Improved constraint weighted least square TDOA / FDOA passive positioning method based on Lagrange multiplier

Abstract

The invention relates to an improved constraint weighted least square TDOA / FDOA passive positioning method based on a Lagrange multiplier, and belongs to the field of passive radar positioning. The method comprises a two-step method and comprises the following steps: firstly, converting a time difference and frequency difference nonlinear positioning equation about a target position and speed in a passive positioning problem into a pseudo-linear equation, on this basis, converting into a quadratic programming problem, simplifying constraint conditions, and solving a position and speed initial solution of target positioning by applying a Lagrangian multiplier method; and then a nonlinear equation set about the target position and the speed is constructed according to the original constraint condition, and the initial solution obtained in the first step is substituted into the constructed nonlinear equation set to further carry out optimization solution on the target position and the speed value. The method has good positioning precision and real-time performance, an estimation RMSE curve of a target positioning parameter can still be close to CRLB in a large-noise environment, and the method has good robustness in a high-noise environment and has good advancement.

Description

Technical Field

[0001] The invention belongs to the field of passive radar positioning, and in particular relates to achieving real-time positioning of a target when relative motion occurs between a receiving station and the target, and specifically relates to an improved constrained weighted least squares TDOA / FDOA passive positioning method based on Lagrange multipliers. Background Art

[0002] As the electromagnetic environment becomes increasingly complex, how to develop an algorithm that can meet the real-time requirements and accurately locate the target in a complex electromagnetic environment is a hot topic. At present, multi-station passive positioning technology has been widely used in military and civilian fields, such as sensor networks, radar, navigation, etc. Multi-station passive positioning technology has been developed for a long time. Traditional technologies are based on a certain observation to achieve target positioning. Later studies found that the positioning technology using the combined observations of different systems can improve the positioning accuracy to a certain extent. The multi-station TDOA / FDOA passive positioning method can simultaneously estimate the position and velocity of the target, which has certain advantages. Therefore, as a relatively classic positioning technology, it has also been the focus of many scholars.

[0003] The positioning algorithm is generally studied based on two methods: iterative method and analytical method. Iterative positioning solution methods include: approximate maximum likelihood estimation method, quadratic constraint method, constrained weighted least squares (CWLS) and Gauss-Newton iterative multi-station TDOA / FDOA positioning algorithm. For the iterative method, an accurate initial positioning value is very important. Inaccurate initial values ​​may fall into the local optimal solution during iteration, resulting in inaccurate positioning results. The iterative method also has the characteristics of complex algorithm structure. It requires multiple iterations to obtain the final positioning result that meets the threshold requirements, so the real-time performance is poor. The analytical method has the characteristics of simple structure, and real-time performance is easy to meet. The more representative analytical method is the two-step weighted least squares method, which has a simple structure and good real-time performance, but the algorithm has a weak ability to adapt to measurement noise. When the TDOA / FDOA measurement noise intensity is at a high level, the positioning effect is poor, and the positioning accuracy will seriously deviate from the Cramer-Rao lower bound (CRLB).

[0004] In summary, both methods have their shortcomings, so how to design a positioning algorithm that can meet the real-time requirements and achieve positioning accuracy close to CRLB under high noise conditions is an urgent problem to be solved. In the prior art, an improved CWLS TDOA / FDOA passive positioning algorithm is proposed. The algorithm has good adaptability to low noise environments and good real-time performance, but the positioning effect deviates from CRLB to a large extent in high noise environments. Therefore, in order to solve the above-mentioned problems and how to design an algorithm that can meet the real-time requirements and has good adaptability to high noise environments, it is necessary to improve the algorithm structure to improve its positioning accuracy and have good adaptability to high noise environments. The improved algorithm should not significantly increase the time it takes to run to ensure that it meets good real-time requirements. Summary of the invention

[0005] The technical problems to be solved by the present invention are:

[0006] In view of the problems of root mean square error (RMSE) and poor ability of positioning deviation to adapt to measurement noise in current time difference / frequency difference passive positioning algorithms, the present invention provides an improved constrained weighted least squares TDOA / FDOA passive positioning method based on Lagrange multipliers.

