阅读说明：本技术 一种分布式mimo雷达系统运动目标定位方法 (Moving target positioning method of distributed MIMO radar system ) 是由 朱健东 李坤 王满喜 戴幻尧 乔会东 陈冬冬 刘海业 崔新风 于 2020-04-15 设计创作，主要内容包括：本发明涉及分布式雷达系统技术领域,公开的一种分布式MIMO雷达系统运动目标定位方法,以双基地距离(BR)作为观测量,包括：分布式MIMO雷达定位场景、约束总体最小二乘定位；首先通过构建辅助向量,将非线性的双基地距离的观测方程进行线性化处理,将定位问题建立为约束总体最小二乘模型,并采用牛顿迭代方法对模型求解,从而得到目标位置估计。本发明利用BR观测,对目标位置和速度进行估计,对于发射单元和接收单元的数量没有额外要求,避免了方程系数矩阵和数据向量中存在的误差,相比于现有的BR定位方法,在测量误差较大时、在定位精度方面具有显著的优势及优越性。(The invention relates to the technical field of distributed radar systems, and discloses a method for positioning a moving target of a distributed MIMO radar system, which takes bistatic distance (BR) as observed quantity and comprises the following steps: a distributed MIMO radar positioning scene and constraint total least square positioning; firstly, constructing an auxiliary vector, carrying out linearization treatment on an observation equation of a nonlinear bistatic distance, establishing a positioning problem as a constrained total least square model, and solving the model by adopting a Newton iteration method, thereby obtaining target position estimation. The method estimates the target position and speed by utilizing BR observation, has no additional requirements on the quantity of the transmitting units and the receiving units, avoids errors existing in an equation coefficient matrix and a data vector, and has remarkable advantages and superiority in positioning precision when the measurement error is larger compared with the conventional BR positioning method.)

1. A distributed MIMO radar system moving target positioning method is characterized in that: the method comprises the following steps: a distributed MIMO radar positioning scene and constraint total least square positioning;

1) distributed MIMO radar positioning scene:

setting a scene to have M transmitting units, N receiving units and a target; position x ═ x, y of target]^TIs a parameter to be estimated; the position of the transmitting unit m isThe position of the receiving unit n isThe distance of the target to the transmitting unit m is then expressed as:

in the formula, | | | | represents a 2-norm; likewise, the distance of the target to the receiving unit n is:

by definition, the BR for the transmitting unit m and the receiving unit n are respectively

Since the BR observation is non-linear with respect to the target position, it is difficult to directly obtain an algebraic solution for the target position; to linearize the BR observation equation, equation (3) is shifted to

The square of the two sides of the formula (4) is finished to obtain

Expression (5) corresponding to the M transmitting units and the N receiving units in the form of a matrix is

Aθ＝b (6)

In the formula:

obviously, the observation equations of MN BR are shared corresponding to M transmitting units and N receiving units; the purpose is to estimate the position of the target through the MN observation equations;

2) and (3) constraining total least square positioning, and estimating the least square of the target position as the least square of the target position when the matrix A in the formula (6) has no error and the error in the vector b is Gaussian white noise with zero mean value

θ_LS＝(A^TA)^-1A^Tb (12)

However, in the application, there will be an error in the matrix a, and the error in the vector b is not gaussian white noise with zero mean, and at this time, the equation (12) is not the optimal estimation of the target position; for this purpose, the target position is estimated by using the CTLS algorithm;

representing BR observations in vector formWherein r is_m＝[r_m,1,r_m,2,...,r_m,N]T; the actual value of BR observation is set asWhereinCorresponding observation error isWherein e_m＝[e_m,1,e_m,1,...,e_m,N]T, the covariance matrix is Q; then

r＝r^o+e (13)

Considering the effect of BR error e on matrix a and vector b, equation (6) is expressed as a function of the observed quantity r:

A(r-e)θ＝b(r-e) (14)

taylor expansion is carried out on A (r-e) and b (r-e) at the measured value r, and second-order error terms and above are ignored to obtain

(A-ΔA)θ＝b-Δb (15)

In the formula

ΔA＝[O_MN×3,F₁e,F₂e,...,F_Me],Δb＝F_M+1e (16)

