CN103701433A - Quantitative filtering method of time-varying target tracking system under condition with multiple measurement loss - Google Patents

Abstract

The invention discloses a quantitative filtering method of a time-varying target tracking system under the condition with multiple measurement loss and relates to a quantitative filtering method of the time-varying target tracking system. By adoption of the quantitative filtering method, the problems that the filtering method adopting the traditional target tracking system can not handle the phenomena of multiple measurement loss and signal quantification under the networked environment and has an influence on the accuracy of signal estimation of the target tracking system. The quantitative filtering method disclosed by the invention has the advantages that the influence of multiple measurement loss and signal quantification on the filtering performance is considered, and the remainder estimation is carried out on the nonlinear linearization process; compared with the existing filtering method of the target tracking system, the filtering method disclosed by the invention has the advantages that the phenomena of multiple measurement loss and output signal quantification can be handled simultaneously, an obtained filter has time-varying and recursive forms, the upper bound of filtering error covariance is optimized, and the signal estimation performance is improved. The quantitative filtering method is applicable to quantitative filtering for the time-varying target tracking system.

Description

A kind of multiple measurement is lost the quantification filtering method that situation lower time becomes Target Tracking System

Technical field

The present invention relates to become when a kind of the quantification filtering method of Target Tracking System.

Background technology

Since network control system is suggested, it become rapidly one of primary study direction of academic and industrial circle.Compare traditional control system, network control system has resource-sharing, remote operation and control, improve the reliability of system, be convenient to the advantages such as I&M of system.Network control system has all obtained practical application widely in a plurality of fields such as military affairs, process automation, traffic system, tele-medicine and intelligent buildings.Yet when communication network is introduced in feedback control loop, it is more complicated that the analysis and design of control system also becomes.Characteristic due to network itself, the network is guided phenomenon (multiple measurement loss, signal quantization etc.) starts to be concerned, they are the one of the main reasons that cause entire system performance to worsen, and the existence of the network is guided phenomenon has proposed new challenge to traditional control method.

In system running, due to reasons such as the polytropy of network environment and unsteadiness, measurement data is lost, signal quantization phenomenon may be often simultaneous, yet mostly existing method is to induce phenomenon to process for a certain particular network, thereby has to a certain extent certain conservative.In addition, the filtering method majority of the current stochastic system about net environment is all to process linear time varying system or non-linear time-invariant system, and Target Tracking System is a typical nonlinear and time-varying system, multiple survey data volume under current filtering method ground reply networked environment is lost and signal quantization phenomenon, affects the Signal estimation accuracy of Target Tracking System.

Summary of the invention

The present invention adopts the filtering method of traditional Target Tracking System can not tackle under networked environment multiple measurement in order to solve is lost and signal quantization phenomenon, affect the problem of the Signal estimation accuracy of Target Tracking System, proposed a kind of multiple measurement and lost the quantification filtering method that situation lower time becomes Target Tracking System.

A kind of multiple measurement of the present invention is lost the quantification filtering method that situation lower time becomes Target Tracking System, and the concrete steps of the method are:

Step 1, set up multiple measurement and lose the dynamic model that situation lower time becomes Target Tracking System:

In the process of tracking target, adopt the radar system with a plurality of transducers to collect metrical information, set up have that multiple measurement is lost and signal quantization situation under, be subject to that multiplicative noise and additive noise affect time become Target Tracking System model:

$\left\{\begin{array}{c}{x}_{k+1}=f\left(x_k\right)+\underset{i=1}{\overset{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000449162480000011.png,HDA0000449162480000021.png"\; state="\{\{state\}\}">n</figure-callout>}}_1}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_{i,k}{A}_{i,k}{x}_k+{\mathrm{\&omega;}}_k\\ y_k={\mathrm{\&Xi;}}_k{C}_k{x}_k+\underset{i=1}{\overset{{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="HDA0000449162480000011.png,HDA0000449162480000021.png"\; state="\{\{state\}\}">m</figure-callout>}}_1}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{i,k}{C}_{i,k}{x}_k+v_k\end{array}\right.---\left(1\right)$

In formula (1):

f(x _k)=Φ _kx _k+G(h(x _k)+H)，

$h\left(x_k\right)=-\frac{\mathrm{g\&rho;}\left(x_{2,k}\right)}{2\mathrm{\&beta;}}\sqrt{{\stackrel{\&CenterDot;}{x}}_{1,\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="HDA0000449162480000021.png"\; state="\{\{state\}\}">k</figure-callout>}}^2+{\stackrel{\&CenterDot;}{x}}_{2,k}^2}\left[\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_{1,k}\\ {\stackrel{\&CenterDot;}{x}}_{2,k}\end{array}\right],$

$\mathrm{\&rho;}\left(x_{2,k}\right)={\mathrm{\&theta;}}_1{e}^{-{\mathrm{\&theta;}}_2{x}_{2,k}}$

${\mathrm{\&Phi;}}_k=\left[\begin{array}{cccc}1& \mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HDA0000449162480000051.png"\; state="\{\{state\}\}">T</figure-callout>}& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& \mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HDA0000449162480000051.png"\; state="\{\{state\}\}">T</figure-callout>}\\ 0& 0& 0& 1\end{array}\right],$

$G=\left[\begin{array}{cc}\frac{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA0000449162480000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^2}{2}& 0\\ \mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HDA0000449162480000051.png"\; state="\{\{state\}\}">T</figure-callout>}& 0\\ 0& \frac{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA0000449162480000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^2}{2}\\ 0& T\end{array}\right],$

$H=\left[\begin{array}{c}0\\ -g\end{array}\right]$

$A_{i,k}=\left[\begin{array}{cccc}0.12\sin \left(k\right)& 0& 0& 0\\ 0& -0.02& 0& 0\\ 0& 0& 0.1\sin \left(2k\right)& 0\\ 0& 0& 0& 0.15\end{array}\right]$

$C_k=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right],$

$C_{i,k}=\left[\begin{array}{cccc}0.15& 0& 0& 0\\ 0& 0& 0.3& 0\end{array}\right]$

X _k+1for the state variable of k+1 Target Tracking System constantly, y _kmeasurement output for k Target Tracking System model constantly; $x_k={\left[\begin{array}{cccc}{x}_{1,k}& {\stackrel{\&CenterDot;}{x}}_{1,k}& x_{2,k}& {\stackrel{\&CenterDot;}{x}}_{2,k}\end{array}\right]}^T$ For the state variable of k Target Tracking System constantly, subscript " ^t" transposition of representing matrix, x _{1, k}for k target abscissa constantly,

for x _{1, k}derivative, x _{2, k}for k target ordinate constantly,

for x _{2, k}derivative; f(x _k) be nonlinear function; T is the sampling period; G is acceleration of gravity; β is ballistic coefficient; ρ () is atmospheric density, works as x _{2, k}during <9144 rice, θ ₁=1.227, θ ₂=1.093 * 10 ^-4, work as x _{2, k}in the time of>=9144 meters, θ ₁=1.754, θ ₂=1.49 * 10 ^-4; α _i,k, β _i,k, ω _kand ν _kbe all the white Gaussian noise of zero-mean, it is 1,1, Q that variance is respectively _k=cdiag{q, q}, c=0.1m ²/ s ³, diag{} represents diagonal matrix,

