A positioning method and system based on fixed rank matrix manifold optimization

Abstract

This invention discloses a positioning method and system based on rank matrix manifold optimization. The method includes: estimating the distance between communication nodes using the maximum likelihood method based on RSSI measurements; constructing a preliminary EDM matrix based on the estimated distances between communication nodes; introducing a non-monotonic Armijo backtracking method and an enhancement algorithm to construct an improved LRM-CG algorithm; and completing and solving the preliminary EDM matrix using the improved LRM-CG algorithm to obtain the user's location. By using this invention and polarization imaging technology, the improved LRM-CG algorithm can be used to solve the EDM matrix, further improving the positioning accuracy and stability of the user node in the wireless communication system. This invention, as a positioning method and system based on rank matrix manifold optimization, can be widely applied in the field of wireless communication technology.

Description

Technical Field

[0001] This invention relates to the field of wireless communication technology, and in particular to a positioning method and system based on rank matrix manifold optimization. Background Technology

[0002] IoT (Internet of Things) is an emerging access scenario. With the development of the Internet and the innovation of wireless communication technology, more and more non-traditional terminals are interconnected through the Internet. In this scenario, each terminal can interact with each other, greatly improving the universality of the Internet while forming a highly distributed device network that communicates with users and other devices. While improving the quality of life, this also brings more complex network edge and access network environments. To provide robust service and response in this complex environment, servers in the network need to obtain the location information of access nodes. Manually configuring the location of each node is an option, but it is usually not feasible for large-scale deployments or frequently moving nodes. Configuring a GNSS (Global Navigation Satellite System) system for nodes is another solution, but GNSS systems require additional hardware and energy support, which contradicts the concept of miniaturization and portability of non-traditional terminals. Furthermore, these non-traditional terminals operate in diverse environments. While providing good line-of-sight (LOS) conditions for positioning satellites can significantly reduce the performance of GNSS systems, requiring assistance from other systems for accurate positioning, current solutions rely on network positioning or sensor network positioning. These solutions assume that only some nodes in the network know their true location, referred to as anchor nodes. All sensor nodes measure the distances between adjacent nodes and transmit this information back to the server. The server then uses the anchor node locations and measured distance information to generate a location map of all nodes. However, for a large number of wireless nodes deployed in various scenarios, the number of nearby nodes a node can detect is limited by the size, energy consumption, distribution characteristics of wireless access nodes, and electromagnetic wave fading characteristics in the environment. The number and location of these reachable nearby nodes are largely determined by the effective transmission range of electromagnetic waves. Therefore, currently, there is a lack of a complete sampling Euclid distance matrix (EDM matrix) for positioning. Summary of the Invention

[0003] To address the aforementioned technical problems, the present invention aims to provide a positioning method and system based on rank matrix manifold optimization, which can further improve the positioning accuracy and stability of user nodes in wireless communication systems by solving the EDM matrix through an improved LRM-CG algorithm.

[0004] The first technical solution adopted in this invention is: a positioning method based on rank matrix manifold optimization, comprising the following steps:

[0005] The estimated distance between communication nodes is estimated using the maximum likelihood method based on RSSI measurements;

[0006] A preliminary EDM matrix is ​​constructed based on the estimated distances between communication nodes;

[0007] An improved LRM-CG algorithm is constructed by introducing the non-monotonic Armijo backtracking method and enhancement algorithms;

[0008] The improved LRM-CG algorithm is used to complete and solve the initial EDM matrix to obtain the user's location.

[0009] Furthermore, the step of constructing a preliminary EDM matrix based on the estimated distance between communication nodes specifically includes:

[0010] The ML estimate of the distance between each node is obtained based on the RSSI measurement values;

[0011] The ML estimates of the distances between nodes are corrected to obtain the unbiased distance values ​​between nodes;

[0012] A preliminary EDM matrix is ​​constructed based on the unbiased distance values ​​between each node.

