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

The invention provides a target positioning method under the condition that the position of an anchor node is uncertain. After vectorization is carried out on the objective function, an anchor node error vector is eliminated by utilizing an S-process, meanwhile, a convex constraint is introduced, and finally, the problem is changed into a convex problem which can be solved through conversion and relaxation. The method can still accurately obtain the estimation of the target position when the positioning system is influenced by the environment and causes larger deviation between the real position of the anchor node and the measured position, and has stronger robustness and practicability.

Description

Target positioning method under condition of uncertain anchor node position
Technical Field
The invention relates to a target positioning method, belongs to the field of signal processing, and is suitable for a self-positioning system for positioning a plurality of target nodes by a plurality of anchor nodes in a wireless sensor network.
Background
The flexibility, broad coverage and ease of deployment characteristics of Wireless Sensor Networks (WSNs) have attracted considerable attention over the past few years. In general, a wireless sensor network is composed of a cluster of low-cost and low-power-consumption sensor nodes distributed in a certain spatial range, which can be used to perform common signal processing tasks, such as detection, positioning, target tracking and monitoring of target state changes. The target positioning is one of the most basic and important tasks in the wireless sensor network, and the acquisition of many physical quantities is premised on the definition of the node positions. Many types of sensor measurements may be used for target location, such as Received Signal Strength (RSS), angle of arrival (AOA), time of arrival (TOA), time difference of arrival (TDOA). In these types of metrology-value-based object location methods, TOA and TDOA-based object location schemes strike a good balance between location performance and computational complexity. The two positioning methods can effectively avoid deploying expensive sensors like the positioning scheme based on AOA, and can also effectively reduce larger positioning errors caused by the positioning method based on RSS.
In the TOA positioning problem, the problem that clocks of an anchor node and a target node to be positioned are not synchronous often exists, and two methods are mainly used for eliminating the influence of time asynchronism, namely, the TOA positioning problem is converted into the TDOA positioning problem; and secondly, using the two-way TOA, namely enabling the anchor node and the target node to carry out two-way information exchange, and solving the positioning problem by utilizing the time stamps corresponding to the anchor node and the target node. In addition, most of the conventional models assume that the position of the anchor node is accurately known, but in practical applications, this condition is difficult to achieve because there is always a certain error between the actual position of the anchor node and the measured position due to the influence of the environment. For example, even if the position of a water buoy node is obtained by using a GPS in advance, the node may drift due to the influence of ocean currents during the positioning process, so that the real position of the node deviates from the measured position. If some measures are not taken, the positioning performance is reduced.
And the difference value between the real position coordinates of the anchor nodes and the measured position coordinates of the anchor nodes is the error vector of the anchor nodes. At present, there are two methods for modeling the error vector of the anchor node, one is to assume that the vector follows a gaussian distribution with a mean value of zero, and the other is not to make any prior assumption on the vector, but only to assume that the maximum value of the modulus of the vector is known. In practical applications, the latter case is more practical because the maximum value of the error model of the anchor node under different environments is easier to estimate than the covariance matrix of the gaussian distribution. However, most of the existing target positioning methods do not consider that the position of the anchor node has errors due to environmental factors, or assume that the error vector of the position of the anchor node follows Gaussian distribution with zero mean. In practice, the statistical distribution of the error is often difficult to obtain, and the maximum value of the error vector mode of the anchor node is easier to obtain by estimation. In the existing method with the known maximum value of the error vector modulus, Xu et al propose a method in which a plurality of anchor nodes locate a single target node and only the known maximum value of the error vector modulus of the anchor nodes is assumed, however, this method only considers the location scene of a single node and needs to add a penalty term, and the penalty factor corresponding to the penalty term needs to be manually adjusted in the actual operation process, which increases the computational complexity and reduces the practicability.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides the target positioning method under the condition that the position of the anchor node is uncertain, only the maximum value of a module of an error vector of the known anchor node is needed, the statistical distribution of the known error is not needed in advance, the causticity degree required by prior information is weakened, meanwhile, a punishment item needing manual adjustment is not set, and the practicability is improved.
The technical scheme adopted by the invention for solving the technical problem comprises the following steps:
firstly, TOA measurement among nodes in a sensor network is obtained, and modeling is carried out on an anchor node error;
secondly, introducing an anchor node error term into original TOA measurement of communication between the anchor node and the target node;
thirdly, converting the error items of the anchor nodes, and eliminating error vectors of the anchor nodes by utilizing an S-process;
fourthly, converting the problem into a convex problem through conversion and relaxation of the objective function and the constraint condition;
and fifthly, solving the convex optimization problem to obtain the estimation of the target position.