$q=\left[\begin{array}{cc}\frac{{\mathrm{<figure-callout\; id="3"\; label="T"\; filenames="HDA0000449162480000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^3}{3}& \frac{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA0000449162480000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^2}{2}\\ \frac{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA0000449162480000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^2}{2}& T\end{array}\right],R_k=100\&CenterDot;\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right];$ N ₁and m ₁all positive integers;

Matrix

the matrix of losing for portraying multiple measurement,

Wherein,

the stochastic variable distributing for obeying Bernoulli Jacob, i=1,2 ..., m; M represents the number of transducer;

Step 2, the measurement output of the Target Tracking System model of step 1 is carried out to logarithmic quantification, obtain the quantized data of the measurement output of Target Tracking System model

Pass through formula:

$q_i\left(y_k^j\right)=\left\{\begin{array}{cc}{u}_i^{\left(j\right)},& \frac{1}{1+{\mathrm{\&delta;}}_j}{u}_i^{\left(j\right)}<y_k^j\&le;\frac{1}{1-{\mathrm{\&delta;}}_j}{u}_i^{\left(j\right)}\\ 0,& y_k^j=0\\ -q_j(-y_k^j),& y_k^j<0\end{array}\right.,j=\mathrm{1,2},...,m---\left(2\right)$

Logarithmic quantification is carried out in measurement output to Target Tracking System model;

In formula,

be that j transducer is at k measured value constantly;

${\mathrm{\&delta;}}_j=\frac{1-{\mathrm{\&chi;}}^{\left(j\right)}}{1+{\mathrm{\&chi;}}^{\left(j\right)}};$

χ ^(j)the constant of value in (0,1);

for quantization resolution, its value is taken from level set Μ _j;

$M_j=\{\&PlusMinus;u_i^{\left(j\right)},u_i^{\left(j\right)}={\left({\mathrm{\&chi;}}^{\left(j\right)}\right)}^i{u}_0^{\left(j\right)},i=0,\&PlusMinus;1,\&PlusMinus;2,...\}\&cup;\left\{0\right\},u_0^{\left(j\right)}>0---\left(3\right)$

Multiple measurement loss of data phenomenon matrix in step 3, calculation procedure one mathematic expectaion, obtain the probability that measurement data is lost;

Step 4, according to the quantized data of the measurement output of measurement data losing probability and tracking system model, multiple measurement is lost to situation lower time and becomes Target Tracking System state and carry out one-step prediction and k+1 state estimation constantly;

Pass through formula:

${\hat{x}}_{k+1|k}=f\left({\hat{x}}_{k|k}\right)---\left(4\right)$

${\hat{x}}_{k+1|k+1}={\hat{x}}_{k+1|k}+G_{k+1}({\tilde{y}}_{k+1}-{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}{\hat{x}}_{k+1|k})---\left(5\right)$

Obtain k one-step prediction value constantly with the k+1 estimation of system mode constantly

In formula,

for x _kstate estimation in k system constantly;

for nonlinear function exists the functional value at place;

G _k+1for filter gain matrix;

for the k+1 information that filter receives constantly,

Wherein, $q\left(y_{k+1}\right)={\left[\begin{array}{cccc}{q}_1\left(y_{k+1}^1\right)& q_2\left(y_{k+1}^2\right)& ...& q_m\left(y_{k+1}^m\right)\end{array}\right]}^T,$

Y _k+1for k+1 measurement output constantly, for the measurement output of k+1 the 1st transducer constantly, for the measurement output of k+1 the 2nd transducer constantly,

measurement output for k+1 m transducer constantly;

for k+1 moment stochastic variable

mathematic expectaion,

for k+1 moment stochastic variable

mathematic expectaion,

for k+1 moment stochastic variable mathematic expectaion;

Step 5, utilize the one-step prediction of the system mode that step 4 obtains

with k+1 state estimation value constantly

to the nonlinear function f (x in step 1 _k) carry out linearization process and remainder estimation, obtain one-step prediction error and the k+1 filtering error constantly of system mode

Formula (1) is deducted to formula (4) and obtain one-step prediction error

${\tilde{x}}_{k+1|k}=f\left(x_k\right)-f\left({\hat{x}}_{k|k}\right)+\underset{i=1}{\overset{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000449162480000011.png,HDA0000449162480000021.png"\; state="\{\{state\}\}">n</figure-callout>}}_1}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_{i,k}{A}_{i,k}{x}_k+D_k{\mathrm{\&omega;}}_k---\left(6\right)$

In formula (6),

for k+1 one-step prediction error constantly;

To nonlinear function f (x _k)

taylor series expansion is done at place:

$f\left(x_k\right)=f\left({\hat{x}}_{k|k}\right)+A_k{\tilde{x}}_{k|k}+o\left(\right|{\tilde{x}}_{k|k}\left|\right)---\left(7\right)$

In formula (7), for k filtering error constantly,

$A_k={\frac{\&PartialD;f\left(x_k\right)}{{\&PartialD;x}_k}|}_{x_k={\hat{x}}_{k|k}},$

for function f (x _k) about x _kpartial derivative,

$o\left(\right|{\tilde{x}}_{k|k}\left|\right)=B_k{\mathrm{\&Theta;}}_k{L}_k{\tilde{x}}_{k|k}$ For the higher order term of Taylor series expansion,

B _kand L _kbounded matrix, matrix Θ _kportray linearisation error, it meets

i is unit matrix;

Step 6, utilize one-step prediction error

covariance recurrence equation and filtering error

covariance recurrence equation, obtain one-step prediction error P _k+1|kwith filtering error P _k+1|k+1;

The covariance of one-step prediction error is passed through formula:

$P_{k+1|k}=E\left\{{\tilde{x}}_{k+1|k}{\tilde{x}}_{k+1|k}^T\right\},$

Obtain, in formula,

for

mathematic expectaion,

By the covariance recurrence equation of one-step prediction error:

$P_{k+1|k}=(A_k+B_k{\mathrm{\&Theta;}}_k{L}_k)P_{k|k}{(A_k+B_k{\mathrm{\&Theta;}}_k{L}_k)}^T+\underset{i=1}{\overset{n_1}{\mathrm{\&Sigma;}}}{A}_{i,k}E\left\{x_k{x}_k^T\right\}{A}_{i,k}^T+Q_k---\left(8\right)$

Obtain one-step prediction error P _k+1|k,

In formula (8),

for

mathematic expectaion,

The covariance of filtering error is passed through formula:

$P_{k|k}=E\left\{{\tilde{x}}_{k|k}{\tilde{x}}_{k|k}^T\right\}$

Obtain, in formula, for

mathematic expectaion,

By the covariance recurrence equation of filtering error:

$\begin{array}{c}{P}_{k+1|k+1}=(I-G_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1})P_{k+1|k}{(I-G_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1})}^T+E\{H_{k+1}+H_{k+1}^T\}\\ +G_{k+1}{\mathrm{\&Delta;}}_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}E\left\{x_{k+1}{x}_{k+1}^T\right\}{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{\mathrm{\&Delta;}}_{k+1}^T{G}_{k+1}^T\\ +G_{k+1}(I+{\mathrm{\&Delta;}}_{k+1})(X_{k+1}+{\mathrm{\&Pi;}}_{k+1}+R_{k+1}){(I+{\mathrm{\&Delta;}}_{k+1})}^T{G}_{k+1}^T\end{array}---\left(9\right)$

