CN112954637B - Target positioning method under condition of uncertain anchor node position - Google Patents

Target positioning method under condition of uncertain anchor node position Download PDF

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CN112954637B
CN112954637B CN202110091536.9A CN202110091536A CN112954637B CN 112954637 B CN112954637 B CN 112954637B CN 202110091536 A CN202110091536 A CN 202110091536A CN 112954637 B CN112954637 B CN 112954637B
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王海燕
杨舸
闫永胜
申晓红
贾天一
王天星
赵晨
张裕昌
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Northwestern Polytechnical University
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Abstract

本发明提供了一种锚节点位置不确定情况下的目标定位方法,在得到锚节点与目标节点以及目标节点之间的TOA量测信息后,列出目标函数,并引入锚节点误差量。在对目标函数进行向量化后,利用S‑过程消除锚节点误差向量,同时引入了一个凸约束,最后再经过转化与松弛将问题变为一个可以解决的凸问题。该方法在定位系统受环境影响,导致锚节点真实位置与量测得到的位置出现较大偏差时仍然能够较为准确地得到目标位置的估计,具有较强的鲁棒性和实用性。

Figure 202110091536

The present invention provides a target positioning method when the anchor node position is uncertain. After obtaining the TOA measurement information between the anchor node and the target node and the target node, the target function is listed, and the error amount of the anchor node is introduced. After the objective function is vectorized, the S-process is used to eliminate the anchor node error vector, and a convex constraint is introduced at the same time. Finally, the problem is transformed into a solvable convex problem through transformation and relaxation. When the positioning system is affected by the environment, resulting in a large deviation between the real position of the anchor node and the measured position, the method can still estimate the target position more accurately, and has strong robustness and practicability.

Figure 202110091536

Description

一种锚节点位置不确定情况下的目标定位方法A target localization method under the uncertainty of anchor node position

技术领域technical field

本发明涉及一种目标定位方法,属于信号处理领域,适合无线传感器网络中多个锚节点定位多个目标节点的自定位系统。The invention relates to a target positioning method, belongs to the field of signal processing, and is suitable for a self-positioning system in which multiple anchor nodes locate multiple target nodes in a wireless sensor network.

背景技术Background technique

无线传感器网络(WSN)的灵活性、广泛的覆盖范围和易于部署的特性在过去的几年中引起了广泛的关注。通常,一个无线传感器网络由一群在一定空间范围内分布的低成本和低功耗的传感器节点组成,它们可以用来执行常见的信号处理任务,例如探测、定位以及目标跟踪和目标状态变化的监测。其中,目标定位是无线传感器网络中最基础也是最重要的任务之一,许多物理量的获得都需要以明确节点位置为前提。许多类型的传感器量测值都可以用来进行目标定位,例如接收信号强度(RSS)、到达角(AOA)、到达时间(TOA)、到达时间差(TDOA)。在基于这些类型的量测值的目标定位方法中,基于TOA和TDOA的目标定位方案在定位性能和计算复杂度之间取得了良好的平衡。这两种定位方法可以有效地避免像基于AOA的定位方案那样部署成本高昂的传感器,也可以有效减小基于RSS的定位方法导致的较大的定位误差。The flexibility, wide coverage, and ease of deployment of wireless sensor networks (WSNs) have attracted extensive attention in the past few years. Generally, a wireless sensor network consists of a group of low-cost and low-power sensor nodes distributed in a certain spatial range, which can be used to perform common signal processing tasks, such as detection, localization, and target tracking and monitoring of target state changes . Among them, target positioning is one of the most basic and important tasks in wireless sensor networks, and the acquisition of many physical quantities needs to be based on the premise of clear node positions. Many types of sensor measurements can be used for target location, such as Received Signal Strength (RSS), Angle of Arrival (AOA), Time of Arrival (TOA), Time Difference of Arrival (TDOA). Among the target localization methods based on these types of measurements, TOA and TDOA-based target localization schemes achieve a good balance between localization performance and computational complexity. These two localization methods can effectively avoid deploying expensive sensors like the AOA-based localization scheme, and can also effectively reduce the large localization error caused by the RSS-based localization method.

在TOA定位问题中,往往存在锚节点与待定位的目标节点时钟不同步的问题,消除时间不同步的影响主要有两种方法,一是将TOA定位问题转化为TDOA定位问题;二是使用双程TOA,即令锚节点和目标节点进行双程信息交换,利用锚节点与目标节点对应的时间戳来求解定位问题。另外,传统模型大多假设锚节点的位置是精确已知的,但是在实际应用中,这个条件很难达到,因为受环境的影响,锚节点的真实位置和通过量测得到的位置总会存在一定的误差。例如水上浮标节点,即使预先使用GPS获取了其位置,在定位过程中,受洋流的影响,节点也会发生漂移,导致其真实位置偏离量测得到的位置。如果不采取一定的措施,就会造成定位性能的下降。In the TOA positioning problem, there is often the problem that the anchor node and the target node to be positioned are out of synchronization. There are two main methods to eliminate the impact of time asynchrony. One is to convert the TOA positioning problem into a TDOA positioning problem; the other is to use dual Process TOA, that is, the anchor node and the target node perform two-way information exchange, and the time stamp corresponding to the anchor node and the target node is used to solve the positioning problem. In addition, most of the traditional models assume that the position of the anchor node is accurately known, but in practical applications, this condition is difficult to achieve, because due to the influence of the environment, the real position of the anchor node and the position obtained by measurement will always exist. error. For example, even if the position of a water buoy node is obtained using GPS in advance, the node will drift due to the influence of ocean currents during the positioning process, causing its real position to deviate from the measured position. If certain measures are not taken, the positioning performance will be degraded.

锚节点真实位置坐标与量测得到的锚节点位置坐标的差值就是锚节点误差向量。目前对锚节点误差向量的建模方法有两种,一种是假设该向量服从均值为零的高斯分布,另一种不对该向量进行任何先验假设,只假设该向量模的最大值已知。在实际应用中,后一种情况更加符合实际,因为在不同的环境下锚节点误差模的最大值比高斯分布的协方差矩阵更易估算。而现有的目标定位方法,大多没有考虑锚节点位置由于环境因素存在误差,或者假设锚节点位置误差向量服从零均值的高斯分布。而在实际中,误差的统计分布往往很难获得,而锚节点误差向量模的最大值较易通过估算得到。而已有的误差向量模的最大值已知的方法中,Xu等人提出了一种多个锚节点定位单个目标节点,且只假设锚节点误差向量模的最大值已知的方法,然而这种方法只考虑了单个节点的定位场景,而且需要添加惩罚项,在实际操作过程中该惩罚项所对应的惩罚因子需要手动调节,增加了计算复杂度,减小了实用性。The difference between the real position coordinates of the anchor node and the measured position coordinates of the anchor node is the anchor node error vector. At present, there are two modeling methods for the anchor node error vector. One is to assume that the vector follows a Gaussian distribution with zero mean. . In practical applications, the latter case is more practical, because the maximum value of the anchor node error norm is easier to estimate than the Gaussian covariance matrix under different circumstances. However, most of the existing target localization methods do not consider the anchor node position error due to environmental factors, or assume that the anchor node position error vector obeys a Gaussian distribution with zero mean. In practice, the statistical distribution of errors is often difficult to obtain, and the maximum value of the anchor node error vector modulus is easier to obtain by estimation. Among the existing methods in which the maximum value of the error vector modulus is known, Xu et al. proposed a method in which multiple anchor nodes locate a single target node and only assume that the maximum value of the error vector modulus of the anchor node is known. The method only considers the positioning scene of a single node, and needs to add a penalty item. In the actual operation process, the penalty factor corresponding to the penalty item needs to be adjusted manually, which increases the computational complexity and reduces the practicability.

发明内容SUMMARY OF THE INVENTION

为了克服现有技术的不足,本发明提供一种锚节点位置不确定情况下的目标定位方法,仅仅需要已知锚节点误差向量的模的最大值,而不需要提前已知误差的统计分布,减弱了先验信息要求的苛求程度,同时没有设置需要手动调节的惩罚项,提升了实用性。In order to overcome the deficiencies of the prior art, the present invention provides a method for locating a target when the position of the anchor node is uncertain, which only needs to know the maximum value of the modulus of the error vector of the anchor node, and does not need to know the statistical distribution of the error in advance, The degree of strictness of prior information requirements is weakened, and there is no penalty item that needs to be manually adjusted, which improves the practicability.

