CN103744058A - Ballistic trajectory formation method based on exponential weighting attenuated memory filtering - Google Patents

Abstract

The invention discloses a ballistic trajectory formation method based on exponential weighting attenuated memory filtering. During the ballistic trajectory formation process, errors generated during the trajectory model linearization process are reduced by a state deviation Kalman filter equation, and filter divergence caused by errors of the trajectory model is inhibited by the exponential weighting attenuated memory filtering. Thus, reliable filter data is provided for a trajectory prediction system. According to the invention, filter divergence can be inhibited effectively, and system stability can be raised.

Description

Ballistic trajectory formation method based on the filtering of exponential weighting Attenuation Memory Recursive

Technical field

The present invention relates to a kind of ballistic trajectory formation method, relate in particular to a kind of inhibition method of dispersing in trajectory extrapolation process based on exponential weighting memory attenuated EKF.

Background technology

Gun Position Radar, after obtaining the movement locus of bullet, utilizes Kalman filtering data to extrapolate and obtain the coordinate in enemy's cannon position ballistic trajectory.The linear model that Kalman filtering adopts minimum mean square error criterion to be white Gauss noise to system noise and process noise carries out dynamic parameter estimation.Because model trajectory is the nonlinear model that utilizes the differential equation to represent, in trajectory extrapolation process, utilize Taylor series expansion that Nonlinear Filtering Problem is converted into an approximate linear filtering problem, then adopt EKF to solve the suboptimal filtering result of Nonlinear Filtering Problem.In filtering, the existence of non-linear factor all has a great impact filter stability and precision of state estimation, if the error of model trajectory in linearization procedure is larger, can cause the result of EKF to be dispersed; In addition, due to meteorologic factor in projectile flight process is familiar with to out of true, institute's model trajectory of building and actual physics process are variant, the process noise covariance of pre-estimating in filtering and measurement noise covariance out of true, in filtering, just produce the accumulation of error, also can cause filtering divergence.Filtering divergence cannot carry out trajectory extrapolation process normally.

Summary of the invention

Technical matters to be solved by this invention is to provide a kind of ballistic trajectory formation method based on the filtering of exponential weighting Attenuation Memory Recursive, by exponential weighting memory attenuated EKF, the filtering divergence that the error that inhibition is produced in linearization procedure by nonlinear model trajectory and system model error cause.

The present invention is for solving the problems of the technologies described above by the following technical solutions:

A kind of ballistic trajectory formation method based on the filtering of exponential weighting Attenuation Memory Recursive, comprises that step is as follows:

Step 1, according to model trajectory, set up trajectory and the observation equation based on state variable, by trajectory and observation equation discretize, linearization, thereby set up the approximately linear equation of trajectory and observation equation;

Step 101, sets up trajectory and observation equation;

In earth axes, set up trajectory:

$\frac{\mathrm{dX}}{\mathrm{dt}}=\begin{array}{c}\left[\begin{array}{c}\frac{\mathrm{dx}}{\mathrm{dt}}\\ \frac{\mathrm{dy}}{\mathrm{dt}}\\ \frac{\mathrm{dz}}{\mathrm{dt}}\\ \frac{{\mathrm{dv}}_x}{\mathrm{dt}}\\ \frac{{\mathrm{dv}}_y}{\mathrm{dt}}\\ \frac{{\mathrm{dv}}_z}{\mathrm{dt}}\end{array}\right]=\left[\begin{array}{c}{v}_x\\ v_y\\ v_z\\ -K_D\frac{{\mathrm{SC}}_x\left(\mathrm{Ma}\right)}{2m}{\mathrm{\ρv}}_r(v_x-w_x)+K_D\frac{{\mathrm{SC}}_x\left(\mathrm{Ma}\right)}{2m}{\mathrm{\ρv}}_r{w}_x^{\′}\\ -K_D\frac{{\mathrm{SC}}_x\left(\mathrm{Ma}\right)}{2m}{\mathrm{\ρv}}_r{v}_y-g\\ -K_D\frac{{\mathrm{SC}}_x\left(\mathrm{Ma}\right)}{2m}{\mathrm{\ρv}}_r(v_z-w_z)+K_L{B}_Z+K_D\frac{{\mathrm{SC}}_x\left(\mathrm{Ma}\right)}{2m}{\mathrm{\ρv}}_r{w}_x^{\′}\end{array}\right]\end{array}$

In formula: X is t state variable constantly; expression is differentiated to time t; X, y, z is respectively distance height, the lateral deviation of bullet; v _x, v _y, v _zbe respectively velocity of shot

three axle components; v _rfor containing wind speed in interior bullet actual speed; w _x, w _zbe respectively range wind and beam wind wind speed, under standard conditions, be all taken as 0; W ' _x, w ' _zbe respectively the random quantity of range wind and beam wind; S is that bullet maximum is blocked area, d is caliber; M is Shell body quality; Ma is Mach number, c _sfor the velocity of sound,

wherein k is specific heat ratio, for air k=1.404; R _dfor dry air gas law constant; τ is virtual temperature, is the correction to temperature when soft air is amounted to into dry air; C _x(Ma) be coefficient of air resistance, it is the function of Mach number Ma; ρ is atmospheric density,

wherein, p is gas pressure intensity; G is the acceleration of gravity at height above sea level y place, g wherein ₀for sea level acceleration of gravity, r ₀for earth radius; K _dfor resistance coincidence coefficient; B _zfor side direction lift acceleration; K _lfor by dynamic equilibrium angle, caused side acceleration time make up the correction factor of error, claim again statical moment coincidence coefficient;

Writ state variable X=[x, y, z, v _x, v _y, v _z, c, ac] ^t=[x ₁, x ₂, x ₃, x ₄, x ₅, x ₆, x ₇, x ₈] ^t, in formula, c is ballistic coefficient, the ballistic coefficient of ac for revising, x ₁～x ₈for 8 states of state variable X, respectively with x, y, z, v _x, v _y, v _z, c, ac is corresponding;

Make x _4r=x ₄-w _xrepresent that bullet is subject to the actual speed after air speed influence in x direction; x _6r=x ₆-w _z, represent that bullet exists, the practical flight speed of z direction,

Get as intermediate variable; The velocity of sound is brought into, Machmeter is shown simultaneously $\mathrm{Ma}=\frac{v_r}{C_s}=\frac{v_r}{20.0468\sqrt{\mathrm{\τ}}},$ Trajectory is rewritten as:

$\frac{\mathrm{dX}}{\mathrm{dt}}=f\left(X\right)+\mathrm{\&Gamma;W}=\left[\begin{array}{c}{x}_4\\ x_5\\ x_6\\ -x_7{\mathrm{Nx}}_{4r}\\ -x_7{\mathrm{Nx}}_5-g\\ -x_7{\mathrm{Nx}}_{6r}+x_8{\mathrm{<figure-callout\; id="0"\; label="B\; z"\; filenames="131224153755.png"\; state="\{\{state\}\}">B</figure-callout>}}_z\\ 0\\ 0\end{array}\right]+\left[\begin{array}{c}{0}_{3\&times;3}\\ I_{3\&times;3}\\ 0_{2\&times;3}\end{array}\right]x_{7}N\left[\begin{array}{c}{w}_x^{\&prime;}\\ 0\\ w_z^{\&prime;}\end{array}\right]$

In formula, f (X) is the function of state variable X; Γ is that noise drives battle array, Γ=[0 _{3 * 3}i _{3 * 3}0 _{2 * 3}] ^tx ₇n; W is system noise, W=[w ' _x0 w ' _z] ^t; 0 _{3 * 3}represent 3 * 3 null matrix; I _{3 * 3}represent 3 * 3 unit matrix; 0 _{2 * 3}represent 2 * 3 null matrix;

Set up observation equation:

$Z=h\left(X\right)+V=\left[\begin{array}{c}\sqrt{x^2+y^2+z^2}\\ \arctan \frac{z}{x}\\ \arctan \frac{y}{\sqrt{x^2+z^2}}\end{array}\right]+V=\left[\begin{array}{c}\sqrt{x_1^2+x_2^2+x_3^2}\\ \arctan \frac{x_3}{x_1}\\ \arctan \frac{x_2}{\sqrt{x_1^2+x_3^2}}\end{array}\right]+V$

In formula, $h\left(X\right)={\left[\begin{array}{ccc}\sqrt{x^2+y^2+z^2}& \arctan \frac{z}{x}& \arctan \frac{y}{\sqrt{x^2+z^2}}\end{array}\right]}^T;$ V is radar measurement noise, and V obedience average is normal distribution zero, that variance is R:

$R=\left[\begin{array}{ccc}{\mathrm{\&sigma;}}_r^2& & \\ & {\mathrm{\&sigma;}}_B^2& \\ & & {\mathrm{\&sigma;}}_e^2\end{array}\right]$

In formula, σ _γ, σ _β, σ _εbe respectively the random meausrement error of Gun Position Radar in antenna coordinate system;

Step 102, to trajectory and observation equation discretize, linearization, obtains the approximately linear equation of trajectory and observation equation;

At sampling interval [t _k, t _k+1] upper by f (X (t)) approximate expansion and by trajectory discretize, obtain:

$X_{k+1}=X_k+f\left(X_k\right)T+A\left(X_k\right)f\left(X_k\right)\frac{T^2}{2}+{\mathrm{\&Gamma;}}_k{q}_k$

In formula, X _kfor k state variable constantly; X _k+1for k+1 state variable constantly; A (X) is that f (X) is about the local derviation of X, plant noise t is the time interval, T=t _k+1-t _k;

Order $F\left(X_k\right)=X_k+f\left(X_k\right)T+A\left(X_k\right)f\left(X_k\right)\frac{T^2}{2},$ ?

X _k+1＝F(X _k)+Γ _kq _k

If taking into account system noise not, system k nominal state is constantly

$X_k^{*}=F\left(X_{k-1}^{*}\right)$

Around nominal state by F (X _k) Taylor expansion, and get first approximation, obtain the approximately linear equation of trajectory:

$X_{k+1}=F\left(X_k^{*}\right)+\frac{\&PartialD;F\left(X\right)}{\&PartialD;X}{|}_{X=X_k^{*}}(X_k-X_k^{*})+{\mathrm{\&Gamma;}}_k{q}_k;$

Due to $F\left(X_k^{*}\right)=X_{k+1}^{*},$ And order ${\mathrm{\&phi;}}_k=\frac{\&PartialD;F\left(X\right)}{\&PartialD;X}{|}_{X=X_k^{*}}=I+A\left(X_k^{*}\right)T+\frac{d\left[A\left(X\right)f\left(X\right)\right]}{\mathrm{dX}}{|}_{X=X_k^{*}}\frac{T^2}{2},$ The approximately linear equation of trajectory is rewritten as:

$X_{k+1}=X_{k+1}^{*}+{\mathrm{\&phi;}}_k(X_k-X_k^{*})+{\mathrm{\&Gamma;}}_k{q}_k;$

By the nonlinear function h (X) in observation equation in nominal state

place is launched into Taylor series, and gets first approximation, obtains the approximately linear equation of observation equation:

$Z_k=Z_k^{*}+\frac{\&PartialD;h\left(X\right)}{\&PartialD;X}{|}_{X=X_k^{*}}[X_k-X_k^{*}]+V_k$

Make observing matrix

the approximately linear equation of observation equation is rewritten as:

$Z_k=Z_k^{*}+H_k(X_k-X_k^{*})+V_k$

In formula, Z _kfor k observational variable constantly,

for k observational variable nominal value constantly, V _kfor k observation noise constantly.

Step 2, state variable and nominal state are made to the poor state deviation that obtains, the Kalman filtering recurrence equation of foundation based on state deviation, one of radar observation section of trajectory parameter is carried out to filtering, in filtering, reduce the error that trajectory produces in linearization procedure;

State variable and nominal state work difference are obtained to state deviation, that is:

$\mathrm{\&Delta;}{X}_k=X_k-X_k^{*}$

$\mathrm{\&Delta;}{X}_k=X_k-X_k^{*}$

Approximately linear equation inference by trajectory in step 1 and observation equation draws:

$\mathrm{\&Delta;}{X}_{k+1}=X_{k+1}-X_{k+1}^{*}={\mathrm{\&phi;}}_k\mathrm{\&Delta;}{X}_k+{\mathrm{\&Gamma;}}_k{q}_k$

$\mathrm{\&Delta;}{Z}_k=Z_k-Z_k^{*}=H_k\mathrm{\&Delta;}{X}_k+V_k$

Obtain thus the Kalman filtering recurrence equation based on state deviation, specific as follows:

State one-step prediction equation:

ΔX _k+1/k＝φ _kΔX _k

State estimation equation:

ΔX _k+1＝ΔX _k+1/k+K _k+1[ΔZ _k+1-H _k+1ΔX _k+1/k]

Filter gain equation:

$K_{k+1}=P_{k+1/k}{H}_{k+1}^T{[H_{k+1}{P}_{k+1/k}{H}_{k+1}^T+R_{k+1}]}^{-1}$

One-step prediction square error equation:

$P_{k+1/k}={\mathrm{\&phi;}}_k{P}_k{\mathrm{\&phi;}}_k^T+{\mathrm{\&Gamma;}}_k{Q}_k{\mathrm{\&Gamma;}}_k^T$

Estimate square error equation:

P _k+1＝[1-K _k+1H _k+1]P _k+1/k

System state equation:

$X_k=X_k^{*}+\mathrm{\&Delta;}{X}_k$

In formula,

one-step prediction value for deviation; K _k+1for filter gain matrix; P _k+1/kone-step prediction value for square error; R _k+1for measuring noise square difference battle array; P _k+1estimated value for square error; Γ _kfor system k noise driving constantly battle array; Q _kfor system noise serial variance battle array.

