CN102590809B - Multiple target tracking method for sequential square root decomposition - Google Patents

Abstract

The invention discloses a multiple target tracking method for sequential square root decomposition, aiming to solve the technical problem of the existing target tracking method that the numerical structure is instable to cause the target tracking failure during the tracking of the radar. According to the technical scheme of the invention, the variance matrix with estimation errors are subjected to the sequential square root decomposition so that a numerical stable structural model is established, and two positive semidefinite matrixes of the variance matrix without estimation errors are subtracted so that symmetric matrixes with negative eigenvalues can not be generated in a processing system of a finite word length. Because the variance matrixes with estimation errors are subjected to the sequential square root decomposition, the numerical stable structural model is established, the subtraction of the positive semidefinite matrixes of the variance matrix with estimation errors is avoided, the numerical divergence can not occur to the processing system of a finite word length, the reliability of the target tracking system is guaranteed hereby, and the target tracking failure during the tracking of the radar and the systematic error of the whole radar are avoided.

Description

The multi-object tracking method that sequence square root decomposes

Technical field

The present invention relates to a kind of Radar Multi Target tracking, the multi-object tracking method that particularly a kind of sequence square root decomposes, belongs to areas of information technology.

Background technology

Multitarget Tracking is all widely used at military and civil area, can be used for aerial target and detects, follows the tracks of and attack, Air Missile defence, air traffic control, harbour and marine surveillance etc.In recent years, along with the change of battlefield surroundings, the development of antagonism and anti-countermeasure techniques, has produced the series of problems such as the strong clutter of background, low signal-to-noise ratio, low detection probability and high false alarm rate, and the precision of multi-object tracking method and accuracy are had higher requirement.

The object of multiple target tracking is by received information source corresponding to measurement of detector, forms different observation set or track, according to track, estimates the number of tracked target and the kinematic parameter of each target, realizes the tracking to a plurality of targets.For the basic filtering method of multiple goal state estimation have that alpha-beta filtering, alpha-beta-γ filtering, Kalman filtering, EKF, gaussian sum are approximate, optimal nonlinear filtering, particle filter and auto adapted filtering etc.Alpha-beta and alpha-beta-γ wave filter are due to simple in structure, and calculated amount is little, and when computer resource is short in early days, application is very wide.Kalman filtering is a kind of basic skills of multiple target tracking, but need to know the mathematical models of system, and is only applicable to linear system, has limited the application of algorithm.EKF expands to non-linear field by kalman filtering theory, is similar to the conditional probability distribution of state by a Gaussian distribution; And when approximate condition does not meet, Gaussian sum filter device is similar to the conditional probability distribution of state by the weighted sum of a Gaussian distribution.Optimal nonlinear filtering is described the dynamic process of target with Makov transition probability, have good characteristic, but calculated amount is larger, is therefore never used widely.Particle filter adopts stochastic sampling, because calculated amount is too large and particle degenerate problem, is not suitable for practical application.In order to improve particle filter, Unscented kalman filtering adopts deterministic sampling, and the particle point number of sampling is reduced, and avoided the particle point degenerate problem in particle filter, so its application is very wide.Adaptive filter method, by the detection to target maneuver, is adjusted the state of filter parameter or increase wave filter in real time, makes wave filter adapt in real time target travel, is particularly suitable for the tracking to maneuvering target; At present, actual radar tracking system the most frequently used be still JPDA (Joint Probabilistic Data Association, JPDA) method (James A.Roecker, A Class of Near Optimal JPDA Algorithms, IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, 1994, VOL.30 (2): 504-510), other method great majority are to simplification of JPDA method etc.Yet, in the variance battle array that the method errors such as JPDA are estimated, there are two positive semidefinite matrixs to subtract each other, in the disposal system of limited wordlength, can produce the symmetric matrix that contains positive and negative eigenwert, cause radar tracking enabling objective to lose and follow and whole radar system mistake.