本发明解决其技术问题所采用的技术方案包括以下步骤:The technical scheme adopted by the present invention to solve its technical problem comprises the following steps:

第一步,获得传感器网络中各节点之间的TOA量测,对锚节点误差进行建模;The first step is to obtain TOA measurements between nodes in the sensor network and model the anchor node error;

第二步,在锚节点与目标节点通信的原始TOA量测中引入锚节点误差项;In the second step, the anchor node error term is introduced into the original TOA measurement of the communication between the anchor node and the target node;

第三步,对锚节点误差项进行转化,利用S-过程消除锚节点误差向量;The third step is to transform the anchor node error term, and use the S-process to eliminate the anchor node error vector;

第四步,通过对目标函数及约束条件的转化和松弛,将问题转化为一个凸问题;The fourth step is to transform the problem into a convex problem by transforming and relaxing the objective function and constraints;

第五步,求解凸优化问题,得到目标位置的估计。The fifth step is to solve the convex optimization problem to obtain an estimate of the target position.

所述的第一步设传感器网络中共M个位置已知但是存在误差的锚节点和N个待定位的目标节点,M≥3,M个锚节点的真实坐标分别为x1,x2,…,xM,N个待估计的目标节点的坐标为y1,y2,…,yN;通过节点之间的通信获得TOA量测集合,包括锚节点与目标节点之间进行通信的TOA量测以及目标节点间进行通信的TOA量测;锚节点与目标节点之间通信获得的TOA量测表达式为

Figure BDA0002912762480000021
其中i,j分别表示编号为i的锚节点与编号为j的目标节点,c为信号传播速度,tij表示锚节点i与目标节点j相互通信所得到的时间量测,rij表示锚节点i与目标节点j之间的距离,eij表示锚节点i与目标节点j之间通信的加性高斯白噪声,eij服从均值为零、方差为
Figure BDA0002912762480000022
的高斯分布;目标节点之间相互通信获得的TOA量测表达式为
Figure BDA0002912762480000023
其中j和j'分别表示编号为j和j'的目标节点,且j≠j',t'jj'表示目标节点j与目标节点j'相互通信所得到的时间量测,r'jj'表示目标节点j与目标节点j之间的距离,e'jj'表示目标节点j与目标节点j之间通信的加性高斯白噪声,e'jj'服从均值为零、方差为
Figure BDA0002912762480000031
的高斯分布;锚节点与目标节点之间的距离与节点位置之间的关系为rij=||xi-xj||2;目标节点与目标节点之间的距离与节点位置之间的关系为r'jj'=||yj-yj'||2;锚节点真实位置与量测得到的位置之间的关系为
Figure BDA0002912762480000032
式中
Figure BDA0002912762480000033
表示量测得到的锚节点位置,ξi是锚节点误差向量,ξi的模的最大值小于等于一个已知的常数ε。In the first step, there are M anchor nodes with known positions but errors and N target nodes to be located in the sensor network, M≥3, and the real coordinates of the M anchor nodes are x 1 , x 2 ,… ,x M , the coordinates of the N target nodes to be estimated are y 1 , y 2 ,...,y N ; the TOA measurement set is obtained through the communication between nodes, including the TOA amount of communication between the anchor node and the target node The TOA measurement of the communication between the anchor node and the target node; the TOA measurement expression obtained from the communication between the anchor node and the target node is:
Figure BDA0002912762480000021
where i and j represent the anchor node numbered i and the target node numbered j respectively, c is the signal propagation speed, tij represents the time measurement obtained by the mutual communication between the anchor node i and the target node j , and rij represents the anchor node The distance between i and the target node j, e ij represents the additive white Gaussian noise of the communication between the anchor node i and the target node j, e ij obeys the mean value of zero and the variance is
Figure BDA0002912762480000022
The Gaussian distribution of ; the TOA measurement expression obtained by mutual communication between target nodes is:
Figure BDA0002912762480000023
where j and j' represent the target nodes numbered j and j' respectively, and j≠j', t'jj' represents the time measurement obtained by the mutual communication between target node j and target node j', r'jj' represents The distance between the target node j and the target node j, e'jj' represents the additive white Gaussian noise of the communication between the target node j and the target node j, e'jj' obeys the mean zero and the variance is
Figure BDA0002912762480000031
The Gaussian distribution of ; the relationship between the distance between the anchor node and the target node and the node position is r ij =||x i -x j || 2 ; The relationship is r'jj' =||y j -y j' || 2 ; the relationship between the real position of the anchor node and the measured position is
Figure BDA0002912762480000032
in the formula
Figure BDA0002912762480000033
Represents the measured anchor node position, ξ i is the anchor node error vector, and the maximum value of the modulus of ξ i is less than or equal to a known constant ε.

所述的第二步利用一阶泰勒展开式将锚节点真实位置与量测位置的关系进一步转化为

Figure BDA0002912762480000034
其中o(||ξi||)表示锚节点误差向量模||ξi||的高阶无穷小量;令
Figure BDA0002912762480000035
则推得|δij|≤ε,其中
Figure BDA0002912762480000036
表示锚节点i与目标节点j之间的距离,与rij不同,这里的使用的锚节点i的坐标是已知的带有误差的锚节点位置,δij表示锚节点i与目标节点j之间的模的误差值;则最终锚节点与目标节点之间通信的TOA量测表示为
Figure BDA0002912762480000037
The second step uses the first-order Taylor expansion to further transform the relationship between the real position of the anchor node and the measured position into
Figure BDA0002912762480000034
where o(||ξ i ||) represents the higher-order infinitesimal of the anchor node error vector modulo ||ξ i ||; let
Figure BDA0002912762480000035
Then it can be deduced that |δ ij |≤ε, where
Figure BDA0002912762480000036
Represents the distance between anchor node i and target node j. Different from r ij , the coordinates of anchor node i used here are the known anchor node positions with errors, and δ ij represents the distance between anchor node i and target node j. The error value of the modulus between the final anchor node and the target node is expressed as
Figure BDA0002912762480000037

所述的第三步采用最大似然估计法来估计目标的位置,设Xu=[y1,y2,…,yN]为未知目标节点坐标的集合,Xa=[x1,x2,…,xK]是所有锚节点坐标的集合,同时定义dij=tij×c,d'jj'=t'jj'×c,dij和d'jj'分别为锚节点i与目标节点j之间,以及目标节点j与j'(j≠j')之间的带噪声的距离量测值;根据已有条件,将待优化的原始目标函数表示为

Figure BDA0002912762480000038
将第二步中的锚节点误差项代入,得到min-max次优化问题
Figure BDA0002912762480000039
经过对目标函数的向量化,以及通过应用S-过程消除锚节点误差向量,原始次优化问题被转化为
Figure BDA0002912762480000041
其中,λ和μ是为了转化锚节点误差向量新引进的常数,≥表示矩阵正定,I表示单位矩阵,Tr(·)表示求括号里面矩阵的迹,d1=[d11,d12,…,dMN]T为锚节点与目标节点通信的带噪声的距离量测,d2=[d'12,d'13,…,d'N,N-1]T为目标节点之间通信的带噪声的距离量测,
Figure BDA0002912762480000042
为所有的带噪声的距离量测集合;定义
Figure BDA0002912762480000043
为锚节点与目标节点之间通信的真实距离量测,r2=[r'12,r'13,…,r'N,N-1]T为目标节点之间通信的真实距离量测,
Figure BDA0002912762480000044
为所有的真实距离量测集合,
Figure BDA0002912762480000045
G=blkdiag(G1,G2),Γ=1(M+N-1)×N,diag{·}表示对角矩阵,矩阵的对角线元素为大括号中的元素,blkdiag{·}表示分块对角矩阵,1(M+N-1)×N表示具有(M+N-1)×N个元素全为1的列向量,γ[·],r[·]表示引用向量γ,r中的元素;同时定义中间变量
Figure BDA0002912762480000046
γjj'以及向量γ12和γ,
Figure BDA00029127624800000411
γ2=[γ1213,…,γN,N-1]T
Figure BDA0002912762480000049
The third step uses the maximum likelihood estimation method to estimate the position of the target. Let Xu =[y 1 ,y 2 ,...,y N ] be the set of unknown target node coordinates, X a =[x 1 ,x 2 ,...,x K ] is the set of coordinates of all anchor nodes, while defining d ij =t ij ×c, d'jj'=t'jj' ×c, d ij and d'jj' are anchor nodes i and d'jj' respectively. The distance measurement value with noise between target nodes j, and between target nodes j and j'(j≠j'); according to the existing conditions, the original objective function to be optimized is expressed as
Figure BDA0002912762480000038
Substitute the anchor node error term in the second step to get the min-max optimization problem
Figure BDA0002912762480000039
After vectorizing the objective function, and removing the anchor node error vector by applying an S-process, the original suboptimization problem is transformed into
Figure BDA0002912762480000041
Among them, λ and μ are newly introduced constants to transform the anchor node error vector, ≥ indicates that the matrix is positive definite, I indicates the identity matrix, Tr( ) indicates the trace of the matrix in the brackets, d 1 =[d 11 ,d 12 ,… ,d MN ] T is the distance measurement with noise for the communication between the anchor node and the target node, d 2 =[d' 12 ,d' 13 ,...,d' N,N-1 ] T is the communication between the target nodes distance measurement with noise,
Figure BDA0002912762480000042
is the set of all noisy distance measurements; define
Figure BDA0002912762480000043
is the real distance measurement of the communication between the anchor node and the target node, r 2 =[r' 12 ,r' 13 ,...,r' N,N-1 ] T is the real distance measurement of the communication between the target nodes,
Figure BDA0002912762480000044
is the set of all true distance measurements,
Figure BDA0002912762480000045
G=blkdiag(G 1 , G 2 ), Γ=1 (M+N-1)×N , diag{·} represents a diagonal matrix, and the diagonal elements of the matrix are the elements in curly brackets, blkdiag{·} Represents a block diagonal matrix, 1 (M+N-1)×N represents a column vector with (M+N-1)×N elements all 1, γ[ ], r[ ] represents the reference vector γ , elements in r; also define intermediate variables
Figure BDA0002912762480000046
γ jj' and the vectors γ 1 , γ 2 and γ,
Figure BDA00029127624800000411
γ 2 =[γ 1213 ,...,γ N,N-1 ] T ,
Figure BDA0002912762480000049

