📄 卡尔曼滤波-来自wiki（2）_mopish ranger.htm
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with the previous state, with the true state at (<EM>k</EM> − 1)
integrated out.</P>
<DL>
<DD><IMG class=tex
alt="p(\textbf{x}_k|\textbf{Z}_{k-1}) = \int p(\textbf{x}_k | \textbf{x}_{k-1}) p(\textbf{x}_{k-1} | \textbf{Z}_{k-1} ) \, d\textbf{x}_{k-1}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/ebaf1314bbc51c25daabff9f3795c722.png"></DD></DL>
<P>The measurement set up to time <EM>t</EM> is</P>
<DL>
<DD><IMG class=tex
alt="\textbf{Z}_{t} = \left \{ \textbf{z}_{1},\dots,\textbf{z}_{t} \right \}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/3176d699be3bcb927199773ed9c4299c.png"></DD></DL>
<P>The probability distribution of updated is proportional to the product
of the measurement likelihood and the predicted state.</P>
<DL>
<DD><IMG class=tex
alt="p(\textbf{x}_k|\textbf{Z}_{k}) = \frac{p(\textbf{z}_k|\textbf{x}_k) p(\textbf{x}_k|\textbf{Z}_{k-1})}{p(\textbf{z}_k|\textbf{Z}_{k-1})}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/29c75286790c43bc2874c628c8977d88.png"></DD></DL>
<P>The denominator</P>
<DL>
<DD><IMG class=tex
alt="p(\textbf{z}_k|\textbf{Z}_{k-1}) = \int p(\textbf{z}_k|\textbf{x}_k) p(\textbf{x}_k|\textbf{Z}_{k-1}) d\textbf{x}_k"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/4f19ec2c4ae37942e46aa59dce2fd70e.png"></DD></DL>
<P>is an unimportant normalization term.</P>
<P>The remaining probability density functions are</P>
<DL>
<DD><IMG class=tex
alt="p(\textbf{x}_k | \textbf{x}_{k-1}) = N(\textbf{x}_k, \textbf{F}_k\textbf{x}_{k-1}, \textbf{Q}_k)"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/c6a952faf0fcebd7290a8854e170d603.png"></DD></DL>
<DL>
<DD><IMG class=tex
alt="p(\textbf{z}_k|\textbf{x}_k) = N(\textbf{z}_k,\textbf{H}_{k}\textbf{x}_k, \textbf{R}_k)"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/89a7ce73ae0c30b4a930f070d9d494fb.png"></DD></DL>
<DL>
<DD><IMG class=tex
alt="p(\textbf{x}_{k-1}|\textbf{Z}_{k-1}) = N(\textbf{x}_{k-1},\hat{\textbf{x}}_{k-1},\textbf{P}_{k-1} )"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/6eab617a2c43f9b5f0fb31b9fbf73118.png"></DD></DL>
<P>Note that the PDF at the previous timestep is inductively assumed to be
the estimated state and covariance. This is justified because, as an
optimal estimator, the Kalman filter makes best use of the measurements,
therefore the PDF for <IMG class=tex alt=\mathbf{x}_k
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/f746f9ef837d783f6918d8a8bff650b7.png">
given the measurements <IMG class=tex alt=\mathbf{Z}_k
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/d9c8f16141a2aedf7b0315e8bd9c0a4a.png">
<EM>is</EM> the Kalman filter estimate.</P>
<P><A name=.E4.BF.A1.E6.81.AF.E6.BB.A4.E6.B3.A2.E5.99.A8></A></P>
<H2><SPAN class=editsection>[<A title=编辑本节
href="http://zh.wikipedia.org/w/index.php?title=%E5%8D%A1%E5%B0%94%E6%9B%BC%E6%BB%A4%E6%B3%A2&action=edit&section=14">编辑</A>]</SPAN>
<SPAN class=mw-headline>信息滤波器</SPAN></H2>
<P>In the <STRONG>information filter</STRONG>, or <STRONG>inverse
covariance filter</STRONG>, the estimated covariance and estimated state
