Multi-station phase difference passive positioning method based on improved whale optimization algorithm

Abstract

The invention belongs to the technical field of passive positioning, and particularly relates to a multi-station phase difference passive positioning method based on an improved whale optimization algorithm. Firstly, a passive phase difference positioning model is established; then, according to the positioning model, establishing a target function based on maximum likelihood estimation; and finally, improving a whale optimization algorithm, improving a probability parameter threshold into a dynamic probability threshold, improving a convergence factor to be adaptively adjusted along with fitness values, solving a target function by using the improved whale optimization algorithm, taking a whale individual position as a target source position, and calculating the fitness value of each whale according to the target function. And adjusting the position of the whale individual according to the fitness value after each iteration, and after the set iteration times are reached, the position of the target source is the position of the whale individual with the minimum fitness value. According to the method, the problem of solving the target source position is converted into the optimization problem with the target source position as a variable, the calculation complexity is reduced while the positioning precision is improved, and the manufacturing cost is saved.

Description

Technical Field

[0001] The present invention belongs to the technical field of passive positioning, and specifically is a multi-station phase difference passive positioning method based on an improved whale optimization algorithm, which can be used to solve the position calculation problem of a passive phase difference positioning system for a target source. Background Art

[0002] With the rapid development of modern science and technology, passive positioning technology has been widely used in many fields. Passive positioning technology refers to estimating the position of a target by receiving signals transmitted by the target and using the signal differences between receiving stations (such as time difference, phase difference, frequency difference, etc.) without relying on the target to actively transmit signals. This technology is widely used in military reconnaissance, communication monitoring, aerospace, ocean detection, environmental monitoring and other fields, with the advantages of strong concealment, low cost, and long-term operation.

[0003] The positioning method based on the phase difference of arrival (PDOA) has been widely studied and applied due to its strong anti-interference ability and high positioning accuracy. The principle is that when the long baseline interferometer (LBI) on the observation station measures the phase difference of arrival, multiple positioning cones with the observation station as the vertex and the straight line where the baseline vector is located as the axis can be obtained by measuring the phase difference through the long baseline interferometer. After these positioning cones intersect with the ground, a series of positioning curves will be formed, and each curve corresponds to a phase ambiguity integer. When multiple observation stations observe together, since the positions of each observation station are different from the baseline vector, the angle between the baseline vector and the incident direction of the signal changes accordingly, and the fuzzy phase differences obtained by different observation stations will form different positioning curve clusters on the ground. In addition to intersecting at the target source position, these curves may also generate multiple fuzzy positioning points at other positions. However, the number of positioning curves that intersect only at the real position point of the target source is the same as the total number of observation stations. Although the target source position can be accurately obtained through multiple observation stations, the phase ambiguity problem causes the relationship between the phase difference measurement value and the target source position to be not only nonlinear but also discontinuous, so it is difficult to obtain an analytical solution to the problem through traditional solution methods. The whale optimization algorithm can effectively solve the nonlinear discontinuity problem in phase difference passive positioning through its strong global search capability, no need for gradient calculation, and adaptability to nonlinear functions.

[0004] Most of the current research converts the positioning problem based on the arrival phase difference into a weighted least squares problem for solution. This method usually includes the following steps: first, the cost function is obtained by using the maximum likelihood estimation based on the baseline vector and the acquired signal of the long baseline interferometer; then, multiple initial positions are selected in the area where the target source may exist, assuming that each initial position is close to the actual position of the target source, and the Gauss-Newton method is used to calculate the maximum likelihood estimate of the position, and the result with the minimum cost function value is taken as the target source position.