Obtain filtering error P _k+1|k+1,

In formula (9):

${\mathrm{\&Pi;}}_{k+1}=\underset{i=1}{\overset{m_1}{\mathrm{\&Sigma;}}}{C}_{i,k+1}E\left\{x_{k+1}{x}_{k+1}^T\right\}{C}_{i,k+1}^T$

$H_{k+1}=-(I-G_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}){\tilde{x}}_{k+1}{x}_{k+1}^T{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{\mathrm{\&Delta;}}_{k+1}^T{G}_{k+1}^T$

" ο " is Hadamard(Adama) Product Operator,

I is unit matrix,

${\mathrm{\&Delta;}}_{k+1}=\mathrm{diag}\left\{\begin{array}{cccc}{\mathrm{\&Delta;}}_{k+1}^{\left(1\right)}& {\mathrm{\&Delta;}}_{k+1}^{\left(2\right)}& ...& {\mathrm{\&Delta;}}_{k+1}^{\left(m\right)}\end{array}\right\}$ For portraying the diagonal matrix of quantization uncertainty,

for matrix △ _k+1the 1st diagonal components,

for matrix △ _k+1the 2nd diagonal components,

for matrix △ _k+1m diagonal components,

$\left|{\mathrm{\&Delta;}}_{k+1}^{\left(1\right)}\right|\&le;{\mathrm{\&delta;}}_1,\left|{\mathrm{\&Delta;}}_{k+1}^{\left(2\right)}\right|\&le;{\mathrm{\&delta;}}_2,\left|{\mathrm{\&Delta;}}_{k+1}^{\left(m\right)}\right|\&le;{\mathrm{\&delta;}}_m,$

for

norm;

The covariance P of step 7, calculation of filtered error _k+1|k+1the upper bound and construct filter gain matrix G _k+1:

Formula is passed through in the upper bound of the covariance of filtering error:

$\begin{array}{c}{\mathrm{\&Sigma;}}_{k+1|k}=A_k{({\mathrm{\&Sigma;}}_{k|k}^{-1}-{\mathrm{\&gamma;}}_{1,k}{L}_k^T{L}_k)}^{-1}{A}_k^T+{\mathrm{\&gamma;}}_{1,k}^T{B}_k{B}_k^T+Q_k\\ +\underset{i=1}{\overset{n_1}{\mathrm{\&Sigma;}}}{A}_{i,k}[(1+{\mathrm{\&epsiv;}}_1){\mathrm{\&Sigma;}}_{k|k}+(1+{\mathrm{\&epsiv;}}_1^{-1}){\hat{x}}_{k|k}{\hat{x}}_{k|k}^T]A_{i,k}^T\end{array}---\left(10\right)$

$\begin{array}{c}{\mathrm{\&Sigma;}}_{k+1|k+1}=(1+{\mathrm{\&epsiv;}}_2)(I-G_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}){\mathrm{\&Sigma;}}_{k+1|k}{(I-G_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1})}^T\\ +G_{k+1}\{(1+{\mathrm{\&epsiv;}}_2^{-1})\mathrm{tr}\left(\mathrm{\&Lambda;}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}{\mathrm{\&Phi;}}_{k+1|k}{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}\mathrm{\&Lambda;}\right)I\\ +\mathrm{tr}\left({\mathrm{\&Psi;}}_{k+1|k}\right)[{(I-{\mathrm{\&gamma;}}_{2,k+1}\mathrm{\&Lambda;\&Lambda;})}^{-1}+{\mathrm{\&gamma;}}_{2,k+1}^{-1}I]\}{G}_{k+1}^T\end{array}---\left(11\right)$

Obtain Σ _k+1|k+1for filtering error P _k+1|k+1upper bound matrix,

Pass through formula

$\begin{array}{c}{G}_{k+1}={\mathrm{\&Sigma;}}_{k+1|k}{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}\{(1+{\mathrm{\&epsiv;}}_2){\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}{\mathrm{\&Sigma;}}_{k+1|k}{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}\\ +(1+{\mathrm{\&epsiv;}}_2^{-1})\mathrm{tr}\left(\mathrm{\&Lambda;}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}{\mathrm{\&Phi;}}_{k+1|k}{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}\mathrm{\&Lambda;}\right)I\\ {+\mathrm{tr}\left({\mathrm{\&Psi;}}_{k+1|k}\right)[{(I-{\mathrm{\&gamma;}}_{2,k+1}\mathrm{\&Lambda;\&Lambda;})}^{-1}+{\mathrm{\&gamma;}}_{2,k+1}^{-1}I]\}}^{-1}\end{array}---\left(12\right)$

Obtain filter gain matrix G _k+1;

In formula (10), (11) and (12),

${\mathrm{\&Phi;}}_{k+1|k}=(1+{\mathrm{\&epsiv;}}_3){\mathrm{\&Sigma;}}_{k+1|k}+(1+{\mathrm{\&epsiv;}}_3^{-1}){\hat{x}}_{k+1|k}{\hat{x}}_{k+1k}^T$

Λ=diag{δ ₁,δ ₂,…,δ _m}

γ _{1, k}, γ _{2, k+1}, ε ₁, ε ₂and ε ₃all normal number,

for matrix Σ _k|kcontrary,

for ε ₁contrary, tr (Ψ _k+1|k) be matrix Ψ _k+1|kmark;

Step 8, the filter gain matrix G that step 7 is obtained _k+1be brought into the formula (5) in step 4, obtain multiple measurement and lose situation lower time and become the state estimation value of Target Tracking System, realize multiple measurement and lose the quantification filtering that situation lower time becomes Target Tracking System.

Filtering method of the present invention has been considered multiple measurement loss and the impact of signal quantization on filtering performance, and non-linear linearization process has been carried out to remainder estimation, compare with the filtering method of existing Target Tracking System, filtering method of the present invention can be processed multiple measurement loss simultaneously and output signal quantizes phenomenon, when having, the filter obtaining becomes, recursive form, and optimized the filtering error covariance upper bound, have and improved Signal estimation performance, be easy to the advantages such as online application, and with existing filtering method ratioing signal estimate accuracy raising more than 10%.

Accompanying drawing explanation

Fig. 1 is the flow chart of the method for the invention;

Fig. 2 is virtual condition track

and estimation track

comparison diagram, in figure, solid line is virtual condition track

curve, dotted line is for estimating track

curve;

Fig. 3 is virtual condition track

and estimation track

comparison diagram, in figure, solid line is virtual condition track

curve, dotted line is for estimating track curve;

Fig. 4 is virtual condition track

and estimation track

comparison diagram, in figure, solid line is virtual condition track

curve, dotted line is for estimating track

curve;

Fig. 5 is virtual condition track

and estimation track

comparison diagram, in figure, solid line is virtual condition track

curve, dotted line is for estimating track

curve.

