CN115372902A - TDOA offset reduction positioning method based on underwater multi-base sonar - Google Patents
TDOA offset reduction positioning method based on underwater multi-base sonar Download PDFInfo
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Abstract
The invention discloses a TDOA offset reduction positioning method based on underwater multi-base sonar, which comprises the steps of firstly, establishing an observation equation by utilizing observed quantities of the multi-base sonar about arrival time differences of a transmitting station and a receiving station, 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 pseudo linear matrix equation; then determining a constant constraint condition, forming 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 the estimation of the target source position according to a first-order Taylor series expansion method, and obtaining a solution of the error equation by using 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 offset of the multi-base sonar TDOA positioning method when the underwater sound velocity has the prior error.
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 positioning an underwater target source. Conventional sonars can be classified into passive sonars and active sonars according to different working modes. The passive sonar directly receives the noise generated by the underwater target mechanical work to find the target, and has better concealment. However, with the deep research of submarine stealth technology, the noise emitted by the submarine is smaller and smaller, and the positioning performance of the passive sonar is obviously reduced. The active sonar automatically transmits sound wave signals and then receives target echoes to position the target. The range is relatively long, but concealment is not strong due to the need to actively transmit the signal. Compared with the two kinds of sonar, the multi-base sonar transceiver devices are separately placed, and the transmitting station can actively transmit signals, so that the multi-base sonar transceiver devices have the advantages of active sonar, and the receiving station of the multi-base sonar transceiver devices is passively operated, so that the multi-base sonar transceiver devices are better in concealment. Because of the advantages of good concealment, strong anti-interference capability, high maneuvering performance and long working distance of the multi-base sonar, the method has become a research hotspot of scholars at home and abroad.
The principle of multi-base sonar positioning is that a single or a plurality of transmitting stations transmit sound wave signals, a plurality of receiving stations receive target echoes, and a target is positioned according to parameter information such as a time domain, a frequency domain, a space domain or an energy domain 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 gain ratio of arrival, among others. Based on the above observation information, more and more positioning algorithms are proposed. The current algorithms can be generally divided into closed-form solution type algorithms and iterative type algorithms. The closed solution type algorithm can obtain a specific formula for solving the target position through formula derivation, and has the advantages of more definite calculation process, simple calculation and smaller calculation amount, so that a plurality of related algorithms are provided; the iterative algorithm solves the target position through an iterative process, and compared with a closed-form solution algorithm, the iterative algorithm is more complex in calculation process and needs to consider the problem of initial value selection.
Although the closed-form solution algorithm has a simple calculation process and low calculation complexity, the formula conversion is generally performed for many times according to the relationship among various physical quantities in the observation equation, and a lot of neglected second-order error terms are often generated in the process, which may cause the increase of the positioning offset. For multi-base sonar positioning, the corresponding observation equation has more physical quantities (including a transmitting station, a receiving station, a TDOA observed quantity and a sound speed), and a formula conversion generates more second-order error terms, so that the system is more sensitive to various error quantities, and therefore, the offset of the algorithm needs to be reduced. Aiming at the problem, the TDOA offset reduction positioning method based on the underwater multi-base sonar is designed to reduce the estimated positioning offset.
Disclosure of Invention
The invention provides a TDOA offset reduction positioning method based on underwater multi-base sonar, aiming at the problem of larger offset in the existing TDOA positioning method of underwater multi-base sonar, and the offset of the existing positioning method can be reduced.
In order to achieve the purpose, the invention adopts the following technical scheme:
the method comprises the steps of firstly, establishing corresponding observation equations by utilizing observed quantities of a transmitting station and a receiving station about arrival time differences in the multi-base sonar, integrating all the observation equations into a matrix function form and deforming the matrix function form to change the observation equations into a pseudo linear matrix equation with unknown quantity including a target position, 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, forming 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 containing the target position estimation. And then determining an equation which is satisfied by the error of the new model for the estimation of the target source position according to a first-order Taylor series expansion method, and obtaining a solution of the new model by using a least square correlation principle. And finally, subtracting the result of the calculated 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 specific steps:
the invention discloses a TDOA offset reduction positioning method based on underwater multi-base sonar, which comprises the following steps:
step 1: 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 changed into a pseudo linear matrix equation with unknown quantity containing the target source position u.
