Motion single station passive positioning method

Abstract

The invention discloses a passive positioning method for a moving single station, and belongs to the technical field of radars, and the method comprises the steps: receiving a periodic pulse signal transmitted by a static radiation source, and obtaining the TOA of the periodic pulse signal; establishing a signal transmission model in a sight distance environment in combination with TOA, and estimating a transmission period of the periodic pulse signal; constructing a nonlinear non-convex optimization problem based on the signal transmission model, relaxing the nonlinear non-convex optimization problem into a convex problem by using a second-order cone programming algorithm, and introducing a penalty term to obtain a positioning model; and after the positioning model is converted into a standard SOCP form, an interior point algorithm is used for solving, and a position estimation result of the static radiation source is obtained. According to the method, the SOCP algorithm is introduced into a single-station positioning problem, the calculation amount is reduced, the method is more suitable for solving a large-scale positioning problem, in addition, the SOCP algorithm is utilized to relax a nonlinear non-convex problem into a convex problem, a penalty term is introduced, and the influence of a convex hull on the SOCP algorithm is improved.

Description

Technical Field

[0001] The present invention belongs to the field of radar technology, and in particular relates to a moving single-station passive positioning method. Background Art

[0002] Single-station passive positioning has become a hot topic of research for technicians in this field due to its good concealment performance, low cost, and no need for clock synchronization.

[0003] Emitter location is a core issue in various applications. Among the commonly used positioning parameters, the time of arrival (TOA) information of the receiving source signal is a widely used option to achieve high positioning accuracy. When multiple receivers are available and the signal transmission time is known, the location of the emitter can be inferred based on the TOA obtained simultaneously at each station, thereby performing trilateration to determine the source position. In the case of a non-cooperative target emitter and the signal transmission time is unknown, the TOA can be converted into a time difference of arrival (TDOA) measurement value, and TDOA positioning can be used to locate the emitter.

[0004] Currently, single-station passive positioning methods primarily include those that use Doppler frequency and direction of arrival (DOA) to estimate the emitter's position, and those that use TOA to measure stationary sources. TOA-based positioning methods typically assume that the target emitter emits multiple pulse signals, which are then intercepted by a mobile receiver to obtain sufficient TOA measurements to estimate the target source's position. Compared to multi-station positioning systems, single-station systems do not require data exchange or synchronization between receivers.

[0005] In recent years, convex optimization theory, a key component of nonlinear programming, has made significant progress in both theoretical and engineering practice. In particular, the introduction of interior-point algorithms has significantly increased the computational speed of convex optimization algorithms, making them valuable for practical applications. Convex optimization algorithms effectively avoid problems such as search step size selection, initial value sensitivity, and trapping in local minima. They also offer advantages such as high precision, strong noise immunity, and robustness, leading to their introduction into solving positioning problems.

[0006] Existing positioning algorithms primarily linearize the objective function through Taylor expansion. However, first-order Taylor expansions often omit some estimated values, introducing errors. Higher-order expansions increase the algorithm's computational complexity and are computationally inefficient. While semidefinite programming (SDP) methods address these issues, their algorithmic complexity is high. When intercepting too many signal pulses, processing takes a long time, making them inefficient in real-time. Summary of the Invention

[0007] In order to solve the above problems existing in the prior art, the present invention provides a moving single-station passive positioning method. The technical problem to be solved by the present invention is achieved through the following technical solutions:

[0008] An embodiment of the present invention provides a moving single-station passive positioning method, comprising:

[0009] Receiving a periodic pulse signal emitted by a stationary radiation source and obtaining a time of arrival (TOA) of the periodic pulse signal;

[0010] Establishing a signal transmission model in a line-of-sight environment in combination with the arrival time TOA, and estimating the transmission period of the periodic pulse signal;

[0011] A nonlinear non-convex optimization problem is constructed based on the signal transmission model, and a second-order cone programming algorithm is further used to relax the nonlinear non-convex optimization problem into a convex problem, and a penalty term is introduced to obtain a positioning model;

[0012] After the positioning model is converted into a standard second-order cone programming (SOCP) form, it is solved using an interior point algorithm to obtain a position estimation result of the stationary radiation source.

