A method for elliptical target positioning with unknown signal propagation speed

Abstract

The present invention discloses an elliptical target positioning method with unknown signal propagation speed. The method constructs a time delay measurement model, regards a clock synchronization error in the time delay measurement model as noise, and equates the signal propagation speed to the form of a nominal value of the signal propagation speed and a residual of the signal propagation speed; divides the time delay measurement model after equivalent transformation into two sub-models, and constructs a constrained weighted least squares problem after squaring both sides of the equation and ignoring the second-order noise term; utilizes a semi-definite relaxation technique to relax the constrained weighted least squares problem into a convex semi-definite programming problem; adds two second-order cone constraints on the basis of the semi-definite programming problem to further tighten the semi-definite programming problem, thereby obtaining a tightened semi-definite programming problem; utilizes an interior point method to solve the tightened semi-definite programming problem, thereby obtaining an optimal estimate of the target position and an optimal estimate of the signal propagation speed; the method has the advantages of simple positioning problem and high target positioning accuracy.

Description

Technical Field

[0001] The present invention relates to a target positioning technology, and in particular to an elliptical target positioning method in which the signal propagation speed is unknown and clock synchronization error exists. Background Art

[0002] In recent decades, with the rapid development of wireless communication technology, target positioning has become a very hot field, and a large number of target positioning technologies have been derived from it. These technologies have been widely used in various fields, such as unmanned driving, disaster relief, precision guidance of missiles, etc. Therefore, it is very meaningful to develop a target positioning technology with high positioning accuracy and applicable to a variety of complex environments.

[0003] Generally speaking, accurate positioning of an unknown target can be achieved by collecting measurement information between sensors at known locations and the unknown target and then applying appropriate positioning algorithms. This measurement information typically includes time of arrival (TOA), time difference of arrival (TDOA), angle of arrival (AOA), received signal strength (RSS), and combinations thereof. Among positioning methods based on various measurement information, those based on time measurement information have been widely studied due to their higher positioning accuracy. The elliptical target positioning method is one such positioning method based on time measurement information. Compared to other positioning methods based on time measurement information, the elliptical target positioning method has the following advantages: For example, compared to positioning methods based on time of arrival (TOA), the elliptical target positioning method does not require cooperation between the target and the sensor, nor does it require time synchronization between sensors. Compared to positioning methods based on time difference of arrival (TDOA), the elliptical target positioning method achieves higher positioning accuracy when all sensors (including transmitters and receivers) are time synchronized. Furthermore, the elliptical target positioning method can be applied to scenarios where all sensors are time-asynchronous, as well as scenarios where some or all sensors are time-synchronized, thus offering greater flexibility. The basic principle of elliptical target positioning is that a transmitter transmits several signals, which are reflected by the target and received by the receiver. During this process, the transmitter and target, and the target and receiver, form several ellipses centered at the transmitter and receiver positions. The target can be located by finding the intersection of these ellipses. Based on this principle, the target's position can be determined in practice simply by using the collected time delay measurement data.

[0004] Generally, under ideal circumstances, when using sensors (including transmitters and receivers) to collect delay measurement data, only the measurement error of the measured value (delay or distance) needs to be considered, and there is no need to consider the clock synchronization error caused by the asynchrony of the sensor clocks, because at this time the sensors are assumed to be clock-synchronized. However, in reality, due to the influence of factors such as production process, temperature changes, environmental changes and aging of sensors (including transmitters and receivers), the clocks between them may be out of sync, which will cause the measurement value to become too large due to the clock synchronization error, thereby affecting the positioning accuracy of the target position. In addition, under normal circumstances, the signal propagation speed is assumed to be precisely known. For example, the propagation speed of electromagnetic waves in air or vacuum is 3×10 8 m / s. Therefore, when modeling related positioning problems, the impact of signal propagation speed is often not considered. Instead, the signal propagation speed is treated as a constant, which greatly simplifies the positioning problem. However, in some extremely harsh environments, the signal propagation speed is often inaccurate or even unknown. For example, underwater, the signal propagation speed is generally 1500m / s. Affected by factors such as water temperature, depth, and salinity, the signal propagation speed often becomes inaccurate or even completely unknown. If the signal propagation speed is still treated as a fixed constant during positioning, the positioning accuracy of the target position will be significantly affected. Therefore, to obtain a high-precision target position, appropriate methods must be used to deal with the problems caused by clock synchronization errors and unknown signal propagation speed.

[0005] Currently, existing methods for elliptical target positioning with unknown signal propagation speed and clock synchronization error typically consider jointly estimating the target position, clock synchronization error, and signal propagation speed. While this approach can achieve target positioning, it presents numerous problems. For one thing, estimating the clock synchronization error as a variable when its statistical distribution is known introduces additional variables, complicating the positioning problem. For another, directly estimating the signal propagation speed often makes it difficult to establish constraints between variables during algorithm design. This results in low positioning accuracy and susceptibility to noise. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to provide an elliptical target positioning method in which the signal propagation speed is unknown and clock synchronization error exists. The method treats the existing clock synchronization error as noise, greatly reducing the number of estimated variables and simplifying the positioning problem. The unknown signal propagation speed is equivalently expressed in the form of the known nominal value of the signal propagation speed and the unknown residual, which greatly facilitates the establishment of the constraint relationship between variables and further tightens the semi-definite programming problem formed, thereby improving the target positioning accuracy.

