阅读说明：本技术 非同步网络中基于toa的鲁棒非视距目标自定位方法 (Robust non-line-of-sight target self-positioning method based on TOA in asynchronous network ) 是由 王刚 朱伟辰 于 2019-11-08 设计创作，主要内容包括：本发明公开了一种非同步网络中基于TOA的鲁棒非视距目标自定位方法,所有锚节点在同一起始传输时间向目标源发送信号,目标源采集各个锚节点发送的信号的到达时间；构建每个锚节点对应的TOA测量模型；指定参考锚节点后将TOA测量模型变换为TDOA测量模型；考虑目标源的时钟偏差、非视距误差的上界,构造最坏情况下的鲁棒最小二乘问题；使用三角不等式得到鲁棒最小二乘问题；将鲁棒最小二乘问题以上镜图的形式进行等价表述；将上镜图的形式放松为初步半正定规划问题；在初步半正定规划问题中添加约束条件得到最终半正定规划问题；求解最终半正定规划问题得到目标源的位置估计值；优点是对目标源和参考路径的非视距误差联合估计,提高了目标定位精度。(The invention discloses a robust non-line-of-sight target self-positioning method based on TOA in an asynchronous network, wherein all anchor nodes send signals to a target source at the same initial transmission time, and the target source collects the arrival time of the signals sent by each anchor node; constructing a TOA measurement model corresponding to each anchor node; after a reference anchor node is appointed, converting the TOA measurement model into a TDOA measurement model; considering the clock deviation of a target source and the upper bound of a non-line-of-sight error, and constructing a robust least square problem under the worst condition; obtaining a robust least square problem by using a triangle inequality; equivalently expressing the robust least square problem in the form of an upper mirror image; relaxing the form of the upper mirror image into a preliminary semi-positive planning problem; adding constraint conditions in the preliminary semi-positive planning problem to obtain a final semi-positive planning problem; solving the final semi-positive definite programming problem to obtain a position estimation value of the target source; the method has the advantages that the non-line-of-sight errors of the target source and the reference path are jointly estimated, and the target positioning precision is improved.)

1. A TOA-based robust non-line-of-sight target self-positioning method in a non-synchronous network is characterized by comprising the following steps:

step 1: selecting a k-dimensional positioning scene, and setting that N +1 anchor nodes and a target source exist in an asynchronous sensor network; recording the coordinate position of the anchor node with the number of i in the k-dimensional positioning scene as s_iRecording the coordinate position of the target source in the k-dimensional positioning scene as x; where k is 2 or 3, i is a positive integer, where 0. ltoreq. i.ltoreq.N, N.gtoreq.2, s_iAnd x are both k-dimensional column vectors;

step 2: in the asynchronous sensor network, all anchor nodes send signals to a target source at the same initial transmission time, and the target source acquires the arrival time of the signals sent by all the anchor nodes; representing the arrival time of the signal sent by each anchor node acquired by the target source by using a TOA measurement model, wherein the TOA measurement model of the arrival time of the signal sent by the anchor node with the number i acquired by the target source is as follows:where 0. ltoreq. i.ltoreq.N, t_iThe TOA measured value of a signal which is acquired by a target source and is sent by an anchor node with the number i, omega represents the clock deviation of the target source, theta represents the clock drift of the target source, and T represents₀Representing the initial transmission time of all anchor nodes sending signals to a target source, the symbol "| | |" is a Euclidean norm symbol, c represents the speed of light, w represents the speed of light_iRepresenting a non-negative non-line-of-sight error on the signal propagation path between the anchor node numbered i and the target source,

and step 3: designating the anchor node with the number of 0 as a reference anchor node, and subtracting the TOA measured value corresponding to the reference anchor node from the TOA measured value corresponding to each anchor node with the number of 1 to N to obtain the corresponding TDOA measured value; then, deriving a TDOA measurement model of TDOA measurement values corresponding to anchor nodes numbered from 1 to N from TOA measurement models of arrival times of signals transmitted by all anchor nodes acquired by the target source, where the TDOA measurement model of the TDOA measurement value corresponding to the anchor node numbered as i acquired by the target source is:

and 4, step 4: let d_i＝c×(t_i-t₀)、

And 5: extracting w according to the TOA measured value of the signal transmitted by the reference anchor node collected by the target source₀Is upper bound of (D), is noted