[0013] Furthermore, the expression for the unbiased distance value between the communication nodes is as follows:

[0014]

[0015] In the above formula, P(d0) represents the signal power received at the reference distance d0, P(d ij ) represents the signal power received at nodes i and j on the link between nodes i and j, γ represents the path loss exponent, and X σ σ represents the variance of the standard normal distribution. 2 a random variable, where η represents the unbiased estimate of the distance between nodes i and j under the RSSI radio propagation model, where η is a constant. This represents the maximum likelihood estimator, expressed through coefficients. It is then corrected to be an unbiased estimator.

[0016] Furthermore, the step of introducing the non-monotonic Armijo backtracking method and the enhancement algorithm to construct the improved LRM-CG algorithm specifically includes:

[0017] Set the maximum number of backtracking steps, initial step size, initial iteration status index, step size inheritance list, and step size indicator for a single line search;

[0018] Determine the state to which the step size inheritance list belongs and calculate the corresponding step size based on the state;

[0019] Perform backtracking calculations to determine the new step size and judge the value of the initial iteration state index based on the calculation results;

[0020] Update the suggested value of the initial step size based on the value of the initial iteration state index;

[0021] Add the updated initial step size recommendation value to the step size inheritance list;

[0022] Introducing β HZ Enhance the algorithm by constructing an improved LRM-CG algorithm.

[0023] Furthermore, the step of completing and solving the initial EDM matrix based on the improved LRM-CG algorithm to obtain the user's location specifically includes:

[0024] The initial user node positions are obtained by solving the completed EDM matrix using the three-point positioning method.

[0025] Noise perturbation is added to the initial user node positions and converted into initial input values ​​for the LRM-CG algorithm;

[0026] The initial value is iteratively solved using the improved LRM-CG algorithm to obtain the solution result;

[0027] The solution results are evaluated. If the number of perturbations in the solution results reaches the preset threshold and the Riemann gradient of the solution results still does not reach the preset gradient value, the user node position is reselected and iterative solution is performed.

[0028] The process continues until the Riemann gradient of the solution is less than a preset threshold. At this point, the solution is output and converted into an EDM (Electronic Data Model). The EDM is then used to calculate the user's location.

[0029] The second technical solution adopted in this invention is: a positioning system based on optimization on a definite-rank matrix manifold, comprising:

[0030] The estimation module estimates the distance between communication nodes using the maximum likelihood method based on RSSI measurements.

[0031] The building module is used to construct a preliminary EDM matrix based on the estimated distance between communication nodes;

[0032] The optimization module is used to introduce the non-monotonic Armijo backtracking method and enhancement algorithms to construct the improved LRM-CG algorithm;

[0033] The solution module completes the initial EDM matrix based on the improved LRM-CG algorithm to obtain the user's location.

[0034] The beneficial effects of the method and system of this invention are as follows: This invention constructs a definite-rank matrix by solving the RSSI radio propagation model using the maximum likelihood estimation method. Furthermore, a non-monotonic Armijo backtracking method is introduced on the definite-rank matrix manifold to ensure the convergence accuracy and speed of matrix solving during the solution process. The β method is also introduced. HZ To enhance the conjugate gradient algorithm's ability to resist iterative congestion, a perturbation and restart module has been added to ensure the stability of the final solution value of the matrix. This allows the recovered EDM matrix to be used to query the distance from each user node to several anchor nodes, and to use a multi-point positioning method to locate the user. Attached Figure Description

[0035] Figure 1 This is a flowchart illustrating the steps of a positioning method based on rank matrix manifold optimization according to the present invention.