The first step is to set M anchor nodes with known positions but errors and N target nodes to be positioned in the sensor network, wherein M is more than or equal to 3, and the real coordinates of the M anchor nodes are x1,x2,…,xMThe coordinates of N target nodes to be estimated are y1,y2,…,yN(ii) a Obtaining a TOA measurement set through communication between nodes, wherein the TOA measurement set comprises TOA measurement for communication between an anchor node and a target node and TOA measurement for communication between the target nodes; the TOA measurement expression obtained by the communication between the anchor node and the target node is
Figure BDA0002912762480000021
Where i, j represent anchor node numbered i and target node numbered j, respectively, c is signal propagation speed, tijRepresenting a time measurement, r, obtained by the anchor node i communicating with the target node jijRepresents the distance between anchor node i and target node j, eijAdditive white Gaussian noise, e, representing communication between anchor node i and target node jijObedience mean value of zero and variance of
Figure BDA0002912762480000022
(ii) a gaussian distribution of; the TOA measurement expression obtained by mutual communication between the target nodes is
Figure BDA0002912762480000023
Where j and j 'represent target nodes numbered j and j', respectively, and j ≠ j ', t'jj'Represents a time measurement, r 'obtained by the mutual communication of the target node j and the target node j'jj'Representing the distance, e ', between the target node j and the target node j'jj'Representing additive white Gaussian noise, e 'of communication between target node j and target node j'jj'Obedience mean value of zero and variance of
Figure BDA0002912762480000031
(ii) a gaussian distribution of; the relation between the distance between the anchor node and the target node and the node position is rij＝||xi-xj||2(ii) a The relation between the distance between the target nodes and the node positions is r'jj'＝||yj-yj'||2(ii) a The relation between the actual position of the anchor node and the measured position is
Figure BDA0002912762480000032
In the formula
Figure BDA0002912762480000033
Indicating measured anchor node position, ξiIs the anchor node error vector, ξiHas a maximum value equal to or less than a known constant epsilon.
In the second step, the relation between the real position and the measured position of the anchor node is further converted into a relation between the real position and the measured position of the anchor node by using a first-order Taylor expansion
Figure BDA0002912762480000034
Where o (| | ξ)i| |) represents the anchor node error vector modulo | | ξiThe high order infinitesimal amount of | l; order to
Figure BDA0002912762480000035
Then the | δ is obtainedij| < epsilon, wherein
Figure BDA0002912762480000036
Represents the distance between the anchor node i and the target node j, and rijIn contrast, the coordinates of anchor node i used herein are the known anchor node locations with errors, δijAn error value representing a modulus between the anchor node i and the target node j; the TOA measurement for communication between the final anchor node and the target node is represented as
Figure BDA0002912762480000037
The third step adopts a maximum likelihood estimation method to estimate the position of the target, and X is setu＝[y1,y2,…,yN]For a set of unknown target node coordinates, Xa＝[x1,x2,…,xK]Is all anchor node coordinatesIs defined at the same time as dij＝tij×c，d'jj'＝t'jj'×c，dijAnd d'jj'Respectively measuring distance values with noise between an anchor node i and a target node j and between the target nodes j and j '(j ≠ j'); according to the existing conditions, the original objective function to be optimized is expressed as
Figure BDA0002912762480000038
Substituting the anchor node error item in the second step to obtain min-max suboptimal problem
Figure BDA0002912762480000039
Through vectorization of the objective function and elimination of the anchor node error vector by applying the S-process, the original sub-optimization problem is transformed into
Figure BDA0002912762480000041
Wherein, λ and μ are constants introduced for transforming anchor node error vector, which is ≧ matrix positive definite, I is identity matrix, Tr (-) is trace of matrix in parentheses, d1＝[d11,d12,…,dMN]TNoisy distance measurement for anchor node to target node communication, d2＝[d'12,d'13,…,d'N,N-1]TNoisy distance measurements for communication between target nodes,
Figure BDA0002912762480000042
aggregating all noisy distance measurements; definition of
Figure BDA0002912762480000043
For true distance measurement of communication between anchor node and target node, r2＝[r'12,r'13,…,r'N,N-1]TFor true distance measurements of communication between target nodes,
Figure BDA0002912762480000044
for all of the set of true distance measurements,
Figure BDA0002912762480000045
G＝blkdiag(G1,G2)，Γ＝1(M+N-1)×Ndiag {. is a diagonal matrix, diagonal elements of the matrix are bracketed elements, blkdiag {. is a block diagonal matrix, 1(M+N-1)×NRepresenting a column vector having (M + N-1) xN elements all 1, gamma [ ·],r[·]Representing elements in a reference vector γ, r; defining intermediate variables simultaneously
Figure BDA0002912762480000046
γjj'And a vector gamma1,γ2And a gamma-ray source,
Figure BDA00029127624800000411
γ2＝[γ12,γ13,…,γN,N-1]T，
Figure BDA0002912762480000049
the fourth step introduces an intermediate matrix
Figure BDA00029127624800000410
The final form of the optimization problem is:
Figure BDA0002912762480000051
wherein Y isu[w,v]Represents the reference matrix YuRow w, column v, w and v must be integers;
Yu[j,N+1:N+l]representation matrix YuAll elements of row j, columns N +1 to N + l, Yu[N+1:N+l,N+1:N+l]Representation matrix YuAll elements of rows N +1 to N + l, columns N +1 to N + l, Xu(: j) represents a matrix XuA column vector consisting of all elements of the jth column of (a).
