Passive Radar Target Localization Method Based on Lagrangian Neural Network

Abstract

The present invention discloses a passive radar target positioning method based on Lagrangian neural network, which solves the problem of stable target positioning under abnormal measurement values. The implementation includes: establishing an observation model; obtaining the time difference measurement values of the target to the main and auxiliary receiving stations; formulating a positioning equation; converting the positioning into a constrained optimization; constructing and reconstructing the Lagrangian multiplier formula; defining two types of neurons for solution; constructing a Lagrangian neural network; and obtaining the target position of passive radar target positioning. The present invention is optimized, and the Lagrangian multiplier formula is obtained by the Lagrangian multiplier method. By taking the derivative to solve and construct the Lagrangian neural network, the traditional method is changed to iteration in the Lagrangian neural network, avoiding linearization loss and realizing stable target positioning under abnormal measurement values. The present invention has higher positioning accuracy under low signal-to-noise ratio and realizes stable positioning. It is used for multi-station passive radar positioning.

Description

Technical Field

[0001] The present invention belongs to the technical field of radar, mainly relates to passive radar positioning, and specifically is a passive radar target positioning method based on Lagrangian neural network. It can be used in multi-passive radar target positioning systems. Background Art

[0002] Technical background: Passive radar positioning refers to a positioning technology in which the radar does not need to radiate electromagnetic wave signals during the positioning process, and can complete the positioning requirements of the target only relying on the radiation signals of the target or the reflected signals of specific known signals. It is a non-cooperative technology. According to the different ways of solving the target position, passive radar positioning algorithms can be divided into the following two categories: indirect positioning algorithms and direct positioning algorithms.

[0003] In direct positioning, most are based on the maximum likelihood ratio criterion at the signal level, and use a two-dimensional search method to find the point that satisfies the maximum likelihood ratio within the search area, which is the target position. This process requires a huge amount of computation. In the indirect method, first, the time difference of arrival measurements are estimated. The passive radar receiving station includes a passive radar receiving main station and a passive radar receiving secondary station. Each time difference of arrival is the difference between the arrival times of the target to the passive radar receiving main station and the passive radar receiving secondary station. By solving a set of hyperbola equations of the time differences of arrival, the target position can be estimated. Since the positioning equation is highly nonlinear, the usual solution is to linearize the hyperbola equation into a set of linear equations, and then use linear least squares or weighted least squares to estimate the target position. However, in the actual measurement process, non-line-of-sight propagation and signal interference may exist. Therefore, there are outlier points in the measured time differences. In this case, if the traditional linear least squares method is used, the positioning performance will be severely reduced.

[0004] In 1992, Shengwei Zhang and A.G. Constantinides proposed a Lagrangian neural network algorithm. This method is based on the Lagrange multiplier theory in optimization and seeks solutions that satisfy the necessary conditions for optimality. The equilibrium point of the network satisfies the Kuhn-Tucker conditions of the problem. Except for some general regularity and convexity conditions, there is no clear restriction on the form of the cost function. The network does not follow the direct search method of penalty functions, but instead looks for points that satisfy the necessary conditions for first-order optimality in the state space whenever possible. There are two types of neurons in the network, variable neurons and Lagrangian neurons, according to their contributions in searching for the optimal solution. Variable neurons seek the minimum point of the cost function and provide solutions at the equilibrium point, while Lagrangian neurons introduce dynamic trajectories into the feasible region, that is, the set of all points that satisfy the constraints.

[0005] Meanwhile, in 2014, Yu Xin, Yu Yan and others applied Lagrangian neural network to the computer field. For the non-smooth optimization problem where the objective function is a locally Lipschitz function and its feasible region consists of a set of equality-constrained smooth convex functions, the non-smooth optimization problem was solved by the smooth Lagrangian neural network.

[0006] In 2017, Wang Xingxing and Li Guocheng applied Lagrangian neural network to the field of signal processing, studied the sparse signal recovery algorithm based on Lagrangian neural network, and proved that the network can converge to the optimal solution quickly and effectively, and then reconstruct the sparse signal. At the same time, in 2018, scholars Zi Fa Han and Chi Sing Leung also studied the sparse signal recovery under compressive sampling based on delayed Lagrangian neural network. In 2018, Hao Wang et al. applied Lagrangian neural network to urban source localization. In 2020, Zhang Lei Shi and Hao Wang applied Lagrangian neural network to MIMO radar localization system again. However, no one has applied Lagrangian neural network to the field of passive radar localization so far.

[0007] In the existing passive radar localization, the commonly used methods for solving the target localization equation are linear least squares method or weighted linear least squares method, which require linearizing the non-linear equation based on passive radar and then solving its analytical solution. However, there will be linearization loss in this process, resulting in a decrease in the final localization accuracy. Due to its least squares property, each measurement value participates in the operation, which means that when there are outliers in the measurement values, it has a great impact on the final result and makes it impossible to perform localization. In passive radar localization, the requirement for the localization accuracy of the target position is relatively high. At the same time, due to non-line-of-sight propagation or some interferences, it is inevitable to obtain outlier measurement value points during the measurement process, so it will directly affect the final target localization accuracy. Summary of the Invention

[0008] The object of the present invention is to propose a passive radar target localization method based on Lagrangian neural network that can still achieve stable localization when there are measurement error outliers, aiming at the deficiencies and problems existing in the prior art.

[0009] The present invention is a passive radar target localization method based on Lagrangian neural network, which is characterized in that it uses a Lagrangian neural network to solve the target position in a passive radar scenario, and includes the following steps:

[0010] (1) Establish an observation model for the target arrival time: First, set the positions of the target, the main receiving station, and the auxiliary receiving stations. Then, taking the main receiving station as the observation center, list the observation equations based on the difference in arrival distances of the target signal at the main and auxiliary receiving stations in this observation scenario to form an observation model for the target arrival time;

[0011] (1a) Set the positions of the target, the main receiving station, and the auxiliary receiving stations: Set the target position (x, y), set (x0, y0) as the position of the main receiving station, and set (x1, y1)(x2, y2)…(x M-1 ,y M-1 ) as the positions of the auxiliary receiving stations, where i represents the i-th auxiliary receiving station, and i = 1, 2, …, M - 1;

[0012] (1b) List the observation equations for the difference in distances from the target to the main and auxiliary receiving stations: According to the target position and the positions of the main and auxiliary receiving stations, taking the main receiving station as the observation center, after the signals are received at each receiving station, perform signal detection and signal processing to obtain the time when the target, i.e., the radiation source signal, arrives at each receiving station. Let the time for the target signal to reach the main receiving station be t0, and the times to the M - 1 auxiliary receiving stations be t1, t2, …, t M-1 , calculate the time differences between the radiation source target signal reaching the main receiving station and the M - 1 auxiliary receiving stations, which are Δt1, Δt2, …, Δt M-1 , and obtain the difference in distances from the target to the main and auxiliary receiving stations as Δt i ·c. Then, the observation equation based on the difference in distances from the target to the main and auxiliary receiving stations is:

