头文件：
/*
* Copyright (c) 2008-2011 Zhang Ming (M. Zhang), zmjerry@163.com
*
* This program is free software; you can redistribute it and/or modify it
* under the terms of the GNU General Public License as published by the
* Free Software Foundation, either version 2 or any later version.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
*
* 1. Redistributions of source code must retain the above copyright notice,
* this list of conditions and the following disclaimer.
*
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* This program is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
* more details. A copy of the GNU General Public License is available at:
* http://www.fsf.org/licensing/licenses
*/
/*****************************************************************************
* kalman.h
*
* Kalman Filter.
*
* The Kalman filter is an efficient recursive filter that estimates the
* internal state of a linear dynamic system from a series of noisy
* measurements. In most applications, the internal state is much larger
* (more degrees of freedom) than the few "observable" parameters which are
* measured. However, by combining a series of measurements, the Kalman
* filter can estimate the entire internal state.
*
* A wide variety of Kalman filters have now been developed, from Kalman's
* original formulation, now called the simple Kalman filter, the Kalman-Bucy
* filter, Schmidt's extended filter, the information filter, and a variety
* of square-root filters that were developed by Bierman, Thornton and so on.
*
* Zhang Ming, 2010-10, Xi'an Jiaotong University.
*****************************************************************************/
#ifndef KALMAN_H
#define KALMAN_H
#include <vector.h>
#include <matrix.h>
#include <inverse.h>
namespace splab
{
template<typename Type>
Vector<Type> kalman( const Matrix<Type>&, const Matrix<Type>&,
const Matrix<Type>&, const Matrix<Type>&,
const Vector<Type>&, const Vector<Type>&,
const Vector<Type>& );
#include <kalman-impl.h>
}
// namespace splab
#endif
// KALMAN_H
实现文件：
/*
* Copyright (c) 2008-2011 Zhang Ming (M. Zhang), zmjerry@163.com
*
* This program is free software; you can redistribute it and/or modify it
* under the terms of the GNU General Public License as published by the
* Free Software Foundation, either version 2 or any later version.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
*
* 1. Redistributions of source code must retain the above copyright notice,
* this list of conditions and the following disclaimer.
*
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* This program is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
* more details. A copy of the GNU General Public License is available at:
* http://www.fsf.org/licensing/licenses
*/
/*****************************************************************************
* kalman-impl.h
*
* Implementation for Kalman Filter.
*
* Zhang Ming, 2010-10, Xi'an Jiaotong University.
*****************************************************************************/
/**
* The simple Kalman filter for one step.
* A ---> system matrix defining linear dynamic system
* C ---> measurement matrix defining relationship between system's state
* and measurements
* Q ---> covariance matrix of process noise in system state dynamics
* R ---> covariance matrix of measurements uncertainty
* y ---> measurements at time t
* xPrev ---> previous estimated state vector of the linear dynamic system
* initDiagV ---> diagonal vector for initializing the covariance matrix of
* state estimation uncertainty
*/
template <typename Type>
Vector<Type> kalman( const Matrix<Type> &A, const Matrix<Type> &C,
const Matrix<Type> &Q, const Matrix<Type> &R,
const Vector<Type> &xPrev, const Vector<Type> &y,
const Vector<Type> &initDiagV )
{
int N = xPrev.size();
// covariance matrix of state estimation uncertainty
static Matrix<Type> V = diag(initDiagV);
// previoused state vector
Vector<Type> xPred = A * xPrev;
// inovation
Vector<Type> alpha = y - C * xPred;
Matrix<Type> CTran = trT( C );
Matrix<Type> VPred = A*V*trT(A) + Q;
// Kalman gain matrix
Matrix<Type> KGain = VPred*CTran * inv(C*VPred*CTran+R);
V = ( eye(N,Type(1.0)) - KGain*C ) * VPred;
// return the estimation of the state vector
return xPred + KGain * alpha;
}
测试代码：
/*****************************************************************************
* kalman.h
*
* Kalman filter testing.
*
* Zhang Ming, 2010-10, Xi'an Jiaotong University.