所述的第四步引入中间矩阵

Figure BDA00029127624800000410
优化问题的最终形式为:The fourth step described introduces the intermediate matrix
Figure BDA00029127624800000410
The final form of the optimization problem is:

Figure BDA0002912762480000051
Figure BDA0002912762480000051

其中,Yu[w,v]表示引用矩阵Yu第w行,第v列的元素,w和v必须是整数;Among them, Yu [w, v] represents the element of the wth row and vth column of the reference matrix Yu, and w and v must be integers;

Yu[j,N+1:N+l]表示矩阵Yu第j行,第N+1到N+l列的所有元素,Yu[N+1:N+l,N+1:N+l]表示矩阵Yu第N+1到N+l行,第N+1到N+l列的所有元素,Xu(:,j)表示矩阵Xu中第j列所有元素组成的列向量。Y u [j,N+1:N+l] represents all elements of the jth row and N+1 to N+l columns of the matrix Yu u , Yu [N+1:N+l,N+1:N +l] represents all elements in the N+1 to N+1 rows and N+1 to N+1 columns of the matrix Yu, and X u ( :,j) represents the column composed of all the elements in the jth column of the matrix X u vector.

本发明的有益效果是:只假设锚节点误差向量的模的最大值已知,这更符合实际情况。在得到锚节点与目标节点以及目标节点之间的TOA量测信息后,列出目标函数,并引入锚节点误差量。在对目标函数进行向量化后,利用S-过程消除锚节点误差向量,同时引入了一个凸约束,最后再经过转化与松弛将问题变为一个可以解决的凸问题。该方法在定位系统受环境影响,导致锚节点真实位置与量测得到的位置出现较大偏差时仍然能够较为准确地得到目标位置的估计,具有较强的鲁棒性和实用性。The beneficial effect of the present invention is that it is only assumed that the maximum value of the modulus of the error vector of the anchor node is known, which is more in line with the actual situation. After obtaining the TOA measurement information between the anchor node and the target node and the target node, the objective function is listed, and the error amount of the anchor node is introduced. After the objective function is vectorized, the S-process is used to eliminate the anchor node error vector, and a convex constraint is introduced at the same time. Finally, the problem is transformed into a solvable convex problem through transformation and relaxation. When the positioning system is affected by the environment, resulting in a large deviation between the real position of the anchor node and the measured position, the method can still estimate the target position more accurately, and has strong robustness and practicability.

附图说明Description of drawings

图1是锚节点位置不确定情况下目标定位方法框图。Figure 1 is a block diagram of the target localization method when the anchor node position is uncertain.

图2是方法性能与TOA量测噪声方差的关系图。Figure 2 is a graph of method performance versus TOA measurement noise variance.

图3是方法性能与锚节点误差向量模的最大值的关系图。Figure 3 is a graph of method performance versus the maximum value of the anchor node error vector norm.

图4是方法性能与锚节点数量的关系图。Figure 4 is a graph of method performance versus the number of anchor nodes.

图5是方法性能与待定位的目标节点的数量关系图。Figure 5 is a graph showing the relationship between method performance and the number of target nodes to be located.

具体实施方式Detailed ways

下面结合附图和实施例对本发明进一步说明,本发明包括但不仅限于下述实施例。The present invention will be further described below with reference to the accompanying drawings and embodiments, and the present invention includes but is not limited to the following embodiments.

针对实际定位问题中锚节点位置受环境影响而导致与量测所得位置出现误差的问题,提出了一种基于锚节点位置存在误差情况下的定位方法。选择l维定位场景,l=2或3。设传感器网络中共M个位置已知但是存在误差的锚节点,M≥3,和N个节点为待定位的目标节点。设M个位置已知的锚节点的真实坐标分别为x1,x2,…,xM,N个待估计的目标节点的坐标为y1,y2,…,yNAiming at the problem that the position of the anchor node is affected by the environment and causes an error with the measured position in the actual positioning problem, a positioning method based on the error of the position of the anchor node is proposed. Select the l-dimensional positioning scene, l=2 or 3. Assume that there are M anchor nodes with known positions but errors in the sensor network, M ≥ 3, and N nodes are the target nodes to be located. Let the real coordinates of the M anchor nodes with known positions be x 1 , x 2 ,...,x M respectively, and the coordinates of the N target nodes to be estimated are y 1 , y 2 ,..., y N .

本发明的主步骤如下:The main steps of the present invention are as follows:

第一步:获得原始TOA量测,对锚节点误差进行建模Step 1: Obtain raw TOA measurements and model anchor node errors

通过节点之间的通信获得TOA量测集合,该集合由两部分组成,分别是锚节点与目标节点之间进行通信的TOA量测以及目标节点间进行通信的TOA量测。锚节点与目标节点之间通信获得的TOA量测表达式为:The TOA measurement set is obtained through the communication between nodes, and the set consists of two parts, namely the TOA measurement of the communication between the anchor node and the target node and the TOA measurement of the communication between the target nodes. The TOA measurement expression obtained from the communication between the anchor node and the target node is:

Figure BDA0002912762480000061
Figure BDA0002912762480000061

其中i,j均为正整数,分别表示编号为i的节点与编号为j的节点。c为信号传播速度,tij表示锚节点i与目标节点j相互通信所得到的时间量测,rij表示锚节点i与目标节点j之间的距离,eij表示锚节点i与目标节点j之间通信的加性高斯白噪声,它服从均值为零,方差为

Figure BDA0002912762480000062
的高斯分布。where i and j are both positive integers, representing the node numbered i and the node numbered j, respectively. c is the speed of signal propagation, t ij represents the time measurement obtained by the mutual communication between anchor node i and target node j, r ij represents the distance between anchor node i and target node j, and e ij represents anchor node i and target node j Additive white Gaussian noise for communication between
Figure BDA0002912762480000062
Gaussian distribution.

目标节点之间相互通信获得的TOA量测表达式为:The TOA measurement expression obtained by the mutual communication between target nodes is:

Figure BDA0002912762480000063
Figure BDA0002912762480000063

其中j和j'分别表示编号为j和j'的目标节点,且j≠j',t'jj'表示目标节点j与目标节点j'相互通信所得到的时间量测,r'jj'表示目标节点j与目标节点j之间的距离,e'jj'表示目标节点j与目标节点j之间通信的加性高斯白噪声,它服从均值为零,方差为

Figure BDA0002912762480000064
的高斯分布。where j and j' represent the target nodes numbered j and j' respectively, and j≠j', t'jj' represents the time measurement obtained by the mutual communication between target node j and target node j', r'jj' represents The distance between the target node j and the target node j, e'jj' represents the additive white Gaussian noise of the communication between the target node j and the target node j, it obeys the mean of zero, and the variance is
Figure BDA0002912762480000064
Gaussian distribution.

锚节点与目标节点之间的距离与节点位置之间的关系为:The relationship between the distance between the anchor node and the target node and the node position is:

rij=||xi-xj||2 r ij =||x i -x j || 2

i=1,2,…,M,j=1,2,…,Ni=1,2,...,M,j=1,2,...,N

其中,|| ||2表示向量的2-范数。where || || 2 represents the 2-norm of the vector.