[0005] For example, in the paper "Multi-station Passive Positioning Method Based on Long Baseline Interferometer Phase Difference", Zhang Min et al. proposed an arrival phase difference positioning method. First, a cost function is constructed through maximum likelihood estimation; then, the maximum fuzzy number of the measured fuzzy phase difference is determined based on the relationship between the baseline length of the long baseline interferometer and the signal wavelength; then, the possible fuzzy numbers of the fuzzy phase difference are traversed to obtain multiple unambiguous arrival phase differences, and a set of direction angles are obtained based on the relationship between the arrival phase difference and the arrival direction angle. According to the direction angles, a target equation group is constructed, and the pseudo-linear least squares method is used to obtain the initial values ​​of the radiation source positions; finally, the Gauss-Newton iterative cost function is calculated for the obtained initial positioning values, and the result with the smallest cost function value is taken as the target source position.

[0006] In his published paper "Multi-hypothesis NLS Positioning Algorithm Based on Rotating Interferometer Fuzzy Phase Difference", Li Teng divided the observation area into uniform square sub-areas, took the center points of the divided sub-areas as the initial values, and then performed Gauss-Newton iteration to solve the maximum likelihood cost function for each positioning initial value, and took the result with the minimum cost function value as the target source position.

[0007] The method of using multiple hypothesized initial positioning points and then iterating through the Gauss-Newton method to obtain the precise target position has a relatively complex solution process and limited positioning accuracy, and faces certain challenges in practical applications.

[0008] Therefore, the present invention proposes a multi-station phase difference passive positioning method based on an improved whale optimization algorithm, which transforms the problem of solving the target position into an optimization problem with the target source coordinates as variables, and applies the improved whale optimization algorithm to solve the objective function. This method simplifies the calculation process, improves the positioning accuracy, and reduces the time required for positioning. Summary of the invention

[0009] In view of the deficiencies in the prior art, the technical problem that the present invention intends to solve is to propose a multi-station phase difference passive positioning method based on an improved whale optimization algorithm.

[0010] The present invention solves the technical problem by adopting the following technical solutions:

[0011] A multi-station phase difference passive positioning method based on an improved whale optimization algorithm, characterized in that the method comprises the following steps:

[0012] Step 1, establishing a passive phase difference positioning model;

[0013] Deploy at least three observatories in three-dimensional space. Each observatory is equipped with a long baseline interferometer for measuring the arrival phase difference. Set the position s of the nth observatory n for:

[0014] s n =[x n ,y n ,z n ] T (1)

[0015] The baseline vector b of the long baseline interferometer installed at the nth observation station n for:

[0016] b n = l n [sinα n ,cosα n ,0] T (2)

[0017] In the formula, l n is the baseline length of the long baseline interferometer, α n are the azimuth angles of the long baseline interferometer, respectively;

[0018] The unambiguous theoretical value of the phase difference from the target source to the i-th observation station is:

[0019]

[0020] In the formula, r n =(s t -s n ) / r n is the unit vector pointing from the nth observation station to the target source, s t is the target source position, r n =‖s t -s n is the straight-line distance from the target source to the nth observation station, λ is the signal wavelength; Φ=[φ 1 ,φ 2 ,…,φ n ,…φ N ] T is the unambiguous theoretical vector of the target arrival phase difference, and the fuzzy PDOA theoretical vector is:

[0021] z=mod 2π [φ 1 ,φ2 ,…,φ n ,…φ N ] T (4)

[0022] In the formula, mod 2π (·) is the 2π modulo operation;

[0023] The PDOA measurement vector is:

[0024]

[0025] z o =z+w n (6)

[0026] In the formula, w n =[w 1 ,w 2 ,…w n ,…,w N ] T represents the noise vector, w n represents the measurement noise of the arrival phase difference at the nth observation station, and N represents the number of observation stations;

[0027] Step 2, establishing the objective function based on maximum likelihood estimation;

[0028] Assume that the measurement noise of the arrival phase difference is w n If it conforms to the Gaussian distribution, then the noise vector w n =z o -z follows a Gaussian distribution with mean 0 and variance σ 2 , then the maximum likelihood function is:

[0029]

[0030] When the number of observation stations is N, the maximum likelihood function is:

[0031]

[0032] The maximum value of equation (9) is the minimum value of equation (10):

[0033] (x,y,z)=arg{min[mod 2π ((zz 0 ) T (zz 0 ))]} (10)

[0034] Then the objective function is:

[0035] J(s t )=mod 2 π((zz 0) T (zz 0 )) (11)

[0036] In the formula, represents the estimated target source position;

[0037] Step 3, improving the whale optimization algorithm to obtain an improved whale optimization algorithm;

[0038] Step 4: Use the improved whale optimization algorithm to solve the objective function, set the position of the target source to the individual in the whale population, calculate the fitness value of the whale individual after each iteration according to the above formula, adjust the whale position according to the fitness value, and after reaching the set number of iterations, the position of the individual with the smallest fitness value is the target source position.