[0036] Figure 2 This is a structural block diagram of a positioning system based on optimization on a rank matrix manifold according to the present invention;

[0037] Figure 3 This is a schematic diagram illustrating the effective communication range of devices in the IoT network of this invention;

[0038] Figure 4 This refers to the communication scenario considered in the specific embodiments of the present invention and the undirected graph abstracted under a certain user distribution in that scenario;

[0039] Figure 5 This is a schematic diagram of the basic framework of the improved LRM-CG algorithm of this invention;

[0040] Figure 6 This is a schematic diagram of the framework of the non-monotonic Armijo backtracking method of the present invention;

[0041] Figure 7 This is a schematic diagram of the framework for introducing the perturbation and restart modules in this invention;

[0042] Figure 8 This is a schematic diagram showing the results of the simulation experiment conducted by this invention;

[0043] Figure 9 This is a schematic diagram showing the results of the simulation experiment conducted by this invention. Detailed Implementation

[0044] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments. The step numbers in the following embodiments are only for ease of explanation and do not limit the order of the steps. The execution order of each step in the embodiments can be adapted according to the understanding of those skilled in the art.

[0045] Reference Figure 1This invention provides a localization method based on rank matrix manifold optimization, which includes the following steps:

[0046] S1. Solve the RSSI radio propagation model using the maximum likelihood estimation method to construct a preliminary EDM matrix;

[0047] Specifically, refer to Figure 3 In the diagram, the devices represent nodes (anchor nodes or users), and the sphere and line r represent the effective communication range. Nodes within this range can communicate with each other. The RSSI scheme is based on the attenuation characteristics of radio signals in the environment. A commonly used radio propagation model is expressed as follows:

[0048]

[0049] In the above formula, P(d) represents the energy received at a distance d from the source, P(d0) represents the energy received at a reference distance d0, γ represents the path loss exponent, and X... σ σ represents the variance of a standard normal distribution. 2 random variables, i.e.

[0050] It is easy to see that the maximum likelihood estimate of the distance between nodes i and j under this model is... As shown below:

[0051]

[0052] In the above formula, Let represent the maximum likelihood estimate of the distance between nodes i and j under the RSSI radio propagation model. Further corrections are made to the estimated distance between the communication nodes, and the expression for the correction is shown below. It is a constant;

[0053] The estimated distance between communication nodes is further corrected, and the corrected expression is shown below:

[0054]

[0055] In the above formula, P(d0) represents the signal power received at the reference distance d0, P(d ij ) represents the signal power received at nodes i and j on the link between nodes i and j, γ represents the path loss exponent, and X σ σ represents the variance of the standard normal distribution. 2 random variables: This represents an unbiased estimate of the distance between nodes i and j under the RSSI radio propagation model. In the above formula Represents the maximum likelihood estimator, expressed by coefficients. It is then corrected to be an unbiased estimator;

[0056] Reference Figure 4 In the graph, the two nodes connected by the edges can communicate with each other, and the distance between them is measured using RSSI. The four corner nodes are anchor nodes, and all other nodes are user nodes. As an estimate of node distance measurement in this scheme, it is further assumed that two nodes with a measured distance less than r can communicate with each other, based on observable... Generate an incomplete EDM matrix and use P Ω This sampling relationship is expressed using a matrix, and the expression for the sampling operation is defined as follows:

[0057]

[0058]

[0059] In the above formula, The symbol representing an operation, P Ω It is a 0-1 matrix, where D represents a hypothetical complete EDM matrix, ⊙ represents the dot product of matrices, and P Ω In ⊙D, a position of 0 represents two nodes that cannot communicate with each other, corresponding to distances in the EDM matrix that cannot be measured. A position that is not 0 represents two nodes that can communicate, corresponding to distances in the EDM matrix that can be measured by RSSI.

[0060] S2, in The EDM matrix can be recovered by solving the following unconstrained optimization problem on the manifold;

[0061] Specifically, in The EDM matrix can be recovered by solving the following unconstrained optimization problem on the manifold, which in turn allows for the localization of user nodes. The localization expression is shown below:

[0062]

[0063]

[0064] In the above formula, f(Y) is the objective function, W represents the weight correction for different measurements, which represents the reliability of these RSSI measurements. In practice, it is specified by the user or taken as a quantity inversely proportional to the noise level. Y is a Gram matrix, which is composed of... An element on a manifold, Map Y to an EDM matrix This represents the estimated user location vector;

[0065] Furthermore, the conjugate gradient algorithm is optimized by gradient flow using the non-monotonic Armijo backtracking method, resulting in the improved LRM-CG algorithm for solving the unconstrained optimization problem in S2.

