CN115372902B - TDOA bias reduction positioning method based on underwater multi-base sonar - Google Patents
TDOA bias reduction positioning method based on underwater multi-base sonar Download PDFInfo
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Abstract
The invention discloses a TDOA bias reduction positioning method based on underwater multi-base sonar, which comprises the steps of firstly, utilizing observed quantity about arrival time difference of a transmitting station and a receiving station in the multi-base sonar to establish an observation equation, integrating and deforming the observation equation into a pseudo-linear matrix equation with unknown quantity including a target position, and determining a cost function for solving the equation and deforming the equation; then determining a constant constraint condition, constructing a new model with a solving model based on a least square criterion, and solving the new model by solving a generalized eigenvector corresponding to the minimum generalized eigenvalue of a matrix bundle to obtain an initial solution; then determining an error equation of the new model for target source position estimation according to a first-order Taylor series expansion method, and obtaining a solution by utilizing least square; and finally, subtracting the estimation error from the estimation of the target position in the initial solution to obtain the estimation result of the final target position. The method can effectively reduce the bias of the multi-base acoustic TDOA positioning method when the prior error exists in the underwater sound velocity.
Description
Technical Field
The invention relates to the technical field of underwater target source positioning, in particular to a TDOA bias reduction positioning method based on underwater multi-base sonar.
Background
Sonar positioning is an important means of underwater target source positioning. The conventional sonar may be classified into passive sonar and active sonar according to the operation mode. The passive sonar directly receives noise emitted by the underwater target machinery to find the target, and has better concealment. However, with the intensive research of submarine stealth technology, noise emitted by submarines is smaller and smaller, and the positioning performance of passive sonar is obviously reduced. The active sonar positions the target by autonomously transmitting an acoustic signal and then receiving a target echo. The range is relatively long, but the concealment is not strong due to the need to actively transmit signals. Compared with the two sonars, the multi-base sonar transceiver is arranged separately, and the transmitting station can actively transmit signals, so that the multi-base sonar transceiver has the advantages of active sonar, and the receiving station of the multi-base sonar transceiver is operated passively, so that the concealment is good. The multi-base sound has the advantages of good concealment, strong anti-interference capability, high maneuvering performance and long acting distance, and becomes a research hotspot for domestic and foreign scholars.
The multi-base sound positioning principle is that a single or a plurality of transmitting stations transmit sound wave signals, a plurality of receiving stations receive target echoes, and targets are positioned according to parameter information such as time domain, frequency domain, space domain or energy domain and the like obtained from the signals. Such information includes time of arrival, time difference of arrival (TDOA), frequency of arrival, frequency difference of arrival, azimuth of arrival, elevation of arrival, received signal strength, and signal to gain ratio, among others. Based on the above observation information, more and more positioning algorithms are proposed. The present algorithms can be generally classified into two kinds of algorithms, namely a closed solution type algorithm and an iterative type algorithm. The algorithm of closed solution is a specific formula which can be used for solving the target position through formula deduction, the calculation process is more clear, the calculation is simple, the calculated amount is smaller, and therefore a plurality of related algorithms are proposed; the iterative algorithm solves the target position through an iterative process, and compared with a closed-form solution algorithm, the iterative algorithm has more complex calculation process and needs to consider the problem of initial value selection.
Although the algorithm of closed solution is simple in calculation process and low in calculation complexity, the algorithm generally needs to convert formulas according to the relationships among various physical quantities in an observation equation for many times, and many ignored second-order error terms are often generated in the process, so that the positioning bias may be increased. For multi-base sonar positioning, the corresponding observation equation has more physical quantities (including a transmitting station, a receiving station, TDOA observables and sound velocity), more second-order error terms are generated when formula conversion is carried out, and the system is more sensitive to various error quantities, so that the algorithm bias is more required to be reduced. To solve the problem, the patent designs a TDOA bias reduction positioning method based on underwater multi-base sonar to reduce the positioning estimation bias.
Disclosure of Invention
Aiming at the problem of larger bias in the existing underwater multi-base sonar TDOA positioning method, the invention provides an underwater multi-base sonar-based TDOA bias reduction positioning method which can reduce the bias of the existing positioning method.
