A signal source positioning method and system based on zero-norm outlier separation

Abstract

The application discloses a signal source positioning method and system based on zero norm outlier separation, and belongs to the technical field of wireless communication and positioning, comprising the following steps: acquiring distance measurement values between a signal source and multiple sensors, and constructing a pseudo-linear measurement model; constructing a projection matrix according to the pseudo-linear measurement model, eliminating source position variables in the pseudo-linear measurement model by using the projection matrix, and constructing an underdetermined observation equation; solving the underdetermined observation equation by using an iterative solving mode, and restoring an outlier deviation by using a hard threshold mode in the solving process; correcting the acquired distance measurement values according to the restored outlier deviation, and constructing an optimization problem with a quadratic equality constraint according to the corrected distance measurement values; and solving the optimization problem with the quadratic equality constraint by using a Lagrange multiplier method, so that the position coordinates of the signal source are obtained. By separating and eliminating sparse outliers, and combining with geometric constraint optimization, the application realizes high-precision and high-robustness signal source positioning.

Description

Technical Field

[0001] This invention belongs to the field of wireless communication and positioning technology, and particularly relates to a signal source localization method and system based on zero-norm outlier separation. Background Technology

[0002] With the rapid development of 5G and IoT technologies, wireless positioning technology is playing an increasingly crucial role in numerous application areas such as intelligent transportation, autonomous driving, emergency rescue, and military monitoring. Among the many positioning technologies, time-of-arrival (TOA) positioning methods have attracted much attention due to their high measurement accuracy.

[0003] Standard time-of-arrival (TOA) positioning systems typically assume that measurement errors are caused solely by zero-mean Gaussian white noise. However, in real-world complex electromagnetic environments or scenarios with adversarial interference, this assumption often fails, presenting two main challenges:

[0004] First, there is non-line-of-sight propagation. In urban canyons or indoor environments, due to building obstruction, multipath effects, or complex geographical conditions, signals often cannot propagate directly through a straight path, resulting in them reaching the receiver only through reflection or diffraction. This introduces a positive measurement bias, making the measured distance significantly greater than the actual distance.

[0005] Second, malicious attacks. In open network environments, positioning systems may also face external deception interference or data injection attacks, causing significant abnormal deviations in the measurement values ​​of some sensor nodes.

[0006] The aforementioned non-line-of-sight propagation errors and attack biases are collectively referred to as outliers. These outliers typically have amplitudes much larger than ordinary Gaussian measurement noise and often exhibit sparse characteristics across all measurement data, meaning that only a small portion of the link's measurements are contaminated. Traditional least squares or maximum likelihood algorithms are extremely sensitive to outliers. If these outlier measurements are directly used for positioning calculations, they often introduce significant errors into the final positioning result, severely reducing the system's reliability.

[0007] To mitigate the impact of outliers, existing research widely employs convex relaxation techniques, utilizing the L1 norm to approximate the sparsity of outliers. While L1 norm optimization transforms a non-convex problem into a convex one, thus guaranteeing a globally optimal solution, the L1 norm method penalizes both the number and magnitude of non-zero elements, introducing unavoidable estimation bias. This can lead to the inability to fully recover the true sparse structure in certain situations. Furthermore, existing methods often fail to adequately consider the geometric constraints between position variables and auxiliary variables in the distance measurement equation when handling outliers, leaving room for further improvement in positioning accuracy.

[0008] In summary, as wireless positioning systems are increasingly used in security-sensitive fields, how to achieve high-precision and robust signal source localization under conditions of abnormal measurement or attack interference has become an important research issue. Summary of the Invention

[0009] To address the aforementioned technical problems, this invention provides a signal source localization method based on zero-norm outlier separation, comprising the following steps:

[0010] Obtain distance measurements between the signal source and multiple sensors, and construct a pseudo-linear measurement model based on the distance measurements;

[0011] A projection matrix is ​​constructed based on the pseudolinear measurement model. The projection matrix is ​​then used to eliminate the source location variables in the pseudolinear measurement model, and an underdetermined observation equation containing only sparse outlier bias vectors is constructed.

[0012] The underdetermined observation equation is solved by iterative solution, and outlier deviations are recovered by using a hard threshold method during the solution process.

[0013] The distance measurement value is corrected based on the recovered outlier deviation, and an optimization problem with quadratic equality constraints is constructed based on the corrected distance measurement value.

[0014] The location coordinates of the signal source are obtained by solving the optimization problem with quadratic equality constraints using the Lagrange multiplier method.

[0015] Optionally, distance measurements between the signal source and multiple sensors are obtained, and a pseudo-linear measurement model is constructed based on the distance measurements, specifically including:

[0016] Obtain the raw distance measurement values ​​between the sensor at each known location and the signal source to be located. The raw distance measurement values ​​include the true Euclidean distance, additive Gaussian noise, and sparse anomaly bias.

[0017] By squaring each of the original distance measurements and ignoring second-order noise terms, multiple linearized equations are obtained.

[0018] By combining all the linearized equations, a compact matrix form is formed that includes the unknown parameter vector, the deviation vector, the coefficient matrix, and the observation vector, which serves as the pseudo-linear measurement model.

[0019] Optionally, a projection matrix is ​​constructed based on the pseudolinear measurement model, and the source location variables in the pseudolinear measurement model are eliminated using the projection matrix to construct an underdetermined observation equation containing only sparse anomaly bias vectors, specifically including:

[0020] Construct the left null space orthogonal basis matrix of the coefficient matrix in the pseudo-linear measurement model as the projection matrix;

[0021] Multiplying both ends of the pseudo-linear measurement model by the left null space orthogonal basis matrix eliminates the source location variables, resulting in a linear equation containing only the bias vector, which serves as the underdetermined observation equation.

[0022] The bias vector in the underdetermined observation equation is modeled as a weighted least squares estimation problem based on zero-norm constraints, and the zero-norm constraints are transformed into functional form by introducing an indicator function.

