CN107526073B - Motion multi-station passive time difference and frequency difference combined positioning method - Google Patents

Motion multi-station passive time difference and frequency difference combined positioning method Download PDF

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CN107526073B
CN107526073B CN201710722272.6A CN201710722272A CN107526073B CN 107526073 B CN107526073 B CN 107526073B CN 201710722272 A CN201710722272 A CN 201710722272A CN 107526073 B CN107526073 B CN 107526073B
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蒋伊琳
刘梦楠
陈涛
郜丽鹏
屈天开
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Harbin Engineering University
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    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
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Abstract

The invention discloses a passive time difference and frequency difference combined positioning method for a plurality of moving stations, and belongs to the technical field of passive positioning. The method comprises the following steps: establishing a time difference positioning model; establishing a frequency difference positioning model; constructing a time difference and frequency difference observation matrix1Designing a fitness function; initializing a population and various parameters; evaluating a fitness function value of each particle; sorting all the particles; and outputting the current global optimal value when the algorithm meets the termination condition: reconstructing a time difference frequency difference matrix2(ii) a Obtaining a weighted least squares solution as θ2And its covariance matrix cov (theta)2) (ii) a And (5) calculating the position and the speed of the radiation source. The invention carries out optimal value solution on the fitness function obtained by the time difference frequency difference observation matrix, combines the particle swarm optimization algorithm and the least square algorithm, can realize high-precision positioning on the target under the condition of four base stations, and simultaneously solves the speed information of the target. The invention can realize high-precision estimation of the position of the radiation source, is not limited by the site layout and has higher positioning estimation precision.

Description

Motion multi-station passive time difference and frequency difference combined positioning method
Technical Field
The invention belongs to the technical field of passive positioning, and particularly relates to a motion multi-station passive time difference and frequency difference combined positioning method.
Background
The passive positioning technology has important application prospect in the field of electronic countermeasure, and draws wide attention in various fields such as navigation, spaceflight, military reconnaissance, global satellite navigation and the like. The passive positioning technology has the characteristics of high positioning precision, long acting distance, strong battlefield survival ability and the like, and can quickly realize the accurate positioning and track tracking of the target. In recent years, the time difference and frequency difference combined positioning technology is rapidly developed and perfected, and compared with other positioning systems, the time difference and frequency difference combined positioning technology is more suitable for a multi-base-station cooperative positioning system so as to complete accurate positioning of a target. For a moving radiation source, information such as time difference of arrival (TDOA) and frequency difference of multiple arrival (FDOA) is included in an information echo received by a base station, the position and the speed of a target can be estimated more accurately by using an increased observed quantity, and the method has the advantages of good real-time performance, wide detection range, high positioning accuracy and the like. At present, a nonlinear equation system in a TDOA/FDOA multi-station cooperative positioning system easily causes deviation of a positioning result, and has the problems of large calculation amount, high calculation complexity and the like, and corresponding opinions are provided for many scholars who prevent the positioning result from falling into local optimization and accelerate the convergence speed of an algorithm. For example, the method proposed by Yangjie et al (the university of Western electronic technology, vol. 42, No. 4, 2015, iterative time difference and frequency difference joint localization algorithm and performance analysis thereof) is to perform Taylor series expansion by using a time difference and frequency difference observation matrix, and obtain a target position and speed by an iterative method when the algorithm meets the convergence condition. The method can effectively reduce the calculated amount, but in the data processing process, the measurement error needs to obey Gaussian distribution, otherwise the positioning accuracy is obviously reduced.
Disclosure of Invention
The invention aims to provide a motion multi-station passive time difference and frequency difference combined positioning method which solves the problem of nonlinear optimization in a motion multi-station passive positioning system, realizes high-precision positioning of a radiation source under the condition of less base stations and can calculate the current speed of the motion radiation source.