[0007] The technical solution adopted by the present invention to solve the above technical problems is: a method for locating an elliptical target with unknown signal propagation speed, characterized by comprising the following steps:

[0008] Step 1: In an underwater positioning scenario, establish a K-dimensional coordinate system as a reference coordinate system, and assume that there are M+N sensors with known real coordinate positions and 1 target with unknown real coordinate position. Assume that each sensor has a clock synchronization error. In this underwater positioning scenario, use M sensors as transmitters for transmitting signals, and the remaining N sensors as receivers for receiving signals. The real coordinate position of the i-th transmitter in the reference coordinate system is recorded as t i , the real coordinate position of the jth receiver in the reference coordinate system is recorded as s j , the real coordinate position of the target in the reference coordinate system is recorded as u o , the clock synchronization error of the i-th transmitter is recorded as The clock synchronization error of the jth receiver is recorded as Wherein, the value of K is 2 or 3, that is, the reference coordinate system is a two-dimensional coordinate system or a three-dimensional coordinate system, M ≥ 2, N ≥ 2, 1 ≤ i ≤ M, 1 ≤ j ≤ N;

[0009] Step 2: Collect the flight time of the signal emitted by the i-th transmitter and then reflected by the target and received by the j-th receiver, and record it as τ i,j , the flight time is the delay measurement value;

[0010] Step 3: According to τ i,j , construct the delay measurement model, which is described as: Then make ε i,j The signal transmitted by the i-th transmitter is regarded as the composite noise on the path received by the j-th receiver after being reflected by the target. The delay measurement model is re-described as: Among them, τ i,j It also represents the delay measurement value on the path from the signal transmitted by the i-th transmitter to the j-th receiver after being reflected by the target, c oIndicates the signal propagation speed, the symbol “|| ||” is the two-norm symbol, ||u o -t i || represents the true distance from the i-th transmitter to the target, ||u o -s j || represents the actual distance from the target to the jth receiver, n i,j represents the measurement noise on the path where the signal transmitted by the i-th transmitter is reflected by the target and received by the j-th receiver, ε i,j is the intermediate amount introduced;

[0011] Step 4: Replace c in the delay measurement model o Equivalently expressed as Then in Multiply both sides by The delay measurement model is equivalently transformed into:

[0012] Where c0 represents the nominal value of the signal propagation velocity, and β is a scaling factor introduced to avoid numerical problems in the solution process. Indicates the signal propagation speed c o The residual difference from the nominal value c0 of the signal propagation speed;

[0013] Step 5: Convert the delay measurement model obtained by equivalent transformation Split into the first sub-model and the second sub-model, the first sub-model is described as: The second sub-model is described as: Then square both sides of the equation for the first submodel and ignore its second-order noise term get: And square both sides of the equation for the second submodel and ignore its second-order noise term get: Among them, τ 1,j represents the time delay measurement value on the path where the signal transmitted by the first transmitter is reflected by the target and received by the jth receiver, t1 represents the real coordinate position of the first transmitter in the reference coordinate system, and ε 1,j It represents the composite noise on the path where the signal transmitted by the first transmitter is reflected by the target and then received by the jth receiver. The superscript “T” represents the transpose of a vector or matrix.