[0023] Optionally, the underdetermined observation equation is solved iteratively, and outlier bias is recovered using a hard thresholding method during the solution process, specifically including:

[0024] By introducing auxiliary and dual variables, an augmented Lagrangian function is constructed based on the underdetermined observation equation.

[0025] Fix the original and dual variables, update the auxiliary variables using a hard threshold method, retain the preset number of elements with the largest absolute value and set the remaining elements to zero;

[0026] By fixing the auxiliary and dual variables, solve the unconstrained quadratic minimization problem to update the original variables;

[0027] Update the dual variable based on the original residual;

[0028] Repeat the above update steps until the convergence condition is met, and output the recovered outlier deviation.

[0029] Optionally, the obtained distance measurement value is corrected based on the recovered outlier deviation, and an optimization problem with quadratic equality constraints is constructed based on the corrected distance measurement value, specifically including:

[0030] The distance measurement value is corrected based on the recovered outlier deviation to obtain the corrected measurement vector;

[0031] Based on the geometric constraint relationship between the position component and the auxiliary component in the unknown parameter vector, construct a quadratic equality constraint;

[0032] Based on the modified measurement vector and the quadratic equality constraint, an optimization problem with quadratic equality constraint is constructed.

[0033] Optionally, the optimization problem with quadratic equality constraints is solved using the Lagrange multiplier method to obtain the position coordinates of the signal source, specifically including:

[0034] By introducing Lagrange multipliers, we construct the Lagrange function for an optimization problem with quadratic equality constraints;

[0035] By differentiating the Lagrange function according to the Kuhn-Tucker conditions and setting the derivative to zero, a closed-form solution for the unknown vector is derived, with the Lagrange multipliers as parameters.

[0036] Substituting the closed-form solution into the quadratic equality constraint, we obtain a univariate nonlinear equation for the Lagrange multipliers.

[0037] The optimal Lagrange multipliers are obtained by solving the univariate nonlinear equation using the bisection method.

[0038] Substituting the optimal Lagrange multiplier into the closed-form solution yields the optimal vector, and the position coordinates of the signal source are extracted based on the optimal vector.

[0039] The present invention also provides a signal source localization system based on zero-norm outlier separation, for implementing the method, comprising:

[0040] The deployment module is used to deploy multiple sensor nodes at known locations within the monitoring area, the sensor nodes being used to receive wireless signals from signal sources at unknown locations;

[0041] A measurement module, the input of which is connected to the output of the deployment module, is used to acquire the propagation time of the wireless signal and convert it into a distance measurement value, and to construct a pseudo-linear measurement model based on the distance measurement value;

[0042] An anomaly decoupling module, whose input is connected to the output of the measurement module, is used to construct a projection matrix based on the pseudo-linear measurement model, eliminate the source location variables in the pseudo-linear measurement model using the projection matrix, construct an underdetermined observation equation containing only sparse anomaly bias vectors, and solve the underdetermined observation equation through an iterative solution method. During the solution process, the outlier bias is recovered using a hard threshold method.

[0043] An optimization construction module is provided, the input of which is connected to the output of the anomaly decoupling module. The module is used to correct the obtained distance measurement value based on the recovered anomaly deviation and to construct an optimization problem with quadratic equality constraints based on the corrected distance measurement value.

[0044] The location solving module, whose input is connected to the output of the optimization construction module, is used to solve the optimization problem with quadratic equality constraints using the Lagrange multiplier method to obtain the location coordinates of the signal source.

[0045] The present invention also provides a computer device, including a memory, a processor, and a computer program stored in the memory, wherein the processor executes the computer program to implement the steps of the method.

[0046] The present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the method.

[0047] The present invention also provides a computer program product, including a computer program that, when executed by a processor, implements the steps of the method.

[0048] Compared with the prior art, the present invention has the following advantages and technical effects:

[0049] This invention constructs a pseudo-linear measurement model based on acquired distance measurements and uses this model to build a projection matrix to eliminate source location variables. It then constructs an underdetermined observation equation containing only sparse outlier deviation vectors, thus decoupling the outlier estimation problem from the source location estimation problem. Based on this, the underdetermined observation equation is solved iteratively, and a hard thresholding method is used to recover outlier deviations during the solution process. This technique directly models the sparsity of outliers, avoiding the estimation bias caused by penalizing outlier amplitudes using L1-norm convex relaxation methods. It achieves accurate identification of outlier indices and unbiased estimation of their amplitudes, providing clean measurement data for subsequent location calculations.

[0050] This invention, after recovering outlier deviations, corrects the obtained distance measurements based on the recovered outlier deviations, eliminating contamination of measurement data caused by non-line-of-sight propagation or malicious attacks. Since this method does not require prior knowledge of the statistical distribution characteristics of outliers, but can achieve effective separation solely based on the sparsity of outliers, it can adapt to various abnormal scenarios such as non-line-of-sight propagation and malicious attacks, significantly improving the robustness of the positioning system in complex electromagnetic environments or under conditions of adversarial interference.

[0051] This invention constructs an optimization problem with quadratic equality constraints based on the corrected distance measurement values ​​and solves it using the Lagrange multiplier method. This technique fully considers the geometric constraint relationship between the position component and auxiliary components in the distance measurement equation, and ensures the structural consistency of the solution through quadratic equality constraints. It avoids the estimation error introduced by traditional methods that ignore this constraint relationship, thereby significantly improving the accuracy of the signal source position calculation.

[0052] This invention eliminates source location variables through a projection matrix, decomposing the original problem into two sub-problems: outlier estimation and location calculation, thus reducing the problem's complexity. In the outlier estimation stage, a combination of iterative solving and hard thresholding is employed, enabling rapid convergence to a sparse solution. In the location calculation stage, the constrained optimization problem is transformed into a root-finding problem of a univariate nonlinear equation using the Lagrange multiplier method, which can be efficiently solved using the bisection method. The entire solution process does not require complex convex optimization iterations, resulting in high computational efficiency and making it suitable for deployment on resource-constrained sensor nodes or edge devices.