are replaced by the <A class=new title="Fisher information matrix"
href="http://zh.wikipedia.org/w/index.php?title=Fisher_information_matrix&action=edit">information
matrix</A> and <A class=new title="Fisher information"
href="http://zh.wikipedia.org/w/index.php?title=Fisher_information&action=edit">information</A>
vector respectively.</P>
<DL>
<DD><IMG class=tex alt="\textbf{Y}_{k|k} \equiv \textbf{P}_{k|k}^{-1}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/624aa62d5874d016ea199920fa622836.png">
<DD><IMG class=tex
alt="\hat{\textbf{y}}_{k|k} \equiv \textbf{P}_{k|k}^{-1}\hat{\textbf{x}}_{k|k}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/0047dbe35e018b690c7260bfdb053b03.png"></DD></DL>
<P>Similarly the predicted covariance and state have equivalent
information forms,</P>
<DL>
<DD><IMG class=tex
alt="\textbf{Y}_{k|k-1} \equiv \textbf{P}_{k|k-1}^{-1}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/a1e0d8f8b7792b2e5f94458f19338838.png">
<DD><IMG class=tex
alt="\hat{\textbf{y}}_{k|k-1} \equiv \textbf{P}_{k|k-1}^{-1}\hat{\textbf{x}}_{k|k-1}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/a584dee58dfe60d1bc8b39ebcc421686.png"></DD></DL>
<P>as have the measurement covariance and measurement vector.</P>
<DL>
<DD><IMG class=tex
alt="\textbf{I}_{k} \equiv \textbf{H}_{k}^{T} \textbf{R}_{k}^{-1} \textbf{H}_{k}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/c5199bfd633af849bcb91ec292f43709.png">
<DD><IMG class=tex
alt="\textbf{i}_{k} \equiv \textbf{H}_{k}^{T} \textbf{R}_{k}^{-1} \textbf{z}_{k}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/ce4861aa690333c24c2821ec4b127062.png"></DD></DL>
<P>The information update now becomes a trivial sum.</P>
<DL>
<DD><IMG class=tex
alt="\textbf{Y}_{k|k} = \textbf{Y}_{k|k-1} + \textbf{I}_{k}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/4d22f6e09f594184932763e3c6731784.png">
<DD><IMG class=tex
alt="\hat{\textbf{y}}_{k|k} = \hat{\textbf{y}}_{k|k-1} + \textbf{i}_{k}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/d12b2227f40d3a7f58112a0681bf4d4a.png"></DD></DL>
<P>The main advantage of the information filter is that <EM>N</EM>
measurements can be filtered at each timestep simply by summing their
information matrices and vectors.</P>
<DL>
<DD><IMG class=tex
alt="\textbf{Y}_{k|k} = \textbf{Y}_{k|k-1} + \sum_{j=1}^N \textbf{I}_{k,j}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/18cb70bf8ed96f40e99f25a67bdbcb21.png">
<DD><IMG class=tex
alt="\hat{\textbf{y}}_{k|k} = \hat{\textbf{y}}_{k|k-1} + \sum_{j=1}^N \textbf{i}_{k,j}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/7efd0553ee382073920e290a95da0dec.png"></DD></DL>
<P>To predict the information filter the information matrix and vector can
be converted back to their state space equivalents, or alternatively the
information space prediction can be used.</P>
<DL>
<DD><IMG class=tex
alt="\textbf{M}_{k} = [\textbf{F}_{k}^{-1}]^{T} \textbf{Y}_{k|k} \textbf{F}_{k}^{-1}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/462faaf7535ba2354a977b44b13a45b6.png"></DD></DL>
<DL>
<DD><IMG class=tex
