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卡尔曼滤波器卡尔曼滤波器（KalmanFilter）是一个最优化自回归数据处理算法（optimalrecursivedataprocessingalgorithm）。对于解决很大部分的问题，他是最优，效率最高甚至是最有用的。他的广泛应用已经超过30年，包括机器人导航，控制，传感器数据融合甚至在军事方面的雷达系统以及导弹追踪等等。近年来更被应用于计算机图像处理，例如头脸识别，图像分割，图像边缘检测等等。最佳线性滤波理论起源于40年代美国科学家Wiener和前苏联科学家Ｋолмогоров等人的研究工作，后人统称为维纳滤波理论。从理论上说，维纳滤波的最大缺点是必须用到无限过去的数据，不适用于实时处理。为了克服这一缺点，60年代Kalman把状态空间模型引入滤波理论，并导出了一套递推估计算法，后人称之为卡尔曼滤波理论。卡尔曼滤波是以最小均方误差为估计的最佳准则，来寻求一套递推估计的算法，其基本思想是：采用信号与噪声的状态空间模型，利用前一时刻地估计值和现时刻的观测值来更新对状态变量的估计，求出现时刻的估计值。它适合于实时处理和计算机运算。现设线性时变系统的离散状态防城和观测方程为：X(k)=F(k,k-1)·X(k-1)+T(k,k-1)·U(k-1)Y(k)=H(k)·X(k)+N(k)其中X(k)和Y(k)分别是k时刻的状态矢量和观测矢量F(k,k-1)为状态转移矩阵U(k)为k时刻动态噪声T(k,k-1)为系统控制矩阵H(k)为k时刻观测矩阵N(k)为k时刻观测噪声则卡尔曼滤波的算法流程为：预估计X(k)^=F(k,k-1)·X(k-1)计算预估计协方差矩阵C(k)^=F(k,k-1)×C(k)×F(k,k-1)'+T(k,k-1)×Q(k)×T(k,k-1)'Q(k)=U(k)×U(k)'计算卡尔曼增益矩阵K(k)=C(k)^×H(k)'×[H(k)×C(k)^×H(k)'+R(k)]^(-1)R(k)=N(k)×N(k)'更新估计X(k)~=X(k)^+K(k)×[Y(k)-H(k)×X(k)^]计算更新后估计协防差矩阵C(k)~=[I-K(k)×H(k)]×C(k)^×[I-K(k)×H(k)]'+K(k)×R(k)×K(k)'X(k+1)=X(k)~C(k+1)=C(k)~重复以上步骤＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊Ｍatlab实现代码[转]*********************************************************************************************************************************%%%%ConstantVelocityModelKalmanFilterSimulation%%%%%==========================================================================clearall;closeall;clc;%%Initialconditionts=1;%Samplingtimet=[0:ts:100];T=length(t);%%Initialstatex=[040020]';x_hat=[0000]';%%Processnoisecovarianceq=5Q=q*eye(2);%%Measurementnoisecovariancer=5R=r*eye(2);%%Processandmeasurementnoisew=sqrt(Q)*randn(2,T);%Processnoisev=sqrt(R)*randn(2,T);%Measurementnoise%%Estimateerrorcovarianceinitializationp=5;P(:,:,1)=p*eye(4);%==========================================================================%%Continuous-timestatespacemodel%{x_dot(t)=Ax(t)+Bu(t)z(t)=Cx(t)+Dn(t)%}A=[0100;0000;0001;0000];B=[00;10;00;01];C=[1000;0010];D=[10;01];%%Discrete-timestatespacemodel%{x(k+1)=Fx(k)+Gw(k)z(k)=Hx(k)+Iv(k)Continuoustodiscreteformbyzoh%}sysc=ss(A,B,C,D);sysd=c2d(sysc,ts,'zoh');[FGHI]=ssdata(sysd);%%Practicestateoftargetfori=1:T-1x(:,i+1)=F*x(:,i);endx=x+G*w;%Statevariablewithnoisez=H*x+I*v;%Measurementvaluewithnoise%==========================================================================%%%KalmanFilterfori=1:T-1%%Predictionphasex_hat(:,i+1)=F*x_hat(:,i);%StateestimatepredictP(:,:,i+1)=F*P(:,:,i)*F'+G*Q*G';%TrackingerrorcovariancepredictP_predicted(:,:,i+1)=P(:,:,i+1);%%KalmangainK=P(:,:,i+1)*H'*inv(H*P(:,:,i+1)*H'+R);%%Updatastepx_hat(:,i+1