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

[0040] The present invention derives the target cost function through the maximum likelihood estimation method based on the arrival phase difference positioning model, transforms the problem of solving the target source position into a cost function optimization problem with the target source position as a variable, and uses the target cost function as the fitness function of the improved whale optimization algorithm. The target source position is solved by the improved whale optimization algorithm, which has high positioning accuracy and reduces the calculation complexity, saves the cost, and has wider applicability in practice.

[0041] This method transforms the problem of solving the target source position into a cost function optimization problem with the target source position as a variable, and uses the target cost function as the fitness function of the improved whale optimization algorithm. By solving the target source position with the improved whale optimization algorithm, the positioning accuracy is improved while the computational complexity is reduced, saving the cost. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 It is the overall flow chart of the present invention;

[0043] Figure 2 is the positioning cone surface obtained based on the fuzzy phase difference;

[0044] Figure 3 It is a schematic diagram of the positioning principle based on PDOA;

[0045] Figure 4 Flowchart for improving the whale algorithm;

[0046] Figure 5 It is the fitness convergence curve of the method of the present invention and the existing whale optimization algorithm;

[0047] Figure 6 This is a comparison chart of the positioning errors of the method of the present invention and the existing whale optimization algorithm and the Cramer-Rao lower bound. DETAILED DESCRIPTION

[0048] Specific embodiments are given below in conjunction with the accompanying drawings. The specific embodiments are only used to introduce the technical solutions of the present invention in detail and are not intended to limit the protection scope of the present application.

[0049] The present invention provides a multi-station phase difference passive positioning method based on an improved whale optimization algorithm (hereinafter referred to as the method, see Figures 1 to 6 ), including the following steps:

[0050] Step 1, establishing a passive phase difference positioning model;

[0051] Deploy at least three observatories in three-dimensional space. Each observatory is equipped with a long baseline interferometer for measuring the arrival phase difference. Set the position s of the nth observatory n for:

[0052] s n =[x n ,y n ,z n ] T (1)

[0053] The baseline vector b of the long baseline interferometer installed at the nth observation station n for:

[0054] b n = l n [sinα n ,cosα n ,0] T (2)

[0055] In the formula, l n is the baseline length of the long baseline interferometer, α n are the azimuth angles of the long baseline interferometer, respectively;

[0056] The unambiguous theoretical value of the phase difference from the target source to the i-th observation station is:

[0057]

[0058] In the formula, r n =(s t -s n ) / r n is the unit vector pointing from the nth observation station to the target source, s t is the target source position, λ is the signal wavelength; Φ=[φ 1 ,φ 2 ,…,φ n ,…φ N ] Tis the unambiguous theoretical vector of the target arrival phase difference. When the length of the baseline of the long baseline interferometer is greater than half the wavelength, the PDOA ambiguity problem occurs. The ambiguity PDOA theoretical vector is:

[0059] z=mod 2π [φ 1 ,φ 2 ,…,φ n ,…φ N ] T (4)

[0060] In the formula, mod 2π (·) is the 2π modulo operation;

[0061] The PDOA measurement vector is:

[0062]

[0063] z o =z+w n (6)

[0064] In the formula, w n =[w 1 ,w 2 ,…w n ,…,w N ] T represents the noise vector, w n represents the measurement noise of the arrival phase difference at the nth observation station; N represents the number of observation stations, that is, the number of long-baseline interferometers; the fuzzy arrival phase difference measured by the long-baseline interferometer is used to obtain the signal incident direction angle θ through the interferometer direction finding principle. n , this direction angle θ n It is described by a long baseline interferometer and a ray pointing from the long baseline interferometer to the target source. The multiple cone surfaces generated around the long baseline interferometer can be determined by the incident direction angle, such as Figure 2 As shown in the figure, the relationship between the interferometer direction finding phase difference and the signal incident angle is:

[0065]

[0066] Generally, the target source is located on the ground. By intersecting the positioning cone with the ground, a series of positioning curves can be obtained. Multiple observation stations form different positioning curve clusters. In addition to intersecting at the target source location, these curves may also produce multiple fuzzy positioning points at other locations. However, only at the target source location, the number of intersecting positioning curves is the same as the total number of observation stations. Therefore, when the number of observation stations is not less than 3, the location of the target source can be determined, such as Figure 3 As shown:

[0067] Step 2, establishing the objective function based on maximum likelihood estimation;

[0068] Assume that the measurement noise of the arrival phase difference is w n If it conforms to the Gaussian distribution, then the noise vector w n =z o -z follows a Gaussian distribution with mean 0 and variance σ 2 , then the maximum likelihood function is:

[0069]

[0070] To simplify the calculation, assume that the noise is independent and identically distributed, and the noise is σ 2 , when the number of observation stations is N, the maximum likelihood function is:

[0071]

[0072] From formula (9), we can get that solving its maximum value is to solve the minimum value of the following formula:

[0073] (x,y,z)=arg{min[mod 2π ((zz 0 ) T (zz 0 ))]} (10)

[0074] Then the objective function is established as:

[0075] J(s t )=mod 2 π((zz 0 ) T (zz 0 )) (11)

[0076] In the formula, represents the estimated target source position;

[0077] Step 3, improving the existing whale optimization algorithm to obtain an improved whale optimization algorithm;

[0078] First, initialize the population, set the maximum number of iterations, and the search range;

[0079] Calculate the initialization individual fitness values ​​of the whales respectively, save the current optimal individual fitness value and position of the whales, and enter the position iterative update process;

[0080] The whale optimization algorithm has three position update mechanisms, namely, encircling the prey, bubble net attack, and random search mechanism:

[0081] (1) Encirclement stage: The whale group takes the best whale position in the group as the target, and other whales update their positions using the following formula:

[0082] X(t+1)=X best (t)-A·D 1 (12)

[0083] Where: t is the number of iterations; D 1 =|C·X best (t)-X(t)| is the bracketing step length; X(t) represents the position vector at the tth iteration; X best (t) represents the optimal position vector of the group at the tth iteration, and other whales will move closer to this position; A and C are coefficient vectors, and the calculation formula is:

[0084] A=2a·ra(13)

[0085] C=2·r(14)

[0086] Where r is a random number in [0,1]; a is the improved adaptive convergence factor. The adaptive convergence factor calculation formula for each individual is as follows:

[0087]

[0088] (2) Bubble Net Attack (Development Phase): The whale approaches its prey along the bubble spiral line, and the position update formula is:

[0089] X(t+1)=D 2 ·e bl ·cos(2πl)+X best (t)

[0090] Where: D 2 =|X best (t)-X(t)|, b is the control helix constant; l is a random value in [-1,1].

[0091] (3) Prey search phase: When the coefficient vector |A|≥1, it indicates that the whale is swimming outside the shrinking encirclement. At this time, the individual whale conducts random search, and the position update formula is:

[0092] X(t+1)=X rand (t)-A·D 3 (16)

[0093] D 3 =|C·X rand (t)-X(t)| (17)

[0094] In the formula, X rand (t) is a position vector randomly selected from the current population.

[0095] Each whale determines which position update mechanism to execute through the probability parameter p and the coefficient vector A. The expression of the improved probability parameter threshold h is as follows:

[0096]

[0097] If p≥h, the position of the whale individual is updated according to the bubble net attack formula; if p<h, it is judged whether the value of |A| is greater than 1; if A≥1, the position of the whale individual is updated according to the random search for prey formula; if A<1, the position of the whale is updated according to the formula for surrounding the prey; then, the fitness is calculated to judge whether the set number of iterations is reached. If so, the result is output; if not, the corresponding position update mechanism is continued according to the probability parameter q and the coefficient vector A.