同理,目标节点与目标节点之间的距离与节点位置之间的关系为:Similarly, the relationship between the distance between the target node and the target node and the node position is:

r'jj'=||yj-yj'||2 r'jj' =||y j -y j' || 2

j=1,2,…,N,j'=1,2,…,Nj=1,2,...,N,j'=1,2,...,N

由于环境因素的影响,量测得到的锚节点位置与其真实位置之间会存在误差,锚节点真实位置与量测得到的位置之间的关系为:Due to the influence of environmental factors, there will be errors between the measured anchor node position and its real position. The relationship between the anchor node real position and the measured position is:

Figure BDA0002912762480000071
Figure BDA0002912762480000071

式中

Figure BDA0002912762480000072
表示量测得到的锚节点位置,ξi是锚节点误差向量,它的模的最大值小于一个已知的常数ε,即in the formula
Figure BDA0002912762480000072
Represents the measured anchor node position, ξ i is the anchor node error vector, and the maximum value of its modulus is less than a known constant ε, that is

||ξi||≤ε||ξ i ||≤ε

第二步:在锚节点与目标节点通信的原始TOA量测中引入锚节点误差项Step 2: Introduce the anchor node error term in the original TOA measurement of the communication between the anchor node and the target node

利用一阶泰勒展开式,可以将锚节点真实位置与量测位置的关系进一步转化为:Using the first-order Taylor expansion, the relationship between the real position of the anchor node and the measured position can be further transformed into:

Figure BDA0002912762480000073
Figure BDA0002912762480000073

i=1,2,…,M,j=1,2,…,Ni=1,2,...,M,j=1,2,...,N

其中o(||ξi||)表示锚节点误差向量模||ξi||的高阶无穷小量。where o(||ξ i ||) represents the higher-order infinitesimal of the anchor node error vector modulo ||ξ i ||.

make

Figure BDA0002912762480000074
Figure BDA0002912762480000074

则可推得:It can be deduced that:

ij|≤εij |≤ε

其中

Figure BDA0002912762480000075
表示锚节点i与目标节点j之间的距离,与rij不同,这里的使用的锚节点i的坐标是已知的带有误差的锚节点位置。δij表示锚节点i与目标节点j之间的模的误差值,符号“||”为求绝对值符号。in
Figure BDA0002912762480000075
Represents the distance between anchor node i and target node j. Different from r ij , the coordinates of anchor node i used here are the known anchor node positions with errors. δ ij represents the error value of the modulus between the anchor node i and the target node j, and the symbol "||" is the symbol for finding the absolute value.

则最终锚节点与目标节点之间通信的TOA量测可以表示为:Then the TOA measurement of the communication between the final anchor node and the target node can be expressed as:

Figure BDA0002912762480000076
Figure BDA0002912762480000076

第三步:对锚节点误差项进行转化Step 3: Transform the anchor node error term

采用最大似然估计法来估计目标的位置。设Xu=[y1,y2,…,yN]为未知目标节点坐标的集合,Xa=[x1,x2,…,xK]是所有锚节点坐标的集合。同时定义:The maximum likelihood estimation method is used to estimate the location of the target. Let X u =[y 1 ,y 2 ,...,y N ] be the set of unknown target node coordinates, and X a =[x 1 ,x 2 ,...,x K ] be the set of all anchor node coordinates. Also define:

dij=tij×c,i=1,2,…,M,j=1,2,…,Nd ij =t ij ×c,i=1,2,...,M,j=1,2,...,N

d'jj'=t'jj'×c,j=1,2,…,N,j'=1,2,…,Nd'jj'=t'jj'×c,j=1,2,…,N,j'=1,2,…,N

dij和d'jj'分别为锚节点i与目标节点j之间,以及目标节点j与j'(j≠j')之间的带噪声的距离量测值。d ij and d'jj' are the noisy distance measurements between anchor node i and target node j, and between target node j and j'(j≠j'), respectively.

根据已有条件,可以将待优化的原始目标函数表示为:According to the existing conditions, the original objective function to be optimized can be expressed as:

Figure BDA0002912762480000081
Figure BDA0002912762480000081

将第二步中的锚节点误差项代入,可得min-max次优化问题:Substitute the anchor node error term in the second step to get the min-max optimization problem:

Figure BDA0002912762480000082
Figure BDA0002912762480000082

subject tosubject to

Figure BDA0002912762480000083
Figure BDA0002912762480000083

r'jj'=||yj-yj'||,j=1,2,…,M,j'=1,2,…,Nr'jj' =||y j -y j' ||,j=1,2,…,M,j'=1,2,…,N

其中,min表示求极小,max表示求极大,∑表示求和。subject to表示“受约束于”。Among them, min represents the minimum, max represents the maximum, and ∑ represents the sum. subject to means "subject to".

上述次优化问题是一个非线性和非凸问题,且目标函数的形式是元素求和的形式,同时目标函数中含有节点误差项δij,都导致该次优化问题十分难于求解。因此,接下来首先对目标函数进行向量化,给出向量化后的目标函数及约束条件,这时锚节点误差项被转化成了一个向量,而只对一个向量进行处理要易于对单个元素进行处理;在向量化完成后,使用S-过程(S-procedure)消除锚节点误差向量,通过引入两个常数变量,消除了锚节点误差项,并将其转化成一个凸约束。The above sub-optimization problem is a nonlinear and non-convex problem, and the form of the objective function is the form of element summation, and the objective function contains the node error term δ ij , which makes the sub-optimization problem very difficult to solve. Therefore, the objective function is firstly vectorized, and the vectorized objective function and constraints are given. At this time, the anchor node error term is converted into a vector, and it is easier to process a single element when processing only one vector. ; After the vectorization is completed, the S-procedure is used to eliminate the anchor node error vector. By introducing two constant variables, the anchor node error term is eliminated and transformed into a convex constraint.

经过对目标函数的向量化,以及通过应用S-过程消除锚节点误差向量,原始次优化问题被转化为:After vectorizing the objective function and eliminating the anchor node error vector by applying an S-process, the original suboptimization problem is transformed into:

Figure BDA0002912762480000084
Figure BDA0002912762480000084

其中,

Figure BDA0002912762480000085
表示开平方根,λ和μ是为了转化锚节点误差向量新引进的常数,≥表示矩阵正定,I表示单位矩阵,Tr(·)表示求括号里面矩阵的迹。且d1=[d11,d12,…,dMN]T为锚节点与目标节点通信的带噪声的距离量测,d2=[d'12,d'13,…,d'N,N-1]T为目标节点之间通信的带噪声的距离量测,
Figure BDA0002912762480000091
为所有的带噪声的距离量测集合;定义
Figure BDA0002912762480000092
为锚节点与目标节点之间通信的真实距离量测,r2=[r'12,r'13,…,r'N,N-1]T为目标节点之间通信的真实距离量测,
Figure BDA0002912762480000093
为所有的真实距离量测集合。
Figure BDA0002912762480000094
G=blkdiag(G1,G2),Γ=1(M+N-1)×N。其中,diag{·}表示对角矩阵,矩阵的对角线元素为大括号中的元素,blkdiag{·}表示分块对角矩阵,1(M+N-1)×N表示具有(M+N-1)×N个元素全为1的列向量,γ[·],r[·]表示引用向量γ,r中的元素。同时定义中间变量
Figure BDA00029127624800000910
γjj'以及向量γ12和γ:in,
Figure BDA0002912762480000085
Indicates the square root, λ and μ are newly introduced constants to transform the anchor node error vector, ≥ indicates that the matrix is positive definite, I indicates the identity matrix, and Tr(·) indicates the trace of the matrix in the parentheses. And d 1 =[d 11 ,d 12 ,...,d MN ] T is the noisy distance measurement between the anchor node and the target node, d 2 =[d' 12 ,d' 13 ,...,d' N, N-1 ] T is the noisy distance measurement for communication between target nodes,
Figure BDA0002912762480000091
is the set of all noisy distance measurements; define
Figure BDA0002912762480000092
is the real distance measurement of the communication between the anchor node and the target node, r 2 =[r' 12 ,r' 13 ,...,r' N,N-1 ] T is the real distance measurement of the communication between the target nodes,
Figure BDA0002912762480000093
Set for all true distance measurements.
Figure BDA0002912762480000094
G=blkdiag(G 1 , G 2 ), Γ=1 (M+N-1)×N . Among them, diag{·} represents a diagonal matrix, and the diagonal elements of the matrix are the elements in the curly brackets, blkdiag{·} represents a block diagonal matrix, and 1 (M+N-1)×N represents a matrix with (M+N-1)×N N-1)×N column vector with all 1 elements, γ[·], r[·] represents the elements in the reference vector γ, r. Also define intermediate variables
Figure BDA00029127624800000910
γ jj' and the vectors γ 1 , γ 2 and γ:

Figure BDA0002912762480000095
Figure BDA0002912762480000095

Figure BDA0002912762480000096
Figure BDA0002912762480000096

Figure BDA0002912762480000097
Figure BDA0002912762480000097

Figure BDA0002912762480000098
Figure BDA0002912762480000098

以往类似的处理只是针对单节点定位问题,而现在的处理是针对多节点定位问题,且处理后的目标函数的形式也与其他方法的形式有所不同,其他类似的方法往往是假设整个目标函数都小于一个常数μ,从而将前半部分Tr[G(ΓγT-2drT)]也放到利用S-过程得到的凸约束里面去,而现在的处理方法是在目标函数中仍然保留前半部分,不放到约束中去。和之前的方法相比,这样处理减弱了松弛程度,优化了目标函数的结构。Similar processing in the past is only for single-node positioning problems, but the current processing is for multi-node positioning problems, and the form of the processed objective function is also different from that of other methods. Other similar methods often assume the entire objective function. are smaller than a constant μ, so the first half of Tr[G(Γγ T -2dr T )] is also put into the convex constraint obtained by the S-process, and the current processing method is to keep the first half of the objective function, Don't put it into constraints. Compared with the previous method, this processing reduces the degree of relaxation and optimizes the structure of the objective function.

