卡尔曼滤波matlab仿真

一、温度测量

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% 程序说明：Kalman滤波用于温度测量的实例(一维）
function main
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%初始化参数
N=120;  %采样点个数
A=1;
B=1;
H=1;
Q=0.01;
R=0.25;
W=sqrt(Q)*randn(1,N);
V=sqrt(R)*randn(1,N);
CON=25;  %温度真实值
%分配空间
X=zeros(1,N);
Z=zeros(1,N);
Xkf=zeros(1,N);
P=zeros(1,N);
X(1)=25.1;
P(1)=0.01;
Z(1)=24.9;
Xkf(1)=Z(1);
I=eye(1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%kalman过程
for k=2:N
    X(k)=A*X(k-1)+B*W(k);
    Z(k)=H*X(k)+V(k);
    X_pre(k)=A*Xkf(k-1);
    P_pre(k)=A*P(k-1)*A'+Q;
    Kg=P_pre(k)/(H*P_pre(k)*H'+R);
    Xkf(k)=X_pre(k)+Kg*(Z(k)-H*X_pre(k));
    P(k)=(I-Kg*H)*P_pre(k);
end
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%误差过程
Err_Messure=zeros(1,N);
Err_Kalman=zeros(1,N);
for k=1:N
    Err_Messure(k)=Z(k)-X(k);
    Err_Kalman(k)=Xkf(k)-X(k);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
t=1:N;
figure;
plot(t,CON*ones(1,N),'-b',t,X,'-r',t,Z,'g+',t,Xkf,'-k');
legend('期望值','真实值','测量值','kalman估计值');
xlabel('采样时间');
ylabel('温度');
title('Kalman Filter Simulation');
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figure;
plot(t,Err_Messure,'-b',t,Err_Kalman,'-k');
legend('测量误差','kalman误差');
xlabel('采样时间');
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二、自由落体追踪（二维x,v）

%kalman滤波用于自由落体运动目标追踪（二维，x,v)
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%初始化参数
N=1000;
Q=[0,0;0,0];
R=0.5;
w=sqrt(Q)*randn(2,N);
v=sqrt(R)*randn(1,N);
A=[1,1;0,1];%状态转移矩阵
B=[0.5;1];%控制矩阵
U=-10;%控制变量
H=[1,0];%观测矩阵
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%分配空间，x,p0,z,xkf
x=zeros(2,N);%物体真实状态
x(:,1)=[95;1];%初始位移和速度
p0=[10,0;0,1];%初始误差
z=zeros(1,N);%观测值
z(1)=H*x(:,1);%观测真实值,第一列的列向量，
xkf=zeros(2,N);%kalman估计状态
xkf(:,1)=x(:,1);%kalman估计状态初始化,第一列的列向量，
err_p=zeros(N,2);
err_p(1,1)=p0(1,1);
err_p(1,2)=p0(2,2);
I=eye(2);%2*2单位矩阵表明二维系统
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%kalman迭代过程
for k=2:N;
    x(:,k)=A*x(:,k-1)+B*U+w(k);%模型
    z(k)=H*x(:,k)+v(k);%模型
    x_pre=A*xkf(:,k-1)+B*U;%①
    p_pre=A*p0*A'+Q;%②
    kg=p_pre*H'/(H*p_pre*H'+R);%③
    xkf(:,k)=x_pre+kg*(z(k)-H*x_pre);%④
    p0=(I-kg*H)*p_pre;%⑤
    err_p(k,1)=p0(1,1);%位移误差均方值
    err_p(k,2)=p0(2,2);%速度误差均方值
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%误差计算
messure_err_x=zeros(1,N);
kalman_err_x=zeros(1,N);
kalman_err_v=zeros(1,N);
for k=1:N
    messure_err_x(k)=z(k)-x(1,k);%位移的测量误差
    kalman_err_x(k)=xkf(1,k)-x(1,k);%kalman估计位移与真实位移之间偏差
    kalman_err_v(k)=xkf(2,k)-x(2,k);%kalman估计速度与真实速度之间偏差
end
%噪声图
figure;
plot(w);xlabel('采样时间');ylabel('过程噪声');
figure;
plot(v);xlabel('采样时间');ylabel('测量噪声');
%位置误差
figure;
hold on,box on;
plot(messure_err_x,'-r.');
plot(kalman_err_x,'-g.');
legend('测量位置','kalman估计位置');
xlabel('采样时间');ylabel('位置偏差');
%kalman速度偏差
figure;
plot(kalman_err_v);
xlabel('采样时间');ylabel('速度偏差');
%均方值
figure;
plot(err_p(:,1));
xlabel('采样时间');ylabel('位移误差均方值');
figure;
plot(err_p(:,1));
xlabel('采样时间');ylabel('速度误差均方值');

三、GPS导航定位

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%  Kalman滤波在船舶GPS导航定位系统中的应用(四维，x方向x,v   ,y方向x,v; ）
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function main
T=1;  %采样周期
N=80/T;  %总的采样次数
F=[1,T,0,0;0,1,0,0;0,0,1,T;0,0,0,1];
H=[1,0,0,0;0,0,1,0];
Q=4e-4  ;
R=100*eye(2);
X=zeros(4,N);
X(:,1)=[100,2,200,20]; %x四维，初始位置（-100,200）水平速度2，垂直速度20
Z=zeros(2,N);
Z(:,1)=[X(1,1),X(3,1)];  %z只观测到位置，没有速度
Xkf=zeros(4,N);
Xkf(:,1)=X(:,1);
P0=eye(4);  %协方差矩阵初始化，4*4单位矩阵
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for t=2:N  %模型
    X(:,t)=F*X(:,t-1)+sqrt(Q)*randn(4,1);%目标真实轨迹  randn(4,1)4*1随机矩阵
    Z(:,t)=H*X(:,t)+sqrt(R)*randn(2,1); %观测轨迹randn(2,1)2*1随机矩阵
end
for i=2:N
    X_pre=F*Xkf(:,i-1);
    P_pre=F*P0*F'+Q;
    K=P_pre*H'/(H*P_pre*H'+R);
    Xkf(:,i)=X_pre+K*(Z(:,i)-H*X_pre);
    P0=(eye(4)-K*H)*P_pre;
end
for i=1:N
    Err_Observation(i)=RMS(X(:,i),Z(:,i));
    Err_KalmanFilter(i)=RMS(X(:,i),Xkf(:,i));
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure;
hold on;box on;
plot(X(1,:),X(3,:),'-k');  %位置轨迹
plot(Z(1,:),Z(2,:),'-b.');  %观测轨迹
plot(Xkf(1,:),Xkf(3,:),'-r+');  %Kalman估计轨迹
legend('真实轨迹','观测轨迹','滤波轨迹')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure;
hold on; box on;
plot(Err_Observation,'-ko','MarkerFace','g')
plot(Err_KalmanFilter,'-ks','MarkerFace','r')
legend('滤波前误差','滤波后误差')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function dist=RMS(X1,X2);
if length(X2)<=2
    dist=sqrt( (X1(1)-X2(1))^2 + (X1(3)-X2(2))^2 );
else
    dist=sqrt( (X1(1)-X2(1))^2 + (X1(3)-X2(3))^2 );
end
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转载于https://blog.csdn.net/try_again_later/article/details/78304691