Embodiment

Embodiment one, in conjunction with Fig. 1, present embodiment is described, a kind of multiple measurement is lost the quantification filtering method that situation lower time becomes Target Tracking System described in present embodiment, and the concrete steps of the method are:

Step 1, set up multiple measurement and lose the dynamic model that situation lower time becomes Target Tracking System:

In the process of tracking target, adopt the radar system with a plurality of transducers to collect metrical information, set up have that multiple measurement is lost and signal quantization situation under, be subject to that multiplicative noise and additive noise affect time become Target Tracking System model:

$\left\{\begin{array}{c}{x}_{k+1}=f\left(x_k\right)+\underset{i=1}{\overset{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000449162480000011.png,HDA0000449162480000021.png"\; state="\{\{state\}\}">n</figure-callout>}}_1}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_{i,k}{A}_{i,k}{x}_k+{\mathrm{\&omega;}}_k\\ y_k={\mathrm{\&Xi;}}_k{C}_k{x}_k+\underset{i=1}{\overset{{\mathrm{<figure-callout\; id="1"\; label="m"\; filenames="HDA0000449162480000011.png,HDA0000449162480000021.png"\; state="\{\{state\}\}">m</figure-callout>}}_1}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{i,k}{C}_{i,k}{x}_k+v_k\end{array}\right.---\left(1\right)$

In formula (1):

f(x _k)=Φ _kx _k+G(h(x _k)+H)，

$h\left(x_k\right)=-\frac{\mathrm{g\&rho;}\left({\mathrm{<figure-callout\; id="2"\; label="x"\; filenames="HDA0000449162480000021.png"\; state="\{\{state\}\}">x</figure-callout>}}_{2,k}\right)}{2\mathrm{\&beta;}}\sqrt{{\stackrel{\&CenterDot;}{x}}_{1,\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="HDA0000449162480000021.png"\; state="\{\{state\}\}">k</figure-callout>}}^2+{\stackrel{\&CenterDot;}{x}}_{2,k}^2}\left[\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_{1,k}\\ {\stackrel{\&CenterDot;}{x}}_{2,k}\end{array}\right],$

$\mathrm{\&rho;}\left({\mathrm{<figure-callout\; id="2"\; label="x"\; filenames="HDA0000449162480000021.png"\; state="\{\{state\}\}">x</figure-callout>}}_{2,k}\right)={\mathrm{\&theta;}}_1{e}^{-{\mathrm{\&theta;}}_2{x}_{2,k}}$

${\mathrm{\&Phi;}}_k=\left[\begin{array}{cccc}1& \mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HDA0000449162480000051.png"\; state="\{\{state\}\}">T</figure-callout>}& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& \mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HDA0000449162480000051.png"\; state="\{\{state\}\}">T</figure-callout>}\\ 0& 0& 0& 1\end{array}\right],$

$G=\left[\begin{array}{cc}\frac{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA0000449162480000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^2}{2}& 0\\ \mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HDA0000449162480000051.png"\; state="\{\{state\}\}">T</figure-callout>}& 0\\ 0& \frac{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA0000449162480000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^2}{2}\\ 0& T\end{array}\right],$

$H=\left[\begin{array}{c}0\\ -g\end{array}\right]$

$A_{i,k}=\left[\begin{array}{cccc}0.12\sin \left(k\right)& 0& 0& 0\\ 0& -0.02& 0& 0\\ 0& 0& 0.1\sin \left(2k\right)& 0\\ 0& 0& 0& 0.15\end{array}\right]$

$C_k=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right],$

$C_{i,k}=\left[\begin{array}{cccc}0.15& 0& 0& 0\\ 0& 0& 0.3& 0\end{array}\right]$

X _k+1for the state variable of k+1 Target Tracking System constantly, y _kmeasurement output for k Target Tracking System model constantly; $x_k={\left[\begin{array}{cccc}{x}_{1,k}& {\stackrel{\&CenterDot;}{x}}_{1,k}& x_{2,k}& {\stackrel{\&CenterDot;}{x}}_{2,k}\end{array}\right]}^T$ For the state variable of k Target Tracking System constantly, subscript " ^t" transposition of representing matrix, x _{1, k}for k target abscissa constantly, for x _{1, k}derivative, x _{2, k}for k target ordinate constantly,

for x _{2, k}derivative; f(x _k) be nonlinear function; T is the sampling period; G is acceleration of gravity; β is ballistic coefficient; ρ () is atmospheric density, works as x _{2, k}during <9144 rice, θ ₁=1.227, θ ₂=1.093 * 10 ^-4, work as x _{2, k}in the time of>=9144 meters, θ ₁=1.754, θ ₂=1.49 * 10 ^-4; α _i,k, β _i,k, ω _kand ν _kbe all the white Gaussian noise of zero-mean, it is 1,1, Q that variance is respectively _k=cdiag{q, q}, c=0.1m ²/ s ³, diag{} represents diagonal matrix,

$q=\left[\begin{array}{cc}\frac{{\mathrm{<figure-callout\; id="3"\; label="T"\; filenames="HDA0000449162480000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^3}{3}& \frac{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA0000449162480000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^2}{2}\\ \frac{{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA0000449162480000021.png"\; state="\{\{state\}\}">T</figure-callout>}}^2}{2}& T\end{array}\right],R_k=100\&CenterDot;\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right];$ N ₁and m ₁all positive integers;

Matrix

the matrix of losing for portraying multiple measurement,

Wherein,

the stochastic variable distributing for obeying Bernoulli Jacob, i=1,2 ..., m; M represents the number of transducer;

Step 2, the measurement output of the Target Tracking System model of step 1 is carried out to logarithmic quantification, obtain the quantized data of the measurement output of Target Tracking System model

Pass through formula:

$q_i\left(y_k^j\right)=\left\{\begin{array}{cc}{u}_i^{\left(j\right)},& \frac{1}{1+{\mathrm{\&delta;}}_j}{u}_i^{\left(j\right)}<y_k^j\&le;\frac{1}{1-{\mathrm{\&delta;}}_j}{u}_i^{\left(j\right)}\\ 0,& y_k^j=0\\ -q_j(-y_k^j),& y_k^j<0\end{array}\right.,j=\mathrm{1,2},...,m---\left(2\right)$

Logarithmic quantification is carried out in measurement output to Target Tracking System model;

In formula,

be that j transducer is at k measured value constantly;

${\mathrm{\&delta;}}_j=\frac{1-{\mathrm{\&chi;}}^{\left(j\right)}}{1+{\mathrm{\&chi;}}^{\left(j\right)}};$

χ ^(j)the constant of value in (0,1);

for quantization resolution, its value is taken from level set Μ _j;

$M_j=\{\&PlusMinus;u_i^{\left(j\right)},u_i^{\left(j\right)}={\left({\mathrm{\&chi;}}^{\left(j\right)}\right)}^i{u}_0^{\left(j\right)},i=0,\&PlusMinus;1,\&PlusMinus;2,...\}\&cup;\left\{0\right\},u_0^{\left(j\right)}>0---\left(3\right)$

Multiple measurement loss of data phenomenon matrix in step 3, calculation procedure one

mathematic expectaion, obtain the probability that measurement data is lost;

