the partial derivative related to g is 0, so that g (lambda) is equal to (A)^TW^-1A+λD)^-1(A^TW^-1b + λ Da); then, g (lambda) is changed to (A)^TW^-1A+λD)^-1(A^TW^-1b + λ Da) into the constraint (g-a) in the standard generalized confidence domain sub-problem^TD (g-a) ═ 0, and defines f (λ) ═ g (λ) -a)^TD (g (. lamda) -a); then searching out an optimal value lambda of lambda by utilizing dichotomy^*So that f (λ)^*) 0; then will lambda^*Substituting g (λ) ═ a^TW^-1A+λD)^-1(A^TW^-1b + λ Da), mixing g (λ)^*) As the optimal value g of g^*(ii) a Finally according to g^*An estimate of u is obtained and is denoted as u^*，u^*＝g^* _(1:K)(ii) a Wherein L () is a Lagrangian function representation form, g in L (g, λ), λ is a parameter of L (), λ represents a Lagrangian multiplier, g (λ) represents a function of g with respect to λ, f (λ) is an introduced intermediate function, u (λ) is a function of L (g, λ), u (u) is a function of L (g, λ) and u (g, λ) is a function of L (g, λ) and u (u) is a function of L (g, λ) and u (a) is a function of L (i) and a parameter of L^*＝g^* _(1:K)Wherein ═ is an assignment symbol, g^* _(1:K)Denotes g ^*1 st to kth element in (1).

In this embodiment, in step 7, an optimal value λ of λ is searched by using a dichotomy^*So that f (λ)^*) In 0, the dichotomy search interval is

Gamma is

The maximum characteristic value can be calculated by utilizing the eig () function of MATLAB software; wherein γ is an introduced intermediate symbol.

In order to verify the feasibility and the effectiveness of the method, the method is subjected to simulation test.

Suppose there are 2 transmitters and 5 receivers in three-dimensional space, their true positions in three-dimensional space s_x,s_y,s_z]^TIs randomly generated, wherein s_x～(-3000,3000)m、s_y- (-3000,3000) m and s_zAnd (1000,2000) m. The true position of the target in three-dimensional space is assumed to be [ x, y, z ]]^TThe true position is also randomly generated, where x- (-4000,4000) m, y- (-4000,4000) m, and z- (3000,6000) m. The measurement noise is assumed to be mean 0 and covariance matrix Q_nThe clock synchronization error of the sensor is also assumed to be mean 0 and covariance matrix Q_△rGaussian noise.

The positioning performance of the method is tested under the conditions of fixed clock synchronous error noise power and fixed measurement noise power respectively, the logarithmic Mean Square Error (MSE) describing the positioning performance under the two conditions is 10 random scenes, and each scene runs the logarithmic MSE obtained by 1000 Monte Carlo experiments. Meanwhile, the existing two-step weighted least square method (TSWLS), the existing maximum likelihood estimation Method (ML) based on the original problem and the Clarmet-Roche lower bound (CRLB) are introduced for comparison. FIG. 3 shows the σ of the present invention method (GTRS) and the existing two-step weighted least squares method (TSWLS), the existing maximum likelihood estimation Method (ML) based on the original problem and the Clarmet-Row lower bound (CRLB)_△rFig. 4 shows a comparison of the logarithmic Mean Square Error (MSE) of the localization of the target position as a function of the measured noise power in the case of 15(m), with the existing two-step weighted least squares method (TSWLS), the existing maximum likelihood estimation Method (ML) based on the original problem and the cramer-lo boundary (CRLB) in the case of the inventive method (GTRS) and the existing two-step weighted least squares method (TSWLS), the existing maximum likelihood estimation Method (ML) based on the original problem and the existing cramer-lo boundary (CRLB)

A comparison graph of the logarithmic Mean Square Error (MSE) for the location of the target position as a function of the clock-synchronous error noise power change. As can be seen from FIG. 3, the noise power when measured as a function of time

When the method is changed, the positioning accuracy of the lower boundary of the Clarit-Luo can be achieved under the condition of small noise, and the positioning accuracy of the lower boundary of the Clarit-Luo cannot be achieved under the condition of small noise by the existing two-step weighted least square method (TSWLS). It can be seen from fig. 4 that the error noise power σ when synchronized with the clock_△rWhen the method is changed, the positioning accuracy of the lower bound of the Claramelo can be achieved under the condition of small noise, but the positioning accuracy can be achieved only under the condition of extremely small noise by the existing two-step weighted least square method (TSWLS), and the positioning accuracy can not be achieved under the condition of other noise power.

The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.