[0007] In order to solve the above technical problems, the technical solution adopted by the present invention is:

[0008] An improved constrained weighted least squares TDOA / FDOA passive positioning method based on Lagrange multipliers, characterized by comprising:

[0009] Obtain TDOA / FDOA measurement values, and establish a target positioning equation based on the TDOA / FDOA measurement values;

[0010] Introducing intermediate redundant variables, constructing a pseudo-linear equation group for target positioning based on the constraint relationship between the target positioning equation and the intermediate redundant variables;

[0011] The pseudo-linear equations are solved using the Lagrange multiplier evaluation idea to obtain the least squares solution of the target positioning value; the solution process is repeated twice to obtain the final solution of the first step of the target positioning solution process;

[0012] Construct a new pseudo linear equation system for target positioning based on the original constraints;

[0013] Substitute the final solution of the first step of the target positioning solution process into the new pseudo-linear equations to solve the least squares solution of the target positioning value; and obtain the optimized target position and velocity estimates.

[0014] A further technical solution of the present invention: The TDOA / FDOA measurement value is specifically:

[0015] α=α o +Δα

[0016] In the formula, represents the true value; r=[r 21 ,r 31 ,…,r M1 ] T , Indicates the measured value of TDOA / FDOA; Δr=[Δr 21 ,Δr 31 ,…,Δr M1 ] T , Represents the measurement noise of TDOA / FDOA.

[0017] A further technical solution of the present invention: the step of introducing intermediate redundant variables and constructing a pseudo-linear equation group for target positioning according to the constraint relationship between the target positioning equation and the intermediate redundant variables; comprises:

[0018] Defining vectors in, And add the intermediate redundant variable r1 o =||v o ||,

[0019] Will and Substituting into the TDOA equation and FDOA equation and removing the second-order noise, we can get

[0020]

[0021] The corresponding matrix expression can be derived from the above formula:

[0022]

[0023] in

[0024]

[0025]

[0026] In the formula,

[0027] Considering actual intermediate redundant variables and There is a constraint relationship between

[0028]

[0029] First, find the WLS solution for the above equation, that is,

[0030] In the formula, is the norm of the weight matrix, W1 is a weight matrix, and is a real symmetric matrix, satisfying W1 = Ε[(BΔα)(BΔα) T ]=BQ α B T ;

[0031] In the formula, Q α is the covariance matrix of the measurement noise vector Δα, and the constraint expression is substituted into Then the WLS solution under the constraints is equivalent to minimizing the cost function under the constraints, that is,

[0032]

[0033] That is, the TDOA / FDOA passive positioning problem is transformed into a quadratic programming problem;

[0034] The constraint condition is expressed according to the parameter The pseudo linear equations are reconstructed, and the constraint condition is

[0035]

[0036] Among them, ∑ 11 =∑ 12 =∑ 21 =diag(1,1,1,-1);∑ is a symmetric reversible matrix;

[0037] Then the WLS solution under the constraint condition is equivalent to making the cost function Get the minimum value This is the optimal solution for the target positioning parameters.

[0038] A further technical solution of the present invention is as follows: the step of using the Lagrange multiplier evaluation idea to solve the pseudo-linear equations to obtain the least squares solution of the target positioning value; repeating the solution process twice to obtain the final solution of the first step of the target positioning solution process includes:

[0039] The Lagrange multiplier is used to solve the extreme value of this condition, that is,

[0040]

[0041] Where λ is the Lagrange multiplier; the extreme point The value of Take the derivative and set it to 0 to obtain:

[0042] Solving the above formula, we can get

[0043]

[0044] In the above equation, the Lagrange multiplier λ is unknown. The solution of the Lagrange multiplier λ can be obtained by substituting the above equation into the constraint condition to obtain; to substitute