*****************************************************************************/
#define BOUNDS_CHECK
#include <iostream>
#include <kalman.h>
using namespace std;
using namespace splab;
typedef double Type;
const int N = 2;
const int M = 2;
const int T = 20;
int main()
{
Matrix<Type> A(N,N), C(M,N), Q(N,N), R(M,M);
A = eye( N, Type(1.0) ); C = eye( N, Type(1.0) );
Q = eye( N, Type(1.0) ); R = eye( N, Type(2.0) );
Vector<Type> x(N,Type(1.0)), y(M), ytInit(M);
ytInit[0] = Type(0.5); ytInit[1] = Type(2.0);
Matrix<Type> yt(M,T);
for( int t=0; t<T; ++t )
yt.setColumn( ytInit, t );
Vector<Type> intV( N, Type(10.0) );
for( int t=0; t<T; ++t )
{
y = yt.getColumn(t);
x = kalman( A, C, Q, R, x, y, intV );
cout << "Estimation of xt at the " << t << "th iteratin: " << x << endl;
}
cout << "The theoretical xt should converge to: " << ytInit << endl;
return 0;
}
运行结果：
Estimation of xt at the 0th iteratin: size: 2 by 1 0.576923 1.84615 Estimation of xt at the 1th iteratin: size: 2 by 1 0.532787 1.93443 Estimation of xt at the 2th iteratin: size: 2 by 1 0.51581 1.96838 Estimation of xt at the 3th iteratin: size: 2 by 1 0.507835 1.98433 Estimation of xt at the 4th iteratin: size: 2 by 1 0.503909 1.99218 Estimation of xt at the 5th iteratin: size: 2 by 1 0.501953 1.99609 Estimation of xt at the 6th iteratin: size: 2 by 1 0.500977 1.99805 Estimation of xt at the 7th iteratin: size: 2 by 1 0.500488 1.99902 Estimation of xt at the 8th iteratin: size: 2 by 1 0.500244 1.99951 Estimation of xt at the 9th iteratin: size: 2 by 1 0.500122 1.99976 Estimation of xt at the 10th iteratin: size: 2 by 1 0.500061 1.99988 Estimation of xt at the 11th iteratin: size: 2 by 1 0.500031 1.99994 Estimation of xt at the 12th iteratin: size: 2 by 1 0.500015 1.99997 Estimation of xt at the 13th iteratin: size: 2 by 1 0.500008 1.99998 Estimation of xt at the 14th iteratin: size: 2 by 1 0.500004 1.99999 Estimation of xt at the 15th iteratin: size: 2 by 1 0.500002 2 Estimation of xt at the 16th iteratin: size: 2 by 1 0.500001 2 Estimation of xt at the 17th iteratin: size: 2 by 1 0.5 2 Estimation of xt at the 18th iteratin: size: 2 by 1 0.5 2 Estimation of xt at the 19th iteratin: size: 2 by 1 0.5 2 The theoretical xt should converge to: size: 2 by 1 0.5 2 Process returned 0 (0x0) execution time : 0.125 s Press any key to continue.
1. 卡尔曼滤波的应用
卡尔曼滤波的一个典型实例是从一组有限的，包含噪声的，对物体位置的观察序列（可能有偏差）预测出物体的位置的 坐标及速度。例如,对于雷达来说，人们感兴趣的是其能够跟踪目标。但目标的位置、速度、加速度的测量值往往在任何时候都有噪声。卡尔曼滤波利用目标的动态信息，设法去掉噪声的影响，得到一个关于目标位置的好的 估计。这个估计可以是对当前目标位置的估计(滤波)，也可以是对于将来位置的估计(预测)，也可以是对过去位置的估计(插值或平滑)。
2. 实例讲解