F_M+1＝2diag{r₁,r₂,...,r_M} (18)

Since the noise components in the model are statistically correlated,when the noise components are statistically correlated, the noise components need to be whitened; cholesky decomposition of Q yields Q ═ E (ee)^T)＝PP^TObtaining a whitened noise vector

＝P^-1e (19)

Let G_m＝F_mP, then Δ A and Δ b are represented by

ΔA＝[O_MN×3,G₁,G₂,...,G_M],Δb＝G_M+1(20)

When formula (20) is substituted for formula (15), formula (15) is represented by

In the formula

Solving the CTLS solution of the target position, namely determining a proper solution vector under the constraint of the formula (21) so that the target function | | | | survival²Minimum; the mathematical expression is as follows:

the formula (22) is a minimization problem of a quadratic function under the constraint of a quadratic constraint equation and is transformed into an unconstrained minimization problem of a minimized variable theta; derived from the constraint of equation (22)

＝G⁺(Aθ-b) (23)

In the formula G⁺＝G^T(GG^T)^-1Moore-Penrose inverse of matrix G; when equation (23) is substituted into the objective function of the formula (22) CTLS model, the CTLS solution of the target location is a variable that satisfies the minimization of the following objective function:

in the process of realizing the linearization of the BR observation equation, introducingAuxiliary vectorIt contains both the target location parameter of interest and M auxiliary parameters; the functional relationship between the target position parameter and the auxiliary parameter is shown as formula (1); in order to eliminate the auxiliary parameters, an objective function only containing the target position is obtained; substituting formula (1) into formula (24) to obtain

F(θ)＝g(x)^TWg(x)＝F(x) (25)

In the formula

Wherein A is_1:2＝[A₁,A₂]，A_mRepresenting the m-th column vector in the matrix A;

due to the nonlinearity of equation (25), it is very difficult to minimize f (x) by analytical methods; solving the solution by adopting a Newton iteration method; assume that an initial estimate x of the target position has been obtained₀Selecting the least squares solution θ in equation (12)_LSAs an initial estimate, F (x) is set at x₀Treating Taylor expansion and neglecting error terms of three orders and above to obtain

Wherein the content of the first and second substances,

the partial derivative of x is obtained by applying the formula (29)

Wherein

In the formulaRepresenting the Kronecker tensor product; f (x) the minimum requirement is that the partial derivative is 0; for this purpose, based on the formula (27)

Solving the formula (34) to obtain a Newton iteration formula of

x＝x₀-H^-1J (35)

Then, the target estimation flow is as follows:

1) and calculating a least squares solution of the target position using equation (12) and using the least squares solution as an initial solution x for Newton iteration₀；

2) Calculating a more accurate target position estimate using equation (35);

3) and (3) taking the solution obtained in the step (2) as an initial solution, and repeating the step (2) until convergence.

Technical Field

The invention relates to the technical field of distributed MIMO radar systems, in particular to a method for positioning a moving target of a distributed MIMO radar system.

Background

The distributed radar adopts a receiving and transmitting split structure, each pair of transmitting and receiving units form an independent bistatic system to observe a target, and then all echo results are transmitted to a fusion center to be processed by signal or information fusion and the like, so that information such as target position, speed and the like is obtained. Compared with the traditional radar, the radar has remarkable advantages in the aspects of improving the target detection performance, processing slow targets, improving the positioning accuracy and the like.

Time delay is a common parameter for positioning moving objects in distributed MIMO radar. The time delay observation directly corresponds to the sum of the propagation distances of the signals from the transmitting unit to the receiving unit after reflection by the target, also known as bistatic distances. Thus, with time-delayed positioning, it is essentially positioning observed with the BR. In the case of radiation source localization problems, the time delay corresponds to the distance difference (RD). Currently, the problem of positioning of radiation sources using RD has been extensively studied. However, the target location problem with BR is relatively less studied than with BR. However, in recent years, as research on the positioning problem of the distributed MIMO radar is gradually focused, some algorithms are developed successively for the positioning problem of the moving target observed by using BR in the distributed MIMO radar.