step 6: order to

and 7: using the triangle inequalityAnd processing a maximized part in the robust least square problem under the worst condition, wherein the right side of the triangle inequality satisfies the following equation:using the equation to the rightAnd replacing a maximized part in the robust least square problem under the worst condition by the result to obtain the robust least square problem, wherein the robust least square problem is described as follows:

and 8: utilizing a semi-positive definite relaxation technology to relax the form of an upper mirror image of the robust least square problem into a preliminary semi-positive definite planning problem, which is described as follows:

and step 9: adding second-order cone constraint conditions in the preliminary semi-positive definite programming problem

step 10: solving the final semi-positive definite planning problem by using interior point method software to obtain an estimated value of the coordinate position x of the target source in the k-dimensional positioning scene, and recording the estimated value as the estimated value

Technical Field

The invention relates to a target self-positioning method, in particular to a robust non-line-of-sight (TOA) -based target self-positioning method in an asynchronous network (namely a sensor network under an asynchronous condition).

Background

In recent years, various positioning techniques have played an important role in people's daily life. Applications of target positioning include navigation, target tracking, rescue, aerospace, and the like. The Time-of-Arrival (TOA) based target positioning method has high positioning accuracy and is very common in practical environments.

The problem of time synchronization in sensor networks under asynchronous conditions has a non-negligible effect on the accuracy of target positioning, and has been studied extensively in recent years. In recent years, researchers have proposed an effective Fractional Programming (FP) method for estimating the location of a target source in an asynchronous sensor network, but a good target positioning effect cannot be obtained in an environment with many obstructions.

The signal may be blocked by some obstacles during propagation, and this phenomenon is called Non-Line-of-Sight (NLOS) signal propagation. Non-line-of-sight signal propagation can result in NLOS errors in TOA measurements of the signal. Various studies have shown that NLOS errors are typically much larger than measurement noise, and such errors have a significant negative impact on target localization performance. Some existing target location methods use the distribution of NLOS errors or statistical information to improve the location accuracy, however, such statistical information is difficult to obtain in a time-varying practical environment. In the robust second-order cone planning and robust semi-positive planning methods, neither path state information nor statistical information of NLOS errors are required, they only need an upper bound of NLOS errors which are easily obtained in an actual environment, and the performance superior to that of the previous non-robust methods is shown. However, the above robust second-order cone planning and robust semi-positive planning methods study the target location problem based on the condition that the sensor network is completely synchronized, and they do not have the function of target self-location. Therefore, it can be seen that the target of NLOS error in the asynchronous sensor network is derived from the positioning problem, which is a new research field and deserves research.

Disclosure of Invention

The technical problem to be solved by the invention is to provide a TOA-based robust non-line-of-sight target self-positioning method in an asynchronous network, which jointly estimates non-line-of-sight errors of a target source and a reference path and effectively improves the target positioning precision.

The technical scheme adopted by the invention for solving the technical problems is as follows: a TOA-based robust non-line-of-sight target self-positioning method in a non-synchronous network is characterized by comprising the following steps:

step 1: selecting a k-dimensional positioning scene, and setting that N +1 anchor nodes and a target source exist in an asynchronous sensor network; recording the coordinate position of the anchor node with the number of i in the k-dimensional positioning scene as s_iRecording the coordinate position of the target source in the k-dimensional positioning scene as x; where k is 2 or 3, i is a positive integer, where 0. ltoreq. i.ltoreq.N, N.gtoreq.2, s_iAnd x are both k-dimensional column vectors;

step 2: in the asynchronous sensor network, all anchor nodes send signals to a target source at the same initial transmission time, and the target source acquires the arrival time of the signals sent by all the anchor nodes; representing the arrival time of the signal sent by each anchor node acquired by the target source by using a TOA measurement model, wherein the TOA measurement model of the arrival time of the signal sent by the anchor node with the number i acquired by the target source is as follows:

where 0. ltoreq. i.ltoreq.N, t_iThe TOA measured value of a signal which is acquired by a target source and is sent by an anchor node with the number i, omega represents the clock deviation of the target source, theta represents the clock drift of the target source, and T represents₀Representing the initial transmission time of all anchor nodes sending signals to a target source, the symbol "| | |" is a Euclidean norm symbol, c represents the speed of light, w represents the speed of light_iRepresenting a non-negative non-line-of-sight error on the signal propagation path between the anchor node numbered i and the target source,