[0013]

[0014] where Δt i is the time difference, Δt i = t i - t0, i = 1, 2, …, M - 1, c = 3×10 8 m / s is the speed of light. At this time, Δt i represents the time difference between the i-th auxiliary receiving station and the main receiving station;

[0015] (2) Obtain the measured value τ i of the time difference between the target and the main and auxiliary receiving stations; Based on the time difference between the target reaching the main and auxiliary receiving stations, obtain the measured value of the arrival time difference of the target signal at the main and auxiliary receiving stations; Let the measured value of the time difference between the target and the main and auxiliary receiving stations be τ i , τ i is composed of the true value Δt i of the time difference plus the measurement noise vector σ i . Then, τ i is:

[0016] τ i=Δt i +σ i

[0017] where σ i is the noise component, and σ i =[σ1,σ2,…,σ M . It is assumed that the noise component is an independent and identically distributed random variable that follows a zero-mean Gaussian distribution;

[0018] (3) Solve the positioning equation for the target position: Based on the measured values of the time differences of arrival from the target to the primary and secondary receiving stations, the positioning equation with measurement errors for solving the target position is listed as follows:

[0019]

[0020] (4) Convert the problem of solving the positioning equation into an optimization problem with constraints: When dealing with the positioning equation with measurement errors, based on the minimum mean square error criterion between the measured values and the true values, the positioning equation with measurement errors is converted into an objective optimization problem with constraints. The objective function and constraint conditions for solving the target positioning in the optimization problem are listed as follows:

[0021] Objective function:

[0022] Constraint condition: r i 2 =(x - x i ) 2 +(y - y i ) 2 , i = 1, 2, …, M - 1

[0023]

[0024] At this time, the form of the objective function is in the form of the second norm; where, d i is the measured value of the distance difference from the target to the primary and secondary receiving stations, r0 is the true distance from the target to the primary receiving station, and r i is the true distance from the target to the secondary receiving station;

[0025] (5) Construct the Lagrangian multiplier form L(r i , r0, x, y, λ i ): Use the objective function in the form of the second norm of the optimization problem in step (4) and its constraints related to the number of primary and secondary receiving stations to construct the Lagrangian multiplier form;

[0026]

[0027] where, r i , r0, x, y, λ iis the variable to be optimized, λ i represents the i-th Lagrange multiplier; in order to make the Lagrangian neural network LPNN reach a balanced and stable state faster, an augmented term is introduced as follows:

[0028]

[0029] where C0 is the augmented term coefficient;

[0030] (6) Transform the objective function in the form of the second norm into the form of the first norm and replace it with the function approximation method: In the actual process of TDOA (Time Difference of Arrival) positioning, due to signal interference or non-line-of-sight propagation, the measured distance differences include outliers, which will have a great impact on the positioning result; in the least mean square error criterion, because the second norm has a square term and is more sensitive to outliers than the first norm, the objective function in the form of the second norm is transformed into the objective function in the form of the first norm, and the objective function in the form of the first norm is replaced with an approximation function to obtain the approximated objective function;

[0031] (7) Reconstruct the Lagrange multiplier formula after function approximation: Based on the objective function in the form of the first norm in step (6), use the Lagrange multiplier method to reconstruct the Lagrange multiplier formula after approximation of the approximation function, so as to prepare for the subsequent construction of the Lagrangian neural network;

[0032] (8) Define and solve two types of neurons: Define the derivative of the variables to be optimized r i , r0, x, y with respect to time as the decision variable neurons; define the derivative of the variable to be optimized λ i with respect to time as the Lagrangian neurons; among them, the decision variable neurons are responsible for finding the minimum point of the original problem, and this neuron will finally give the balance point of the neural network, so as to make a further judgment for finding the minimum point of the original problem; the Lagrangian neurons are responsible for quickly introducing the dynamic trajectory into the feasible region; then solve the two types of neurons based on the Lagrange multiplier formula, so as to further prepare for the construction of the Lagrangian neural network;

[0033] (9) Construct the Lagrangian neural network: First, set the initial values of the variables to be optimized r i , r0, x, y, λ i and different learning rates for each variable; then, set the termination iteration conditions for each variable to be optimized, including the change threshold and the maximum number of iterations; at the same time, each variable to be optimized performs learning iterations according to its own learning rate. When the change amount of each iteration is less than the threshold or exceeds the maximum number of iterations, the iteration stops and the network reaches dynamic balance;

[0034] (10) Obtain the target position for passive radar target positioning: Through continuous learning of each decision variable neuron and Lagrange neuron, when the iteration stops and the network reaches dynamic equilibrium, obtain the target position for target positioning in the passive radar scenario according to the neuron information output at the equilibrium point.

[0035] The present invention solves the technical problem of achieving stable target positioning in the case of abnormal measurement values.

[0036] Compared with the prior art, the present invention has the following advantages:

[0037] At low signal-to-noise ratio, it has higher positioning accuracy: In traditional passive radar positioning methods, when solving the positioning equation, the least squares method or weighted least squares algorithm is used. These least squares algorithms need to be linearized when solving the solution of the equation, and there will be a linearization loss during this process, resulting in lower positioning accuracy, especially at low signal-to-noise ratio, which is more obvious. The present invention is based on constructing a Lagrange neural network to solve the solution of the positioning equation with errors, and adopts the way of learning and iterating with the initial value. When the network is in a stable state, the target position is output by the corresponding variable neuron, which can avoid the linear error introduced by the linearization of the traditional linear least squares method, so that the present invention has higher positioning accuracy, and can also have higher stability when positioning at low signal-to-noise ratio.

[0038] Achieve stable positioning in the presence of abnormal measurement values: In traditional least squares passive radar positioning algorithms, due to the form of their analytical solutions, the presence of abnormal measurement values has a great impact on the final analytical solution of the target position, resulting in inaccurate final positioning and unable to achieve stable positioning. The present invention uses a Lagrange neural network to learn and iterate the initial value, and finally obtains an iterative solution. In the actual process of solving the non-linear measurement equation with errors, the present invention can achieve a more stable positioning effect compared with traditional methods in the case of abnormal values. Brief Description of the Drawings

[0039] Figure 1 is a block diagram of the present invention;

[0040] Figure 2 is a flowchart of the solution implementation based on the Lagrange neural network;

[0041] Figures 3(a) and 3(b) in Figure 3 are RMSE comparison diagrams between the present invention and traditional positioning methods without introducing abnormal values under two station layout methods, where:

[0042] Figure 3(a) shows the RMSE comparison diagram between the present invention and traditional positioning methods without introducing abnormal values in the case of a diamond layout of four radar receiving stations;

[0043] Figure 3(b) shows the comparison chart of RMSE between the present invention and the traditional positioning method without introducing outliers in the case of Y-shaped layout of four radar receiving stations;