The purpose of the invention is realized by the following technical scheme:
a passive time difference and frequency difference combined positioning method for a plurality of moving stations comprises the following steps:
(1) establishing a time difference positioning model, respectively determining the coordinate positions of the main station and the auxiliary station, and calculating the real distance d between the target and each base stationiI 1,2, M, the distance d between the target and the master station is derived1And the distance between the target and the secondary stationFrom di(i ≠ 1), M base stations can form M-1 group distance differences di1,(i≠1)；
(2) Establishing a frequency difference positioning model, and determining the real distance d between the target and each base stationiObtaining the distance change rate by derivation, obtaining the time derivative in the time difference equation to obtain the distance difference change rate, the noise obedience mean value is 0, and the variance is sigma2(ii) a gaussian distribution of;
(3) introducing an auxiliary variable d1And
Figure BDA0001385229070000021
constructing a time difference and frequency difference observation matrix1Considering the influence of the error on the positioning precision and designing a fitness function;
(4) initializing a population and various parameters, setting the scale of the population, the dimension of a problem, a maximum iteration acceleration constant and a search area range, and randomly generating the speed and the position of a particle:
according to the above, the dimension D of the problem is set to 6, the position and velocity of the particles are initialized randomly in the search space of the target potential solution, the number of the particles is set to n 80, the maximum number of iterations m is 100, and the acceleration constant c is set to 801＝c2Maximum weight factor w of 1max0.8, minimum weight factor wmin＝0.2；
(5) Evaluating the fitness function value of each particle, and setting a parameter w of linear decreasing weightmaxAnd wminMeanwhile, the speed and the position of the particles are updated, the fitness function values of all the particles are compared with the fitness function value of the best particle, an individual extreme value and a global extreme value are obtained according to the fitness function values, and the global extreme value is updated;
(6) all particles are ordered according to a natural selection mechanism, the speed and position of the worst half of the particles in the population are replaced by the speed and position of the best half of the particles in the population, while the individual memory of each particle is retained:
the natural selection is that the fitness function values of all the particles are sorted from low to high, the speed and the position of the front 1/2 better particle in the population are replaced by the speed and the position of the rear 1/2 worse particle, and meanwhile, the historical optimal value of each particle is reserved;
(7) and outputting the current global optimal value when the algorithm meets the termination condition:
when the algorithm reaches the maximum iteration times or meets the preset precision, stopping searching and outputting a result, otherwise, returning to the step (5) to continue searching the target position;
(8) using the searched target position as the initial target position theta, and using the covariance matrix QαConstructing a weighting matrix W for known conditions1And covariance matrix cov (theta)1) Reconstructing a time difference frequency difference matrix according to the obtained radiation source estimated value theta2；
(9) It is known that2Is about theta2To obtain a weighted least squares solution of theta2And its covariance matrix cov (theta)2)；
(10) From the resulting weighted least squares solution to θ2And (5) calculating the position and the speed of the radiation source.
The invention has the beneficial effects that:
the method and the device solve the optimal value of the fitness function obtained by the time difference frequency difference observation matrix, and can effectively avoid the problems of large calculation amount and high calculation complexity of the time difference frequency difference observation matrix.
The invention combines the particle swarm optimization algorithm and the least square algorithm, can realize high-precision positioning of the target under the condition of four base stations, and simultaneously obtains the speed information of the target.
The invention can realize high-precision estimation of the position of the radiation source, can quickly approach to the global optimal solution without relying on the prior condition of the initial target position, and simultaneously accelerates the convergence speed of the algorithm, thereby being difficult to have the situations of positioning ambiguity and no solution.
The invention is not limited by the site layout, can carry out global search on the target area under the specific site layout, and has higher positioning estimation precision compared with the traditional method.
Drawings
FIG. 1 is a time difference/Doppler frequency difference positioning schematic;
FIG. 2 is a plot comparing the position and velocity root mean square error of the method of the present invention with a particle swarm optimization algorithm and a standard particle swarm algorithm;
FIG. 3 is a graph comparing the position and velocity deviations of the method of the present invention with a particle swarm optimization algorithm and a standard particle swarm algorithm;
figure 4 is a graph comparing the mean square position and velocity deviation for three different site layouts according to the method of the present invention.