And 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.
And 4, step 4: and solving a constant constraint condition based on the deformed cost function J, and forming an optimized solving model with unknown quantity including the position u of the underwater target source with a solving model based on a least square rule.
And 5: by solving the matrix bundle (A) T W 1 A, omega) to obtain an initial solution containing target position estimation.
Step 6: and determining an equation which is satisfied by the error of the optimization solution model for the estimation of the target source position according to a first-order Taylor series expansion method, and obtaining a solution of the optimization solution model by using a least square correlation principle.
And 7: and 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 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 position of the jth receiving station is 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 speed of sound is c; then the associated time difference of arrival (TDOA) τ ij Is as follows
In the formula
A priori value, σ, representing the speed of sound c A priori error representing the speed of sound correspondence;
Further, in step 2, the pseudolinear form of the TDOA observation equation after deformation is
Wherein B is 1 Error matrix representing the equation, ε represents all εs ij Form of vectors of composition, h 1 Representing an observation vector, G 1 A representation of an observation matrix is shown,
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 M An identity matrix of dimension M;
representing the Kronecker product of the matrix; b 11 Represents about B 1 The sub-diagonal matrix of (a); epsilon i A subvector representing epsilon; h is 1i Represents h 1 The sub-vectors of (a); g 1i Represents G 1 The sub-matrix of (2).
Wherein
B 11 ＝2diag([||u-s 1 || ||u-s 2 || … ||u-s N ||])
ε i ＝[ε i1 ε i2 … ε iN ] T
Wherein diag (x) represents a matrix with each element in the vector as a diagonal; o < O > of a compound i×j A zero matrix with a number of rows i and a number of columns j is shown.
Further, in step 3, the cost function required to solve step 2 is as follows
In the formula W 1 Representing the corresponding weighting matrix, in particular as
E(εε T ) Is a covariance matrix of e, expressed in particular as
Wherein Q τ An error covariance matrix representing the TDOA observations; τ represents a matrix form of 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 ]And dimension-expanded vector
The cost function J can be morphed to
J＝V T A T W 1 AV
Further, in step 4, a constant constraint condition needs to be determined, and is specifically derived as follows:
the augmentation matrix a may be decomposed into a = a ο +. DELTA A, wherein A ο Representing the precisely known part of the augmentation matrix A, and Δ A representing the error part of A, respectively
△A＝[-△G 1 △h 1 ]
In the formula
Represents G 1 Is free of the error term portion of (a),
represents h 1 Can be expressed as
△G 1 Represents G 1 Part of the error term of (c), Δ h 1 Denotes h 1 Can be expressed as
In the formula
△h 1i ＝D i1 △τ i +D i2 σ c
In the formula D i1 Indicates the relation Δ τ i Error correlation matrix of D i2 Represents a correlation of c Is expressed specifically 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, the expectation is obtained for two 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
With the second term in E (J) as a constant constraint, a new relationship can be constructed as follows
Is optimized and solved model
Where Ω represents a second order error correlation term and k represents an arbitrary constant.
Omega can be specifically expressed as
Formula mid omega 1 、Ω 2 And Ω 3 Different sub-matrices, each representing Ω, may be represented as
Wherein omega 1 (m 1 :m 2 In that) the representation matrix omega 1 M of 1 ～m 2 A sub-matrix of rows; omega 1 (m 1 :m 2 ,n 1 :n 2 ) M < th > of the representation matrix 1 ～m 2 Line and nth 1 ～n 2 A sub-matrix of columns; omega 2 (m: n) represents a sub-vector formed by m to n columns of the vector; omega 2 (m) represents a vector Ω 2 The mth element of (1); trace (×) represents the trace of the matrix.