In order to achieve the above purpose, the present invention adopts the following technical scheme:
firstly, establishing a corresponding observation equation by utilizing observables of arrival time differences of a transmitting station and a receiving station in multi-base sonar, integrating all the observation equations into a matrix function form, deforming the matrix function form to change the matrix function into a pseudo-linear matrix equation with unknown quantity containing target positions, determining a cost function for solving the matrix equation, and deforming the cost function by introducing an augmentation matrix and an expansion vector. And then determining a constant constraint condition, constructing a new model with a solving model based on a least square criterion, and solving the new model by solving a generalized eigenvector corresponding to the minimum generalized eigenvalue of a matrix beam to obtain an initial solution containing the target position estimation. And then determining an equation satisfied by the new model for the error of the target source position estimation according to a first-order Taylor series expansion method, and obtaining a solution by utilizing a least square correlation principle. And finally, subtracting the obtained estimation error from the estimation of the target position in the initial solution to obtain the final estimation result of the target position. The method comprises the following steps:
the invention discloses a TDOA bias reduction positioning method based on underwater multi-base sonar, which comprises the following steps:
step 1: the respective TDOA observation equations are established using the respective TDOA observations for the M transmitting stations and the N receiving stations.
Step 2: all TDOA observation equations are integrated and converted into a pseudo-linear matrix equation whose unknowns contain the target source position u.
Step 3: and (3) determining a cost function J for solving the pseudo-linear equation in the step (2), and introducing an augmentation matrix A and an expansion vector V to deform the cost function J.
Step 4: and solving a constant constraint condition based on the deformed cost function J, and forming an optimization solving model with unknown quantity comprising the underwater target source position u with the solving model based on the least square criterion.
Step 5: by solving for matrix bundles (A T W 1 A, Ω) a generalized eigenvector corresponding to the minimum generalized eigenvalue, to obtain an initial solution comprising the target position estimate.
Step 6: and determining an equation satisfied by the error of the optimization solving model for the target source position estimation according to a first-order Taylor series expansion method, and obtaining a solution by utilizing a least square correlation principle.
Step 7: and (3) subtracting the result obtained in the step (6) from the estimation of the target position in the initial solution obtained in the step (5) to obtain the final target position estimation.
Further, in the step 1, the position of the ith transmitting station is t i Wherein i is more than or equal to 1 and less than or equal to M, and M represents the number of transmitting stations; the j-th receiving station has a position s j Wherein j is more than or equal to 1 and less than or equal to N, and j represents the number of receiving stations; u represents an underwater target source; the sound velocity is c; then the associated time difference of arrival (TDOA) τ ij The observation equation of (2) is
In the middle of Representing a priori value, σ, of sound velocity c Representing a priori error corresponding to the speed of sound;
indicating the TDOA observed quantity with error corresponding to the ith transmitting station and the jth receiving station, tau ij Indicating the TDOA observed quantity corresponding to the ith transmitting station and the jth receiving station, and Deltaτ ij Indicating TDOA observed errors corresponding to the ith transmitting station and the jth receiving station; epsilon ij ＝c△τ ij +τ ij σ c Representing the corresponding error value.
Further, in the step 2, the pseudo-linear form of the deformed TDOA observation equation is as follows
Wherein B is 1 Error matrix representing equation, ε represents all ε ij Form of vectors of constitution, h 1 Representing the observation vector, G 1 Representing the observation matrix of the image of the object,representing the unknowns in the equation.
The specific expression of each element in the formula is
ε＝[(ε 1 ) T (ε 2 ) T … (ε M ) T ] T
Wherein I is M Representing an identity matrix with dimension M;kronecker product representing a matrix; b (B) 11 Representing the relation B 1 Is a sub-diagonal matrix of (a); epsilon i A subvector representing epsilon; h is a 1i Represents h 1 Is a sub-vector of (2); g 1i Represents G 1 Is a sub-matrix of (c).
Wherein the method comprises the steps of
B 11 ＝2diag([||u-s 1 || ||u-s 2 || … ||u-s N ||])
ε i ＝[ε i1 ε i2 … ε iN ] T
Wherein diag is expressed as a vector* Each element in the matrix is a diagonal matrix; o (O) i×j Representing a zero matrix with a number i of rows and j of columns.
