A convex optimization target positioning method for quantifying AOA
By employing convex optimization methods and Taylor series approximation in underwater acoustic sensor networks, combined with semidefinite relaxation techniques, the local convergence problem of AOA target localization in underwater acoustic sensor networks is solved, achieving high-precision target localization, which is suitable for underwater acoustic environments with limited resources.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- NORTHWESTERN POLYTECHNICAL UNIV
- Filing Date
- 2023-12-13
- Publication Date
- 2026-05-29
AI Technical Summary
In underwater acoustic sensor networks, traditional AOA target localization methods tend to converge to local optima, resulting in insufficient localization accuracy. Furthermore, suboptimal communication resources and channels lead to a decline in localization performance. Existing convex optimization methods are not widely applicable in practical underwater acoustic environments.
A convex optimization method combined with Taylor series approximation and semidefinite relaxation technique is adopted. The target is located by quantizing the AOA measurement value. The convex optimization problem is solved using MATLAB's CVX toolbox to ensure the acquisition of the global optimal solution.
In a resource-constrained underwater acoustic sensor network, high-precision target localization is achieved without relying on initial values, demonstrating high accuracy and practicality.
Smart Images
Figure CN117686974B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of signal processing, and in particular to a target localization method suitable for a localization system in which multiple sensor nodes in a wireless sensor network locate a single target node. Background Technology
[0002] In underwater acoustic sensor networks, it is difficult to maintain precise time synchronization between sensor nodes, making distance measurement challenging. In contrast, AOA-based target localization methods only require array antennas to be deployed on sensor nodes, making AOA measurement more suitable for the actual underwater acoustic environment. However, in researching AOA-based target localization methods, traditional methods transform the target node position estimation process into a maximum likelihood estimation optimization problem or a least squares optimization problem. Both of these optimization problems have highly nonlinear and nonconvex objective functions. During iterative solving, the localization problem eventually converges to a local optimum, reducing the accuracy of target node localization. Patent 201710986504.9 provides an AOA localization method that combines closed-form solutions with iterative algorithms. It first uses a one-step weighted least squares estimation method to obtain a closed-form solution for target position estimation, then uses this as the initial value for iterative calculations to further refine the target position. However, the localization accuracy of this method depends on the initial value, and if the iteration results diverge, the closed-form solution becomes the final solution, resulting in poor accuracy. Convex optimization methods can converge to the global optimum and do not require an initial point, achieving better localization performance than iterative methods under high noise conditions. However, there is currently little research on convex optimization target localization methods based on AOA.
[0003] Underwater acoustic sensor networks have strict limitations on communication resources and bandwidth, making it difficult for the fusion center to obtain accurate raw AOA measurements with noise. Furthermore, the imperfect communication channel between the sensor nodes and the fusion center leads to a decrease in the performance of the target position estimation method. Therefore, in practical underwater acoustic environments, it is necessary to consider both quantization transmission and transmission errors caused by imperfect channels during communication. Summary of the Invention
[0004] To overcome the shortcomings of existing technologies, this invention provides a convex optimization target localization method based on quantified AOA. This method utilizes AOA measurements, which are more readily available underwater, and investigates the localization problem in underwater acoustic sensor networks with limited resources and energy, making it more consistent with actual underwater acoustic environments and improving its practicality. Furthermore, it employs a convex optimization method to solve the target localization problem, which converges to the global optimum and is independent of the selection of the initial point, thus improving accuracy.
[0005] The technical solution adopted by this invention to solve its technical problem includes the following steps:
[0006] The first step is to obtain the raw AOA measurements between the sensor nodes in the sensor network;
[0007] The sensor network consists of N sensor nodes with known locations and 1 target node to be located. The true coordinates of the N sensor nodes are as follows: The coordinates of one target node to be located are Then the AOA measurement model between each sensor node is obtained;
[0008] The second step involves using Taylor series approximation to process the arctangent function. .
