中财网关于遇见的优美句子:卡尔曼滤波_附：算法实现代码
来源：百度文库 编辑：中财网 时间：2024/04/28 02:03:07
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
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Ｍatlab实现代码 [转]
*********************************************************************************************************************************
%%%% Constant Velocity Model Kalman Filter Simulation %%%%
%==========================================================================
clear all; close all; clc;
%% Initial condition
ts = 1; % Sampling time
t = [0:ts:100];
T = length(t);
%% Initial state
x = [0 40 0 20]';
x_hat = [0 0 0 0]';
%% Process noise covariance
q = 5
Q = q*eye(2);
%% Measurement noise covariance
r = 5
R = r*eye(2);
%% Process and measurement noise
w = sqrt(Q)*randn(2,T); % Process noise
v = sqrt(R)*randn(2,T); % Measurement noise
%% Estimate error covariance initialization
p = 5;
P(:,:,1) = p*eye(4);
%==========================================================================
%% Continuous-time state space model
%{
x_dot(t) = Ax(t)+Bu(t)
z(t) = Cx(t)+Dn(t)
%}
A = [0 1 0 0;
0 0 0 0;
0 0 0 1;
0 0 0 0];
B = [0 0;
1 0;
0 0;
0 1];
C = [1 0 0 0;
0 0 1 0];
D = [1 0;
0 1];
%% Discrete-time state space model
%{
x(k+1) = Fx(k)+Gw(k)
z(k) = Hx(k)+Iv(k)
Continuous to discrete form by zoh
%}
sysc = ss(A,B,C,D);
sysd = c2d(sysc, ts, 'zoh');
[F G H I] = ssdata(sysd);
%% Practice state of target
for i = 1:T-1
x(:,i+1) = F*x(:,i);
end
x = x+G*w; % State variable with noise
z = H*x+I*v; % Measurement value with noise
%==========================================================================
%%% Kalman Filter
for i = 1:T-1
%% Prediction phase
x_hat(:,i+1) = F*x_hat(:,i);
% State estimate predict
P(:,:,i+1) = F*P(:,:,i)*F'+G*Q*G';
% Tracking error covariance predict
P_predicted(:,:,i+1) = P(:,:,i+1);
%% Kalman gain
K = P(:,:,i+1)*H'*inv(H*P(:,:,i+1)*H'+R);
%% Updata step
x_hat(:,i+1) = x_hat(:,i+1)+K*(z(:,i+1)-H*x_hat(:,i+1));
% State estimate update
P(:,:,i+1) = P(:,:,i+1)-K*H*P(:,:,i+1);
% Tracking error covariance update
P_updated(:,:,i+1) = P(:,:,i+1);
end
%==========================================================================
%% Estimate error
x_error = x-x_hat;
%% Graph 1 practical and tracking position
figure(1)
plot(x(1,:),x(3,:),'r');
hold on;
plot(x_hat(1,:),x_hat(3,:),'g.');
title('2D Target Position')
legend('Practical Position','Tracking Position')
xlabel('X axis [m]')
ylabel('Y axis [m]')
hold off;
%% Graph 2
figure(2)
plot(t,x(1,:)),grid on;
hold on;
plot(t,x_hat(1,:),'r'),grid on;
title('Practical and Tracking Position on X axis')
legend('Practical Position','Tracking Position')
xlabel('Time [sec]')
ylabel('Position [m]')
hold off;
%% Graph 3
figure(3)
plot(t,x_error(1,:)),grid on;
title('Position Error on X axis')
xlabel('Time [sec]')
ylabel('Position RMSE [m]')
hold off;
%% Graph 4
figure(4)
plot(t,x(2,:)),grid on;
hold on;
plot(t,x_hat(2,:),'r'),grid on;
title('Practical and Tracking Velocity on X axis')
legend('Practical Velocity','Tracking Velocity')
xlabel('Time [sec]')
ylabel('Velocity [m/sec]')
hold off;
%% Graph 5
figure(5)
plot(t,x_error(2,:)),grid on;
title('Velocity Error on X axis')
xlabel('Time [sec]')
ylabel('Velocity RMSE [m/sec]')
hold off;
%==========================================================================
*********************************************************************************************************************************
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c语言实现代码 [转]
-----------------------------------------------------------------------------------------------------------------------------
#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
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 (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［转］
--------------------------------------------------------------------------------------------------------------------------------------------------------
// 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 "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
/* 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 k
P'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 );
/* adjust Kalman filter state */
/* Kk=P'k鈥 T鈥?H鈥 'k鈥 T+R)-1
xk=x'k+Kk鈥?zk-H鈥 'k)
Pk=(I-Kk鈥 )鈥 'k */
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;
}