Step 3, employing exponential weighting Attenuation Memory Recursive filtering method are to suppress in step 2 to the filtering divergence producing in trajectory parametric filtering process, for trajectory extrapolation provides reliable filtering data;

By obtaining in step 2, the filter gain equation of Kalman filtering is:

$K_{k+1}=P_{k+1/k}{H}_{k+1}^T{[H_{k+1}{P}_{k+1/k}{H}_{k+1}^T+R_{k+1}]}^{-1}$

Get k=M, $K_M=P_{M/M-1}{H}_M^T{[H_M{P}_{M/M-1}{H}_M^T+R_M]}^{-1};$

Due to R _kand K _kthe relation that is inversely proportional to, in order to reduce the K of M before constantly _kvalue, should make the R away from more and more constantly from M _kby

Gradual change is large; Q _kwith R _kdo identical variation; If R _k, Q _k, P ₀all increase k doubly, the gain matrix K of stationary filter _k

Remain unchanged;

Take the way of exponential weighting, by P ₀; R ₁, R ₂, R ₃..., R _m; Q ₁, Q ₂, Q ₃..., Q _mbecome respectively lower column matrix:

$\left\{\begin{array}{cccccc}{e}^{\underset{i=0}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{P}_0& & & & & \\ e^{\underset{i=0}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{R}_1,& e^{\underset{i=1}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{R}_2,& e^{\underset{i=2}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{R}_3,& ...,& e^{{\mathrm{\&lambda;}}_{M-1}}{R}_M,& R_M\\ e^{\underset{i=0}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{Q}_1,& e^{\underset{i=1}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{Q}_2,& e^{\underset{i=2}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{Q}_3,& ...,& e^{{\mathrm{\&lambda;}}_{M-1}}{Q}_M,& Q_M\end{array}\right.$

In formula, λ _ifor positive number; I is natural number, i=1, and 2,3 ..., M;

Exponential weighting, is equivalent to state X _kwhile asking optimal filtering, P ₀with the noise variance matrix R of M before the moment ₁, R ₂, R ₃..., R _m; Q ₁, Q ₂, Q ₃..., Q _mwith above formula, replace; One-step prediction square error equation, through arranging, obtains following weighted equation:

$P_{k+1/k}={\mathrm{\&phi;}}_k{P}_k{\mathrm{\&phi;}}_k^T{e}^{{\mathrm{\&lambda;}}_k}+{\mathrm{\&Gamma;}}_k{Q}_k{\mathrm{\&Gamma;}}_k^T$

In formula, for Attenuation Memory Recursive filtering factor;

By Kalman filtering divergence criterion,

$e^{{\mathrm{\&lambda;}}_k}=\frac{{\tilde{Z}}_k^T{\tilde{Z}}_k-\mathrm{tr}(H_k{Q}_{k-1}^{*}{H}_k^T+R_k)}{\mathrm{tr}\left[H_k{\mathrm{\&phi;}}_k{P}_{k-1}{\mathrm{\&phi;}}_k^T{H}_k^T\right]}$

?

${\mathrm{\&lambda;}}_k=\ln \left(\frac{{\tilde{Z}}_k^T{\tilde{Z}}_k-\mathrm{tr}(H_k{Q}_{k-1}^{*}{H}_k^T+R_k)}{\mathrm{tr}\left[H_k{\mathrm{\&phi;}}_k{P}_{k-1}{\mathrm{\&phi;}}_k^T{H}_k^T\right]}\right)$

In formula,

for residual sequence, ${\tilde{Z}}_k=\mathrm{\&Delta;}{Z}_k-H_k\mathrm{\&Delta;}{X}_{k/k-1};$ for plant noise variance matrix, $Q_{k-1}^{*}={\mathrm{\&Gamma;}}_{k-1}{Q}_{k-1}{\mathrm{\&Gamma;}}_{k-1}^T;$

In filtering, arrange one and disperse judgment criterion, if

${\tilde{Z}}_k^T{\tilde{Z}}_k>r\&CenterDot;E\left(k\right)$

Carrying out above Attenuation Memory Recursive exponential weighting processes;

In formula, $\left(k\right)=\mathrm{tr}[H_k{P}_{k/k-1}{H}_k^T+R_k];$ R is reserve factor, r >=1;

If above formula is set up, the r that the actual estimated error that filtering is described is more than or equal to calculated value doubly; If r=1 is the strictest convergence basis for estimation condition.

Step 4, according to filtered trajectory parameter, utilize model trajectory extrapolation ballistic impact, form ballistic trajectory.

The present invention adopts above technical scheme compared with prior art, first adopts state deviation Kalman filter equation, has reduced the generation error of non-linear trajectory system in linearization procedure; Use the filtering of exponential weighting Attenuation Memory Recursive simultaneously, suppress the filtering divergence phenomenon in filtering, improved greatly filtering accuracy, strengthened system stability, guaranteed the reliability of trajectory extrapolation.

Accompanying drawing explanation

Fig. 1 is process flow diagram of the present invention.

Fig. 2 is the inhibition situation design sketchs of different filtering methods to Divergent Phenomenon, and wherein, (a) figure is the design sketch that does not use the filtering processing of method of weighting; (b) figure is the design sketch that uses the deviation kalman filter method processing that the present invention is based on exponential weighting.

Embodiment

Below in conjunction with accompanying drawing, technical scheme of the present invention is described in further detail:

A kind of ballistic trajectory formation method based on the filtering of exponential weighting Attenuation Memory Recursive, as shown in Figure 1, comprises that step is as follows:

Step 1, according to model trajectory, set up trajectory and the observation equation based on state variable, by trajectory and observation equation discretize, linearization, thereby set up the approximately linear equation of trajectory and observation equation;

Step 101, sets up trajectory and observation equation;

In earth axes, set up trajectory:

$\frac{\mathrm{dX}}{\mathrm{dt}}=\begin{array}{c}\left[\begin{array}{c}\frac{\mathrm{dx}}{\mathrm{dt}}\\ \frac{\mathrm{dy}}{\mathrm{dt}}\\ \frac{\mathrm{dz}}{\mathrm{dt}}\\ \frac{{\mathrm{dv}}_x}{\mathrm{dt}}\\ \frac{{\mathrm{dv}}_y}{\mathrm{dt}}\\ \frac{{\mathrm{dv}}_z}{\mathrm{dt}}\end{array}\right]=\left[\begin{array}{c}{v}_x\\ v_y\\ v_z\\ -K_D\frac{{\mathrm{SC}}_x\left(\mathrm{Ma}\right)}{2m}{\mathrm{\&rho;v}}_r(v_x-w_x)+K_D\frac{{\mathrm{SC}}_x\left(\mathrm{Ma}\right)}{2m}{\mathrm{\&rho;v}}_r{w}_x^{\&prime;}\\ -K_D\frac{{\mathrm{SC}}_x\left(\mathrm{Ma}\right)}{2m}{\mathrm{\&rho;v}}_r{v}_y-g\\ -K_D\frac{{\mathrm{SC}}_x\left(\mathrm{Ma}\right)}{2m}{\mathrm{\&rho;v}}_r(v_z-w_z)+K_L{B}_Z+K_D\frac{{\mathrm{SC}}_x\left(\mathrm{Ma}\right)}{2m}{\mathrm{\&rho;v}}_r{w}_x^{\&prime;}\end{array}\right]\end{array}$

In formula: X is t state variable constantly; expression is differentiated to time t; X, y, z be respectively the distance of bullet, highly, lateral deviation; v _x, v _y, v _zbe respectively velocity of shot

three axle components; v _rfor containing wind speed in interior bullet actual speed; w _x, w _zbe respectively range wind and beam wind wind speed, under standard conditions, be all taken as 0; W' _x, w' _zbe respectively the random quantity of range wind and beam wind; S is that bullet maximum is blocked area,

d is caliber; M is Shell body quality; Ma is Mach number, c _sfor the velocity of sound,

wherein k is specific heat ratio, for air k=1.404; R _dfor dry air gas law constant; τ is virtual temperature, is the correction to temperature when soft air is amounted to into dry air; C _x(Ma) be coefficient of air resistance, it is the function of Mach number Ma; ρ is atmospheric density,

wherein, p is gas pressure intensity; G is the acceleration of gravity at height above sea level y place,

g wherein ₀for sea level acceleration of gravity, r ₀for earth radius; K _dfor resistance coincidence coefficient; B _zfor side direction lift acceleration; K _lfor by dynamic equilibrium angle, caused side acceleration time make up the correction factor of error, claim again statical moment coincidence coefficient;

Writ state variable X=[x, y, z, v _x, v _y, v _z, c, ac] ^t=[x ₁, x ₂, x ₃, x ₄, x ₅, x ₆, x ₇, x ₈] ^t, in formula, c is ballistic coefficient, the ballistic coefficient of ac for revising, x ₁～x ₈for 8 states of state variable X, respectively with x, y, z, v _x, v _y, v _z, c, ac is corresponding;

Make x _4r=x ₄-w _xrepresent that bullet is subject to the actual speed after air speed influence in x direction; x _6r=x ₆-w _z, represent that bullet exists, the practical flight speed of z direction,

Get

as intermediate variable; The velocity of sound is brought into, Machmeter is shown simultaneously $\mathrm{Ma}=\frac{v_r}{C_s}=\frac{v_r}{20.0468\sqrt{\mathrm{\&tau;}}},$ Trajectory is rewritten as:

$\frac{\mathrm{dX}}{\mathrm{dt}}=f\left(X\right)+\mathrm{\&Gamma;W}=\left[\begin{array}{c}{x}_4\\ x_5\\ x_6\\ -x_7{\mathrm{Nx}}_{4r}\\ -x_7{\mathrm{Nx}}_5-g\\ -x_7{\mathrm{Nx}}_{6r}+x_8{\mathrm{<figure-callout\; id="0"\; label="B\; z"\; filenames="131224153755.png"\; state="\{\{state\}\}">B</figure-callout>}}_z\\ 0\\ 0\end{array}\right]+\left[\begin{array}{c}{0}_{3\&times;3}\\ I_{3\&times;3}\\ 0_{2\&times;3}\end{array}\right]x_{7}N\left[\begin{array}{c}{w}_x^{\&prime;}\\ 0\\ w_z^{\&prime;}\end{array}\right]$

In formula, f (X) is the function of state variable X; Γ is that noise drives battle array, Γ=[0 _{3 * 3}i _{3 * 3}0 _{2 * 3}] ^tx ₇n; W is system noise, W=[w ' _x0 w ' _z] ^t; 0 _{3 * 3}represent 3 * 3 null matrix; I _{3 * 3}represent 3 * 3 unit matrix; 0 _{2 * 3}represent 2 * 3 null matrix;

Set up observation equation:

$Z=h\left(X\right)+V=\left[\begin{array}{c}\sqrt{x^2+y^2+z^2}\\ \arctan \frac{z}{x}\\ \arctan \frac{y}{\sqrt{x^2+z^2}}\end{array}\right]+V=\left[\begin{array}{c}\sqrt{x_1^2+x_2^2+x_3^2}\\ \arctan \frac{x_3}{x_1}\\ \arctan \frac{x_2}{\sqrt{x_1^2+x_3^2}}\end{array}\right]+V$

In formula, $h\left(X\right)={\left[\begin{array}{ccc}\sqrt{x^2+y^2+z^2}& \arctan \frac{z}{x}& \arctan \frac{y}{\sqrt{x^2+z^2}}\end{array}\right]}^T;$ V is radar measurement noise, and V obedience average is normal distribution zero, that variance is R:

$R=\left[\begin{array}{ccc}{\mathrm{\&sigma;}}_r^2& & \\ & {\mathrm{\&sigma;}}_B^2& \\ & & {\mathrm{\&sigma;}}_e^2\end{array}\right]$

In formula, σ _γ, σ _β, σ _εbe respectively the random meausrement error of Gun Position Radar in antenna coordinate system;

Step 102, to trajectory and observation equation discretize, linearization, obtains the approximately linear equation of trajectory and observation equation;

At sampling interval [t _k, t _k+1] upper by f (X (t)) approximate expansion and by trajectory discretize, obtain:

$X_{k+1}=X_k+f\left(X_k\right)T+A\left(X_k\right)f\left(X_k\right)\frac{T^2}{2}+{\mathrm{\&Gamma;}}_k{q}_k$

In formula, X _kfor k state variable constantly; X _k+1for k+1 state variable constantly; A (X) is that f (X) is about the local derviation of X,

plant noise

t is the time interval, T=t _k+1-t _k;

Order $F\left(X_k\right)=X_k+f\left(X_k\right)T+A\left(X_k\right)f\left(X_k\right)\frac{T^2}{2},$ ?