[0013] In one embodiment of the present invention, the signal transmission model is expressed as follows:

[0014]

[0015] Wherein, t0 represents the time when the stationary radiation source starts to emit periodic pulse signals, u represents the unknown position of the stationary radiation source, and s i Indicates the position of the receiving station when receiving the i-th cycle pulse signal, N i Indicates the receiving sequence number of the i-th cycle pulse signal, T s Indicates the emission period of the periodic pulse signal, Δt i Indicates s i The error when receiving the i-th cycle pulse signal, Δt i It has a mean of zero and a variance of Gaussian distribution, t i represents the arrival time of the i-th periodic pulse signal, c represents the propagation speed of electromagnetic waves, i=1, 2,…, M, and M represents the number of periodic pulse signals emitted by the stationary radiation source.

[0016] In one embodiment of the present invention, the step of constructing a nonlinear non-convex optimization problem based on the signal transmission model includes:

[0017] The signal transmission model is preprocessed and converted into a pseudorange model:

[0018] ri =t0c+N i T s c+||us i ||+n i ;

[0019] Where r i Indicates receiving station s i The pseudorange, r i =t i c,n i =Δt i c;

[0020] Based on the pseudorange model, a nonlinear non-convex optimization problem is constructed:

[0021]

[0022] In one embodiment of the present invention, before the step of further using a second-order cone programming algorithm to relax the nonlinear non-convex optimization problem into a convex problem and introducing a penalty term to obtain a positioning model, the method further includes:

[0023] The nonlinear non-convex optimization problem is expressed as a vector form:

[0024]

[0025] Where, d=[d1,d2,…,d M ] T ,r=[r1,r2,…,r M ] T ,q=[N1,N2,…,N M ] T ,d1,d2,…,d M are the distances from the stationary radiation source to the receiving station that receives the 1st, 2nd, ..., Mth periodic pulse signals, r1, r2, ..., r M They represent the pseudoranges of the receiving stations receiving the 1st, 2nd, ..., Mth periodic pulse signals, N1, N2, ..., N M Respectively represent the receiving sequence numbers of the 1st, 2nd, ..., Mth periodic pulse signals, I M represents the M-dimensional identity matrix, T represents the transpose, and Q represents the noise covariance matrix.

[0026] In one embodiment of the present invention, the step of further using a second-order cone programming algorithm to relax the nonlinear non-convex optimization problem into a convex problem and introducing a penalty term to obtain a positioning model includes:

[0027] In the vector form of the nonlinear non-convex optimization problem, let h = [t0, T s ] T 、F=[1M ,q], the nonlinear non-convex problem is relaxed into a convex problem using the SOCP algorithm, and a penalty term is introduced:

[0028]

[0029] Where, δ represents the preset penalty coefficient, τ i1 =t i -t1-(N i -N1)T s , i=2,…,M.

[0030] In one embodiment of the present invention, the step of converting the positioning model into a standard second-order cone programming (SOCP) form includes:

[0031] The positioning model is converted into the following form:

[0032]

[0033] By defining the k-dimensional second-order cone The positioning model in standard SOCP form is obtained:

[0034] min v T χ;

[0035] st

[0036]

[0037]

[0038] Where α represents the width of the cone in the second-order cone programming, β represents the height of the cone in the second-order cone programming, and the coefficient vector 0 1×(M+2) Represents a zero matrix of dimension 1×(M+2), vector ξ′,η′,ω′,u′ are all optimization variables, Represents a p-dimensional column vector whose qth element is 1 and the rest are 0.

[0039] Compared with the prior art, the present invention has the following beneficial effects:

[0040] The present invention provides a moving single-station passive positioning method, which introduces the SOCP algorithm into the single-station positioning problem. Since the computational complexity of the SOCP algorithm is much smaller than that of the existing SDP algorithm, it is more suitable for solving large-scale positioning problems. In addition, the present invention uses the SOCP algorithm to relax the nonlinear non-convex problem into a convex problem and introduces a penalty term. The relaxed constraints are closer to the constraints of the original problem, thus effectively improving the influence of the convex hull on the SOCP algorithm.

[0041] In addition, the present invention can also provide an initial iteration point for other iterative algorithms, thereby effectively improving the positioning accuracy and convergence speed of the iterative algorithms.