[0098] Step 4: Solve the objective function using the improved whale optimization algorithm, set the solution range of the target source position, set the target source position as the whale individual in each iteration of the improved whale optimization algorithm, substitute it into Equation (11) to calculate the individual fitness value, and iteratively update the individual position according to the position update formula. After reaching the set number of iterations, the target source position is the position coordinate of the whale individual with the lowest fitness value.

[0099] Embodiment

[0100] In this embodiment, taking 4 observation stations (equipped with long baseline interferometers) as an example, the multi-station phase difference passive positioning method based on the improved whale optimization algorithm is described.

[0101] Taking any point on the ground as the origin, a three-dimensional coordinate system is established. The positions of the four long baseline interferometers are respectively: s1 = [198.2, -151.3, 21.8] T 、s2 = [33.1, 264.6, -87.3] T 、s3 = [35.1, -94.6, 13.9] T and s4 = [162.8, 13.9, 152.1] T ; The lengths of the baselines of the four long baseline interferometers are all set to 1, and the direction vectors are respectively: l 1 = [-0.1023, 1.1019, 0] T 、l 2 = [0.9758, -0.0014, 0] T 、l 3 = [0.4639, -0.7215, 0] T and l 4 = [0.1992, -0.9675, 0] T The carrier frequency of the target signal is 3.5 GHz; assume that the upper bound of the unknown source position is [1000, 1000, 1000] T, the lower bound is [-300,-300,-300] T , the coordinates of the target position s t =[617,395,0] T ,The number of Monte Carlo simulations is set to 1000 times.

[0102] The present invention uses the improved whale optimization algorithm and the existing whale optimization algorithm for simulation experiment comparison; the whale group size is 100, and the maximum number of iterations is set to 300 times. The whale group position is initialized within the target source setting range, and the individual fitness value is calculated by formula (11). The individual position is iteratively updated according to the position update formula. After reaching the set number of iterations, the optimal result of the target source position is obtained, and the positioning is completed. The flowchart of the improved whale optimization algorithm is shown in the figure. Figure 4 , the fitness convergence curve of the algorithm is as follows Figure 5 As shown in the figure, it can be seen that the improved whale optimization algorithm converges to the optimal fitness in fewer iterations than the whale optimization algorithm.

[0103] The Cramer-Rao lower bound (CRLB) of the positioning model is derived as a test indicator, as follows:

[0104]

[0105] Where Q represents the covariance matrix of measurement noise;

[0106] z pairs (s t ) T The partial derivative of is expressed as:

[0107]

[0108] The target source position is obtained by Monte Carlo simulation, and different noise disturbances are added. The root mean square error (RMSE) and Cramer-Rao lower bound of the target source position obtained by the improved whale optimization algorithm and the whale optimization algorithm are compared. Figure 6 As shown in the figure, it can be seen that under different noise conditions, the improved whale optimization algorithm can reach the Cramer-Rao lower bound, while the existing whale optimization algorithm cannot reach the Cramer-Rao lower bound under the condition of 25° noise or above. Figure 5 and Figure 6 It can be seen that the present invention can converge to the optimal fitness value within a relatively small number of iterations and has good positioning performance under different noise conditions.

[0109] Any matters not described in the present invention are applicable to the prior art.