[0053] In summary, this invention solves the problems of low positioning accuracy, poor robustness, and high computational complexity of existing technologies under outlier interference by using a series of technical means, such as outlier and location decoupling estimation, hard threshold sparse recovery, correction of measurement values ​​and construction of quadratic equation constraint optimization, and closed-loop solution using the Lagrange multiplier method. It achieves high-precision, high-robustness, and high-efficiency signal source positioning. Attached Figure Description

[0054] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. In the drawings:

[0055] Figure 1 The overall flowchart of the signal source localization method based on zero-norm outlier separation provided by the present invention is shown below.

[0056] Figure 2 This is a schematic diagram illustrating the positioning of a two-dimensional wireless sensor network in a malicious attack scenario according to an embodiment of the present invention;

[0057] Figure 3 This is a schematic diagram illustrating the positioning of a three-dimensional wireless sensor network in a non-line-of-sight propagation scenario according to an embodiment of the present invention;

[0058] Figure 4 This is a structural diagram of the signal source localization system based on zero-norm outlier separation in an embodiment of the present invention;

[0059] Figure 5 This is a structural block diagram of the positioning processing device in an embodiment of the present invention. Detailed Implementation

[0060] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.

[0061] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.

[0062] Example 1

[0063] like Figure 1 As shown, this embodiment provides a signal source localization method based on zero-norm outlier separation, which includes the following steps:

[0064] Obtain distance measurements between the signal source and multiple sensors, and construct a pseudo-linear measurement model based on the distance measurements;

[0065] A projection matrix is ​​constructed based on the pseudolinear measurement model. The projection matrix is ​​then used to eliminate the source location variables in the pseudolinear measurement model, and an underdetermined observation equation containing only sparse outlier bias vectors is constructed.

[0066] The underdetermined observation equation is solved by iterative solution, and outlier deviations are recovered by using a hard threshold method during the solution process.

[0067] The distance measurement value is corrected based on the recovered outlier deviation, and an optimization problem with quadratic equality constraints is constructed based on the corrected distance measurement value.

[0068] The location coordinates of the signal source are obtained by solving the optimization problem with quadratic equality constraints using the Lagrange multiplier method.

[0069] Furthermore, the process of obtaining distance measurements containing outliers and observation noise, and transforming them into a pseudo-linear measurement model in matrix form, specifically includes:

[0070] The known first The coordinates of the sensors are as follows The location of the signal source to be located is denoted as By using the time-of-arrival method, the distance between the signal source and the first... The distance measurement value between the sensors is:

[0071] ;

[0072] in, Indicates signal source With the Distance measurements between sensors Indicates signal source With the The true Euclidean distance between the sensors It is additive Gaussian noise. To address the sparsity anomaly bias, Stacked into vectors ,satisfy The symbols The L0-norm symbol represents the number of non-zero elements in a vector. This represents the maximum number of measurements that are contaminated.

[0073] By squaring and ignoring second-order noise terms, the equation is linearized and rearranged into a compact matrix form:

[0074] ;

[0075] in, Represents an unknown parameter vector, its specific form is: , For the deviation vector, Represents the coefficient matrix. For the observation vector, This is the residual term caused by noise.

[0076] The specific definitions of each matrix and vector are shown in the following equations: ;

[0077] Furthermore, in eliminating source location variables using the null projection matrix of the coefficient matrix and constructing an underdetermined observation equation containing only sparse anomaly bias vectors, the specific steps include:

[0078] Constructing a matrix Left null space orthogonal basis matrix ,satisfy Multiply by the left side of both sides of the pseudolinear measurement model. The decoupled linear equations are obtained as follows:

[0079] ;

[0080] make , At this point, the problem is transformed into solving an underdetermined linear system. Solving the deviation vector .because It is sparse, and can be modeled as a constrained weighted least squares estimation problem based on the L0 norm:

[0081] ;

[0082] in This represents the upper limit of the estimated number of outliers. Due to the highly non-convex nature of the L0-norm constraint, direct solution is extremely difficult. Therefore, an indicator function will be introduced. The original constraints It can be expressed in the form of a function as follows:

[0083] ;

[0084] The problem can now be rewritten as:

[0085] ;

[0086] Furthermore, the process of iteratively solving the underdetermined equations using the Alternating Direction Multiplier Method (ADMM) and recovering outlier biases by introducing a hard threshold operator specifically includes:

[0087] (1) Introducing auxiliary variables and dual variables (1) Construct the augmented Lagrangian function; (2) Update the auxiliary variables. : Fix the original variable and dual variables Solve the minimization subproblem, and retain the one with the largest absolute value using the hard threshold operator. (3) Update the original variable. Fixed auxiliary variables and dual variables Solving the unconstrained quadratic minimization problem yields information about... (3) Solve the system of linear equations; (4) Update the dual variables. : Update using gradient ascent based on the original residuals.

[0088] Furthermore, in correcting the original distance measurement vector using the estimated bias and constructing a squared distance least squares optimization problem (SR-LS) with quadratic equality constraints, the specific steps include:

[0089] Based on the system bias estimate obtained in the first step, the original measurement data is corrected to obtain the corrected measurement vector. Its calculation expression is: .

[0090] Due to unknown parameters The elements in the text are not completely independent; specifically, Front two-dimensional position components With the third auxiliary component The coupling relationship between them can be expressed as The relationship between them can be represented by quadratic form constraints, which can be obtained in the following form:

[0091] ;

[0092] in , Therefore, the source localization problem is modeled as a quadratic constraint squared distance least squares (SR-LS) optimization problem:

[0093] ;

[0094] Furthermore, in deriving the closed-form solution to the constrained optimization problem using the Lagrange multiplier method to obtain the precise location coordinates of the signal source, the specific steps include:

[0095] Introducing Lagrange multipliers Construct the Lagrange function Its expression is as follows:

[0096] ;

[0097] Specifically, based on the KKT conditions, by differentiating the Lagrange function and setting it to zero, the relationship with the unknown vector is derived. Closed-form solution :

[0098] ;

[0099] The closed solution depends on the Lagrange multipliers. Substituting the closed-form solution into the quadratic constraint equation, we obtain the following about Univariate nonlinear equations :

[0100] ;

[0101] To ensure that the obtained stationary point is the global minimum, right The Hessian matrix must be positive semi-definite. This is achieved through... Taking the second derivative, we can obtain:

[0102] ;

[0103] In order to establish , It must lie within a specific feasible interval, the range of which is determined by the distribution of the generalized eigenvalues. This interval is defined as... :

[0104] ;

[0105] use In the interval To determine the monotonicity, the optimal value is obtained by numerically searching using the bisection method. Ultimately, Substituting into the closed-form solution formula, the optimal vector is obtained. The first two elements are the final source location estimate. .