A positioning method based on least square is provided for distributed MIMO radar target positioning based on BR, firstly, BR observation is converted into RD observation through selecting a reference station, and then positioning of a target is realized by utilizing the RD observation. But the positioning accuracy of the RD-based positioning method is lower than that of the BR-based positioning method. There is also a weighted least squares based target location method, which first linearizes the BR observation for a single transmit unit, then extends it to multiple transmit units by constructing an auxiliary vector, and finally determines the position of the target using a weighted least squares algorithm. However, this method ignores the constraint relationship between the auxiliary parameter and the target position parameter in the auxiliary vector, and therefore, it is not a theoretically optimal method. And also, by taking the classic two-step weighted least square) idea as reference, aiming at the problem of positioning a moving target in a distributed MIMO radar system, a two-step weighted least square (Group-2SWLS) method based on grouping is provided, the basic idea is that the observation of the system is divided into a plurality of groups according to a transmitting unit, then target position parameters are independently estimated from each Group by using the two-step weighted least square idea, and finally the estimation results of each Group are fused to obtain the final estimation of the target position and speed. The Group-2SWLS method is based on independent observation errors of all groups, but in practical application, the assumption is generally difficult to satisfy, so that the positioning performance of the algorithm is reduced.

There is also a two-step weighted least squares method that does not require grouping. CRLB can be achieved when the measurement error is small, but its performance degradation is severe when the measurement error is large. Therefore, further research is necessary to address the problem of target location in distributed MIMO radar.

In recent years, the constrained total least squares method has been successfully applied to the RD-based radiation source localization problem, resulting in a localization accuracy superior to two-step weighted least squares.

Disclosure of Invention

Aiming at the problem of target positioning in the distributed MIMO radar, the invention provides a method for positioning a moving target of a distributed MIMO radar system, which is suitable for a constrained total least square method for positioning the target of the distributed MIMO radar. The target position and velocity are estimated using BR observations. The constrained total least square concept is applied to the positioning model, and compared with the existing algorithm, the method has remarkable advantages in positioning accuracy.

In order to achieve the purpose, the invention adopts the following technical scheme:

a method for positioning a moving target of a distributed MIMO radar system comprises the following steps:

1. distributed MIMO radar positioning scenarios:

suppose there are M transmitting units, N receiving units, a target in the scene; position x ═ x, y of target]^TIs a parameter to be estimated; the position of the transmitting unit m isThe position of the receiving unit n isThen, the distance of the target to the transmitting unit m can be expressed as:

in the formula, | | | | represents a 2-norm; likewise, the distance of the target to the receiving unit n is:

by definition, the BR for the transmitting unit m and the receiving unit n are respectively

Since the BR observation is non-linear with respect to the target position, it is difficult to directly obtain an algebraic solution for the target position; to linearize the BR observation equation, equation (3) is shifted to

The square of the two sides of the formula (4) is finished to obtain

Expression (5) corresponding to M transmitting units and N receiving units is expressed in matrix form as a θ ═ b (6)

In the formula:

obviously, the observation equations of MN BR are shared corresponding to M transmitting units and N receiving units; the primary object of the present invention is to,

it is through the MN observation equations that the position of the target is estimated.

2. Constraining the global least squares positioning, when there is no error in matrix A in equation (6) and the error in vector b is white Gaussian noise with zero mean, the least squares estimate of the target position is

θ_LS＝(A^TA)^-1A^Tb (12)

However, in practical applications, there are inevitable errors in the matrix a, and the error in the vector b is not white gaussian noise with zero mean, and at this time, the equation (12) is not the optimal estimation of the target position; for this purpose, the target position is estimated by using the CTLS algorithm;

representing BR observations in vector formWherein r is_m＝[r_m,1,r_m,2,...,r_m,N]^T(ii) a Assuming that the actual value of its BR observation isWhereinCorresponding observation error isWherein e_m＝[e_m,1,e_m,1,...,e_m,N]^TThe covariance matrix is Q; then

r＝r^o+e (13)

Considering the effect of BR error e on matrix a and vector b, equation (6) is expressed as a function of the observed quantity r:

A(r-e)θ＝b(r-e) (14)

taylor expansion is carried out on A (r-e) and b (r-e) at the measured value r, and second-order error terms and above are ignored to obtain

