CN110972094A - L0 norm positioning method under indoor mixed sparse LOS/NLOS scene - Google Patents

L0 norm positioning method under indoor mixed sparse LOS/NLOS scene Download PDF

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CN110972094A
CN110972094A CN201911242243.5A CN201911242243A CN110972094A CN 110972094 A CN110972094 A CN 110972094A CN 201911242243 A CN201911242243 A CN 201911242243A CN 110972094 A CN110972094 A CN 110972094A
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何成文
袁运斌
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Institute of Geodesy and Geophysics of CAS
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Abstract

The invention provides an L0 norm positioning method under an indoor mixed sparse LOS/NLOS scene, which comprises the steps of selecting an LOS reference station according to the known fixed base station coordinates and the distance observation value of a mobile station in a TOA mode, and then establishing a sparse L0 norm problem; in order to solve the sparse L0 norm problem, all possible mobile stations are arranged and combined by adopting an exhaustion method, so that the sparse L0 norm problem is relaxed into a plurality of groups of L2 norm problems, and a least square method is adopted to obtain the solution of each group of L2 norm problems; and substituting the solution of each group of L2 norm problems into the residual errors obtained in the sparse L0 norm problem by applying an L1 norm criterion to obtain the sum of absolute values of the residual errors of the solution of each group of L2 norm problems, and then taking the solution of the group of L2 norm problems with the minimum sum of absolute values as the final positioning solution of the sparse L0 norm problem. The invention provides an L0 norm minimization method, which is applicable to a mixed sparse LOS/NLOS environment in a TOA positioning mode and can reduce the precision of NLOS errors to a positioning system.