Step 4, according to the quantized data of the measurement output of measurement data losing probability and tracking system model, multiple measurement is lost to situation lower time and becomes Target Tracking System state and carry out one-step prediction and k+1 state estimation constantly;

Pass through formula:

${\hat{x}}_{k+1|k}=f\left({\hat{x}}_{k|k}\right)---\left(4\right)$

${\hat{x}}_{k+1|k+1}={\hat{x}}_{k+1|k}+G_{k+1}({\tilde{y}}_{k+1}-{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}{\hat{x}}_{k+1|k})---\left(5\right)$

Obtain k one-step prediction value constantly

with the k+1 estimation of system mode constantly

In formula,

for x _kstate estimation in k system constantly; for nonlinear function exists

the functional value at place;

G _k+1for filter gain matrix;

for the k+1 information that filter receives constantly,

Wherein, $q\left(y_{k+1}\right)={\left[\begin{array}{cccc}{q}_1\left(y_{k+1}^1\right)& q_2\left(y_{k+1}^2\right)& ...& q_m\left(y_{k+1}^m\right)\end{array}\right]}^T,$

Y _k+1for k+1 measurement output constantly,

for the measurement output of k+1 the 1st transducer constantly, for the measurement output of k+1 the 2nd transducer constantly,

measurement output for k+1 m transducer constantly;

for k+1 moment stochastic variable

mathematic expectaion,

for k+1 moment stochastic variable

mathematic expectaion,

for k+1 moment stochastic variable

mathematic expectaion;

Step 5, utilize the one-step prediction of the system mode that step 4 obtains

with k+1 state estimation value constantly

to the nonlinear function f (x in step 1 _k) carry out linearization process and remainder estimation, obtain one-step prediction error and the k+1 filtering error constantly of system mode

Formula (1) is deducted to formula (4) and obtain one-step prediction error

${\tilde{x}}_{k+1|k}=f\left(x_k\right)-f\left({\hat{x}}_{k|k}\right)+\underset{i=1}{\overset{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA0000449162480000011.png,HDA0000449162480000021.png"\; state="\{\{state\}\}">n</figure-callout>}}_1}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_{i,k}{A}_{i,k}{x}_k+D_k{\mathrm{\&omega;}}_k---\left(6\right)$

In formula (6), for k+1 one-step prediction error constantly;

To nonlinear function f (x _k)

taylor series expansion is done at place:

$f\left(x_k\right)=f\left({\hat{x}}_{k|k}\right)+A_k{\tilde{x}}_{k|k}+o\left(\right|{\tilde{x}}_{k|k}\left|\right)---\left(7\right)$

In formula (7),

for k filtering error constantly,

$A_k={\frac{\&PartialD;f\left(x_k\right)}{{\&PartialD;x}_k}|}_{x_k={\hat{x}}_{k|k}},\frac{\&PartialD;f\left(x_k\right)}{{\&PartialD;x}_k}$ For function f (x _k) about x _kpartial derivative,

$o\left(\right|{\tilde{x}}_{k|k}\left|\right)=B_k{\mathrm{\&Theta;}}_k{L}_k{\tilde{x}}_{k|k}$ For the higher order term of Taylor series expansion,

B _kand L _kbounded matrix, matrix Θ _kportray linearisation error, it meets

i is unit matrix;

Step 6, utilize one-step prediction error covariance recurrence equation and filtering error

covariance recurrence equation, obtain one-step prediction error P _k+1|kwith filtering error P _k+1|k+1;

The covariance of one-step prediction error is passed through formula:

$P_{k+1|k}=E\left\{{\tilde{x}}_{k+1|k}{\tilde{x}}_{k+1|k}^T\right\},$

Obtain, in formula, for

mathematic expectaion,

By the covariance recurrence equation of one-step prediction error:

$P_{k+1|k}=(A_k+B_k{\mathrm{\&Theta;}}_k{L}_k)P_{k|k}{(A_k+B_k{\mathrm{\&Theta;}}_k{L}_k)}^T+\underset{i=1}{\overset{n_1}{\mathrm{\&Sigma;}}}{A}_{i,k}E\left\{x_k{x}_k^T\right\}{A}_{i,k}^T+Q_k---\left(8\right)$

Obtain one-step prediction error P _k+1|k,

In formula (8),

for

mathematic expectaion,

The covariance of filtering error is passed through formula:

$P_{k|k}=E\left\{{\tilde{x}}_{k|k}{\tilde{x}}_{k|k}^T\right\}$

Obtain, in formula,

for

mathematic expectaion,

By the covariance recurrence equation of filtering error:

$\begin{array}{c}{P}_{k+1|k+1}=(I-G_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1})P_{k+1|k}{(I-G_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1})}^T+E\{H_{k+1}+H_{k+1}^T\}\\ +G_{k+1}{\mathrm{\&Delta;}}_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}E\left\{x_{k+1}{x}_{k+1}^T\right\}{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{\mathrm{\&Delta;}}_{k+1}^T{G}_{k+1}^T\\ +G_{k+1}(I+{\mathrm{\&Delta;}}_{k+1})(X_{k+1}+{\mathrm{\&Pi;}}_{k+1}+R_{k+1}){(I+{\mathrm{\&Delta;}}_{k+1})}^T{G}_{k+1}^T\end{array}---\left(9\right)$

Obtain filtering error P _k+1|k+1,

In formula (9):

${\mathrm{\&Pi;}}_{k+1}=\underset{i=1}{\overset{m_1}{\mathrm{\&Sigma;}}}{C}_{i,k+1}E\left\{x_{k+1}{x}_{k+1}^T\right\}{C}_{i,k+1}^T$

$H_{k+1}=-(I-G_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}){\tilde{x}}_{k+1}{x}_{k+1}^T{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{\mathrm{\&Delta;}}_{k+1}^T{G}_{k+1}^T$

" ο " is Hadamard(Adama) Product Operator,

I is unit matrix,

${\mathrm{\&Delta;}}_{k+1}=\mathrm{diag}\left\{\begin{array}{cccc}{\mathrm{\&Delta;}}_{k+1}^{\left(1\right)}& {\mathrm{\&Delta;}}_{k+1}^{\left(2\right)}& ...& {\mathrm{\&Delta;}}_{k+1}^{\left(m\right)}\end{array}\right\}$ For portraying the diagonal matrix of quantization uncertainty,

for matrix △ _k+1the 1st diagonal components,

for matrix △ _k+1the 2nd diagonal components,

for matrix △ _k+1m diagonal components,

$\left|{\mathrm{\&Delta;}}_{k+1}^{\left(1\right)}\right|\&le;{\mathrm{\&delta;}}_1,\left|{\mathrm{\&Delta;}}_{k+1}^{\left(2\right)}\right|\&le;{\mathrm{\&delta;}}_2,\left|{\mathrm{\&Delta;}}_{k+1}^{\left(m\right)}\right|\&le;{\mathrm{\&delta;}}_m,$

for

norm;

The covariance P of step 7, calculation of filtered error _k+1|k+1the upper bound and construct filter gain matrix G _k+1:

Formula is passed through in the upper bound of the covariance of filtering error:

$\begin{array}{c}{\mathrm{\&Sigma;}}_{k+1|k}=A_k{({\mathrm{\&Sigma;}}_{k|k}^{-1}-{\mathrm{\&gamma;}}_{1,k}{L}_k^T{L}_k)}^{-1}{A}_k^T+{\mathrm{\&gamma;}}_{1,k}^T{B}_k{B}_k^T+Q_k\\ +\underset{i=1}{\overset{n_1}{\mathrm{\&Sigma;}}}{A}_{i,k}[(1+{\mathrm{\&epsiv;}}_1){\mathrm{\&Sigma;}}_{k|k}+(1+{\mathrm{\&epsiv;}}_1^{-1}){\hat{x}}_{k|k}{\hat{x}}_{k|k}^T]A_{i,k}^T\end{array}---\left(10\right)$

$\begin{array}{c}{\mathrm{\&Sigma;}}_{k+1|k+1}=(1+{\mathrm{\&epsiv;}}_2)(I-G_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}){\mathrm{\&Sigma;}}_{k+1|k}{(I-G_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1})}^T\\ +G_{k+1}\{(1+{\mathrm{\&epsiv;}}_2^{-1})\mathrm{tr}\left(\mathrm{\&Lambda;}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}{\mathrm{\&Phi;}}_{k+1|k}{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}\mathrm{\&Lambda;}\right)I\\ +\mathrm{tr}\left({\mathrm{\&Psi;}}_{k+1|k}\right)[{(I-{\mathrm{\&gamma;}}_{2,k+1}\mathrm{\&Lambda;\&Lambda;})}^{-1}+{\mathrm{\&gamma;}}_{2,k+1}^{-1}I]\}{G}_{k+1}^T\end{array}---\left(11\right)$

Obtain Σ _k+1|k+1for filtering error P _k+1|k+1upper bound matrix,

Pass through formula

$\begin{array}{c}{G}_{k+1}={\mathrm{\&Sigma;}}_{k+1|k}{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}\{(1+{\mathrm{\&epsiv;}}_2){\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}{\mathrm{\&Sigma;}}_{k+1|k}{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}\\ +(1+{\mathrm{\&epsiv;}}_2^{-1})\mathrm{tr}\left(\mathrm{\&Lambda;}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}{\mathrm{\&Phi;}}_{k+1|k}{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}\mathrm{\&Lambda;}\right)I\\ {+\mathrm{tr}\left({\mathrm{\&Psi;}}_{k+1|k}\right)[{(I-{\mathrm{\&gamma;}}_{2,k+1}\mathrm{\&Lambda;\&Lambda;})}^{-1}+{\mathrm{\&gamma;}}_{2,k+1}^{-1}I]\}}^{-1}\end{array}---\left(12\right)$

Obtain filter gain matrix G _k+1;

In formula (10), (11) and (12),

${\mathrm{\&Phi;}}_{k+1|k}=(1+{\mathrm{\&epsiv;}}_3){\mathrm{\&Sigma;}}_{k+1|k}+(1+{\mathrm{\&epsiv;}}_3^{-1}){\hat{x}}_{k+1|k}{\hat{x}}_{k+1k}^T$

Λ=diag{δ ₁,δ ₂,…,δ _m}

γ _{1, k}, γ _{2, k+1}, ε ₁, ε ₂and ε ₃all normal number,

for matrix Σ _k|kcontrary, for ε ₁contrary, tr (Ψ _k+1|k) be matrix Ψ _k+1|kmark;

Step 8, the filter gain matrix G that step 7 is obtained _k+1be brought into the formula (5) in step 4, obtain multiple measurement and lose situation lower time and become the state estimation value of Target Tracking System, realize multiple measurement and lose the quantification filtering that situation lower time becomes Target Tracking System.

Adopt the method for the invention to carry out emulation:

Adopt parameter:

1) acceleration of gravity: g=9.81m/s ²

2) ballistic coefficient: β=4 * 10 ⁴kg/ms ²

3) atmospheric density parameter: work as x _{2, k}during <9144 rice, θ ₁=1.227, θ ₂=1.093 * 10 ^-4

Work as x _{2, k}in the time of>=9144 meters, θ ₁=1.754, θ ₂=1.49 * 10 ^-4

4) sampling period: T=1s

5) state initial value: x ₀=10 ³* [300 4 90 3] ^t

6) estimate initial value: ${\stackrel{\&OverBar;}{x}}_0={10}^3\&times;{\left[\begin{array}{cccc}270& 4.01& 95& 2.9\end{array}\right]}^T$

Filter gain solves:

Solve formula (10), (11) and (12), obtain filter gain matrix.

Filter effect is as shown in Figures 2 to 5:

From Fig. 2 to Fig. 5, metrical information experience lose and the situation of signal quantization under, the quantification filtering method of inventing is estimating target state effectively, realizes multiple measurement is lost to the quantification filtering that situation lower time becomes Target Tracking System.

Claims (1)

1. multiple measurement is lost the quantification filtering method that situation lower time becomes Target Tracking System, it is characterized in that, the concrete steps of the method are:

Step 1, set up multiple measurement and lose the dynamic model that situation lower time becomes Target Tracking System:

In the process of tracking target, adopt the radar system with a plurality of transducers to collect metrical information, set up have that multiple measurement is lost and signal quantization situation under, be subject to that multiplicative noise and additive noise affect time become Target Tracking System model:

$\left\{\begin{array}{c}{x}_{k+1}=f\left(x_k\right)+\underset{i=1}{\overset{n_1}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_{i,k}{A}_{i,k}{x}_k+{\mathrm{\&omega;}}_k\\ y_k={\mathrm{\&Xi;}}_k{C}_k{x}_k+\underset{i=1}{\overset{m_1}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_{i,k}{C}_{i,k}{x}_k+v_k\end{array}\right.---\left(1\right)$

In formula (1):

f(x _k)=Φ _kx _k+G(h(x _k)+H)，

$h\left(x_k\right)=-\frac{\mathrm{g\&rho;}\left(x_{2,k}\right)}{2\mathrm{\&beta;}}\sqrt{{\stackrel{\&CenterDot;}{x}}_{1,k}^2+{\stackrel{\&CenterDot;}{x}}_{2,k}^2}\left[\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_{1,k}\\ {\stackrel{\&CenterDot;}{x}}_{2,k}\end{array}\right],$

$\mathrm{\&rho;}\left(x_{2,k}\right)={\mathrm{\&theta;}}_1{e}^{-{\mathrm{\&theta;}}_2{x}_{2,k}}$

${\mathrm{\&Phi;}}_k=\left[\begin{array}{cccc}1& T& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& T\\ 0& 0& 0& 1\end{array}\right],$

$G=\left[\begin{array}{cc}\frac{T^2}{2}& 0\\ T& 0\\ 0& \frac{T^2}{2}\\ 0& T\end{array}\right],$

$H=\left[\begin{array}{c}0\\ -g\end{array}\right]$

$A_{i,k}=\left[\begin{array}{cccc}0.12\sin \left(k\right)& 0& 0& 0\\ 0& -0.02& 0& 0\\ 0& 0& 0.1\sin \left(2k\right)& 0\\ 0& 0& 0& 0.15\end{array}\right]$