Summary of the invention

In order to solve the unstable technological deficiency that causes radar tracking enabling objective to lose and follow of existing method for tracking target value structure, the invention provides the multi-object tracking method that a kind of sequence square root decomposes, the method is in the time and measurement renewal of multiple target tracking, by the sequence square root to estimation error variance battle array, decompose, set up numerical stability structural model, not having has two positive semidefinite matrixs to subtract each other in the variance battle array of estimation of error, in the disposal system of limited wordlength, can guarantee can not produce the symmetric matrix that contains negative eigenwert, can avoid radar tracking enabling objective to lose follows and whole radar system mistake.

The present invention solves the technical scheme that its technical matters adopts, the multi-object tracking method that a kind of sequence square root decomposes, and its feature comprises the following steps:

1, in N target following of definition, the discretization model of i target is

x _i(k+1)＝Φ(k+1，k)x _i(k)+Λω _i(k)，

Wherein:

for state vector, (x, y, z) is the position coordinates of target under ground reference rectangular coordinate system, ω _i(k) represent process noise vector, Φ (k+1, k)=Φ=diag[Φ ₁, Φ ₁, Φ ₁] be state-transition matrix, $\mathrm{\Λ}={\∫}_{\mathrm{kT}}^{(k+1)T}\mathrm{\Φ}(k+1,\mathrm{\τ})\mathrm{\Γ}\left(\mathrm{\τ}\right)\mathrm{d\τ}=\left[\begin{array}{ccc}{\mathrm{\Λ}}_1& 0& 0\\ 0& {\mathrm{\Λ}}_1& 0\\ 0& 0& {\mathrm{\Λ}}_1\end{array}\right],$ Γ (t) is matrix of coefficients, $\mathrm{\Γ}=\left[\begin{array}{ccc}{\mathrm{\Γ}}_1& 0& 0\\ 0& {\mathrm{\Γ}}_1& 0\\ 0& 0& {\mathrm{\Γ}}_1\end{array}\right],$ Γ ₁＝[0?0?1] ^T， ${\mathrm{\Φ}}_1=\left[\begin{array}{ccc}1& T& \frac{1}{2}{T}^2\\ 0& 1& T\\ 0& 0& 1\end{array}\right],$ ${\mathrm{\Λ}}_1={\left[\begin{array}{ccc}\frac{1}{6}{T}^3& \frac{1}{2}{T}^2& T\end{array}\right]}^T,$ T is the sampling period;

The time of i target is updated to:

x _i(k/k-1)＝Φx _i(k-1/k-1)

${\stackrel{\‾}{S}}_i(k/k-1)=\mathrm{\Φ}{S}_i(k-1/k-1)$

$F_i=\left[\begin{array}{cccc}{f}_1& f_2& \·\·\·& f_9\end{array}\right]=\left\{f_{\mathrm{jl}}\right\}={\stackrel{\‾}{S}}_i^{-T}(k/k-1)\mathrm{\Λ}$

$S_0=\left[\begin{array}{cccc}{S}_{01}& S_{02}& \·\·\·& S_{0n}\end{array}\right]={\stackrel{\‾}{S}}^{-1}(k/k-1)$

Q _i(k-1)＝diag[Q _i1，Q _i2，Q _i3]

S ^-1(k/k-1)＝[S ₁?S ₂?…?S _n]

Wherein: n=9, x _i(k/k-1) be that i target is to kT one-step prediction value constantly, Q _i(k) be ω _i(k) variance,

for the variance battle array of corresponding one-step prediction error, S _i(k/k-1) be upper triangular matrix, C _ljfor S _i(k+1/k) the capable j column element of l; Starting condition is x _iand S (0/0) _i(0/0);

2, i target observation equation is: z _i(k)=g _i[x _i(k)]+v _i(k)

Wherein: z _i(k) be the r dimension observation vector to i target, g _i[x _i(k)] be corresponding output, v _i(k) represent to measure noise; Calculate