[0014] Step 6: According to and Construct a weighted least squares problem with constraints, described as: Among them, min() is the minimization function, st means "subject to...", y is the optimization variable, and u represents the target position variable, δ cIndicates the signal propagation speed c o The residual variable between the nominal value c0 of the signal propagation speed, is the variable δ c The square of y (K+2) represents the K+2th element in y, y (K+3) represents the K+3th element in y, y (K+4) represents the K+4th element in y, y (1:K) represents the column vector consisting of the 1st to Kth elements in y, (b-Ay) T W -1 (b-Ay) is the objective function, b, b1 and b2 are all introduced coefficient vectors, s1 represents the real coordinate position of the first receiver in the reference coordinate system, s N represents the true coordinate position of the Nth receiver in the reference coordinate system, t1 represents the true coordinate position of the first transmitter in the reference coordinate system, t2 represents the true coordinate position of the second transmitter in the reference coordinate system, and t M represents the true coordinate position of the Mth transmitter in the reference coordinate system, τ 1,1 It represents the time delay measurement value of the path from the signal transmitted by the first transmitter to the signal received by the first receiver after being reflected by the target, τ 1,N It represents the time delay measurement value of the path from the first transmitter to the Nth receiver after the signal is reflected by the target, τ 2,1 It represents the time delay measurement value of the path from the signal transmitted by the second transmitter to the signal received by the first receiver after being reflected by the target, τ 2,N It represents the time delay measurement value of the path from the second transmitter to the Nth receiver after the signal is reflected by the target, τ M,1 It represents the time delay measurement value of the path from the signal transmitted by the Mth transmitter to the first receiver after being reflected by the target, τ M,N It represents the delay measurement value on the path from the signal transmitted by the Mth transmitter to the signal received by the Nth receiver after being reflected by the target. A2(k,:)=[2(t1-t i ) T ,-2βc0(τ i,j -τ 1,j ) 2 ,-β 2 (τ i,j -τ 1,j ) 2 ,-2c0(τ i,j -τ 1,j ),-2β(τ i,j -τ 1,j)], and k=(i-2)+j, j=1,...,N, i=2,...,M, A, A1 and A2 are all introduced coefficient matrices, A1(j,:) represents the j-th row element of A1, A2(k,:) represents the k-th row element of A2, W -1 Represents the inverse of W, W is the introduced weight matrix, W = FQ ε F T , F is the introduced intermediate coefficient matrix, B is the introduced coefficient matrix, B1 and B2 are the introduced coefficient matrices, B1 = 2P1T1, B2 = 2P2T2, T1, T2, P1 and P2 are the introduced intermediate coefficient matrices, T1 = [I N ,0 N×(M-1)N ], P1=diag(||u o -s1||,...,||u o -s N ||), 1 (M-1) represents a vector of all 1s with a dimension of (M-1)×1, 0 N×(M-1)N Represents an all-zero matrix with dimension N×(M-1)N, I N Represents the identity matrix of dimension N×N, I (M-1)N Represents the identity matrix of dimension (M-1)N×(M-1)N, symbol is the Kronecker product operator, diag() is the element diagonal operation function, ||u o -s1|| represents the actual distance from the target to the first receiver, ||u o -s N || represents the actual distance from the target to the Nth receiver, ||u o -t2|| represents the true distance from the second transmitter to the target, ||u o -t M || represents the true distance from the Mth transmitter to the target, Q ε represents the covariance matrix of ε, ε represents the composite noise vector, ε=[ε 1,1 ,...,ε 1,N ,...,ε M,1 ,...,ε M,N ] T , ε 1,1 It represents the composite noise on the path where the signal transmitted by the first transmitter is reflected by the target and received by the first receiver, ε 1,N It represents the composite noise on the path where the signal transmitted by the first transmitter is reflected by the target and then received by the Nth receiver, ε M,1represents the composite noise on the path where the signal transmitted by the Mth transmitter is reflected by the target and received by the first receiver, ε M,N It represents the composite noise on the path where the signal transmitted by the Mth transmitter is reflected by the target and received by the Nth receiver. ε has a mean of 0 and a covariance matrix of Q. ε Gaussian distribution;

[0015] Step 7: Introduce a new auxiliary matrix variable Y, Y = yy T ; Then, after considering the relationship between the elements in y, the semi-positive relaxation technique is used to relax the constrained weighted least squares problem into a convex semi-positive programming problem, which can be described as: in, is the objective function of the semi-definite programming problem, Φ is the introduced intermediate coefficient matrix, tr{} is the trace operation of matrix elements, Y and y are both optimization variables of the semi-positive programming problem, Y (K+1,K+1) Represents the element corresponding to the K+1th row and K+1th column of Y, Y (K+3,K+3) Represents the element corresponding to the K+3th row and K+3th column of Y, Y (K+1,K+3) Represents the element corresponding to the K+1th row and K+3th column of Y, Y (K+1,K+4) Represents the element corresponding to the K+1th row and K+4th column of Y, Y (K+2,K+3) represents the element corresponding to the K+2th row and K+3th column of Y, Y (1:K,1:K) represents the submatrix consisting of the elements corresponding to the 1st to the Kth row and the 1st to the Kth column of Y, is a matrix of dimension (K+5)×(K+5), Representation matrix is positive semidefinite;

[0016] Step 8: Based on the semidefinite programming problem, add ||y (1:K) -t1||≤y (K+3) and ||Y (1:K,K+2) -y (K+2) t1||≤Y (K+1,K+4) Two second-order cone constraints are used to further tighten the semi-definite programming problem, and the tightened semi-definite programming problem is described as: Among them, Y (1:K,K+2) represents the vector consisting of the elements corresponding to the 1st to Kth rows and the K+2th columns of Y;

[0017] Step 9: Use the interior point method to solve the tightened semi-definite programming problem to obtain the optimal estimate of the target position and the optimal estimate of the signal propagation speed, which are denoted as u * and c * , Among them, y* is the best estimate of y, represents y * The 1st to Kth elements of represents y * The K+1th element of .

[0018] In step 2, τ i,j The acquisition method is as follows: the signal transmitted by the i-th transmitter has a timestamp, the signal transmitted by the i-th transmitter is reflected by the target and received by the j-th receiver, and τ is calculated based on the timestamp of the received signal recorded by the j-th receiver. i,j .

[0019] Compared with the prior art, the advantages of the present invention are:

[0020] 1) The method of the present invention treats the clock synchronization error of the sensor as noise, avoiding the estimation of the clock synchronization error, thereby greatly reducing the number of variables to be estimated, reducing the computational complexity, and improving the operating efficiency.

