graham hesketh information engineering group rolls-royce strategic research centre

NCAF Manchester 2000 16 July 2000

Graham HeskethInformation Engineering Group

Rolls-Royce Strategic Research Centre


Basic Kalman Filter Equations

)1()1|1()1|(

)1|1(ˆ)1|(ˆ

kkkkk

kkkkT QAAPP

xAxPrediction

Updating (correction)

)1|()1|()|()]1|(ˆ)([)1|(ˆ)|(ˆ

)()1|()1|( 1

kkkkkkkkkkkkkkkkkk TT

KCPPPxCyKxx

]RC[CPCPK


Definitions

System state: mean estimatex - vector of size qe.g. for a single channel,constant acceleration modelx1 = positionx2 = velocityx3 = acceleration


Definitions

System state: covariance (uncertainty)P - matrix of size q x qe.g.

P11 = uncertainty (variance) in positionP22 = uncertainty in velocityP12 = covariance in position/velocity. . .

333231

232221

131211

PPPPPPPPP

P


Definitions

State dynamics: state transition matrixA - matrix of size q x q

2)1(ˆ

)1(ˆ)1(ˆ)1|(ˆ

...

3211

1001102

111

10010

212

kxkxkxkkx

TtrwT

TTA

)1(ˆ)1|(ˆ kkk xAxe.g. for the constant acceleration example


Definitions

State dynamics: process noise (uncertainty)Q - matrix of size q x qe.g. (modelling acceleration as constant with a long duration; 3rd derivative is random and instantaneous)

T interval sampling the tow.r.t.

changeson accelerati random of variance2

00000000

2

a

a

Q


Definitions

Measurement space: observation matrixC - matrix of size r x q

)1|(ˆ)1|(ˆ kkkk xCy

)1|()1|(ˆ 1 kkxkky

e.g. for the simple single measurementconstant acceleration model:C = [1 0 0]


Definitions

Measurement space: innovation sequence

• e - error between observation and prediction• a.k.a. the innovation sequence• should be zero-mean, random white noise

if not, there is a problem with either» the model, or» a sensor

)1|(ˆ)( kkk xCye


Definitions

Measurement space: measurement noise matrixR - matrix of size r x re.g. for the single measurement case, R = 2

R is the covariance matrix of the measurementnoise. It is usually diagonal (i.e. uncorrelatedsensor noise).

233

222

211

000000

R


Definitions

Correction phase: Kalman gain matrixK - matrix of size q x r

K is the optimal correction factor to obtainthe Minimum Mean Squared Error estimateof the system state mean and covariance

1)()1|()1|( ]RC[CPCPK kkkkk TT


Definitions

Confidence: Relative contributions to uncertainty

Predicted measurement - uncertainty due to model

Actual measurement - uncertainty due to sensor noise)(kR

Tkk CCP )1|(

Relative weighting for the innovation error1)()1|()1|( ]RC[CPCCP kkkkk TT


Definitions

Correction phase: system stiffness

Model uncertainty >> Measurement noise Believe Sensor (e.g. Gain is high)

Low system stiffness, low lagModel uncertainty << Measurement noise Believe Model (e.g. Gain is low)

High system stiffness, high lag

Relative weighting (Gain) for the innovation error

21

1

1

form theof is

)()1|()1|(

]RC[CPCCP kkkkk TT


Definitions

Correction phase: updating state estimates

“Best” estimates of the system state mean andcovariance, at time k, conditioned on allmeasurements up to and including time k

)1|()1|()|()]1|(ˆ)([)1|(ˆ)|(ˆ

)()1|()1|( 1

kkkkkkkkkkkkkkkkkk TT

KCPPPxCyKxx

]RC[CPCPK


Comparison with Fixed-Coefficient Filters

If the A, C, Q and R matrices do not vary with kthen the single channel CA Kalman filter settlesdown have steady-state Kalman gains which are equivalent to a standard -filter, but withoptimum coefficients selected automatically.

This approach is known as Weiner filtering.It is computationally much simpler than a Kalmanfilter, but the steady-state assumptions may notbe valid in many practical cases.


Advantages of Kalman Filters

• Copes with variable A, C, Q and R matrices• Copes with the large uncertainty of the initialisation phase• Copes with missing data• Provides a convenient measure of estimation accuracy (via the covariance matrix P)• Fuses information from multiple-sensors


Disadvantages of Kalman Filters

• Computationally complex (especially for large numbers of sensor channels r) • Requires conditional independence of the measurement errors, sample-to-sample• Requires linear models for state dynamics and observation processes (A and C)• Getting a suitable value for Q (a.k.a. tuning) can be something of a black art


Dealing with the disadvantages [1]

• Computationally complex (especially for large numbers of sensor channels r)

Computers are getting faster all the timeNew algorithms for fast matrix inversion



• Requires conditional independence of the measurement errors, sample-to-sample

The independence criterion can be removed byusing Covariance Intersection updating

AB

Ca,b,c


Covariance Intersection vs Kalman updates

Standard Kalman filter update:

)(

)(11

111

bBaACcBAC

Covariance Intersection update:

))1((

))1((11

111

bBaACcBAC

Kalman filter covariance (alternative form):11

211

21

21 )( BAC



• Requires linear models for state dynamics and observation processes (A and C)

This can be overcome by using the EKF(Extended Kalman Filter), but only at theexpense of your sanity and all your free time

A superior alternative to EKF is theUnscented Filter


Unscented Filter - General Problem

n-dimensional vector random variable x with mean andcovariance xxPx,

We want to predict the distribution

][xy g

yyPy,

of an m-dimensional vector random variable y, wherey is related to x by the non-linear transformation

In filtering there are two such transformations:» the state transition» the observation

)]1(ˆ[)1|(ˆ 1 kgkk xx)]1|(ˆ[)1|(ˆ 2 kkgkk xy


Unscented Filter - Solution

Compute the set of 2n points from the rows or columnsof the matrices PnThis set is zero mean with covariance P. Compute a set ofpoints with the same covariance, but with mean as

x ii x

Transform each point as ][ ii g

yCompute and by computing the mean and covarianceof the 2n points in the set }{ i

yyP

This is computationally much simpler and significantlymore accurate (lower MSE) than the corresponding EKF



• Getting a suitable value for Q (a.k.a. tuning) can be something of a black art

There are many heuristics for dealing with processnoise (mainly to keep the filter stable); one methodis to switch between alternative models

We have Matlab toolboxes for learning Q (and allthe other matrices) from empirical data, using atechnique called Expectation Maximisation (EM)


Summary

• The Kalman Filter is the way to fuse multiple data sources, providing that their errors are conditionally independent• Covariance Intersection is the conservative way to deal with inter-dependence in distributed systems• The Unscented Filter is universally superior to the EKF for all cases involving non-linear transformations• Many heuristics exist to extend and improve the basic Kalman filter for tracking and sensor bias detection


End of Talk

Unless there are any questions – make ‘em brief

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