X _k+1＝F(X _k)+Γ _kq _k

If taking into account system noise not, system k nominal state is constantly

Around nominal state by F (X _k) Taylor expansion, and get first approximation, obtain the approximately linear equation of trajectory:

$X_{k+1}=F\left(X_k^{*}\right)+\frac{\&PartialD;F\left(X\right)}{\&PartialD;X}{|}_{X=X_k^{*}}(X_k-X_k^{*})+{\mathrm{\&Gamma;}}_k{q}_k;$

Due to $F\left(X_k^{*}\right)=X_{k+1}^{*},$ And order ${\mathrm{\&phi;}}_k=\frac{\&PartialD;F\left(X\right)}{\&PartialD;X}{|}_{X=X_k^{*}}=I+A\left(X_k^{*}\right)T+\frac{d\left[A\left(X\right)f\left(X\right)\right]}{\mathrm{dX}}{|}_{X=X_k^{*}}\frac{T^2}{2},$ The approximately linear equation of trajectory is rewritten as:

$X_{k+1}=X_{k+1}^{*}+{\mathrm{\&phi;}}_k(X_k-X_k^{*})+{\mathrm{\&Gamma;}}_k{q}_k;$

By the nonlinear function h (X) in observation equation in nominal state

place is launched into Taylor series, and gets first approximation, obtains the approximately linear equation of observation equation:

$Z_k=Z_k^{*}+\frac{\&PartialD;h\left(X\right)}{\&PartialD;X}{|}_{X=X_k^{*}}[X_k-X_k^{*}]+V_k$

Make observing matrix

the approximately linear equation of observation equation is rewritten as:

$Z_k=Z_k^{*}+H_k(X_k-X_k^{*})+V_k$

In formula, Z _kfor k observational variable constantly, for k observational variable nominal value constantly, V _kfor k observation noise constantly.

Step 2, state variable and nominal state are made to the poor state deviation that obtains, the Kalman filtering recurrence equation of foundation based on state deviation, one of radar observation section of trajectory parameter is carried out to filtering, in filtering, reduce the error that trajectory produces in linearization procedure;

State variable and nominal state work difference are obtained to state deviation, that is:

$\mathrm{\&Delta;}{X}_k=X_k-X_k^{*}$

$\mathrm{\&Delta;}{Z}_k=Z_k-Z_k^{*}$

Approximately linear equation inference by trajectory in step 1 and observation equation draws:

$\mathrm{\&Delta;}{X}_{k+1}=X_{k+1}-X_{k+1}^{*}={\mathrm{\&phi;}}_k\mathrm{\&Delta;}{X}_k+{\mathrm{\&Gamma;}}_k{q}_k$

$\mathrm{\&Delta;}{Z}_k=Z_k-Z_k^{*}=H_k\mathrm{\&Delta;}{X}_k+V_k$

Obtain thus the Kalman filtering recurrence equation based on state deviation, specific as follows:

State one-step prediction equation:

ΔX _k+1/k＝φ _kΔX _k

State estimation equation:

ΔX _k+1＝ΔX _k+1/k+K _k+1[ΔZ _k+1-H _k+1ΔX _k+1/k]

Filter gain equation:

$K_{k+1}=P_{k+1/k}{H}_{k+1}^T{[H_{k+1}{P}_{k+1/k}{H}_{k+1}^T+R_{k+1}]}^{-1}$

One-step prediction square error equation:

$P_{k+1/k}={\mathrm{\&phi;}}_k{P}_k{\mathrm{\&phi;}}_k^T+{\mathrm{\&Gamma;}}_k{Q}_k{\mathrm{\&Gamma;}}_k^T$

Estimate square error equation:

P _k+1＝[1-K _k+1H _k+1]P _k+1/k

System state equation:

$X_k=X_k^{*}+\mathrm{\&Delta;}{X}_k$

In formula, one-step prediction value for deviation; K _k+1for filter gain matrix; P _k+1/kone-step prediction value for square error; R _k+1for measuring noise square difference battle array; P _k+1estimated value for square error; Γ _kfor system k noise driving constantly battle array; Q _kfor system noise serial variance battle array.

Step 3, employing exponential weighting Attenuation Memory Recursive filtering method are to suppress in step 2 to the filtering divergence producing in trajectory parametric filtering process, for trajectory extrapolation provides reliable filtering data;

By obtaining in step 2, the filter gain equation of Kalman filtering is:

$K_{k+1}=P_{k+1/k}{H}_{k+1}^T{[H_{k+1}{P}_{k+1/k}{H}_{k+1}^T+R_{k+1}]}^{-1}$

Get k=M, $K_{k+1}=P_{k+1/k}{H}_{k+1}^T{[H_{k+1}{P}_{k+1/k}{H}_{k+1}^T+R_{k+1}]}^{-1}$

For M Kalman filtering constantly, in order to suppress dispersing of filtering, should relatively give prominence to M K constantly _mvalue, and due to R _kand K _kthe relation that is inversely proportional to, in order to reduce the K of M before constantly _kvalue, should make the R away from more and more constantly from M _kbecome gradually large; Q _kwith R _kdo identical variation; If R _k, Q _k, P ₀all increase k doubly, the gain matrix K of stationary filter _kremain unchanged;

Take the way of exponential weighting, by P ₀; R ₁, R ₂, R ₃..., R _m; Q ₁, Q ₂, Q ₃..., Q _mbecome respectively lower column matrix:

$\left\{\begin{array}{cccccc}{e}^{\underset{i=0}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{P}_0& & & & & \\ e^{\underset{i=0}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{R}_1,& e^{\underset{i=1}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{R}_2,& e^{\underset{i=2}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{R}_3,& ...,& e^{{\mathrm{\&lambda;}}_{M-1}}{R}_M,& R_M\\ e^{\underset{i=0}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{Q}_1,& e^{\underset{i=1}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{Q}_2,& e^{\underset{i=2}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{Q}_3,& ...,& e^{{\mathrm{\&lambda;}}_{M-1}}{Q}_M,& Q_M\end{array}\right.$

In formula, λ _ifor positive number; I is natural number, i=1, and 2,3 ..., M;

Exponential weighting, is equivalent to state X _kwhile asking optimal filtering, P ₀with the noise variance matrix R of M before the moment ₁, R ₂, R ₃..., R _m; Q ₁, Q ₂, Q ₃..., Q _mwith above formula, replace; One-step prediction square error equation, through arranging, obtains following weighted equation:

$P_{k+1/k}={\mathrm{\&phi;}}_k{P}_k{\mathrm{\&phi;}}_k^T{e}^{{\mathrm{\&lambda;}}_k}+{\mathrm{\&Gamma;}}_k{Q}_k{\mathrm{\&Gamma;}}_k^T$

In formula,

for Attenuation Memory Recursive filtering factor;

By Kalman filtering divergence criterion,

$e^{{\mathrm{\&lambda;}}_k}=\frac{{\tilde{Z}}_k^T{\tilde{Z}}_k-\mathrm{tr}(H_k{Q}_{k-1}^{*}{H}_k^T+R_k)}{\mathrm{tr}\left[H_k{\mathrm{\&phi;}}_k{P}_{k-1}{\mathrm{\&phi;}}_k^T{H}_k^T\right]}$

?

