The Intuition of Kalman Filter
Weijie Chen
Department of Political and Economic Studies
University of Helsinki
August20,2011
Abstract
This tutorial helps you grasp the core idea of Kalmanfilter intuitively.This note is bad on Maybeck(1979)and some graphs are borrowed from the book.
1Introduction
中国大学生就业网Before wetting feet in daunting mathematics,we could have some casual talk about the history of Kalmanfilter.Last century,from1950s to1960s,the Air Force Office of Scientific Rearch sponsored some far-reaching rearches on control theory applied in high-speed aircrafts(suchfighters,bombers), aerospace vehicles.One of projects were leading by Dr.Rudolph Kalman, who is the main contributor of Kalmanfilter,thus entitled after his name. Later on,NASA ud Kalmanfilter to calculate the satellite orbits,and more significant applications were on Ranger,Mariner,and Apollo missions of1960s.Due to its outstanding performance on engineering,the Kalman filter started to be extensively made u of in missile tracking system,sonar ranging,Global Positioning System(GPS)and etc..
The esntial idea of Kalmanfiltering is‘weighting’.Suppo you have two measurements,x1and x2,you are eking the best estimate of x bad on the two measurements.So the question is how
much weight you intend to put on either of them.This is a straightforward question,we prefer to putting more weight on the measurement with smaller variance.Thus we can easily t up a formula,
ˆx=
ao系统
σ22
σ21+σ22
x1+
胡润榜
成人用品网购σ21
σ21+σ22
x2(1)
愁乡Ifσ22is larger thanσ21,naturally we shall weight more on x1sinceσ21is smaller.To illustrate how this works,we can u a simple weather science example to give intuitions.
We launched an airship11into sky to collect data of weather,the nsors 1Picture borrowed from The 89多大
Fifth Column bulletin.
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一类生字Figure1:Airship
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22installed on the spaceship will transmit a vector of data x1,elements such as wind speed,wind direction,humidity,temperature,coordinates, etc.,to computer system,and vector x2is the last forecast of the state of atmosphere.In real rocket science or meteorology,the dimension of state vectors x2can be as large as millions.Then we naturally u covariance matrices to replaceσ21andσ22.So the job of Kalmanfilter is tofind the optimal weight between the forecast and nsor data at each time period, say every10conds.
产品销售合同1.1Insight of Kalman Filter
The rest of this ction will give more insight and intuition about Kalman filter,this is extremely important to understand the mathematical derivation later on.
As we can e from the meteorology example above,Kalmanfilter is an algorithm to decide weight on nsor data and last forecast at each time instance.In general,Kalmanfilter will absorb all information,such as data from nsors and forecast results,to generate the an overall best estimates. To give another concrete example,to measure the velocity of an vehicle,we have veral ways:using Doppler radar,inertial navigation system,pitot-static tube and windflow in air data system,all the data will be made u of,regardless of their precision.The Kalmanfilter will asss all information you have in hand and make the overall best estimate.
But one thing to notice,since Kalmanfilter is a recursive algorithm, 2Picture borrowed from uk/pb100.htm,a website of supplying hydrogenomatics equipment.
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which does not process all previous data together when a new measurement comes.In the long run,Kalmanfilter stands out among all availablefilters, we could say Kalmanfilter is an optimalfilter.
2Three Assumptions
To formulate Kalmanfilter,there are three assumptions we need:linearity, whiteness,Gaussian.
Linearity
Linear models,compare with nonlinear counterparts,are usually the pri-orities we would like to work on.Even if we don’t have a linear model, we can always lineari the nonlinear one around somefixed point,such as linearising a nonlinear differential equations around its steady-state value. For reasons of doing this,first,we have a full t of tools to handle linear dynamic systems,such as differential and difference equations;cond,they are easier to be handled by computer than nonlinear ones.
Whiteness
‘Whiteness’means error term follows a white noi process,which is inde-pendent of time.The Fourier transform of white noi process shows that the noi has equal power at all frequencies,which does not exist in real life. The reason we need to u white noi is mainly from a physical point of view,efigure3.‘Bandpass’is a frequency range which a certain physical system can respond,for instance,human ears cannot hear the sound with
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