Claims (3)

1. An ellipse target positioning method with clock synchronization error is characterized by comprising the following steps:

step 1: in a multi-input multi-output radar system, establishing a K-dimensional coordinate system as a reference coordinate system, setting a sensor with M + N known real positions and a target with 1 unknown real position, and setting that each sensor has a clock synchronization error; then taking M sensors as transmitters for transmitting signals, taking the rest N sensors as receivers for receiving signals, and recording the real position of the ith transmitter in a reference coordinate system as t_iThe real position of the jth receiver in the reference coordinate system is denoted as s_jThe real position of the target in the reference coordinate system is recorded as u^oThe clock synchronization error of the ith transmitter is recorded as

The clock synchronization error existing in the jth receiver is recorded as

The signal propagation distance error caused by the clock synchronization error existing in the ith transmitter is recalculated and recorded as

And calculating the signal propagation distance error caused by the clock synchronization error existing in the jth receiver, and recording as

Wherein, the value of K is 2 or 3, M is more than or equal to 2, N is more than or equal to 2, i is more than or equal to 1 and less than or equal to M, j is more than or equal to 1 and less than or equal to N, c represents the speed of signal propagation and is the speed of light;

step 2: calculating the sum of the measured distances from the ith transmitter to the target and from the target to the jth receiver, namely the distance measurement value of the reflection path from the ith transmitter to the jth receiver after the signal transmitted by the ith transmitter is reflected by the target and received by the jth receiver, and recording as r_i,j，r_i,j＝τ_i,jX c; wherein, tau_i,jRepresenting the flight time of the signal transmitted by the ith transmitter after being reflected by the target and received by the jth receiver;

and step 3: according to r_i,jAnd constructing a reflection path distance measurement model, which is described as follows:

then, all distance measurement values in the reflection path distance measurement model, signal propagation distance errors caused by all clock synchronization errors and all measurement noises are expressed in a vector form respectively, and the correspondence is recorded as r, delta r and n, wherein r is [ r ═ r_1,1,…,r_1,N,…,r_M,1,…,r_M,N]^T，

n＝[n_1,1,...,n_1,N,...,n_M,1,...,n_M,N]^T(ii) a Wherein the symbol "| | |" is a two-norm symbol, | | | | u | ", which is a binary norm symbol^o-t_i| represents the real distance from the ith transmitter to the target, and | u^o-s_jI represents the real distance from the target to the jth receiver, n_i,jRepresenting the measurement noise on the reflected path of the signal transmitted by the ith transmitter reflected by the target and received by the jth receiver]"is a vector or matrix representing a symbol, the superscript" T "representing a transpose, r_1,1Distance measurement, r, representing the reflection path of a signal transmitted by a 1 st transmitter reflected by a target and received by a 1 st receiver_1,NDistance measurement, r, representing the reflection path of a signal transmitted by a 1 st transmitter reflected by a target and received by an Nth receiver_M,1Distance measurement, r, representing the reflection path of the signal transmitted by the Mth transmitter reflected by the target and received by the 1 st receiver_M,NA distance measurement representing a reflected path of a signal transmitted by the mth transmitter reflected by the target and received by the nth receiver,

indicating the signal propagation distance error caused by the clock synchronization error present at the 1 st transmitter,

indicating a signal propagation distance error caused by a clock synchronization error present at the mth transmitter,

indicating the signal propagation distance error caused by the clock synchronization error present at the 1 st receiver,

representing the error in the propagation distance of the signal caused by the clock synchronization error present in the Nth receiver, N_1,1Representing the measurement noise on the reflected path of the signal transmitted by the 1 st transmitter after reflection by the target and received by the 1 st receiver, n_1,NRepresenting the measurement noise, N, on the reflected path of the signal transmitted by the 1 st transmitter after reflection by the target and received by the Nth receiver_M,1Representing the measurement noise on the reflected path of the signal transmitted by the Mth transmitter after reflection by the target and received by the 1 st receiver, n_M,NRepresenting the measurement noise on the reflection path of the Mth transmitter reflected by the target and received by the Nth receiver, the mean value of the Deltar obeys 0, and the covariance matrix is Q_△rWith n obeying a mean of 0 and a covariance matrix of Q_nIs in a Gaussian distribution of (a) and (b) are independent of each other, Q_△rCovariance matrix, Q, representing Deltar_nA covariance matrix representing n;

and 4, step 4: model the distance measurement of the reflection path

Split into a first sub-model

And a second submodel

(ii) a Then squaring both sides of the equation for the first submodel and ignoring its second order noise term

To obtain

And squaring both sides of the equation of the second submodel and ignoring its second order noise term

To obtain

Wherein r is_1,jDistance measurement, t, representing the reflection path of a signal transmitted by the 1 st transmitter reflected by a target and received by the jth receiver₁Representing the true position of the 1 st transmitter in the reference frame, n_1,jRepresenting the measurement noise on the reflected path of the signal transmitted by the 1 st transmitter reflected by the target and received by the jth receiver,

representing the true distance of the reflected path of the signal transmitted by the 1 st transmitter after reflection by the target and received by the jth receiver,

the real distance of a reflection path which represents that a signal transmitted by the ith transmitter is reflected by a target and then is received by the jth receiver is represented;

and 5: according to

And

constructing a constrained weighted least squares problem, described as:

where min () is the minimization function, s.t. denotes "constrained to … …", g is the optimization variable, g ═ u^T,||u-t₁||]^TU denotes a position variable of the object, g_(K+1)Denotes the K +1 th element in g, (b-Ag)^TW^-1(b-Ag) is an objective function,

b、b₁and b₂Are all introduced coefficient vectors, s₁Representing the true position, s, of the 1 st receiver in the reference frame_NRepresenting the true position, t, of the Nth receiver in the reference frame₂Representing the true position, t, of the 2 nd transmitter in the reference frame_MRepresenting the true position of the Mth transmitter in the reference coordinate system, r_2,1Distance measurement, r, representing the reflection path of the signal transmitted by the 2 nd transmitter reflected by the target and received by the 1 st receiver_2,NA distance measurement representing a reflected path of a signal transmitted by the 2 nd transmitter reflected by the target and received by the nth receiver,