${\mathrm{\&lambda;}}_k=\ln \left(\frac{{\tilde{Z}}_k^T{\tilde{Z}}_k-\mathrm{tr}(H_k{Q}_{k-1}^{*}{H}_k^T+R_k)}{\mathrm{tr}\left[H_k{\mathrm{\&phi;}}_k{P}_{k-1}{\mathrm{\&phi;}}_k^T{H}_k^T\right]}\right)$

In formula, for residual sequence, ${\tilde{Z}}_k=\mathrm{\&Delta;}{Z}_k-H_k\mathrm{\&Delta;}{X}_{k/k-1};$

for plant noise variance matrix, $Q_{k-1}^{*}={\mathrm{\&Gamma;}}_{k-1}{Q}_{k-1}{\mathrm{\&Gamma;}}_{k-1}^T;$ In filtering, arrange one and disperse judgment criterion, if

${\tilde{Z}}_k^T{\tilde{Z}}_k>r\&CenterDot;E\left(k\right)$

Carrying out above Attenuation Memory Recursive exponential weighting processes;

In formula, $\left(k\right)=\mathrm{tr}[H_k{P}_{k/k-1}{H}_k^T+R_k];$ R is reserve factor, r >=1;

If above formula is set up, the r that the actual estimated error that filtering is described is more than or equal to calculated value doubly; If r=1 is the strictest convergence basis for estimation condition.

Step 4, according to filtered trajectory parameter, utilize model trajectory extrapolation ballistic impact, form ballistic trajectory.This step is ordinary skill in the art means, does not repeat them here.

Suppose that radar is 10m at the noise variance of range direction, the noise variance of deflection and the angle of pitch is 0.0015rad, utilize the measurement noise of the random function simulation generation radar of computing machine, and by the be added to upper (A of distance R, position angle A and angle of pitch E of projectile flight of these noises, R, E is respectively the flight parameter of bullet in radar rectangular coordinate system).At radar front normal direction position angle, be-45 °, antenna superelevation angle is 0 °, and antenna fore-and-aft tilt angle is under the condition of 45 °, simulation bullet flight path.

Owing to can being subject to various interference in projectile flight process, so its flight path can not be the para-curve of standard.If in serious interference, in the situation that noise ratio is larger, bullet may depart from original track, and may occur Divergent Phenomenon in extrapolation process, causing extrapolation is launching site less than Fall Of Shot.As shown in (a) figure in Fig. 2, do not use the filtering of method of weighting to process, bullet has departed from original track at x direction nearly 900 meters of of flying, and causes serious filtering divergence, make trajectory extrapolation can not get ballistic trajectory accurately, such extrapolation has just lost original meaning; As shown in (b) figure in Fig. 2, use the deviation kalman filter method based on exponential weighting effectively to suppress filtering divergence phenomenon, for trajectory extrapolation provides reliable filtering data, thereby obtain ballistic trajectory accurately.

The above; it is only the embodiment in the present invention; but protection scope of the present invention is not limited to this; any people who is familiar with this technology is in the disclosed technical scope of the present invention; can understand conversion or the replacement expected; all should be encompassed in of the present invention comprise scope within, therefore, protection scope of the present invention should be as the criterion with the protection domain of claims.

Claims (4)

1. the ballistic trajectory formation method based on the filtering of exponential weighting Attenuation Memory Recursive, is characterized in that, comprises that step is as follows:

Step 1, according to model trajectory, set up trajectory and the observation equation based on state variable, by trajectory and observation equation discretize, linearization, thereby set up the approximately linear equation of trajectory and observation equation;

Step 2, state variable and nominal state are made to the poor state deviation that obtains, the Kalman filtering recurrence equation of foundation based on state deviation, one of radar observation section of trajectory parameter is carried out to filtering, in filtering, reduce the error that trajectory produces in linearization procedure;

Step 3, employing exponential weighting Attenuation Memory Recursive filtering method are to suppress in step 2 to the filtering divergence producing in trajectory parametric filtering process, for trajectory extrapolation provides reliable filtering data;

Step 4, according to filtered trajectory parameter, utilize model trajectory extrapolation ballistic impact, form ballistic trajectory.

2. a kind of ballistic trajectory formation method based on the filtering of exponential weighting Attenuation Memory Recursive according to claim 1, is characterized in that, the approximately linear equation of setting up based on state deviation described in step 1 is specific as follows:

Step 101, sets up trajectory and observation equation;

In earth axes, set up trajectory:

$\frac{\mathrm{dX}}{\mathrm{dt}}=\begin{array}{c}\left[\begin{array}{c}\frac{\mathrm{dx}}{\mathrm{dt}}\\ \frac{\mathrm{dy}}{\mathrm{dt}}\\ \frac{\mathrm{dz}}{\mathrm{dt}}\\ \frac{{\mathrm{dv}}_x}{\mathrm{dt}}\\ \frac{{\mathrm{dv}}_y}{\mathrm{dt}}\\ \frac{{\mathrm{dv}}_z}{\mathrm{dt}}\end{array}\right]=\left[\begin{array}{c}{v}_x\\ v_y\\ v_z\\ -K_D\frac{{\mathrm{SC}}_x\left(\mathrm{Ma}\right)}{2m}{\mathrm{\&rho;v}}_r(v_x-w_x)+K_D\frac{{\mathrm{SC}}_x\left(\mathrm{Ma}\right)}{2m}{\mathrm{\&rho;v}}_r{w}_x^{\&prime;}\\ -K_D\frac{{\mathrm{SC}}_x\left(\mathrm{Ma}\right)}{2m}{\mathrm{\&rho;v}}_r{v}_y-g\\ -K_D\frac{{\mathrm{SC}}_x\left(\mathrm{Ma}\right)}{2m}{\mathrm{\&rho;v}}_r(v_z-w_z)+K_L{B}_Z+K_D\frac{{\mathrm{SC}}_x\left(\mathrm{Ma}\right)}{2m}{\mathrm{\&rho;v}}_r{w}_x^{\&prime;}\end{array}\right]\end{array}$