The beneficial effects of the invention are: it is more realistic to assume that only the maximum of the modulus of the anchor node error vector is known. And after TOA measurement information among the anchor node, the target node and the target node is obtained, listing a target function and introducing an error amount of the anchor node. After vectorization is carried out on the objective function, an anchor node error vector is eliminated by utilizing an S-process, meanwhile, a convex constraint is introduced, and finally, the problem is changed into a convex problem which can be solved through conversion and relaxation. The method can still accurately obtain the estimation of the target position when the actual position of the anchor node and the measured position have larger deviation due to the influence of the environment on the positioning system, and has stronger robustness and practicability.
Drawings
Fig. 1 is a block diagram of a target positioning method in the case of an uncertain anchor node position.
FIG. 2 is a graph of method performance versus TOA measurement noise variance.
FIG. 3 is a graph of method performance versus maximum value of the anchor node error vector norm.
FIG. 4 is a graph of method performance versus anchor node number.
Fig. 5 is a graph of method performance versus the number of target nodes to be located.
Detailed Description
The present invention will be further described with reference to the following drawings and examples, which include, but are not limited to, the following examples.
Aiming at the problem that the anchor node position is influenced by the environment to cause errors with the measured position in the actual positioning problem, the positioning method based on the anchor node position with errors is provided. And selecting an l-dimensional positioning scene, wherein l is 2 or 3. And setting M anchor nodes with known positions but errors in the sensor network, wherein M is more than or equal to 3, and N nodes are target nodes to be positioned. The real coordinates of M anchor nodes with known positions are respectively set as x1,x2,…,xMThe coordinates of N target nodes to be estimated are y1,y2,…,yN。
The invention comprises the following main steps:
the first step is as follows: obtaining original TOA measurement, modeling anchor node error
A TOA measurement set is obtained by communication between nodes, and the set is composed of two parts, namely TOA measurement for communication between the anchor node and the target node and TOA measurement for communication between the target nodes. The TOA measurement expression obtained by the communication between the anchor node and the target node is as follows:
Figure BDA0002912762480000061
wherein i and j are positive integers which respectively represent a node numbered i and a node numbered j. c is the signal propagation velocity, tijRepresents a time measurement, r, obtained by the anchor node i and the target node j communicating with each otherijRepresents the distance between anchor node i and target node j, eijAdditive white Gaussian noise representing communication between anchor node i and target node j obeys a mean of zero and a variance of
Figure BDA0002912762480000062
Gaussian distribution of (a).
The TOA measurement expression obtained by mutual communication between the target nodes is as follows:
Figure BDA0002912762480000063
where j and j 'represent target nodes numbered j and j', respectively, and j ≠ j ', t'jj'Represents a time measurement, r 'obtained by the mutual communication of the target node j and the target node j'jj'Representing the distance, e ', between the target node j and the target node j'jj'Additive white Gaussian noise representing communication between a target node j and the target node j, obeys a mean of zero and a variance of
Figure BDA0002912762480000064
Gaussian distribution of (a).
The relationship between the distance between the anchor node and the target node and the node position is as follows:
rij＝||xi-xj||2
i＝1,2,…,M,j＝1,2,…,N
wherein | | | purple hair2Representing the 2-norm of the vector.