[0021] 2) The method of the present invention adopts a method similar to TDOA difference processing, and selects the first transmitter as the reference, so that after a clever transformation, the processed model only contains ||u o -t1||A two-norm variable, which once again reduces the number of variables to be estimated and further reduces the computational complexity.

[0022] 3) The method of the present invention innovatively expresses the unknown signal propagation speed equivalently in the form of the nominal value of the signal propagation speed and its residual. This change effectively establishes the constraint relationship between the unknown variables, thus laying a solid foundation for the subsequent construction of a tight semi-definite programming problem.

[0023] 4) The method of the present invention uses a "relaxation + tightening" strategy to deal with this problem, which further improves the positioning accuracy of the method of the present invention.

[0024] 5) The experimental results of the method of the present invention show that when the noise is not too large, it can achieve the target positioning accuracy of the Cramer-Rao lower bound (CRLB) and has stable performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 This is a block diagram of the overall implementation of the method of the present invention;

[0026] Figure 2 Schematic diagram of a system model with M transmitters and N receivers;

[0027] Figure 3The method of the present invention (semi-positive definite relaxation method) and the Cramer-Rao lower bound (CRLB) in σ △τ =0.001(s) when the logarithmic mean square error (MSE) of the target position is compared with the measurement noise power;

[0028] Figure 4 The method of the present invention (semi-positive definite relaxation method) and the Cramer-Rao lower bound (CRLB) are and σ △τ =0.001(s) using two transmitters to locate the target position, and the comparison chart of the logarithmic mean square error (MSE) as the number of receivers changes. DETAILED DESCRIPTION

[0029] The present invention will be described in further detail below with reference to the accompanying drawings and embodiments.

[0030] The present invention proposes an elliptical target positioning method with unknown signal propagation speed, and its overall implementation block diagram is as follows: Figure 1 As shown, it includes the following steps:

[0031] Step 1: In an underwater positioning scenario, establish a K-dimensional coordinate system as a reference coordinate system, and assume that there are M+N sensors with known real coordinate positions and 1 target with unknown real coordinate position. Assume that each sensor has a clock synchronization error. In this underwater positioning scenario, use M sensors as transmitters for transmitting signals, and the remaining N sensors as receivers for receiving signals. The real coordinate position of the i-th transmitter in the reference coordinate system is recorded as t i , the real coordinate position of the jth receiver in the reference coordinate system is recorded as s j , the real coordinate position of the target in the reference coordinate system is recorded as u o , the clock synchronization error of the i-th transmitter is recorded as The clock synchronization error of the jth receiver is recorded as Wherein, the value of K is 2 or 3, that is, the reference coordinate system is a two-dimensional coordinate system or a three-dimensional coordinate system, M ≥ 2, such as taking M = 2, N ≥ 2, such as taking N = 6, 1 ≤ i ≤ M, 1 ≤ j ≤ N; Figure 2 A schematic diagram of the system model with M transmitters and N receivers is given.

[0032] Step 2: Collect the flight time of the signal emitted by the i-th transmitter and then reflected by the target and received by the j-th receiver, and record it as τ i,j , and the flight time is the delay measurement value.

[0033] In this embodiment, in step 2, τ i,jThe acquisition method is as follows: the signal transmitted by the i-th transmitter has a timestamp, the signal transmitted by the i-th transmitter is reflected by the target and received by the j-th receiver, and τ is calculated based on the timestamp of the received signal recorded by the j-th receiver. i,j .

[0034] Step 3: According to τ i,j , construct the delay measurement model, which is described as: Then make ε i,j The signal transmitted by the i-th transmitter is regarded as the composite noise on the path received by the j-th receiver after being reflected by the target. The delay measurement model is re-described as: Among them, τ i,j It also represents the delay measurement value on the path from the signal transmitted by the i-th transmitter to the j-th receiver after being reflected by the target, c o Indicates the signal propagation speed, the symbol “|| ||” is the two-norm symbol, ||u o -t i || represents the true distance from the i-th transmitter to the target, ||u o -s j || represents the actual distance from the target to the jth receiver, n i,j represents the measurement noise on the path where the signal transmitted by the i-th transmitter is reflected by the target and received by the j-th receiver, ε i,j is the intermediate amount introduced.