第四步:将问题转化为一个凸问题Step 4: Transform the problem into a convex problem

经过第三步的转化后,锚节点误差项被消除,但这时问题仍然不是一个可解的凸问题。这时通过对目标函数及约束条件的一系列转化和松弛,使问题转化为一个凸问题。采用的松弛方法是半正定松弛方法,最终,目标函数被转化为一个凸函数,约束条件被转化为凸约束条件,从而使该问题变得可解。After the third step of transformation, the anchor node error term is eliminated, but the problem is still not a solvable convex problem. At this time, through a series of transformation and relaxation of the objective function and constraints, the problem is transformed into a convex problem. The relaxation method used is a positive semi-definite relaxation method. Finally, the objective function is transformed into a convex function, and the constraints are transformed into convex constraints, thereby making the problem solvable.

为了将问题转化为凸问题,引入中间矩阵YuTo transform the problem into a convex problem, an intermediate matrix Yu is introduced :

Figure BDA0002912762480000099
Figure BDA0002912762480000099

通过该中间矩阵与待优化变量γ,Xu建立联系,并进行适当的松弛,就将所有的约束转化为凸约束,而目标函数也是凸函数,问题转化为了一个可解的凸问题。Through the connection between the intermediate matrix and the variables to be optimized γ, X u , and appropriate relaxation, all constraints are transformed into convex constraints, and the objective function is also a convex function, and the problem is transformed into a solvable convex problem.

优化问题的最终形式为:The final form of the optimization problem is:

Figure BDA0002912762480000101
Figure BDA0002912762480000101

其中,Yu[w,v]表示引用矩阵Yu第w行,第v列的元素,w和v必须是整数;Yu[j,N+1:N+l]表示矩阵Yu第j行,第N+1到N+l列的所有元素,Yu[N+1:N+l,N+1:N+l]表示矩阵Yu第N+1到N+l行,第N+1到N+l列的所有元素,Xu(:,j)表示矩阵Xu中第j列所有元素组成的列向量。Among them, Yu [w,v] refers to the element in the wth row and vth column of the reference matrix Yu, and w and v must be integers; Yu [j, N+1:N+l] means the matrix Yu jth Row, all elements in columns N+1 to N+l, Yu [N+1:N+l,N+1:N+l] represents matrix Yu Row N+1 to N+l, Nth All elements from +1 to N+l columns, X u (:,j) represents a column vector composed of all elements in the jth column of the matrix X u .

之前类似的算法中,往往在目标函数中加入了惩罚项,即加上一个很小的正数(惩罚因子)乘以某个待优化矩阵的所有元素和或者迹,这种方法需要手动选择惩罚因子,惩罚因子选择不合适,会导致算法性能的下降,因而实用性不强,而本发明没有加入惩罚项,从而避免了手动选择惩罚因子,增加了实用性。In similar algorithms before, a penalty term is often added to the objective function, that is, a small positive number (penalty factor) is multiplied by all elements and or traces of a matrix to be optimized. This method requires manual selection of the penalty. The inappropriate selection of the penalty factor will lead to a decrease in the performance of the algorithm, so the practicability is not strong, and the present invention does not add a penalty item, thereby avoiding the manual selection of the penalty factor and increasing the practicability.

第五步:求解凸优化问题,得到目标位置的估计Step 5: Solve the convex optimization problem to get an estimate of the target position

将第四步得到的凸优化问题用MATLAB中的CVX凸优化工具箱求解,从而得到目标位置的估计。The convex optimization problem obtained in the fourth step is solved by the CVX convex optimization toolbox in MATLAB to obtain the estimation of the target position.

本发明的实施例考虑二维空间的定位问题,设在一个1200m×1200m的区域内有4个位置已知的锚节点与3个待定位的目标节点,即M=4,N=3。设4个位置已知的锚节点的真实坐标分别为x1,x2,x3,x4,N个待估计的目标节点的坐标为y1,y2,y3In the embodiment of the present invention, the positioning problem in two-dimensional space is considered, and in an area of 1200m×1200m, there are 4 anchor nodes with known positions and 3 target nodes to be positioned, ie M=4, N=3. Let the real coordinates of the four anchor nodes with known positions be x 1 , x 2 , x 3 , and x 4 respectively, and the coordinates of the N target nodes to be estimated are y 1 , y 2 , and y 3 .

第一步:获得原始TOA量测,对锚节点误差进行建模Step 1: Obtain raw TOA measurements and model anchor node errors

通过节点之间的通信获得TOA量测集合。其中,锚节点与目标节点之间通信获得的TOA量测表达式为:The TOA measurement set is obtained through communication between nodes. Among them, the TOA measurement expression obtained from the communication between the anchor node and the target node is:

Figure BDA0002912762480000111
Figure BDA0002912762480000111

i,j均为正整数,分别表示编号为i的节点与编号为j的节点。c为信号传播速度,tij表示锚节点i与目标节点j相互通信所得到的时间量测,rij表示锚节点i与目标节点j之间的距离,eij表示锚节点i与目标节点j之间通信的加性高斯白噪声,它服从均值为零,方差为

Figure BDA0002912762480000112
的高斯分布。i and j are both positive integers, representing the node numbered i and the node numbered j, respectively. c is the speed of signal propagation, t ij represents the time measurement obtained by the mutual communication between anchor node i and target node j, r ij represents the distance between anchor node i and target node j, and e ij represents anchor node i and target node j Additive white Gaussian noise for communication between
Figure BDA0002912762480000112
Gaussian distribution.

目标节点之间相互通信获得的TOA量测表达式为:The TOA measurement expression obtained by the mutual communication between target nodes is:

Figure BDA0002912762480000113
Figure BDA0002912762480000113

其中j和j'分别表示编号为j和j'的目标节点,且j≠j',t'jj'表示目标节点j与目标节点j'相互通信所得到的时间量测,r'jj'表示目标节点j与目标节点j之间的距离,e'jj'表示目标节点j与目标节点j之间通信的加性高斯白噪声,它服从均值为零,方差为

Figure BDA0002912762480000114
的高斯分布。where j and j' represent the target nodes numbered j and j' respectively, and j≠j', t'jj' represents the time measurement obtained by the mutual communication between target node j and target node j', r'jj' represents The distance between the target node j and the target node j, e'jj' represents the additive white Gaussian noise of the communication between the target node j and the target node j, it obeys the mean of zero, and the variance is
Figure BDA0002912762480000114
Gaussian distribution.

锚节点与目标节点之间的距离与节点位置之间的关系为:The relationship between the distance between the anchor node and the target node and the node position is:

rij=||xi-xj||2 r ij =||x i -x j || 2

i=1,2,3,4,j=1,2,3i=1,2,3,4,j=1,2,3

其中,|| ||2表示向量的2-范数。where || || 2 represents the 2-norm of the vector.

同理,目标节点与目标节点之间的距离与节点位置之间的关系为:Similarly, the relationship between the distance between the target node and the target node and the node position is:

r'jj'=||yj-yj'||2,j=1,2,3,j'=1,2,3r'jj' =||y j -y j' || 2 ,j=1,2,3,j'=1,2,3

由于环境因素的影响,量测得到的锚节点位置与其真实位置之间会存在误差,锚节点真实位置与量测得到的位置之间的关系为:Due to the influence of environmental factors, there will be errors between the measured anchor node position and its real position. The relationship between the anchor node real position and the measured position is:

Figure BDA0002912762480000115
Figure BDA0002912762480000115

式中

Figure BDA0002912762480000116
表示量测得到的锚节点位置,ξi是锚节点误差向量,它的模的最大值小于一个已知的常数ε,即in the formula
Figure BDA0002912762480000116
Represents the measured anchor node position, ξ i is the anchor node error vector, and the maximum value of its modulus is less than a known constant ε, that is

||ξi||≤ε||ξ i ||≤ε

这里将ε设置为0.5m。Here ε is set to 0.5m.

