copyright 2004 aaron lanterman the kalman filter prof. aaron d. lanterman school of electrical &...

Copyright 2004 Aaron Lanterman

The Kalman FilterThe Kalman FilterThe Kalman FilterThe Kalman Filter

Prof. Aaron D. Lanterman

School of Electrical & Computer EngineeringGeorgia Institute of Technology

AL: 404-385-2548<[email protected]>

ECE 7251: Spring 2004Lecture 17

2/16/04


State equation

Measurement eqn.

Process “noise” covariance

Measurement noise covariance

Initial guess, before taking any data

Covariance indicating confidence of initial guess

are uncorrelated with each otherand for different k

k1[m1] A[mm ]k

[m1] Uk[m1]

Yk[n1] C[nm ]k

[m1] Wk[n1]

Uk N(0,KU )

Wk N(0,KW )

1 N( ˆ 1|0,P1|0)

Uk , Wk , 1

The Setup for Kalman FilteringThe Setup for Kalman FilteringThe Setup for Kalman FilteringThe Setup for Kalman Filtering


Constant Velocity ModelConstant Velocity ModelConstant Velocity ModelConstant Velocity Model

• For small sample intervals T, the following model is commonly used (Blackman & Popoli, Sec. 4.2.2):

k

kx

x

kx

x Uv

pT

v

p

10

1

1

TT

TTqKU

2/

2/3/2

23

• Bar-Shalom & Li’s rule of thumb for selecting q

1

2max_acceleration q max_acceleration


Constant Acceleration ModelConstant Acceleration ModelConstant Acceleration ModelConstant Acceleration Model

• For small sample intervals T, another common model is (Blackman & Popoli, p. 202):

k

kx

x

x

kx

x

x

U

a

v

p

T

TT

a

v

p

100

1021

2

1

TTT

TTT

TTT

qKU

2/6/

2/3/8/

6/8/20/

23

234

345


Singer Maneuver ModelSinger Maneuver ModelSinger Maneuver ModelSinger Maneuver Model

• Often a good idea to encourage the acceration process to tend back towards zero (Blackman & Popoli, Sec. 4.2.1):

k

kx

x

x

kx

x

x

U

a

v

p

TT

TT

a

v

p

00

)2/1(10

2/1 2

1

0 1where

• R.A. Singer, “Estimating Optimal Tracking Filter Performance for Manned Maneuvering Targets,” IEEE Trans. On Aerospace and Electronic Systems, vol. 5, pp. 473-483, July 1970.


Extending to Higher DimensionsExtending to Higher DimensionsExtending to Higher DimensionsExtending to Higher Dimensions

• Easily extended to two or more dimensions by making processes in the different coordinates independent:

k

ky

y

x

x

ky

y

x

x

U

v

p

v

p

T

T

v

p

v

p

1000

100

0010

001

1

• This is just a convenient mathematical model– Of course, for real aircraft, the motion in the different

coordinates is not independent!


A Tale of Two SystemsA Tale of Two SystemsA Tale of Two SystemsA Tale of Two Systems

A

+ +

kw

Delay

C

Delay

C

A +

+L-+

k

k 1

uk

k

ˆ k|k

ˆ k 1|k 1

ˆ k|k 1

yk

ˆ y k|k 1

(based on a lecture by J.A. O’Sullivan)


Step 2A: State UpdateStep 2A: State UpdateStep 2A: State UpdateStep 2A: State Update

• Consider predicted data

Conditioned on all data up to k

• Recall

Kalman Gain k̂Ak̂A

Yk CWk

Yk1|k N(C ˆ k1|k,CPk1|kCT KW )

)ˆ()(ˆ|11

1|1|1|1 kkkW

Tkk

Tkkkk CyKCCPCP

],Cov[][ˆ|1|1|11|1 kkkkkkkk YE

])[]([Cov |11|1-1

kkkkk YEyY


Step 2B: Covariance UpdateStep 2B: Covariance UpdateStep 2B: Covariance UpdateStep 2B: Covariance Update

KkWT

kkT

kkkk CPKCCPCPP |11

|1|1|1 )(

APk|k A KU

],Cov[]Cov[ |11|11|1 kkkkkkk YP

],Cov[][Cov 1|1|11

kkkkk YY

• Covariance matrix tells us how much confidence we should have in our state estimates


The Innovation SequenceThe Innovation SequenceThe Innovation SequenceThe Innovation Sequence

• The innovations are defined as:

• Let the prediction of the data given the previous data be given by

• Innovations process is white:

• When trying on real data, testing innovations for “whiteness” tells you how accurate your models are

• Innovations are orthogonal to the data:

ˆ y k1|k E[Yk1 | y1,,yk ] CA ˆ k

kkkkkk yyCAy |1111 ˆˆ

0][ kTkYE

lkE lTk for 0][


Assorted TidbitsAssorted TidbitsAssorted TidbitsAssorted Tidbits• For non-Gaussian statistics (process and measurement

noise), the Kalman filter is the best linear MMSE estimator

• Combining covariance update steps yield the discrete Riccati equation:

• Under some conditions, DRE has a fixed point and ; in this case, Kalman filter acts likea Wiener filter for large k

• tells us our confidence in out state estimates. If it is small, then is small, then the filter is saturated; we

pay little attention to measurements.

Pk1|k1 messy function of Pk|k

P

Pk P

Pk|k

Lk|k


Pseudomeasurement ApproachPseudomeasurement ApproachPseudomeasurement ApproachPseudomeasurement Approach

)(sin

cosPOLCAR yg

r

r

y

xy

• Problem with Kalman filter applied to radar: data is rarely a linear function of the parameters• Canonical example: radar measuring range and angle

– Here we consider a 2-D scenario, and consider just the azimuth angle; could extend to include an elevation angle

• One possible solution: convert raw polar data into Cartesian coordinate “pseudomeasurements”, then process using Kalman filter

• Need covariance on the new data – use CRB-type transformation formulas


Trouble with PseudomeasurementsTrouble with PseudomeasurementsTrouble with PseudomeasurementsTrouble with Pseudomeasurements• Accuracy of covariance transformation depends on accuracy of state estimate

– Can have problems with significant bias for large cross-range errors (Bar-Shalom & Li, Sec. 1.6.2)– Bar-Shalom & Li offer an improved approach with “debiased conversion”

• Original measurement errors may be Gaussian, but converted measurement errors are not– Kalman filter only truly optimal for Gaussian measurement errors

• In more general problems, we may not always have a nice invertible mapping from the original parameter space to the sensor space!

copyright 2004 aaron lanterman the kalman filter prof. aaron d. lanterman school of electrical &...

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