[0066] Reference Figure 5 The basic framework of the improved LRM-CG algorithm is presented as an extension of the traditional conjugate gradient method (CG) to the feasible region of the manifold. Its process is as follows: Figure 5 As shown in the diagram, in the frame, ||·|| represents the F-norm, Y k p represents the current iteration point. k The iteration direction represents the current point, and the Riemann gradient at the k-th iteration point is represented by gradf(Y). k ) is expressed by Y k Euclid gradient To its tangent plane projection Obtain, defined in Y k α at the location k p k The vector is withdrawn by R(Y) k p k α k This expression is equivalent to a line search in the Euclid space, while requiring the use of vector transitions. Map vectors belonging to two tangent planes onto the same tangent plane;

[0067] As is well known, a CG should include the following steps: initialization, step size calculation, update, calculation of the gradient at the new iteration point, calculation of β, and calculation of the new search direction using a linear combination of the previous search direction and the gradient at the new iteration point. These steps correspond to... Figure 5 The diagrams show the various blocks in the text. As mentioned earlier, we used the non-monotonic Armijo backtracking method to calculate the step size α at each iteration. k Using R(Y) k p k α k Complete the update of the iteration point using β. HZ+ As a β-value selection strategy in the improved LRM-CG algorithm, it has the characteristics described in S22; Figure 5 The bottommost box represents the calculation of the new search direction;

[0068] The framework introduces a non-mono-Armijo backtracking method, as shown in the diagram, which achieves faster convergence speed while maintaining convergence accuracy; β is introduced. HZ+ To enhance the algorithm's resilience to iterative congestion, perturbation and restart modules were added to ensure the algorithm's numerical stability. Figure 7 As shown.

[0069] S21. First, the step size of the algorithm is optimized based on the non-monotonic Armijo backtracking method;

[0070] Specifically, a non-monotonic Armijo backtracking method is used instead of the traditional Armijo backtracking, using the method to generate the previous step size α. k-1 The number of backtracking iterations encountered determines the current iteration's state and guides the next step size α. k Initial step size α during generation init The selection of step size allows for the use of step sizes much larger than 1 when the iteration state is good, thus accelerating the convergence speed.

[0071] Specifically, it has the following characteristics: Figure 6 As shown, the α generated during multiple backtrackings is stored in an AlphaList, and a flag variable is added to record the number of backtrackings. At the beginning of each backtracking, the α generated in the previous backtracking is retrieved. init It is stored at the end of AlphaList, and the value of flag is used to determine the next step α after backtracking is completed. init Selection strategy: When the linear search performs well and encounters a small number of backtracking attempts, boldly increase α. init =2α, but be careful when choosing α when encountering many backtrackings. init =α, when encountering a very large number of backtrackings, the α output of the current step will be very small, and will not have much significance for the effective decrease in subsequent iterations. Therefore, it is necessary to appropriately increase α. init These three options correspond to Figure 6 The final choice branch in the algorithm; the role of the SL variable in the diagram is to store the step size corresponding to the current feasible α. When SL is sufficiently small, the LRM-CG algorithm will trigger the stopping condition.

[0072] S22, based on β HZ+ The search direction is updated;

[0073] Specifically, β HZ+ The specific forms of expression are in Figure 5 The value δ is given in the text. k It is used to ensure β HZ+ The constant for sufficient descent, and its specific value, can be adjusted by the user using δ. Note that the difference between two adjacent Riemann gradients is used in its expression; this is relevant when encountering iteration congestion. k The term will become quite small, directly leading to β HZ+ The β value approaches 0, thus ensuring gradient descent in the current step, thereby avoiding incorrect search directions and improving iteration congestion. Furthermore, it can be proven that this β...HZ+ Achieving the theoretically guaranteed effective decrease can be considered as β HS Strategy improvement;

[0074] β HZ+ The design aims to separate the sufficient descent condition from the precision of the line search, ensuring a valid descent direction is obtained even when using an imprecise line search, thus guaranteeing sufficient descent in each iteration. Both experimental verification and theoretical derivation show that β... HZ+ The strategy has a stronger ability to resist iterative congestion and approach local minima.