Similarly, the relationship between the distance between the target node and the node position is as follows:
r'jj'＝||yj-yj'||2
j＝1,2,…,N,j'＝1,2,…,N
due to the influence of environmental factors, an error exists between the measured anchor node position and the actual position of the anchor node, and the relationship between the actual position of the anchor node and the measured position is as follows:
Figure BDA0002912762480000071
in the formula
Figure BDA0002912762480000072
Indicating measured anchor node position, ξiIs an anchor node error vector whose maximum value of its modulus is less than a known constant epsilon, i.e.
||ξi||≤ε
The second step is that: anchor node error term is introduced into original TOA measurement of anchor node and target node communication
The relation between the real position of the anchor node and the measured position can be further converted into the following relation by using a first-order Taylor expansion formula:
Figure BDA0002912762480000073
i＝1,2,…,M,j＝1,2,…,N
where o (| | ξ)i| |) represents the anchor node error vector modulo | | ξiThe higher order of | is infinitesimally small.
Order to
Figure BDA0002912762480000074
Then it can be deduced that:
|δij|≤ε
wherein
Figure BDA0002912762480000075
Represents the distance between the anchor node i and the target node j, and rijIn contrast, the coordinates of anchor node i as used herein is the known anchor node location with error. Delta. for the preparation of a coatingijAn error value representing a modulus between the anchor node i and the target node j, and the symbol "|" is a sign of solving an absolute value.
The TOA measurement communicated between the final anchor node and the target node may be expressed as:
Figure BDA0002912762480000076
the third step: transforming anchor node error terms
The position of the target is estimated using maximum likelihood estimation. Let Xu＝[y1,y2,…,yN]For a set of unknown target node coordinates, Xa＝[x1,x2,…,xK]Is the set of coordinates for all anchor nodes. Simultaneously defining:
dij＝tij×c,i＝1,2,…,M,j＝1,2,…,N
d'jj'＝t'jj'×c,j＝1,2,…,N,j'＝1,2,…,N
dijand d'jj'The noisy distance measurements between anchor node i and target node j, and between target nodes j and j '(j ≠ j'), respectively.
According to existing conditions, the original objective function to be optimized can be expressed as:
Figure BDA0002912762480000081
substituting the anchor node error term in the second step can obtain a min-max suboptimal problem:
Figure BDA0002912762480000082
subject to
Figure BDA0002912762480000083
r'jj'＝||yj-yj'||,j＝1,2,…,M,j'＝1,2,…,N
wherein min represents minimum, max represents maximum, and Σ represents summation. The subject to indicates "constrained to".
The sub-optimization problem is a non-linear and non-convex problem, and the objective function is in the form of element summation, and the objective function contains a node error term deltaijThis sub-optimization problem is very difficult to solve. Therefore, firstly, the objective function is vectorized, the vectorized objective function and the constraint conditions are given, then the error item of the anchor node is converted into a vector, and only one vector is processed to easily process a single element; after the vectorization is completed, an S-process (S-procedure) is used for eliminating an anchor node error vector, an anchor node error term is eliminated by introducing two constant variables, and the anchor node error term is converted into a convex constraint.
Through vectorization of the objective function and elimination of the anchor node error vector by applying the S-process, the original sub-optimization problem is transformed into:
Figure BDA0002912762480000084
wherein,
Figure BDA0002912762480000085
the root of the square is expressed, lambda and mu are constants which are newly introduced for transforming the error vector of the anchor node, and are equal to or more than positive definite of the matrix, and I is an identity matrixTr (-) indicates finding the trace of the matrix inside the parenthesis. And d is1＝[d11,d12,…,dMN]TNoisy distance measurement for communication of anchor node and target node, d2＝[d'12,d'13,…,d'N,N-1]TNoisy distance measurements for communication between target nodes,
Figure BDA0002912762480000091
aggregating all noisy distance measurements; definition of
Figure BDA0002912762480000092
A true distance measure, r, for communication between the anchor node and the target node2＝[r'12,r'13,…,r'N,N-1]TFor true distance measurements of communication between target nodes,
Figure BDA0002912762480000093
all the true distance measurements are aggregated.
Figure BDA0002912762480000094
G＝blkdiag(G1,G2)，Γ＝1(M+N-1)×N. Wherein diag {. is } represents a diagonal matrix, diagonal elements of the matrix are in parenthesis, blkdiag {. is } represents a block diagonal matrix, 1(M+N-1)×NRepresenting a column vector having (M + N-1) xN elements all 1, gamma [ ·],r[·]Representing the elements in the reference vector gamma, r. Defining intermediate variables simultaneously
Figure BDA00029127624800000910
γjj'And a vector gamma1,γ2And γ:
Figure BDA0002912762480000095
Figure BDA0002912762480000096
Figure BDA0002912762480000097
Figure BDA0002912762480000098
the former similar processing only aims at the problem of single-node positioning, the current processing aims at the problem of multi-node positioning, the form of the processed objective function is different from that of other methods, and other similar methods always assume that the whole objective function is smaller than a constant mu, so that the first half Tr [ G (gamma) is dividedT-2drT)]And also put into the convex constraint obtained by the S-process, while the current processing method still keeps the first half part in the objective function and does not put into the constraint. This process reduces the degree of relaxation and optimizes the structure of the objective function compared to previous methods.