[0045] h T W1 -T G(G T W1 -1 G+λ∑) -1 ∑(G T W1 -1 G+λ∑) -1 G T W1 -1 h=h T W1 -T G∑ -1 [(G T W1 -1 G+λΣ)Σ -1 ] -1 ·[(G T W1 -1 G+λ∑)∑ -1 ] -1 G T W1 -1 h=h T W1 -T G∑ -1 (G T W1 -1 G∑ -1 +λI8) -1 ·(G T W1 -1 G∑ -1 +λI8) -1 G T W1 -1 h=0

[0046] In the formula, I8 is an 8×8 unit matrix, ∑ is an 8×8 matrix; for the matrix G T W1 -1 G∑ -1 Performing eigenvalue decomposition, we can get:

[0047] G T W1 -1 G∑ -1 =UΛU -1

[0048] In the formula, Λ=diag{η1,η2,…,η8} is a diagonal matrix; substituting the equation after eigenvalue decomposition back into the calculation formula for λ, we can get the 14th-order equation about λ:

[0049]

[0050] Where p = [p1, p2, …, p8] T =U T ∑ -T G T W1 -1 h;q=[q1,q2,…,q8] T =U -1 G T W1 -1 h, the above formula can be simplified through convolution operation to obtain a 14th-order standard polynomial equation about λ, and then all solutions of this standard polynomial can be solved by polynomial root-finding operation. Considering the actual situation, the Lagrange multiplier λ should be a real number, so only real roots are retained in all solutions;

[0051] Substitute the obtained real roots into the formula one by one Get the corresponding positioning estimate solution Then Substitute into the cost function one by one The estimated solution that minimizes the cost function This is the optimal solution for the target positioning estimate. The covariance matrix of the first step estimation error is

[0052] Using the optimal solution Update the weight matrix W1 and repeat the solution twice to get the final solution of the first step of the target positioning solution process.

[0053] A further technical solution of the present invention is as follows: the step of constructing a new pseudo-linear equation set for target positioning according to the original constraint conditions comprises:

[0054] Assumptions The system of equations is as follows:

[0055] In the formula as well as

[0056]

[0057] A further technical solution of the present invention is to substitute the final solution of the first step of the target positioning solution process into the new pseudo-linear equation group to solve the least squares solution of the target positioning value; and the step of obtaining the optimized target position and speed estimation value includes:

[0058] By Can get The weighted least squares solution of is

[0059] The weight matrix W2 is By Calculated;

[0060] Utilization optimization The final estimated values ​​of the target position and velocity can be obtained as and in Indicated by The column vector consisting of the p-th to q-th elements in .

[0061] A computer system, characterized in that it comprises: one or more processors, and a computer-readable storage medium for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors implement the above-mentioned method.

[0062] A computer-readable storage medium, characterized in that it stores computer-executable instructions, which are used to implement the above method when executed.

[0063] A computer program product, characterized by comprising computer executable instructions, wherein the instructions are used to implement the above method when executed.

[0064] The beneficial effects of the present invention are:

[0065] The improved constrained weighted least squares TDOA / FDOA passive positioning method based on Lagrange multipliers provided by the present invention is divided into two steps. First, the time difference and frequency difference nonlinear positioning equations about the target position and speed in the passive positioning problem are converted into pseudo-linear equations, and then converted into a quadratic programming problem on this basis, and the constraints are simplified, and the Lagrange multiplier method is applied to solve the initial solution of the position and speed of the target positioning; then, according to the original constraints, a nonlinear equation group about the target position and speed is constructed, and the initial solution obtained in the first step is substituted into the constructed nonlinear equation group to further optimize and solve the target position and speed values. The method has good positioning accuracy and real-time performance, and the estimated RMSE curve of the target positioning parameters in a relatively large noise environment is still close to CRLB, has good robustness for a high noise environment, and has good advanced performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] The drawings are only for the purpose of illustrating particular embodiments and are not to be considered limiting of the present invention. Like reference symbols denote like components throughout the drawings.

[0067] Figure 1 Schematic diagram of the three-dimensional multi-station TDOA / FDOA positioning model.