Description

L0 norm positioning method under indoor mixed sparse LOS/NLOS scene
Technical Field
The invention belongs to the technical field of wireless positioning, and particularly relates to an L0 norm positioning method in an indoor mixed sparse LOS/NLOS scene.
Background
In the indoor environment propagation process of a radio signal, if the radio signal meets the shielding of an obstacle, a non line of sight (NLOS) error is generated, the NLOS error can cause the precision of a positioning system to be reduced, and if the NLOS error is serious, the positioning result can be diverged. In general, the number of mixed line-of-sight (LOS) paths in an indoor positioning system is often greater than the number of NLOS paths.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the L0 norm positioning method under the indoor mixed sparse LOS/NLOS scene is provided, and the positioning accuracy of an indoor positioning system under the mixed LOS/NLOS scene is improved.
The technical scheme adopted by the invention for solving the technical problems is as follows: an L0 norm positioning method under an indoor mixed sparse LOS/NLOS scene is characterized in that: the method comprises the following steps:
s1, selecting an LOS reference station according to the known fixed base station coordinates and the distance observation value of the mobile station in the TOA mode, and then establishing a sparse L0 norm problem;
s2, in order to solve the sparse L0 norm problem, an exhaustion method is adopted to arrange and combine all possible mobile stations, so that the sparse L0 norm problem is relaxed into a plurality of groups of L2 norm problems, and a least square method is adopted to obtain the solution of each group of L2 norm problems;
s3, substituting the solutions of each group of L2 norm problems into the residual errors obtained in the sparse L0 norm problem by applying an L1 norm criterion to obtain the sum of the absolute values of the residual errors of the solutions of each group of L2 norm problems, and then taking the solution of the group of L2 norm problems corresponding to the minimum sum of the absolute values as the final positioning solution of the sparse L0 norm problem.
According to the method, the S1 specifically comprises the following steps: the basic positioning equation between the mobile station and the fixed base station in the TOA mode is:
Figure BDA0002306591340000011
wherein, ηiAnd εiRespectively represent the mean value of the ith fixed base station as 0 and the variance as
Figure BDA0002306591340000012
White gaussian noise and NLOS errors much larger than white noise; i ∈ φLRepresents a LOS path, i ∈ φNLThen the NLOS path is represented; (x)i,yi) Is the ith fixedDetermining the known coordinates of the base station, wherein (x, y) is the coordinate to be solved of the mobile station; diIs the distance observed value between the ith fixed base station and the mobile station;
then, assuming that the jth fixed base station is an LOS base station, that is, the basic positioning equation corresponding to the jth fixed base station is used as a reference equation, and a new positioning expression is obtained by making a difference between each basic positioning equation and the reference equation:
Figure BDA0002306591340000021
dJobserved distance between the J-th fixed base station and the mobile station ηJFor the J-th fixed base station, the mean value is 0 and the variance is
Figure BDA0002306591340000022
White gaussian noise of (1);
since gaussian white noise is small, the square term of white noise in equation (2) is omitted and then converted to the standard form of least squares AX-b ═ E, then:
Figure BDA0002306591340000023
X＝[x,y]T(4)
Figure BDA0002306591340000024
E＝[E1… EN]T(6)
wherein the content of the first and second substances,
Figure BDA0002306591340000025
x1-xNknown abscissas, y, for the 1 st to N fixed base stations1-yNKnown ordinates of the 1 st to N th fixed base stations, E1-ENE in the corresponding least squares criterion form for 1 st to N fixed base stations; n is the total number of the fixed base stations;
based on the above analysis, an L0 norm model was thus constructed:
min||AX-b||0(8)
i.e. the sparse L0 norm problem.
According to the method, the S2 specifically comprises the following steps:
the matrix A and the vector b in the formula (8) are decomposed into the sub-matrix A by adopting an exhaustion methodiAnd the subvector biThe possibility of decomposition is always common
Figure BDA0002306591340000026
Seed growing;
relaxing the L0 norm problem to the L2 norm problem, then
min||AiXi-bi||2,(i＝1,...,M) (9)
With the least squares algorithm, the above equation is solved as:
Figure BDA0002306591340000031
according to the method, S3 specifically comprises the following steps: substituting equation (10) into equation (1) one by one, thereby obtaining residual vectors for each set of solutions, and then calculating the L1 norm of the residual vectors;
and finally, selecting the group of solutions corresponding to the L1 norm value minimum as the final positioning solution of the L0 norm problem.
The invention has the beneficial effects that: the L0 norm minimization method is provided, the application condition is a mixed sparse LOS/NLOS environment in a TOA positioning mode, the method is applied to the field of wireless positioning taking measuring time or distance as an observed value, the UWB and 5G positioning technology is included, the method has a great promotion effect on reducing the influence of NLOS errors on the accuracy of a positioning system, and the method has a positive and important reference value for industrial popularization and market application.
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FIG. 1 is a graph of the positioning error of the present invention under 1 NLOS path.
Detailed Description
The invention is further illustrated by the following specific examples and figures.
The invention provides an L0 norm positioning method under an indoor mixed sparse LOS/NLOS scene, which comprises the following steps:
s1, selecting an LOS reference station according to the known fixed base station coordinates and the distance observation value of the mobile station in the TOA mode, and then establishing a sparse L0 norm problem.
The basic positioning equation between the mobile station and the fixed base station in the TOA mode is:
Figure BDA0002306591340000032
wherein, ηiAnd εiRespectively represent the mean value of the ith fixed base station as 0 and the variance as
Figure BDA0002306591340000033
White gaussian noise and NLOS errors much larger than white noise; i ∈ φLRepresents a LOS path, i ∈ φNLThen the NLOS path is represented; (x)i,yi) Known coordinates of the ith fixed base station, (x, y) the coordinates to be found of the mobile station; diIs the distance observed value between the ith fixed base station and the mobile station;
then, assuming that the jth fixed base station is an LOS base station, that is, the basic positioning equation corresponding to the jth fixed base station is used as a reference equation, and a new positioning expression is obtained by making a difference between each basic positioning equation and the reference equation:
Figure BDA0002306591340000034
dJobserved distance between the J-th fixed base station and the mobile station ηJFor the J-th fixed base station, the mean value is 0 and the variance is
Figure BDA0002306591340000035
White gaussian noise of (1);
the squared term of white noise, abbreviated as (2), is then converted to the standard form of least squares AX-b ═ E, then:
Figure BDA0002306591340000041
X＝[x,y]T(4)
Figure BDA0002306591340000042
E＝[E1… EN]T(6)
wherein the content of the first and second substances,
Figure BDA0002306591340000043
x1-xNknown abscissas, y, for the 1 st to N fixed base stations1-yNKnown ordinates of the 1 st to N th fixed base stations, E1-ENE in the corresponding least squares criterion form for 1 st to N fixed base stations; and N is the total number of the fixed base stations.
Based on the above analysis, an L0 norm model was thus constructed:
min||AX-b||0(8)
i.e. the sparse L0 norm problem.
And S2, in order to solve the sparse L0 norm problem, arranging and combining all possible mobile stations by adopting an exhaustion method, so that the sparse L0 norm problem is relaxed into a plurality of groups of L2 norm problems, and a least square method is adopted to obtain the solution of each group of L2 norm problems.
The matrix A and the vector b in the formula (8) are decomposed into the sub-matrix A by adopting an exhaustion methodiAnd the subvector biThe possibility of decomposition is always common
Figure BDA0002306591340000044
And (4) seed preparation. Firstly, the above equations (3) and (5) need to be decomposed into blocks to obtain AiAnd biThus, relaxing the L0 norm problem to the L2 norm problem, then
min||AiXi-bi||2,(i＝1,...,M) (9)
With the least squares algorithm, the above equation is solved as:
Figure BDA0002306591340000045
s3, substituting the solutions of each group of L2 norm problems into the residual errors obtained in the sparse L0 norm problem by applying an L1 norm criterion to obtain the sum of the absolute values of the residual errors of the solutions of each group of L2 norm problems, and then taking the solution of the group of L2 norm problems corresponding to the minimum sum of the absolute values as the final positioning solution of the sparse L0 norm problem.
The method specifically comprises the following steps: substituting equation (10) into equation (1) one by one, thereby obtaining residual vectors for each set of solutions, and then calculating the L1 norm of the residual vectors; and finally, selecting the group of solutions corresponding to the L1 norm value minimum as the final positioning solution of the L0 norm problem.
The experimental results of fig. 1 derive from the experimental conditions that given 10 fixed base stations, their coordinates are located at points (0,0), (8,0), (16,3), (16,11), (16,16), (9,16), (0,12) and (0, 6). η, respectivelyiObeying a mean of 0 and a variance of
Figure BDA0002306591340000051
The error range of the NLOS is uniformly distributed from 6 max (η) to 15 max (η). the position of the mobile station is randomly generated in an area surrounded by fixed base stations, and the simulation time of Monte Carlo is 500 under each noise, wherein, the figure 1 is a positioning accuracy graph of the system containing 1 NLOS base station.
As can be seen from fig. 1, when the gaussian white noise variance values are 0.01, 0.06, 0.1, 0.6 and 1, the corresponding positioning accuracies are 0.0089m, 0.0568m, 0.0898m, 0.5617m and 0.9113m, respectively.
According to the method, firstly, a difference equation is constructed by selecting a reference base station according to a basic positioning equation between a fixed base station and a mobile station, so that an L0 norm problem is constructed; decomposing the L0 norm original problem into a plurality of L2 norm subproblems by an exhaustion method, and further obtaining a plurality of groups of positioning solutions; and performing performance evaluation on each group of positioning solutions so as to complete the screening process of the optimal solution. The method has the main function of reducing the influence of NLOS errors on the accuracy of the indoor positioning system, so that the method has higher positioning accuracy in a sparse LOS/NLOS environment. The method has simple principle, convenient realization and better engineering value in practical application.
The above embodiments are only used for illustrating the design idea and features of the present invention, and the purpose of the present invention is to enable those skilled in the art to understand the content of the present invention and implement the present invention accordingly, and the protection scope of the present invention is not limited to the above embodiments. Therefore, all equivalent changes and modifications made in accordance with the principles and concepts disclosed herein are intended to be included within the scope of the present invention.