$F_i=\left[\begin{array}{cccc}{f}_1& f_2& \·\·\·& f_9\end{array}\right]=\left\{f_{\mathrm{jl}}\right\}=S_i^T(k/k-1)H^T\left(k\right)$

S(k/k-1)＝S ₀＝[S ₀₁?S ₀₂?…?S _0n]

R _i＝diag[R ₁，R ₂，…，R _r]

$\stackrel{\‾}{S}(k/k)=\left[\begin{array}{cccc}{S}_1& S_2& \·\·\·& S_n\end{array}\right]$

G _i(k)＝B _rn/α _rn

$x_i(k/k)=x_i(k/k-1)+G_i\left(k\right)\{\underset{j=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\λ}}_{\mathrm{ij}}\left(k\right)z_{\mathrm{ij}}\left(k\right)-g_i[x_i(k/k-1)\left]\right\}$

Wherein: n=9,

for upper triangular matrix, z _ij(k) be radar to the j of i target (j=1,2 ..., m) individual echo, x _i(k/k) be i target kT filter value constantly, R _i(k) be v _i(k) variance, λ _ij(k) be weight coefficient, and: $\underset{j=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\λ}}_{i,j}\left(k\right)=1,$ $H_i\left(k\right)=\frac{\∂g_i\left[x_i\left(k\right)\right]}{\∂x_i\left(k\right)}{|}_{x_i\left(k\right)=x_i(k/k-1)};$

3, i tracking method of estimation is:

$F_i=\left[\begin{array}{cccc}{f}_1& f_2& \·\·\·& f_9\end{array}\right]=\left\{f_{\mathrm{jl}}\right\}={\stackrel{\‾}{S}}_i^{-T}(k/k)G_i(k+1)d^T(I-\mathrm{\Ωu}{u}^T)$

$S_0=\left[\begin{array}{cccc}{S}_{01}& S_{02}& \·\·\·& S_{0n}\end{array}\right]={\stackrel{\‾}{S}}^{-1}(k/k)$

Ω(k)＝diag[λ _i，1，λ _i，2，…，λ _i，m]

S ^-1(k/k)＝[S ₁?S ₂?…?S _n]

Wherein: n=9, S _i(k/k) be upper triangular matrix,

it is i target kT variance of estimaion error battle array constantly;

$u=\left[\begin{array}{c}1\\ 1\\ \·\\ \·\\ \·\\ 1\end{array}\right],$ $d=\left[\begin{array}{c}{\mathrm{\Δ}}_{i,1}^T\left(k\right)\\ {\mathrm{\Δ}}_{i,2}^T\left(k\right)\\ \·\\ \·\\ \·\\ {\mathrm{\Δ}}_{i,m}^T\left(k\right)\end{array}\right];$

Δ _{i, j}(k) be j candidate's echo information vector,

Δ _i，j(k)＝z _i，j(k`)-g _i[x _i(k/k-1)]。

Useful result of the present invention is: the method is in the time and measurement renewal of multiple target tracking, by the three subsequence square roots to estimation error variance battle array, decompose, set up the multiple target tracking structural model of numerical stability, avoided two positive semidefinite matrixs in the variance battle array of estimation of error to subtract each other, in the disposal system of limited wordlength, there will not be numerical value to disperse, thus guaranteed multi-object tracking method reliability, avoided radar tracking enabling objective lose with and whole radar system mistake.

Below in conjunction with example, the present invention is elaborated.