representing the measurement noise on the signal propagation path between the anchor node numbered i and the target source,

obedience mean is zero and variance is

The distribution of the gaussian component of (a) is,

and step 3: designating the anchor node with the number of 0 as a reference anchor node, and subtracting the TOA measured value corresponding to the reference anchor node from the TOA measured value corresponding to each anchor node with the number of 1 to N to obtain the corresponding TDOA measured value; then, deriving a TDOA measurement model of TDOA measurement values corresponding to anchor nodes numbered from 1 to N from TOA measurement models of arrival times of signals transmitted by all anchor nodes acquired by the target source, where the TDOA measurement model of the TDOA measurement value corresponding to the anchor node numbered as i acquired by the target source is:

where 1. ltoreq. i.ltoreq.N, t₀TOA measurement, s, representing a signal transmitted by a reference anchor node acquired by a target source₀Representing the coordinate position, w, of a reference anchor node in a k-dimensional positioning scenario₀Representing a non-negative non-line-of-sight error on a signal propagation path between the reference anchor node and the target source,

representing measurement noise on the signal propagation path between the reference anchor node and the target source,

obedience mean is zero and variance is

(ii) a gaussian distribution of;

and 4, step 4: let d_i＝c×(t_i-t₀)、

Will be provided withChange to d_i＝ω×(||x-s_i||-||x-s₀||+w_i-w₀+n_i) (ii) a Then, let ω be 1+ δ, δ be a random variable much smaller than 1 and obey the interval(-δ_max,δ_max) Is uniformly distributed so that

Is established and let | | | x-s_i||＝r_i，||x-s₀||＝r₀D is mixing_i＝ω×(||x-s_i||-||x-s₀||+w_i-w₀+n_i) Change to d_i≈r_i-r₀+w_i-w₀+n_i+δd_i(ii) a Then let ∈ e_i＝n_i+δd_iD is mixing_i≈r_i-r₀+w_i-w₀+n_i+δd_iChange to d_i≈r_i-r₀+w_i-w₀+∈_i(ii) a Wherein, i is not less than 1 and not more than N, d_i、n_iFor the introduced intermediate variable, n_iObeying a gaussian distribution with mean zero and covariance matrix Q,

representing diagonal elements of

The diagonal matrix of (a) is,

representing measurement noise on the signal propagation path between anchor node numbered 1 and the target sourceThe variance of the gaussian distribution to which it is submitted,

representing measurement noise on the signal propagation path between anchor node numbered N and the target source

The variance of the gaussian distribution to which it is submitted,

representing measurement noise on a signal propagation path between a reference anchor node and a target sourceVariance of obeyed Gaussian distribution, 1_N×NRepresenting an N x N dimensional matrix of elements all 1, r_i、r₀、∈_iFor the introduction of intermediate variables, δ_maxIs a known constant, δ_max＞0；

And 5: extracting w according to the TOA measured value of the signal transmitted by the reference anchor node collected by the target source₀Is upper bound of (D), is noted

And extracting w according to TOA measured values of signals which are acquired by a target source and transmitted by each anchor node with the number from 1 to N_iIs upper bound of (D), is noted

Then to d_i≈r_i-r₀+w_i-w₀+∈_iEquation of (2) is equally subtracted on both sides

To obtainThen order

Will be provided withIs rewritten into

Then will be

R in_iMove to the left of the equation and square the two sides to obtain

Second order term in the formula

Has been omitted; wherein, i is more than or equal to 1 and less than or equal to N,

are all intermediate variables introduced, ()^TRepresenting transposing the vector;

step 6: order to

And according to

Constructing a worst case robust least squares problem, described as:

wherein, i is not less than 1 and not more than N, a_i、b_i、

y are all intermediate variables introduced, a_iAnd y are column vectors of dimension N + k +4, 0_1×(i-1)A vector of dimension 1X (i-1) representing elements all 0, 0_1×(N-i+2)A vector of dimension 1 × (N-i +2) representing elements all 0, r₁And r_NIs given by | | x-s_i||＝r_iCalculated, the symbol "|" is an absolute value symbol, "s.t." means "Is constrained to … … ",

represents satisfaction

Under the condition of

The maximum value of (a) is,

representing the solution obtained under the condition that y is taken as a variable