Further, in the step 3, the cost function of the step 2 needs to be solved as follows
W in the formula 1 Representing a corresponding weighting matrix, in particular expressed as
E(εε T ) Is a covariance matrix with respect to ε, specifically expressed as
Wherein Q is τ An error covariance matrix representing TDOA observables; τ represents the matrix form of the TDOA observations, where
τ＝[(τ 1 ) T (τ 2 ) T … (τ M ) T ] T
τ i ＝[τ i1 τ i2 … τ iN ] T
Introducing an augmentation matrix A= [ -G 1 h 1 ]Dimension-expanding vectorThe cost function J can be deformed into
J＝V T A T W 1 AV
Further, in the step 4, a constant constraint condition needs to be determined, which is specifically deduced as follows:
the augmentation matrix a may be decomposed into a=a ο A + [ delta ] A, wherein A ο Representing the exact known portion of the augmentation matrix a, Δa represents the error portion of aRespectively expressed as
△A＝[-△G 1 △h 1 ]
In the middle ofRepresents G 1 Error term free part of->Represents h 1 Is free of error term and can be expressed as
△G 1 Represents G 1 Error term part of Deltah 1 Represents h 1 Can be expressed as
In the middle of
△h 1i ＝D i1 △τ i +D i2 σ c
D in i1 Representing the relevant Deltaτ i Error correlation matrix, D i2 Representation of sigma c Is specifically expressed as
D i1 ＝2diag([c||t i -s 1 ||+c 2 τ i1 c||t i -s 2 ||+c 2 τ i2 … c||t i -s N ||+c 2 τ iN ])
The cost function J can be expressed as
J＝V T (A ο +△A) T W 1 (A ο +△A)V
＝V T A οT W 1 A ο V+2V T A οT W 1 △AV+V T △A T W 1 △AV
Then make expectations available for both sides of the cost function
E(J)＝V T A οT W 1 A ο V+V T E(△A T W 1 △A)V
Taking the second term in E (J) as a constant constraint, a new term can be constructed as followsIs an optimized solution model of (a)
Where Ω denotes a second order error correlation term and k denotes an arbitrary constant.
Omega may be specifically expressed as
Omega in 1 、Ω 2 And omega 3 The different sub-matrices, each representing Ω, may be represented as
Wherein Ω 1 (m 1 :m 2 Representing the matrix omega 1 Is the m < th > of 1 ～m 2 A sub-matrix of rows; omega shape 1 (m 1 :m 2 ,n 1 :n 2 ) Represents the mth of the matrix 1 ～m 2 Row and nth 1 ～n 2 A sub-matrix of columns; omega shape 2 (m: n) represents a sub-vector formed by the m-n-th columns of vectors; omega shape 2 (m) represents a vector Ω 2 The m-th element of (2); trace represents the trace of matrix.
Further, in the step 5, if the estimation result of the optimization solution model with respect to V is recorded asThen->Estimate of +.>Can be expressed as
Further, in the step 6, according to the first-order taylor series expansion method,the individual elements of (a) may be represented as
Representation->According to ∈then>It can be derived that
If the error of the optimization solving model to the target source position estimation is recorded asFrom the above set of equations, the following matrix equation can be obtained
H in 2 Represents h 1 With respect toIs converted into form G 2 Represents G 1 About->Is respectively expressed as
According to the least squares correlation theory,estimate of +.>Can be expressed as
W in the formula 2 Representing a corresponding weighting matrix
Representation->Is used for the estimation of covariance matrix.
Further, in the step 7, the final estimation result of the target source positionCan be expressed as
Compared with the prior art, the invention has the beneficial effects that:
the invention utilizes various known quantities and priori information in a positioning system, solves the generalized eigenvector corresponding to the minimum generalized eigenvalue, and positions the underwater target source through the basic idea of least square, thereby effectively carrying out offset reduction on the TDOA positioning method of the underwater multi-base sonar.
Drawings
FIG. 1 is a basic flow chart of a TDOA bias clipping positioning method based on underwater multi-base sonar according to an embodiment of the invention;
FIG. 2 is a geometric schematic of multi-base sonar positioning;
FIG. 3 is a graph of the positioning result and an elliptical error curve with TDOA observation error of 0.001 s;
FIG. 4 is a graph showing the variation of the estimated root mean square of the target location with TDOA observation error;
FIG. 5 is a plot of estimated bias of target position as a function of TDOA observed error;
FIG. 6 is a target position (90,90,90) T m is a positioning result graph and an error elliptic curve;
FIG. 7 is a graph showing the variation of the estimated root mean square of a target location with different target locations;
FIG. 8 is a graph of estimated bias of target position versus target position.