[0009] arctangent function At the initial point Performing a Taylor series expansion at a given point, and retaining only polynomials of degree two and below, results in the expression: ,in:
[0010] (1)
[0011] (2)
[0012] (3)
[0013] (4)
[0014] (5)
[0015] ; (6)
[0016] Simplifying equations (1) to (6), we get ,in , , , , , The final AOA measurement model, after Taylor series approximation, is expressed as follows: ;
[0017] The third step is to quantize the original AOA measurement values and then send the quantized AOA measurement values to the fusion center after transmission through a non-ideal channel.
[0018] Quantization result at the i-th sensor for Where L is the series, For use in quantization series The preset quantization threshold value of the bit quantizer; at the fusion center, the estimated target position is obtained based on the maximum likelihood estimator;
[0019] Fourth step, due to the polynomial , , The existence of makes equation (7) a non-convex optimization problem. By using positive semidefinite relaxation, the objective function of equation (7) can be transformed into a convex function.
[0020] definition ,in , , , , ,definition , , , ,in , , ,definition , , The objective function of the optimization problem can then be transformed into:
[0021] (8)
[0022] The fifth step is to transform and relax the constraints, converting the problem into a convex problem. The specific implementation process is as follows:
[0023] By applying semidefinite relaxation, the equation , relaxation ,because Then the implicit constraint is obtained as , , , If the five implicit constraints hold, then the final form of this convex optimization problem is:
[0024] (9)
[0025] The sixth step is to solve the convex optimization problem of formula (9) in the fifth step to obtain the estimated value of the target position.
[0026] In the first step, the AOA measurement model between each sensor node is represented as follows: ,in , This represents the AOA measurement value acquired by the sensor node. This represents the actual value of AOA, where i indicates the index. Sensor nodes, The AOA measurement error is a value with a mean of zero and a variance of . Gaussian white noise.
[0027] In the third step, the expression for estimating the target position based on the maximum likelihood estimator is:
[0028] (7)
[0029] in For the estimated coordinates of the target nodes, , , For the i-th sensor, from the value Become a center of integration The transition probability, It is a quantized value.
[0030] In the sixth step, the convex optimization problem of formula (9) is solved using MATLAB's CVX toolbox.
[0031] The beneficial effects of this invention are that it uses AOA measurements, which are easier to obtain underwater, and can still obtain a relatively accurate estimate of the target position in a water acoustic sensor network positioning system with limited resources and energy. Furthermore, it uses a convex optimization method to solve the target positioning problem, which can converge to the global optimal solution and does not depend on the selection of the initial point, thus having strong practicality and high accuracy. Attached Figure Description
[0032] Figure 1 This is a graph showing the relationship between method performance and the variance of quantized AOA measurement noise.
[0033] Figure 2 This is a graph showing the relationship between method performance and quantization series.
[0034] Figure 3 This is a graph showing the relationship between method performance and bit error rate. Detailed Implementation
[0035] The present invention will be further described below with reference to the accompanying drawings and embodiments.
[0036] Suppose there are 10 sensor nodes with known locations and 1 target node to be located in the sensor network. Let the true coordinates of the 10 sensor nodes with known locations be... The coordinates of one target node to be estimated are: The positioning scene is two-dimensional.
[0037] The specific implementation steps are as follows:
[0038] Step 1: Obtain the raw AOA measurements between each sensor node in the sensor network.
[0039] The original AOA measurement expression is obtained through communication between the sensor node and the target node:
[0040]
[0041] in , Indicates the number is Sensor nodes, It is a value with a mean of zero and a variance of . Gaussian white noise.
[0042] Step 2: Due to the arctangent function It is highly nonlinear and can be approximated using Taylor series. .
[0043] arctangent function At the initial point Performing a Taylor series expansion at this point, and retaining only polynomials of degree two and below, results in:
[0044]
[0045] in:
[0046]
[0047]
[0048]
[0049]
[0050]
[0051]
[0052] Approximate processing Simplifying, we can obtain another form, which is expressed as:
[0053]
[0054] in:
[0055]
[0056]
[0057]
[0058] , ,
[0059] The final AOA measurement model, after Taylor series approximation, can be expressed as:
[0060]
[0061] Step 3: Quantize the original AOA measurement values, transmit them through a non-ideal channel, and then send the quantized AOA measurement values to the fusion center.