第二步:在锚节点与目标节点通信的原始TOA量测中引入锚节点误差项Step 2: Introduce the anchor node error term in the original TOA measurement of the communication between the anchor node and the target node

利用一阶泰勒展开式,可以将锚节点真实位置与量测位置的关系进一步转化为:Using the first-order Taylor expansion, the relationship between the real position of the anchor node and the measured position can be further transformed into:

Figure BDA0002912762480000121
Figure BDA0002912762480000121

i=1,2,3,4,j=1,2,3i=1,2,3,4,j=1,2,3

其中o(||ξi||)表示锚节点误差向量模||ξi||的高阶无穷小量。where o(||ξ i ||) represents the higher-order infinitesimal of the anchor node error vector modulo ||ξ i ||.

make

Figure BDA0002912762480000122
Figure BDA0002912762480000122

则有then there are

Figure BDA0002912762480000123
Figure BDA0002912762480000123

which is

ij|≤εij |≤ε

其中

Figure BDA0002912762480000124
表示锚节点i与目标节点j之间的距离,与rij不同,这里的使用的锚节点i的坐标是已知的带有误差的锚节点位置。δij表示锚节点i与目标节点j之间的模的误差值,符号“||”为求绝对值符号。in
Figure BDA0002912762480000124
Represents the distance between anchor node i and target node j. Different from r ij , the coordinates of anchor node i used here are the known anchor node positions with errors. δ ij represents the error value of the modulus between the anchor node i and the target node j, and the symbol "||" is the symbol for finding the absolute value.

则最终锚节点与目标节点之间通信的TOA量测可以表示为:Then the TOA measurement of the communication between the final anchor node and the target node can be expressed as:

Figure BDA0002912762480000125
Figure BDA0002912762480000125

第三步:对锚节点误差项进行转化Step 3: Transform the anchor node error term

采用最大似然估计法来估计目标的位置。设Xu=[y1,y2,y3]为未知目标节点坐标的集合,Xa=[x1,x2,x3,x4]是所有锚节点坐标的集合。同时定义:The maximum likelihood estimation method is used to estimate the location of the target. Let X u =[y 1 , y 2 , y 3 ] be the set of unknown target node coordinates, and X a =[x 1 , x 2 , x 3 , x 4 ] be the set of all anchor node coordinates. Also define:

dij=tij×c,i=1,2,3,4,j=1,2,3d ij =t ij ×c,i=1,2,3,4,j=1,2,3

d'jj'=t'jj'×c,j=1,2,3,j'=1,2,3d'jj'=t'jj'×c,j=1,2,3,j'=1,2,3

dij和d'jj'分别为锚节点i与目标节点j之间,以及目标节点j与j'(j≠j')之间的带噪声的距离量测值。d ij and d'jj' are the noisy distance measurements between anchor node i and target node j, and between target node j and j'(j≠j'), respectively.

根据已有条件,可以将待优化的原始目标函数表示为:According to the existing conditions, the original objective function to be optimized can be expressed as:

Figure BDA0002912762480000126
Figure BDA0002912762480000126

将第二步中的锚节点误差项代入,可得min-max次优化问题:Substitute the anchor node error term in the second step to get the min-max optimization problem:

Figure BDA0002912762480000131
Figure BDA0002912762480000131

subject tosubject to

Figure BDA0002912762480000132
Figure BDA0002912762480000132

r'jj'=||yj-yj'||,j=1,2,3,j'=1,2,3r'jj' =||y j -y j' ||,j=1,2,3,j'=1,2,3

其中,min表示求极小,max表示求极大,∑表示求和。subject to表示“受约束于”。Among them, min represents the minimum, max represents the maximum, and ∑ represents the sum. subject to means "subject to".

接下来,对目标函数进行向量化。对于单个TOA量测,锚节点误差满足|δij|≤ε,则向量化后的锚节点误差向量满足:Next, vectorize the objective function. For a single TOA measurement, the anchor node error satisfies |δ ij |≤ε, then the vectorized anchor node error vector satisfies:

Figure BDA0002912762480000133
Figure BDA0002912762480000133

其中δ=[δ1112,…,δ43]T是M个锚节点与N个目标节点通信的锚节点模误差的集合,

Figure BDA0002912762480000134
表示开平方根。同样地,定义d1=[d11,d12,…,d43]T为锚节点与目标节点通信的带噪声的距离量测,d2=[d'12,d'13,…,d'32]T为目标节点之间通信的带噪声的距离量测,
Figure BDA0002912762480000135
为所有的带噪声的距离量测集合;定义
Figure BDA0002912762480000136
为锚节点与目标节点之间通信的真实距离量测,r2=[r'12,r'13,…,r3'2]T为目标节点之间通信的真实距离量测,
Figure BDA0002912762480000137
为所有的真实距离量测集合。同时定义中间变量
Figure BDA0002912762480000138
γjj'以及向量γ12和γ:where δ=[δ 1112 ,...,δ 43 ] T is the set of anchor node modulo errors of M anchor nodes communicating with N target nodes,
Figure BDA0002912762480000134
represents the square root. Similarly, define d 1 =[d 11 ,d 12 ,...,d 43 ] T is the noisy distance measurement between the anchor node and the target node, d 2 =[d' 12 ,d' 13 ,...,d '32 ] T is a noisy distance measurement for communication between target nodes,
Figure BDA0002912762480000135
is the set of all noisy distance measurements; define
Figure BDA0002912762480000136
is the real distance measurement of the communication between the anchor node and the target node, r 2 =[r' 12 ,r' 13 ,...,r 3 ' 2 ] T is the real distance measurement of the communication between the target nodes,
Figure BDA0002912762480000137
Set for all true distance measurements. Also define intermediate variables
Figure BDA0002912762480000138
γ jj' and the vectors γ 1 , γ 2 and γ:

Figure BDA0002912762480000139
Figure BDA0002912762480000139

Figure BDA00029127624800001310
Figure BDA00029127624800001310

Figure BDA00029127624800001311
Figure BDA00029127624800001311

Figure BDA00029127624800001312
Figure BDA00029127624800001312

则原次优化问题可以转化为:Then the original suboptimization problem can be transformed into:

Figure BDA00029127624800001313
Figure BDA00029127624800001313

subject tosubject to

γ[3(i-1)+j]=r2[3(i-1)+j],i=1,2,3,4,j=1,2,3γ[3(i-1)+j]=r 2 [3(i-1)+j], i=1,2,3,4,j=1,2,3

γ[12+3(j-1)+j']=r2[12+3(j-1)+j'],j=1,2,3,j'=1,2,3γ[12+3(j-1)+j']=r 2 [12+3(j-1)+j'],j=1,2,3,j'=1,2,3

Figure BDA00029127624800001314
Figure BDA00029127624800001314

r[12+3(i-1)+j]=||yj-yj'||,j=1,2,3,j'=1,2,3r[12+3(i-1)+j]=||y j -y j' ||,j=1,2,3,j'=1,2,3

其中

Figure BDA00029127624800001315
G=blkdiag(G1,G2),Γ=1(3+4-1)×3。其中,diag{·}表示对角矩阵,矩阵的对角线元素为大括号中的元素,blkdiag{·}表示分块对角矩阵,1(3+4-1)×3表示具有18个元素全为1的列向量,γ[·],r[·]表示引用向量γ,r中的元素。in
Figure BDA00029127624800001315
G=blkdiag(G 1 , G 2 ), Γ=1 (3+4-1)×3 . Among them, diag{·} represents a diagonal matrix, the diagonal elements of the matrix are the elements in the curly brackets, blkdiag{·} represents a block diagonal matrix, and 1 (3+4-1)×3 means that there are 18 elements A column vector of all 1s, γ[·], r[·] represents the element in the reference vector γ, r.

通过引入常数因子μ消除min-max优化问题中的max,即对于

Figure BDA0002912762480000141
有如下结论成立:Eliminate max in min-max optimization problem by introducing a constant factor μ, i.e. for
Figure BDA0002912762480000141
The following conclusions are established:

δTG1δ-2δTG1(d1-r1)≤μδ T G 1 δ-2δ T G 1 (d 1 -r 1 )≤μ

即:which is:

Figure BDA0002912762480000142
Figure BDA0002912762480000142

其中

Figure BDA0002912762480000143
表示充分条件。in
Figure BDA0002912762480000143
represents a sufficient condition.