Detailed Description
The following further describes embodiments of the present invention with reference to the accompanying drawings:
a passive time difference and frequency difference combined positioning method for a plurality of moving stations comprises the following steps:
step 1: and establishing a time difference positioning model, and respectively determining the coordinate positions of the main station and the auxiliary station. Calculating the real distance d between the target and each base stationiI 1,2, M, the distance d between the target and the master station is derived1And the distance d between the target and the secondary stationi(i ≠ 1), M base stations can form M-1 group distance differences di1,(i≠1)。
Fig. 1 shows a time difference/doppler difference positioning model, where M motion receivers are placed in the same three-dimensional space, where the coordinate position and velocity of the target to be estimated are u ═ x, y, z, respectively]TAnd
Figure BDA0001385229070000031
the coordinate position of the receiver being si＝[xi,yi,zi]TAt a speed of
Figure BDA0001385229070000032
T is the transpose of the matrix, representing the true value of the measured variable (°). The case of 4 receivers is considered here, but the positions of 4 receivers cannot be located on one plane or the same straight line so as to ensure the uniqueness of the solution target position.
Selecting the first receiver as the reference receiver, the radiation source u and the receiver siIs a distance of
Figure BDA0001385229070000033
In the formula, | | is 2 norm, M receivers can form M-1 group distance difference, and then the target and the receiver siAnd target and reference receivers s1The difference in distance between
di1＝cτi1＝di-d1(2)
Where c is the propagation velocity of the light, τi1Representing the time difference of arrival. After the formula (2) is subjected to term shifting, the method is obtained
di1+d1＝di(3)
Substituting the formula (1) into the formula (3), and simultaneously performing square shift terms on two sides to obtain a new group of time difference equations
di1 2+2di1d1＝si Tsi-s1 Ts1-2(si-s1)Tu (4)
Step 2: establishing a frequency difference positioning model, and determining the real distance d between the target and each base stationiObtaining the distance change rate by derivation, obtaining the time derivative in the time difference equation to obtain the distance difference change rate, the noise obedience mean value is 0, and the variance is sigma2Gaussian distribution of (c).
The expression for deriving the rate of change of distance from equation (1) is
Figure BDA0001385229070000041
Then the target goes to the receiver siAnd receiver s1The rate of change of the distance difference therebetween is expressed as
Figure BDA0001385229070000042
Therefore, the time derivative in equation (4) is obtained
Figure BDA0001385229070000043
Let d ═ d21,d31,...,dM1,]TFor the distance difference containing the noise,
Figure BDA0001385229070000044
for the rate of change of the range difference including noise, do＝[d21 o,d31 o,...,dM1 o]TThe true value of the distance difference.
Figure BDA0001385229070000045
For the true value of the rate of change of the range difference, the TDOA-related and FDOA-related measurement models can be expressed as
d＝cτ＝do+n (8)
Figure BDA0001385229070000046
Wherein n ═ n21,n31,...,nM1]TAnd
Figure BDA0001385229070000047
respectively the corresponding measurement noise.
τ＝[τ21,τ31,...,τM1]TFor time difference of arrival, Fd＝[fd21,fd31,...,fdM1,]TDoppler frequency difference, f0Is the radiation source frequency. Two groups of noise variables are independently distributed and the mean value is zero, and the two groups of noise variables are recorded
Figure BDA0001385229070000048
Having a covariance matrix of
E[ααT]＝Qα(10)
And step 3: introducing an auxiliary variable d1And
Figure BDA0001385229070000049
constructing a time difference and frequency difference observation matrix1And considering the influence of the error on the positioning precision and designing a fitness function.
D is known from the equation of time difference and frequency differencei1°＝di1-ni1Thus, it can be obtained from the formula (4)
t,i＝di1 2+2di1d1-si Tsi+s1 Ts1+2(si-s1)Tu (11)
Note the bookt＝[t,1,t,2,...,t,M]TConstructing a moveout equation matrix using equation (11)
t＝ht-Gtθ1(12)
Wherein
Figure BDA0001385229070000051
Figure BDA0001385229070000052
Figure BDA0001385229070000053
Where 0 is a zero matrix of 3 rows and 1 column, θ1Including the position u and velocity of the target to be estimated
Figure BDA0001385229070000054
The equation of time difference and frequency difference location can be known
Figure BDA0001385229070000055
Thus, according to the formula (7), it can be obtained
Figure BDA0001385229070000056
Note the bookf＝[f,1,f,2,...,f,M]TConstructing a matrix of equations relating to frequency differences using equation (16)
f＝hf-Gfθ1(17)
Wherein
Figure BDA0001385229070000057
Figure BDA0001385229070000058
H is obtained from the equations (12) and (17)1＝[ht,hf]T，G1＝[Gt,Gf]TThe time difference and frequency difference joint estimation observation matrix can be expressed as
Figure BDA0001385229070000059
In the formula (20)1The matrix is a matrix with 6 rows and 1 column, in order to estimate the initial position and the speed of the target by utilizing the particle swarm optimization algorithm, the numerical value of a fitness function in the particle swarm optimization algorithm needs to be obtained, and therefore the obtained fitness function is
fit＝||h1-G1θ1|| (21)
And 4, step 4: initializing a population and various parameters, setting the scale of the population, the dimension of a problem, the maximum iteration acceleration constant and the search area range, and randomly generating the speed and the position of the particles.