Detailed Description
The invention is further illustrated by the following description of specific embodiments in conjunction with the accompanying drawings:
as shown in FIG. 1, the method for positioning the TDOA offset reduction based on the underwater multi-base sonar comprises the following specific implementation steps:
step 1: the respective TDOA observation equations are established using the respective TDOA observations for the M transmitting stations and the N receiving stations.
Step 2: all TDOA observation equations are integrated and converted into a pseudo-linear matrix equation whose unknowns contain the target source position u.
Step 3: and (3) determining a cost function J for solving the pseudo-linear equation in the step (2), and introducing an augmentation matrix A and an expansion vector V to deform the cost function J.
Step 4: and solving a constant constraint condition based on the deformed cost function J, and forming an optimization solving model with unknown quantity comprising the underwater target source position u with the solving model based on the least square criterion.
Step 5: by solving for matrix bundles (A T W 1 A, Ω) a generalized eigenvector corresponding to the minimum generalized eigenvalue, to obtain an initial solution comprising the target position estimate.
Step 6: and determining an equation satisfied by the error of the optimization solving model for the target source position estimation according to a first-order Taylor series expansion method, and obtaining a solution by utilizing a least square correlation principle.
Step 7: and (3) subtracting the result obtained in the step (6) from the estimation of the target position in the initial solution obtained in the step (5) to obtain the final target position estimation.
Further, in the step 1, the position of the ith transmitting station is t i Wherein i is more than or equal to 1 and less than or equal to M, and M represents the number of transmitting stations; the j-th receiving station has a position s j Wherein j is more than or equal to 1 and less than or equal to N, and j represents the number of receiving stations; u represents an underwater target source; the sound velocity is c; then the associated time difference of arrival (TDOA) τ ij The observation equation of (2) is
In the middle of Representing a priori value, σ, of sound velocity c Representing a priori error corresponding to the speed of sound;
indicating the TDOA observed quantity with error corresponding to the ith transmitting station and the jth receiving station, tau ij Indicating the TDOA observed quantity corresponding to the ith transmitting station and the jth receiving station, and Deltaτ ij Indicating TDOA observed errors corresponding to the ith transmitting station and the jth receiving station; epsilon ij ＝c△τ ij +τ ij σ c Representing the corresponding error value.
A geometric diagram of the multi-base sonar positioning is shown in fig. 2.
Further, in the step 2, the pseudo-linear form of the deformed TDOA observation equation is as follows
Wherein B is 1 Error matrix representing equation, ε represents all ε ij Form of vectors of constitution, h 1 Representing the observation vector, G 1 Representing the observation matrix of the image of the object,representing the unknowns in the equation.
The specific expression of each element in the formula is
ε＝[(ε 1 ) T (ε 2 ) T … (ε M ) T ] T
Wherein I is M Representing an identity matrix with dimension M;kronecker product representing a matrix; b (B) 11 Representing the relation B 1 Is a sub-diagonal matrix of (a); epsilon i A subvector representing epsilon; h is a 1i Represents h 1 Is a sub-vector of (2); g 1i Represents G 1 Is a sub-matrix of (c).
Wherein the method comprises the steps of
B 11 ＝2diag([||u-s 1 || ||u-s 2 || … ||u-s N ||])
ε i ＝[ε i1 ε i2 … ε iN ] T
Wherein diag represents a matrix diagonal to the elements in the vector; o (O) i×j Representing a zero matrix with a number i of rows and j of columns.
Further, in the step 3, the cost function of the step 2 needs to be solved as follows
W in the formula 1 Representing a corresponding weighting matrix, in particular expressed as
E(εε T ) Is a covariance matrix with respect to ε, specifically expressed as
Wherein Q is τ An error covariance matrix representing TDOA observables; τ represents the matrix form of the TDOA observations, where
τ＝[(τ 1 ) T (τ 2 ) T … (τ M ) T ] T
τ i ＝[τ i1 τ i2 … τ iN ] T
Introducing an augmentation matrix A= [ -G 1 h 1 ]Dimension-expanding vectorThe cost function J can be deformed into
J＝V T A T W 1 AV
Further, in the step 4, a constant constraint condition needs to be determined, which is specifically deduced as follows:
the augmentation matrix a may be decomposed into a=a ο A + [ delta ] A, wherein A ο Representing the exact known portion of the augmentation matrix a, Δa represents the error portion of a, respectively expressed as
△A＝[-△G 1 △h 1 ]
In the middle ofRepresents G 1 Error term free part of->Represents h 1 Is free of errorsA difference term portion, which can be expressed as
△G 1 Represents G 1 Error term part of Deltah 1 Represents h 1 Can be expressed as
In the middle of
△h 1i ＝D i1 △τ i +D i2 σ c
D in i1 Representing the relevant Deltaτ i Error correlation matrix, D i2 Representation of sigma c Is specifically expressed as
D i1 ＝2diag([c||t i -s 1 ||+c 2 τ i1 c||t i -s 2 ||+c 2 τ i2 … c||t i -s N ||+c 2 τ iN ])
The cost function J can be expressed as
J＝V T (A ο +△A) T W 1 (A ο +△A)V
＝V T A οT W 1 A ο V+2V T A οT W 1 △AV+V T △A T W 1 △AV
Then make expectations available for both sides of the cost function
E(J)＝V T A οT W 1 A ο V+V T E(△A T W 1 △A)V
Taking the second term in E (J) as a constant constraint, a new term can be constructed as followsIs an optimized solution model of (a)
Where Ω denotes a second order error correlation term and k denotes an arbitrary constant.