The fourth step: convert the problem into a convex problem
After the transformation in the third step, the error term of the anchor node is eliminated, but the problem is still not a solvable convex problem. The problem is then transformed into a convex problem by a series of transformations and relaxations of the objective function and constraints. The relaxation method used is a semi-positive definite relaxation method, and finally the objective function is transformed into a convex function and the constraints are transformed into convex constraints, thereby making the problem solvable.
To convert the problem into a convex one, an intermediate matrix Y is introducedu：
Figure BDA0002912762480000099
By the intermediate matrix and the variable gamma, X to be optimizeduEstablishing a connection and taking appropriate slack transforms all constraints into convex constraints, and the objective function is also a convex function, the problem is transformed intoA solvable convex problem.
The final form of the optimization problem is:
Figure BDA0002912762480000101
wherein, Yu[w,v]Represents the reference matrix YuRow w, column v, w and v must be integers; y isu[j,N+1:N+l]Representation matrix YuAll elements of row j, columns N +1 to N + l, Yu[N+1:N+l,N+1:N+l]Representation matrix YuAll elements of rows N +1 to N + l and columns N +1 to N + l, Xu(j) represents a matrix XuA column vector consisting of all elements of the jth column of (1).
In the previous similar algorithms, penalty terms are often added into an objective function, namely a very small positive number (penalty factor) is added to multiply all elements and/or traces of a certain matrix to be optimized, the method needs to manually select the penalty factors, the penalty factor is improperly selected, the performance of the algorithm is reduced, and therefore the practicability is not strong, and the penalty terms are not added in the method, so that the manual selection of the penalty factors is avoided, and the practicability is increased.
The fifth step: solving a convex optimization problem to obtain an estimate of the target position
And solving the convex optimization problem obtained in the fourth step by using a CVX convex optimization toolbox in the MATLAB so as to obtain the estimation of the target position.
The embodiment of the present invention considers the problem of positioning in two-dimensional space, and there are 4 anchor nodes with known positions and 3 target nodes to be positioned in one 1200M × 1200M area, i.e., M is 4 and N is 3. Let the real coordinates of 4 anchor nodes with known positions be x1,x2,x3,x4The coordinates of N target nodes to be estimated are y1,y2,y3。
The first step is as follows: obtaining original TOA measurement, modeling anchor node error
A TOA measurement set is obtained through communication between nodes. Wherein, the TOA measurement expression obtained by the communication between the anchor node and the target node is:
Figure BDA0002912762480000111
i and j are positive integers, which respectively represent a node numbered i and a node numbered j. c is the signal propagation velocity, tijRepresents a time measurement, r, obtained by the anchor node i and the target node j communicating with each otherijRepresents the distance between the anchor node i and the target node j, eijAdditive white Gaussian noise representing communication between anchor node i and target node j obeys a mean of zero and a variance of
Figure BDA0002912762480000112
Gaussian distribution of (a).
The TOA measurement expression obtained by mutual communication between the target nodes is as follows:
Figure BDA0002912762480000113
where j and j 'represent target nodes numbered j and j', respectively, and j ≠ j ', t'jj'Represents a time measurement, r 'obtained by the mutual communication of the target node j and the target node j'jj'Representing the distance, e ', between the target node j and the target node j'jj'An additive white Gaussian noise representing the communication between the target node j and the target node j, which obeys a mean of zero and a variance of
Figure BDA0002912762480000114
Gaussian distribution of (a).
The relationship between the distance between the anchor node and the target node and the node position is as follows:
rij＝||xi-xj||2
i＝1,2,3，4,j＝1,2,3
wherein | | | calving2Representing the 2-norm of the vector.
Similarly, the relationship between the distance between the target node and the node position is as follows:
r'jj'＝||yj-yj'||2,j＝1,2,3,j'＝1,2,3
due to the influence of environmental factors, an error exists between the measured anchor node position and the actual position of the anchor node, and the relationship between the actual position of the anchor node and the measured position is as follows:
Figure BDA0002912762480000115
in the formula
Figure BDA0002912762480000116
Indicating measured anchor node position, ξiIs an anchor node error vector whose maximum value of its modulus is less than a known constant epsilon, i.e.
||ξi||≤ε
Here, epsilon is set to 0.5 m.