Embodiment

1, in N target following of definition, the discretization model of i target is

x _i(k+1)＝Φ(k+1，k)x _i(k)+Λω _i(k)，

Wherein:

for state vector, (x, y, z) is the position coordinates of target under ground reference rectangular coordinate system, ω _i(k) represent process noise vector, Φ (k+1, k)=Φ=diag[Φ ₁, Φ ₁, Φ ₁] be state-transition matrix, $\mathrm{\Λ}={\∫}_{\mathrm{kT}}^{(k+1)T}\mathrm{\Φ}(k+1,\mathrm{\τ})\mathrm{\Γ}\left(\mathrm{\τ}\right)\mathrm{d\τ}=\left[\begin{array}{ccc}{\mathrm{\Λ}}_1& 0& 0\\ 0& {\mathrm{\Λ}}_1& 0\\ 0& 0& {\mathrm{\Λ}}_1\end{array}\right],$ Γ (t) is matrix of coefficients, $\mathrm{\Γ}=\left[\begin{array}{ccc}{\mathrm{\Γ}}_1& 0& 0\\ 0& {\mathrm{\Γ}}_1& 0\\ 0& 0& {\mathrm{\Γ}}_1\end{array}\right],$ Γ ₁＝[0?0?1] ^T， ${\mathrm{\Φ}}_1=\left[\begin{array}{ccc}1& T& \frac{1}{2}{T}^2\\ 0& 1& T\\ 0& 0& 1\end{array}\right],$ ${\mathrm{\Λ}}_1={\left[\begin{array}{ccc}\frac{1}{6}{T}^3& \frac{1}{2}{T}^2& T\end{array}\right]}^T,$ T is the sampling period;

The time of i target is updated to:

x _i(k/k-1)＝Φx _i(k-1/k-1)

${\stackrel{\‾}{S}}_i(k/k-1)=\mathrm{\Φ}{S}_i(k-1/k-1)$

$F_i=\left[\begin{array}{cccc}{f}_1& f_2& \·\·\·& f_9\end{array}\right]=\left\{f_{\mathrm{jl}}\right\}={\stackrel{\‾}{S}}_i^{-T}(k/k-1)\mathrm{\Λ}$

$S_0=\left[\begin{array}{cccc}{S}_{01}& S_{02}& \·\·\·& S_{0n}\end{array}\right]={\stackrel{\‾}{S}}^{-1}(k/k-1)$

Q _i(k-1)＝diag[Q _i1，Q _i2，Q _i3]

S ^-1(k/k-1)＝[S ₁?S ₂?…?S _n]

Wherein: n=9, x _i(k/k-1) be that i target is to kT one-step prediction value constantly, Q _i(k) be ω _i(k) variance,

for the variance battle array of corresponding one-step prediction error, S _i(k/k-1) be upper triangular matrix, C _ljfor S _i(k+1/k) the capable j column element of l; Starting condition is x _iand S (0/0) _i(0/0);

2, i target observation equation is: z _i(k)=g _i[x _i(k)]+v _i(k)

Wherein: z _i(k) be the r dimension observation vector to i target, g _i[x _i(k)] be corresponding output, v _i(k) represent to measure noise; Calculate

$F_i=\left[\begin{array}{cccc}{f}_1& f_2& \·\·\·& f_9\end{array}\right]=\left\{f_{\mathrm{jl}}\right\}=S_i^T(k/k-1)H^T\left(k\right)$

S(k/k-1)＝S ₀＝[S ₀₁?S ₀₂?…?S _0n]

R _i＝diag[R ₁，R ₂，…，R _r]

$\stackrel{\‾}{S}(k/k)=\left[\begin{array}{cccc}{S}_1& S_2& \·\·\·& S_n\end{array}\right]$

G _i(k)＝B _rn/α _rn

$x_i(k/k)=x_i(k/k-1)+G_i\left(k\right)\{\underset{j=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\λ}}_{\mathrm{ij}}\left(k\right)z_{\mathrm{ij}}\left(k\right)-g_i[x_i(k/k-1)\left]\right\}$

Wherein: n=9,

for upper triangular matrix, z _ij(k) be radar to the j of i target (j=1,2 ..., m) individual echo, x _i(k/k) be i target kT filter value constantly, R _i(k) be v _i(k) variance, λ _ij(k) be weight coefficient, and:

for example get g _i[x _i(k)]=[r _i(k) α _i(k) β _i(k)] ^t, r _ifor radar can be measured oblique distance, α _ifor angular altitude, β _iposition angle, and