[0044] In Figure 4 Figure 4(a) , 4(b) , 4(c) and 4(d) are the comparison charts of RMSE between the present invention and the traditional positioning method when outliers are introduced in four passive radar receiving stations. Among them:

[0045] Figure 4(a) shows the comparison chart of RMSE between the present invention and the traditional positioning method when exponentially distributed outliers are introduced in the case of diamond layout of four radar receiving stations;

[0046] Figure 4(b) respectively shows the comparison chart of RMSE between the present invention and the traditional positioning method when uniformly distributed outliers are introduced in the case of diamond layout of four radar receiving stations;

[0047] Figure 4(c) respectively shows the comparison chart of RMSE between the present invention and the traditional positioning method when exponentially distributed outliers are introduced in the case of Y-shaped layout of four radar receiving stations;

[0048] Figure 4(d) respectively shows the comparison chart of RMSE between the present invention and the traditional positioning method when uniformly distributed outliers are introduced in the case of Y-shaped layout of four radar receiving stations;

[0049] Figure 5 is the hyperbolic function image used by the present invention for objective function approximation. Detailed implementation manners

[0050] Example 1:

[0051] In the existing passive radar target positioning method, the least squares method or the weighted least squares method is usually used when solving the positioning equation. When the least square-based algorithm is used to solve the target position of the analytical solution, the non-linear positioning equation needs to be linearized to solve its analytical solution, and there will be a certain loss in this process, resulting in a loss of the final positioning accuracy. Especially at low signal-to-noise ratios, the positioning accuracy is particularly low. At the same time, due to its least square characteristics, each measurement value participates in the operation, which makes the presence of outliers in the measurement values have a great impact on the final result, making it impossible to perform positioning. There are abnormal measurement value points in passive radar positioning, and due to large positioning errors, traditional algorithms cannot perform actual positioning.

[0052] In recent years, under the condition that both the constraints and the objective function are differentiable, some scholars have also been exploring the optimal solutions that satisfy the constraint conditions, and some related scholars have applied Lagrangian neural networks to MIMO radar positioning or source positioning in cities. Aiming at the problem that abnormal measurement values cannot be located, the present invention uses a Lagrangian neural network to solve the target position through iterative cycles to solve the above problems, and proposes a passive radar target positioning method based on a Lagrangian neural network.

[0053] The present invention is a passive radar target positioning method based on a Lagrangian neural network, including the establishment of an observation model to the solution of a positioning equation, see Figure 1 , Figure 1 which is the flow chart of the present invention. Using a Lagrangian neural network to solve the target position in a passive radar scenario, it includes the following steps:

[0054] (1) Observation model based on time of arrival of the target: First, set the position coordinates of the target and the main and auxiliary receiving stations, and then, with the main receiving station as the observation center, list the observation equations based on the distance differences of the target signal arriving at the main and auxiliary receiving stations in this observation scenario to form an observation model of the time of arrival of the target.

[0055] (1a) Set the positions of the target and the main and auxiliary receiving stations: The present invention sets the target position as (x, y), that is, the coordinates of the target in the observation model are (x, y), sets the position of the main receiving station as (x0, y0), that is, the coordinates of the main receiving station in the observation model are (x0, y0), and sets (x1, y1)(x2, y2)…(x M-1 , y M-1 ) as the positions of the auxiliary receiving stations, that is, the coordinates of the auxiliary receiving stations in the observation model are (x1, y1)(x2, y2)…(x M-1 , y M-1 ), where i represents the i-th auxiliary receiving station, i = 1, 2, …, M - 1. Under the observation model of the present invention, it mainly involves the target position and the positions of each receiving station, so the specific coordinates of the target and the main and auxiliary receiving stations need to be set.

[0056] (1b) Based on the specific position coordinates of the target and the main and auxiliary receiving stations modeled above, list the observation equations of the distance differences of the target arriving at the main and auxiliary receiving stations. The observation equation is the primary premise for the target positioning equation. According to the target position and the positions of the main and auxiliary receiving stations in step (1a), with the main receiving station as the observation center, after each receiving station receives the signal, signal detection and signal processing are carried out to obtain the time when the target, that is, the radiation source signal, arrives at each receiving station. Let the time of the target signal arriving at the main receiving station be t0, and the times of arriving at the M - 1 auxiliary receiving stations be t1, t2, …, t M-1, and calculate the time differences between the target signal of the radiation source and the main receiving station and M - 1 auxiliary receiving stations from this, which are Δt1, Δt2, …, Δt M-1 , and from this, the distance differences between the target and the main and auxiliary receiving stations can be obtained as Δt i ·c. Then, the observation equation based on the distance differences between the target and the main and auxiliary receiving stations is:

[0057]

[0058] Among them, Δt i is the time difference, Δt i = t i - t0, i = 1, 2, …, M - 1, c = 3×10 8 m / s is the speed of light. At this time, Δt i represents the time difference between the i-th auxiliary receiving station and the main receiving station.

[0059] In order to better elaborate on the idea and steps of solving the target position in the passive radar scenario of the present invention, it is necessary to model the target and each receiving station in the scenario based on the observation model to more accurately and clearly illustrate the specific process of solving the target position.

[0060] (2) According to the modeled target coordinates, the coordinates of each receiving station, and the observation equation, obtain the measured value τ of the time difference between the target and the main and auxiliary receiving stations i . Here, the measured value of the time difference is the sum of the true value of the time difference and the estimated value of the time difference, that is, the noise vector. Based on the time differences between the target and the main and auxiliary receiving stations, obtain the measured value of the arrival time difference of the target signal at the main and auxiliary receiving stations; let the measured value of the time difference between the target and the main and auxiliary receiving stations be τ i , τ i is composed of the true value Δt i of the time difference plus the measurement noise vector σ i , as shown in the following formula: Obtain τ i as:

[0061] τ i = Δt i + σ i

[0062] Among them, σ i is the noise component, σ i = [σ1, σ2, …, σ M , and it is assumed that the noise component is an independent and identically distributed random variable that follows a zero-mean Gaussian distribution.

[0063] (3) Establish the positioning equation for the target position: For the passive radar target positioning method based on the time difference of arrival, based on the measured values of the time difference between the target and the main and auxiliary receiving stations obtained in step (2), the positioning equation with measurement errors for solving the target position is listed as follows:

[0064]

[0065] The positioning equation is a necessary prerequisite for solving the target position. If there is no error in the ideal situation, it can be directly solved. However, the current positioning equation has measurement errors. Therefore, when solving this equation, the least squares method has always been used to solve the solution of the equation. The disadvantages will not be described here. The present invention converts the method of solving the positioning equation into a constrained optimization problem to eliminate the disadvantages of the traditional method.

