CN112954637A - Target positioning method under condition of uncertain anchor node position - Google Patents

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 position. 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, the conventional model mostly assumes that the position of the anchor node is accurately known, but in practical application, 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 the water buoy node is obtained by using the 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 no measures are 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 anchor node errors;

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 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.

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,…,x_MThe coordinates of N target nodes to be estimated are y₁,y₂,…,y_N(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

Wherein i and j respectively represent an anchor node with the number i and a target node with the number j, c is the signal propagation speed, and t_ijRepresenting a time measurement, r, obtained by the anchor node i communicating with the target node j_ijRepresents the distance between anchor node i and target node j, e_ijAdditive white Gaussian noise, e, representing communication between anchor node i and target node j_ijObedience mean value of zero and variance of

(ii) a gaussian distribution of; the TOA measurement expression obtained by mutual communication between the target nodes is

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'Obedience mean value of zero and variance of

(ii) a gaussian distribution of; the relation between the distance between the anchor node and the target node and the node position is r_ij＝||x_i-x_j||₂(ii) a The relation between the distance between the target nodes and the node positions is r'_jj'＝||y_j-y_j'||₂(ii) a The relation between the actual position of the anchor node and the measured position is

In the formula

Indicating measured anchor node position, ξ_iIs the anchor node error vector, ξ_iHas 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 by utilizing a first-order Taylor expansion

Where o (| | ξ)_i| |) represents the anchor node error vector modulo | | ξ_iThe high order infinitesimal amount of | l; order to

Then get | δ_ij| < epsilon, wherein

Represents the distance between the anchor node i and the target node j, and r_ijInstead, as used herein, the coordinates of anchor node i are the known locations of anchor nodes with errors, δ_ijTo representAn error value of 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

The third step adopts a maximum likelihood estimation method to estimate the position of the target, and X is set_u＝[y₁,y₂,…,y_N]For a set of unknown target node coordinates, X_a＝[x₁,x₂,…,x_K]Is a set of coordinates of all anchor nodes, defining d simultaneously_ij＝t_ij×c，d'_jj'＝t'_jj'×c，d_ijAnd 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

Substituting the anchor node error item in the second step to obtain min-max suboptimal problem

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

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,…,d_MN]^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,

for all that isThe noisy distance measurement set of (a); definition of

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,

for all the sets of real distance measurements,

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

γ_jj'And a vector gamma₁,γ₂And a gamma-ray source,

γ₂＝[γ₁₂,γ₁₃,…,γ_N,N-1]^T，

the fourth step introduces an intermediate matrix

The final form of the optimization problem is:

wherein, Y_u[w,v]Representing a reference matrix Y_uRow w, column v, w and v must be integers;

Y_u[j,N+1:N+l]representation matrix Y_uAll elements of row j, columns N +1 to N + l, Y_u[N+1:N+l,N+1:N+l]Representation matrix Y_uAll elements of rows N +1 to N + l and columns N +1 to N + l, X_u(j) represents a matrix X_uA column vector consisting of all elements of the jth column of (a).

The invention has the beneficial effects that: 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 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.

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 number of anchor nodes.

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 providedThe method is carried out. 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. Let the real coordinates of M anchor nodes with known positions be x respectively₁,x₂,…,x_MThe coordinates of N target nodes to be estimated are y₁,y₂,…,y_N。

The method 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 communication between the anchor node and the target node is:

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, t_ijRepresenting a time measurement, r, obtained by the anchor node i communicating with the target node j_ijRepresents the distance between anchor node i and target node j, e_ijAdditive white Gaussian noise representing communication between anchor node i and target node j obeys a mean of zero and a variance of

A gaussian distribution of (a).

The TOA measurement expression obtained by mutual communication between the target nodes is as follows:

wherein j and j 'represent target nodes numbered j and j', respectively, and j ≠ j ', t'_jj'Representing the time taken for the target node j and the target node j' to communicate with each otherMeasure r'_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

A gaussian distribution of (a).