[0106] On the other hand, this embodiment also provides a signal source localization system based on zero-norm outlier separation, used to implement the method, including:

[0107] The deployment module is used to deploy multiple sensor nodes at known locations within the monitoring area, the sensor nodes being used to receive wireless signals from signal sources at unknown locations;

[0108] A measurement module, the input of which is connected to the output of the deployment module, is used to acquire the propagation time of the wireless signal and convert it into a distance measurement value, and to construct a pseudo-linear measurement model based on the distance measurement value;

[0109] An anomaly decoupling module, whose input is connected to the output of the measurement module, is used to construct a projection matrix based on the pseudo-linear measurement model, eliminate the source location variables in the pseudo-linear measurement model using the projection matrix, construct an underdetermined observation equation containing only sparse anomaly bias vectors, and solve the underdetermined observation equation through an iterative solution method. During the solution process, the outlier bias is recovered using a hard threshold method.

[0110] An optimization construction module is provided, the input of which is connected to the output of the anomaly decoupling module. The module is used to correct the obtained distance measurement value based on the recovered anomaly deviation and to construct an optimization problem with quadratic equality constraints based on the corrected distance measurement value.

[0111] The location solving module, whose input is connected to the output of the optimization construction module, is used to solve the optimization problem with quadratic equality constraints using the Lagrange multiplier method to obtain the location coordinates of the signal source.

[0112] Example 2

[0113] To mitigate the impact of outliers, existing research widely employs convex relaxation techniques, utilizing the L1-norm to approximate the sparsity of outliers. While L1-norm optimization transforms a non-convex problem into a convex one, thus guaranteeing a globally optimal solution, it is merely a relaxed approximation of the L1-norm. The L1-norm method tends to penalize both the number and magnitude of non-zero elements, introducing unavoidable estimation bias and making it impossible to fully recover the true sparse structure in some cases.

[0114] In contrast, optimization methods directly based on the L0 norm exhibit significant theoretical and performance advantages in handling sparse outliers. First, the L0 norm directly corresponds to the number of non-zero elements, perfectly aligning with the physical characteristics of NLOS propagation or malicious node attacks. Second, unlike the L1 norm, the L0 norm does not penalize the magnitude of outliers, thus avoiding "shrinkage bias." After correctly identifying outlier indices, it can achieve unbiased estimation of normal measurements, much like an Oracle estimator. Although L0 norm optimization is inherently an NP-hard problem, by introducing algorithms such as the Alternating Direction Multiplier Method (ADMM), it can be decomposed into a series of easily solvable subproblems, thereby achieving superior robustness and localization accuracy compared to convex relaxation methods while maintaining computational efficiency.

[0115] This embodiment simulates a high-security wireless positioning scenario (such as military area monitoring or critical infrastructure protection), in which some sensor nodes may be subject to malicious third-party cyberattacks (such as data tampering or fake data injection), resulting in significant human-induced deviations in ranging data.

[0116] like Figure 1 The diagram shown is a schematic representation of the overall process provided in this embodiment. The specific steps are as follows:

[0117] S100: Obtain distance measurements containing outliers and observation noise, and convert them into a pseudo-linear measurement model in matrix form;

[0118] S101: Constructing a two-dimensional positioning system model:

[0119] This embodiment first defines a two-dimensional wireless positioning scenario. (Refer to...) Figure 2 This is a schematic diagram illustrating a usage scenario in this embodiment. The example includes... The sensor node with a known location will be the first The coordinates of the sensors are as follows and an unknown signal source to be located. , in Corresponding signal source The x and y coordinates are plotted in a two-dimensional coordinate system. The distance between the signal source and the i-th sensor is obtained by measuring the time *t* it takes for the signal to travel from the signal source to the receiver, multiplying this time by the signal propagation speed *c*.

[0120] ;

[0121] in Indicates signal source With the Distance measurements between sensors Indicates signal source With the The true Euclidean distance between the sensors For measurement noise that follows a zero-mean Gaussian distribution, the variance is... That is, satisfying In this embodiment, The measurement bias caused by an attack is represented by the attack bias vector, which is typically generated because attackers can only control a small number of nodes in the network. It exhibits sparsity properties. Specifically, in Of the sensors, only a small portion (at most) The measurements from one sensor were contaminated, while those from the remaining sensors... ,Right now satisfy , where the symbol The L0-norm symbol represents the number of non-zero elements in a vector. This represents the maximum number of measurements that are contaminated.

[0122] If these non-zero values ​​are not processed Directly substituting into the traditional least squares algorithm, because... Typically much larger than noise This can lead to significant errors in the positioning results. Therefore, the core of this embodiment lies in... The unknown parameters are estimated and eliminated to achieve robust localization.

[0123] S102: Linearization of the measurement equation:

[0124] Because the original distance equation is highly nonlinear, the processor first linearizes it. In Moving the equation to the left side and squaring both sides while ignoring the second-order noise term, we obtain an approximate linear equation:

[0125] ;

[0126] By defining an unknown vector Sum of deviation vectors ,in Solving the equations for all sensors simultaneously yields a compact matrix form:

[0127] ;

[0128] The matrices and vectors are defined as follows:

[0129] , , ;

[0130] S200: Eliminate source location variables by using the null projection matrix of the coefficient matrix and construct an underdetermined observation equation that contains only sparse anomaly bias vectors;

[0131] S201: Transform the system of equations using the null projection matrix of the matrix:

[0132] The goal of the first phase is to operate without knowing the source location. In this case, first estimate the abnormal deviation. .