[0075] S3. Based on the improved LRM-CG algorithm, the initial EDM matrix is ​​completed and solved to obtain the user's location.

[0076] Specifically, perturbation and restart modules have been added, such as... Figure 7 As shown, during the actual testing of the LRM-CG algorithm, we noticed that when the number of nodes is large, a single LRM-CG iteration loop often fails to achieve good convergence upon exiting. This means that the algorithm stops when the Riemann gradient is not sufficiently small. In this case, adding a small amount of noise to the initial solution and solving it again can alleviate the problem of the algorithm stopping at local minima or saddle points to some extent.

[0077] Specific operation reference Figure 7 The EDM D obtained from the first solution is used in the middle. sol Calculate the position of each UE, then add small noise to the position to obtain the perturbation initial value Y1, and put Y1 into the LRM-CG loop again to solve. Repeat this process until the number of perturbations is too large or the Riemann gradient of the new solution is sufficiently small.

[0078] When the number of perturbations exceeds a certain threshold, the current solution is abandoned, a new random user position is generated as the initial value, and Y is generated from it. new Y new The perturbation loop is repeated until too many restarts are performed or the Riemann gradient of the new solution is sufficiently small. When the Riemann gradient cannot be sufficiently small, the algorithm outputs the solution with the smallest Riemann gradient among multiple iterations.

[0079] Reference Figure 8 and Figure 9 This is a simulation experiment data diagram of the present invention. The distances from each user node to several anchor nodes are queried using the recovered EDM, and the user location is completed using a multi-point positioning method. Figure 8 , 9 The middle section was 100m 2The impact of different numbers of users and effective communication distance on algorithm performance in a square field, as well as the decline in algorithm positioning ability as noise level increases, are investigated. Mean square positioning error (MSLE) is used to measure the user positioning error under different conditions.

[0080] Specifically, Figure 8 The diagram shows four scenarios with varying levels of user attention (50 / 100 / 150 / 200 users distributed across 100m). 2 The percentage of events with MSLE < 1m in 100 Monte Carlo simulations in a square field varies with the effective communication distance between nodes. Figure 9 The figure shows how the 80th percentile of MSLE in 100 Monte Carlo simulations varies with the noise variance σ in the distance measurement in four specific scenarios (50 users - 40m communication distance / 100UE - 35m / 150UE - 35m / 200UE - 30m).

[0081] Reference Figure 2 A positioning system based on optimization on a rank matrix manifold includes:

[0082] The estimation module estimates the distance between communication nodes using the maximum likelihood method based on RSSI measurements.

[0083] The building module is used to construct a preliminary EDM matrix based on the estimated distance between communication nodes;

[0084] The optimization module is used to introduce the non-monotonic Armijo backtracking method and enhancement algorithms to construct the improved LRM-CG algorithm;

[0085] The solution module completes the initial EDM matrix based on the improved LRM-CG algorithm to obtain the user's location.

[0086] The content of the above method embodiments is applicable to this system embodiment. The specific functions implemented in this system embodiment are the same as those in the above method embodiments, and the beneficial effects achieved are also the same as those achieved in the above method embodiments.

[0087] The above is a detailed description of the preferred embodiments of the present invention. However, the present invention is not limited to the embodiments described. Those skilled in the art can make various equivalent modifications or substitutions without departing from the spirit of the present invention. All such equivalent modifications or substitutions are included within the scope defined by the claims of this application.