[0066] (4) To construct the Lagrangian multiplier formula, the problem of solving the positioning equation needs to be converted into a constrained optimization problem: Refer to Figure 2 , Figure 2 is the implementation flowchart for solving based on the Lagrangian neural network. When the present invention processes the positioning equation with measurement errors, based on the least mean square error criterion between the measured value and the true value, the positioning equation with measurement errors listed in step (3) is converted into a constrained target optimization problem. The objective function and constraint conditions for target positioning solution in the optimization problem are listed as follows:

[0067] Objective function:

[0068] Constraint conditions: r i 2 =(x - x i ) 2 +(y - y i ) 2 , i = 1, 2, …, M - 1

[0069]

[0070] At this time, the form of the objective function is in the form of the second norm; among them, d i is the measured value of the distance difference between the target and the main and auxiliary receiving stations, r0 is the true distance from the target to the main receiving station, and r i is the true distance from the target to the auxiliary receiving station.

[0071] (5) Based on the form of the constrained optimization problem, in order to obtain each neuron of the Lagrangian neural network, thereby constructing the Lagrangian multiplier formula L(r i , r0, x, y, λ i):In step (4), the objective function is listed based on the least mean square error criterion. Therefore, the objective function in step (4) is in the form of the second norm. In this step, the Lagrangian multiplier expression is constructed based on the Lagrange multiplier method. Therefore, the Lagrangian multiplier expression is constructed using the objective function in the form of the second norm of the optimization problem in step (4) and its constraints related to the number of main and auxiliary receiving stations:

[0072]

[0073] where r i , r0, x, y, λ i are variables to be optimized. The variables to be optimized are divided into decision variables and Lagrangian variables. r i , r0, x, y are decision variables, and λ i represents the i-th Lagrangian multiplier; in order to make the Lagrangian neural network LPNN reach a balanced and stable state faster, an augmented term is introduced as:

[0074]

[0075] where C0 is the augmented term coefficient.

[0076] (6) Since the second norm and the first norm have different sensitivities to outliers, when dealing with outliers, the objective function in the form of the second norm needs to be changed to the form of the first norm and replaced with an approximation function: In the actual process of measuring time difference for positioning, due to signal interference or non-line-of-sight propagation, the measured range difference includes outliers, which has a great impact on the positioning result; in the least mean square error criterion, because the second norm has a square term and is more sensitive to outliers than the first norm, the objective function in the form of the second norm is changed to the objective function in the form of the first norm. Since the objective function of the optimization problem is required to be differentiable when solving each neuron of the Lagrangian, an approximation function is used to replace the objective function in the form of the first norm here to obtain the approximated objective function, where the approximation function is detailed in Embodiment 2.

[0077] (7) After modifying the form of the objective function, in order to construct a Lagrangian neural network that can handle abnormal measurement values, the present invention reconstructs the Lagrangian multiplier expression in the form of the first norm: Based on the objective function in the form of the first norm in step (6), using the Lagrange multiplier method, the Lagrangian multiplier expression after replacement with the approximation function is reconstructed to prepare for the subsequent construction of the Lagrangian neural network.

[0078] (8) Define and solve two types of neurons: Define the derivatives of the variables to be optimized r i , r0, x, y with respect to time as decision variable neurons; define the derivative of the variable to be optimized λ iThe derivative with respect to time is the Lagrangian neuron; among them, the decision variable neuron is responsible for finding the minimum point of the original problem, and this neuron will ultimately give the equilibrium point of the neural network, so as to make a further judgment for finding the minimum point of the original problem; the Lagrangian neuron is responsible for quickly introducing the dynamic trajectory into the feasible region; then, based on the Lagrangian multiplier formula, the two types of neurons are solved to further prepare for constructing the Lagrangian neural network. The Lagrangian neural network is constructed by defining and solving the two types of neurons.

[0079] (9) Construct the Lagrangian neural network and solve the target position in an iterative manner, avoiding the influence of the existence of outliers on the positioning accuracy: First, set the variables to be optimized r i , r0, x, y, λ i 's initial values, as well as different learning rates for each variable; then, set the termination iteration conditions for each variable to be optimized, including the change threshold and the maximum number of iterations; at the same time, each variable to be optimized learns and iterates according to its own learning rate. When the change amount of each iteration is less than the threshold or exceeds the maximum number of iterations, the iteration stops and the network reaches dynamic equilibrium.

[0080] The present invention adopts the way of learning and iterating with initial values. When the network is in a stable state, the target position is output by the corresponding variable neurons, which can avoid the linear error introduced by the linearization of the traditional linear least squares method, so that the present invention has higher positioning accuracy, and also has higher stability when positioning at low signal-to-noise ratios.

[0081] (10) Obtain the target position of passive radar target positioning through the information output by neurons: Through continuous learning of each decision variable neuron and Lagrangian neuron, when the iteration stops and the network reaches dynamic equilibrium, the information of each neuron can be output. According to the neuron information output at the equilibrium point, the target position of passive radar observation model scenario target positioning can be obtained, and at the same time, the Lagrangian multiplier value at the target position can also be obtained, completing the passive radar target positioning based on Lagrangian.

[0082] The original purpose of the Lagrangian neural network was to solve non-smooth convex optimization problems. In recent years, many scholars have also applied this algorithm to solve non-smooth convex optimization mathematical problems. When solving the passive radar positioning equation, if the position of the initial point is selected well, so that the local optimal point is the global optimal point, the highly non-linear non-convex positioning equation can be solved by the Lagrangian neural network method. This method is to find the optimal solution that satisfies the constraint conditions based on the Lagrangian multiplier theory in optimization, where the constraint conditions can be either equalities or inequalities, and the equilibrium point of the neural network satisfies the algorithm conditions. In this improved neural network, there are two types of neurons: variable neurons and Lagrangian neurons. Among them, the variable neurons are responsible for finding the minimum point of the objective function and providing the equilibrium point to solve the problem; while the Lagrangian neurons are responsible for quickly introducing the dynamic trajectory into the feasible region. In the present invention, the optimization problem is limited to the case where both the objective function and the constraint function are smooth.

[0083] The technical idea of the present invention is: list the target positioning equation with errors based on the observation model, then convert the positioning equation into a constrained target optimization problem, and then convert the optimization problem into a Lagrangian function equation, construct the Lagrangian multiplier formula, and finally solve the corresponding neurons of the Lagrangian neural network according to the Lagrangian multiplier formula, so as to establish a Lagrangian network according to the corresponding neurons and solve the target position.

[0084] Through steps (4) to (10) of the present invention, the method of linearly solving the positioning equation by the traditional least squares method is converted into iterative solution by the Lagrangian neural network, avoiding the linearization loss, so that at low signal-to-noise ratio, the present invention has higher positioning accuracy.

[0085] The present invention solves the problem of inaccurate positioning in the presence of abnormal measurement values in the traditional algorithm and the inability to achieve stable positioning in the presence of abnormal measurement values. Compared with the traditional algorithm, it has high positioning accuracy, ensuring that the passive radar can still achieve stable positioning in some special environments or bad weather, and after successfully positioning the target position, related operations such as target monitoring and target recognition can be carried out.