The relationship between the distance between the anchor node and the target node and the node position is as follows:

r_ij＝||x_i-x_j||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'＝||y_j-y_j'||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:

in the formula

Indicating measured anchor node position, ξ_iIs an anchor node error vector whose maximum value of its modulus is less than a known constant epsilon, i.e.

||ξ_i||≤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:

i＝1,2,…,M,j＝1,2,…,N

where o (| | ξ)_i| |) represents the anchor node error vector modulo | | ξ_iThe higher order of | is an infinitesimal quantity.

Order to

Then it can be deduced that:

|δ_ij|≤ε

wherein

Represents the distance between the anchor node i and the target node j, and r_ijIn contrast, the coordinates of anchor node i as used herein is the known anchor node location with error. Delta_ijAn error value representing a modulus between the anchor node i and the target node j, and the symbol "|" is a sign for solving an absolute value.

The TOA measurement communicated between the final anchor node and the target node may be expressed as:

the third step: transforming anchor node error terms

The position of the target is estimated using maximum likelihood estimation. Let X_u＝[y₁,y₂,…,y_N]For a set of unknown target node coordinates, X_a＝[x₁,x₂,…,x_K]Is the set of coordinates of all anchor nodes. Simultaneously defining:

d_ij＝t_ij×c,i＝1,2,…,M,j＝1,2,…,N

d'_jj'＝t'_jj'×c,j＝1,2,…,N,j'＝1,2,…,N

d_ijand 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:

substituting the anchor node error term in the second step can obtain a min-max suboptimal problem:

subject to

r'_jj'＝||y_j-y_j'||,j＝1,2,…,M,j'＝1,2,…,N

wherein min represents minimum, max represents maximum, and Σ represents summation. subject to represents "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 delta_ijThis 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:

wherein the content of the first and second substances,

the square root is shown, λ and μ are constants introduced for transforming the anchor node error vector, ≧ represents the matrix positive definite, I represents the identity matrix, and Tr (-) represents the trace of the matrix inside the bracket. And d is₁＝[d₁₁,d₁₂,…,d_MN]^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,

aggregating all noisy distance measurements; definition of

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,

all the true distance measurements are aggregated.

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)×NRepresents a column vector having (M + N-1) x N elements all 1, gamma [ ·],r[·]Representing the elements in the reference vector gamma, r. Defining intermediate variables simultaneously

γ_jj'And a vector gamma₁,γ₂And γ:

the former similar processing only aims at the single-node positioning problem, the current processing aims at the multi-node positioning problem, 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 divided into two parts^T-2dr^T)]And also put into the convex constraint obtained by the S-process, and the current processing method is to still keep the first half part in the objective function and 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 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 introduced_u：

By the intermediate matrix and the variable gamma, X to be optimized_uEstablishing a connection and performing appropriate relaxation converts all constraints into convex constraints, and 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:

wherein, Y_u[w,v]Representing a reference matrix Y_uRow w, column v, w and v must be integers; y is_u[j,N+1:N+l]Representation matrix Y_uAll elements of row j, columns N +1 to N + l, Y_u[N+1:N+l,N+1:N+l]Representation matrix Y_uAll elements of rows N +1 to N + l and columns N +1 to N + l, X_u(j) represents a matrix X_uA column vector consisting of all elements of the jth column of (a).

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 tool box in MATLAB so as to obtain the estimation of the target position.

The embodiment of the invention considers the positioning problem of two-dimensional space and is arranged at oneThere are 4 anchor nodes with known positions and 3 target nodes to be located in a 1200 mx 1200M area, i.e., M is 4 and N is 3. Let the real coordinates of the 4 anchor nodes with known positions be x respectively₁,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:

i and j are positive integers, which respectively represent a node numbered i and a node numbered j. c is the signal propagation velocity, t_ijRepresenting a time measurement, r, obtained by the anchor node i communicating with the target node j_ijRepresents the distance between anchor node i and target node j, e_ijAdditive white Gaussian noise representing communication between anchor node i and target node j obeys a mean of zero and a variance of

A gaussian distribution of (a).

