卡尔曼滤波程序.doc

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;aii=0.0;for(kk=0;kk=n-1;kk+)aii=aii+pi*n+kk*fj*n+kk;for(i=0;i=n-1;i+)for(j=0;j=n-1;j+)ii=i*n+j;pii=qii;for(kk=0;kk=n-1;kk+)pii=pii+fi*n+kk*akk*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;ajj=0.0;for(kk=0;kk=n-1;kk+)ajj=ajj+pi*n+kk*hj*n+kk;for(i=0;i=m-1;i+)for(j=0;j=m-1;j+)jj=i*m+j;ejj=rjj;for(kk=0;kk=n-1;kk+)ejj=ejj+hi*n+kk*akk*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;gjj=0.0;for(kk=0;kk=m-1;kk+)gjj=gjj+ai*l+kk*ej*m+kk;for(i=0;i=n-1;i+)jj=(ii-1)*n+i;xjj=0.0;for(j=0;j=n-1;j+)xjj=xjj+fi*n+j*x(ii-2)*n+j;for(i=0;i=m-1;i+)jj=i*l;bjj=y(ii-1)*m+i;for(j=0;j=n-1;j+)bjj=bjj-hi*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+)xjj=xjj+gi*m+j*bj*l;if(iik)for(i=0;i=n-1;i+)for(j=0;j=n-1;j+)jj=i*l+j;ajj=0.0;for(kk=0;kk=m-1;kk+)ajj=ajj-gi*m+kk*hkk*n+j;if(i=j)ajj=1.0+ajj;for(i=0;i=n-1;i+)for(j=0;j=n-1;j+)jj=i*l+j;bjj=0.0;for(kk=0;kk=n-1;kk+)bjj=bjj+ai*l+kk*pkk*n+j;for(i=0;i=n-1;i+)for(j=0;j=n-1;j+)jj=i*l+j;ajj=0.0;for(kk=0;kk=n-1;kk+)ajj=ajj+bi*l+kk*fj*n+kk;for(i=0;i=n-1;i+)for(j=0;j=n-1;j+)jj=i*n+j;pjj=qjj;for(kk=0;kk1000#pragmaonce#endif/_MSC_VER1000#include#includecv.hclasskalmanpublic: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);/virtualkalman();#endif/!defined(AFX_KALMAN_H_ED3D740F_01D2_4616_8B74_8BF57636F2C0_INCLUDED_)=kalman.cpp=#includekalman.h#include/*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.fl0=x;state-data.fl1=xv;state-data.fl2=y;state-data.fl3=yv;cvkalman-state_post-data.fl0=x;cvkalman-state_post-data.fl1=xv;cvkalman-state_post-data.fl2=y;cvkalman-state_post-data.fl3=yv;cvRandSetRange(&rng,0,sqrt(cvkalman-process_noise_cov-data.fl0),0);cvRand(&rng,process_noise);CvPoint2D32fkalman:get_predict(floatx,floaty)/*updatestatewithcurrentposition*/state-data.fl0=x;state-data.fl2=y;/*predictpointposition*/*xk=A鈥k+B鈥kPk=A鈥k-1*AT+Q*/cvRandSetRange(&rng,0,sqrt(cvkalman-measurement_noise_cov-data.fl0),0);cvRand(&rng,measurement);/*xk=A?xk-1+B?uk+wk*/cvMatMulAdd(cvkalman-transition_matrix,state,process_noise,cvkalman-state_post);/*zk=H?xk+vk*/cvMatMulAdd(cvkalman-measurement_matrix,cvkalman-state_post,measurement,measurement);/*adjustKalmanfilterstate*/*Kk=Pk鈥T鈥?H鈥k鈥T+R)-1xk=xk+Kk鈥?zk-H鈥k)Pk=(I-Kk鈥)鈥k*/cvKalmanCorrect(cvkalman,measurement);floatmeasured_value_x=measurement-data.fl0;floatmeasured_value_y=measurement-data.fl2;constCvMat*prediction=cvKalmanPredict(cvkalman,0);floatpredict_value_x=prediction-data.fl0;floatpredict_value_y=prediction-data.fl2;return(cvPoint2D32f(predict_valu