卡尔曼滤波matlab仿真

一、温度测量

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  % 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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%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');  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%figure;  plot(t,Err_Messure,'-b',t,Err_Kalman,'-k');  legend('测量误差','kalman误差');           xlabel('采样时间');  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%empt

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)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%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,xkfx=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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %  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  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%