Minimum value of (d);

and 7: using the triangle inequality

And processing a maximized part in the robust least square problem under the worst condition, wherein the right side of the triangle inequality satisfies the following equation:

and replacing the maximized part in the robust least square problem under the worst condition by using the right result of the equation to obtain the robust least square problem, which is described as follows:

then, the robust least square problem is equivalently expressed in the form of an upper mirror image, and the robust least square problem is expressed as follows:

wherein the content of the first and second substances,

represents satisfactionUnder the condition ofThe maximum value of (a) is,

representing the solution obtained under the condition that y is taken as a variable

The minimum value of (a) is determined,

representing the solution obtained under the condition that y and η are used as variables

η is an intermediate variable introduced, η ═ η₁,…,η_N]^T，η₁Is the 1 st element in η, η_NIs the Nth element in η, η_iη for the ith element;

and 8: utilizing a semi-positive definite relaxation technology to relax the form of an upper mirror image of the robust least square problem into a preliminary semi-positive definite planning problem, which is described as follows:wherein the content of the first and second substances,

indicating that Y, η, Y are variables

Y, C is minimized_i、c_i、v_i、

ψ_iFor the intermediate variables introduced, Y ═ yy^T，

v_i＝[0_1×(N+k),1,1,0,0]^T，0_1×(N+k)A vector of dimension 1 x (N + k) representing elements all 0,

Y_(1:k,1:k)a submatrix consisting of 1 st row to k th row and 1 st column to k column representing Y_(1:k)A sub-vector consisting of the 1 st to k th elements of y, tr () is a trace of the matrix, and the symbol "≧" is a semidefinite symbol, a]^TRepresenting transposing the vector;

and step 9: adding second-order cone constraint conditions in the preliminary semi-positive definite programming problemAccording to known conditionsAnd adding constraint conditions in the preliminary semi-positive planning problem by using the internal relation between the Y and the Y elementsY_{(N+k+2,N+k+2)}＝y_(N+k+3)、Y_{(N+k+1,N+k+2)}＝y_(N+k+4)(ii) a Adding an additional constraint condition formed by applying a reconstruction linearization technique in the preliminary semi-positive planning problem

And d_iy_(N+k+2)-Y_(2+i,N+k+2)+y_(N+k+4)+y_(N+k+3)Not less than 0; and obtaining a final semi-positive definite planning problem after adding the constraint conditions, wherein the description is as follows:

wherein, the symbol

Is an equivalent symbol, y_(N+k+1)N + k +1 th element representing y, y_(N+k+2)N + k +2 th element representing y, y_(N+k+3)N + k +3 th element representing y, y_(N+k+4)N + k +4 th element representing y, y_(k+i)The k + i th element representing y, y_(2+i)2+ i th element representing Y, Y_{(N+k+1,N+k+2)}Elements of row N + k +1 and column N + k +2 of Y, Y_{(N+k+2,N+k+2)}Elements of row N + k +2 and column N + k +2 of Y, Y_(2+i,N+k+2)An element representing the 2+ i th row of Y and the N + k +2 th column;

step 10: solving the final semi-positive definite planning problem by using interior point method software to obtain an estimated value of the coordinate position x of the target source in the k-dimensional positioning scene, and recording the estimated value as the estimated value

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

1) in the method, a plurality of anchor nodes send signals to the target source at the same unknown time, the target source measures the TOA by depending on the received signals, and the target source does not send signals to the anchor nodes, so that the energy of the target source is saved, the battery life of the target source is prolonged, and the self-positioning function of the target source is realized.

2) In the method, the TOA measurement model is converted into the TDOA measurement model, so that part of unknown parameters, namely initial transmission time and clock drift, are eliminated, and reasonable approximation is further utilized to combine a clock deviation item of a target source and a measurement noise item, so that the problem of parameter interference caused by time deviation in an asynchronous sensor network is solved, and the target positioning precision is improved.

3) In the method, the problem of overlarge robust upper bound range of the non-line-of-sight error is analyzed and solved, the non-line-of-sight error of the target source and the reference path is jointly estimated, the upper bound of the non-line-of-sight error is reduced to a reasonable range, and the target positioning precision is further improved.