According to the above, the dimension D of the problem is set to 6, the position and velocity of the particles are initialized randomly in the search space of the target potential solution, the number of the particles is set to n 80, the maximum number of iterations m is 100, and the acceleration constant c is set to 801＝c2Maximum weight factor w of 1max0.8, minimum weight factor wmin＝0.2。
And 5: evaluating fitness function for each particleValue, setting the parameter w of linearly decreasing weightmaxAnd wminAnd updating the speed and the position of the particles, comparing the fitness function values of all the particles with the best fitness function value of the particles, obtaining an individual extreme value and a global extreme value according to the fitness function values, and updating the global extreme value.
Substituting the initialized position of each particle into a fitness function fit, wherein each corresponding particle corresponds to a fitness function value, and storing the fitness value and the current position of each particle into an individual extreme value pbestIn the method, the best individual position and fitness value in all the individual extrema are stored into a global extremum gbestIn (1). Defining a global extreme value among all the individual extreme values, and then combining the fitness function value of each particle with the global extreme value gbestComparing if the fitness function fit (i) of the ith particle is better than gbestThen the global extreme value g is setbestReplaced with fit (i).
The position and velocity of each particle is updated using equation (22).
Figure BDA0001385229070000061
In the formula pbestFor individual optimum, gbestFor a global optimum, vi,jFlight speed, x, representing the particle swarmi,jRepresenting the current position of the particle. The expression of linearly decreasing inertial weight is
w＝wmax-(t)×(wmax-wmin)/m (23)
Step 6: all particles are ordered according to a natural selection mechanism, replacing the speed and position of the worst half of the particles in the population with the speed and position of the best half of the particles in the volume. While preserving individual memory of each particle.
The natural selection is that the fitness function values of all the particles are sorted from low to high, the speed and the position of the front 1/2 better particle in the population are replaced by the speed and the position of the rear 1/2 worse particle, and meanwhile, the historical optimal value of each particle is reserved.
And 7: and outputting the current global optimal value when the algorithm meets the termination condition.
And when the algorithm reaches the maximum iteration times or meets the preset precision, stopping searching and outputting a result. Otherwise, returning to the step 5 to continue searching the target position.
And 8: using the searched target position as the initial target position theta, and using the covariance matrix QαConstructing a weighting matrix W for known conditions1And covariance matrix cov (theta)1) Reconstructing a time difference frequency difference matrix according to the obtained radiation source estimated value theta2。
When Q isαWeighting matrix W for known conditions1Is expressed as
Figure BDA0001385229070000071
Wherein
Figure BDA0001385229070000072
In the formula (26)
Figure BDA0001385229070000073
P is a zero matrix of appropriate dimensions.
Thus, according to G1And W1Can obtain theta1Has a covariance matrix of
cov(θ1)＝(G1 TW1G1)-1(26)
Reconstructing a time difference frequency difference matrix according to the obtained radiation source estimated value theta to obtain
2＝h2-G2θ2(27)
Figure RE-GDA0001430253610000074
Figure BDA0001385229070000075
Figure RE-GDA0001430253610000076
In formula (29), I is a unit matrix of 3 rows and 3 columns, O is an all-zero matrix of 3 rows and 3 columns, 1 is an all-1 matrix of 3 rows and 1 column, and 0 is an all-zero matrix of 3 rows and 1 column, which indicates dot multiplication.