[0086] Embodiment 2:

[0087] The passive radar target positioning method based on Lagrangian is the same as that in Embodiment 1. The transformation of the objective function in the form of the second norm into the form of the first norm in step (6) includes the following steps:

[0088] (6a) The transformation of the objective function into the form of the first norm is:

[0089] Objective function:

[0090] Constraint condition: r i2 =(x - x i ) 2 +(y - y i ) 2

[0091]

[0092] However, in LPNN, since the sensitivity of the first norm to outliers is less than that of the second norm, the objective function is changed to the form of the first norm to solve the target position under abnormal measurement values. At the same time, LPNN requires that both the constraint and the objective function must be differentiable. Obviously, the objective function in the form of the first norm does not satisfy the differentiability condition at the zero point; where i represents the i-th auxiliary receiving station, i = 1, 2, …, M - 1; therefore, an approximation function is introduced here to replace the first norm to handle the non-differentiable objective function in the form of the first norm.

[0093] (6b) obtains the objective function after adopting the approximation function: In mathematics, hyperbolic functions are a class of common trigonometric functions, also known as circular functions, and can satisfy differentiability at all points. When the coefficient μ of the hyperbolic function is large enough, the graph of the hyperbolic function is approximately the same as that of the first norm. Therefore, approximating the first norm with a hyperbolic function can well solve this problem;

[0094] Using the similarity between the hyperbolic function and the first norm function, let the approximation function be:

[0095]

[0096] where x is the independent variable and μ is the coefficient of the hyperbolic function, and the objective function after adopting the approximation function is obtained:

[0097] Objective function:

[0098] Constraint condition: r i 2 =(x - x i ) 2 +(y - y i ) 2

[0099]

[0100] where i represents the i-th auxiliary receiving station, i = 1, 2, …, M - 1.

[0101] In the present invention, due to the different sensitivities of the norm forms to outliers, the objective function does not satisfy the requirement of differentiability everywhere in the Lagrangian neural network after being changed to the form of the first norm. Since the above hyperbolic function is approximated to the first norm function, as Figure 5As shown, it is the hyperbolic function image for function approximation, where the horizontal axis is the independent variable and the vertical axis is the dependent variable. From bottom to top are the function images when μ is 10, 20, 40, and infinity respectively. It can be seen that the larger the coefficient μ of the function, the closer the image is to the one-norm function. Therefore, in step 6 of the present invention, the function approximation method is used to perform an approximate function substitution on the objective function in the form of the one-norm, so that the objective function is differentiable everywhere and meets the requirements for using the Lagrangian neural network.

[0102] Embodiment 3:

[0103] The Lagrangian-based passive radar target positioning method is the same as in Embodiments 1-2. The reconstruction of the Lagrangian multiplier formula after adopting the approximate function in step (7) includes the following steps:

[0104] The reconstructed Lagrangian multiplier formula is:

[0105]

[0106] Among them, in order to make the LPNN network reach a balanced and stable state faster, an augmented term is still introduced as:

[0107]

[0108] Among them, C0 is the augmented term coefficient, which is a constant. Its value is usually related to the mathematical model. When C0 is large enough, it can ensure that the Lagrangian network quickly reaches stability.

[0109] After the present invention reconstructs the Lagrangian multiplier formula, it solves the neurons of the new Lagrangian neural network, and then constructs the Lagrangian neural network, so that in the process of passive radar target positioning, the influence of abnormal measurement values on the final positioning accuracy can be avoided. According to the objective function after the approximate function substitution in step 6, the Lagrangian multiplier formula is reconstructed in step 7, which is prepared for the next step of solving the two types of neurons. Because after the objective function changes, the corresponding Lagrangian multiplier formula also changes, and then the Lagrangian multiplier formula needs to be reconstructed to take the derivative of different variables to be optimized to obtain the two types of neurons, including decision variable neurons and Lagrangian neurons.

[0110] Embodiment 4:

[0111] The Lagrangian-based passive radar target positioning method is the same as in Embodiments 1-3. The definition and solution of the two types of neurons in step (8) include the following steps:

[0112] (8a) Define each decision variable neuron, where the decision variable neuron regarding the true distances from the target to the main and auxiliary receiving stations is:

[0113]

[0114] The decision variable neurons for the target position are as follows:

[0115]

[0116] According to the organized Lagrangian function, the solution of each decision variable neuron is respectively:

[0117]

[0118]

[0119]

[0120]

[0121] (8b) The Lagrangian neurons defining the Lagrange multipliers are as follows:

[0122]

[0123] According to the organized Lagrangian function, the solution of the constructed Lagrangian neurons for passive radar target positioning is:

[0124]

[0125] In this way, the decision variable neurons and Lagrangian neurons are obtained.

[0126] Based on the reconstructed Lagrange multiplier formula in step 7, in step 8, two types of neurons are solved to prepare for the construction of the Lagrangian neural network in the next step. First, two types of neurons are defined respectively, including decision variable neurons and Lagrangian neurons, and then based on the derivative of each decision variable and Lagrangian variable with respect to time, the two types of neurons are obtained respectively. After obtaining the two types of neurons, the Lagrangian neural network is further constructed.

[0127] Example 5:

[0128] The Lagrangian-based passive radar target positioning method is the same as that in Examples 1-4. The construction of the Lagrangian neural network for passive radar target positioning described in step (9) includes the following steps:

[0129] (9a) Set the initial values of the variables to be optimized: The initial values include the initial values of the target position (x, y), the initial value of the true position r0 of the target from the main receiving station, the initial value of the true position r i of the target from the auxiliary receiving station, the initial value λ0 of the Lagrange multiplier formula related to the main receiving station, and the i initial values λ i of the Lagrange multipliers related to the number i of auxiliary receiving stations, where i = 1, 2,..., M - 1;

[0130] (9b) Set the learning rate l of each variable to be optimized t , and set the termination conditions: including the threshold ε and the maximum number of iterations N, each variable is iterated according to a different learning rate;

[0131] (9c) Lagrangian neural network iteration under passive radar target positioning: The specific iteration process is: t is the tth variable to be learned, t=1,2,…,2M+2, k is the number of iterations, M is the number of passive radar receiving stations; where X 1 The variables x and X to be optimized are the target positions. 2 y and X are the variables to be optimized at the target position 3 ~X M+2 The variables r0 and r1 to be optimized are the actual position of the target from the primary and auxiliary receiving stations. i , i=1,2,…,M-1,X M+3 ~X 2M+2 is λ0 and the initial value λ of the i Lagrange multipliers i , i=1,2,…,M-1; ΔX t is the neuron corresponding to the tth variable to be optimized. When ΔX t When ΔX is greater than ε or the number of iterations is less than N, the iteration continues; t When it is less than ε or the number of iterations is greater than N, the iteration stops and the network is dynamically balanced. When the network is dynamically balanced, the neuron information output at the equilibrium point is the target position information.