Drawings

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

FIG. 2 shows the results when N is 4, sigma is 1.6,

Comparing the performance of the method with the performance of the existing fractional programming, the existing robust second-order cone programming and the existing robust semi-definite programming;

FIG. 3 shows the results when N is 4, sigma is 1.6,

Comparing the performance of the method with the performance of the existing fractional programming, the existing robust second-order cone programming and the existing robust semi-definite programming;

FIG. 4 shows the increase of N from 4 to 8, sigma being 1.6,

The method of the invention is compared with the performances of the existing fractional programming, the existing robust second-order cone programming and the existing robust semi-definite programming.

Detailed Description

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

The invention provides a TOA-based robust non-line-of-sight target self-positioning method in an asynchronous network, the general implementation flow diagram of which is shown in figure 1, and the method comprises the following steps:

step 1: selecting a k-dimensional positioning scene, and setting that N +1 anchor nodes and a target source exist in an asynchronous sensor network; recording the coordinate position of the anchor node with the number of i in the k-dimensional positioning scene as s_iRecording the coordinate position of the target source in the k-dimensional positioning scene as x; wherein k is 2 or 3, i is a positive integer, i is 0 ≦ N, N ≧ 2, such as N ≦ 4, s_iAnd x are both k-dimensional column vectors.

Step 2: in an unsynchronized sensor network, all anchor nodes are co-locatedThe initial transmission time sends a signal to a target source, and the target source acquires the arrival time of the signal sent by each anchor node; representing the arrival time of the signal sent by each anchor node acquired by the target source by using a TOA measurement model, wherein the TOA measurement model of the arrival time of the signal sent by the anchor node with the number i acquired by the target source is as follows:

where 0. ltoreq. i.ltoreq.N, t_iThe TOA measured value of a signal which is acquired by a target source and is sent by an anchor node with the number i, omega represents the clock deviation of the target source, theta represents the clock drift of the target source, and T represents₀Representing the initial transmission time of all anchor nodes sending signals to a target source, the symbol "| | |" is a Euclidean norm symbol, c represents the speed of light, w represents the speed of light_iRepresenting a non-negative non-line-of-sight error on the signal propagation path between the anchor node numbered i and the target source,

representing the measurement noise on the signal propagation path between the anchor node numbered i and the target source,

obedience mean is zero and variance is

The distribution of the gaussian component of (a) is,the value of (A) is set by itself, in this embodiment

And step 3: in order to eliminate excessive unknown variables in the TOA measurement model, an anchor node with the number of 0 is designated as a reference anchor node, and the TOA measurement value corresponding to the reference anchor node is subtracted from the TOA measurement value corresponding to each anchor node with the numbers of 1 to N to obtain a corresponding TDOA (time difference of arrival) measurement value; all anchors then acquired by the target sourceDeriving a TDOA measurement model of TDOA measured values corresponding to anchor nodes with the numbers from 1 to N by using a TOA measurement model of the arrival time of signals sent by the nodes, wherein the TDOA measurement model of the TDOA measured values corresponding to the anchor nodes with the numbers i collected by a target source is as follows:

where 1. ltoreq. i.ltoreq.N, t₀TOA measurement, s, representing a signal transmitted by a reference anchor node acquired by a target source₀Representing the coordinate position, w, of a reference anchor node in a k-dimensional positioning scenario₀Representing a non-negative non-line-of-sight error on a signal propagation path between the reference anchor node and the target source,

representing measurement noise on the signal propagation path between the reference anchor node and the target source,

obedience mean is zero and variance is

The distribution of the gaussian component of (a) is,

the value of (2) is set by itself.

And 4, step 4: let d_i＝c×(t_i-t₀)、Will be provided withChange to d_i＝ω×(||x-s_i||-||x-s₀||+w_i-w₀+n_i) (ii) a Since the value of the clock deviation ω of the target source is usually about 1, then it is set that ω is 1+ δ, and δ is a random variable much smaller than 1 and obeys a range of (- δ)_max,δ_max) Is uniformly distributed so that

Is established and let | | | x-s_i||＝r_i，||x-s₀||＝r₀D is mixing_i＝ω×(||x-s_i||-||x-s₀||+w_i-w₀+n_i) Change to d_i≈r_i-r₀+w_i-w₀+n_i+δd_i(ii) a Then let ∈ e_i＝n_i+δd_iD is mixing_i≈r_i-r₀+w_i-w₀+n_i+δd_iChange to d_i≈r_i-r₀+w_i-w₀+∈_i(ii) a Wherein, i is not less than 1 and not more than N, d_i、n_iFor the introduced intermediate variable, n_iObeying a gaussian distribution with mean zero and covariance matrix Q, representing diagonal elements ofThe diagonal matrix of (a) is,representing measurement noise on the signal propagation path between anchor node numbered 1 and the target source

