% 程序说明：目标跟踪程序，实现运动弹头对运动物体的三维跟踪，主函数
% 状态方程： x(t)=Ax(t-1)+Bu(t-1)+w(t）
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clc;
clear;
close all;
delta_t=0.01; %测量周期，采样周期
longa=10000; %机动时间常数的倒数，即机动频率
tf=6;
T=tf/delta_t; %时间长度3.7s，一共采用T=370次
%状态转移矩阵，用F表示
F=[eye(3),delta_t*eye(3),(exp(-1*longa*delta_t)+...
longa*delta_t-1)/longa^2*eye(3);
zeros(3),eye(3),(1-exp(-1*longa*delta_t))/longa*eye(3);
zeros(3),zeros(3),exp(-1*longa*delta_t)*eye(3)];
%控制量驱动矩阵gama
G=[-1*0.5*delta_t^2*eye(3);-1*delta_t*eye(3);zeros(3)];
N=3; %导航比 (制导率)
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u=10*ones(3,T);
for i=1:80 %做50次蒙特卡罗仿真
x=zeros(9,T);
x(:,1)=[3500,1500,1000,-1100,-150,-50,0,0,0]'; %初始状态X(0)
ex=zeros(9,T);
ex(:,1)=[3000,1200,960,-800,-100,-100,0,0,0]'; %滤波器状态Xekf(0)
cigema=sqrt(0.1);
w=[zeros(6,T);cigema*randn(3,T)]; %过程噪声
Q=[zeros(6),zeros(6,3);zeros(3,6),cigema^2*eye(3)];
z=zeros(2,T); %观测值
z(:,1)=[atan( x(2,1)/sqrt(x(1,1)^2+x(3,1)^2) ), atan(-1*x(3,1)/x(1,1))]';
v=zeros(2,T); %观测噪声
for k=2:T-3
tgo=tf-k*0.01+0.0000000000000001;
c1=N/tgo^2; %制导率的系数
c2=N/tgo; %制导率的系数
c3=N*(exp(-longa*tgo)+longa*tgo-1)/(longa*tgo)^2; %制导率的系数
%三个方向的导弹加速度
u(1,k-1)=[c1,c2,c3]*[x(1,k-1),x(4,k-1),x(7,k-1)]';
u(2,k-1)=[c1,c2,c3]*[x(2,k-1),x(5,k-1),x(8,k-1)]';
u(3,k-1)=[c1,c2,c3]*[x(3,k-1),x(6,k-1),x(9,k-1)]';
x(:,k)=F*x(:,k-1)+G*u(:,k-1)+w(:,k-1); %目标状态方程
d=sqrt(x(1,k)^2+x(2,k)^2+x(3,k)^2);
D=[d,0;0,d]; %参考书中的公式
R=inv(D)*0.1*eye(2)*inv(D)'; %观测噪声方差
v(:,k)=sqrtm(R)*randn(2,1); %观测噪声模拟
%目标观测方程
z(:,k)=[atan( x(2,k)/sqrt(x(1,k)^2+x(3,k)^2) ), ...
atan(-1*x(3,k)/x(1,k))]'+v(:,k);
end
%下面根据观测值开始滤波
P0=[10^4*eye(6),zeros(6,3);zeros(3,6),10^2*eye(3)]; %协方差初始化
eP0=P0;
stop=0.5/0.01;
span=1/0.01;
for k=2:T-3
dd=sqrt(ex(1,k-1)^2+ex(2,k-1)^2+ex(3,k-1)^2);
DD=[dd,0;0,dd];
RR=0.1*eye(2)/(DD*DD');
tgo=tf-k*0.01+0.0000000000000001;
c1=N/tgo^2;
c2=N/tgo;
c3=N*(exp(-longa*tgo)+longa*tgo-1)/(longa*tgo)^2;
u(1,k-1)=[c1,c2,c3]*[ex(1,k-1),ex(4,k-1),ex(7,k-1)]';
u(2,k-1)=[c1,c2,c3]*[ex(2,k-1),ex(5,k-1),ex(8,k-1)]';
u(3,k-1)=[c1,c2,c3]*[ex(3,k-1),ex(6,k-1),ex(9,k-1)]';
%调用扩展Kalman算法子函数
[ex(:,k),eP0]=ekf(F,G,Q,RR,eP0,u(:,k-1),z(:,k),ex(:,k-1));
end
for t=1:T-3 %求每个时间点误差的平方
Ep_ekfx(i,t)=sqrt((ex(1,t)-x(1,t))^2);
Ep_ekfy(i,t)=sqrt((ex(2,t)-x(2,t))^2);
Ep_ekfz(i,t)=sqrt((ex(3,t)-x(3,t))^2);
Ep_ekf(i,t)=sqrt( (ex(1,t)-x(1,t))^2+(ex(2,t)-x(2,t))^2+(ex(3,t)-x(3,t))^2 );
Ev_ekf(i,t)=sqrt( (ex(4,t)-x(4,t))^2+(ex(5,t)-x(5,t))^2+(ex(6,t)-x(6,t))^2 );
Ea_ekf(i,t)=sqrt( (ex(7,t)-x(7,t))^2+(ex(8,t)-x(8,t))^2+(ex(9,t)-x(9,t))^2 );
end
for t=1:T-3 %求误差的均值，即RMS
error_x(t)=mean(Ep_ekfx(:,t));
error_y(t)=mean(Ep_ekfy(:,t));
error_z(t)=mean(Ep_ekfz(:,t));
error_r(t)=mean(Ep_ekf(:,t));
error_v(t)=mean(Ev_ekf(:,t));
error_a(t)=mean(Ea_ekf(:,t));
end
end
t=0.01:0.01:5.97;
figure %轨迹图
hold on;box on;grid on;
plot3(x(1,:),x(2,:),x(3,:),'-k.')
plot3(ex(1,:),ex(2,:),ex(3,:),'-r*','MarkerFace','r')
legend('真实值','EKF滤波值');
view(3)
xlabel('x/m');
ylabel('y/m');
zlabel('z/m');
title('position')
figure %位置偏差图
hold on;box on;grid on;
plot(t,error_r,'y*');
xlabel('飞行时间/s');
ylabel('弹-目相对距离估计误差/m');
title('位置偏差图')
figure %速度偏差图
hold on;box on;grid on;
plot(t,error_v,'-r*');
xlabel('飞行时间/s');
ylabel('速度估计误差m/s');
title('速度偏差图')
figure %加速度偏差图
hold on;box on;grid on;
plot(t,error_a,'-g*');
xlabel('飞行时间/s');
ylabel('加速度估计误差m^2/s');
title('加速度偏差图')