$\left\{\begin{array}{c}{r}_i=\sqrt{x_i^2+y_i^2+z_i^2}\\ {\mathrm{\α}}_i={\tan }^{-1}\frac{z_i}{\sqrt{x_i^2+y_i^2}}\\ {\mathrm{\β}}_i={\tan }^{-1}\frac{x_i}{y_i}\end{array}\right.$

$H_i\left(k\right)=\frac{\∂g_i\left[x_i\left(k\right)\right]}{\∂x_i\left(k\right)}{|}_{x_i\left(k\right)=x_i(k/k-1)}$

3, i tracking method of estimation is:

$F_i=\left[\begin{array}{cccc}{f}_1& f_2& \·\·\·& f_9\end{array}\right]=\left\{f_{\mathrm{jl}}\right\}={\stackrel{\‾}{S}}_i^{-T}(k/k)G_i(k+1)d^T(I-\mathrm{\Ωu}{u}^T)$

$S_0=\left[\begin{array}{cccc}{S}_{01}& S_{02}& \·\·\·& S_{0n}\end{array}\right]={\stackrel{\‾}{S}}^{-1}(k/k)$

Ω(k)＝diag[λ _i，1，λ _i，2，…，λ _i，m]

S ^-1(k/k)＝[S ₁?S ₂?…?S _n]

Wherein: n=9, S _i(k/k) be upper triangular matrix,

it is i target kT variance of estimaion error battle array constantly;

$u=\left[\begin{array}{c}1\\ 1\\ \·\\ \·\\ \·\\ 1\end{array}\right],$ $d=\left[\begin{array}{c}{\mathrm{\Δ}}_{i,1}^T\left(k\right)\\ {\mathrm{\Δ}}_{i,2}^T\left(k\right)\\ \·\\ \·\\ \·\\ {\mathrm{\Δ}}_{i,m}^T\left(k\right)\end{array}\right];$

Δ _{i, j}(k) be j candidate's echo information vector, Δ _{i, j}(k)=z _{i, j}(k`)-g _i[x _i(k/k-1)].

Claims (1)

1. the multi-object tracking method that sequence square root decomposes, is characterized in that comprising the following steps:

(1), in N target following of definition, the discretization model of i target is

x _i(k+1)=Φ(k+1,k)x _i(k)+Λω _i(k)，

Wherein:

for state vector, (x, y, z) is the position coordinates of target under ground reference rectangular coordinate system, ω _i(k) represent process noise vector, Φ (k+1, k)=Φ=diag[Φ ₁, Φ ₁, Φ ₁] be state-transition matrix, $\mathrm{\Λ}={\∫}_{\mathrm{kT}}^{(k+1)T}\mathrm{\Φ}(k+1,\mathrm{\τ})\mathrm{\Γd\τ}=\left[\begin{array}{ccc}{\mathrm{\Λ}}_1& 0& 0\\ 0& {\mathrm{\Λ}}_1& 0\\ 0& 0& {\mathrm{\Λ}}_1\end{array}\right],$ $\mathrm{\Γ}=\left[\begin{array}{ccc}{\mathrm{\Γ}}_1& 0& 0\\ 0& {\mathrm{\Γ}}_1& 0\\ 0& 0& {\mathrm{\Γ}}_1\end{array}\right],$ Γ ₁=[0?0?1] ^T， ${\mathrm{\Φ}}_1=\left[\begin{array}{ccc}1& T& \frac{1}{2}{T}^2\\ 0& 1& T\\ 0& 0& 1\end{array}\right],$ ${\mathrm{\Λ}}_1={\left[\begin{array}{ccc}\frac{1}{6}{T}^3& \frac{1}{2}{T}^2& T\end{array}\right]}^T,$ T is the sampling period;

The time of i target is updated to:

x _i(k/k-1)=Φx _i(k-1/k-1)