[0035] The delay measurement value vector is recorded as τ, τ = [τ 1,1 ,…,τ 1,N ,…,τ M,1 ,…,τ M,N ] T , τ 1,1 It represents the time delay measurement value of the path from the signal transmitted by the first transmitter to the signal received by the first receiver after being reflected by the target, τ 1,N It represents the time delay measurement value of the path from the first transmitter to the Nth receiver after the signal is reflected by the target, τ M,1 It represents the time delay measurement value of the path from the signal transmitted by the Mth transmitter to the first receiver after being reflected by the target, τ M,N represents the delay measurement value on the path from the signal transmitted by the Mth transmitter to the signal received by the Nth receiver after being reflected by the target; the clock synchronization error vector is recorded as △τ, Indicates the clock synchronization error of the first transmitter. Indicates the clock synchronization error of the Mth transmitter, Indicates the clock synchronization error of the first receiver. Indicates the clock synchronization error of the Nth receiver, △τ obeys the mean of 0 and the covariance matrix is ​​Q △τ Gaussian distribution, Q △τ Denotes the covariance matrix of △τ; the measurement noise vector is recorded as n, n = [n 1,1 ,...,n 1,N ,...,n M,1 ,...,n M,N ] T , n 1,1 It represents the measurement noise on the path where the signal transmitted by the first transmitter is reflected by the target and received by the first receiver, n 1,N It represents the measurement noise on the path where the signal transmitted by the first transmitter is reflected by the target and received by the Nth receiver, n M,1 n represents the measurement noise on the path where the signal transmitted by the Mth transmitter is reflected by the target and received by the first receiver, M,N It represents the measurement noise on the path where the signal transmitted by the Mth transmitter is reflected by the target and received by the Nth receiver. n has a mean of 0 and a covariance matrix of Q n Gaussian distribution, Q n represents the covariance matrix of n; △τ and n are independent of each other.

[0036] Step 4: Replace c in the delay measurement model o Equivalently expressed as Then in Multiply both sides by The delay measurement model is equivalently transformed into:

[0037] Wherein, c0 represents the nominal value of the signal propagation speed, for example, the nominal value of the signal propagation speed underwater is 1500 m / s, β is a scaling factor introduced to avoid numerical problems in the solution process, and in this embodiment, β=1000. Indicates the signal propagation speed c o The residual from the nominal value c0 of the signal propagation velocity.

[0038] Step 5: Convert the delay measurement model obtained by equivalent transformation Split into the first sub-model and the second sub-model, the first sub-model is described as: The second sub-model is described as: Then square both sides of the equation for the first submodel and ignore its second-order noise term get: And square both sides of the equation for the second submodel and ignore its second-order noise term get: Among them, τ 1,jrepresents the time delay measurement value on the path where the signal transmitted by the first transmitter is reflected by the target and received by the jth receiver, t1 represents the real coordinate position of the first transmitter in the reference coordinate system, and ε 1,j It represents the composite noise on the path where the signal transmitted by the first transmitter is reflected by the target and then received by the jth receiver. The superscript “T” represents the transpose of a vector or matrix.

[0039] Step 6: According to and Construct a weighted least squares problem with constraints, described as: Among them, min() is the minimization function, st means "subject to...", y is the optimization variable, and u represents the target position variable, δ c Indicates the signal propagation speed c o The residual variable between the nominal value c0 of the signal propagation speed, is the variable δ c The square of y (K+2) represents the K+2th element in y, y (K+3) represents the K+3th element in y, y (K+4) represents the K+4th element in y, y (1:K) represents the column vector consisting of the 1st to Kth elements in y, (b-Ay) T W -1 (b-Ay) is the objective function, b, b1 and b2 are all introduced coefficient vectors, s1 represents the real coordinate position of the first receiver in the reference coordinate system, s N represents the true coordinate position of the Nth receiver in the reference coordinate system, t1 represents the true coordinate position of the first transmitter in the reference coordinate system, t2 represents the true coordinate position of the second transmitter in the reference coordinate system, and t M represents the true coordinate position of the Mth transmitter in the reference coordinate system, τ 1,1 It represents the time delay measurement value of the path from the signal transmitted by the first transmitter to the signal received by the first receiver after being reflected by the target, τ 1,N It represents the time delay measurement value of the path from the first transmitter to the Nth receiver after the signal is reflected by the target, τ 2,1 It represents the time delay measurement value of the path from the signal transmitted by the second transmitter to the signal received by the first receiver after being reflected by the target, τ 2,N It represents the time delay measurement value of the path from the second transmitter to the Nth receiver after the signal is reflected by the target, τ M,1 It represents the time delay measurement value of the path from the signal transmitted by the Mth transmitter to the first receiver after being reflected by the target, τ M,NIt represents the delay measurement value on the path from the signal transmitted by the Mth transmitter to the signal received by the Nth receiver after being reflected by the target. A2(k,:)=[2(t1-t i ) T ,-2βc0(τ i,j -τ 1,j ) 2 ,-β 2 (τ i,j -τ 1,j ) 2 ,-2c0(τ i,j -τ 1,j ),-2β(τ i,j -τ 1,j )], and k=(i-2)+j, j=1,...,N, i=2,...,M, A, A1 and A2 are all introduced coefficient matrices, A1(j,:) represents the j-th row element of A1, A2(k,:) represents the k-th row element of A2, W -1 Represents the inverse of W, W is the introduced weight matrix, W = FQ ε F T , F is the introduced intermediate coefficient matrix, B is the introduced coefficient matrix, B1 and B2 are the introduced coefficient matrices, B1 = 2P1T1, B2 = 2P2T2, T1, T2, P1 and P2 are the introduced intermediate coefficient matrices, T1 = [I N ,0 N×(M-1)N ], P1=diag(||u o -s1||,...,||u o -s N ||), 1 (M-1) represents a vector of all 1s with a dimension of (M-1)×1, 0 N×(M-1)N Represents an all-zero matrix with dimension N×(M-1)N, I N Represents the identity matrix of dimension N×N, I (M-1)N Represents the identity matrix of dimension (M-1)N×(M-1)N, symbol is the Kronecker product operator, diag() is the element diagonal operation function, ||u o -s1|| represents the actual distance from the target to the first receiver, ||u o -s N || represents the actual distance from the target to the Nth receiver, ||u o -t2|| represents the true distance from the second transmitter to the target, ||u o -t M|| represents the true distance from the Mth transmitter to the target, Q ε represents the covariance matrix of ε, ε represents the composite noise vector, ε=[ε 1,1 ,...,ε 1,N ,...,ε M,1 ,...,ε M,N ] T , ε 1,1 It represents the composite noise on the path where the signal transmitted by the first transmitter is reflected by the target and received by the first receiver, ε 1,N It represents the composite noise on the path where the signal transmitted by the first transmitter is reflected by the target and then received by the Nth receiver, ε M,1 represents the composite noise on the path where the signal transmitted by the Mth transmitter is reflected by the target and received by the first receiver, ε M,N It represents the composite noise on the path where the signal transmitted by the Mth transmitter is reflected by the target and received by the Nth receiver. ε has a mean of 0 and a covariance matrix of Q. ε Gaussian distribution.