Further, in the step 5, if the estimation result of the optimization solution model with respect to V is recorded as V
Then
Is estimated value of
Can be expressed as
Further, in the step 6, according to the first-order Taylor series expansion method,
can be represented as
If the error of the optimization solution model for the target source position estimation is recorded as
From the above set of equations, the following matrix equation can be obtained
In the formula h 2 Denotes h 1 About
Of a transition form of (C), G 2 Represents G 1 About
Respectively expressed as
According to the theory of correlation of least squares,
is estimated value of
Can be expressed as
In the formula W 2 Representing the corresponding weighting matrix
Further, in the step 7, the final estimation result of the position of the target source
Can be expressed as
Compared with the prior art, the invention has the following beneficial effects:
the method utilizes various known quantities and prior information in a positioning system, solves the generalized characteristic vector corresponding to the minimum generalized characteristic value, and positions the underwater target source through the basic idea of least square, so that the bias reduction can be effectively carried out on the TDOA positioning method of the underwater multi-base sonar.
Drawings
FIG. 1 is a basic flowchart of a TDOA offset reduction positioning method based on an underwater multi-base sonar according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of a multi-base sonar positioning geometry;
FIG. 3 is a diagram of a TDOA observed error as a 0.001s positioning result and an error ellipse curve;
FIG. 4 is a plot of the estimated root mean square of the target location as a function of TDOA observed error;
FIG. 5 is a plot of estimated offset of target location versus TDOA observation error;
FIG. 6 shows the target positions as (90, 90) T m positioning result graph and error elliptic curve;
FIG. 7 is a plot of the estimated root mean square of the target location versus different target locations;
FIG. 8 is a graph of estimated bias for target locations as a function of different target locations.
Detailed Description
The invention is further illustrated by the following examples in conjunction with the accompanying drawings:
as shown in fig. 1, the specific implementation steps of the TDOA offset reduction positioning method based on underwater multi-base sonar of the present invention are as follows:
step 1: respective TDOA observation equations are established using respective TDOA observations for the M transmitting stations and the N receiving stations.
Step 2: and integrating all TDOA observation equations, and changing the TDOA observation equations into a pseudo linear matrix equation of which the unknown quantity comprises the position u of the target source.
And step 3: and (3) determining a cost function J of the pseudo linear equation obtained in the step (2), and introducing an augmentation matrix A and an dimension expansion vector V to deform the cost function J.
And 4, step 4: and solving a constant constraint condition based on the deformed cost function J, and forming an optimized solving model of which the unknown quantity comprises the position u of the underwater target source together with a solving model based on a least square rule.
And 5: by solving the matrix bundle (A) T W 1 A, omega) to obtain an initial solution containing target position estimation.
And 6: and determining an equation which is satisfied by the error of the optimization solution model for the estimation of the target source position according to a first-order Taylor series expansion method, and obtaining the solution of the optimization solution model by using a least square correlation principle.
And 7: and 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 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 position of the jth receiving station is 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 speed of sound is c; then the associated time difference of arrival (TDOA) τ ij Is as follows
In the formula
A priori value, σ, representing the speed of sound c A priori error representing the speed of sound correspondence;
A schematic diagram of the multi-base sonar positioning geometry is shown in fig. 2.
Further, in step 2, the pseudolinear form after the deformation of the TDOA observation equation is
Wherein B is 1 Error matrix representing equation, ε represents all εs ij Form of vectors of composition, h 1 Representing an observation vector, G 1 Which represents the observation matrix, is shown,
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
In which I M An identity matrix representing a dimension M;
representing the Kronecker product of the matrix; b 11 Indicates a relation of B 1 The sub-diagonal matrix of (a); epsilon i A subvector representing ε; h is a total of 1i Denotes h 1 The subvectors of (1); g 1i Represents G 1 The sub-matrix of (2).