alt="\textbf{C}_{k} = \textbf{M}_{k} [\textbf{M}_{k}+\textbf{Q}_{k}^{-1}]^{-1}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/baba972b698cfcfce00c6fdb6c3c52e9.png">
<DD><IMG class=tex alt="\textbf{L}_{k} = I - \textbf{C}_{k}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/2d6cd5e39afabb2682fb1cb827342364.png">
<DD><IMG class=tex
alt="\textbf{Y}_{k|k-1} = \textbf{L}_{k} \textbf{M}_{k} \textbf{L}_{k}^{T} + \textbf{C}_{k} \textbf{Q}_{k}^{-1} \textbf{C}_{k}^{T}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/4b8c7c4e169b7da56671671c5f986c9c.png"></DD></DL>
<DL>
<DD><IMG class=tex
alt="\hat{\textbf{y}}_{k|k-1} = \textbf{L}_{k} [\textbf{F}_{k}^{-1}]^{T}\hat{\textbf{y}}_{k|k}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/0995a9549f9eda92c179c5833690b88b.png"></DD></DL>
<P>Note that if <EM>F</EM> and <EM>Q</EM> are time invariant these values
can be cached. Note also that <EM>F</EM> and <EM>Q</EM> need to be
invertible.</P>
<P><A name=.E9.9D.9E.E7.BA.BF.E6.80.A7.E6.BB.A4.E6.B3.A2.E5.99.A8></A></P>
<H2><SPAN class=editsection>[<A title=编辑本节
href="http://zh.wikipedia.org/w/index.php?title=%E5%8D%A1%E5%B0%94%E6%9B%BC%E6%BB%A4%E6%B3%A2&action=edit&section=15">编辑</A>]</SPAN>
<SPAN class=mw-headline>非线性滤波器</SPAN></H2>
<P>基 本卡尔曼滤波器（The basic Kalman
filter）是限制在线性的假设之下。然而，大部份非平凡的（non-trial）的系统都是非线性系统。其中的“非线性性质”（non-
linearity ）可能是伴随存在过程模型（process model）中或观测模型（observation
model）中，或者两者兼有之。</P>
<P><A
name=.E6.89.A9.E5.B1.95.E5.8D.A1.E5.B0.94.E6.9B.BC.E6.BB.A4.E6.B3.A2.E5.99.A8></A></P>
<H3><SPAN class=editsection>[<A title=编辑本节
href="http://zh.wikipedia.org/w/index.php?title=%E5%8D%A1%E5%B0%94%E6%9B%BC%E6%BB%A4%E6%B3%A2&action=edit&section=16">编辑</A>]</SPAN>
<SPAN class=mw-headline>扩展卡尔曼滤波器</SPAN></H3>
<P>在扩展卡尔曼滤波器（EKF）中状态转换和观测模型不需要是状态的线性函数，可替换为（可微的）函数。</P>
<DL>
<DD><IMG class=tex
alt="\textbf{x}_{k} = f(\textbf{x}_{k-1}, \textbf{u}_{k}, \textbf{w}_{k})"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/86d37a1bebece0c49c1f00808f08a66a.png"></DD></DL>
<DL>
<DD><IMG class=tex
alt="\textbf{z}_{k} = h(\textbf{x}_{k}, \textbf{v}_{k})"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/142e774484f50ba1bd9e34d4751c3121.png"></DD></DL>
<P>函数 <EM>f</EM> 可以用来从过去的估计值中计算预测的状态，相似的，函数
<EM>h</EM>可以用来以预测的状态计算预测的测量值。然而 <EM>f</EM> 和 <EM>h</EM>
不能直接的应用在协方差中，取而代之的是计算偏导矩阵(<A class=new title=Jacobian
href="http://zh.wikipedia.org/w/index.php?title=Jacobian&action=edit">Jacobian</A>)。</P>
<P>在每一步中使用当前的估计状态计算Jacobian矩阵，这几个矩阵可以用在卡尔曼滤波器的方程中。这个过程，实质上将非线性的函数在当前估计值处线性化了。</P>
<P>这样一来，卡尔曼滤波器的等式为：</P>
<P><STRONG>预测</STRONG></P>
<DL>
<DD><IMG class=tex
alt="\hat{\textbf{x}}_{k|k-1} = f(\textbf{x}_{k-1}, \textbf{u}_{k}, 0)"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/0985a740083d96f4679d0f789109f325.png"></DD></DL>
<DL>
<DD><IMG class=tex
alt="\textbf{P}_{k|k-1} = \textbf{F}_{k} \textbf{P}_{k-1|k-1} \textbf{F}_{k}^{T} + \textbf{Q}_{k}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/e0e2f106fd766bb418e0960256489ec8.png"></DD></DL>
<P><STRONG>使用Jacobians矩阵更新模型</STRONG></P>
<DL>
<DD><IMG class=tex
alt="\textbf{F}_{k} = \left . \frac{\partial f}{\partial \textbf{x} } \right \vert _{\hat{\textbf{x}}_{k-1|k-1},\textbf{u}_{k}}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/e4a0477a7e3d71c13ae93729bd7c2b26.png"></DD></DL>