The TOA measurement expression obtained by mutual communication between the target nodes is as follows:

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'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

A gaussian distribution of (a).

The relationship between the distance between the anchor node and the target node and the node position is as follows:

r_ij＝||x_i-x_j||₂

i＝1,2,3，4,j＝1,2,3

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'＝||y_j-y_j'||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:

in the formula

Indicating measured anchor node position, ξ_iIs an anchor node error vector whose maximum value of its modulus is less than a known constant epsilon, i.e.

||ξ_i||≤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:

i＝1,2,3，4,j＝1,2,3

where o (| | ξ)_i| |) represents the anchor node error vector modulo | | ξ_iThe higher order of | is an infinitesimal quantity.

Order to

Then there is

Namely, it is

|δ_ij|≤ε

Wherein

Represents the distance between the anchor node i and the target node j, and r_ijIn contrast, the coordinates of anchor node i as used herein is the known anchor node location with error. Delta_ijAn error value representing a modulus between the anchor node i and the target node j, and the symbol "|" is a sign for solving an absolute value.

The TOA measurement communicated between the final anchor node and the target node may be expressed as:

the third step: transforming anchor node error terms

The position of the target is estimated using maximum likelihood estimation. Let X_u＝[y₁,y₂,y₃]For a set of unknown target node coordinates, X_a＝[x₁,x₂,x₃,x₄]Is the set of coordinates of all anchor nodes. Simultaneously defining:

d_ij＝t_ij×c,i＝1,2,3,4,j＝1,2,3

d'_jj'＝t'_jj'×c,j＝1,2,3,j'＝1,2,3

d_ijand 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:

substituting the anchor node error term in the second step can obtain a min-max suboptimal problem:

subject to

r'_jj'＝||y_j-y_j'||,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 | δ_ijIf | ≦ ε, the vectorized anchor node error vector satisfies:

wherein δ is [ δ ═ δ₁₁,δ₁₂,…,δ₄₃]^TIs a set of anchor node modulo errors for M anchor nodes communicating with N target nodes,

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,

aggregating all noisy distance measurements; definition of

For true distance measurement of communication between anchor node and target node, r₂＝[r'₁₂,r'₁₃,…,r₃'₂]^TFor true distance measurements of communication between target nodes,

all the true distance measurements are aggregated. Defining intermediate variables simultaneously

γ_jj'And a vector gamma₁,γ₂And γ:

the original sub-optimization problem can be translated into:

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

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

wherein

G＝blkdiag(G₁,G₂)，Γ＝1_(3+4-1)×3. Wherein diag {. is } represents a diagonal matrix, diagonal elements of the matrix are in parenthesis, blkdiag {. is } 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

The following conclusion holds:

δ^TG₁δ-2δ^TG₁(d₁-r₁)≤μ

namely:

wherein

Indicating the sufficient condition.

The above formula can be further expressed as:

by applying the S-process, the above recursive relationship is finally converted into a convex constraint:

namely:

where lambda and mu are constants newly introduced to translate the anchor node error vector,

indicating "presence" ≧ which indicates the matrix semi-positive definite.

Meanwhile, the objective function to be optimized may be converted into:

Tr[G(Γγ^T-2dr^T)]+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 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 introduced_u：

Then the matrix Y_uThe element in (1) and the existing constraint relation are as follows:

i＝1,2,3,4,j＝1,2,3

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'＝1,2,3

wherein, Y_u[j,j]Representing a reference matrix Y_uRow jth and column jth. The two constraints are converted into convex constraints. Simultaneously, 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 Y_uThe other convex constraints hidden in (1) are:

y_j＝[Y_u[j,4],Y_u[j,5]]^T,j＝1,2,3

Y_u≥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:

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 toolkit of MATLAB, the tool being Sedumi. Programming is carried out in an MATLAB environment, and an estimation of the target position can be obtained by inputting an objective function and a constraint condition.

