卡尔曼滤波应用

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方程和观测方程为：

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

卡尔曼滤波C程序：

  #include "stdlib.h"
  #include "rinv.c"
  int kalman(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++实现程序如下

============================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>

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;
    
    
     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) );

    
    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){
    

    
    state->data.fl[0]=x;
    state->data.fl[2]=y;

    
    
    
    cvRandSetRange( &rng, 0, sqrt(cvkalman->measurement_noise_cov->data.fl[0]), 0 );
    cvRand( &rng, measurement );
    
     
    cvMatMulAdd( cvkalman->transition_matrix, state, process_noise, cvkalman->state_post );
    
    
    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;
}

注：卡尔曼全名Rudolf Emil Kalman，匈牙利数学家，1930年出生于匈牙利首都布达佩斯。1953，1954年于麻省理工学院分别获得电机工程学士及硕士学位。1957年于哥伦比亚大学获得博士学位。我们现在要学习的卡尔曼滤波器，正是源于他的博士论文和1960年发表的论文《A New Approach to Linear Filtering and Prediction Problems》（线性滤波与预测问题的新方法）。如果对这编论文有兴趣，可以到这里的地址下载：http://www.cs.unc.edu/~welch/media/pdf/Kalman1960.pdf

还有个帖子，http://dzone.com/snippets/simple-kalman-filter-c，外国人写的简易卡曼滤波的C程序，值得参考。