$C_k=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right],$

$C_{i,k}=\left[\begin{array}{cccc}0.15& 0& 0& 0\\ 0& 0& 0.3& 0\end{array}\right]$

X _k+1for the state variable of k+1 Target Tracking System constantly, y _kmeasurement output for k Target Tracking System model constantly; $x_k={\left[\begin{array}{cccc}{x}_{1,k}& {\stackrel{\&CenterDot;}{x}}_{1,k}& x_{2,k}& {\stackrel{\&CenterDot;}{x}}_{2,k}\end{array}\right]}^T$ For the state variable of k Target Tracking System constantly, subscript " ^t" transposition of representing matrix, x _{1, k}for k target abscissa constantly,

for x _{1, k}derivative, x _{2, k}for k target ordinate constantly, for x _{2, k}derivative; f(x _k) be nonlinear function; T is the sampling period; G is acceleration of gravity; β is ballistic coefficient; ρ () is atmospheric density, works as x _{2, k}during <9144 rice, θ ₁=1.227, θ ₂=1.093 * 10 ^-4, work as x _{2, k}in the time of>=9144 meters, θ ₁=1.754, θ ₂=1.49 * 10 ^-4; α _i,k, β _i,k, ω _kand ν _kbe all the white Gaussian noise of zero-mean, it is 1,1, Q that variance is respectively _k=cdiag{q, q}, c=0.1m ²/ s ³, diag{} represents diagonal matrix,

$q=\left[\begin{array}{cc}\frac{T^3}{3}& \frac{T^2}{2}\\ \frac{T^2}{2}& T\end{array}\right],R_k=100\&CenterDot;\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right];$ N ₁and m ₁all positive integers;

Matrix

the matrix of losing for portraying multiple measurement,

Wherein,

the stochastic variable distributing for obeying Bernoulli Jacob, i=1,2 ..., m; M represents the number of transducer;

Step 2, the measurement output of the Target Tracking System model of step 1 is carried out to logarithmic quantification, obtain the quantized data of the measurement output of Target Tracking System model

Pass through formula:

$q_i\left(y_k^j\right)=\left\{\begin{array}{cc}{u}_i^{\left(j\right)},& \frac{1}{1+{\mathrm{\&delta;}}_j}{u}_i^{\left(j\right)}<y_k^j\&le;\frac{1}{1-{\mathrm{\&delta;}}_j}{u}_i^{\left(j\right)}\\ 0,& y_k^j=0\\ -q_j(-y_k^j),& y_k^j<0\end{array}\right.,j=\mathrm{1,2},...,m---\left(2\right)$

Logarithmic quantification is carried out in measurement output to Target Tracking System model;

In formula, be that j transducer is at k measured value constantly;

${\mathrm{\&delta;}}_j=\frac{1-{\mathrm{\&chi;}}^{\left(j\right)}}{1+{\mathrm{\&chi;}}^{\left(j\right)}};$

χ ^(j)the constant of value in (0,1);

for quantization resolution, its value is taken from level set Μ _j;

$M_j=\{\&PlusMinus;u_i^{\left(j\right)},u_i^{\left(j\right)}={\left({\mathrm{\&chi;}}^{\left(j\right)}\right)}^i{u}_0^{\left(j\right)},i=0,\&PlusMinus;1,\&PlusMinus;2,...\}\&cup;\left\{0\right\},u_0^{\left(j\right)}>0---\left(3\right)$

Multiple measurement loss of data phenomenon matrix in step 3, calculation procedure one

mathematic expectaion, obtain the probability that measurement data is lost;

Step 4, according to the quantized data of the measurement output of measurement data losing probability and tracking system model, multiple measurement is lost to situation lower time and becomes Target Tracking System state and carry out one-step prediction and k+1 state estimation constantly;

Pass through formula:

${\hat{x}}_{k+1|k}=f\left({\hat{x}}_{k|k}\right)---\left(4\right)$

${\hat{x}}_{k+1|k+1}={\hat{x}}_{k+1|k}+G_{k+1}({\tilde{y}}_{k+1}-{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}{\hat{x}}_{k+1|k})---\left(5\right)$

Obtain k one-step prediction value constantly

with the k+1 estimation of system mode constantly

In formula,

for x _kstate estimation in k system constantly;

for nonlinear function exists

the functional value at place;

G _k+1for filter gain matrix;

for the k+1 information that filter receives constantly,

Wherein, $q\left(y_{k+1}\right)={\left[\begin{array}{cccc}{q}_1\left(y_{k+1}^1\right)& q_2\left(y_{k+1}^2\right)& ...& q_m\left(y_{k+1}^m\right)\end{array}\right]}^T,$

Y _k+1for k+1 measurement output constantly,

for the measurement output of k+1 the 1st transducer constantly, for the measurement output of k+1 the 2nd transducer constantly,

measurement output for k+1 m transducer constantly;

for k+1 moment stochastic variable mathematic expectaion,

for k+1 moment stochastic variable mathematic expectaion,

for k+1 moment stochastic variable

mathematic expectaion;

Step 5, utilize the one-step prediction of the system mode that step 4 obtains

with k+1 state estimation value constantly

to the nonlinear function f (x in step 1 _k) carry out linearization process and remainder estimation, obtain one-step prediction error and the k+1 filtering error constantly of system mode;

Formula (1) is deducted to formula (4) and obtain one-step prediction error

${\tilde{x}}_{k+1|k}=f\left(x_k\right)-f\left({\hat{x}}_{k|k}\right)+\underset{i=1}{\overset{n_1}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_{i,k}{A}_{i,k}{x}_k+D_k{\mathrm{\&omega;}}_k---\left(6\right)$

In formula (6),

for k+1 one-step prediction error constantly;

To nonlinear function f (x _k)

taylor series expansion is done at place:

$f\left(x_k\right)=f\left({\hat{x}}_{k|k}\right)+A_k{\tilde{x}}_{k|k}+o\left(\right|{\tilde{x}}_{k|k}\left|\right)---\left(7\right)$

In formula (7),

for k filtering error constantly,

$A_k={\frac{\&PartialD;f\left(x_k\right)}{{\&PartialD;x}_k}|}_{x_k={\hat{x}}_{k|k}},\frac{\&PartialD;f\left(x_k\right)}{{\&PartialD;x}_k}$ For function f (x _k) about x _kpartial derivative,

$o\left(\right|{\tilde{x}}_{k|k}\left|\right)=B_k{\mathrm{\&Theta;}}_k{L}_k{\tilde{x}}_{k|k}$ For the higher order term of Taylor series expansion,

B _kand L _kbounded matrix, matrix Θ _kportray linearisation error, it meets

i is unit matrix;

Step 6, utilize one-step prediction error

covariance recurrence equation and filtering error

covariance recurrence equation, obtain one-step prediction error P _k+1|kwith filtering error P _k+1|k+1;

The covariance of one-step prediction error is passed through formula:

$P_{k+1|k}=E\left\{{\tilde{x}}_{k+1|k}{\tilde{x}}_{k+1|k}^T\right\},$

Obtain, in formula,

for

mathematic expectaion,

By the covariance recurrence equation of one-step prediction error:

$P_{k+1|k}=(A_k+B_k{\mathrm{\&Theta;}}_k{L}_k)P_{k|k}{(A_k+B_k{\mathrm{\&Theta;}}_k{L}_k)}^T+\underset{i=1}{\overset{n_1}{\mathrm{\&Sigma;}}}{A}_{i,k}E\left\{x_k{x}_k^T\right\}{A}_{i,k}^T+Q_k---\left(8\right)$

Obtain one-step prediction error P _k+1|k,

In formula (8),

for

mathematic expectaion,

The covariance of filtering error is passed through formula:

$P_{k|k}=E\left\{{\tilde{x}}_{k|k}{\tilde{x}}_{k|k}^T\right\}$

Obtain, in formula,

for

mathematic expectaion,

By the covariance recurrence equation of filtering error:

$\begin{array}{c}{P}_{k+1|k+1}=(I-G_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1})P_{k+1|k}{(I-G_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1})}^T+E\{H_{k+1}+H_{k+1}^T\}\\ +G_{k+1}{\mathrm{\&Delta;}}_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}E\left\{x_{k+1}{x}_{k+1}^T\right\}{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{\mathrm{\&Delta;}}_{k+1}^T{G}_{k+1}^T\\ +G_{k+1}(I+{\mathrm{\&Delta;}}_{k+1})(X_{k+1}+{\mathrm{\&Pi;}}_{k+1}+R_{k+1}){(I+{\mathrm{\&Delta;}}_{k+1})}^T{G}_{k+1}^T\end{array}---\left(9\right)$

Obtain filtering error P _k+1|k+1,

In formula (9):

${\mathrm{\&Pi;}}_{k+1}=\underset{i=1}{\overset{m_1}{\mathrm{\&Sigma;}}}{C}_{i,k+1}E\left\{x_{k+1}{x}_{k+1}^T\right\}{C}_{i,k+1}^T$

$H_{k+1}=-(I-G_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}){\tilde{x}}_{k+1}{x}_{k+1}^T{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{\mathrm{\&Delta;}}_{k+1}^T{G}_{k+1}^T$

" ο " is Hadamard(Adama) Product Operator,

I is unit matrix,

${\mathrm{\&Delta;}}_{k+1}=\mathrm{diag}\left\{\begin{array}{cccc}{\mathrm{\&Delta;}}_{k+1}^{\left(1\right)}& {\mathrm{\&Delta;}}_{k+1}^{\left(2\right)}& ...& {\mathrm{\&Delta;}}_{k+1}^{\left(m\right)}\end{array}\right\}$ For portraying the diagonal matrix of quantization uncertainty,

for matrix △ _k+1the 1st diagonal components,

for matrix △ _k+1the 2nd diagonal components,

for matrix △ _k+1m diagonal components,

$\left|{\mathrm{\&Delta;}}_{k+1}^{\left(1\right)}\right|\&le;{\mathrm{\&delta;}}_1,\left|{\mathrm{\&Delta;}}_{k+1}^{\left(2\right)}\right|\&le;{\mathrm{\&delta;}}_2,\left|{\mathrm{\&Delta;}}_{k+1}^{\left(m\right)}\right|\&le;{\mathrm{\&delta;}}_m,$

for norm;

The covariance P of step 7, calculation of filtered error _k+1|k+1the upper bound and construct filter gain matrix G _k+1:

Formula is passed through in the upper bound of the covariance of filtering error:

$\begin{array}{c}{\mathrm{\&Sigma;}}_{k+1|k}=A_k{({\mathrm{\&Sigma;}}_{k|k}^{-1}-{\mathrm{\&gamma;}}_{1,k}{L}_k^T{L}_k)}^{-1}{A}_k^T+{\mathrm{\&gamma;}}_{1,k}^T{B}_k{B}_k^T+Q_k\\ +\underset{i=1}{\overset{n_1}{\mathrm{\&Sigma;}}}{A}_{i,k}[(1+{\mathrm{\&epsiv;}}_1){\mathrm{\&Sigma;}}_{k|k}+(1+{\mathrm{\&epsiv;}}_1^{-1}){\hat{x}}_{k|k}{\hat{x}}_{k|k}^T]A_{i,k}^T\end{array}---\left(10\right)$

$\begin{array}{c}{\mathrm{\&Sigma;}}_{k+1|k+1}=(1+{\mathrm{\&epsiv;}}_2)(I-G_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}){\mathrm{\&Sigma;}}_{k+1|k}{(I-G_{k+1}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1})}^T\\ +G_{k+1}\{(1+{\mathrm{\&epsiv;}}_2^{-1})\mathrm{tr}\left(\mathrm{\&Lambda;}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}{\mathrm{\&Phi;}}_{k+1|k}{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}\mathrm{\&Lambda;}\right)I\\ +\mathrm{tr}\left({\mathrm{\&Psi;}}_{k+1|k}\right)[{(I-{\mathrm{\&gamma;}}_{2,k+1}\mathrm{\&Lambda;\&Lambda;})}^{-1}+{\mathrm{\&gamma;}}_{2,k+1}^{-1}I]\}{G}_{k+1}^T\end{array}---\left(11\right)$

Obtain Σ _k+1|k+1for filtering error P _k+1|k+1upper bound matrix,

Pass through formula

$\begin{array}{c}{G}_{k+1}={\mathrm{\&Sigma;}}_{k+1|k}{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}\{(1+{\mathrm{\&epsiv;}}_2){\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}{\mathrm{\&Sigma;}}_{k+1|k}{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}\\ +(1+{\mathrm{\&epsiv;}}_2^{-1})\mathrm{tr}\left(\mathrm{\&Lambda;}{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}{C}_{k+1}{\mathrm{\&Phi;}}_{k+1|k}{C}_{k+1}^T{\stackrel{\&OverBar;}{\mathrm{\&Xi;}}}_{k+1}\mathrm{\&Lambda;}\right)I\\ {+\mathrm{tr}\left({\mathrm{\&Psi;}}_{k+1|k}\right)[{(I-{\mathrm{\&gamma;}}_{2,k+1}\mathrm{\&Lambda;\&Lambda;})}^{-1}+{\mathrm{\&gamma;}}_{2,k+1}^{-1}I]\}}^{-1}\end{array}---\left(12\right)$

Obtain filter gain matrix G _k+1;

In formula (10), (11) and (12),

${\mathrm{\&Phi;}}_{k+1|k}=(1+{\mathrm{\&epsiv;}}_3){\mathrm{\&Sigma;}}_{k+1|k}+(1+{\mathrm{\&epsiv;}}_3^{-1}){\hat{x}}_{k+1|k}{\hat{x}}_{k+1k}^T$

Λ=diag{δ ₁,δ ₂,…,δ _m}

γ _{1, k}, γ _{2, k+1}, ε ₁, ε ₂and ε ₃all normal number,

for matrix Σ _k|kcontrary,

for ε ₁contrary, tr (Ψ _k+1|k) be matrix Ψ _k+1|kmark;

Step 8, the filter gain matrix G that step 7 is obtained _k+1be brought into the formula 5 in step 4, obtain multiple measurement and lose situation lower time and become the state estimation value of Target Tracking System, realize multiple measurement and lose the quantification filtering that situation lower time becomes Target Tracking System.

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