And step 9: it is known that2Is about theta2To obtain a weighted least squares solution of theta2And its covariance matrix cov (theta)2)。
It is known that the equations (27) to (30) are a function of θ2Of a weighted least squares solution of
Figure BDA0001385229070000078
Thus with respect to theta2Has a covariance matrix of
Figure BDA0001385229070000081
Wherein the weighting matrix W2Is expressed as
Figure BDA0001385229070000082
B2Is expressed as
Figure BDA0001385229070000083
Step 10: from the resulting weighted least squares solution to θ2And (5) calculating the position and the speed of the radiation source.
According to equation (31), the target position and velocity of the radiation source are defined as
Figure BDA0001385229070000084
Figure BDA0001385229070000085
Wherein pi ═ diag { sgn (θ)1(1:3)-s1) To avoid sign ambiguity resulting from square root operations,/denotes dot division.
Example (b):
FIG. 2 is a plot comparing the root mean square error of position and velocity for the method of the present invention with particle swarm optimization and standard particle swarm optimization, where the time error follows a Gaussian distribution. As can be seen from the figure, with the increase of the measurement error, the method has higher positioning accuracy, can achieve higher positioning accuracy without the estimation of the target initial position, can avoid falling into the local optimum, and approaches to the global optimum solution with faster convergence speed.
FIG. 3 is a graph comparing the position and velocity deviations of the method of the present invention with a particle swarm optimization algorithm and a standard particle swarm optimization algorithm, wherein the time error follows a Gaussian distribution. The graph shows that the method still has higher positioning accuracy of the position and speed deviation of the target under the condition of high signal-to-noise ratio compared with the other two methods.
Figure 4 is a graph comparing the root mean square error of the mean square position and velocity for three different site layouts. As can be seen from the figure, the position positioning accuracy of the star shape and the inverted triangle is basically kept consistent with the increase of the measurement noise, and the position positioning accuracy of the parallelogram is optimal. The change of the site layout can effectively improve the positioning precision of the position of the radiation source without greatly influencing the speed of the radiation source.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (8)

1. A passive time difference and frequency difference combined positioning method for a plurality of moving stations is characterized by comprising the following steps:
(1) establishing a time difference positioning model, respectively determining the coordinate positions of the main station and the auxiliary station, and calculating the real distance d between the target and each base stationiI 1,2, M, the distance d between the target and the master station is derived1And the distance d between the target and the secondary stationiI ≠ 1, M base stations can form M-1 group distance differences di1,i≠1；
(2) Establishing a frequency difference positioning model, and determining the real distance d between the target and each base stationiObtaining the distance change rate by derivation, obtaining the time derivative in the time difference equation to obtain the distance difference change rate, the noise obedience mean value is 0, and the variance is sigma2(ii) a gaussian distribution of;
(3) introducing an auxiliary variable d1And
Figure FDA0002571666550000011
constructing a time difference and frequency difference observation matrix1Considering the influence of the error on the positioning precision and designing a fitness function;
(4) initializing a population and various parameters, setting the scale of the population, the dimension of a problem, a maximum iteration acceleration constant and a search area range, and randomly generating the speed and the position of a particle;
according to the above, the dimension D of the problem is set to 6, the position and velocity of the particles are initialized randomly in the search space of the target potential solution, the number of the particles is set to n 80, the maximum number of iterations m is 100, and the acceleration constant c is set to 801＝c2Maximum weight factor w ═ 1max0.8, minimum weight factor wmin＝0.2；
(5) Evaluating the fitness function value of each particle, and setting a parameter w of linear decreasing weightmaxAnd wminMeanwhile, the speed and the position of the particles are updated, the fitness function values of all the particles are compared with the fitness function value of the best particle, an individual extreme value and a global extreme value are obtained according to the fitness function values, and the global extreme value is updated;
(6) all particles are sorted according to a natural selection mechanism, the speed and position of the worst half of the particles in the population are replaced by the speed and position of the best half of the particles in the population, while the individual memory of each particle is retained:
the natural selection is that the fitness function values of all the particles are sorted from low to high, the speed and the position of the front 1/2 better particle in the population are replaced by the speed and the position of the rear 1/2 worse particle, and meanwhile, the historical optimal value of each particle is reserved;
(7) and outputting the current global optimal value when the algorithm meets the termination condition:
when the algorithm reaches the maximum iteration times or meets the preset precision, stopping searching and outputting a result, otherwise, returning to the step (5) to continue searching the target position;
(8) using the searched target position as the initial target position theta, and using the covariance matrix QαConstructing a weighting matrix W for known conditions1And covariance matrix cov (theta)1) Based on the obtained radiation source estimation value theta1Reconstructing a time difference frequency difference matrix2；
(9) It is known that2Is about theta2To obtain a weighted least squares solution of theta2And its covariance matrix cov (theta)2)；
(10) From the resulting weighted least squares solution to θ2And (5) calculating the position and the speed of the radiation source.