<DL>
<DD><IMG class=tex
alt="\textbf{H}_{k} = \left . \frac{\partial h}{\partial \textbf{x} } \right \vert _{\hat{\textbf{x}}_{k|k-1}}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/8292be8664995891e2323d6d0a3bc6fd.png"></DD></DL>
<P><STRONG>更新</STRONG></P>
<DL>
<DD><IMG class=tex
alt="\tilde{\textbf{y}}_{k} = \textbf{z}_{k} - h(\hat{\textbf{x}}_{k|k-1}, 0)"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/52219b9a7a01ff37d2b3d21267d193fa.png"></DD></DL>
<DL>
<DD><IMG class=tex
alt="\textbf{S}_{k} = \textbf{H}_{k}\textbf{P}_{k|k-1}\textbf{H}_{k}^{T} + \textbf{R}_{k}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/fd02b643652906100408999815819d1d.png"></DD></DL>
<DL>
<DD><IMG class=tex
alt="\textbf{K}_{k} = \textbf{P}_{k|k-1}\textbf{H}_{k}^{T}\textbf{S}_{k}^{-1}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/cb1f0cfcdb55e9e4d4187149c9d5895c.png"></DD></DL>
<DL>
<DD><IMG class=tex
alt="\hat{\textbf{x}}_{k|k} = \hat{\textbf{x}}_{k|k-1} + \textbf{K}_{k}\tilde{\textbf{y}}_{k}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/3f5ba84ca462d08131c1679cdf94190d.png"></DD></DL>
<DL>
<DD><IMG class=tex
alt="\textbf{P}_{k|k} = (I - \textbf{K}_{k} \textbf{H}_{k}) \textbf{P}_{k|k-1}"
src="卡尔曼滤波-来自Wiki（2）_Mopish Ranger.files/3e270acc11bb3ed2769f9205181c775f.png"></DD></DL>
<P><A name=Unscented_Kalman_filter></A></P><SPAN class=editsection>[<A
title=编辑本节
href="http://zh.wikipedia.org/w/index.php?title=%E5%8D%A1%E5%B0%94%E6%9B%BC%E6%BB%A4%E6%B3%A2&action=edit&section=18">编辑</A>]</SPAN>
<SPAN class=mw-headline>应用</SPAN>
<UL>
<LI><A class=new title=自动驾驶仪
href="http://zh.wikipedia.org/w/index.php?title=%E8%87%AA%E5%8A%A8%E9%A9%BE%E9%A9%B6%E4%BB%AA&action=edit">自动驾驶仪</A>
<LI><A class=new title=动态定位系统
href="http://zh.wikipedia.org/w/index.php?title=%E5%8A%A8%E6%80%81%E5%AE%9A%E4%BD%8D%E7%B3%BB%E7%BB%9F&action=edit">动态定位系统</A>
<LI><A title=经济学
href="http://zh.wikipedia.org/w/index.php?title=%E7%BB%8F%E6%B5%8E%E5%AD%A6&variant=zh-cn">经济学</A>,
特别是<A title=宏观经济学
href="http://zh.wikipedia.org/w/index.php?title=%E5%AE%8F%E8%A7%82%E7%BB%8F%E6%B5%8E%E5%AD%A6&variant=zh-cn">宏观经济学</A>，
<A class=new title=时间序列模型
href="http://zh.wikipedia.org/w/index.php?title=%E6%97%B6%E9%97%B4%E5%BA%8F%E5%88%97%E6%A8%A1%E5%9E%8B&action=edit">时间序列模型</A>,
and <A class=new title=经济计量学
href="http://zh.wikipedia.org/w/index.php?title=%E7%BB%8F%E6%B5%8E%E8%AE%A1%E9%87%8F%E5%AD%A6&action=edit">经济计量学</A>
<LI><A class=new title=惯性引导系统
href="http://zh.wikipedia.org/w/index.php?title=%E6%83%AF%E6%80%A7%E5%BC%95%E5%AF%BC%E7%B3%BB%E7%BB%9F&action=edit">惯性引导系统</A>
<LI><A class=new title=雷达跟踪器
href="http://zh.wikipedia.org/w/index.php?title=%E9%9B%B7%E8%BE%BE%E8%B7%9F%E8%B8%AA%E5%99%A8&action=edit">雷达跟踪器</A>
<LI><A title=卫星导航系统
href="http://zh.wikipedia.org/w/index.php?title=%E5%8D%AB%E6%98%9F%E5%AF%BC%E8%88%AA%E7%B3%BB%E7%BB%9F&variant=zh-cn">卫星导航系统</A>
</LI></UL>
<P><A name=.E5.8F.82.E8.A7.81></A></P>
<H2><SPAN class=editsection>[<A title=编辑本节