[0068] Figure 2 Flowchart of the implementation of the algorithm proposed in the present invention.

[0069] Figure 3 Comparison of the root mean square error of near-field target velocity estimation.

[0070] Figure 4 Comparison of the root mean square error of near-field target position estimation.

[0071] Figure 5 Comparison of the root mean square error of far-field target velocity estimation.

[0072] Figure 6 Comparison of the root mean square error of far-field target position estimation. DETAILED DESCRIPTION

[0073] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0074] It should be noted that the terms "first", "second", etc. in the specification and claims of the present invention and the above-mentioned drawings are used to distinguish similar objects, and are not necessarily used to describe a specific order or sequence. It should be understood that the terms used in this way can be interchangeable where appropriate, so that the embodiments of the present invention described herein can be implemented in an order other than those illustrated or described herein. In addition, the terms "including" and "having" and any variations thereof are intended to cover non-exclusive inclusions, for example, a process, method, system, product or device that includes a series of steps or units is not necessarily limited to those steps or units that are clearly listed, but may include other steps or units that are not clearly listed or inherent to these processes, methods, products or devices.

[0075] The present invention proposes an improved constrained weighted least squares method based on Lagrange multipliers, comprising the following steps:

[0076] Step 1: Definition Obtain TDOA and FDOA measurement values, and establish positioning equations based on the measured information;

[0077] Step 2: Introduce intermediate redundant variables to transform the time-frequency-difference nonlinear positioning equation problem of solving the target position and speed in the passive positioning problem into a quadratic programming problem;

[0078] Step 3: Simplify the original constraints so that solving the quadratic programming problem is equivalent to finding the cost function under the simplified constraints. Conditional extreme value problem;

[0079] Step 4: Use convolution operation to convert f(λ) into a standard polynomial, and solve the polynomial equation by root-finding operation. Since the Lagrange multiplier λ is a real number, the solution of the equation only retains the real roots;

[0080] Step 5: Substitute the real root λ obtained in step 4 into the formula Get the corresponding solution Then the solutions of each real root Substitute into the cost function respectively Make Smallest This is the optimal solution;

[0081] Step 6: Use the optimal solution obtained in step 5 Update the weight matrix W1 and repeat steps 2 and 3 twice to get the final solution of the first step of the target positioning solution process.

[0082] Step 7: Substitute the final solution of step 6 into the formula The new pseudo-linear equations derived from the original constraints are used to solve the second step of the target positioning solution process. untie That is, the final solution in the first step The optimization solution based on

[0083] Step 8: Put the result of step 7 Substitution and Calculate the final estimated values ​​of the target position and velocity during the entire positioning process.

[0084] In order to enable those skilled in the art to better understand the present invention, the present invention is described in detail below in conjunction with specific embodiments.

[0085] Example 1

[0086] The embodiment of the present invention provides an improved constrained weighted least squares TDOA / FDOA passive positioning method based on Lagrange multipliers, which specifically includes:

[0087] Step 1: Discuss the three-dimensional space objectives. The three-dimensional space positioning diagram is as follows: Figure 1 As shown in the figure, the positioning scenario is that M receiving stations perform passive positioning on the target. Assume u o =[x,y,z] T , are the position and velocity of the target respectively. The positions and velocities of the M receiving stations are known, respectively And when the target is located in three-dimensional space, the target can be located only when the number of receiving stations M ≥ 4. For the convenience of discussion and without loss of generality, the first receiving station is set as the reference station.

[0088] The distance between the i-th receiving station and the target to be measured can be expressed as

[0089]

[0090] In the formula, ||*|| is the 2-norm operation.