[0042] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 1 is a flow chart of a moving single-station passive positioning method provided by an embodiment of the present invention;

[0044] Figure 2 1 is a schematic diagram of a hexagonal moving trajectory simulation of a receiving station provided by an embodiment of the present invention;

[0045] Figure 3 The receiving station provided by the embodiment of the present invention is Figure 2 Comparison of the position estimation results of a stationary radiation source when moving along the trajectory shown. DETAILED DESCRIPTION

[0046] The present invention will be further described in detail below with reference to specific examples, but the embodiments of the present invention are not limited thereto.

[0047] Figure 1 This is a flow chart of the single-station passive positioning method provided by an embodiment of the present invention. Figure 1 The embodiment of the present invention provides a moving single-station passive positioning method, comprising:

[0048] S1. Receive a periodic pulse signal emitted by a stationary radiation source and obtain the arrival time TOA of the periodic pulse signal.

[0049] It should be noted that this embodiment of the present invention considers a two-dimensional positioning scenario, where a stationary radiation source transmits a periodic pulse signal that is intercepted by a mobile receiver. During the observation interval, the stationary radiation source transmits a total of M periodic pulse signals. At the receiving end, there is a certain probability that the received periodic pulse signals will be lost due to factors such as antenna rotation and noise.

[0050] S2. Based on the arrival time TOA, a signal transmission model is established in a line-of-sight environment, and the transmission period of the periodic pulse signal is estimated.

[0051] Exemplarily, the signal transmission model is expressed as follows:

[0052]

[0053] where \(t_0\) represents the start time of the periodic pulse signal, that is, the time when the stationary radiation source starts to emit the periodic pulse signal, \(u = [x t ,y t T represents the unknown position where the stationary radiation source is located, \(s i represents the position of the receiving station when receiving the \(i\)-th periodic pulse signal, \(c\) represents the electromagnetic wave propagation speed, represents the signal propagation time, \(N i represents the reception sequence number of the \(i\)-th periodic pulse signal, \(T s represents the emission period of the periodic pulse signal, \(\Delta t i represents the error of \(s i when receiving the \(i\)-th periodic pulse signal, \(\Delta t i obeys a Gaussian distribution with a mean of zero and a variance of , and different \(\Delta t i are independent of each other, \(t i represents the arrival time of the \(i\)-th periodic pulse signal, \(i = 1, 2, \cdots, M\), and \(M\) represents the number of periodic pulse signals emitted by the stationary radiation source.

[0054] In order to extract \(T s from the TOA pulse train, the following analysis is carried out according to the above signal transmission model: \(N i is a positive integer, and \(N i >N j , \(j < i\), and at the same time \(t_0 + N i T s represents the time when the stationary radiation source emits the \(i\)-th periodic pulse signal. Due to the situation that some periodic pulse signals are not detected, for example, the antenna of the mobile receiver cannot receive all the periodic pulse signals during the rotation process, so \(N i does not satisfy \(N i =N i-1 + 1, and in some scenarios, \(N i -N i-1 may be very large, which will lead to cycle ambiguity. At this time, there is \(N i T s =(N i / N)\(\cdot\)(NT s ), where \(N\) is the greatest common divisor of \(N i . However, this scaling ambiguity does not affect the validity of the signal transmission model, and thus does not affect the applicability of the positioning algorithm.

[0055] It should be understood that the signal transmission time and the propagation error \(\Delta t i ​, we can selectively use the MEA (Modified Euclidean Algorithm) or the ZnLLS (the integer lattice line search) algorithm to estimate the emission period of the periodic pulse signal. Then the emission period can be described as the following optimization problem:

[0056]

[0057] By solving the above optimization problem, we can get the emission period T of the periodic pulse signal s .

[0058] S3. Based on the signal transmission model, a nonlinear non-convex optimization problem is constructed. The second-order cone programming algorithm is further used to relax the nonlinear non-convex optimization problem into a convex problem, and a penalty term is introduced to obtain a positioning model.