(A-ΔA)θ＝b-Δb (15)

In the formula

ΔA＝[O_MN×3,F₁e,F₂e,...,F_Me],Δb＝F_M+1e (16)

F_M+1＝2diag{r₁,r₂,...,r_M} (18)

Because the noise components in the model are statistically correlated, when the noise components are statistically correlated, the noise components need to be whitened; cholesky decomposition of Q yields Q ═ E (ee)^T)＝PP^TObtaining a whitened noise vector

＝P^-1e (19)

Let G_m＝F_mP, then Δ A and Δ b are represented by

ΔA＝[O_MN×3,G₁,G₂,...,G_M],Δb＝G_M+1(20)

When formula (20) is substituted into formula (15), formula (15) is expressed as

In the formula

Solving the CTLS solution of the target position, namely determining a proper solution vector under the constraint of the formula (21) so that the target function | | | | survival²And minimum. The mathematical expression is as follows:

the formula (22) is a minimization problem of a quadratic function under the constraint of a quadratic constraint equation and is transformed into an unconstrained minimization problem of a minimized variable theta; derived from the constraint of equation (22)

＝G⁺(Aθ-b) (23)

In the formula G⁺＝G^T(GG^T)^-1Moore-Penrose inverse of matrix G; when equation (23) is substituted into the objective function of the formula (22) CTLS model, the CTLS solution of the target location is a variable that satisfies the minimization of the following objective function:

in the process of realizing the linearization of the BR observation equation, an auxiliary vector is introducedIt contains both the target location parameter of interest and the M auxiliary parameters. The functional relationship between the target position parameter and the auxiliary parameter is shown as formula (1); in order to eliminate the auxiliary parameters, an objective function only containing the target position is obtained; substituting formula (1) into formula (24) to obtain

F(θ)＝g(x)^TWg(x)＝F(x) (25)

In the formula

Wherein A is₁:₂＝[A₁,A₂]，A_mRepresenting the m-th column vector in the matrix A;

due to the nonlinearity of equation (25), it is very difficult to minimize f (x) by analytical methods; solving the solution by adopting a Newton iteration method; assume that an initial estimate x of the target position has been obtained₀Selecting the least squares solution θ in equation (12)_LSAs an initial estimate of the position of the target,

f (x) is in x₀Treating Taylor expansion and neglecting error terms of three orders and above to obtain

Wherein the content of the first and second substances,

the partial derivative of x is obtained by applying the formula (29)

Wherein

In the formulaRepresenting the Kronecker tensor product; f (x) the minimum requirement is that the partial derivative is 0; for this purpose, based on the formula (27),

to obtain

Solving the formula (34) to obtain a Newton iteration formula of

x＝x₀-H^-1J (35)

Then, the summary of the target estimation procedure is as follows:

1. the least squares solution for the target position is calculated using equation (12) and taken as the initial solution x for the newton iteration₀；

2. Calculating a more accurate target position estimate using equation (35);

3. and (3) taking the solution obtained in the step (2) as an initial solution, and repeating the step (2) until convergence.

Due to the adoption of the technical scheme, the invention has the following advantages:

the invention provides a method for positioning a moving target of a distributed MIMO radar system by taking bistatic distance (BR) as an observed quantity aiming at the problem of target positioning in the distributed MIMO radar system. Firstly, an auxiliary vector is constructed, a nonlinear bistatic distance observation equation is subjected to linearization treatment, a positioning problem is established as a constraint total least square model in consideration of errors in an equation coefficient matrix and a data vector, and the model is solved by a Newton iteration method, so that target position estimation is obtained. In a simulation experiment, the method is compared with the existing method, and the superiority of the positioning method is verified. Compared with the prior BR

The positioning method has obvious advantages in positioning precision when the measurement error is large. And the positioning method of the invention has no additional requirement on the number of the transmitting units and the receiving units.