2. The method for passive moveout-of-frequency joint positioning according to claim 1, wherein the step (1) comprises:
placing M motion receivers in the same three-dimensional space, wherein the coordinate position and the speed of an object to be estimated are respectively u ═ x, y, z]TAnd
Figure FDA0002571666550000021
the coordinate position of the receiver being si＝[xi,yi,zi]TAt a speed of
Figure FDA0002571666550000022
T is momentTransposing of arrays, using ()οThe true value representing the measured variable (#);
consider the case of 4 receivers, but the positions of the 4 receivers cannot lie on a plane or on the same straight line;
selecting the first receiver as a reference receiver, the coordinate position u of the object to be estimated and the coordinate position s of the receiveriIs a distance of
Figure FDA0002571666550000023
In the formula, if | is | · | | is a 2 norm, M receivers can form M-1 group distance differences, and then the coordinate position s of the target and the receiveriAnd target and reference receivers s1The difference in distance between
di1＝cτi1＝di-d1(2)
Where c is the propagation velocity of the light, τi1Expressing the arrival time difference, and obtaining the result after shifting the term of the formula (2)
di1+d1＝di(3)
Substituting the formula (1) into the formula (3), and simultaneously performing square shift terms on two sides to obtain a new group of time difference equations
di1 2+2di1d1＝si Tsi-s1 Ts1-2(si-s1)Tu (4)。
3. The method for the passive moveout-of-time frequency difference joint positioning of the moving multi-station as claimed in claim 1 or 2, wherein the step (2) specifically comprises:
the expression for deriving the rate of change of distance from equation (1) is
Figure FDA0002571666550000024
The coordinate position s of the object to the receiveriAnd reference receptionMachine s1The rate of change of the distance difference therebetween is expressed as
Figure FDA0002571666550000025
Therefore, the time derivative in equation (4) is obtained
Figure FDA0002571666550000026
Let d ═ d21,d31,...,dM1,]TFor the distance difference containing the noise,
Figure FDA0002571666550000027
for the rate of change of range difference including noise, do＝[d21 o,d31 o,...,dM1 o]TAs the true value of the distance difference,
Figure FDA0002571666550000028
for the true value of the rate of change of the range difference, the TDOA-related and FDOA-related measurement models can be expressed as
d＝cτ＝do+n (8)
Figure FDA0002571666550000031
Wherein n ═ n21,n31,...,nM1]TAnd
Figure FDA0002571666550000032
respectively corresponding measurement noise, [ tau ] - [ tau ]21,τ31,...,τM1]TFor time difference of arrival, Fd＝[fd21,fd31,...,fdM1,]TDoppler frequency difference, f0Is the radiation source frequency; two groups of noise variables are independently distributed and the mean value is zero, and the two groups of noise variables are recorded
Figure FDA0002571666550000039
Having a covariance matrix of
E[ααT]＝Qα(10)。
4. A method for passive moveout-of-frequency-difference joint positioning of multiple stations in motion according to claim 1 or 3, wherein the step (3) comprises:
d is known from the equation of time difference and frequency differencei1 ο＝di1-ni1Thus, it can be obtained from the formula (4)
t,i＝di1 2+2di1d1-si Tsi+s1 Ts1+2(si-s1)Tu (11)
Note the bookt＝[t,1,t,2,...,t,M]TConstructing a moveout equation matrix using equation (11)
t＝ht-Gtθ1(12)
Wherein
Figure FDA0002571666550000033
Figure FDA0002571666550000034
Figure FDA0002571666550000035
Where 0 is a zero matrix of 3 rows and 1 column, θ1Including the coordinate position u and the velocity of the target to be estimated
Figure FDA0002571666550000036
The equation of time difference and frequency difference location can be known
Figure FDA0002571666550000037
Thus, according to the formula (7), it can be obtained
Figure FDA0002571666550000038
Note the bookf＝[f,1,f,2,...,f,M]TConstructing a matrix of equations relating to frequency differences using equation (16)