[0132] In step (9), the present invention fully describes the whole process of constructing the Lagrangian neural network, including setting the initial value of the variable to be optimized, setting the learning rate of each variable to be optimized and the iteration termination condition, and describing the specific iteration process. It is ensured that each variable to be optimized continuously learns iteratively according to its own learning rate until the iteration termination condition is met, the iteration stops, and the network is balanced. Based on the output of the neurons when the network is balanced, the target position is obtained.

[0133] The present invention discloses a passive radar target positioning method based on Lagrangian neural network. It mainly solves the problems of low positioning accuracy when the measurement error is large and inaccurate positioning when there are abnormal measurement values in the prior art. The solution is as follows: Based on the observed measurement values, a target positioning equation with measurement error is listed, and then the positioning equation is converted into a constrained target optimization problem for solution. After that, the optimization problem is converted into a Lagrangian function equation, so as to construct a Lagrangian multiplier formula. Finally, the corresponding neurons of the Lagrangian neural network are solved according to the Lagrangian multiplier formula, and then the Lagrangian network is established based on the corresponding neurons to solve the target position. At the same time, when it comes to abnormal measurement values, a function approximation method is proposed to approximate the one-norm, which satisfies the characteristic that the Lagrangian neural network must process a cost function that is differentiable everywhere. The present invention has high positioning accuracy, good robustness and strong engineering applicability, and can be used in multi-station passive radar positioning systems.

[0134] The technical effects of the present invention are described below in combination with simulation experiments:

[0135] Example 6:

[0136] The passive radar target positioning method based on Lagrangian is the same as that in Examples 1-5.

[0137] Simulation conditions:

[0138] Four passive radar receiving stations and one target are set. The station layout methods are diamond layout and Y-shaped layout. Among them, one of the four passive radars is the main receiving station and the other three are auxiliary receiving stations. Among them, in the diamond layout, the coordinates of the main station are [5, 5] km, and the coordinates of the three auxiliary stations are [5, -5] km, [-5, -5] km, [-5, 5] km. The true position of the target is set at a position 5 kilometers away from the main station. In the Y-shaped layout, the position of the main receiving station is [0, 0], and the coordinates of the three auxiliary stations are [-25, 0] km, and the target is set at a position 40 km away from the main receiving station. In the simulation environment, the measurement error is set to satisfy a Gaussian distribution with zero mean.

[0139] Simulation content:

[0140] Simulation 1: Under the above simulation conditions, without introducing outliers, the root mean square error (RMSE) of positioning of three methods, namely the linear least squares method (LLS) in the prior art, the Lagrangian neural network two-norm (LPNN_2), and the Lagrangian neural network one-norm (LPNN_1) of the present invention, are compared under different station layout methods of the four passive radar receiving stations, as shown in Figure 3.

[0141] Analysis of simulation results:

[0142] Referring to FIGS. 3(a) and 3(b), the horizontal axes of FIGS. 3(a) and 3(b) are both the standard deviation of the measurement error in meters (m), and the vertical axes are both the root mean square error (RMSE) of the positioning result in meters (m).

[0143] FIG. 3(a) is a comparison chart of the RMSE between the present invention and traditional positioning methods without introducing outliers in the case of a rhombus layout of four radar receiving stations; the curve with square markers represents the positioning root mean square error of the LPNN_1 norm method of the present invention, the curve with triangular markers represents the positioning root mean square error of the LPNN_2 norm method, and the curve with circular markers represents the positioning root mean square error of the traditional LLS method. From the comparison Figure 3a of the curves of the three methods, it can be obtained that when the passive radar receiving stations are in a rhombus distribution and there are no abnormal measurement values, as the measurement error continuously increases, before the error distribution reaches 100 m, the positioning root mean square errors of the curves of the three methods are approximately the same. After the error distribution reaches 100 m, the LPNN_1 norm method has higher positioning accuracy compared to the other two methods. When the error distribution is 10,000 m, the positioning root mean square error of the present invention is 24 m, while the positioning root mean square error of the LPNN_2 norm method is 72 m, and the positioning root mean square error of the LLS method is 86 m. The experimental data shows that at low signal-to-noise ratios, the LPNN_1 norm method of the present invention has a positioning accuracy with a root mean square error at least 40 m higher than the other two algorithms.

[0144] FIG. 3(b) is a comparison chart of the RMSE between the present invention and traditional positioning methods without introducing outliers in the case of a Y-shaped layout of four radar receiving stations; the curve with square markers represents the positioning root mean square error of the LPNN_1 norm method of the present invention, the curve with triangular markers represents the positioning root mean square error of the LPNN_2 norm method, and the curve with circular markers represents the positioning root mean square error of the traditional LLS method. From Figure 3b the comparison of the curves of the three methods, it can be obtained that when the passive radar receiving stations are in a Y-shaped distribution and there are no abnormal measurement values, as the measurement error continuously increases, the LPNN_1 norm method of the present invention always has the highest positioning accuracy compared to the other two methods, and there is no significant change compared to the root mean square error when the error distribution is 0. Therefore, from FIG. 3(b), it can be obtained that when the error distribution gradually increases, that is, at low signal-to-noise ratios, the LPNN_1 norm method of the present invention has higher positioning accuracy.

[0145] Regardless of whether it is a rhombus layout or a Y-shaped layout, the LPNN_1 norm method of the present invention can achieve higher positioning accuracy compared to traditional algorithms at low signal-to-noise ratios.

[0146] Example 8:

[0147] The Lagrange-based passive radar target localization method is the same as in Embodiments 1-6, and the simulation conditions are as in Embodiment 7.

[0148] Simulation 2: Under the above simulation conditions, an outlier subject to exponential distribution and uniform distribution is introduced respectively. The horizontal axis is the change in the mean intensity of the outlier, and the vertical axis is the root mean square error RMSE. The root mean square errors of the localization of the above three methods are compared again, as shown in Figure 4.

[0149] Simulation content:

[0150] In Figure 4, Figure 4(a) shows that in the case of diamond-shaped station layout, an outlier subject to exponential distribution is introduced. Among them, the curve with square marks represents the root mean square error of the localization of the LPNN_1 norm method, the curve with triangular marks represents the root mean square error of the localization of the LPNN_2 norm method, and the curve with circular marks represents the root mean square error of the localization of the traditional LLS method. Figure 4(b) shows that in the case of diamond-shaped station layout, an outlier subject to uniform distribution is introduced. Among them, the curve with square marks represents the root mean square error of the localization of the LPNN_1 norm method, the curve with triangular marks represents the root mean square error of the localization of the LPNN_2 norm method, and the curve with circular marks represents the root mean square error of the localization of the traditional LLS method. Figure 4(c) shows that in the case of Y-shaped station layout, an outlier subject to exponential distribution is introduced. Among them, the curve with square marks represents the root mean square error of the localization of the LPNN_1 norm method, the curve with triangular marks represents the root mean square error of the localization of the LPNN_2 norm method, and the curve with circular marks represents the root mean square error of the localization of the traditional LLS method. Figure 4(d) shows that in the case of Y-shaped station layout, an outlier subject to uniform distribution is introduced. Among them, the curve with square marks represents the root mean square error of the localization of the LPNN_1 norm method, the curve with triangular marks represents the root mean square error of the localization of the LPNN_2 norm method, and the curve with circular marks represents the root mean square error of the localization of the traditional LLS method.