Claims (5)

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;

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 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 for positioning the target under the condition of uncertain anchor node positions as claimed in claim 1, wherein the first step of setting the sensor network comprises M anchor nodes with known positions but errors and N target nodes to be positioned, wherein M is larger than or equal to 3, and the real coordinates of the M anchor nodes are x respectively₁,x₂,…,x_MThe coordinates of N target nodes to be estimated are y₁,y₂,…,y_N(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

Wherein i and j respectively represent an anchor node with the number i and a target node with the number j, c is the signal propagation speed, and t_ijRepresenting a time measurement, r, obtained by the anchor node i communicating with the target node j_ijRepresenting the distance between anchor node i and target node j,e_ijadditive white Gaussian noise, e, representing communication between anchor node i and target node j_ijObedience mean value of zero and variance of

(ii) a gaussian distribution of; the TOA measurement expression obtained by mutual communication between the target nodes is

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'Obedience mean value of zero and variance of

(ii) a gaussian distribution of; the relation between the distance between the anchor node and the target node and the node position is r_ij＝||x_i-x_j||₂(ii) a The relation between the distance between the target nodes and the node positions is r'_jj'＝||y_j-y_j'||₂(ii) a The relation between the actual position of the anchor node and the measured position is

In the formula

Indicating measured anchor node position, ξ_iIs the anchor node error vector, ξ_iHas a maximum value equal to or less than a known constant epsilon.

3. The method of claim 2, wherein the second step utilizes first-order taylor's expansionFurther converting the relation between the real position and the measured position of the anchor node into an open type

Where o (| | ξ)_i| |) represents the anchor node error vector modulo | | ξ_iThe high order infinitesimal amount of | l; order to

Then get | δ_ij| < epsilon, wherein

Represents the distance between the anchor node i and the target node j, and r_ijInstead, as used herein, the coordinates of anchor node i are the known locations of anchor nodes with errors, δ_ijAn 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

4. The method of claim 3, wherein the third step employs a maximum likelihood estimation method to estimate the position of the target, where X is the number_u＝[y₁,y₂,…,y_N]For a set of unknown target node coordinates, X_a＝[x₁,x₂,…,x_K]Is a set of coordinates of all anchor nodes, defining d simultaneously_ij＝t_ij×c，d′_jj'＝t′_jj'×c，d_ijAnd 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

Error of anchor node in the second stepItem substitution to get min-max suboptimal problem

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

Wherein, lambda and mu are constants which are newly introduced for converting the error vector of the anchor node,

indicating the positive definite matrix, I indicating the identity matrix, Tr (-) indicating the trace of the matrix inside the bracket, d₁＝[d₁₁,d₁₂,…,d_MN]^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,

aggregating all noisy distance measurements; definition of

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,

for all the sets of real distance measurements,

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

γ_jj′And a vector gamma₁,γ₂And a gamma-ray source,

γ₂＝[γ₁₂,γ₁₃,…,γ_N,N-1]^T，

5. the method for positioning the target of claim 4, wherein the fourth step introduces an intermediate matrix

The final form of the optimization problem is:

subject to

γ[(i-1)N+j]≥r²[(i-1)N+j]

γ[MN+(j-1)N+j']≥r²[MN+(j-1)N+j']

γ[MN+(j-1)N+j']＝Y_u[j,j]+Y_u[j',j']-Y_u[j,j']-Y_u[j',j]

X_u(:,j)＝Y_u[j,N+1:N+l]

Y_u[N+1:N+l,N+1:N+l]＝I_l

wherein, Y_u[w,v]Representing a reference matrix Y_uRow w, column v, w and v must be integers; y is_u[j,N+1:N+l]Representation matrix Y_uAll elements of row j, columns N +1 to N + l, Y_u[N+1:N+l,N+1:N+l]Representation matrix Y_uAll elements of rows N +1 to N + l and columns N +1 to N + l, X_u(j) represents a matrix X_uA column vector consisting of all elements of the jth column of (a).

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