[0062] quantization result at the i-th sensor for
[0063]
[0064] Where L is the quantization series. It is used for The preset quantization threshold value of the bit quantizer.
[0065] At the fusion center, the target location is estimated based on the maximum likelihood estimator, and its expression is:
[0066]
[0067] in , , For the i-th sensor, from the value Become a center of integration The transition probability, It is a specific quantized value.
[0068] Step 4: Due to the polynomial , , The existence of makes the above equation a non-convex optimization problem. We can use positive semidefinite relaxation to transform the above objective function into a convex function.
[0069] definition
[0070]
[0071] ,
[0072] ,
[0073]
[0074] ,
[0075] in , , , , , , , .
[0076] The objective function of the optimization problem obtained in the third step is then transformed and expressed as:
[0077]
[0078] Step 5: Transform and relax the constraints to convert the problem into a convex problem. The specific implementation process is as follows.
[0079] By applying positive semidefinite relaxation, non-convex equality constraints are transformed. , relaxation
[0080]
[0081] By observing the formula, we can obtain:
[0082]
[0083]
[0084]
[0085]
[0086]
[0087] Therefore, the above five implicit constraints are valid.
[0088] The final form of this convex optimization problem is:
[0089]
[0090] Step 6: Solve the convex optimization problem to obtain an estimate of the target position.
[0091] The convex optimization problem obtained in step 5 is solved using MATLAB's CVX toolbox, ultimately yielding an estimate of the target position.
Claims
1. A convex optimization target localization method for quantified AOA, characterized in that... Includes the following steps: The first step is to obtain the raw AOA measurements between the sensor nodes in the sensor network; The sensor network consists of N sensor nodes with known locations and 1 target node to be located. The true coordinates of the N sensor nodes are as follows: The coordinates of one target node to be located are Then the AOA measurement model between each sensor node is obtained; The second step involves using Taylor series approximation to process the arctangent function. ; arctangent function At the initial point Performing a Taylor series expansion at a given point, and retaining only polynomials of degree two and below, results in the expression: ,in: （1） （2） （3） （4） （5） ； （6） Simplifying equations (1) to (6), we get ,in , , , , , The final AOA measurement model, after Taylor series approximation, is expressed as follows: ; The AOA measurement error is a value with a mean of zero and a variance of . Gaussian white noise; The third step is to quantize the original AOA measurement values and then send the quantized AOA measurement values to the fusion center after transmission through a non-ideal channel. Quantization result at the i-th sensor for Where L is the series, For use in quantization series The preset quantization threshold value of the bit quantizer; at the fusion center, the estimated target position is obtained based on the maximum likelihood estimator; The expression for estimating the target location based on the maximum likelihood estimator is: （7） in The estimated coordinates of the target node. , , For the i-th sensor, from the value Become a center of integration The transition probability, It is a quantized value; Fourth step, due to the polynomial , , The existence of makes equation (7) a non-convex optimization problem. By using positive semidefinite relaxation, the objective function of equation (7) can be transformed into a convex function. definition ,in , , , , ,definition , , , ,in , , ,definition , , The objective function of the optimization problem can then be transformed into: （8） The fifth step is to transform and relax the constraints, converting the problem into a convex problem. The specific implementation process is as follows: By applying semidefinite relaxation, the equation , relaxation ,because Then the implicit constraint is obtained as , , , If the five implicit constraints hold, then the final form of this convex optimization problem is: （9） The sixth step is to solve the convex optimization problem of formula (9) in the fifth step to obtain the estimated value of the target position.
2. The convex optimization target localization method for quantized AOA according to claim 1, characterized in that: In the first step, the AOA measurement model between each sensor node is represented as follows: ,in , This represents the AOA measurement value acquired by the sensor node. This represents the true AOA value, where i represents the sensor node number. The AOA measurement error is a value with a mean of zero and a variance of . Gaussian white noise.
3. The convex optimization target localization method for quantized AOA according to claim 1, characterized in that: In the sixth step, the convex optimization problem of formula (9) is solved using MATLAB's CVX toolbox.