[0151] Analysis of simulation results:

[0152] Comparing the localization performance of the above three methods, it can be seen from Figure 4(a) that when using the diamond-shaped station layout and introducing an exponentially distributed outlier measurement value, the LPNN_1 norm method of the present invention does not change much with the increase in the intensity of the outlier, and the root mean square error of the localization has been around 10m, while the root mean square errors of the other two algorithms have reached 50m and stable localization cannot be achieved. It can be seen from Figure 4(b) that when using the diamond-shaped station layout and introducing a uniformly distributed outlier measurement value, the LPNN_1 norm method of the present invention still does not change much with the increase in the intensity of the outlier, and the root mean square error of the localization has been around 10m, while the root mean square errors of the other two algorithms have collapsed and stable localization cannot be achieved.

[0153] As can be seen from Fig. 4(c), when the Y-shaped station layout is adopted and exponential distribution abnormal measurement values are introduced, with the increase of the abnormal value intensity, the LPNN_1 norm method of the present invention does not change much, the positioning mean square error has been very stable, while the positioning mean square errors of the other two algorithms have collapsed. In particular, the RMSE of the LLS algorithm has reached 2000 m, and stable positioning can no longer be achieved. As can be seen from Fig. 4(d), when the Y-shaped station layout is adopted and uniform distribution abnormal measurement values are introduced, with the increase of the abnormal value intensity, the LPNN_1 norm method of the present invention still does not change much, and the positioning mean square error has been around 150 m, achieving stable positioning.

[0154] For each station layout method and the introduction of abnormal measurement values with different distribution methods, the positioning performance of the LPNN_1 norm method of the present invention is the best, and its stability is better than that of the other two algorithms. This enables stable positioning in the actual measurement process when there are non-line-of-sight propagation and signal interference and there are outliers in the measured time difference. And it avoids the linearization loss brought by the traditional least squares algorithm. The present invention uses a Lagrangian neural network to learn and iterate the initial value, and finally obtains an iterative solution. In the actual process of solving the non-linear measurement equation with errors, the present invention can achieve a more stable positioning effect compared with the traditional method in the case of abnormal values.

[0155] In summary, the present invention is a passive radar target positioning method based on a Lagrangian neural network, belonging to the technical field of passive radar positioning, and solves the technical problem of achieving stable target positioning in the case of abnormal measurement values. The implementation includes: establishing an observation model of the time of arrival of the target; obtaining the measured value τ of the time difference between the target and the main and auxiliary receiving stations i ; listing the positioning equation for solving the target position; converting the problem of solving the positioning equation into a problem of solving an optimization problem with constraints; constructing a Lagrangian multiplier formula L(r i , r0, x, y, λ i) Transform the objective function in the form of the second norm into the form of the first norm; reconstruct the Lagrangian multiplier expression in the form of the first norm; define and solve two types of neurons; construct a Lagrangian neural network; obtain the target position of passive radar target positioning. The present invention transforms the problem of solving the positioning equation into a constrained optimization problem, obtains the Lagrangian multiplier expression through the Lagrangian multiplier method, and then solves the Lagrangian neurons by means of derivation, thereby constructing a Lagrangian neural network, changing the traditional method of solving the analytical solution of the positioning equation using the least squares algorithm into iteration in the Lagrangian neural network, avoiding the loss of least squares linearization and realizing stable target positioning in the case of abnormal measurement values. The present invention has higher positioning accuracy in the case of low signal-to-noise ratio; realizes stable positioning in the presence of abnormal measurement values. It is applied to a multi-station passive radar positioning system.

