graham hesketh information engineering group rolls-royce strategic research centre
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NCAF Manchester 2000 16 July 2000
Graham HeskethInformation Engineering Group
Rolls-Royce Strategic Research Centre
NCAF Manchester 2000 26 July 2000
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
NCAF Manchester 2000 36 July 2000
Definitions
System state: mean estimatex - vector of size qe.g. for a single channel,constant acceleration modelx1 = positionx2 = velocityx3 = acceleration
NCAF Manchester 2000 46 July 2000
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
NCAF Manchester 2000 56 July 2000
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
NCAF Manchester 2000 66 July 2000
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
NCAF Manchester 2000 76 July 2000
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]
NCAF Manchester 2000 86 July 2000
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
NCAF Manchester 2000 96 July 2000
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
NCAF Manchester 2000 106 July 2000
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
NCAF Manchester 2000 116 July 2000
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
NCAF Manchester 2000 126 July 2000
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
NCAF Manchester 2000 136 July 2000
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
NCAF Manchester 2000 146 July 2000
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.
NCAF Manchester 2000 156 July 2000
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
NCAF Manchester 2000 166 July 2000
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
NCAF Manchester 2000 176 July 2000
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
NCAF Manchester 2000 186 July 2000
Dealing with the disadvantages [2]
• Requires conditional independence of the measurement errors, sample-to-sample
The independence criterion can be removed byusing Covariance Intersection updating
AB
Ca,b,c
NCAF Manchester 2000 196 July 2000
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
NCAF Manchester 2000 206 July 2000
Dealing with the disadvantages [3]
• 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
NCAF Manchester 2000 216 July 2000
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
NCAF Manchester 2000 226 July 2000
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
NCAF Manchester 2000 236 July 2000
Dealing with the disadvantages [4]
• 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)
NCAF Manchester 2000 246 July 2000
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
NCAF Manchester 2000 256 July 2000
End of Talk
Unless there are any questions – make ‘em brief
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