[0133] Therefore, the above matrix equations are processed using matrix... Left null space orthogonal basis matrix Decoupling is performed to satisfy Multiply by the left side of both sides of the equation. Eliminate unknown parameters This yields a result containing only the deviation vector. The linear equation:

[0134] ;

[0135] make , At this point, the problem is transformed into solving underdetermined linear equations. Solving the deviation vector .

[0136] S201: Introduce indicator functions to eliminate L0-norm constraints:

[0137] because It is sparse, and can be modeled as a constrained weighted least squares estimation problem based on the L0 norm:

[0138] ;

[0139] in This represents the upper limit of the estimated number of outliers. Due to the highly non-convex nature of the L0-norm constraint, direct solution is extremely difficult. Therefore, an indicator function will be introduced. The original constraints It can be expressed in the form of a function as follows:

[0140] ;

[0141] The problem can now be rewritten as:

[0142] ;

[0143] S300: The underdetermined model is solved using the Alternating Direction Multiplier Method (ADMM), and the sparse large numerical measurement bias is recovered iteratively through the hard threshold operator;

[0144] S301: Constructing the augmented Lagrangian function:

[0145] This embodiment uses ADMM for iterative solution. By introducing auxiliary variables... , making By splitting the variables, the original problem is reconstructed into the following constrained optimization form:

[0146] ;;

[0147] Accordingly, we construct the augmented Lagrangian function for this problem. as follows:

[0148] ;

[0149] in For the dual variables corresponding to the equality constraints, This is a penalty parameter used to control the convergence speed of the algorithm and the severity of the penalty for constraint violations.

[0150] The ADMM algorithm uses alternating minimization... about and The subproblem, and update the dual variable. Let's find a solution. The specific derivation steps for the next iteration are as follows:

[0151] S302: Update auxiliary variables :

[0152] First, fix the variables. and dual variables By minimizing To update auxiliary variables Ignore irrelevant terms. The update can be achieved by solving the following subproblems:

[0153] ;

[0154] in For indicator functions With the introduction of non-convex constraints, this subproblem has a closed-form solution based on the hard threshold operator. Specifically, the optimal estimate is... The update rule is to preserve the vector. The largest absolute value By setting one element to zero and setting the remaining elements to zero, the mathematical expression is:

[0155] ;

[0156] in, for The largest median value A set of indices for elements.

[0157] S303: Update the deviation vector renew:

[0158] Utilizing the updated auxiliary variables and fixed dual variables Update the original variable Due to the hard threshold operator and Irrelevant, variable The update involves solving an unconstrained quadratic minimization problem:

[0159] ;

[0160] Given that the objective function is related to It is strictly convex and differentiable, and the optimal solution must satisfy the condition that the gradient is zero, that is:

[0161] ;

[0162] Rearranging the above equations, we obtain the following about The system of linear equations is as follows:

[0163] ;

[0164] S304: Update the dual vector :

[0165] Update the Lagrange multipliers based on the original residuals. This step can be viewed as a gradient ascent process, used to strengthen the constraints. :

[0166] ;

[0167] The algorithm iterates until the convergence condition is met, and the original residual is usually defined. With dual residual When the norms of both are less than a preset threshold When the iteration stops, output the final attack bias estimation vector. .

[0168] S400: Correct the original distance measurement vector using the estimated bias and construct a squared distance least squares optimization problem with quadratic equality constraints (SR-LS):

[0169] In obtaining attack bias estimation Then, the original measurement equation is corrected. The corrected measurement vector is then obtained. Its calculation expression is: Representing a new least squares problem:

[0170] ;

[0171] because The elements in the solution are not independent; additional structural constraints must be introduced to ensure the consistency of the physical meaning of the solution. Specifically, The first two positional components With the third auxiliary component Geometric constraints must be satisfied between them. This embodiment models the relationship as about Quadratic form equality constraints:

[0172] ;

[0173] in, , Based on the above constraints, this embodiment transforms the source localization problem into a quadratic constraint squared distance least squares (SR-LS) optimization problem:

[0174] ;

[0175] S500: Using the Lagrange multiplier method, the closed-form solution to the constrained optimization problem is derived, yielding the precise location coordinates of the signal source.

[0176] To solve the aforementioned constrained optimization problem, this embodiment introduces Lagrange multipliers. Construct the Lagrange function as follows:

[0177] ;

[0178] According to the KKT conditions, Find the first-order partial derivative and set it to zero, then derive the expression for... The linear equation is:

[0179] ;

[0180] From this, we can obtain information about Lagrange multipliers. Unknown parameters Closed-form solution:

[0181] ;

[0182] At this point, solving the original problem is transformed into finding the optimal solution. , the above Substituting the analytical expression back into the original quadratic constraint equation, we eliminate the variables. , get about Univariate nonlinear equations ,Right now:

[0183] ;

[0184] To ensure that the obtained stationary point is the global minimum, right The Hessian matrix must be positive semi-definite. This is achieved through... Taking the second derivative, we can obtain:

[0185] ;

[0186] In order to establish , It must lie within a specific feasible interval, the range of which is determined by the distribution of the generalized eigenvalues. This interval is defined as... :

[0187] ;

[0188] because In the interval To address the monotonicity of the solution, this embodiment employs a bisection method for numerical root finding, rapidly converging to the optimal solution through iterative calculation. Ultimately, Substituting into the closed-form solution formula, the optimal vector is calculated. The first two elements are the final source location estimate. .

[0189] Example 3

[0190] This embodiment discloses a robust positioning method applicable to complex indoor environments such as large-scale warehousing and logistics centers, underground parking lots, or industrial plants. In these application scenarios, there are numerous obstacles such as high-rise shelves, load-bearing walls, metal equipment, or moving vehicles. These obstacles prevent some wireless signals from propagating in a straight line, forcing them to reach the receiver through reflection, refraction, or diffraction, thus generating non-line-of-sight errors. If these errors are not identified and suppressed, this positive time delay error will significantly reduce positioning accuracy, leading to navigation failure of logistics robots or automated guided vehicles.