f＝hf-Gfθ1(17)
Wherein
Figure FDA0002571666550000041
Figure FDA0002571666550000042
H is obtained from the equations (12) and (17)1＝[ht,hf]T，G1＝[Gt,Gf]TThe time difference and frequency difference joint estimation observation matrix can be expressed as
Figure FDA0002571666550000043
In the formula (20)1The matrix is a matrix with 6 rows and 1 column, in order to estimate the initial position and the speed of the target by utilizing the particle swarm optimization algorithm, the numerical value of a fitness function in the particle swarm optimization algorithm needs to be obtained, and therefore the obtained fitness function is
fit＝||h1-G1θ1|| (21)。
5. The method for the passive moveout-of-time frequency difference joint positioning of the moving multi-station as claimed in claim 1 or 4, wherein the step (5) comprises:
the position of each particle after initialization is substituted into the fitness function fit, and each particle corresponds to oneA fitness function value, storing the fitness value and the current position of each particle into an individual extreme value pbestIn the method, the optimal individual position and fitness value in all the individual extrema are stored into a global extremum gbestPerforming the following steps;
defining a global extreme value among all the individual extreme values, and then combining the fitness function value of each particle with the global extreme value gbestComparing if the fitness function fit (i) of the ith particle is better than gbestThen the global extreme value g is setbestReplaced with fit (i);
the position and velocity of each particle is updated using equation (22):
Figure FDA0002571666550000044
the expression of linearly decreasing inertial weight is
w＝wmax-(t)×(wmax-wmin)/m (23)
In the formula vi,jFlight speed, x, representing the particle swarmi,jRepresenting the current position of the particle.
6. A method for passive moveout-of-frequency-difference joint positioning of multiple stations in motion according to claim 1 or 5, wherein the step (8) comprises:
when Q isαWeighting matrix W for known conditions1Is expressed as
Figure FDA0002571666550000051
Wherein
Figure FDA0002571666550000052
Wherein B is 2diag (d) in the formula (26)2 o,d3 o,...,dM o)，
Figure FDA0002571666550000053
P is a zero matrix of appropriate dimensions;
thus, according to G1And W1Can obtain theta1Has a covariance matrix of
cov(θ1)＝(G1 TW1G1)-1(26)
According to the obtained radiation source estimated value theta1Reconstructing a time difference and frequency difference matrix to obtain
2＝h2-G2θ2(27)
Figure FDA0002571666550000054
Figure FDA0002571666550000055
Figure FDA0002571666550000056
In formula (29), I is a unit matrix of 3 rows and 3 columns, O is an all-zero matrix of 3 rows and 3 columns, 1 is an all-1 matrix of 3 rows and 1 column, and 0 is an all-zero matrix of 3 rows and 1 column, which indicates dot multiplication.
7. The method for the passive moveout-of-time frequency difference joint positioning of the moving multi-station as claimed in claim 1 or 6, wherein the step (9) comprises:
it is known that the equations (27) to (30) are a function of θ2Of a weighted least squares solution of
Figure FDA0002571666550000057
Thus with respect to theta2Has a covariance matrix of
Figure FDA0002571666550000061
Wherein the weighting matrix W2Is expressed as
Figure FDA0002571666550000062
B2Is expressed as
Figure FDA0002571666550000063
8. The method for the passive moveout-of-time frequency difference joint positioning of the moving multi-station as claimed in claim 1 or 7, wherein the step (10) specifically comprises:
according to equation (31), the target position and velocity of the radiation source are defined as
Figure FDA0002571666550000064
Figure FDA0002571666550000065
Wherein ═ diag { sgn (θ)1(1:3)-s1) To avoid sign ambiguity resulting from square root operations,/denotes dot division.
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