[0040] Step 7: Introduce a new auxiliary matrix variable Y, Y = yy T ; Then, after considering the relationship between the elements in y, the semi-positive relaxation technique is used to relax the constrained weighted least squares problem into a convex semi-positive programming problem, which can be described as: in, is the objective function of the semi-definite programming problem, Φ is the introduced intermediate coefficient matrix, tr{} is the trace operation of matrix elements, Y and y are both optimization variables of the semi-positive programming problem, Y (K+1,K+1) Represents the element corresponding to the K+1th row and K+1th column of Y, Y (K+3,K+3) Represents the element corresponding to the K+3th row and K+3th column of Y, Y (K+1,K+3) Represents the element corresponding to the K+1th row and K+3th column of Y, Y (K+1,K+4) Represents the element corresponding to the K+1th row and K+4th column of Y, Y (K+2,K+3) represents the element corresponding to the K+2th row and K+3th column of Y, Y (1:K,1:K) represents the submatrix consisting of the elements corresponding to the 1st to the Kth row and the 1st to the Kth column of Y, is a matrix of dimension (K+5)×(K+5), Representation matrix is positive semidefinite.

[0041] Step 8: Based on the semidefinite programming problem, add ||y (1:K) -t1||≤y (K+3) and ||Y (1:K,K+2) -y(K+2) t1||≤Y (K+1,K+4) Two second-order cone constraints are used to further tighten the semi-definite programming problem, and the tightened semi-definite programming problem is described as: Among them, Y (1:K,K+2) Represents the vector consisting of the elements corresponding to rows 1 to K and columns K+2 of Y.

[0042] Step 9: Use the interior point method to solve the tightened semi-definite programming problem to obtain the optimal estimate of the target position and the optimal estimate of the signal propagation speed, which are denoted as u * and c * , Among them, y * is the best estimate of y, represents y * The 1st to Kth elements of represents y * The K+1th element of .

[0043] In order to verify the feasibility and effectiveness of the method of the present invention, a simulation test was carried out on the method of the present invention.

[0044] Assume that there are 2 transmitters and 6 receivers in a three-dimensional space, i.e., a three-dimensional coordinate system. Their real Cartesian coordinate positions in the three-dimensional space are [s x ,s y ,s z ] T is randomly generated, where s x ~(-3000,3000)m,s y ~(-3000,3000) meters and s z ~(1000,2000) meters. The target’s real Cartesian coordinate position in three-dimensional space is assumed to be [u x ,u y ,u z ] T , the real coordinate position is also randomly generated, where u x ~(-4000,4000) meters, u y ~(-4000,4000) meters and u z ~(3000,6000) meters. The nominal value of the signal propagation velocity c0 is set to 1500 meters per second, and the true value of the signal propagation velocity residual is The speed is randomly generated in the range [-100, 100] m / s, and the scaling factor β is set to 1000. The measurement noise is assumed to have a mean of 0 and a covariance matrix of The Gaussian noise of the sensor is also assumed to have a mean of 0 and a covariance matrix of Gaussian noise, where represents the variance of the measurement noise, represents the variance of clock synchronization error, I MN is the identity matrix of dimension MN×MN, I M+N is the identity matrix of dimension (M+N)×(M+N).