The variance of the gaussian distribution to which it is submitted,

representing measurement noise on the signal propagation path between anchor node numbered N and the target source

The variance of the gaussian distribution to which it is submitted,representing measurement noise on a signal propagation path between a reference anchor node and a target source

Variance of obeyed Gaussian distribution, 1_N×NRepresenting an N x N dimensional matrix of elements all 1, r_i、r₀、∈_iFor the introduction of intermediate variables, δ_maxIs a known constant, δ_max> 0, in this example taken from δ_max＝0.005。

And 5: extracting w according to the TOA measured value of the signal transmitted by the reference anchor node collected by the target source₀Is upper bound of (D), is noted

And extracting w according to TOA measured values of signals which are acquired by a target source and transmitted by each anchor node with the number from 1 to N_iIs upper bound of (D), is noted

In order to solve the problem that the value range of the non-line-of-sight error is unreasonable, the unknown w is considered₀Jointly estimating the coordinate position of the variable and the target source, and then carrying out d_i≈r_i-r₀+w_i-w₀+∈_iEquation of (2) is equally subtracted on both sidesTo obtainThen order

Will be provided with

Is rewritten intoThen will be

R in_iMove to the left of the equation and square the two sides to obtain

Second order term in the formula

Has been omitted because

Is much smaller thanWherein, i is more than or equal to 1 and less than or equal to N,

are all intermediate variables introduced, ()^TIndicating transposing the vector.

Step 6: order to

And according to

Constructing a worst case robust least squares problem, described as:

wherein, i is not less than 1 and not more than N, a_i、b_i、

y are all intermediate variables introduced, a_iAnd y are column vectors of dimension N + k +4, 0_1×(i-1)A vector of dimension 1X (i-1) representing elements all 0, 0_1×(N-i+2)A vector of dimension 1 × (N-i +2) representing elements all 0, r₁And r_NIs given by | | x-s_i||＝r_iCalculated, the symbol "|" is an absolute value symbol, "s.t." means "constrained to … …",

represents satisfaction

Under the condition of

The maximum value of (a) is,

representing the solution obtained under the condition that y is taken as a variable

Is measured.

And 7: using the triangle inequality

And processing a maximized part in the robust least square problem under the worst condition, wherein the right side of the triangle inequality satisfies the following equation:

and replacing the maximized part in the robust least square problem under the worst condition by using the right result of the equation to obtain the robust least square problem, which is described as follows:

then, the robust least square problem is equivalently expressed in the form of an upper mirror image, and the robust least square problem is expressed as follows:

wherein the content of the first and second substances,

represents satisfaction

Under the condition of

The maximum value of (a) is,

representing the solution obtained under the condition that y is taken as a variableThe minimum value of (a) is determined,

representing the solution obtained under the condition that y and η are used as variables

η is an intermediate variable introduced, η ═ η₁,…,η_N]^T，η₁Is the 1 st element in η, η_NIs the Nth element in η, η_iWhich is the ith element in η.

And 8: since the form of the upper mirror image of the robust least squares problem is still a non-convex problem and is difficult to solve, the form of the upper mirror image of the robust least squares problem is relaxed into a preliminary semi-positive definite programming (SDP) problem by using a semi-positive definite relaxation technique, which is described as:

wherein the content of the first and second substances,indicating that Y, η, Y are variables

Y, C is minimized_i、c_i、v_i、

ψ_iFor the intermediate variables introduced, Y ═ yy^T，

v_i＝[0_1×(N+k),1,1,0,0]^T，0_1×(N+k)A vector of dimension 1 x (N + k) representing elements all 0,

Y_(1:k,1:k)a submatrix consisting of 1 st row to k th row and 1 st column to k column representing Y_(1:k)A sub-vector consisting of the 1 st to k th elements of y, tr () is a trace of the matrix, and the symbol "≧" is a semidefinite symbol, a]^TIndicating transposing the vector.