[0191] The specific implementation steps of this embodiment are basically the same as those of Embodiment 2, in conjunction with the appendix. Figure 1 As shown, the specific steps are described below:

[0192] S100: Obtain distance measurements containing outliers and observation noise, and transform them into a pseudo-linear measurement model in matrix form.

[0193] S101: Establishment of a 3D non-line-of-sight measurement model:

[0194] This embodiment extends the monitoring area to a three-dimensional plane, such as Figure 3 As shown, M sensor nodes are deployed at known locations. The coordinates of each sensor are represented as follows: The location of the signal source to be located is denoted as... The system obtains the distance measurements between the signal source and each sensor by multiplying the signal's time of flight by its propagation speed.

[0195] ;

[0196] in Indicates signal source With the The true Euclidean distance between the sensors For measurement noise that follows a zero-mean Gaussian distribution, the variance is... That is, satisfying .

[0197] In this embodiment, Specifically, this refers to measurement bias caused by NLOS propagation. Unlike the malicious attack in Example 2, NLOS bias is caused by physical path elongation, and therefore has a non-negative property. Meanwhile, since there are usually some direct paths in the warehouse environment, only a portion of the base stations are affected by NLOS, therefore the deviation vector... It still exhibits sparsity. Assume the maximum number of paths contaminated by NLOS is... Then the vector satisfy The symbols The L0-norm symbol represents the number of non-zero elements in a vector. This represents the maximum number of measurements that are contaminated.

[0198] S102: Linearization of the measurement equation:

[0199] Because the original distance equation is highly nonlinear, the processor first linearizes it. In Moving the equation to the left side and squaring both sides while ignoring the second-order noise term, we obtain an approximate linear equation:

[0200] ;

[0201] By defining an unknown vector Sum of deviation vectors ,in Solving the equations for all sensors simultaneously yields a compact matrix form:

[0202] ;

[0203] The matrices and vectors are defined as follows:

[0204] , , ;

[0205] S200: Transform the equations using the null projection matrix of the matrix to eliminate source location variables and construct an underdetermined observation model that contains only sparse measurement biases:

[0206] S201: Transform the system of equations using the null projection matrix of the matrix:

[0207] In order to avoid involving unknown source locations In the case of deviation, using a matrix Left null space orthogonal basis matrix Decoupling is performed to satisfy Multiply by the left side of both sides of the equation. Eliminate unknown parameters This yields a result containing only the deviation vector. The linear equation:

[0208] ;

[0209] make , At this point, the problem is transformed into solving an underdetermined system of linear equations. .

[0210] S201: Introduce indicator functions to eliminate L0-norm constraints:

[0211] because It is sparse, and can be modeled as a constrained weighted least squares estimation problem based on the zero norm as follows:

[0212] ;

[0213] in This represents the upper limit of the estimated number of outliers. Due to the high non-convexity of the zero norm constraint, direct solution is extremely difficult. Therefore, an indicator function will be introduced. The original constraint can be expressed as a function in the following form:

[0214] ;

[0215] The problem can now be rewritten as:

[0216] ;

[0217] S300: The alternating direction multiplier method (ADMM) is used to solve the underdetermined equations, and a hard threshold operator is introduced to recover outlier deviations;

[0218] S301: Constructing the augmented Lagrangian function:

[0219] This embodiment uses ADMM for iterative solution. By introducing auxiliary variables... , making By splitting the variables, the original problem is reconstructed into the following constrained optimization form:

[0220] ;

[0221] Accordingly, we construct the augmented Lagrangian function for this problem. as follows:

[0222] ;

[0223] in For the dual variables corresponding to the equality constraints, This is a penalty parameter used to control the convergence speed of the algorithm and the severity of the penalty for constraint violations.

[0224] The ADMM algorithm uses alternating minimization... about and The subproblem, and update the dual variable. Let's find a solution. The specific derivation steps for the next iteration are as follows:

[0225] S302: Update auxiliary variables :

[0226] First, fix the variables. and dual variables By minimizing To update auxiliary variables Ignore irrelevant terms. The update can be achieved by solving the following subproblems:

[0227] ;

[0228] in For indicator functions With the introduction of non-convex constraints, this subproblem has a closed-form solution based on the hard threshold operator. Specifically, the optimal estimate is... The update rule is to preserve the vector. The largest absolute value By setting one element to zero and setting the remaining elements to zero, the mathematical expression is:

[0229] ;

[0230] in, for The largest median value A set of indices for elements.

[0231] S303: Update the deviation vector renew:

[0232] Utilizing the updated auxiliary variables and fixed dual variables Update the original variable Due to the hard threshold operator and Irrelevant, variable The update involves solving an unconstrained quadratic minimization problem:

[0233] ;

[0234] Given that the objective function is related to It is strictly convex and differentiable, and the optimal solution must satisfy the condition that the gradient is zero, that is:

[0235] ;

[0236] Rearranging the above equations, we obtain the following about The system of linear equations is as follows:

[0237] ;

[0238] S304: Update the dual vector :

[0239] Update the Lagrange multipliers based on the original residuals. This step can be viewed as a gradient ascent process, used to strengthen the constraints. :

[0240] ;

[0241] The algorithm iterates until the convergence condition is met, and the original residual is usually defined. With dual residual When the norms of both are less than a preset threshold When the iteration stops, output the final attack bias estimation vector. .

