卡尔曼滤波

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[mw_shl_code=applescript,true]%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  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）
%  详细原理介绍及中文注释请参考：
%  《卡尔曼滤波原理及应用-MATLAB仿真》，电子工业出版社，黄小平著。
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function main
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delta_t=0.01;
longa=10000;
tf=3.7;   
T=tf/delta_t;
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)];
G=[-1*0.5*delta_t^2*eye(3);-1*delta_t*eye(3);zeros(3)];
N=3;
for i=1:50
    x=zeros(9,TT);   % 此处有误，请修改为P99页一致即可运行
    x(:,1)=[3500,1500,1000,-1100,-150,-50,0,0,0]';
    ex=zeros(9,T);   
    ex(:,1)=[3000,1200,960,-800,-100,-100,0,0,0]';
    cigema=sqrt(0.1);  
    w=[zeros(6,T);cigema*randn(3,T)];
    Q=[zeros(6),zeros(6,3);zeros(3,6),cigema^2*eye(3)];  
    z=zeros(2,T);
    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);
    for k=2:T-3
        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]*[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);
        d=sqrt(x(1,k)^2+x(2,k)^2+x(3,k)^2);
        D=[d,0;0,d];
        R=inv(D)*0.1*eye(2)*inv(D)';
        v(:,k)=sqrtm(R)*randn(2,1);
        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
    P0=[10^4*eye(6),zeros(6,3);zeros(3,6),10^2*eye(3)];
    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)]';
        [ex(:,k),eP0]=ekf(F,G,Q,RR,eP0,u(:,k-1),z(:,k),ex(:,k-1));
    end
    for t=1:T-3
        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
        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=0.01:0.01:3.67;
figure
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('real','ekf');
view(3)
title('position')
figure
hold on;box on;grid on;
plot(t,error_r,'b');
xlabel('飞行时间/s');
ylabel('弹目相对距离估计误差/m');
figure
hold on;box on;grid on;
plot(t,error_v,'b');
xlabel('飞行时间/s');
ylabel('速度估计误差m/s');
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [ex,P0]=ekf(F,G,Q,R,P0,u,z,ex)
Xn=F*ex+G*u;
Zn=[atan( Xn(2)/sqrt(Xn(1)^2+Xn(3)^2) ),atan(-1*Xn(3)/Xn(1))]';
P=F*P0*F'+Q;
dh1_dx=-1*Xn(1)*Xn(2)/(Xn(1)^2+Xn(2)^2+Xn(3)^2)/sqrt(Xn(1)^2+Xn(3)^2);
dh1_dy=sqrt(Xn(1)^2+Xn(3)^2)/(Xn(1)^2+Xn(2)^2+Xn(3)^2);
dh1_dz=-1*Xn(2)*Xn(3)/(Xn(1)^2+Xn(2)^2+Xn(3)^2)/sqrt(Xn(1)^2+Xn(3)^2);
dh2_dx=Xn(3)/(Xn(1)^2+Xn(3)^2);
dh2_dy=0;
dh2_dz=-1*Xn(1)/(Xn(1)^2+Xn(3)^2);
H=[dh1_dx,dh1_dy,dh1_dz,0,0,0,0,0,0;dh2_dx,dh2_dy,dh2_dz,0,0,0,0,0,0];
K=P*H'/(H*P*H'+R);
ex=Xn+K*(z-Zn);
P0=(eye(9)-K*H)*P;
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