上式可进一步表示为:The above formula can be further expressed as:

Figure BDA0002912762480000144
Figure BDA0002912762480000144

通过应用S-过程,上述递推关系最终被转化为一个凸约束:By applying the S-process, the above recurrence relation is finally transformed into a convex constraint:

Figure BDA0002912762480000145
Figure BDA0002912762480000145

即:which is:

Figure BDA0002912762480000146
Figure BDA0002912762480000146

式中λ和μ是为了转化锚节点误差向量新引进的常数,

Figure BDA0002912762480000148
表示“存在”,≥表示矩阵半正定。where λ and μ are newly introduced constants to transform the anchor node error vector,
Figure BDA0002912762480000148
means "exist", and ≥ means that the matrix is positive semi-definite.

同时,待优化的目标函数可转化为:At the same time, the objective function to be optimized can be transformed into:

Tr[G(ΓγT-2drT)]+μTr[G(Γγ T -2dr T )]+μ

式中,Tr(·)表示求括号里面矩阵的迹。In the formula, Tr(·) means to find the trace of the matrix in the brackets.

第四步:将问题转化为一个凸问题Step 4: Transform the problem into a convex problem

经过第三步的转化后,锚节点误差项被消除,但这时问题仍然不是一个凸问题。这时通过对目标函数及约束条件的一系列转化和松弛,使问题转化为一个凸问题。采用的松弛方法是半正定松弛方法,最终,目标函数被转化为一个凸函数,约束条件被转化为凸约束条件,从而使该问题变得可解。After the third step of transformation, the anchor node error term is eliminated, but the problem is still not a convex problem. At this time, through a series of transformation and relaxation of the objective function and constraints, the problem is transformed into a convex problem. The relaxation method used is a positive semi-definite relaxation method. Finally, the objective function is transformed into a convex function, and the constraints are transformed into convex constraints, thereby making the problem solvable.

为了将问题转化为凸问题,引入中间矩阵YuTo transform the problem into a convex problem, an intermediate matrix Yu is introduced :

Figure BDA0002912762480000147
Figure BDA0002912762480000147

则矩阵Yu中的元素与已有约束的关系为:Then the relationship between the elements in the matrix Yu u and the existing constraints is:

Figure BDA0002912762480000151
Figure BDA0002912762480000151

i=1,2,3,4,j=1,2,3i=1,2,3,4,j=1,2,3

γ[12+3(j-1)+j']=Yu[j,j]+Yu[j',j']-Yu[j,j']-Yu[j',j]γ[12+3(j-1)+j']=Y u [j,j]+Y u [j',j']-Y u [j,j']-Y u [j',j]

j=1,2,3,j'=1,2,3j=1,2,3,j'=1,2,3

其中,Yu[j,j]表示引用矩阵Yu第j行第j列的元素。上述两个约束被转化成了凸约束。同时,将另外两个等式约束松弛成不等式约束,得:Among them, Yu [j, j] represents the element of the jth row and the jth column of the reference matrix Yu. The above two constraints are transformed into convex constraints. At the same time, relax the other two equality constraints into inequality constraints, we get:

γ[3(i-1)+j]≥r2[3(i-1)+j],i=1,2,3,4,j=1,2,3γ[3(i-1)+j]≥r 2 [3(i-1)+j], i=1,2,3,4,j=1,2,3

γ[12+3(j-1)+j']≥r2[12+3(j-1)+j'],j=1,2,3,j'=1,2,3γ[12+3(j-1)+j']≥r 2 [12+3(j-1)+j'],j=1,2,3,j'=1,2,3

矩阵Yu中隐藏的其他凸约束为:The other convex constraints hidden in the matrix Yu are:

yj=[Yu[j,4],Yu[j,5]]T,j=1,2,3y j =[Y u [j,4],Y u [j,5]] T ,j=1,2,3

Figure BDA0002912762480000152
Figure BDA0002912762480000152

Yu≥03+2 Y u ≥0 3+2

这样就将所有的约束转化为凸约束,目标函数也是凸函数,问题转化为了一个可解的凸问题。This converts all constraints into convex constraints, the objective function is also a convex function, and the problem is transformed into a solvable convex problem.

优化问题的最终形式为:The final form of the optimization problem is:

Figure BDA0002912762480000153
Figure BDA0002912762480000153

第五步:求解凸优化问题,得到目标位置的估计Step 5: Solve the convex optimization problem to get an estimate of the target position

由第四步得到的凸优化问题可以由MATLAB的CVX工具箱解决,工具是Sedumi。在MATLAB环境下进行编程,输入目标函数与约束条件,就可以得到目标位置的估计。The convex optimization problem obtained by the fourth step can be solved by MATLAB's CVX toolbox, the tool is Sedumi. By programming in the MATLAB environment and inputting the objective function and constraints, the estimation of the target position can be obtained.

Claims (3)