[0242] S400: Correct the original distance measurement vector using the estimated bias and construct a squared distance least squares optimization problem with quadratic equality constraints (SR-LS):

[0243] In obtaining attack bias estimation Then, the original measurement equation is modified. The modified measurement vector is defined. Representing a new least squares problem:

[0244] ;

[0245] because The elements in the solution are not independent; additional structural constraints must be introduced to ensure the consistency of the physical meaning of the solution. Specifically, The first two positional components With the third auxiliary component The coupling relationship between them can be expressed as Write it as about The quadratic form of can be used to obtain the following equality constraints:

[0246] ;

[0247] in , Based on the above constraints, this embodiment transforms the source localization problem into a quadratic constraint squared distance least squares (SR-LS) optimization problem:

[0248] ;

[0249] S500: Using the Lagrange multiplier method, the closed-form solution to the constrained optimization problem is derived, yielding the precise location coordinates of the signal source.

[0250] To solve the aforementioned constrained optimization problem, this embodiment introduces Lagrange multipliers. Construct the Lagrange function as follows:

[0251] ;

[0252] According to the KKT conditions, Taking the first-order partial derivative and setting it to zero, we can obtain the following about The linear equation is:

[0253] ;

[0254] From this, we can deduce the Lagrange multipliers. about The closed-form solution is:

[0255] ;

[0256] At this point, solving the original problem is transformed into finding the optimal solution. , the above Substituting the analytical expression back into the original quadratic constraint equation and eliminating the variables , get about Univariate nonlinear equations ,Right now:

[0257] ;

[0258] To ensure that the obtained stationary point is the global minimum, right The Hessian matrix must be positive semi-definite. This is achieved through... Taking the second derivative, we can obtain:

[0259] ;

[0260] In order to establish , It must lie within a specific feasible interval, the range of which is determined by the distribution of the generalized eigenvalues. This interval is defined as... :

[0261] ;

[0262] because In the interval To address the monotonicity of the solution, this embodiment employs a bisection method for numerical root finding, rapidly converging to the optimal solution. .Will Substituting into the closed-form solution formula, the optimal augmented vector is calculated. .

[0263] Ultimately, the system from The first two elements are extracted to obtain the precise coordinates of the signal source. These coordinates are then transmitted in real time to the robot's navigation control module for path planning and obstacle avoidance, effectively overcoming the positioning drift problem caused by shelf obstruction and achieving robust positioning in a warehouse environment.

[0264] To address the distance localization problem with sparse outlier bias, this embodiment proposes a robust source localization method based on zero-norm outlier decomposition. Compared to existing technologies, it has significant technical advantages:

[0265] High robustness: By introducing a sparse bias vector By employing L0-norm constraints, this method can explicitly identify and remove outliers from measurement data. During the iteration of the ADMM algorithm, a hard threshold operator is used to dynamically filter out large attack errors or NLOS errors, ensuring that subsequent location calculations are not severely affected by contaminated data. This is particularly important for security-sensitive wireless sensor networks.

[0266] High accuracy: Traditional methods often ignore the algebraic relationship between the squared distance terms, leading to biased results. This method uses the SR-LS algorithm in the second stage, strictly incorporating the coupling relationship between the position component and the auxiliary component as a quadratic constraint into the optimization problem, and uses the Lagrange multiplier method to obtain a closed-form solution, avoiding local optima and significantly improving positioning accuracy.

[0267] Computational efficiency optimization: Although the problem itself is non-convex, the above embodiments decompose it using a two-step strategy. The first step, the ADMM algorithm, decomposes the complex problem into simpler subproblems, resulting in low computational complexity. The second step, the SR-LS algorithm, transforms the multidimensional optimization into a root-finding problem with one-dimensional variables, which converges quickly using a bisection method. This makes the algorithm suitable for operation on embedded sensor nodes or edge gateways with limited computing power.

[0268] No prior distribution information required: Unlike methods such as Bayesian estimation or Kalman filtering, this device does not require prior knowledge of the specific probability distribution of abnormal deviations. It only needs to assume that the deviations are sparse, which enhances the adaptability of the algorithm in real-world complex environments.

[0269] Example 4

[0270] Based on the above method, this embodiment also provides a signal source localization system based on zero-norm outlier separation, such as... Figure 4 As shown: It includes a deployment module 501, a measurement module 502, and a processing module 503.

[0271] Deployment module 501 is configured to deploy sensor nodes at known locations within the monitoring area to receive wireless signals transmitted from a signal source at an unknown location, wherein the location coordinates of the signal source are parameters to be estimated.

[0272] The measurement module 502 is configured to acquire the propagation time of the signal from the signal source to each sensor node, convert it into a distance measurement value, and construct a nonlinear distance measurement model containing sparse abnormal deviations based on the distance measurement values. The abnormal deviations are caused by non-line-of-sight propagation or malicious attacks.

[0273] Processing module 503 is configured to convert the nonlinear distance measurement model into a pseudo-linear system of equations and perform a two-step robust positioning solution:

[0274] The first stage involves constructing an orthogonal projection matrix to eliminate source location variables in the pseudo-linear equation system, establishing an L0-norm-constrained least squares problem for the outlier, and using the alternating direction multiplier method (ADMM) to iteratively solve for the estimated value of the sparse outlier.

[0275] The second stage involves using the estimated value of the abnormal deviation to correct the original distance measurement value, constructing a squared distance weighted least squares (SR-LS) optimization problem with quadratic equality constraints, and solving for the precise location coordinates of the signal source by combining the Lagrange multiplier method with the bisection method.

[0276] refer to Figure 5 This embodiment also provides a positioning processing device, including a processor, a memory, and a communication bus: the communication bus is used to realize the connection and communication between the processor, the memory, and several sensor interfaces; the processor may be one or more central processing units (CPU), microprocessors, or application-specific integrated circuits (ASICs), which are configured to execute computer programs stored in the memory, and implement the above-described method when executing the computer programs.

[0277] On the other hand, this embodiment also provides a computer device, including a memory, a processor, and a computer program stored in the memory, wherein the processor executes the computer program to implement the steps provided in any one of Embodiments 1, 2, and 3.

[0278] On the other hand, this embodiment also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps provided in any one of Embodiments 1, 2, and 3.

[0279] On the other hand, this embodiment also provides a computer program product, including a computer program that, when executed by a processor, implements the steps provided in any one of embodiments one, two, and three.