${\stackrel{\‾}{S}}_i(k/k-1)=\mathrm{\Φ}{S}_i(k-1/k-1)$

$F_i=\left[\begin{array}{cccc}{f}_1& f_2& ...& f_9\end{array}\right]=\left\{f_{\mathrm{jl}}\right\}={\stackrel{\‾}{S}}_i^{-T}(k/k-1)\mathrm{\Λ}$

$S_0=\left[\begin{array}{cccc}{S}_{01}& S_{02}& ...& S_{0n}\end{array}\right]={\stackrel{\‾}{S}}_i^{-1}(k/k-1)$

Q _i(k-1)=diag[Q _i1,Q _i2,Q _i3]

$\left.\begin{array}{c}{\mathrm{\α}}_{l0}=Q_{\mathrm{il}}\\ B_{l0}=0\\ \left.\begin{array}{c}{\mathrm{\α}}_{\mathrm{lj}}={\mathrm{\α}}_{l(j-1)+f_{\mathrm{jl}}}\\ M_{\mathrm{lj}}={({\mathrm{\α}}_{l(j-1)}/{\mathrm{\α}}_{\mathrm{lj}})}^{\frac{1}{2}}\\ N_{\mathrm{lj}}=f_{\mathrm{jl}}/{\left({\mathrm{\α}}_{l(j-1)}{\mathrm{\α}}_{\mathrm{lj}}\right)}^{\frac{1}{2}}\\ S_j\←M_{\mathrm{lj}}{S}_{(l-1)j}-B_{l(j-1)N_{\mathrm{lj}}}\\ B_{\mathrm{lj}}=B_{l(j-1)}+S_{(l-1)j}{f}_{\mathrm{jl}}\end{array}\right\}j=\mathrm{1,2},...,n\end{array}\right\}l=\mathrm{1,2},...,3$

$S_i^{-1}(k/k-1)=\left[\begin{array}{cccc}{S}_1& S_2& ...& S_n\end{array}\right]$

Wherein: n=9, x _i(k/k-1) be that i target is to kT one-step prediction value constantly, Q _i(k) be ω _i(k) variance,

for the variance battle array of corresponding one-step prediction error, S _i(k/k-1) be upper triangular matrix, C _l,jfor S _i(k+1/k) the capable j column element of l; Starting condition is x _iand S (0/0) _i(0/0);

(2), i target observation equation is: z _i(k)=g _i[x _i(k)]+v _i(k)

Wherein: z _i(k) be the r dimension observation vector to i target, g _i[x _i(k)] be corresponding output, v _i(k) represent to measure noise; Calculate

$F_i=\left[\begin{array}{cccc}{f}_1& f_2& ...& f_9\end{array}\right]=\left\{f_{\mathrm{jl}}\right\}=S_i^T(k/k-1)H_i^T\left(k\right)$

$\stackrel{\‾}{S}(k/k-1)=S_0=\left[\begin{array}{cccc}{S}_{01}& S_{02}& ...& S_{0n}\end{array}\right]$

R _i(k)=diag[R ₁,R ₂,…,R _r]

$\left.\begin{array}{c}{\mathrm{\α}}_{l0}=R_l\\ B_{l0}=0\\ \left.\begin{array}{c}{\mathrm{\α}}_{\mathrm{lj}}={\mathrm{\α}}_{l(j-1)+f_{\mathrm{jl}}}\\ M_{\mathrm{lj}}={({\mathrm{\α}}_{l(j-1)}/{\mathrm{\α}}_{\mathrm{lj}})}^{\frac{1}{2}}\\ N_{\mathrm{lj}}=f_{\mathrm{jl}}/{\left({\mathrm{\α}}_{l(j-1)}{\mathrm{\α}}_{\mathrm{lj}}\right)}^{\frac{1}{2}}\\ S_j\←M_{\mathrm{lj}}{S}_{(l-1)j}-B_{l(j-1)N_{\mathrm{lj}}}\\ B_{\mathrm{lj}}=B_{l(j-1)}+S_{(l-1)j}{f}_{\mathrm{jl}}\end{array}\right\}j=\mathrm{1,2},...,n\end{array}\right\}l=\mathrm{1,2},...,r$