1.一种锚节点位置不确定情况下的目标定位方法,其特征在于,包括以下步骤:1. a target positioning method under the uncertain situation of anchor node position, is characterized in that, comprises the following steps: 第一步,获得传感器网络中各节点之间的TOA量测,对锚节点误差进行建模;The first step is to obtain TOA measurements between nodes in the sensor network and model the anchor node error; 所述的第一步设传感器网络中共M个位置已知但是存在误差的锚节点和N个待定位的目标节点,M≥3,M个锚节点的真实坐标分别为x1,x2,…,xM,N个待估计的目标节点的坐标为y1,y2,…,yN;通过节点之间的通信获得TOA量测集合,包括锚节点与目标节点之间进行通信的TOA量测以及目标节点间进行通信的TOA量测;锚节点与目标节点之间通信获得的TOA量测表达式为
Figure FDA0003675161820000011
其中i,j分别表示编号为i的锚节点与编号为j的目标节点,c为信号传播速度,tij表示锚节点i与目标节点j相互通信所得到的时间量测,rij表示锚节点i与目标节点j之间的距离,eij表示锚节点i与目标节点j之间通信的加性高斯白噪声,eij服从均值为零、方差为
Figure FDA0003675161820000012
的高斯分布;目标节点之间相互通信获得的TOA量测表达式为
Figure FDA0003675161820000013
其中j和j'分别表示编号为j和j'的目标节点,且j≠j',t'jj'表示目标节点j与目标节点j'相互通信所得到的时间量测,r′jj'表示目标节点j与目标节点j之间的距离,e'jj'表示目标节点j与目标节点j之间通信的加性高斯白噪声,e'jj'服从均值为零、方差为
Figure FDA0003675161820000014
的高斯分布;锚节点与目标节点之间的距离与节点位置之间的关系为rij=||xi-xj||2;目标节点与目标节点之间的距离与节点位置之间的关系为r′jj'=||yj-yj'||2;锚节点真实位置与量测得到的位置之间的关系为
Figure FDA0003675161820000015
式中
Figure FDA0003675161820000016
表示量测得到的锚节点位置,ξi是锚节点误差向量,ξi的模的最大值小于等于一个已知的常数ε。
In the first step, there are M anchor nodes with known positions but errors and N target nodes to be located in the sensor network, M≥3, and the real coordinates of the M anchor nodes are x 1 , x 2 ,… ,x M , the coordinates of the N target nodes to be estimated are y 1 , y 2 ,...,y N ; the TOA measurement set is obtained through the communication between nodes, including the TOA amount of communication between the anchor node and the target node The TOA measurement of the communication between the anchor node and the target node; the TOA measurement expression obtained from the communication between the anchor node and the target node is:
Figure FDA0003675161820000011
where i and j represent the anchor node numbered i and the target node numbered j respectively, c is the signal propagation speed, tij represents the time measurement obtained by the mutual communication between the anchor node i and the target node j , and rij represents the anchor node The distance between i and the target node j, e ij represents the additive white Gaussian noise of the communication between the anchor node i and the target node j, e ij obeys the mean value of zero and the variance is
Figure FDA0003675161820000012
The Gaussian distribution of ; the TOA measurement expression obtained by mutual communication between target nodes is:
Figure FDA0003675161820000013
where j and j' represent the target nodes numbered j and j' respectively, and j≠j', t'jj' represents the time measurement obtained by the mutual communication between target node j and target node j', r'jj' represents The distance between the target node j and the target node j, e'jj' represents the additive white Gaussian noise of the communication between the target node j and the target node j, e'jj' obeys the mean zero and the variance is
Figure FDA0003675161820000014
The Gaussian distribution of ; the relationship between the distance between the anchor node and the target node and the node position is r ij =||x i -x j || 2 ; The relationship is r'jj' =||y j -y j' || 2 ; the relationship between the real position of the anchor node and the measured position is
Figure FDA0003675161820000015
in the formula
Figure FDA0003675161820000016
Represents the measured anchor node position, ξ i is the anchor node error vector, and the maximum value of the modulus of ξ i is less than or equal to a known constant ε.
第二步,在锚节点与目标节点通信的原始TOA量测中引入锚节点误差项;In the second step, the anchor node error term is introduced into the original TOA measurement of the communication between the anchor node and the target node; 第三步,对锚节点误差项进行转化,利用S-过程消除锚节点误差向量;The third step is to transform the anchor node error term, and use the S-process to eliminate the anchor node error vector; 所述的第三步采用最大似然估计法来估计目标的位置,设Xu=[y1,y2,…,yN]为未知目标节点坐标的集合,Xa=[x1,x2,…,xK]是所有锚节点坐标的集合,同时定义dij=tij×c,d'jj'=t'jj'×c,dij和d'jj'分别为锚节点i与目标节点j之间,以及目标节点j与j'(j≠j')之间的带噪声的距离量测值;根据已有条件,将待优化的原始目标函数表示为
Figure FDA0003675161820000021
将第二步中的锚节点误差项代入,得到min-max次优化问题
Figure FDA0003675161820000022
经过对目标函数的向量化,以及通过应用S-过程消除锚节点误差向量,原始次优化问题被转化为
Figure FDA0003675161820000023
其中,λ和μ是为了转化锚节点误差向量新引进的常数,
Figure FDA00036751618200000211
表示矩阵正定,I表示单位矩阵,Tr(·)表示求括号里面矩阵的迹,d1=[d11,d12,…,dMN]T为锚节点与目标节点通信的带噪声的距离量测,d2=[d′12,d′13,…,d'N,N-1]T为目标节点之间通信的带噪声的距离量测,
Figure FDA0003675161820000024
为所有的带噪声的距离量测集合;定义
Figure FDA0003675161820000025
为锚节点与目标节点之间通信的真实距离量测,r2=[r′12,r′13,…,r'N,N-1]T为目标节点之间通信的真实距离量测,
Figure FDA0003675161820000026
为所有的真实距离量测集合,
Figure FDA0003675161820000027
G=blkdiag(G1,G2),Γ=1(M+N-1)×N,diag{·}表示对角矩阵,矩阵的对角线元素为大括号中的元素,blkdiag{·}表示分块对角矩阵,1(M+N-1)×N表示具有(M+N-1)×N个元素全为1的列向量,γ[·],r[·]表示引用向量γ,r中的元素;同时定义中间变量
Figure FDA0003675161820000028
γjj'以及向量γ12和γ,
Figure FDA0003675161820000029
γ2=[γ1213,…,γN,N-1]T
Figure FDA00036751618200000210
The third step uses the maximum likelihood estimation method to estimate the position of the target. Let Xu =[y 1 ,y 2 ,...,y N ] be the set of unknown target node coordinates, X a =[x 1 ,x 2 ,...,x K ] is the set of coordinates of all anchor nodes, while defining d ij =t ij ×c, d'jj'=t'jj' ×c, d ij and d'jj' are anchor nodes i and d'jj' respectively. The distance measurement value with noise between target nodes j, and between target nodes j and j'(j≠j'); according to the existing conditions, the original objective function to be optimized is expressed as
Figure FDA0003675161820000021
Substitute the anchor node error term in the second step to get the min-max optimization problem
Figure FDA0003675161820000022
After vectorizing the objective function, and removing the anchor node error vector by applying an S-process, the original suboptimization problem is transformed into
Figure FDA0003675161820000023
where λ and μ are newly introduced constants to transform the anchor node error vector,
Figure FDA00036751618200000211
Indicates that the matrix is positive definite, I represents the identity matrix, Tr(·) represents the trace of the matrix in the brackets, d 1 =[d 11 ,d 12 ,...,d MN ] T is the distance between the anchor node and the target node with noise for communication d 2 =[d' 12 ,d' 13 ,...,d' N,N-1 ] T is the distance measurement with noise for communication between target nodes,
Figure FDA0003675161820000024
is the set of all noisy distance measurements; define
Figure FDA0003675161820000025
is the real distance measurement of the communication between the anchor node and the target node, r 2 =[r′ 12 ,r′ 13 ,...,r' N,N-1 ] T is the real distance measurement of the communication between the target nodes,
Figure FDA0003675161820000026
is the set of all true distance measurements,
Figure FDA0003675161820000027
G=blkdiag(G 1 , G 2 ), Γ=1 (M+N-1)×N , diag{·} represents a diagonal matrix, and the diagonal elements of the matrix are the elements in curly brackets, blkdiag{·} Represents a block diagonal matrix, 1 (M+N-1)×N represents a column vector with (M+N-1)×N elements all 1, γ[ ], r[ ] represents the reference vector γ , elements in r; also define intermediate variables
Figure FDA0003675161820000028
γ jj' and the vectors γ 1 , γ 2 and γ,
Figure FDA0003675161820000029
γ 2 =[γ 1213 ,...,γ N,N-1 ] T ,
Figure FDA00036751618200000210
第四步,通过对目标函数及约束条件的转化和松弛,将问题转化为一个凸问题;The fourth step is to transform the problem into a convex problem by transforming and relaxing the objective function and constraints; 第五步,求解凸优化问题,得到目标位置的估计。The fifth step is to solve the convex optimization problem to obtain an estimate of the target position.
2.根据权利要求1所述的锚节点位置不确定情况下的目标定位方法,其特征在于,所述的第二步利用一阶泰勒展开式将锚节点真实位置与量测位置的关系进一步转化为
Figure FDA0003675161820000031
其中o(||ξi||)表示锚节点误差向量模||ξi||的高阶无穷小量;令
Figure FDA0003675161820000032
则推得|δij|≤ε,其中
Figure FDA0003675161820000033
表示锚节点i与目标节点j之间的距离,与rij不同,这里的使用的锚节点i的坐标是已知的带有误差的锚节点位置,δij表示锚节点i与目标节点j之间的模的误差值;则最终锚节点与目标节点之间通信的TOA量测表示为
Figure FDA0003675161820000034
2. the target positioning method under the uncertain situation of anchor node position according to claim 1, is characterized in that, described second step utilizes first-order Taylor expansion formula to further transform the relationship between anchor node real position and measurement position for
Figure FDA0003675161820000031
where o(||ξ i ||) represents the higher-order infinitesimal of the anchor node error vector modulo ||ξ i ||; let
Figure FDA0003675161820000032
Then it can be deduced that |δ ij |≤ε, where
Figure FDA0003675161820000033
Represents the distance between anchor node i and target node j. Different from r ij , the coordinates of anchor node i used here are the known anchor node positions with errors, and δ ij represents the distance between anchor node i and target node j. The error value of the modulus between the final anchor node and the target node is expressed as
Figure FDA0003675161820000034
3.根据权利要求1所述的锚节点位置不确定情况下的目标定位方法,其特征在于,所述的第四步引入中间矩阵
Figure FDA0003675161820000035
优化问题的最终形式为:
3. the target positioning method under the uncertain situation of anchor node position according to claim 1, is characterized in that, described 4th step introduces intermediate matrix
Figure FDA0003675161820000035
The final form of the optimization problem is:
Figure FDA0003675161820000036
Figure FDA0003675161820000036
subject tosubject to
Figure FDA0003675161820000037
Figure FDA0003675161820000037
γ[(i-1)N+j]≥r2[(i-1)N+j]γ[(i-1)N+j]≥r 2 [(i-1)N+j] γ[MN+(j-1)N+j']≥r2[MN+(j-1)N+j']γ[MN+(j-1)N+j']≥r 2 [MN+(j-1)N+j']
Figure FDA0003675161820000038
Figure FDA0003675161820000038
γ[MN+(j-1)N+j']=Yu[j,j]+Yu[j',j']-Yu[j,j']-Yu[j',j]γ[MN+(j-1)N+j']=Y u [j,j]+Y u [j',j']-Y u [j,j']-Y u [j',j] Xu(:,j)=Yu[j,N+1:N+l]X u (:,j)=Y u [j,N+1:N+l] Yu[N+1:N+l,N+1:N+l]=Il Y u [N+1:N+1,N+1:N+ 1 ]=I1
Figure FDA0003675161820000039
Figure FDA0003675161820000039
其中,Yu[w,v]表示引用矩阵Yu第w行,第v列的元素,w和v必须是整数;Yu[j,N+1:N+l]表示矩阵Yu第j行,第N+1到N+l列的所有元素,Yu[N+1:N+l,N+1:N+l]表示矩阵Yu第N+1到N+l行,第N+1到N+l列的所有元素,Xu(:,j)表示矩阵Xu中第j列所有元素组成的列向量。Among them, Yu [w,v] refers to the element in the wth row and vth column of the reference matrix Yu, and w and v must be integers; Yu [j, N+1:N+l] means the matrix Yu jth Row, all elements in columns N+1 to N+l, Yu [N+1:N+l,N+1:N+l] represents matrix Yu Row N+1 to N+l, Nth All elements from +1 to N+l columns, X u (:,j) represents a column vector composed of all elements in the jth column of the matrix X u .
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