[0280] The above are merely preferred embodiments of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A signal source localization method based on zero-norm outlier separation, characterized in that, Includes the following steps: Obtain distance measurements between the signal source and multiple sensors, and construct a pseudo-linear measurement model based on the distance measurements; A projection matrix is ​​constructed based on the pseudolinear measurement model. The projection matrix is ​​then used to eliminate the source location variables in the pseudolinear measurement model, and an underdetermined observation equation containing only sparse outlier bias vectors is constructed. The underdetermined observation equation is solved by iterative solution, and outlier deviations are recovered by using a hard threshold method during the solution process. The distance measurement value is corrected based on the recovered outlier deviation, and an optimization problem with quadratic equality constraints is constructed based on the corrected distance measurement value. The location coordinates of the signal source are obtained by solving the optimization problem with quadratic equality constraints using the Lagrange multiplier method.

2. The method according to claim 1, characterized in that, Acquire distance measurements between the signal source and multiple sensors, and construct a pseudo-linear measurement model based on the distance measurements, specifically including: Obtain the raw distance measurement values ​​between the sensor at each known location and the signal source to be located. The raw distance measurement values ​​include the true Euclidean distance, additive Gaussian noise, and sparse anomaly bias. By squaring each of the original distance measurements and ignoring second-order noise terms, multiple linearized equations are obtained. By combining all the linearized equations, a compact matrix form is formed that includes the unknown parameter vector, the deviation vector, the coefficient matrix, and the observation vector, which serves as the pseudo-linear measurement model.

3. The method according to claim 1, characterized in that, A projection matrix is ​​constructed based on the pseudolinear measurement model. This projection matrix is ​​then used to eliminate the source location variables in the pseudolinear measurement model, resulting in an underdetermined observation equation containing only sparse anomaly bias vectors. Specifically, this includes: Construct the left null space orthogonal basis matrix of the coefficient matrix in the pseudo-linear measurement model as the projection matrix; Multiplying both ends of the pseudo-linear measurement model by the left null space orthogonal basis matrix eliminates the source location variables, resulting in a linear equation containing only the bias vector, which serves as the underdetermined observation equation. The bias vector in the underdetermined observation equation is modeled as a weighted least squares estimation problem based on zero-norm constraints, and the zero-norm constraints are transformed into functional form by introducing an indicator function.

4. The method according to claim 1, characterized in that, The underdetermined observation equation is solved iteratively, and outlier bias is recovered using a hard thresholding method during the solution process. Specifically, this includes: By introducing auxiliary and dual variables, an augmented Lagrangian function is constructed based on the underdetermined observation equation. Fix the original and dual variables, update the auxiliary variables using a hard threshold method, retain the preset number of elements with the largest absolute value and set the remaining elements to zero; By fixing the auxiliary and dual variables, solve the unconstrained quadratic minimization problem to update the original variables; Update the dual variable based on the original residual; Repeat the above update steps until the convergence condition is met, and output the recovered outlier deviation.

5. The method according to claim 1, characterized in that, The distance measurement value is corrected based on the recovered outlier deviation, and an optimization problem with quadratic equality constraints is constructed based on the corrected distance measurement value, specifically including: The distance measurement value is corrected based on the recovered outlier deviation to obtain a corrected measurement vector; Based on the geometric constraint relationship between the position component and the auxiliary component in the unknown parameter vector, construct a quadratic equality constraint; Based on the modified measurement vector and the quadratic equality constraint, an optimization problem with quadratic equality constraint is constructed.

6. The method according to claim 1, characterized in that, The optimization problem with quadratic equality constraints is solved using the Lagrange multiplier method to obtain the position coordinates of the signal source, specifically including: By introducing Lagrange multipliers, we construct the Lagrange function for an optimization problem with quadratic equality constraints; By differentiating the Lagrange function according to the Kuhn-Tucker conditions and setting the derivative to zero, a closed-form solution for the unknown vector is derived, with the Lagrange multipliers as parameters. Substituting the closed-form solution into the quadratic equality constraint, we obtain a univariate nonlinear equation for the Lagrange multipliers. The optimal Lagrange multipliers are obtained by solving the univariate nonlinear equation using the bisection method. Substituting the optimal Lagrange multiplier into the closed-form solution yields the optimal vector, and the position coordinates of the signal source are extracted based on the optimal vector.

7. A signal source localization system based on zero-norm outlier separation, characterized in that, For implementing the method of any one of claims 1-6, comprising: The deployment module is used to deploy multiple sensor nodes at known locations within the monitoring area, the sensor nodes being used to receive wireless signals from signal sources at unknown locations; A measurement module, the input of which is connected to the output of the deployment module, is used to acquire the propagation time of the wireless signal and convert it into a distance measurement value, and to construct a pseudo-linear measurement model based on the distance measurement value; An anomaly decoupling module, the input of which is connected to the output of the measurement module, is used to construct a projection matrix based on the pseudo-linear measurement model, eliminate the source location variables in the pseudo-linear measurement model using the projection matrix, construct an underdetermined observation equation containing only sparse anomaly bias vectors, and solve the underdetermined observation equation through an iterative solution method. During the solution process, the outlier bias is recovered using a hard threshold method. An optimization construction module is provided, the input of which is connected to the output of the anomaly decoupling module. The module is used to correct the obtained distance measurement value based on the recovered anomaly deviation and to construct an optimization problem with quadratic equality constraints based on the corrected distance measurement value. The location solving module, whose input is connected to the output of the optimization construction module, is used to solve the optimization problem with quadratic equality constraints using the Lagrange multiplier method to obtain the location coordinates of the signal source.

8. A computer device comprising a memory, a processor, and a computer program stored in the memory, characterized in that, The processor executes the computer program to implement the steps of the method according to any one of claims 1-6.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that, When executed by a processor, the computer program implements the steps of the method according to any one of claims 1-6.

10. A computer program product, comprising a computer program, characterized in that, When executed by a processor, the computer program implements the steps of the method according to any one of claims 1-6.

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