Claims

1. A passive radar target localization method based on Lagrangian neural network, characterized in that, Using a Lagrangian neural network to solve the target position in a passive radar scenario, the steps are as follows: (1) Establish an observation model for the time of arrival of the target: First, set the positions of the target, the main receiving station, and the auxiliary receiving stations. Then, with the main receiving station as the observation center, list the observation equations based on the difference in arrival distances of the target signal at the main and auxiliary receiving stations in the observation scenario, forming an observation model for the time of arrival of the target; (1a)Set the target and the positions of the main and auxiliary receiving stations: Set the target position (x, y), set (x0, y0) as the position of the main receiving station, and set (x1, y1)(x2, y2)…(x M-1 , y M-1 ) as the positions of the auxiliary receiving stations, where i represents the i-th auxiliary receiving station, and i = 1, 2, …, M - 1; (1b) List the observation equations for the distance difference between the target and the main and auxiliary receiving stations: Based on the target position and the positions of the main and auxiliary receiving stations, with the main receiving station as the observation center, after the signals are received at each receiving station, signal detection and signal processing are carried out to obtain the time when the target, i.e., the radiation source signal, arrives at each receiving station. Let the time for the target signal to reach the main receiving station be t0, and the times for it to reach the M - 1 auxiliary receiving stations be t1, t2, …, t M-1 , calculate the time differences between the radiation source target signal reaching the main receiving station and the M - 1 auxiliary receiving stations, which are Δt1, Δt2, …, Δt M-1 , and obtain the distance differences between the target and the main and auxiliary receiving stations as Δt i ·c. Then the observation equation based on the distance difference between the target and the main and auxiliary receiving stations is: Among them, Δt i is the time difference, and Δt i = t i - t0, where i = 1, 2, …, M - 1, and c = 3×10 8 m / s is the speed of light. At this time, Δt i represents the time difference between the i-th auxiliary receiving station and the main receiving station; (2) Obtain the measurement value τ of the time difference between the target and the primary and secondary receiving stations i ; Based on the time difference between the target's arrival at the primary and secondary receiving stations, obtain the measurement value of the arrival time difference of the target signal at the primary and secondary receiving stations; Let the measurement value of the time difference between the target and the primary and secondary receiving stations be τ i , τ i is composed of the true value Δt of the time difference i plus the measurement noise vector σ i , and obtain τ i as: τ i = Δt i + σ i where σ i is the noise component, and σ i = [σ1, σ2, …, σ M . It is assumed that the noise component is an independent and identically distributed random variable following a zero-mean Gaussian distribution; (3) List the positioning equation for solving the target position: From the measured values of the time differences between the target and the main and auxiliary receiving stations, list the positioning equation with measurement errors for solving the target position as: (4) Convert the problem of solving the positioning equation into a constrained optimization problem: When dealing with the positioning equation with measurement errors, based on the minimum mean square error criterion between the measured value and the true value, convert the positioning equation with measurement errors into a constrained target optimization problem, and list the objective function and constraint conditions for target positioning solution in the optimization problem as: Constraint: r i 2 =(x - x i ) 2 +(y - y i ) 2 , i = 1, 2, …, M - 1 At this time, the form of the objective function is in the form of the second norm; where d i is the measured value of the distance difference between the target and the master and slave receiving stations, r0 is the true distance from the target to the master receiving station, and r i is the true distance from the target to the slave receiving station; (5) Construct the Lagrangian multiplier expression \(L(r i , r_0, x, y, \lambda i ) in the form of the second norm: Construct the Lagrangian multiplier expression using the objective function in the form of the second norm of the optimization problem and its constraints related to the number of primary and secondary receiving stations; where r i , r0, x, y, λ i are variables to be optimized, and λ i represents the i-th Lagrange multiplier; in order to make the Lagrangian neural network LPNN reach a balanced and stable state faster, an augmented term is introduced as: where C0 is the augmented term coefficient; (6) Convert the objective function in the form of the second norm into the form of the first norm and replace it with the function approximation method: In the actual process of measuring time difference for positioning, due to signal interference or non-line-of-sight propagation, the measured distance difference includes outliers, which will have a great impact on the positioning result; in the minimum mean square error criterion, because the second norm has a square term and is more sensitive to outliers than the first norm, convert the objective function in the form of the second norm into the objective function in the form of the first norm, and replace the objective function in the form of the first norm with an approximate function to obtain the approximated objective function; (7) Reconstruct the Lagrangian multiplier formula after function approximation: Use the Lagrangian multiplier method to reconstruct the Lagrangian multiplier formula after replacing it with the approximate function, preparing for the subsequent construction of the Lagrangian neural network; (8) Define and solve two types of neurons: Define the variables to be optimized \(r\) in the Lagrangian multiplier formula i , the derivatives of \(r_0\), \(x\), and \(y\) with respect to time are decision variable neurons; Define the variable to be optimized \(\lambda\) in the Lagrangian multiplier formula i The derivatives with respect to time are Lagrangian neurons; among them, the decision variable neurons are responsible for finding the minimum point of the original problem. This neuron will ultimately give the equilibrium point of the neural network to further judge for finding the minimum point of the original problem; the Lagrangian neurons are responsible for quickly introducing the dynamic trajectory into the feasible region; then solve the two types of neurons based on the Lagrangian multiplier formula to further prepare for constructing the Lagrangian neural network; (9)Construct a Lagrangian neural network: First, set the variable r to be optimized i , r0, x, y, λ i with their initial values, as well as different learning rates for each variable; then, set the termination iteration conditions for each variable to be optimized, including the change threshold and the maximum number of iterations; at the same time, each variable to be optimized performs learning iterations according to its own learning rate. When the change amount in each iteration is less than the threshold or exceeds the maximum number of iterations, the iteration stops and the network reaches dynamic equilibrium; (10) Obtain the target position of passive radar target positioning: Through continuous learning of each decision variable neuron and Lagrangian neuron, when the iteration stops and the network reaches dynamic balance, according to the neuron information output at the equilibrium point, obtain the target position of passive radar target positioning.

2. The passive radar target positioning method based on Lagrangian neural network according to claim 1, wherein The conversion of the objective function in the form of the second norm into the form of the first norm described in step (6) includes the following steps: (6a) The objective function is transformed into the form of the first norm as: Objective function: Constraint: r i 2 = (x - x i ) 2 + (y - y i ) 2 where i represents the i-th auxiliary receiving station, i = 1, 2,..., M - 1; introduce an approximate function to replace the first norm and handle the non-differentiable objective function in the form of the first norm; (6b) Obtain the objective function after using the approximate function: Utilize the similarity between the hyperbolic function and the first norm function, and set the approximate function as: where x is the independent variable and μ is the coefficient of the hyperbolic function, to obtain the objective function after using the approximate function: Objective function: Constraint: r i 2 =(x - x i ) 2 +(y - y i ) 2 where i represents the i-th auxiliary receiving station, i = 1, 2,..., M - 1.

3. The passive radar target positioning method based on Lagrangian neural network according to claim 1, characterized in that: The reconstruction of the Lagrangian multiplier formula after using the approximate function described in step (7) is to reconstruct the Lagrangian multiplier formula as: where, in order to enable the LPNN network to reach a balanced and stable state faster, still introduce the augmented term as: where C0 is the augmented term coefficient.

4. The passive radar target positioning method based on Lagrangian neural network according to claim 1, characterized in that: The definition and solution of two types of neurons described in step (8) include the following steps: (8a) Define each decision variable neuron, where the decision variable neurons regarding the true distances from the target to the primary and secondary receiving stations are as follows: The decision variable neurons regarding the target location are as follows: According to the organized Lagrangian function, solve each decision variable neuron respectively as follows: (8b) Define the Lagrangian neurons of the Lagrange multipliers as follows: According to the organized Lagrangian function, solve the Lagrangian neurons of the constructed passive radar target positioning as follows: Thus, obtain the decision variable neurons and the Lagrangian neurons.

5. The passive radar target positioning method based on Lagrangian neural network according to claim 1, characterized in that: The construction of the Lagrangian neural network described in step (9) includes the following steps: (9a) Set the initial values of the variables to be optimized: The initial values include the initial values of the target position (x, y), the initial value of the true position r0 of the target from the main receiving station, and the initial value of the true position r of the target from the auxiliary receiving station i , the initial value of the Lagrangian multiplier formula related to the main receiving station λ0, and the initial values of i Lagrangian multipliers related to the number of auxiliary receiving stations λ i , where i = 1, 2,..., M - 1; (9b) Set the learning rate l of each variable to be optimized t , and set the termination iteration conditions: including the threshold ε and the maximum number of iterations N, and each variable is iterated according to different learning rates; (9c)Iteration of the Lagrangian neural network for passive radar target positioning: The specific iteration process is as follows: t is the t-th variable to be learned, where t = 1, 2, …, 2M + 2, k is the number of iterations, and M is the number of passive radar receiving stations; among them, X 1 X 2 are the variables x, y to be optimized for the target position, X 3 ~X M+2 are the variables r0 and r to be optimized for the true positions of the target from the main and auxiliary receiving stations i , where i = 1, 2, …, M - 1, X M+3 ~X 2M+2 are the initial values λ0 and the i-th Lagrange multiplier λ i , where i = 1, 2, …, M - 1; ΔX i is the neuron corresponding to the i-th variable to be optimized. When ΔX i is greater than ε or the number of iterations is less than N, the iteration continues; when ΔX i is less than ε or the number of iterations is greater than N, the iteration stops and the network dynamic balance is completed.

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