[0059] Specifically, step S3 includes:

[0060] S301. Preprocess the signal transmission model by multiplying both sides of the equal sign by the electromagnetic wave propagation speed c to obtain a pseudorange model:

[0061] r i =t0c+N i T s c+||us i ||+n i ;

[0062] Where r i Indicates receiving station s i The pseudorange, r i =t i c,n i =Δt i c, Δt i It still obeys a zero-mean Gaussian distribution.

[0063] S302. Construct a nonlinear non-convex optimization problem based on the pseudo-range model:

[0064]

[0065] In the above formula, T s It has been obtained by MEA or ZnLLS algorithm, u and t0 are unknown, so this is a nonlinear and non-convex problem, which is difficult to solve by maximum likelihood estimation.

[0066] S303. Express the nonlinear non-convex optimization problem in vector form:

[0067]

[0068] Where, d=[d1,d2,…,d M ] T ,r=[r1,r2,…,r M ] T ,q=[N1,N2,…,N M ] T ,d1,d2,…,d M are the distances from the stationary radiation source to the receiving station that receives the 1st, 2nd, ..., Mth periodic pulse signals, r1, r2, ..., r M They represent the pseudoranges of the receiving stations receiving the 1st, 2nd, ..., Mth periodic pulse signals, N1, N2, ..., N M Respectively represent the receiving sequence numbers of the 1st, 2nd, ..., Mth periodic pulse signals, I M represents the M-dimensional identity matrix, T represents the transpose, and Q represents the noise covariance matrix.

[0069] S304. In the vector form of the nonlinear non-convex optimization problem, let h = [t0, T s ] T 、F=[1 M ,q], the nonlinear non-convex problem is relaxed into a convex problem using the SOCP algorithm, and a penalty term is introduced:

[0070]

[0071] Where, δ represents the preset penalty coefficient, τ i1 =t i -t1-(N i -N1)T s , i=2,…,M.

[0072] In step S303, the nonlinear non-convex optimization problem is expressed as a quadratic problem. Since the constraints are nonlinear, it is still a non-convex problem. Therefore, this embodiment introduces the use of the SOCP algorithm to relax the non-convex constraints into convex constraints before solving the problem.

[0073] Specifically, in step S304, the non-convex problem is relaxed into a convex problem by using the SOCP algorithm, and a penalty term is added to restrict the inequality constraint. The equality constraint in step S303 is now relaxed into an inequality constraint, and a penalty term δd is introduced. T d restricts the inequality constraints, then the original non-convex problem is transformed into the following convex problem:

[0074]

[0075] Here, δ is a penalty coefficient, typically a small positive constant. The purpose of adding a penalty term is to constrain the constraints so that the relaxed constraints are as close as possible to the equality constraints of the original problem. As can be seen from the above formula, without the penalty term, adding an arbitrary constant to all elements in d simultaneously will not change the minimum value of the objective function. However, if the values of the elements in d are very large, many estimated coordinates that are not the true coordinates of the radiation source will satisfy the constraints, thus preventing the SOCP algorithm from estimating the correct position of the stationary radiation source. With the penalty term, if the values of the elements in d are too large, the objective function value increases, and the corresponding estimates will be eliminated.

[0076] However, the above formula has a convex hull problem, that is, when using the SOCP algorithm to solve the positioning problem, the relaxed constraints are not completely equivalent to the original constraints. The algorithm will limit the search area of the radiation source position according to the position of each reconnaissance station. This area is the convex hull. When the radiation source position is within the convex hull formed by the receiving station, the SOCP algorithm is feasible; when the target position is outside the convex hull formed by the receiving station, the above formula cannot effectively estimate the radiation source position. In view of this, it is necessary to reasonably restrict the constraints in the above formula so that they are as equivalent as possible to the constraints of the original problem, so as to effectively improve the impact of the convex hull on solving the positioning problem using SOCP. To this end, this embodiment adds a new penalty term to further restrict the relaxation of the inequality.

[0077] When constructing a new penalty term, two points need to be noted: first, the constructed penalty term must maintain the convex property, otherwise the modified model will be non-convex and cannot be solved using the SOCP algorithm; second, the structure of the new penalty term should be simple and easy to convert to the standard SOCP form.