Wherein
B 11 ＝2diag([||u-s 1 || ||u-s 2 || … ||u-s N ||])
ε i ＝[ε i1 ε i2 … ε iN ] T
Wherein diag (x) represents a matrix with each element in the vector as a diagonal; o i×j A zero matrix with a number of rows i and a number of columns j is shown.
Further, in step 3, the cost function required to solve step 2 is as follows
In the formula W 1 Representing the corresponding weighting matrix, in particular as
E(εε T ) Is a covariance matrix of e, expressed in particular as
Wherein Q τ An error covariance matrix representing the TDOA observations; τ represents a matrix form of 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 ]And dimension vector
The cost function J can be morphed to
J＝V T A T W 1 AV
Further, in step 4, a constant constraint condition needs to be determined, and is specifically derived as follows:
the augmentation matrix a may be decomposed into a = a ο +. DELTA.A, wherein A ο Representing the precisely known part of the augmentation matrix A, and Δ A representing the error part of A, respectively
△A＝[-△G 1 △h 1 ]
In the formula
Represents G 1 Is free of the error term part of (a),
represents h 1 Can be expressed as
△G 1 Represents G 1 Part of error term of,. DELTA.h 1 Denotes h 1 Can be expressed as
In the formula
△h 1i ＝D i1 △τ i +D i2 σ c
In the formula D i1 Indicates a correlation of i Error correlation matrix of D i2 Is expressed in relation to c Is expressed specifically 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, the expectation is obtained for two 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
With the second term in E (J) as a constant constraint, a new relationship can be constructed as follows
Optimization solution model of (2)
Where Ω represents a second order error correlation term and k represents an arbitrary constant.
Omega can be specifically expressed as
Formula mid omega 1 、Ω 2 And Ω 3 Respectively represent different sub-matrices of omega, respectively represented as
Wherein omega 1 (m 1 :m 2 In that) the representation matrix omega 1 M of 1 ～m 2 A sub-matrix of rows; omega 1 (m 1 :m 2 ,n 1 :n 2 ) M < th > of the representation matrix 1 ～m 2 Line and nth 1 ～n 2 A sub-matrix of columns; omega 2 (m: n) represents a sub-vector formed by m to n columns of the vector; omega 2 (m) represents a vector Ω 2 The mth element of (1); trace (×) represents the trace of the matrix.
Further, in the step 5, if the estimation result of the optimization solution model about V is recorded as V
Then
Is estimated value of
Can be expressed as
Further, in the step 6, according to the first-order Taylor series expansion method,
can be represented as
If the error of the optimization solution model for the target source position estimation is recorded as
From the above set of equations, the following matrix equation can be obtained
In the formula h 2 Denotes h 1 About
Of a transition form of (C), G 2 Represents G 1 About
Respectively, are represented as
According to the theory of correlation of the least squares,
is estimated by
Can be expressed as
In the formula W 2 Representing the corresponding weighting matrix
Further, in the step 7, the final estimation result of the position of the target source
Can be expressed as
To verify the effect of the present invention, the following specific examples are performed:
suppose 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;5 receiving stationsThe position ratio of (B) is (-500, 600) T m，(600,-600,600) T m，(700,700,-600) T m，(-600,-600,600) T m，(750,-600,-700) T And m is selected. The value of the speed of sound is 1500m/s underwater.
(1) The sound velocity is assumed to correspond to the prior error of 0.5m/s, and the TDOA observation error is assumed to be a dependent argument sigma 1 Value of variation 0.001 σ 1 s, fig. 3 shows a positioning result graph and an error elliptic curve of the offset reduction positioning method disclosed in the present patent when the TDOA observation error is 0.001s, and the basic situation of the positioning result of the method disclosed in the present patent can be visually seen; fig. 4 shows a variation curve of the estimated root mean square of the target position along with the observed error of TDOA, and it can be seen that, compared with the existing multi-step weighting algorithm and the taylor series method, the estimated mean square error of the offset reduction positioning method disclosed by the present patent does not change, and there is no case that the initial value selection of the taylor series method is not suitable for estimating that the mean square error is significantly increased; fig. 5 shows a variation curve of the estimated bias of the target position along with the observed error of TDOA, and it can be seen that compared with the existing multi-step weighting algorithm, the bias reduction positioning method disclosed by the patent is obviously reduced, and is close to the bias of the taylor series method with the initial value being the true value, and meanwhile, there is no problem that the estimated bias is increased when the initial value is improperly selected.