$\stackrel{\‾}{S}(k/k)=\left[\begin{array}{cccc}{S}_1& S_2& ...& S_n\end{array}\right]$

G _i(k)=B _rn/α _rn

$x_i(k/k)=x_i(k/k-1)+G_i\left(k\right)\{\underset{j=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\λ}}_{i,j}\left(k\right)z_{i,j}\left(k\right)-g_i[x_i(k/k-1)\left]\right\}$

Wherein: n=9,

for upper triangular matrix, z _i,j(k) be j the echo of radar to i target, j=1,2 ..., m, x _i(k/k) be i target kT filter value constantly, R _i(k) be v _i(k) variance, λ _i,j(k) be weight coefficient, and: $\underset{j=1}{\overset{m}{\mathrm{\Σ}}}{\mathrm{\λ}}_{i,j}\left(k\right)=1,$ $H_i\left(k\right)=\frac{{\∂g}_i\left[x_i\left(k\right)\right]}{{\∂x}_i\left(k\right)}{|}_{x_i\left(k\right)=x_i(k/k-1)};$

(3), i target following method of estimation is:

$F_i=\left[\begin{array}{cccc}{f}_1& f_2& ...& f_9\end{array}\right]=\left\{f_{\mathrm{jl}}\right\}={\stackrel{\‾}{S}}_i^{-T}(k/k)G_i(k+1)d^T(I-{\mathrm{\Ωuu}}^T)$

$S_0=\left[\begin{array}{cccc}{S}_{01}& S_{02}& ...& S_{0n}\end{array}\right]={\stackrel{\‾}{S}}^{-1}(k/k)$

Ω(k)=diag[λ _i,1(k),λ _i,2(k),…,λ _i,m(k)]

$\left.\begin{array}{c}{\mathrm{\α}}_{l0}={\mathrm{\λ}}_{i,l\left(k\right)}\\ B_{l0}=0\\ \left.\begin{array}{c}{\mathrm{\α}}_{\mathrm{lj}}={\mathrm{\α}}_{l(j-1)+f_{\mathrm{jl}}}\\ M_{\mathrm{lj}}={({\mathrm{\α}}_{l(j-1)}/{\mathrm{\α}}_{\mathrm{lj}})}^{\frac{1}{2}}\\ N_{\mathrm{lj}}=f_{\mathrm{jl}}/{\left({\mathrm{\α}}_{l(j-1)}{\mathrm{\α}}_{\mathrm{lj}}\right)}^{\frac{1}{2}}\\ S_j\←M_{\mathrm{lj}}{S}_{(l-1)j}-B_{l(j-1)N_{\mathrm{lj}}}\\ B_{\mathrm{lj}}=B_{l(j-1)}+S_{(l-1)j}{f}_{\mathrm{jl}}\end{array}\right\}j=\mathrm{1,2},...,n\end{array}\right\}l=\mathrm{1,2},...,m$

$S_i^{-1}(k/k)=\left[\begin{array}{cccc}{S}_1& S_2& ...& S_n\end{array}\right]$

Wherein: n=9, S _i(k/k) be upper triangular matrix,

it is i target kT variance of estimaion error battle array constantly;

$u=\begin{array}{}\end{array}\left[\begin{array}{c}1\\ 1\\ .\\ .\\ .\\ 1\end{array}\right],$ $d=\left[\begin{array}{c}{\mathrm{\Δ}}_{i,1}^T\left(k\right)\\ {\mathrm{\Δ}}_{i,2}^T\left(k\right)\\ .\\ .\\ .\\ {\mathrm{\Δ}}_{i,m}^T\left(k\right)\end{array}\right];$

△ _i,j(k) be j candidate's echo information vector,

△ _i,j(k)=z _i,j(k)-g _i[x _i(k/k-1)]。