[0078] Based on the above two points, the penalty term constructed in this embodiment is where τ i1 =t i -t1-(N i -N1)T s , expand it and write it into matrix form:

[0079]

[0080] in,

[0081] By adding the constructed penalty term to the model, we can get the positioning model:

[0082]

[0083] S4. After converting the positioning model into the standard second-order cone programming SOCP form, the interior point algorithm is used to solve it and the position estimation result of the stationary radiation source is obtained.

[0084] Since the positioning model is a quadratic form, the positioning model can be converted into a SOCP problem and solved. In the positioning model, the noise covariance matrix Q is a symmetric positive definite matrix. The Cholesky decomposition of it can be obtained as Q = R T R, then the positioning model can be transformed into the following form:

[0085]

[0086] It can be seen that the constraint term in the above formula is a quadratic constraint, which is not easy to convert into the standard SOCP form. Therefore, the above formula is further converted to an equivalent form to obtain:

[0087]

[0088] Next, define the k-dimensional second-order cone If and only if k = 1, define a one-dimensional second-order cone as Then the original positioning problem model can be transformed into a positioning model in the standard SOCP form:

[0089] min v T χ;

[0090] st

[0091]

[0092] Where α represents the width of the cone in the second-order cone programming, β represents the height of the cone in the second-order cone programming, and the coefficient vector 0 1×(M+2) Represents a zero matrix of dimension 1×(M+2), vector ξ′,η′,ω′,u′ are all optimization vectors, Represents a p-dimensional column vector whose qth element is 1 and the rest are 0.

[0093] Furthermore, the global optimal solution obtained by the interior point algorithm is the position estimation result of the stationary radiation source.

[0094] The moving single-station passive positioning method provided by the present invention is further illustrated below through simulation.

[0095] Figure 2 : is a schematic diagram of a hexagonal moving trajectory simulation of a receiving station provided by an embodiment of the present invention, Figure 3 The receiving station provided by the embodiment of the present invention is Figure 2 Comparison of the position estimation results of a stationary radiation source when moving along the trajectory shown. Figures 2-3 , Figure 2 The horizontal and vertical axes represent the horizontal and vertical coordinates of the receiving station respectively. The blue square represents the stationary radiation source. Figure 2When the hexagonal trajectory shown in FIG moves, the position of the stationary radiation source is estimated by using the moving single-station passive positioning method and the semi-definite programming (SDP) algorithm provided by the present invention. The results are as follows: Figure 3 As shown in the figure, the red circle represents the stationary emitter position estimation result obtained using the moving single-station passive positioning method provided by the present invention, and the green circle represents the stationary emitter position estimation result obtained using the SDP algorithm. CRLB is the Cramer-Rao Lower Bound, which is also the theoretical minimum error and represents the positioning accuracy of the stationary emitter. Clearly, within the error range shown, the present invention can achieve the Cramer-Rao Lower Bound and outperforms the SDP algorithm in the case of high error.

[0096] It can be seen from the above embodiments that the beneficial effects of the present invention are:

[0097] The present invention provides a moving single-station passive positioning method, which introduces the SOCP algorithm into the single-station positioning problem. Since the computational complexity of the SOCP algorithm is much smaller than that of the existing SDP algorithm, it is more suitable for solving large-scale positioning problems. In addition, the present invention uses the SOCP algorithm to relax the nonlinear non-convex problem into a convex problem and introduces a penalty term. The relaxed constraints are closer to the constraints of the original problem, thus effectively improving the influence of the convex hull on the SOCP algorithm.

[0098] In addition, the present invention can also provide an initial iteration point for other iterative algorithms, thereby effectively improving the positioning accuracy and convergence speed of the iterative algorithms.

[0099] In the description of the present invention, reference to the terms "one embodiment," "some embodiments," "examples," "specific examples," or "some examples" means that the specific features, structures, materials, or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials, or characteristics described can be combined in any suitable manner in any one or more embodiments or examples. In addition, those skilled in the art can combine and combine different embodiments or examples described in this specification.

[0100] The above is a further detailed description of the present invention in conjunction with specific preferred embodiments, and the specific implementation of the present invention should not be considered to be limited to these descriptions. For those skilled in the art of the present invention, without departing from the concept of the present invention, several simple deductions or substitutions can be made, which should be considered to fall within the scope of protection of the present invention.