Drawings

FIG. 1 is a geometric profile of a positioning system;

FIG. 2(a) is a schematic diagram of the positioning accuracy of the algorithm for the near-field target under the measurement error condition that the Bias varies with the measurement error;

FIG. 2(b) is a schematic diagram of the positioning accuracy of the algorithm on the near-field target under the measurement error condition that the RMSE varies with the measurement error;

fig. 3(a) is a graph showing the influence of the RMSE on the positioning accuracy, where N is 5 and σ is 10m, depending on the number of transmitting units;

fig. 3(b) is a graph showing the influence of the RMSE on the positioning accuracy, where N is 5 and σ is 10 m.

Detailed Description

As shown in fig. 1 to fig. 3(b), a method for locating a moving object in a distributed MIMO radar system includes:

1. distributed MIMO radar positioning scenarios:

suppose there are M transmitting units, N receiving units, a target in the scene; position x ═ x, y of target]^TIs a parameter to be estimated;

the position of the transmitting unit m isThe position of the receiving unit n isThen, the distance of the target to the transmitting unit m can be expressed as:

in the formula, | | | | represents a 2-norm; likewise, the distance of the target to the receiving unit n is:

by definition, the BR for the transmitting unit m and the receiving unit n are respectively

Since the BR observation is non-linear with respect to the target position, it is difficult to directly obtain an algebraic solution for the target position; to linearize the BR observation equation, equation (3) is shifted to

The square of the two sides of the formula (4) is finished to obtain

Expression (5) corresponding to M transmitting units and N receiving units is expressed in matrix form as a θ ═ b (41)

In the formula:

obviously, the observation equations of MN BR are shared corresponding to M transmitting units and N receiving units; the primary object of the present invention is to,

it is through the MN observation equations that the position of the target is estimated.

2. Constraining the global least squares positioning, when there is no error in matrix A in equation (6) and the error in vector b is white Gaussian noise with zero mean, the least squares estimate of the target position is

θ_LS＝(A^TA)^-1A^Tb (47)

However, in practical applications, there are inevitable errors in the matrix a, and the error in the vector b is not white gaussian noise with zero mean, and at this time, the equation (12) is not the optimal estimation of the target position; for this purpose, the target position is estimated by using the CTLS algorithm;

representing BR observations in vector formWherein r is_m＝[r_m,1,r_m,2,...,r_m,N]^T(ii) a Assuming that the actual value of its BR observation isWhereinCorresponding observation error isWherein e_m＝[e_m,1,e_m,1,...,e_m,N]^TThe covariance matrix is Q; then

r＝r^o+e (48)

Considering the effect of BR error e on matrix a and vector b, equation (6) is expressed as a function of the observed quantity r:

A(r-e)θ＝b(r-e) (49)

taylor expansion is carried out on A (r-e) and b (r-e) at the measured value r, and second-order error terms and above are ignored to obtain

(A-ΔA)θ＝b-Δb (50)

In the formula

ΔA＝[O_MN×3,F₁e,F₂e,...,F_Me],Δb＝F_M+1e (51)

F_M+1＝2diag{r₁,r₂,...,r_M} (53)

Because the noise components in the model are statistically correlated, when the noise components are statistically correlated, the noise components need to be whitened; cholesky decomposition of Q yields Q ═ E (ee)^T)＝PP^TObtaining a whitened noise vector

＝P^-1e (54)

Let G_m＝F_mP, then Δ A and Δ b are represented by

ΔA＝[O_MN×3,G₁,G₂,...,G_M],Δb＝G_M+1(55)

When formula (20) is substituted into formula (15), formula (15) is expressed as

In the formula

Solving the CTLS solution of the target position, namely determining a proper solution vector under the constraint of the formula (21) so that the target function | | | | survival²And minimum. The mathematical expression is as follows:

the formula (22) is a minimization problem of a quadratic function under the constraint of a quadratic constraint equation and is transformed into an unconstrained minimization problem of a minimized variable theta; derived from the constraint of equation (22)

＝G⁺(Aθ-b) (58)

In the formula G⁺＝G^T(GG^T)^-wMoore-Penrose inverse of matrix G; by substituting equation (23) into the objective function of equation (22) CTLS model, the target positionThe CTLS solution is a minimized variable that satisfies the following objective function:

in the process of realizing the linearization of the BR observation equation, an auxiliary vector is introducedIt contains both the target location parameter of interest and the M auxiliary parameters. The functional relationship between the target position parameter and the auxiliary parameter is shown as formula (1); in order to eliminate the auxiliary parameters, an objective function only containing the target position is obtained; substituting formula (1) into formula (24) to obtain