The second step is that: anchor node error term is introduced into original TOA measurement of anchor node and target node communication
The relation between the real position of the anchor node and the measured position can be further converted into the following relation by using a first-order Taylor expansion formula:
Figure BDA0002912762480000121
i＝1,2,3，4,j＝1,2,3
wherein o (| | xi)i| |) represents the anchor node error vector modulo | | ξiThe higher order of | is an infinitesimal quantity.
Order to
Figure BDA0002912762480000122
Then there is
Figure BDA0002912762480000123
Namely that
|δij|≤ε
Wherein
Figure BDA0002912762480000124
Represents the distance between the anchor node i and the target node j, and rijIn contrast, the coordinates of anchor node i as used herein is the known anchor node location with error. Delta. for the preparation of a coatingijAn error value representing a modulus between the anchor node i and the target node j, and the symbol "|" is a sign of solving an absolute value.
The TOA measurement communicated between the final anchor node and the target node may be expressed as:
Figure BDA0002912762480000125
the third step: transforming anchor node error terms
The position of the target is estimated using maximum likelihood estimation. Let Xu＝[y1,y2,y3]For a set of unknown target node coordinates, Xa＝[x1,x2,x3,x4]Is the set of coordinates of all anchor nodes. Simultaneously defining:
dij＝tij×c,i＝1,2,3,4,j＝1,2,3
d'jj'＝t'jj'×c,j＝1,2,3,j'＝1,2,3
dijand d'jj'The noisy distance measurements between anchor node i and target node j, and between target nodes j and j '(j ≠ j'), respectively.
According to the existing conditions, the original objective function to be optimized can be represented as:
Figure BDA0002912762480000126
substituting the anchor node error term in the second step into the second step can obtain a min-max suboptimal problem:
Figure BDA0002912762480000131
subject to
Figure BDA0002912762480000132
r'jj'＝||yj-yj'||,j＝1,2,3,j'＝1,2,3
wherein min represents minimum, max represents maximum, and Σ represents summation. subject to represents "constrained to".
Next, the objective function is vectorized. For a single TOA measurement, the anchor node error satisfies | δijIf | < epsilon, the error vector of the vectorized anchor node meets the following conditions:
Figure BDA0002912762480000133
wherein δ is [ δ ]11,δ12,…,δ43]TIs a set of anchor node modulo errors for M anchor nodes communicating with N target nodes,
Figure BDA0002912762480000134
indicating the open square root. Likewise, define d1＝[d11,d12,…,d43]TNoisy distance measurement for communication of anchor node and target node, d2＝[d'12,d'13,…,d'32]TNoisy distance measurements for communication between target nodes,
Figure BDA0002912762480000135
aggregating all noisy distance measurements; definition of
Figure BDA0002912762480000136
Is an anchor node andtrue distance measurement, r, of communication between target nodes2＝[r'12,r'13,…,r3'2]TFor true distance measurements of communication between target nodes,
Figure BDA0002912762480000137
all the true distance measurements are aggregated. Defining intermediate variables simultaneously
Figure BDA0002912762480000138
γjj'And a vector gamma1,γ2And γ:
Figure BDA0002912762480000139
Figure BDA00029127624800001310
Figure BDA00029127624800001311
Figure BDA00029127624800001312
the original sub-optimization problem can be translated into:
Figure BDA00029127624800001313
subject to
γ[3(i-1)+j]＝r2[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'＝1,2,3
Figure BDA00029127624800001314
r[12+3(i-1)+j]＝||yj-yj'||,j＝1,2,3,j'＝1,2,3
wherein
Figure BDA00029127624800001315
G＝blkdiag(G1,G2)，Γ＝1(3+4-1)×3. Wherein diag {. DEG } represents a diagonal matrix, diagonal elements of the matrix are elements in braces, blkdiag {. DEG represents a block diagonal matrix, 1(3+4-1)×3Representing a column vector having 18 elements all of 1, gamma [ ·],r[·]Representing the elements in the reference vector gamma, r.
Eliminating max in min-max optimization problem by introducing constant factor μ, i.e. for
Figure BDA0002912762480000141
The following is true:
δTG1δ-2δTG1(d1-r1)≤μ
namely:
Figure BDA0002912762480000142
wherein
Figure BDA0002912762480000143
Indicating a sufficient condition.