(2) Assuming that the sound velocity corresponds to a priori error of 0.5m/s, the target position can be expressed as (50 +40 σ) 2 ,50+40σ 2 ,50+40σ 2 ) T m, FIG. 6 shows the offset reduction 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 visually seen through a positioning result graph and an error elliptic curve at m; fig. 7 shows a variation curve of the estimated root mean square of the target position along with different target positions, and it can be seen that the estimated mean square error of the bias reduction positioning method disclosed by the present invention does not change compared with the existing multi-step weighting algorithm and the taylor series method, and there is no case that the initial value selection of the taylor series method is not suitable for estimating the mean square error and is significantly increased; FIG. 8 shows the variation of the estimated bias of the target position with different target positions, and it can be seen that this patent disclosure providesCompared with the existing multi-step weighting algorithm, the bias is obviously reduced, the bias is close to the bias of the Taylor series method with the initial value as the true value, and meanwhile, the problem that the bias is estimated to be increased when the initial value is not properly selected is solved.
In summary, the invention utilizes various known quantities and prior information in the positioning system, and by solving the generalized eigenvector corresponding to the minimum generalized eigenvalue and by the basic idea of least square, the underwater target source is positioned, and offset reduction can be effectively performed on the TDOA positioning method of the underwater multi-base sonar.
The above shows only the preferred embodiments of the present invention, and it should be noted that it is obvious to those skilled in the art that various modifications and improvements can be made without departing from the principle of the present invention, and these modifications and improvements should also be considered as the protection scope of the present invention.
Claims (8)
1. A TDOA offset reduction positioning method based on underwater multi-base sonar is characterized by comprising the following steps:
step 1: establishing respective TDOA observation equations using the 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;
and 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;
and 4, step 4: solving a constant constraint condition based on the deformed cost function J, and constructing an optimized solving model of which unknown quantity comprises the position u of the underwater target source;
and 5: by solving the matrix bundle (A) T W 1 A, omega) to obtain the generalized eigenvector corresponding to the minimum generalized eigenvalue; wherein W 1 Representing a weighting matrix, and omega represents a second-order error correlation term;
step 6: determining an equation which is satisfied by the error of the optimization solution model for the estimation of the target source position according to a first-order Taylor series expansion method, and obtaining the solution of the optimization solution model by using a least square method;
and 7: a final target position estimate is derived based on the solutions obtained in step 5 and step 6.
2. The method for TDOA-offset-based TDOA-biased reduction positioning based on underwater multi-base sonar of claim 1, wherein in the step 1, the TDOA observation equation is established as follows:
wherein
A priori value representing the speed of sound, c being the speed of sound, σ c Representing the prior error corresponding to the sound speed;
representing TDOA observations with errors corresponding to the ith and jth transmitting and receiving stations ij Represents TDOA observed quantity, delta tau, corresponding to the ith transmitting station and the jth receiving station ij Representing TDOA observed quantity 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 j Indicating the location of the jth 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; j is more than or equal to 1 and less than or equal to N, and j represents the number of receiving stations.