Claims

1. A multi-station phase difference passive positioning method based on an improved whale optimization algorithm, characterized in that: The method includes the following steps: Step 1, establish a passive phase difference positioning model; Deploy at least three observatories in three-dimensional space. Each observatory is equipped with a long baseline interferometer for measuring the arrival phase difference. Set the position s of the nth observatory n for: s n =[x n ,y n ,z n ] T (1) The baseline vector b of the long baseline interferometer installed at the nth observation station n for: b n =l n [sinα n ,cosα n ,0] T (2) In the formula, l n is the baseline length of the long baseline interferometer, α n are the azimuth angles of the long baseline interferometer, respectively; The unambiguous theoretical value of the phase difference from the target source to the i-th observation station is: In the formula, r n =(s t -s n ) / r n is the unit vector pointing from the nth observation station to the target source, s t is the target source position, r n =||s t -s n || is the straight-line distance from the target source to the nth observation station, λ is the signal wavelength; Φ=[φ1,φ2,…,φ n …φ N ] T is the unambiguous theoretical vector of the target arrival phase difference, and the fuzzy PDOA theoretical vector is: z=mod 2π [φ1,φ2,…,φ n …f N ] T (4) In the formula, mod 2π (·) is the 2π modulo operation; The PDOA measurement vector is: With o =z+w n (6) In the formula, w n =[w1,w2,…w n ,…,w N ] T represents the noise vector, w n represents the measurement noise of the arrival phase difference at the nth observation station, and N represents the number of observation stations; Step 2, establish an objective function based on maximum likelihood estimation; Assume that the measurement noise of the arrival phase difference is w n If it conforms to Gaussian distribution, then the noise vector w n =z o -z follows a Gaussian distribution with mean 0 and variance σ 2 , then the maximum likelihood function is: When the number of observation stations is N, the maximum likelihood function is: Solving the maximum value of Equation (9) is to solve the minimum value of Equation (10): (x,y,z)=arg{min[mod 2π ((zz 0 ) T (z 0 ))]} (10) Then the objective function is: J(s t )=mod 2π ((z-z 0 ) T (z-z 0 )) (11) In the formula, represents the estimated target source position; Step 3, improve the whale optimization algorithm to obtain an improved whale optimization algorithm; Step 4, use the improved whale optimization algorithm to solve the objective function. Set the position of the target source as an individual in the whale population. Calculate the fitness value of each whale individual after each iteration according to the above formula. Adjust the whale position according to the fitness value. After reaching the set number of iterations, the position of the individual with the minimum fitness value is the position of the target source.

2. According to claim 1, the multi-station phase difference passive positioning method based on the improved whale optimization algorithm is characterized in that: The improved whale optimization algorithm includes the following: (1) Encircling prey stage: The whale group takes the position of the optimal whale in the population as the target, and other whales update their positions through the following formula: X(t+1)=X best (t)-A·D1(12) Where: t is the number of iterations; D1 = |C·X best (t)-X(t)| is the bracketing step length; X(t) represents the position vector at the tth iteration; X best (t) represents the optimal position vector of the group at the tth iteration, and other whales will move closer to this position; A and C are coefficient vectors, and the calculation formula is: A = 2a·r - a (13) C＝2·r(14) Where: r is a random number in [0, 1]; a is the improved adaptive convergence factor, and the calculation formula for the adaptive convergence factor of each individual is as follows: (2) Bubble net attack: The whale approaches the prey along the bubble spiral, and the position update formula is: X(t+1)=D2·e bl ·cos(2πl)+X best (t) Where: D2=|X best (t)-X(t)|, b is the control helix constant; l is a random value in [-1,1]; (3) Searching for prey stage: When the coefficient vector |A| ≥ 1, it indicates that the whale swims outside the shrinking encirclement. At this time, the whale individual conducts a random search, and the position update formula is X(t+1)=X rand (t)-A·D3 (16) D3=|C·X rand (t)-X(t)| (17) Where, X rand (t) is a position vector randomly selected from the current population; Each whale decides which position update mechanism to execute through the probability parameter p and the coefficient vector A. Then the expression of the improved probability parameter threshold h is: If p ≥ h, update the whale individual position according to the bubble net attack formula; if p < h, judge whether the value of |A| is greater than 1; if A ≥ 1, update the whale individual position according to the random search for prey formula; if A < 1, update the whale position according to the encircling prey formula; then, calculate the fitness and judge whether the set number of iterations is reached. If so, output the result; if not, continue to perform the corresponding position update mechanism according to the probability parameter q and the coefficient vector A.

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