In formula: X is t state variable constantly;

expression is differentiated to time t; X, y, z be respectively the distance of bullet, highly, lateral deviation; v _x, v _y, v _zbe respectively velocity of shot

three axle components; v _rfor containing wind speed in interior bullet actual speed; w _x, w _zbe respectively range wind and beam wind wind speed, under standard conditions, be all taken as 0; W ' _x, w ' _zbe respectively the random quantity of range wind and beam wind; S is that bullet maximum is blocked area, d is caliber; M is Shell body quality; Ma is Mach number,

c _sfor the velocity of sound,

wherein k is specific heat ratio, for air k=1.404; R _dfor dry air gas law constant; τ is virtual temperature, is the correction to temperature when soft air is amounted to into dry air; C _x(Ma) be coefficient of air resistance, it is the function of Mach number Ma; ρ is atmospheric density, wherein, p is gas pressure intensity; G is the acceleration of gravity at height above sea level y place,

g wherein ₀for sea level acceleration of gravity, r ₀for earth radius; K _dfor resistance coincidence coefficient; B _zfor side direction lift acceleration; K _lfor by dynamic equilibrium angle, caused side acceleration time make up the correction factor of error, claim again statical moment coincidence coefficient;

Writ state variable X=[x, y, z, v _x, v _y, v _z, c, ac] ^t=[x ₁, x ₂, x ₃, x ₄, x ₅, x ₆, x ₇, x ₈] ^t, in formula, c is ballistic coefficient, the ballistic coefficient of ac for revising, x ₁～x ₈for 8 states of state variable X, respectively with x, y, z, v _x, v _y, v _z, c, ac is corresponding;

Make x _4r=x ₄-w _xrepresent that bullet is subject to the actual speed after air speed influence in x direction; x _6r=x ₆-w _z, representing bullet, ball is in the practical flight speed of z direction,

Get as intermediate variable; The velocity of sound is brought into, Machmeter is shown simultaneously $\mathrm{Ma}=\frac{v_r}{C_s}=\frac{v_r}{20.0468\sqrt{\mathrm{\&tau;}}},$ Trajectory is rewritten as:

$\frac{\mathrm{dX}}{\mathrm{dt}}=f\left(X\right)+\mathrm{\&Gamma;W}=\left[\begin{array}{c}{x}_4\\ x_5\\ x_6\\ -x_7{\mathrm{Nx}}_{4r}\\ -x_7{\mathrm{Nx}}_5-g\\ -x_7{\mathrm{Nx}}_{6r}+x_8{B}_z\\ 0\\ 0\end{array}\right]+\left[\begin{array}{c}{0}_{3\&times;3}\\ I_{3\&times;3}\\ 0_{2\&times;3}\end{array}\right]x_{7}N\left[\begin{array}{c}{w}_x^{\&prime;}\\ 0\\ w_z^{\&prime;}\end{array}\right]$

In formula, f (X) is the function of state variable X; Γ is that noise drives battle array, Γ=[0 _{3 * 3}i _{3 * 3}0 _{2 * 3}] ^tx ₇n; W is system noise, W=[w ' _x0 w ' _z] ^t; 0 _{3 * 3}represent 3 * 3 null matrix; I _{3 * 3}represent 3 * 3 unit matrix; 0 _{2 * 3}represent 2 * 3 null matrix;

Set up observation equation:

$Z=h\left(X\right)+V=\left[\begin{array}{c}\sqrt{x^2+y^2+z^2}\\ \arctan \frac{z}{x}\\ \arctan \frac{y}{\sqrt{x^2+z^2}}\end{array}\right]+V=\left[\begin{array}{c}\sqrt{x_1^2+x_2^2+x_3^2}\\ \arctan \frac{x_3}{x_1}\\ \arctan \frac{x_2}{\sqrt{x_1^2+x_3^2}}\end{array}\right]+V$

In formula, $h\left(X\right)={\left[\begin{array}{ccc}\sqrt{x^2+y^2+z^2}& \arctan \frac{z}{x}& \arctan \frac{y}{\sqrt{x^2+z^2}}\end{array}\right]}^T;$ V is radar measurement noise, and V obedience average is normal distribution zero, that variance is R:

$R=\left[\begin{array}{ccc}{\mathrm{\&sigma;}}_r^2& & \\ & {\mathrm{\&sigma;}}_B^2& \\ & & {\mathrm{\&sigma;}}_e^2\end{array}\right]$

In formula, σ _γ, σ _β, σ _εbe respectively the random meausrement error of Gun Position Radar in antenna coordinate system;

Step 102, to trajectory and observation equation discretize, linearization, obtains the approximately linear equation of trajectory and observation equation;

At sampling interval [t _k, t _k+1] upper by f (X (t)) approximate expansion and by trajectory discretize, obtain:

$X_{k+1}=X_k+f\left(X_k\right)T+A\left(X_k\right)f\left(X_k\right)\frac{T^2}{2}+{\mathrm{\&Gamma;}}_k{q}_k$

In formula, X _kfor k state variable constantly; X _k+1for k+1 state variable constantly; A (X) is that f (X) is about the local derviation of X,

plant noise

t is the time interval, T=t _k+1-t _k;

Order $F\left(X_k\right)=X_k+f\left(X_k\right)T+A\left(X_k\right)f\left(X_k\right)\frac{T^2}{2},$ ?

X _k+1＝F(X _k)+Γ _kq _k

If taking into account system noise not, system k nominal state is constantly

$X_k^{*}=F\left(X_{k-1}^{*}\right)$

Around nominal state

by F (X _k) Taylor expansion, and get first approximation, obtain the approximately linear equation of trajectory:

$X_{k+1}=F\left(X_k^{*}\right)+\frac{\&PartialD;F\left(X\right)}{\&PartialD;X}{|}_{X=X_k^{*}}(X_k-X_k^{*})+{\mathrm{\&Gamma;}}_k{q}_k;$

Due to $F\left(X_k^{*}\right)=X_{k+1}^{*},$ And order ${\mathrm{\&phi;}}_k=\frac{\&PartialD;F\left(X\right)}{\&PartialD;X}{|}_{X=X_k^{*}}=I+A\left(X_k^{*}\right)T+\frac{d\left[A\left(X\right)f\left(X\right)\right]}{\mathrm{dX}}{|}_{X=X_k^{*}}\frac{T^2}{2},$ The approximately linear equation of trajectory is rewritten as:

$X_{k+1}=X_{k+1}^{*}+{\mathrm{\&phi;}}_k(X_k-X_k^{*})+{\mathrm{\&Gamma;}}_k{q}_k;$

By the nonlinear function h (X) in observation equation in nominal state

place is launched into Taylor series, and gets first approximation, obtains the approximately linear equation of observation equation:

$Z_k=Z_k^{*}+\frac{\&PartialD;h\left(X\right)}{\&PartialD;X}{|}_{X=X_k^{*}}[X_k-X_k^{*}]+V_k$

Make observing matrix the approximately linear equation of observation equation is rewritten as:

$Z_k=Z_k^{*}+H_k(X_k-X_k^{*})+V_k$

In formula, Z _kfor k observational variable constantly,

for k observational variable nominal value constantly, V _kfor k observation noise constantly.