[0091] The difference between the distance between the target to be measured and the receiving station i and the distance between the target to be measured and the reference receiving station can be expressed as

[0092]

[0093] Pair Taking the derivative with respect to time, we can get the distance change rate equation as follows:

[0094]

[0095] Pair Taking the derivative with respect to time, we can get the distance difference change rate equation as follows:

[0096]

[0097] In practical applications, the vector form of the TDOA / FDOA observation equation containing measurement noise can be expressed as

[0098] α=α o +Δα

[0099] In the formula, represents the true value; r=[r 21 ,r 31 ,…,r M1 ] T , Indicates the measured value of TDOA / FDOA; Δr=[Δr 21 ,Δr 31 ,…,Δr M1 ] T , It represents the measurement noise of TDOA / FDOA. Assume that the Gaussian noise vector Δα has a mean of zero and a covariance of Q α , where Q α =diag(Q t ,Q f). Assume that the covariance matrices of TDOA and FDOA measurement noise are Q t =E[ΔrΔr T ]=σ 2 R, Among them, σ 2 To measure the variance of the noise, the main diagonal elements of the matrix R are all 1, and the rest are all 0.5. Similarly, the position and speed of the receiving station are also expressed in vector form, that is,

[0100] r i o and r1 o Substitute the expression into Then square both sides of the equation to obtain the TDOA equation:

[0101]

[0102] Differentiating the above equation with respect to time, we can obtain the FDOA equation as follows:

[0103]

[0104] Step 2: Define the vector in, And add the intermediate redundant variable r1 o =||v o ||,

[0105] Will and Substituting into the TDOA equation and FDOA equation and removing the second-order noise, we can get

[0106]

[0107] The corresponding matrix expression can be derived from the above formula:

[0108]

[0109] in

[0110]

[0111] In the formula,

[0112] Consider the actual intermediate redundant variable r1 o , and There is a constraint relationship between

[0113]

[0114] First, find the WLS solution for the above equation, that is,

[0115] In the formula, is the norm of the weight matrix, W1 is a weight matrix, and is a real symmetric matrix, satisfying W1 = Ε[(BΔα)(BΔα) T ]=BQ α B T .

[0116] In the formula, Q α is the covariance matrix of the measurement noise vector Δα, and the constraint expression is substituted into Then the WLS solution under the constraints is equivalent to minimizing the cost function under the constraints, that is,

[0117]

[0118] That is, the TDOA / FDOA passive positioning problem is transformed into a quadratic programming problem.

[0119] Step 3: Formulate the constraint conditions according to the parameters The pseudo linear equations are reconstructed, and the constraint condition is:

[0120]

[0121] Among them, ∑ 11 =∑ 12 =∑ 21 =diag(1,1,1,-1);∑ is a symmetric reversible matrix.

[0122] Then the WLS solution under the constraint condition is equivalent to making the cost function Get the minimum value This is the optimal solution for the target positioning parameters.

[0123] Step 4: Use Lagrange multipliers to solve the extreme value of this condition, that is

[0124]

[0125] Where λ is the Lagrange multiplier. The value of Take the derivative and set it to 0 to obtain:

[0126] Solving the above formula, we can get

[0127]

[0128] In the above equation, the Lagrange multiplier λ is unknown. The solution of the Lagrange multiplier λ can be obtained by substituting the above equation into the constraint condition Substituting into the equation, we get

[0129] h T W1 -T G(G T W1 -1 G+λ∑) -1 ∑(G T W1 -1 G+λ∑) -1 G T W1 -1 h=h T W1 -T G∑ -1 [(G T W1 -1 G+λ∑)∑ -1 ] -1 ·[(G T W1 -1 G+λ∑)∑ -1 ] -1 G T W1 -1 h=h T W1 -T G∑ -1 (G T W1 -1 G∑ -1 +λI8) -1 ·(G T W1 -1 G∑ -1 +λI8) -1 G T W1 -1 h=0

[0130] In the formula, I8 is an 8×8 unit matrix, and ∑ is an 8×8 matrix. T W1 -1 G∑ -1 Performing eigenvalue decomposition, we can get:

[0131] G T W1 -1 G∑ -1 =UΛU -1

[0132] In the formula, Λ=diag{η1,η2,…,η8} is a diagonal matrix. Substituting the equation after eigenvalue decomposition back into the calculation formula for λ, we can get the 14th-order equation about λ:

[0133]

[0134] Where p = [p1, p2, …, p8] T =U T ∑ -T G T W1 -1 h;q=[q1,q2,…,q8] T =U -1 G T W1 -1 h, the above formula can be simplified through convolution operation to obtain a 14th-order standard polynomial equation about λ, and then the root-finding operation of the polynomial is used to solve all the solutions of this standard polynomial. Considering the actual situation, the Lagrange multiplier λ should be a real number, so only real roots are retained in all solutions.

