WHAT IS A KALMAN FILTER An recursive analyical echnique o esimae ime dependen physical parameers in he presence of noise processes Example of a ime and frequency applicaion: Offse beween wo clocks
PREDICTORS, FILTERS AND SMOOTHING ALGORITHMS A predicor provides esimaes of he physical parameer s curren values using previous measuremens A filer provides esimaes of he physical parameer s using previous and curren measuremens A smoohing algorihm provides esimaes of he physical parameer s curren values using previous, curren and fuure measuremens
WHY USE A KALMAN FILTER? Provides opimal esimaes when noise model is exac Compuaionally efficien, updae filer when adding new measuremens o he exising daa se Very simple o implemen Recursive algorihm
WHY USE A KALMAN FILTER? May be used as a predicor Easy o verify performance using simulaed daa Uncerainy esimaes are provided as par of he filer May be easily modified o provide a seering algorihm or oher applicaion
ELEMENTS OF THE KALMAN FILTER Physical parameers being esimaed represened by he componens of he sae vecor. xˆ is he sae vecor wih he curren measuremens include xˆ + is he sae vecor exrapolaed o nex daa poin where he nex se of measuremens have no ye been included Example of he elemens of a sae vecor (Phase offse, Normalised frequency offse, Linear frequency drif
DETERMINISTIC CLOCK CHARACTERISTICS (ACTIVE HYDROGEN MASERS 1 0 0 (Clock - Composie(ns 2 0 0 1 5 0 1 0 0 b5 0 - H M 1 - C o m p o s ie H M 2 - C o m p o s ie H M 3 - C o m p o s ie 0-5 0 5 3 0 2 0 5 3 0 6 0 5 3 1 0 0 5 3 1 4 0 5 3 1 8 0 M J D
ELEMENTS OF THE KALMAN FILTER y is he measuremen vecor. H is he design marix ha relaes he measuremens o he physical parameers being esimaed y =Hx. In our simple example we have phase measuremens and H = (1,0,0. K is he Kalman gain, he higher he weighing of curren measuremens used o esimaes he physical parameers. K is deermined by he Kalman filer.
STATE PROPAGATION MATRIX Sae propagaion marix is used o exrapolae curren esimae: Φ = 1 0 0 τ 1 0 τ 2 τ 1 2
COVARIANCE MATRICES Process covariance marix Q: Describes he process noise added a each measuremen. This is he clock noise R is he measuremen noise covariance marix. Noise is assumed o be whie and may be correlaed. P is he parameer covariance marix, describes he uncerainies in he sae vecor esimaes. P + and P - is he marix afer and before he measuremen updae
SIMPLE MEASUREMENT SET CURRENT POINT = 3 250 200 Measuremen 150 100 50 0 0 1 2 3 4 5 6 7 8 Poin No
HOW IT ALL WORKS Sae vecor exrapolaion (1 x ˆ ( = Φ( τ xˆ( 1 n n Parameer covariance marix exrapolaion (2 + P ( = Φ( τ P( Φ ( τ + Q( n n 1 n + T
HOW IT ALL WORKS Kalman gain deerminaion (3 Incorporaing curren measuremens (4 Parameer covariance marix updae (5 [ ] 1 ( ( ( ( + = n T n T n n R H H P H P K [ ] ˆ( ( ( ˆ( ˆ( + + = n n n n n H x y K x x [ ] ( ( ( + = n n n P H K I P
WHAT ARE THE PITFALLS? Lieraure is full of descripions of very badly consruced Kalman filers (including ime and frequency lieraure Require a reasonably accurae model of underlying physical processes Require a reasonably accurae model of he noise processes Physical parameers mus be observable Clock ensemble algorihm physical parameers are only parly observable. Problem repored 1987, problem solved 2002
PROPERTIES OF THE KALMAN FILTER Equaions 2, 3 and 5 may be run independenly of he res, and do no include acual measuremens Uncerainy of he sae vecor deerminaion depends only on he covariance marices Q and R, he measuremen marix H and he number of ieraions Kalman gain has similar properies
ASSUMPTIONS MADE WHEN USING THE KALMAN FILTER Process noise added on successive epochs does no correlae Measuremen noise is assumed o be whie, zero mean and disribued normally
WHERE DO WE USE KALMAN FILTER CLOCK ALGORITHMS? Clock ensemble algorihm Inpu from hree or more clock, oupus a composie ha should be more sable han any individual clock Clock predicor Predics he fuure offse beween wo clocks based on curren and previous measuremens
WHERE DO WE USE KALMAN FILTER CLOCK ALGORITHMS? Clock seering algorihm Seers a hardware or sofware clock o say close o a reference imescale. Trade-off of frequency sabiliy and ime offse Time ransfer combining algorihm Combines measuremens from several ime ransfer links beween he same wo clocks o form an opimal composie
APPLICATION TO NPL S CLOCK ENSEMBLE ALGORITHM Log 10 (σ y -1 4.2-1 4.4-1 4.6-1 4.8-1 5-1 5.2-1 5.4-1 5.6 C lo c k 1 (F F M C lo c k 2 (W F M C lo c k 3 (W F M C o m p o s ie S im p le C o m p o s ie O p im a l C o m p o s ie -1 5.8 2 2.5 3 3.5 4 4.5 5 L o g 1 0 (τ
APPLICATION TO NPL S CLOCK ENSEMBLE ALGORITHM -1 4-1 4.5 P lo s o f L o g ( σ a g a in s L o g ( τ (s 1 0 y 1 0 C lo c k 1 (F F M C lo c k 2 (W F M C lo c k 3 (W F M C o m p o s ie S im p le C o m p o s ie O p im a l C o m p o s ie Log 10 (σ y -1 5-1 5.5-1 6 2 2.5 3 3.5 4 4.5 5 L o g 1 0 ( τ
CLOCK PREDICTOR APPLICATION Predicion Error Variance 7 6 5 4 3 2 1 C lo c k P re d ic o r E rro r V a ria n c e O p im a l P E V K a lm a n file r P E V S im p le lin e a r p re d ic o r M S E K a lm a n file r p re d ic o r M S E S im p le lin e a r p re d ic o r P E V 0 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 P re d ic io n le n g h (s
CONCLUSIONS Kalman filers may be used as a very effecive in ime and frequency analysis Mus undersand heir limiaions Choice of noise models and he sae vecor componens is criical