3. a kind of ballistic trajectory formation method based on the filtering of exponential weighting Attenuation Memory Recursive according to claim 2, it is characterized in that, the Kalman filtering recurrence equation of setting up described in step 2 based on state deviation comprises state one-step prediction equation, state estimation equation, filter gain equation, one-step prediction square error equation, estimates square error equation, system state equation, specific as follows:

State variable and nominal state work difference are obtained to state deviation, that is:

$\mathrm{\&Delta;}{X}_k=X_k-X_k^{*}$

$\mathrm{\&Delta;}{Z}_k=Z_k-Z_k^{*}$

Approximately linear equation inference by trajectory in step 1 and observation equation draws:

$\mathrm{\&Delta;}{X}_{k+1}=X_{k+1}-X_{k+1}^{*}={\mathrm{\&phi;}}_k\mathrm{\&Delta;}{X}_k+{\mathrm{\&Gamma;}}_k{q}_k$

$\mathrm{\&Delta;}{Z}_k=Z_k-Z_k^{*}=H_k\mathrm{\&Delta;}{X}_k+V_k$

Obtain thus the Kalman filtering recurrence equation based on state deviation, specific as follows:

State one-step prediction equation:

ΔX _k+1/k＝φ _kΔX _k

State estimation equation:

ΔX _k+1＝ΔX _k+/1k+K _k+1[ΔZ _k+1-H _k+1ΔX _k+1/k]

Filter gain equation:

$K_{k+1}=P_{k+1/k}{H}_{k+1}^T{[H_{k+1}{P}_{k+1/k}{H}_{k+1}^T+R_{k+1}]}^{-1}$

One-step prediction square error equation:

$P_{k+1/k}={\mathrm{\&phi;}}_k{P}_k{\mathrm{\&phi;}}_k^T+{\mathrm{\&Gamma;}}_k{Q}_k{\mathrm{\&Gamma;}}_k^T$

Estimate square error equation:

P _k+1＝[1-K _k+1H _k+1]P _k+1/k

System state equation:

$X_k=X_k^{*}+\mathrm{\&Delta;}{X}_k$

In formula, Δ X _k+1/kone-step prediction value for state deviation; K _k+1for filter gain matrix; P _k+1/kone-step prediction value for square error; R _k+1for measuring noise square difference battle array; P _k+1estimated value for square error; Γ _kfor system k noise driving constantly battle array; Q _kfor system noise serial variance battle array.

4. a kind of ballistic trajectory formation method based on the filtering of exponential weighting Attenuation Memory Recursive according to claim 3, is characterized in that, the Attenuation Memory Recursive of exponential weighting described in step 3 filtering method is specific as follows:

By obtaining in step 2, the filter gain equation of Kalman filtering is:

$K_{k+1}=P_{k+1/k}{H}_{k+1}^T{[H_{k+1}{P}_{k+1/k}{H}_{k+1}^T+R_{k+1}]}^{-1}$

Get k=M, $K_M=P_{M/M-1}{H}_M^T{[H_M{P}_{M/M-1}{H}_M^T+R_M]}^{-1};$

For M Kalman filtering constantly, in order to suppress dispersing of filtering, should relatively give prominence to M K constantly _mvalue, and due to R _kand K _kthe relation that is inversely proportional to, in order to reduce the K of M before constantly _kvalue, should make the R away from more and more constantly from M _kbecome gradually large; Q _kwith R _kdo identical variation; If R _k, Q _k, P ₀all increase k doubly, the gain matrix K of stationary filter _kremain unchanged;

Take the way of exponential weighting, by P ₀; R ₁, R ₂, R ₃..., R _m; Q ₁, Q ₂, Q ₃..., Q _mbecome respectively lower column matrix:

$\left\{\begin{array}{cccccc}{e}^{\underset{i=0}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{P}_0& & & & & \\ e^{\underset{i=0}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{R}_1,& e^{\underset{i=1}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{R}_2,& e^{\underset{i=2}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{R}_3,& ...,& e^{{\mathrm{\&lambda;}}_{M-1}}{R}_M,& R_M\\ e^{\underset{i=0}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{Q}_1,& e^{\underset{i=1}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{Q}_2,& e^{\underset{i=2}{\overset{M-1}{\mathrm{\&Sigma;}}}{\mathrm{\&lambda;}}_i}{Q}_3,& ...,& e^{{\mathrm{\&lambda;}}_{M-1}}{Q}_M,& Q_M\end{array}\right.$

In formula, λ _ifor positive number; I is natural number, i=1, and 2,3 ..., M;

Exponential weighting, is equivalent to state X _kwhile asking optimal filtering, P ₀with the noise variance matrix R of M before the moment ₁, R ₂, R ₃..., R _m; Q ₁, Q ₂, Q ₃..., Q _mwith above formula, replace; One-step prediction square error equation, through arranging, obtains following weighted equation:

$P_{k+1/k}={\mathrm{\&phi;}}_k{P}_k{\mathrm{\&phi;}}_k^T{e}^{{\mathrm{\&lambda;}}_k}+{\mathrm{\&Gamma;}}_k{Q}_k{\mathrm{\&Gamma;}}_k^T$

In formula,

for Attenuation Memory Recursive filtering factor;

By Kalman filtering divergence criterion,

$e^{{\mathrm{\&lambda;}}_k}=\frac{{\tilde{Z}}_k^T{\tilde{Z}}_k-\mathrm{tr}(H_k{Q}_{k-1}^{*}{H}_k^T+R_k)}{\mathrm{tr}\left[H_k{\mathrm{\&phi;}}_k{P}_{k-1}{\mathrm{\&phi;}}_k^T{H}_k^T\right]}$

?

${\mathrm{\&lambda;}}_k=\ln \left(\frac{{\tilde{Z}}_k^T{\tilde{Z}}_k-\mathrm{tr}(H_k{Q}_{k-1}^{*}{H}_k^T+R_k)}{\mathrm{tr}\left[H_k{\mathrm{\&phi;}}_k{P}_{k-1}{\mathrm{\&phi;}}_k^T{H}_k^T\right]}\right)$

In formula,

for residual sequence,

for plant noise variance matrix, $Q_{k-1}^{*}={\mathrm{\&Gamma;}}_{k-1}{Q}_{k-1}{\mathrm{\&Gamma;}}_{k-1}^T;$

In filtering, arrange one and disperse judgment criterion, if

${\tilde{Z}}_k^T{\tilde{Z}}_k>r\&CenterDot;E\left(k\right)$

Carrying out above Attenuation Memory Recursive exponential weighting processes;

In formula,

r is reserve factor, r>=1;

If above formula is set up, the r that the actual estimated error that filtering is described is more than or equal to calculated value doubly; If r=1 is the strictest convergence basis for estimation condition.

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