Claims

1. A positioning method based on rank-constrained manifold optimization, characterized in that, The method comprises the following steps: estimating the estimated distance between communication nodes based on RSSI measurement values by using a maximum likelihood method; constructing a preliminary EDM matrix according to the estimated distance between the communication nodes; introducing a non-monotonic Armijo backtracking method and an enhanced algorithm to construct an improved LRM-CG algorithm; completing and solving the preliminary EDM matrix based on the improved LRM-CG algorithm to obtain the position of the user; the step of constructing the preliminary EDM matrix according to the estimated distance between the communication nodes specifically comprises: obtaining the ML estimation value of the distance between each node according to the RSSI measurement value; correcting the ML estimation value of the distance between each node to obtain the unbiased distance value between each node; constructing the preliminary EDM matrix according to the unbiased distance value between each node; the expression of the unbiased distance value between the communication nodes is as follows: In the above formula, P(d0) represents the signal power received at the reference distance d0, P(d ij ) represents the signal power received at the i, j nodes for the link between the i, j nodes, γ represents the path loss exponent, X σ represents a random variable of the standard normal distribution with variance σ 2 , where represents an unbiased estimate of the distance between the i, j nodes under the RSSI radio propagation model, and η represents a constant, represents the maximum likelihood estimator, which is modified to an unbiased estimator by the coefficient .

2. The positioning method based on the fixed-rank matrix manifold optimization according to claim 1, characterized in that, the step of introducing the non-monotonic Armijo backtracking method and the enhanced algorithm to construct the improved LRM-CG algorithm specifically comprises: setting the maximum backtracking number of single line search, the initial step length, the initial iteration state index, the step length inheritance list and the step length indication index; judging the belonging state of the step length inheritance list and calculating the corresponding step length according to the belonging state; carrying out backtracking calculation of a new step length and judging the value of the initial iteration state index according to the calculation result; updating the recommended value of the initial step length according to the value of the initial iteration state index; adding the updated initial step length recommendation value to the step length inheritance list; Introducing β HZ , and construct the improved LRM-CG algorithm.

3. The positioning method based on the fixed-rank matrix manifold optimization according to claim 2, characterized in that, the step of completing and solving the preliminary EDM matrix based on the improved LRM-CG algorithm to obtain the position of the user specifically comprises: solving the completed EDM matrix by a three-point positioning method to obtain the initial user node position; adding noise disturbance to the initial user node position and converting it into the input initial value of the LRM-CG algorithm; iteratively solving the initial value by the improved LRM-CG algorithm to obtain the solution result; judging the solution result, reselecting the user node position for iterative solution when the solution result perturbation frequency reaches a preset threshold and the Riemann gradient of the solution result still does not reach a preset gradient value; until the Riemann gradient of the solution result is less than the preset threshold, outputting the solution at this time and converting it into an EDM, and then calculating the position of the user by using the EDM.

4. A positioning system based on optimization on a manifold of constant-rank matrices, characterized in that, The method comprises the following modules: an estimation module for estimating the estimated distance between communication nodes based on RSSI measurement values by using a maximum likelihood method; a construction module for constructing a preliminary EDM matrix according to the estimated distance between the communication nodes; an optimization module for introducing a non-monotonic Armijo backtracking method and an enhanced algorithm to construct an improved LRM-CG algorithm; a solving module for completing and solving the preliminary EDM matrix based on the improved LRM-CG algorithm to obtain the position of the user; the step of constructing the preliminary EDM matrix according to the estimated distance between the communication nodes specifically comprises: obtaining the ML estimation value of the distance between each node according to the RSSI measurement value; correcting the ML estimation value of the distance between each node to obtain the unbiased distance value between each node; According to the unbiased distance values between the nodes, a preliminary EDM matrix is constructed; The expression of the unbiased distance values between the communication nodes is as follows: In the above formula, P(d0) represents the signal power received at the reference distance d0, P(d ij ) represents the signal power received at the i, j nodes of the link between the i, j nodes, γ represents the path loss exponent, X σ represents a random variable of the standard normal distribution with variance σ 2 , where represents the unbiased estimate of the distance between the i, j nodes under the RSSI radio propagation model, and η represents a constant, represents the maximum likelihood estimator, which is modified to an unbiased estimator by the coefficient .

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