A Simplified Derivation of Scalar Kalman Filter using Bayesian Probability Theory

Transcription

1 6th Iteratioal Workshop o Aalysis of Dyamic Measuremets Jue -3 0 Göteorg Swede A Simplified Derivatio of Scalar Kalma Filter usig Bayesia Proaility Theory Gregory Kyriazis Imetro - Electrical Metrology Divisio

2 6th Iteratioal Workshop o Aalysis of Dyamic Measuremets Jue -3 0 Göteorg Swede. Itroductio. Liear dyamical models 3. Estimatio 4. Coclusios

3 6th Iteratioal Workshop o Aalysis of Dyamic Measuremets Jue -3 0 Göteorg Swede. Itroductio 960 R. E. Kalma - A ew approach to liear filterig ad predictio prolems 963 R. E. Kalma - New methods i Wieer filterig theory Kalma receives Natioal Medal of Sciece Octoer I may estimatio prolems especially those ivolvig dyamical systems oservatios are made sequetially i time ad up-to-date parameter estimates are required. The recursive solutio to the discrete-time liear estimatio prolem was first pulished y Kalma. The estimatio algorithm is called Kalma filter.

4 6th Iteratioal Workshop o Aalysis of Dyamic Measuremets Jue -3 0 Göteorg Swede Kalma filter The Kalma filterig algorithm made it possile to avigate precisely over log distaces ad time spas. Kalma s algorithm is used extesively i all avigatio systems for deep-space exploratio. It has also ee applied to forecastig (P. J. Harriso ad C. F. Steves 976). Historically this sigal estimatio prolem was viewed as filterig arrowad sigals from widead oise; hece the ame filterig for sigal estimatio. Kalma used orthogoality to derive his filterig algorithm (R. E. Kalma 960). Kalma s equatios have ofte ee derived usig iovatios. The cocept of iovatios or the upredictale part of oservatios was itroduced y Kailath (968). Kalma s equatios were derived usig a Bayesia approach for the first time i NASA TR R-35 (96). A similar approach was used y Y. C. Ho ad R. C. K. Lee (964). A simplified derivatio is preseted here.

5 6th Iteratioal Workshop o Aalysis of Dyamic Measuremets Jue -3 0 Göteorg Swede. Liear dyamical models Liear dyamical models are state-space models whose state upredictale vagaries with time are descried proailistically X Z X Z a - w a Z ~ N 0 Z w w ~ N 0 Oservatio equatio System equatio oservatio sequece Gauss-Markov sequece of ukow process states kow series of costats (liear relatioship) kow series of costats (first-order differece equatio) oise sequece system oise drivig fuctio Both w ad are white ad mutually ucorrelated.

6 6th Iteratioal Workshop o Aalysis of Dyamic Measuremets Jue -3 0 Göteorg Swede. Estimatio X Z ukow values of the correspodig quatities possile values of those ukows Let d = {d X = x } with d 0 descriig the iitial availale iformatio icludig the values of a - ad. All the iformatio d aout the ukow state is ecoded y the posterior PDF at ad used to derive the ew posterior oce the data sample X = x is received at. It is show i the sequel how to evolve from the posterior PDF at to the posterior at. It is assumed that iitially at time = iformatio cocerig the state Z 0 was descried as 0 N 0 0 Z ~ ν (mea ad variace kow)

7 6th Iteratioal Workshop o Aalysis of Dyamic Measuremets Jue -3 0 Göteorg Swede Prior PDF for the process state at ~ exp Z d p ~ a a This follows immediately from the system equatio ad the properties of the Gaussia distriutio. Posterior PDF for the process state at exp p Z d

8 6th Iteratioal Workshop o Aalysis of Dyamic Measuremets Jue -3 0 Göteorg Swede A samplig distriutio with ukow locatio ad scale parameters is assiged that descries the prior kowledge aout the oise. As a fuctio of those parameters the samplig distriutio is the termed the likelihood of the parameters give the oserved data. Likelihood for Z give that X = x l x exp x ; Noise ad prior iformatio are comied at each time with Bayes theorem i.e. posterior prior likelihood. O comiig the prior ad the likelihood completig the square ad lumpig the terms that do ot deped o ito the proportioality costat

9 6th Iteratioal Workshop o Aalysis of Dyamic Measuremets Jue -3 0 Göteorg Swede exp Z d p Posterior PDF for the process state at ~ x The posterior mea is the weighted average of the prior mea ad x / with the weights eig ad / respectively. For fast recursive estimatio oe is just iterested i the estimate ad the associated ucertaity. I dyamical prolems each state ca e estimated from the last previous estimate ad the ew data sample received. Thus oly the last estimate ad associated ucertaity eed to e stored.

10 6th Iteratioal Workshop o Aalysis of Dyamic Measuremets Jue -3 0 Göteorg Swede σ We ow have all the equatios required to recursively geerate the solutio to the estimatio prolem: Recursive equatios ν x a a It is assumed that the state estimate at a ad x are all kow at. est estimate of the state at ucertaity associated with the estimate

11 6th Iteratioal Workshop o Aalysis of Dyamic Measuremets Jue -3 0 Göteorg Swede Scalar Kalma filter a a x a If a - does ot vary with ad w ad are oth statioary that is ad are oth costats the oth ad will approach limits as approaches ifiity. By defiig the Kalma gai as the state estimate at ad the associated ucertaity ecome

12 Tale I. Parameter estimates for a sigal cotaiig five frequecies. 6th Iteratioal Workshop o Aalysis of Dyamic Measuremets Jue -3 0 Göteorg Swede 4. Coclusio The Bayesia approach preseted here provides a simpler ad more direct derivatio of the solutio to the prolem of recursive estimatio of the state of a firstorder liear dyamical system. It was show that as expected the solutio is Kalma s filterig algorithm.