[0045] Based on the above parameter settings, simulation experiments tested the performance of the proposed method under varying measurement noise power and numbers of receivers. The logarithmic mean square error (MSE) describing positioning performance in both cases was calculated by running 1000 Monte Carlo experiments for each of 10 random scenarios. Furthermore, the Cramer-Rao lower bound (CRLB) was introduced for comparison. Figure 3 The comparison between the proposed method (semidefinite relaxation method) and the Cramer-Rao lower bound (CRLB) in σ is given. △τ = 0.001 seconds for the target position when the logarithmic mean square error (MSE) changes with the measurement noise power, Figure 4 The method of the present invention (semidefinite relaxation method) and the Cramer-Rao lower bound (CRLB) are given. and σ △τ = 0.001 seconds using two transmitters to locate the target position compared with the number of receivers. Figure 3 As can be seen, when the measured noise power is 10log 10 When the value of changes, the method of the present invention can achieve the positioning accuracy of the Cramer-Rao lower bound under the condition of small noise. Figure 4 It can be seen from the figure that the method of the present invention can achieve the positioning accuracy of the Cramer-Rao lower bound even when only 5 receivers are used. In addition, as the number of receivers N increases, the MSE of the method of the present invention decreases significantly, which means that when the number of receivers increases, the positioning accuracy of the method of the present invention is significantly improved.