假设我们要研究的对象是一个房间的温度。(1) 系统预测值。根据你的经验判断，这个房间的温度是恒定的，也就是下一分钟的温度等于现在这一分钟的温度(我们以一分钟为一个单位)，且假设你的经验存在高斯白噪声(White Gaussian Noise)。(2) 测量值。我们在房间里放一个温度计，但是这个温度计也不准确的，测量值会比实际值偏差。我们也把这些偏差看成是高斯白噪声。
现在我们要估算k时刻的实际温度值。因为你相信温度是恒定的，所以你会得到k时刻的温度预测值是跟k-1时刻一样的，假设是23度，同时该值的高斯噪声的偏差是5度（5是这样得到的：如果 k-1时刻估算出的最优温度值的偏差是3，你对自己预测的不确定度是4度，他们平方相加再开方，就是5）。然后，你从温度计那里得到了k时刻的温度值，假设是25度，同时该值的偏差是4度。
由于我们用于估算k时刻的实际温度有两个温度值，分别是23度和25度。究竟实际温度是多少呢？相信自己还是相信温度计呢？究竟相信谁多一点，我 们可以用他们的covariance(协方差)来判断。因为Kg^2=5^2/(5^2+4^2)，所以Kg=0.78(Kg:卡尔曼增益,Kalman Gain)，我们可以估算出k时刻的实际温度值 是：23+0.78*(25-23)=24.56度。可以看出，因为温度计的covariance比较小（比较相信温度计），所以估算出的最优温度值偏向温度计的值。
现在我们已经得到k时刻的最优温度值了，下一步就是要进入k+1时刻，进行新的最优估算。到现在为止，好像还没看到什么自回归的东西出现。对了，在进入 k+1时刻之前，我们还要算出k时刻那个最优值（24.56度）的偏差。算法如下：((1-Kg)*5^2)^0.5=2.35。这里的5就是上面的k时 刻你预测的那个23度温度值的偏差，得出的2.35就是进入k+1时刻以后k时刻估算出的最优温度值的偏差（对应于上面的3）。
就是这样，卡尔曼滤波器就不断的把covariance递归，从而估算出最优的温度值。他运行的很快，而且它只保留了上一时刻的covariance。
3. 数学描述
首先，我们先要引入一个离散控制过程的系统。该系统可用一个线性随机微分方程（Linear Stochastic Difference equation）来描述：
X(k)=A X(k-1)+B U(k)+W(k)再加上系统的测量值：
Z(k)=H X(k)+V(k)上 两式子中，X(k)是k时刻的系统状态，U(k)是k时刻对系统的控制量。A和B是系统参数，对于多模型系统，他们为矩阵。Z(k)是k时刻的测量值，H 是测量系统的参数，对于多测量系统，H为矩阵。W(k)和V(k)分别表示过程和测量的噪声。他们被假设成高斯白噪声(White Gaussian Noise)，他们的covariance 分别是Q，R（这里我们假设他们不随系统状态变化而变化）。
首先我们要利用系统的过程模型，来预测下一状态的系统。假设现在的系统状态是k，根据系统的模型，可以基于系统的上一状态而预测出现在状态：
X(k|k-1)=A X(k-1|k-1)+B U(k) ……….. (1)式(1)中，X(k|k-1)是利用上一状态预测的结果，X(k-1|k-1)是上一状态最优的结果，U(k)为现在状态的控制量，如果没有控制量，它可以为0。
到现在为止，我们的系统结果已经更新了，可是，对应于X(k|k-1)的covariance还没更新。我们用P表示covariance：
P(k|k-1)=A P(k-1|k-1) A’+Q ……… (2)式 (2)中，P(k|k-1)是X(k|k-1)对应的covariance，P(k-1|k-1)是X(k-1|k-1)对应的 covariance，A’表示A的转置矩阵，Q是系统过程的covariance。式子1，2就是卡尔曼滤波器5个公式当中的前两个，也就是对系统的预 测。
现在我们有了现在状态的预测结果，然后我们再收集现在状态的测量值。结合预测值和测量值，我们可以得到现在状态(k)的最优化估算值X(k|k)：
X(k|k)= X(k|k-1)+Kg(k) (Z(k)-H X(k|k-1)) ……… (3)其中Kg为卡尔曼增益(Kalman Gain)：
Kg(k)= P(k|k-1) H’ / (H P(k|k-1) H’ + R) ……… (4)到现在为止，我们已经得到了k状态下最优的估算值X(k|k)。但是为了要另卡尔曼滤波器不断的运行下去直到系统过程结束，我们还要更新k状态下X(k|k)的covariance：
P(k|k)=（I-Kg(k) H）P(k|k-1) ……… (5)其中I 为1的矩阵，对于单模型单测量，I=1。当系统进入k+1状态时，P(k|k)就是式子(2)的P(k-1|k-1)。这样，算法就可以自回归的运算下去。
卡尔曼滤波器的原理基本描述了，式子1，2，3，4和5就是他的5 个基本公式。根据这5个公式，可以很容易的实现计算机的程序。
4. 再述房间温度问题
把房间看成一个系统，然后对这个系统建模。当然，我们见的模型不需要非常地精确。我们所知道的这个房间的温度是跟前一时刻的温度相同的，所以A=1。没有控制量，所以U(k)=0。因此得出：
X(k|k-1)=X(k-1|k-1) ……….. (6)式子（2）可以改成：
P(k|k-1)=P(k-1|k-1) +Q ……… (7)因为测量的值是温度计的，跟温度直接对应，所以H=1。式子3，4，5可以改成以下：
X(k|k)= X(k|k-1)+Kg(k) (Z(k)-X(k|k-1)) ……… (8)
Kg(k)= P(k|k-1) / (P(k|k-1) + R) ……… (9)
P(k|k)=（1-Kg(k)）P(k|k-1) ……… (10)为了令卡尔曼滤波器开始工作，我们需要告诉卡尔曼两个零时刻的初始值，是X(0|0)和P(0|0)。他们的值不用太在意，随便给一个就可以了，因为随着 卡尔曼的工作，X会逐渐的收敛。但是对于P，一般不要取0，因为这样可能会令卡尔曼完全相信你给定的X(0|0)是系统最优的，从而使算法不能收敛。我选 了X(0|0)=1度，P(0|0)=10。
引用:
carpa.bokee.com/4725695.html
http://zh.wikipedia.org/zh/%E5%8D%A1%E5%B0%94%E6%9B%BC%E6%BB%A4%E6%B3%A2
扫一扫
被折叠的 条评论
为什么被折叠?
到【灌水乐园】发言