A、A₁and A₂Are all an introduced coefficient matrix, W^-1Denotes the inverse of W, W being the introduced weight matrix,

F₁and F₂Are all an introduced intermediate coefficient matrix, F₁＝2HP₁，F₂＝2HP₂，H、P₁、P₂Are all introduced intermediate coefficient matrices, H ═ blkdiag (H)₁,H₂)，H₁、H₂Are all the introduced intermediate coefficient matrixes,

representing the true distance of the reflected path of the signal transmitted by the 1 st transmitter reflected by the target and received by the 1 st receiver,

representing the true distance of the reflected path of the signal transmitted by the 2 nd transmitter reflected by the target and received by the 1 st receiver,

representing the true distance of the reflected path of the signal transmitted by the 1 st transmitter reflected by the target and received by the nth receiver,

representing the true distance of the reflected path that the mth transmitter transmits after reflection by the target and is received by the nth receiver, blkdiag () is a block diagonal operation function, diag () is an element diagonal operation function,

I_Mrepresenting an identity matrix of dimension M x M,

a column vector of dimension M × 1, representing the 1 st element as 1 and the remaining elements as 0_MRepresenting all 1 vectors, symbols, of dimension Mx 1

Is a symbol of a kronecker product operation, I_NIdentity matrix with dimension NxN, 1_NRepresenting a full 1 vector of dimension N × 1;

step 6: the constrained weighted least squares problem is transformed into a standard generalized confidence domain subproblem, described as:

wherein, a is the introduced coefficient vector,

d is an introduced coefficient matrix, and D is blkdiag (I)_K,-1)，I_KAn identity matrix having a dimension of K × K;

and 7: adopting GTRS algorithm to solve the standard generalized confidence domain subproblem, the concrete process is as follows: according to the generalized confidence domain subproblem of the standard, a Lagrangian function is obtained, and is marked as L (g, lambda), and L (g, lambda) is (b-Ag)^TW^-1(b-Ag)+λ(g-a)^TD (g-a); then, for L (g, lambda) ═ b-Ag^TW^-1(b-Ag)+λ(g-a)^TD (g-a) obtains a partial derivative related to g, and the partial derivative related to g is 0, so that g (lambda) is equal to (A)^TW^-1A+λD)^-1(A^TW^-1b + λ Da); then, g (lambda) is changed to (A)^TW^-1A+λD)^-1(A^TW^-1b + λ Da) into the constraint (g-a) in the standard generalized confidence domain sub-problem^TD (g-a) ═ 0, and defines f (λ) ═ g (λ) -a)^TD (g (. lamda) -a); then searching out an optimal value lambda of lambda by utilizing dichotomy^*So that f (λ)^*) 0; then will lambda^*Substituting g (λ) ═ a^TW^-1A+λD)^-1(A^TW^-1b + λ Da), mixing g (λ)^*) As the optimal value g of g^*(ii) a Finally according to g^*An estimate of u is obtained and is denoted as u^*，u^*＝g^* _(1:K)(ii) a Wherein L () is a Lagrangian function representation form, g in L (g, λ), λ is a parameter of L (), λ represents a Lagrangian multiplier, g (λ) represents a function of g with respect to λ, f (λ) is an introduced intermediate function, u (λ) is a function of L (g, λ), u (u) is a function of L (g, λ) and u (g, λ) is a function of L (g, λ) and u (u) is a function of L (g, λ) and u (a) is a function of L (i) and a parameter of L^*＝g^* _(1:K)Wherein ═ is an assignment symbol, g^* _(1:K)Denotes g^*1 st to kth element in (1).

2. The method as claimed in claim 1, wherein τ in step 2 is a measure of the elliptical target location with clock synchronization error_i,jThe acquisition mode is as follows: the signal transmitted by the ith transmitter is time-stamped, and the signal transmitted by the ith transmitter is reflected by the target and then received by the jth receiverCalculating τ from the time stamp of the received signal recorded by the jth receiver_i,j。

3. An elliptical target location method with clock synchronization error as defined in claim 1 or 2, wherein in step 7, an optimal value λ of λ is searched by using dichotomy^*So that f (λ)^*) In 0, the dichotomy search interval is

Gamma is

The maximum eigenvalue of (d); wherein γ is an introduced intermediate symbol.

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