[0046] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for locating an elliptical target with unknown signal propagation speed, characterized in that The following steps are involved: Step 1: In an underwater positioning scenario, establish a K-dimensional coordinate system as a reference coordinate system, and assume that there are M+N sensors with known real coordinate positions and 1 target with unknown real coordinate position. Assume that each sensor has a clock synchronization error. In this underwater positioning scenario, use M sensors as transmitters for transmitting signals, and the remaining N sensors as receivers for receiving signals. The real coordinate position of the i-th transmitter in the reference coordinate system is recorded as t i , the real coordinate position of the jth receiver in the reference coordinate system is recorded as s j , the real coordinate position of the target in the reference coordinate system is recorded as u o , the clock synchronization error of the i-th transmitter is recorded as The clock synchronization error of the jth receiver is recorded as Wherein, the value of K is 2 or 3, that is, the reference coordinate system is a two-dimensional coordinate system or a three-dimensional coordinate system, M ≥ 2, N ≥ 2, 1 ≤ i ≤ M, 1 ≤ j ≤ N; Step 2: Collect the flight time of the signal emitted by the i-th transmitter and then reflected by the target and received by the j-th receiver, and record it as τ i,j , the flight time is the delay measurement value; Step 3: According to τ i,j , construct the delay measurement model, which is described as: Then make ε i,j The signal transmitted by the i-th transmitter is regarded as the composite noise on the path received by the j-th receiver after being reflected by the target. The delay measurement model is re-described as: Among them, τ i,j It also represents the delay measurement value on the path from the signal transmitted by the i-th transmitter to the j-th receiver after being reflected by the target, c o Indicates the signal propagation speed, the symbol "|| ||" is the symbol of the two-norm, ||u o -t i || represents the true distance from the i-th transmitter to the target, ||u o -s j || represents the actual distance from the target to the jth receiver, n i,j represents the measurement noise on the path where the signal transmitted by the i-th transmitter is reflected by the target and received by the j-th receiver, ε i,j is the intermediate amount introduced; Step 4: Replace c in the delay measurement model o Equivalently expressed as Then Multiply both sides by The delay measurement model is equivalently transformed into: Where c0 represents the nominal value of the signal propagation velocity, and β is a scaling factor introduced to avoid numerical problems in the solution process. Indicates the signal propagation speed c o The residual difference from the nominal value c0 of the signal propagation speed; Step 5: Convert the delay measurement model obtained by equivalent transformation Split into the first sub-model and the second sub-model, the first sub-model is described as: The second sub-model is described as: Then square both sides of the equation for the first submodel and ignore its second-order noise term get: And square both sides of the equation for the second submodel and ignore its second-order noise term get: Among them, τ 1,j represents the time delay measurement value on the path where the signal transmitted by the first transmitter is reflected by the target and received by the jth receiver, t1 represents the real coordinate position of the first transmitter in the reference coordinate system, and ε 1,j It represents the composite noise on the path where the signal transmitted by the first transmitter is reflected by the target and then received by the jth receiver. The superscript "T" represents the transpose of a vector or matrix. Step 6: According to and Construct a weighted least squares problem with constraints, described as: Among them, min() is the minimization function, st means "subject to...", y is the optimization variable, and u represents the target position variable, δ c Indicates the signal propagation speed c o The residual variable between the nominal value c0 of the signal propagation speed, is the variable δ c The square of y (K+2) represents the K+2th element in y, y (K+3) represents the K+3th element in y, y (K+4) represents the K+4th element in y, y (1:K) represents the column vector consisting of the 1st to Kth elements in y, (b-Ay) T W -1 (b-Ay) is the objective function, b, b1 and b2 are all introduced coefficient vectors, s1 represents the real coordinate position of the first receiver in the reference coordinate system, s N represents the true coordinate position of the Nth receiver in the reference coordinate system, t1 represents the true coordinate position of the first transmitter in the reference coordinate system, t2 represents the true coordinate position of the second transmitter in the reference coordinate system, and t M represents the true coordinate position of the Mth transmitter in the reference coordinate system, τ 1,1 It represents the time delay measurement value of the path from the signal transmitted by the first transmitter to the signal received by the first receiver after being reflected by the target, τ 1,N It represents the time delay measurement value of the path from the first transmitter to the Nth receiver after the signal is reflected by the target, τ 2,1 It represents the time delay measurement value of the path from the signal transmitted by the second transmitter to the signal received by the first receiver after being reflected by the target, τ 2,N It represents the time delay measurement value of the path from the second transmitter to the Nth receiver after the signal is reflected by the target, τ M,1 It represents the time delay measurement value of the path from the signal transmitted by the Mth transmitter to the first receiver after being reflected by the target, τ M,N It represents the delay measurement value on the path from the signal transmitted by the Mth transmitter to the signal received by the Nth receiver after being reflected by the target. A2(k,:)=[2(t1-t i ) T ,-2βc0(τ i,j -τ 1,j ) 2 ,-β 2 (τ i,j -τ 1,j ) 2 ,-2c0(τ i,j -τ 1,j ),-2β(τ i,j -τ 1,j )], and k=(i-2)+j, j=1,...,N, i=2,...,M, A, A1 and A2 are all introduced coefficient matrices, A1(j,:) represents the j-th row element of A1, A2(k,:) represents the k-th row element of A2, W -1 Represents the inverse of W, W is the introduced weight matrix, W = FQ ε F T , F is the introduced intermediate coefficient matrix, B is the introduced coefficient matrix, B1 and B2 are the introduced coefficient matrices, B1 = 2P1T1, B2 = 2P2T2, T1, T2, P1 and P2 are the introduced intermediate coefficient matrices, T1 = [I N ,0 N×(M-1)N ], P1=diag(||u o -s1||,...,||u o -s N ||), 1 (M-1) represents a vector of all 1s with a dimension of (M-1)×1, 0 N×(M-1)N Represents an all-zero matrix with dimension N×(M-1)N, I N Represents the identity matrix of dimension N×N, I (M-1)N Represents the identity matrix of dimension (M-1)N×(M-1)N, symbol is the Kronecker product operator, diag() is the element diagonal operation function, ||u o -s1|| represents the actual distance from the target to the first receiver, ||u o -s N || represents the actual distance from the target to the Nth receiver, ||u o -t2|| represents the true distance from the second transmitter to the target, ||u o -t M || represents the true distance from the Mth transmitter to the target, Q ε represents the covariance matrix of ε, ε represents the composite noise vector, ε=[ε 1,1 ,...,ε 1,N ,...,ε M,1 ,...,ε M,N ] T , ε 1,1 It represents the composite noise on the path where the signal transmitted by the first transmitter is reflected by the target and then received by the first receiver, ε 1,N It represents the composite noise on the path where the signal transmitted by the first transmitter is reflected by the target and received by the Nth receiver, ε M,1 represents the composite noise on the path where the signal transmitted by the Mth transmitter is reflected by the target and received by the first receiver, ε M,N It represents the composite noise on the path where the signal transmitted by the Mth transmitter is reflected by the target and received by the Nth receiver. ε has a mean of 0 and a covariance matrix of Q. ε Gaussian distribution; Step 7: Introduce a new auxiliary matrix variable Y, Y = yy T ; Then, after considering the relationship between the elements in y, the semi-definite relaxation technique is used to relax the constrained weighted least squares problem into a convex semi-definite programming problem, which can be described as: in, is the objective function of the semi-definite programming problem, Φ is the introduced intermediate coefficient matrix, tr{} is the trace operation of matrix elements, Y and y are both optimization variables of the semi-positive programming problem, Y (K+1,K+1) Represents the element corresponding to the K+1th row and K+1th column of Y, Y (K+3,K+3) Represents the element corresponding to the K+3th row and K+3th column of Y, Y (K+1,K+3) Represents the element corresponding to the K+1th row and K+3th column of Y, Y (K+1,K+4) Represents the element corresponding to the K+1th row and K+4th column of Y, Y (K+2,K+3) represents the element corresponding to the K+2th row and K+3th column of Y, Y (1:K,1:K) represents the submatrix consisting of the elements corresponding to the 1st to the Kth row and the 1st to the Kth column of Y, is a matrix of dimension (K+5)×(K+5), Representation matrix is positive semidefinite; Step 8: Based on the semidefinite programming problem, add ||y (1:K) -t1||≤y (K+3) and ||Y (1:K,K+2) -y (K+2) t1||≤Y (K+1,K+4) Two second-order cone constraints are used to further tighten the semi-definite programming problem, and the tightened semi-definite programming problem is described as: Among them, Y (1:K,K+2) represents the vector consisting of the elements corresponding to the 1st to Kth rows and the K+2th columns of Y; Step 9: Use the interior point method to solve the tightened semi-definite programming problem to obtain the optimal estimate of the target position and the optimal estimate of the signal propagation speed, which are denoted as u * and c * , Among them, y * is the best estimate of y, represents y * The 1st to Kth elements of represents y * The K+1th element of .

2. The method for locating an elliptical target with unknown signal propagation speed according to claim 1, characterized in that In step 2, τ i,j The acquisition method is as follows: the signal transmitted by the i-th transmitter has a timestamp, the signal transmitted by the i-th transmitter is reflected by the target and received by the j-th receiver, and τ is calculated based on the timestamp of the received signal recorded by the j-th receiver. i,j .

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