Claims

1. A moving single-station passive positioning method, characterized in that: include: Receiving a periodic pulse signal emitted by a stationary radiation source and obtaining a time of arrival (TOA) of the periodic pulse signal; Establishing a signal transmission model in a line-of-sight environment in combination with the arrival time TOA, and estimating the transmission period of the periodic pulse signal; A nonlinear non-convex optimization problem is constructed based on the signal transmission model, and a second-order cone programming algorithm is further used to relax the nonlinear non-convex optimization problem into a convex problem, and a penalty term is introduced to obtain a positioning model; After the positioning model is converted into a standard second-order cone programming (SOCP) form, it is solved using an interior point algorithm to obtain a position estimation result of the stationary radiation source.

2. The moving single-station passive positioning method according to claim 1, characterized in that: The signal transmission model is expressed as follows: Wherein, t0 represents the time when the stationary radiation source starts to emit the periodic pulse signal, u represents the unknown position of the stationary radiation source, and s i Indicates the position of the receiving station when receiving the i-th cycle pulse signal, N i Indicates the receiving sequence number of the i-th cycle pulse signal, T s Indicates the emission period of the periodic pulse signal, Δt i Indicates s i The error when receiving the i-th cycle pulse signal, Δt i It has a mean of zero and a variance of Gaussian distribution, t i represents the arrival time of the i-th periodic pulse signal, c represents the propagation speed of electromagnetic waves, i=1, 2,…, M, and M represents the number of periodic pulse signals emitted by the stationary radiation source.

3. The moving single-station passive positioning method according to claim 2, characterized in that: The step of constructing a nonlinear non-convex optimization problem based on the signal transmission model includes: The signal transmission model is preprocessed and converted into a pseudorange model: r i =t0c+N i T s c+||u-s i ||+n i ; Where r i Indicates receiving station s i The pseudorange, r i =t i c,n i =Δt i c; Based on the pseudorange model, a nonlinear non-convex optimization problem is constructed:

4. The moving single-station passive positioning method according to claim 3, characterized in that: Before further using a second-order cone programming algorithm to relax the nonlinear non-convex optimization problem into a convex problem and introducing a penalty term to obtain a positioning model, the method further includes: The nonlinear non-convex optimization problem is expressed as a vector form: Where, d=[d1,d2,…,d M ] T ,r=[r1,r2,…,r M ] T ,q=[N1,N2,…,N M ] T ,d1,d2,…,d M are the distances from the stationary radiation source to the receiving station that receives the 1st, 2nd, ..., Mth periodic pulse signals, r1, r2, ..., r M They represent the pseudoranges of the receiving stations receiving the 1st, 2nd, ..., Mth periodic pulse signals, N1, N2, ..., N M Respectively represent the receiving sequence numbers of the 1st, 2nd, ..., Mth periodic pulse signals, I M represents the M-dimensional identity matrix, T represents the transpose, and Q represents the noise covariance matrix.

5. The moving single-station passive positioning method according to claim 4, characterized in that: The steps of further using a second-order cone programming algorithm to relax the nonlinear non-convex optimization problem into a convex problem and introducing a penalty term to obtain a positioning model include: In the vector form of the nonlinear non-convex optimization problem, let h = [t0, T s ] T 、F=[1 M ,q], the nonlinear non-convex problem is relaxed into a convex problem using the SOCP algorithm, and a penalty term is introduced: Where, δ represents the preset penalty coefficient, τ i1 =t i -t1-(N i -N1)T s , i=2,…,M.

6. The moving single-station passive positioning method according to claim 5, characterized in that: The steps of converting the positioning model into a standard second-order cone programming SOCP form include: The positioning model is converted into the following form: By defining the k-dimensional second-order cone The positioning model in standard SOCP form is obtained: min v T x; st Where α represents the width of the cone in the second-order cone programming, β represents the height of the cone in the second-order cone programming, and the coefficient vector 0 1×(M+2) Represents a zero matrix of dimension 1×(M+2), vector ξ′,η′,ω′,u′ are all optimization variables, Represents a p-dimensional column vector whose qth element is 1 and the rest are 0.

Images