[0135] Step 5: Then substitute each real root into the formula one by one Get the corresponding positioning estimate solution Then Substitute into the cost function one by one The estimated solution that minimizes the cost function This is the optimal solution for the target location estimation value. At this point, the first step of the method of the present invention is basically completed. The covariance matrix of the first step estimation error is

[0136] Step 6: Use the optimal solution obtained in step 5 Update the weight matrix W1 and repeat steps 2 and 3 twice to get the final solution of the first step of the target positioning solution process.

[0137] Step 7: Analyze the constraints It is found that the constraints are not fully utilized in the first step of the target positioning value solution. Therefore, in the second step, the original constraints are used to reconstruct the positioning pseudo-linear equation to solve the target positioning estimate. This achieves the estimated value calculated in the first step Optimize. The specific analysis is as follows:

[0138] The constraints Expand to get

[0139]

[0140] From the above formula, we can see that r1 o2 =v oT v o , It can be deduced And by Unable to deduce r1 o2 =v oT v o , That is, the constraint With the original constraint r1 o2 =v oT v o , are not completely equivalent. Therefore, it is necessary to make fuller use of the original constraint r1 in the second step o2 =v oT v o , right The value is further optimized.

[0141] General and Substitute into the above formula and Ignoring the second-order noise term, we assume and Available

[0142]

[0143] The target position and speed estimated by the first step of the solution process always have the following two relationships with the target's actual position and speed:

[0144] Combining the above two equations, a new pseudo-linear system of equations is reconstructed. This system of equations is also derived by making full use of the original constraints. Assume The system of equations is as follows:

[0145] In the formula as well as

[0146]

[0147] By Can get The weighted least squares solution of is

[0148] The weight matrix W2 is By Calculated.

[0149] Step 8: Utilize the formula optimization The final estimated values ​​of the target position and velocity can be obtained as and in Indicated by The column vector consisting of the p-th to q-th elements in .

[0150] The effects of this application are described in detail below in conjunction with simulation.

[0151] In order to further illustrate the performance of the method of the present invention in estimating the target parameter value, the following simulation experiment is carried out. The number of receiving stations M is set to 5, and the position and velocity information of each receiving station is shown in Table 1. The unit of the target position in the table is m, and the unit of the velocity is m / s. Then two experimental scenarios are set to carry out, where 5 receiving stations locate near-field targets and 5 receiving stations locate far-field targets. The position and velocity information of the near-field targets are u and m, respectively. o =[280,325,275] T and The position and velocity information of the far-field target are u o =[2000,2500,3000] T and

[0152] Table 1 Position and velocity of receiving station

[0153]

[0154] In both scenarios, the estimated root mean square error is used to analyze the positioning performance, and CRLB is used as a standard to measure the accuracy of positioning estimation. At the same time, the positioning performance of this algorithm is compared with the CWLS algorithm in the existing technology, WLS based on error correction, improved TSWLS, traditional TSWLS algorithm and least squares method using Gaussian iteration.

[0155] Figure 3 and Figure 4 , Figure 5 and Figure 6 It reflects the root mean square error of the algorithm in locating and estimating the target parameters in the two scenarios, and can also reflect the root mean square error of other comparison algorithms in the two scenarios. From these result graphs, it can be seen that the method proposed in the present invention shows good positioning performance in both near-field and far-field scenarios. For higher noise environments, the root mean square error of the positioning of the algorithm can still be close to CRLB, and it has good adaptability to high noise environments. Moreover, by comparing the positioning root mean square error curves of the method proposed in the present invention and the CWLS algorithm, it can be found that the improvement of the CWLS algorithm by the present invention is still very effective, and good positioning performance is obtained, and it still has good robustness in high noise environments.