Omega may be specifically expressed as
Omega in 1 、Ω 2 And omega 3 The different sub-matrices, each representing Ω, may be represented as
Wherein Ω 1 (m 1 :m 2 Representing the matrix omega 1 Is the m < th > of 1 ～m 2 A sub-matrix of rows; omega shape 1 (m 1 :m 2 ,n 1 :n 2 ) Represents the mth of the matrix 1 ～m 2 Row and nth 1 ～n 2 A sub-matrix of columns; omega shape 2 (m: n) represents a sub-vector formed by the m-n-th columns of vectors; omega shape 2 (m) represents a vector Ω 2 The m-th element of (2); trace represents the trace of matrix.
Further, in the step 5, if the estimation result of the optimization solution model with respect to V is recorded asThen->Estimate of +.>Can be expressed as
Further, in the step 6, according to the first-order taylor series expansion method,the individual elements of (a) may be represented as
Representation->According to ∈then>It can be derived that
If the error of the optimization solving model to the target source position estimation is recorded asFrom the above set of equations, the following matrix equation can be obtained
H in 2 Represents h 1 With respect toIs converted into form G 2 Represents G 1 About->Is respectively expressed as
According to the least squares correlation theory,estimate of +.>Can be expressed as
/>
W in the formula 2 Representing a corresponding weighting matrix
Representation->Is used for the estimation of covariance matrix.
Further, in the step 7, the final estimation result of the target source positionCan be expressed as
To verify the effect of the present invention, the following specific examples are performed:
assuming that there are 2 transmitting stations and 5 receiving stations in the multi-base sonar, the position ratio of the 2 transmitting stations is (600,900,600) T m，(600,700,800) T m; the location percentage of 5 receiving stations is (-500,500,600) T m，(600,-600,600) T m，(700,700,-600) T m，(-600,-600,600) T m，(750,-600,-700) T m. The sound velocity under water has a value of 1500m/s.
(1) Assuming that the prior error corresponding to the sound velocity is 0.5m/s, the TDOA observation error is assumed to be a dependent variable sigma 1 Value of 0.001 sigma of change 1 s, FIG. 3The basic situation of the positioning result of the method disclosed by the patent can be intuitively seen by giving a positioning result graph and an error elliptic curve of the method disclosed by the patent when the TDOA observation error is 0.001 s; FIG. 4 shows a variation curve of the estimated root mean square of the target position along with the TDOA observation error, and can be seen that compared with the existing multi-step weighting algorithm and the Taylor series method, the offset reduction positioning method disclosed by the patent has the advantages that the estimated root mean square error is not changed, and the condition that the initial value selection of the Taylor series method is not suitable for obviously increasing the estimated root mean square error is avoided; FIG. 5 shows a variation curve of the estimated bias of the target position along with the TDOA observation error, and can be seen that compared with the existing multi-step weighting algorithm, the bias reduction positioning method disclosed by the patent has the advantages that the bias is obviously reduced, the bias is close to the bias of the Taylor series method with the initial value being the true value, and meanwhile, the problem that the estimated bias is increased when the initial value is inappropriately selected is avoided.