F(θ)＝g(x)^TW_g(x)＝F(x) (60)

In the formula

Wherein A is₁:₂＝[A₁,A₂]，A_mRepresenting the m-th column vector in the matrix A;

due to the nonlinearity of equation (25), it is very difficult to minimize f (x) by analytical methods; solving the solution by adopting a Newton iteration method; assume that an initial estimate x of the target position has been obtained₀Selecting the least squares solution θ in equation (12)_LSAs an initial estimate of the position of the target,

f (x) is in x₀Treating Taylor expansion and neglecting error terms of three orders and above to obtain

Wherein the content of the first and second substances,

the partial derivative of x is obtained by applying the formula (29)

Wherein

In the formulaRepresenting the Kronecker tensor product; f (x) the minimum requirement is that the partial derivative is 0; for this purpose, based on the formula (27),

to obtain

Solving the formula (34) to obtain a Newton iteration formula of

x＝x₀-H^-1J (70)

Then, the summary of the target estimation procedure is as follows:

1. the least squares solution for the target position is calculated using equation (12) and taken as the initial solution x for the newton iteration₀；

2. Calculating a more accurate target position estimate using equation (35);

3. and (3) taking the solution obtained in the step (2) as an initial solution, and repeating the step (2) until convergence.

The invention evaluates the positioning performance of the algorithm through a simulation experiment, and the scene design of the simulation experiment is as followsThe following: there are 9 transmit units, 9 receive units, and 1 object in a 2-dimensional scene, whose positions are shown in fig. 1. The measurement error of bistatic distance is set to be Gaussian distribution with zero mean value, and the covariance matrix is Q ═ sigma²And R, wherein the elements on the main diagonal of the matrix R are 1, and the rest are 0.5. The positioning Error of the algorithm is the Root Mean Square Error (RMSE) and the Bias (Bias) of 3000 monte carlo simulations.

Firstly, in order to highlight the positioning performance of the method, the method is utilized to carry out a simulation positioning experiment under the condition of different measurement errors, the root mean square error and the deviation of the algorithm are counted and compared with the algorithm proposed by Noroozi, the algorithm proposed by Einemo, the algorithm proposed by Park and CRLB. The target is located by using 5 transmitting units Tx 1-Tx 5 and 5 receiving units Rx.1-Rx.5.

FIG. 2(a) shows that the BR measurement error is 0.1m to 10⁴m, the positioning deviation condition of the algorithm. It can be seen that the bias of the algorithm proposed by Noroozi is lower than that of Einemo and Park, but greater than that of the present algorithm. Furthermore, due to the non-linearity of the positioning model, the deviation of all four positioning algorithms increases with the increase of the measurement error. FIG. 2(b) shows that the BR measurement error is 0.1m to 10⁴m, the root mean square error condition of the algorithm positioning. As can be seen from fig. 2(b), in terms of RMSE index, the algorithm proposed by Noroozi fails to reach CRLB because it ignores the constraint relationship between the auxiliary parameter and the target position parameter. The positioning performance of the other three algorithms is equivalent, and when the measurement error is small, the CRLB can be approached. However, as can be seen from the enlarged view, the algorithm is closest to CRLB. When the measurement error exceeds 10^2.5When m is reached, the algorithm proposed by Einemo and the algorithm proposed by Park deviate from the CRLB rapidly, and under the condition of the measurement error, the method can still obtain a solution approaching the accuracy of the CRLB.

In order to analyze the influence of the number of the transmitting units and the receiving units on the positioning effect, an algorithm is utilized to carry out a simulation positioning experiment under the condition of different numbers of the transmitting units and the receiving units. The simulation results are shown in fig. 3.

Fig. 3 shows the RMSE for algorithmic positioning for different numbers of transmit and receive units. It can be seen that as the number of transmit and receive units increases, the RMSE of several algorithms decreases. Compared with the other three algorithms, the RMSE of the algorithm can reach CRLB, and the RMSE is positioned to be superior to the other three algorithms.