[0156] The calculation time required for 10,000 MATLAB simulation experiments of the six algorithms was compared later. The traditional TSWLS algorithm took 2.588 seconds, the improved TSWLS algorithm took 2.925 seconds, the error-corrected WLS algorithm took 2.898 seconds, the WLS algorithm using Gauss-Newton iteration took 4.930 seconds, the CWLS algorithm took 9.144 seconds, and the algorithm proposed in the present invention took 9.634 seconds. The average execution time of the six algorithms is shown in Table 2. The comparative experiment shows that the time required for the present invention is the longest, mainly because the first step of the present invention method increases the computational complexity by obtaining the Lagrange multiplier through matrix decomposition, polynomial rooting and other methods. Despite this, the time spent by the present invention method for one positioning is only increased by 0.704ms compared to the TSWLS algorithm with the least time. While meeting the real-time performance, it can still maintain good positioning performance in a high noise environment and has relatively good robustness.

[0157] Table 2 Average operation time of different algorithms

[0158]

[0159] In summary, the present invention aims at the problem that the positioning performance of the current time difference / frequency difference passive positioning algorithm has poor ability to adapt to measurement noise, and proposes an improved constrained weighted least squares method based on Lagrange multipliers. The algorithm is a two-step positioning method. First, the nonlinear positioning equation of the time difference and frequency difference about the target position and speed in the passive positioning problem is converted into a pseudo-linear equation, and then converted into a quadratic programming problem on this basis, and the constraints are simplified. The Lagrange multiplier method is used to solve the initial solution of the target positioning, and then the nonlinear equation group about the target position and speed is constructed according to the original constraints. The initial solution obtained in the first step is substituted into the constructed nonlinear equation group to further optimize the target position and speed values ​​to obtain the final value. Finally, the simulation experiment proves that the present invention has good positioning accuracy and real-time performance. The estimated RMSE curve of the target positioning parameters in a large noise environment can also be close to CRLB, and has good robustness.

[0160] The above description is only a specific implementation mode of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can easily think of various equivalent modifications or substitutions within the technical scope disclosed in the present invention, and these modifications or substitutions should be included in the protection scope of the present invention.

Claims

1. An improved constrained weighted least squares TDOA / FDOA passive positioning method based on Lagrange multipliers, characterized in that: include: Obtain TDOA / FDOA measurement values, and establish a target positioning equation based on the TDOA / FDOA measurement values; Introducing intermediate redundant variables, constructing a pseudo-linear equation group for target positioning based on the constraint relationship between the target positioning equation and the intermediate redundant variables; The pseudo-linear equations are solved using the Lagrange multiplier evaluation idea to obtain the least squares solution of the target positioning value; Repeat the solution process twice to obtain the final solution of the first step of the target positioning solution process; Construct a new pseudo linear equation system for target positioning based on the original constraints; Substitute the final solution of the first step of the target positioning solution process into the new pseudo-linear equation system to obtain the least squares solution of the target positioning value; Obtain optimized target position and velocity estimates.

2. The improved constrained weighted least squares TDOA / FDOA passive positioning method based on Lagrange multipliers according to claim 1, characterized in that: The TDOA / FDOA measurement values ​​are specifically: a = a o +Da In the formula, represents the true value; Indicates the measured value of TDOA / FDOA; Represents the measurement noise of TDOA / FDOA.

