卡尔曼滤波基础知识

卡尔曼滤波马尔可夫过程：在随机理论中，把在某时刻的事件受在这之前事件的影响，其影响范围有限的随机过程，称为马尔可夫过程。

一个事件受在它之前的事件的影响的深远程度，通常用在它之前的事件作为条件的概率来表达。

受前一个事件的影响，简称为马尔可夫过程；受前两个事件的影响，称为二阶马尔可夫过程；受前三个事件的影响，称为三阶马尔可夫过程！卡尔曼滤波简介＋算法实现代码（转）：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：12预估计X(k)^= F(k,k-1)·X(k-1)3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)'4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)'5更新估计X(k)~=X(k)^+K(k)×[Y(k)-H(k)×X(k)^]6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)'7X(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：#include "stdlib.h"#include "rinv.c"int lman(n,m,k,f,q,r,h,y,x,p,g)int n,m,k;double f[],q[],r[],h[],y[],x[],p[],g[];{ int i,j,kk,ii,l,jj,js;double *e,*a,*b;e=malloc(m*m*sizeof(double));l=m;if (l<n) 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 (ii<k){ 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: interface for the kalman class.////////////////////////////////////////////////////////////////////////#if !defined(AFX_KALMAN_H__ED3D740F_01D2_4616_8B74_8BF57636F2C0__INCLUDED_) #define AFX_KALMAN_H__ED3D740F_01D2_4616_8B74_8BF57636F2C0__INCLUDED_#if _MSC_VER > 1000#pragma once#endif // _MSC_VER > 1000#include <math.h>#include "cv.h"class kalman{public:void init_kalman(int x,int xv,int y,int yv);CvKalman* cvkalman;CvMat* state;CvMat* process_noise;CvMat* measurement;const CvMat* prediction;CvPoint2D32f get_predict(float x, float y);kalman(int x=0,int xv=0,int y=0,int yv=0);//virtual ~kalman();};#endif // !defined(AFX_KALMAN_H__ED3D740F_01D2_4616_8B74_8BF57636F2C0__INCLUDED_) ============================kalman.cpp================================#include "kalman.h"#include <stdio.h>/* tester de printer toutes les valeurs des vecteurs*//* tester de changer les matrices du noises *//* replace state by cvkalman->state_post ??? */CvRandState rng;const double T = 0.1;kalman::kalman(int x,int xv,int y,int yv){cvkalman = cvCreateKalman( 4, 4, 0 );state = cvCreateMat( 4, 1, CV_32FC1 );process_noise = cvCreateMat( 4, 1, CV_32FC1 );measurement = cvCreateMat( 4, 1, CV_32FC1 );int code = -1;/* create matrix data */const float A[] = {1, T, 0, 0,0, 1, 0, 0,0, 0, 1, T,0, 0, 0, 1};const float H[] = {1, 0, 0, 0,0, 0, 0, 0,0, 0, 1, 0,0, 0, 0, 0};const float P[] = {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)};const float Q[] = {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};const float R[] = {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) );/* choose initial state */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_cov->data.fl[0]), 0 );cvRand( &rng, process_noise );}CvPoint2D32f kalman::get_predict(float x, float y){/* update state with current position */state->data.fl[0]=x;state->data.fl[2]=y;/* predict point position *//* x'k=A鈥k+B鈥kP'k=A鈥k-1*AT + Q */cvRandSetRange( &rng, 0, sqrt(cvkalman->measurement_noise_cov->data.fl[0]), 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 );cvKalmanCorrect( cvkalman, measurement );float measured_value_x = measurement->data.fl[0];float measured_value_y = measurement->data.fl[2];const CvMat* prediction = cvKalmanPredict( cvkalman, 0 );float predict_value_x = prediction->data.fl[0];float predict_value_y = prediction->data.fl[2];return(cvPoint2D32f(predict_value_x,predict_value_y));}void kalman::init_kalman(int x,int xv,int y,int yv){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;}白噪声是的特点是功率谱为常数，也就是各个频率成分幅值相同，无法用传统的低通、高通、带通、带阻滤波器去除。