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卡尔曼滤波简介与算法实现代码

2007-01-13

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

其c语言实现代码如下：

#include \ #include \

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

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

{ 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 #include \

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 \#include

/* 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 );