The above formula can be further expressed as:
Figure BDA0002912762480000144
by applying the S-process, the above recursive relationship is finally converted into a convex constraint:
Figure BDA0002912762480000145
namely:
Figure BDA0002912762480000146
where lambda and mu are newly introduced constants for transforming the anchor node error vector,
Figure BDA0002912762480000148
indicating "presence" ≧ which indicates the matrix semi-positive definite.
Meanwhile, the objective function to be optimized may be converted into:
Tr[G(ΓγT-2drT)]+μ
in the formula, Tr (. cndot.) represents finding the trace of the matrix inside the parentheses.
The fourth step: convert the problem into a convex problem
After the transformation in the third step, the error term of the anchor node is eliminated, but the problem is still not a convex problem. The problem is then transformed into a convex problem by a series of transformations and relaxations of the objective function and constraints. The relaxation method used is a semi-positive definite relaxation method, and finally the objective function is transformed into a convex function and the constraints are transformed into convex constraints, thereby making the problem solvable.
To convert the problem into a convex one, an intermediate matrix Y is introducedu：
Figure BDA0002912762480000147
Then matrix YuThe relationship between the elements in (1) and the existing constraints is as follows:
Figure BDA0002912762480000151
i＝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]
j＝1,2,3,j'＝1,2,3
wherein, Yu[j,j]Representing a reference matrix YuRow jth, column jth. The two constraints are converted into convex constraints. Meanwhile, relaxing the other two equality constraints into inequality constraints to obtain:
γ[3(i-1)+j]≥r2[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'＝1,2,3
matrix YuThe other convex constraints hidden in (1) are:
yj＝[Yu[j,4],Yu[j,5]]T,j＝1,2,3
Figure BDA0002912762480000152
Yu≥03+2
thus, all constraints are converted into convex constraints, the objective function is also a convex function, and the problem is converted into a solvable convex problem.
The final form of the optimization problem is:
Figure BDA0002912762480000153
the fifth step: solving a convex optimization problem to obtain an estimate of the target position
The convex optimization problem resulting from the fourth step can be solved by the CVX toolbox of MATLAB, the tool being Sedumi. Programming under the MATLAB environment, and inputting an objective function and a constraint condition to obtain the estimation of the target position.

Claims (3)

1. A target positioning method under the condition that the position of an anchor node is uncertain is characterized by comprising the following steps:
firstly, TOA measurement among nodes in a sensor network is obtained, and modeling is carried out on anchor node errors;
the first step is set with the position of M in the sensor network known butThe target nodes are the anchor nodes with errors and N target nodes to be positioned, M is more than or equal to 3, and the real coordinates of the M anchor nodes are x1,x2,…,xMThe coordinates of N target nodes to be estimated are y1,y2,…,yN(ii) a Obtaining a TOA measurement set through communication between nodes, wherein the TOA measurement set comprises TOA measurement of communication between an anchor node and a target node and TOA measurement of communication between the target nodes; the TOA measurement expression obtained by the communication between the anchor node and the target node is
Figure FDA0003675161820000011
Where i, j represent anchor node numbered i and target node numbered j, respectively, c is signal propagation speed, tijRepresenting a time measurement, r, obtained by the anchor node i communicating with the target node jijRepresents the distance between anchor node i and target node j, eijAdditive white Gaussian noise, e, representing communication between anchor node i and target node jijObey mean of zero and variance of
Figure FDA0003675161820000012
(ii) a gaussian distribution of; the TOA measurement expression obtained by mutual communication between the target nodes is
Figure FDA0003675161820000013
Wherein j and j 'represent target nodes numbered j and j', respectively, and j ≠ j ', t'jj'Represents a time measurement, r ', of a target node j and a target node j ' in communication with each other 'jj'Representing the distance, e ', between the target node j and the target node j'jj'Representing additive white Gaussian noise, e 'of communication between target node j and target node j'jj'Obey mean of zero and variance of
Figure FDA0003675161820000014
(ii) a gaussian distribution of; the relation between the distance between the anchor node and the target node and the node position is rij＝||xi-xj||2(ii) a Target node and target nodeThe relation between the distance between and the node position is r'jj'＝||yj-yj'||2(ii) a The relation between the actual position of the anchor node and the measured position is
Figure FDA0003675161820000015
In the formula
Figure FDA0003675161820000016
Indicating measured anchor node position, ξiIs the anchor node error vector, ξiHas a maximum value equal to or less than a known constant epsilon.