And step 9: the relation among the optimization variables is mined and combined with known conditions, and some constraint conditions can be added to further improve the accuracy of the problem to be solved, so that second-order cone constraint conditions are added to the preliminary semi-positive definite programming problem

According to known conditions

And adding constraint conditions in the preliminary semi-positive planning problem by using the internal relation between the Y and the Y elements

Y_{(N+k+2,N+k+2)}＝y_(N+k+3)、Y_{(N+k+1,N+k+2)}＝y_(N+k+4)(ii) a Since the NLOS error is much larger than the measurement noise and δ is small, the condition | ∈ is_i|＜＜w_iIs established, and thus can be in the preliminary stageAn additional constraint condition formed by applying a reconstruction-linearization technique (RLT) is added in the semi-definite programming problem

And d_iy_(N+k+2)-Y_(2+i,N+k+2)+y_(N+k+4)+y_(N+k+3)Not less than 0; and obtaining a final semi-positive definite planning problem after adding the constraint conditions, wherein the description is as follows:

wherein, the symbol

Is an equivalent symbol, y_(N+k+1)N + k +1 th element representing y, y_(N+k+2)N + k +2 th element representing y, y_(N+k+3)N + k +3 th element representing y, y_(N+k+4)N + k +4 th element representing y, y_(k+i)The k + i th element representing y, y_(2+i)2+ i th element representing Y, Y_{(N+k+1,N+k+2)}Elements of row N + k +1 and column N + k +2 of Y, Y_{(N+k+2,N+k+2)}Elements of row N + k +2 and column N + k +2 of Y, Y_(2+i,N+k+2)The element representing the 2+ i th row of Y and the N + k +2 th column.

Step 10: solving the final semi-definite programming problem by using interior point method software (such as CVX), obtaining the estimated value of the coordinate position x of the target source in the k-dimensional positioning scene, and recording the estimated value as the estimated value

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

The coordinate positions of the anchor nodes in the 2-dimensional positioning scene are given in table 1, and the coordinate positions of the target sources in the 2-dimensional positioning scene are randomly generated from the square regions of [ -50,50] × [ -50,50 ].

TABLE 1 coordinate position of each Anchor node in a 2-dimensional positioning scenario

Numbering of anchor nodes	0	1	2	3	4	5	6	7	8
										X axis coordinate	0	40	-40	40	-40	40	-40	0	0
Y-axis coordinate	0	-40	40	40	-40	0	0	40	-40

During simulation, TOA measurement values are generated according to the TOA measurement model in step 2, wherein NLOS errors are generated by uniform distribution, i.e.

And

is uniformly distributed. The clock bias ω of the target source is 1+ δ, δ being subject to a uniform distribution

δ_max0.005. Initial transmission time T₀And the clock drift of the target source theta can be set arbitrarily because they do not affect the result.

Root Mean Square Error (RMSE) was used to evaluate performance by

Obtaining a mixture of, in which,

represents the estimated position of the target source of the j-th simulation experiment,

showing the real position of the target source in the jth simulation experiment, and M showing Monte Carlo (MC) transportThe number of rows. In the simulation experiment, M is set to 3000. Let the standard deviation of Gaussian noise be

Parameter α is used to control the magnitude of the NLOS error variation.

The performance of the method of the invention under different simulation scenes is tested. FIG. 2 shows the values of N-4, sigma-1.6,

Comparing the performance of the method with the existing fractional programming, the existing robust second-order cone programming and the existing robust semi-definite programming, it can be seen from fig. 2 that the performance of the method of the invention is better than the existing method as a whole, and the advantage is more obvious along with the increase of α, fig. 3 shows that the performance of the method is more obvious when N is 4, sigma is 1.6,

The performance of the method of the present invention is compared with the performance of the existing fractional programming, the existing robust second order cone programming and the existing robust semi-definite programming, and as can be seen from fig. 3, the method of the present invention is hardly affected by the NLOS error on the reference path (which refers to the signal propagation path between the reference anchor node and the target source), because the method of the present invention uses w₀As an estimated variable, the NLOS error is processed by a robust method, namely the method is only subjected to w_iMagnitude of magnitude. FIG. 4 shows the increase in N from 4 to 8, σ ═ 1.6,

The performance of the method of the present invention is compared with the performance of the existing fractional programming, the existing robust second order cone programming and the existing robust semi-definite programming, and as can be seen from fig. 4, the performance of the method of the present inventionGradually increases with the increase of the number of the anchor nodes, and still has advantages in positioning effect.