(2) Assuming that the a priori error corresponding to the speed of sound is 0.5m/s, the target position can be expressed as (50+40 sigma) 2 ,50+40σ 2 ,50+40σ 2 ) T m, FIG. 6 shows the bias clipping positioning method of the present patent disclosure when the target position is (50,50,50) T The basic situation of the positioning result of the method disclosed by the patent can be intuitively seen through the positioning result graph and the error elliptic curve in m; FIG. 7 shows the variation curves of the estimated root mean square of the target position along with different target positions, and it can be seen that compared with the existing multi-step weighting algorithm and Taylor series method, the offset reduction positioning method disclosed by the patent has the advantages that the estimated mean square error is not changed, and the situation that the estimated mean square error is obviously increased due to improper selection of the initial value of the Taylor series method does not exist; fig. 8 shows a variation curve of estimated bias of a target position along with different target positions, and it can be seen that compared with the existing multi-step weighting algorithm, the bias reduction positioning method disclosed by the patent has the advantages that the bias is obviously reduced, the bias is close to the bias of the taylor series method with the initial value being a true value, and meanwhile, the problem that the estimated bias is increased when the initial value is inappropriately selected is avoided.
In summary, the invention utilizes various known quantities and priori information in a positioning system, solves the generalized eigenvector corresponding to the minimum generalized eigenvalue, and positions the underwater target source through the basic idea of least square, thereby effectively carrying out offset reduction on the TDOA positioning method of the underwater multi-base sonar.
The foregoing is merely illustrative of the preferred embodiments of this invention, and it will be appreciated by those skilled in the art that changes and modifications may be made without departing from the principles of this invention, and it is intended to cover such modifications and changes as fall within the true scope of the invention.
Claims (8)
1. The TDOA bias reduction positioning method based on the underwater multi-base sonar is characterized by comprising the following steps of:
step 1: establishing respective TDOA observation equations using respective TDOA observations for the M transmitting stations and the N receiving stations;
step 2: integrating all TDOA observation equations and converting the TDOA observation equations into a pseudo-linear equation with unknown quantity including the position u of the underwater target source;
step 3: determining a cost function J for solving the pseudo linear equation in the step 2, and introducing an augmentation matrix A and an expansion vector V to deform the cost function J;
step 4: solving a constant constraint condition based on the deformed cost function J, and constructing an optimization solving model with unknown quantity comprising the underwater target source position u;
step 5: by solving for matrix bundles (A T W 1 A, omega) the generalized eigenvector corresponding to the minimum generalized eigenvalue is solved, and an initial solution containing target position estimation is obtained by the optimized solution model constructed in the step 4; wherein W is 1 Representing a weighting matrix, wherein Ω represents a second order error correlation term;
step 6: determining an equation satisfied by an optimization solution model for the error of the target source position estimation according to a first-order Taylor series expansion method, and obtaining a solution by using a least square method;
step 7: and (3) subtracting the result obtained in the step (6) from the estimation of the target position in the initial solution obtained in the step (5) to obtain the final target position estimation.
2. The method for TDOA bias reduction positioning based on underwater multi-base sonar according to claim 1, wherein in step 1, the TDOA observation equation is established as follows:
wherein the method comprises the steps ofRepresenting a priori the speed of sound, c being the speed of sound, σ c Representing a priori error corresponding to the speed of sound;indicating the TDOA observed quantity with error corresponding to the ith transmitting station and the jth receiving station, tau ij Indicating the TDOA observed quantity corresponding to the ith transmitting station and the jth receiving station, and Deltaτ ij Indicating TDOA observed errors corresponding to the ith transmitting station and the jth receiving station; epsilon ij ＝c△τ ij +τ ij σ c Representing a corresponding error value; t is t i Indicating the location of the ith transmitting station; s is(s) j Indicating the location of the j-th receiving station; i is more than or equal to 1 and less than or equal to M, wherein M represents the number of transmitting stations; and j is more than or equal to 1 and less than or equal to N, wherein N represents the number of receiving stations.
3. The method for TDOA bias curtailment positioning based on underwater multi-base sonar according to claim 2, wherein in step 2, the pseudo-linear equation is:
wherein B is 1 Error matrix representing equation, ε represents all ε ij Form of vectors of constitution, h 1 Representing the observation vector, G 1 Representing the observation matrix of the image of the object,representing an unknown quantity in the equation;
in the middle of
ε＝[(ε 1 ) T (ε 2 ) T … (ε M ) T ] T
Wherein I is M Representing an identity matrix with dimension M;kronecker product representing a matrix; b (B) 11 Representing the relation B 1 Is a sub-diagonal matrix of (a); epsilon i A subvector representing epsilon; h is a 1i Represents h 1 Is a sub-vector of (2); g 1i Represents G 1 Is a sub-matrix of (a);
wherein the method comprises the steps of
B 11 ＝2diag([||u-s 1 || ||u-s 2 || … ||u-s N ||])
ε i ＝[ε i1 ε i2 … ε iN ] T
Wherein diag represents a matrix diagonal to the elements in the vector; o (O) i×j Representing a zero matrix with a number i of rows and j of columns.