Secondly, introducing an anchor node error term into original TOA measurement of communication between the anchor node and the target node;
thirdly, converting the error items of the anchor nodes, and eliminating error vectors of the anchor nodes by utilizing an S-process;
the third step adopts a maximum likelihood estimation method to estimate the position of the target, and X is setu＝[y1,y2,…,yN]For a set of unknown target node coordinates, Xa＝[x1,x2,…,xK]Is a set of coordinates of all anchor nodes, defining d simultaneouslyij＝tij×c，d'jj'＝t'jj'×c，dijAnd d'jj'Respectively measuring distance values with noise between an anchor node i and a target node j and between the target nodes j and j '(j ≠ j'); according to the existing conditions, the original objective function to be optimized is expressed as
Figure FDA0003675161820000021
Substituting the anchor node error item in the second step to obtain min-max suboptimal problem
Figure FDA0003675161820000022
Through vectorization of the objective function and elimination of the anchor node error vector by applying the S-process, the original sub-optimization problem is transformed into
Figure FDA0003675161820000023
Wherein, lambda and mu are constants which are newly introduced for converting the error vector of the anchor node,
Figure FDA00036751618200000211
indicating the positive definite matrix, I indicating the identity matrix, Tr (-) indicating the trace of the matrix inside the bracket, d1＝[d11,d12,…,dMN]TNoisy distance measurement for anchor node to target node communication, d2＝[d′12,d′13,…,d'N,N-1]TNoisy distance measurements for communication between target nodes,
Figure FDA0003675161820000024
aggregating all noisy distance measurements; definition of
Figure FDA0003675161820000025
For true distance measurement of communication between anchor node and target node, r2＝[r′12,r′13,…,r'N,N-1]TFor true distance measurements of communication between target nodes,
Figure FDA0003675161820000026
for all the sets of real distance measurements,
Figure FDA0003675161820000027
G＝blkdiag(G1,G2)，Γ＝1(M+N-1)×Ndiag {. is a diagonal matrix, diagonal elements of the matrix are bracketed elements, blkdiag {. is a block diagonal matrix, 1(M+N-1)×NRepresents a column vector having (M + N-1) x N elements all 1, gamma [ ·],r[·]Representing elements in a reference vector γ, r; defining intermediate variables simultaneously
Figure FDA0003675161820000028
γjj'And toQuantity gamma1,γ2And a gamma-ray source,
Figure FDA0003675161820000029
γ2＝[γ12,γ13,…,γN,N-1]T，
Figure FDA00036751618200000210
fourthly, converting the problem into a convex problem by converting and relaxing the objective function and the constraint condition;
and fifthly, solving the convex optimization problem to obtain the estimation of the target position.
2. The method as claimed in claim 1, wherein the second step further transforms the relationship between the true position and the measured position of the anchor node into a relationship between the true position and the measured position of the anchor node using a first order Taylor expansion
Figure FDA0003675161820000031
Wherein o (| | xi)i| |) represents the anchor node error vector modulo | | ξiThe high order infinitesimal amount of | l; order to
Figure FDA0003675161820000032
Then get | δij| ≦ epsilon, wherein
Figure FDA0003675161820000033
Represents the distance between the anchor node i and the target node j, and rijInstead, as used herein, the coordinates of anchor node i are the known locations of anchor nodes with errors, δijAn error value representing a modulus between anchor node i and target node j; the TOA measurement for the communication between the final anchor node and the target node is expressed as
Figure FDA0003675161820000034
3. The method for positioning the target of claim 1, wherein the fourth step introduces an intermediate matrix
Figure FDA0003675161820000035
The final form of the optimization problem is:
Figure FDA0003675161820000036
subject to
Figure FDA0003675161820000037
γ[(i-1)N+j]≥r2[(i-1)N+j]
γ[MN+(j-1)N+j']≥r2[MN+(j-1)N+j']
Figure FDA0003675161820000038
γ[MN+(j-1)N+j']＝Yu[j,j]+Yu[j',j']-Yu[j,j']-Yu[j',j]
Xu(:,j)＝Yu[j,N+1:N+l]
Yu[N+1:N+l,N+1:N+l]＝Il
Figure FDA0003675161820000039
wherein Y isu[w,v]Representing a reference matrix YuRow w, column v, w and v must be integers; y isu[j,N+1:N+l]Representation matrix YuAll elements of row j, columns N +1 to N + l, Yu[N+1:N+l,N+1:N+l]Representation matrix YuAll elements of rows N +1 to N + l and columns N +1 to N + l, Xu(j) represents a matrix XuRow (j) ofA column vector composed of all elements.
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