href="http://zh.wikipedia.org/w/index.php?title=%E5%8D%A1%E5%B0%94%E6%9B%BC%E6%BB%A4%E6%B3%A2&action=edit&section=19">编辑</A>]</SPAN>
<SPAN class=mw-headline>参见</SPAN></H2>
<UL>
<LI><A class=new title=快速卡尔曼滤波
href="http://zh.wikipedia.org/w/index.php?title=%E5%BF%AB%E9%80%9F%E5%8D%A1%E5%B0%94%E6%9B%BC%E6%BB%A4%E6%B3%A2&action=edit">快速卡尔曼滤波</A>
<LI>比较: <A title=维纳滤波
href="http://zh.wikipedia.org/w/index.php?title=%E7%BB%B4%E7%BA%B3%E6%BB%A4%E6%B3%A2&variant=zh-cn">维纳滤波</A>及
the multimodal <A class=new title="Particle filter"
href="http://zh.wikipedia.org/w/index.php?title=Particle_filter&action=edit">Particle
filter</A> estimator. </LI></UL>
<P><A name=.E5.A4.96.E9.83.A8.E8.BF.9E.E6.8E.A5></A></P>
<H2><SPAN class=editsection>[<A title=编辑本节
href="http://zh.wikipedia.org/w/index.php?title=%E5%8D%A1%E5%B0%94%E6%9B%BC%E6%BB%A4%E6%B3%A2&action=edit&section=20">编辑</A>]</SPAN>
<SPAN class=mw-headline>外部连接</SPAN></H2>
<UL>
<LI><A class="external text"
title=http://www.cs.unc.edu/~tracker/media/pdf/SIGGRAPH2001_CoursePack_08.pdf
href="http://www.cs.unc.edu/~tracker/media/pdf/SIGGRAPH2001_CoursePack_08.pdf"
rel=nofollow>An Introduction to the Kalman Filter</A>, SIGGRAPH 2001
Course, Greg Welch and Gary Bishop
<LI><A class="external text"
title=http://www.cs.unc.edu/~welch/media/pdf/maybeck_ch1.pdf
href="http://www.cs.unc.edu/~welch/media/pdf/maybeck_ch1.pdf"
rel=nofollow>Kalman filtering chapter</A> from <EM>Stochastic Models,
Estimation</EM>, by Peter Maybeck
<LI><A class="external text" title=http://www.cs.unc.edu/~welch/kalman/
href="http://www.cs.unc.edu/~welch/kalman/" rel=nofollow>Kalman
Filter</A> webpage, with lots of links
<LI><A class="external text"
title=http://www.innovatia.com/software/papers/kalman.htm
href="http://www.innovatia.com/software/papers/kalman.htm"
rel=nofollow>Kalman Filtering</A>
<LI><A class="external text"
title=http://ieeexplore.ieee.org/xpl/abs_free.jsp?arNumber=882463
href="http://ieeexplore.ieee.org/xpl/abs_free.jsp?arNumber=882463"
rel=nofollow>The unscented Kalman filter for nonlinear estimation</A>
<LI><A class="external text" title=http://www.negenborn.net/kal_loc/
href="http://www.negenborn.net/kal_loc/" rel=nofollow><EM>Kalman
Filters</EM>, thorough introduction to several types, together with
applications to <EM>Robot Localization</EM></A> </LI></UL>
<P><A name=.E5.8F.82.E8.80.83.E6.96.87.E7.8C.AE></A></P>
<H2><SPAN class=editsection>[<A title=编辑本节
href="http://zh.wikipedia.org/w/index.php?title=%E5%8D%A1%E5%B0%94%E6%9B%BC%E6%BB%A4%E6%B3%A2&action=edit&section=21">编辑</A>]</SPAN>
<SPAN class=mw-headline>参考文献</SPAN></H2>
<UL>
<LI>Gelb A., editor. Applied optimal estimation. MIT Press, 1974.
</LI></UL>
<UL>
<LI>Kalman, R. E. A New Approach to Linear Filtering and Prediction
Problems, <EM>Transactions of the ASME - Journal of Basic
Engineering</EM> Vol. 82: pp. 35-45 (1960) </LI></UL>
<UL>
<LI>Kalman, R. E., Bucy R. S., New Results in Linear Filtering and
Prediction Theory, <EM>Transactions of the ASME - Journal of Basic
Engineering</EM> Vol. 83: pp. 95-107 (1961) </LI></UL>
<UL>
<LI>[JU97] Julier, Simon J. and Jeffery K. Uhlmann. A New Extension of
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