3. The method for TDOA offset reduction positioning based on underwater multi-base sonar according to claim 2, wherein in the step 2, the pseudo linear equation is as follows:
wherein B is 1 Error matrix representing equation, ε represents all εs ij Form of vectors of composition, h 1 Representing an observation vector, G 1 A representation of an observation matrix is shown,
representing the unknowns in the equation;
in the formula
ε＝[(ε 1 ) T (ε 2 ) T … (ε M ) T ] T
Wherein I M An identity matrix representing a dimension M;
representing the Kronecker product of the matrix; b is 11 Indicates a relation of B 1 The sub-diagonal matrix of (a); epsilon i Denotes the sub-direction of epsilonAn amount; h is 1i Represents h 1 The subvectors of (1); g 1i Represents G 1 A sub-matrix of (a);
wherein
B 11 ＝2diag([||u-s 1 || ||u-s 2 || … ||u-s N ||])
ε i ＝[ε i1 ε i2 … ε iN ] T
Wherein diag (x) represents a matrix with each element in the vector as a diagonal; o i×j A zero matrix with a number of rows i and a number of columns j is shown.
4. The method for TDOA-bias reduction location based on underwater multi-base sonar according to claim 3, wherein the step 3 comprises:
determining a cost function J for solving the pseudowire equation of step 2
In the formula W 1 Represents a corresponding weighting matrix, represented as
E(εε T ) Is a covariance matrix on ε, expressed as
Wherein Q τ An error covariance matrix representing the TDOA observations; τ represents a matrix form of 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 ]And dimension vector
Transforming the cost function J into
J＝V T A T W 1 AV。
5. The method for TDOA-bias reduction location based on underwater multi-base sonar according to claim 4, wherein the step 4 comprises:
decomposing the augmentation matrix A into A = A ο +. DELTA.A, wherein A ο Representing the precisely known part of the augmentation matrix A, and Δ A representing the error part of A, respectively
A ο ＝[-G 1 ο h 1 ο ]
△A＝[-△G 1 △h 1 ]
In the formula
Represents G 1 Is free of the error term portion of (a),
denotes h 1 Is expressed as
△G 1 Represents G 1 Part of the error term of (c), Δ h 1 Denotes h 1 Is represented as
In the formula
△h 1i ＝D i1 △τ i +D i2 σ c
In the formula D i1 Indicates a correlation of i Error correlation matrix of D i2 Represents a correlation of c Is expressed as an error correlation vector of
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, the expectation is obtained for two 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 an expectation of a cost function J;
with the second term in E (J) as a constant constraint, construct the following about
Is optimized and solved model
Wherein Ω represents a second order error correlation term, and k represents an arbitrary constant;
in the formula of omega 1 、Ω 2 And Ω 3 Different sub-matrices, each representing Ω, each represented as
Wherein Ω is 1 (m 1 :m 2 In (b) represents a matrix omega 1 M of 1 、m 2 A sub-matrix of rows; omega 1 (m 1 :m 2 ,n 1 :n 2 ) M < th > of the representation matrix 1 、m 2 Line and nth 1 、n 2 A sub-matrix of columns; omega 2 (m: n) represents a sub-vector composed of m-th to n-th columns of the vector; omega 2 (m) represents a vector Ω 2 The mth element of (1); trace (, denotes the trace of the matrix).
6. The method for TDOA-bias-based TDOA-biased reduction positioning based on underwater multi-base sonar according to claim 5, wherein in the step 5, the estimation result of the optimization solution model about V is recorded as V
Then
Is estimated value of
Is shown as
7. The method for TDOA offset-based TDOA offset-reduced positioning based on underwater multi-base sonar according to claim 6, wherein the step 6 comprises the following steps:
according to the first-order Taylor series expansion method,
each element in (1) is represented as
The error of the optimization solution model to the target source position estimation is recorded as
From the above set of equations, the following matrix equation can be obtained
In the formula h 2 Denotes h 1 About
Of a transition form of (C), G 2 Represents G 1 About
Respectively, are represented as
According to the method of least squares,
is estimated value of
Is shown as
In the formula W 2 Representing the corresponding weighting matrix
8. The method for TDOA-bias-reduction positioning based on underwater multi-base sonar according to claim 7, wherein in the step 7, the final estimation result of the target source location is
Is shown as
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