4. A TDOA bias curtailment positioning method based on underwater multi-base sonar according to claim 3, wherein said step 3 comprises:
determining a cost function J for solving the pseudo-linear equation of step 2
W in the formula 1 Representing the corresponding weighting matrix, expressed as
E(εε T ) Is a covariance matrix with respect to ε, expressed as
Wherein Q is τ An error covariance matrix representing TDOA observables; τ represents the matrix form of the TDOA observations, where
τ＝[(τ 1 ) T (τ 2 ) T … (τ M ) T ] T
τ i ＝[τ i1 τ i2 … τ iN ] T
Introducing an augmentation matrix A= [ -G 1 h 1 ]Dimension-expanding vectorTransforming the cost function J into
J＝V T A T W 1 AV。
5. The TDOA offset curtailment positioning method based on underwater multi-base sonar of claim 4, wherein said step 4 comprises:
decomposing the augmentation matrix a into a=a ο A + [ delta ] A, wherein A ο Representing the exact known portion of the augmentation matrix a, Δa represents the error portion of a, respectively expressed as
△A＝[-△G 1 △h 1 ]
In the middle ofRepresents G 1 Error term free part of->Represents h 1 Is represented as
△G 1 Represents G 1 Error term part of Deltah 1 Represents h 1 Is expressed as
In the middle of
△h 1i ＝D i1 △τ i +D i2 σ c
D in i1 Representing the relevant Deltaτ i Error correlation matrix, D i2 Representation of sigma c Is expressed as an error correlation vector of (a)
D i1 ＝2diag([c||t i -s 1 ||+c 2 τ i1 c||t i -s 2 ||+c 2 τ i2 …c||t i -s N ||+c 2 τ iN ])
The cost function J is deformed into
J＝V T (A ο +△A) T W 1 (A ο +△A)V
＝V T A οT W 1 A ο V+2V T A οT W 1 △AV+V T △A T W 1 △AV
Then make expectations available for both sides of the cost function
E(J)＝V T A οT W 1 A ο V+V T E(△A T W 1 △A)V
Wherein E (J) represents taking the expectation of the cost function J;
the second term in E (J) is used as a constant constraint, constructed as followsIs an optimized solution model of (a)
Wherein Ω represents a second order error correlation term, and k represents an arbitrary constant;
omega in 1 、Ω 2 And omega 3 Different sub-matrices, denoted Ω, respectively
Wherein Ω 1 (m 1 :m 2 Representing the matrix omega 1 Is the m < th > of 1 、m 2 A sub-matrix of rows; omega shape 1 (m 1 :m 2 ,n 1 :n 2 ) Represents the mth of the matrix 1 、m 2 Row and nth 1 、n 2 A sub-matrix of columns; omega shape 2 (m: n) represents the directionSub-vectors formed by m-n columns of the quantity; omega shape 2 (m) represents a vector Ω 2 The m-th element of (2); trace represents the trace of matrix.
6. The method for TDOA bias curtailment positioning based on underwater multi-base sonar of claim 5, wherein in said step 5, the estimation result of the optimization solving model with respect to V is recorded asThen->Estimate of +.>Represented as
7. The method for TDOA bias curtailment positioning based on underwater multi-base sonar of claim 6, wherein step 6 comprises:
according to the first-order taylor series expansion method,each element in (a) is expressed as
Representation->Error of estimation of ∈10->Deriving
The error of the optimization solving model to the target source position estimation is recorded asFrom the above set of equations, the following matrix equation can be obtained
H in 2 Represents h 1 With respect toIs converted into form G 2 Represents G 1 About->Is respectively expressed as
According to the least-squares method of the method,estimate of +.>Represented as
W in the formula 2 Representing a corresponding weighting matrix
Representation->Is used for the estimation of covariance matrix.
8. The method for TDOA bias curtailment positioning based on underwater multi-base sonar of claim 7, wherein in said step 7, the final estimation result of the target source positionRepresented as
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