)=x_hat(:,i+1)+K*(z(:,i+1)-H*x_hat(:,i+1));%StateestimateupdateP(:,:,i+1)=P(:,:,i+1)-K*H*P(:,:,i+1);%TrackingerrorcovarianceupdateP_updated(:,:,i+1)=P(:,:,i+1);end%==========================================================================%%Estimateerrorx_error=x-x_hat;%%Graph1practicalandtrackingpositionfigure(1)plot(x(1,:),x(3,:),'r');holdon;plot(x_hat(1,:),x_hat(3,:),'g.');title('2DTargetPosition')legend('PracticalPosition','TrackingPosition')xlabel('Xaxis[m]')ylabel('Yaxis[m]')holdoff;%%Graph2figure(2)plot(t,x(1,:)),gridon;holdon;plot(t,x_hat(1,:),'r'),gridon;title('PracticalandTrackingPositiononXaxis')legend('PracticalPosition','TrackingPosition')xlabel('Time[sec]')ylabel('Position[m]')holdoff;%%Graph3figure(3)plot(t,x_error(1,:)),gridon;title('PositionErroronXaxis')xlabel('Time[sec]')ylabel('PositionRMSE[m]')holdoff;%%Graph4figure(4)plot(t,x(2,:)),gridon;holdon;plot(t,x_hat(2,:),'r'),gridon;title('PracticalandTrackingVelocityonXaxis')legend('PracticalVelocity','TrackingVelocity')xlabel('Time[sec]')ylabel('Velocity[m/sec]')holdoff;%%Graph5figure(5)plot(t,x_error(2,:)),gridon;title('VelocityErroronXaxis')xlabel('Time[sec]')ylabel('VelocityRMSE[m/sec]')holdoff;%==========================================================================*********************************************************************************************************************************＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊c语言实现代码[转]-----------------------------------------------------------------------------------------------------------------------------#includestdlib.h#includerinv.cintlman(n,m,k,f,q,r,h,y,x,p,g)intn,m,k;doublef[],q[],r[],h[],y[],x[],p[],g[];{inti,j,kk,ii,l,jj,js;double*e,*a,*b;e=malloc(m*m*sizeof(double));l=m;if(ln)l=n;a=malloc(l*l*sizeof(double));b=malloc(l*l*sizeof(double));for(i=0;i=n-1;i++)for(j=0;j=n-1;j++){ii=i*l+j;a[ii]=0.0;for(kk=0;kk=n-1;kk++)a[ii]=a[ii]+p[i*n+kk]*f[j*n+kk];}for(i=0;i=n-1;i++)for(j=0;j=n-1;j++){ii=i*n+j;p[ii]=q[ii];for(kk=0;kk=n-1;kk++)p[ii]=p[ii]+f[i*n+kk]*a[kk*l+j];}for(ii=2;ii=k;ii++){for(i=0;i=n-1;i++)for(j=0;j=m-1;j++){jj=i*l+j;a[jj]=0.0;for(kk=0;kk=n-1;kk++)a[jj]=a[jj]+p[i*n+kk]*h[j*n+kk];}for(i=0;i=m-1;i++)for(j=0;j=m-1;j++){jj=i*m+j;e[jj]=r[jj];for(kk=0;kk=n-1;kk++)e[jj]=e[jj]+h[i*n+kk]*a[kk*l+j];}js=rinv(e,m);if(js==0){free(e);free(a);free(b);return(js);}for(i=0;i=n-1;i++)for(j=0;j=m-1;j++){jj=i*m+j;g[jj]=0.0;for(kk=0;kk=m-1;kk++)g[jj]=g[jj]+a[i*l+kk]*e[j*m+kk];}for(i=0;i=n-1;i++){jj=(ii-1)*n+i;x[jj]=0.0;for(j=0;j=n-1;j++)x[jj]=x[jj]