Claims (4)

1. An L0 norm positioning method under an indoor mixed sparse LOS/NLOS scene is characterized in that: the method comprises the following steps:
s1, selecting an LOS reference station according to the known fixed base station coordinates and the distance observation value of the mobile station in the TOA mode, and then establishing a sparse L0 norm problem;
s2, in order to solve the sparse L0 norm problem, an exhaustion method is adopted to arrange and combine all possible mobile stations, so that the sparse L0 norm problem is relaxed into a plurality of groups of L2 norm problems, and a least square method is adopted to obtain the solution of each group of L2 norm problems;
s3, substituting the solutions of each group of L2 norm problems into the residual errors obtained in the sparse L0 norm problem by applying an L1 norm criterion to obtain the sum of the absolute values of the residual errors of the solutions of each group of L2 norm problems, and then taking the solution of the group of L2 norm problems corresponding to the minimum sum of the absolute values as the final positioning solution of the sparse L0 norm problem.
2. The method of claim 1, wherein: the S1 specifically includes: the basic positioning equation between the mobile station and the fixed base station in the TOA mode is:
Figure FDA0002306591330000011
wherein, ηiAnd εiRespectively represent the mean value of the ith fixed base station as 0 and the variance as
Figure FDA0002306591330000012
White gaussian noise and NLOS errors much larger than white noise; i ∈ φLRepresents a LOS path, i ∈ φNLThen the NLOS path is represented; (x)i,yi) Known coordinates of the ith fixed base station, (x, y) the coordinates to be found of the mobile station; diIs the distance observed value between the ith fixed base station and the mobile station;
then, assuming that the jth fixed base station is an LOS base station, that is, the basic positioning equation corresponding to the jth fixed base station is used as a reference equation, and a new positioning expression is obtained by making a difference between each basic positioning equation and the reference equation:
Figure FDA0002306591330000013
dJobserved distance between the J-th fixed base station and the mobile station ηJFor the J-th fixed base station, the mean value is 0 and the variance is
Figure FDA0002306591330000014
White gaussian noise of (1);
the squared term of white noise, abbreviated as (2), is then converted to the standard form of least squares AX-b ═ E, then:
Figure FDA0002306591330000015
X＝[x,y]T(4)
Figure FDA0002306591330000021
E＝[E1… EN]T(6)
wherein the content of the first and second substances,
Figure FDA0002306591330000022
x1-xNknown abscissas, y, for the 1 st to N fixed base stations1-yNKnown ordinates of the 1 st to N th fixed base stations, E1-ENE in the corresponding least squares criterion form for 1 st to N fixed base stations; n is the total number of the fixed base stations;
based on the above analysis, an L0 norm model was thus constructed:
min||AX-b||0(8)
i.e. the sparse L0 norm problem.
3. The method of claim 2, wherein: the S2 specifically includes:
the matrix A and the vector b in the formula (8) are decomposed into the sub-matrix A by adopting an exhaustion methodiAnd the subvector biThe possibility of decomposition is always common
Figure FDA0002306591330000023
Seed growing;
relaxing the L0 norm problem to the L2 norm problem, then
min||AiXi-bi||2,(i＝1,...,M) (9)
With the least squares algorithm, the above equation is solved as:
Figure FDA0002306591330000024
4. the method of claim 3, wherein: s3 specifically includes: substituting equation (10) into equation (1) one by one, thereby obtaining residual vectors for each set of solutions, and then calculating the L1 norm of the residual vectors;
and finally, selecting the group of solutions corresponding to the L1 norm value minimum as the final positioning solution of the L0 norm problem.
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