卡尔曼滤波简介

卡尔曼滤波简介＋算法实现代码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：1.预估计X(k)^=F(k,k-1)·X(k-1)2.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)'3.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)'4.更新估计X(k)~=X(k)^+K(k)×[Y(k)-H(k)×X(k)^]5.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)'6.X(k+1)=X(k)~C(k+1)=C(k)~重复以上步骤其c语言实现代码如下：#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]+f[i*n+j]*x[(ii-2)*n+j];}for(i=0;i=m-1;i++){jj=i*l;b[jj]=y[(ii-1)*m+i];for(j=0;j=n-1;j++)b[jj]=b[jj]-h[i*n+j]*x[(ii-1)*n+j];}for(i=0;i=n-1;i++){jj=(ii-1)*n+i;for(j=0;j=m-1;j++)x[jj]=x[jj]+g[i*m+j]*b[j*l];}if(iik){for(i=0;i=n-1;i++)for(j=0;j=n-1;j++){jj=i*l+j;a[jj]=0.0;for(kk=0;kk=m-1;kk++)a[jj]=a[jj]-g[i*m+kk]*h[kk*n+j];if(i==j)a[jj]=1.0+a[jj];}for(i=0;i=n-1;i++)for(j=0;j=n-1;j++){jj=i*l+j;b[jj]=0.0;for(kk=0;kk=n-1;kk++)b[jj]=b[jj]+a[i*l+kk]*p[kk*n+j];}for(i=0;i=n-1;i++)for(j=0;j=n-1;j++){jj=i*l+j;a[jj]=0.0;for(kk=0;kk=n-1;kk++)a[jj]=a[jj]+b[i*l+kk]*f[j*n+kk];}for(i=0;i=n-1;i++)for(j=0;j=n-1;j++){jj=i*n+j;p[jj]=q[jj];for(kk=0;kk=n-1;kk++)p[jj]=p[jj]+f[i*n+kk]*a[j*l+kk];}}}free(e);free(a);free(b);return(js);}C++实现代码如下：============================kalman.h================================//kalman.h:interfaceforthekalmanclass.////////////////////////////////////////////////////////////////////////#if!defined(AFX_KALMAN_H__ED3D740F_01D2_4616_8B74_8BF57636F2C0__INCLUDED_)#defineAFX_KALMAN_H__ED3D740F_01D2_4616_8B74_8BF57636F2C0__INCLUDED_#if_MSC_VER1000#pragmaonce#endif//_MSC_VER1000#includemath.h#includecv.hclasskalman{public:voidinit_kalman(intx,intxv,inty,intyv);CvKalman*cvkalman;CvMat*state;CvMat*process_noise;CvMat*measurement;constCvMat*prediction;CvPoint2D32fget_predict(floatx,floaty);kalman(intx=0,intxv=0,inty=0,intyv=0);//virtual~kalman();};#endif//!defined(AFX_KALMAN_H__ED3D740F_01D2_4616_8B74_8BF57636F2C0__INCLUDED_)============================kalman.cpp================================#includekalman.h#includestdio.h/*testerdeprintertouteslesvaleursdesvecteurs*//*testerdechangerlesmatricesdunoises*//*replacestatebycvkalman-state_post???*/CvRandStaterng;constdoubleT=0.1;kalman::kalman(intx,intxv,inty,intyv){cvkalman=cvCreateKalman(4,4,0);state=cvCreateMat(4,1,CV_32FC1);process_noise=cvCreateMat(4,1,CV_32FC1);measurement=cvCreateMat(4,1,CV_32FC1);intcode=-1;/*creatematrixdata*/constfloatA[]={1,T,0,0,0,1,0,0,0,0,1,T,0,0,0,1};constfloatH[]={1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0};constfloatP[]={pow(320,2),pow(320,2)/T,0,0,pow(320,2)/T,pow(320,2)/pow(T,2),0,0,0,0,pow(240,2),pow(240,2)/T,0,0,pow(240,2)/T,pow(240,2)/pow(T,2)};constfloatQ[]={pow(T,3)/3,pow(T,2)/2,0,0,pow(T,2)/2,T,0,0,0,0,pow(T,3)/3,pow(T,2)/2,0,0,pow(T,2)/2,T};constfloatR[]={1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0};cvRandInit(&rng,0,1,-1,CV_RAND_UNI);cvZero(measurement);cvRandSetRange(&rng,0,0.1,0);rng.disttype=CV_RAND_NORMAL;cvRand(&rng,state);memcpy(cvkalman-transition_matrix-data.fl,A,sizeof(A));memcpy(cvkalman-measurement_matrix-data.fl,H,sizeof(H));memcpy(cvkalman-process_noise_cov-data.fl,Q,sizeof(Q));memcpy(cvkalman-error_cov_post-data.fl,P,sizeof(P));memcpy(cvkalman-measurement_noise_cov-data.fl,R,sizeof(R));//cvSetIdentity(cvkalman-process_noise_cov,cvRealScalar(1e-5));//cvSetIdentity(cvkalman-error_cov_post,cvRealScalar(1));//cvSetIdentity(cvkalman-measurement_noise_cov,cvRealScalar(1e-1));/*chooseinitialstate*/state-data.fl[0]=x;state-data.fl[1]=xv;state-data.fl[2]=y;state-data.fl[3]=yv;cvkalman-state_post-data.fl[0]=x;cvkalman-state_post-data.fl[1]=xv;cvkalman-state_post-data.fl[2]=y;cvkalman-state_post-data.fl[3]=yv;cvRandSetRange(&rng,0,sqrt(cvkalman-process_noise_