3. The improved constrained weighted least squares TDOA / FDOA passive positioning method based on Lagrange multipliers according to claim 2, characterized in that: In the step of introducing intermediate redundant variables and constructing a pseudo-linear equation group for target positioning according to the constraint relationship between the target positioning equation and the intermediate redundant variables; include: Defining vectors in, And add intermediate redundant variables Will and Substituting into the TDOA equation and FDOA equation and removing the second-order noise, we can get The corresponding matrix expression can be derived from the above formula: in In the formula, Considering actual intermediate redundant variables and There is a constraint relationship between First, find the WLS solution for the above equation, that is, In the formula, is the norm of the weight matrix, W1 is a weight matrix, and is a real symmetric matrix, satisfying W1 = Ε[(BΔα)(BΔα) T ]=BQ α B T ; In the formula, Q α is the covariance matrix of the measurement noise vector Δα, and the constraint expression is substituted into Then the WLS solution under the constraints is equivalent to minimizing the cost function under the constraints, that is, That is, the TDOA / FDOA passive positioning problem is transformed into a quadratic programming problem; The constraint condition is expressed according to the parameter θ1 o The pseudo linear equations are reconstructed, and the constraint condition is Among them, ∑ 11 =∑ 12 =∑ 21 =diag(1,1,1,-1);∑ is a symmetric reversible matrix; Then the WLS solution under the constraint condition is equivalent to making the cost function Get the minimum value This is the optimal solution for the target positioning parameters.

4. The improved constrained weighted least squares TDOA / FDOA passive positioning method based on Lagrange multipliers according to claim 3, characterized in that: The pseudo-linear equations are solved by using the Lagrange multiplier evaluation idea to obtain the least squares solution of the target positioning value; Repeat the solution process twice to obtain the final solution of the first step of the target positioning solution process, including: The Lagrange multiplier is used to solve the extreme value of this condition, that is, Where λ is the Lagrange multiplier; the extreme point The value of Take the derivative and set it to 0 to obtain: Solving the above formula, we can get In the above equation, the Lagrange multiplier λ is unknown. The solution of the Lagrange multiplier λ can be obtained by substituting the above equation into the constraint condition to obtain; to substitute Where I8 is an 8×8 unit matrix, ∑ is an 8×8 matrix; for the matrix Performing eigenvalue decomposition, we can get: In the formula, Λ=diag{η1,η2,…,η8} is a diagonal matrix; substituting the equation after eigenvalue decomposition back into the calculation formula for λ, we can get the 14th-order equation about λ: Where p = [p1, p2, …, p8] T =U T ∑ -T G T W1 -1 h;q=[q1,q2,…,q8] T =U -1 G T W1 -1 h, the above formula can be simplified through convolution operation to obtain a 14th-order standard polynomial equation about λ, and then all solutions of this standard polynomial can be solved by polynomial root-finding operation. Considering the actual situation, the Lagrange multiplier λ should be a real number, so only real roots are retained in all solutions; Substitute the obtained real roots into the formula one by one Get the corresponding positioning estimate solution Then Substitute into the cost function one by one The estimated solution that minimizes the cost function This is the optimal solution for the target positioning estimate. The covariance matrix of the first step estimation error is Using the optimal solution Update the weight matrix W1 and repeat the solution twice to get the final solution of the first step of the target positioning solution process.

5. The improved constrained weighted least squares TDOA / FDOA passive positioning method based on Lagrange multipliers according to claim 4, characterized in that: The step of constructing a new pseudo-linear equation set for target positioning according to the original constraint conditions includes: Assumptions The system of equations is as follows: In the formula as well as 6. The improved constrained weighted least squares TDOA / FDOA passive positioning method based on Lagrange multipliers according to claim 5, characterized in that: Substitute the final solution of the first step of the target positioning solution process into the new pseudo-linear equation system to obtain the least squares solution of the target positioning value; The steps of obtaining the optimized target position and velocity estimates include: By Can get The weighted least squares solution of is The weight matrix W2 is By Calculated; Utilization optimization The final estimated values ​​of the target position and velocity can be obtained as and in Indicated by The column vector consisting of the p-th to q-th elements in .

7. A computer system, characterized in that include: One or more processors, and a computer-readable storage medium for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors are enabled to implement the method of claim 1.

8. A computer-readable storage medium, characterized in that Computer executable instructions are stored, and when the instructions are executed, they are used to implement the method of claim 1.

9. A computer program product, characterized in that The invention comprises computer executable instructions, which are used to implement the method of claim 1 when being executed.

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