L13 KF Intuitive Intro

8/8/2019 L13 KF Intuitive Intro

1/32

Intuitive Introduction to theIntuitive Introduction to theKalman FilterKalman Filter


2/32

Historical Development ofHistorical Development of

Optimal Filtering TheoryOptimal Filtering Theory! Norbert Wiener filter: developed during 1940's, prompted by needs of US military.

Secret report considered to be very difficult theoretically and had a yellow cover.Led to name among military as Yellow Peril.

! After war years: filter applied successfully in analog signal processing applicationsbut the mathematics surrounded the technique in mystery for many potential users.

! In later years as computers became available: original work of Wiener was thoughtto be difficult to program; therefore relatively neglected.

! In 1960s Rudi Kalman developed state-space approach to optimal filtering with hisco-worker Bucy. Digital form of filter involved a recursive algorithm; particularlysuitable for digital estimation work inspired by space industry.

! Kalman filter was widely adopted in the aerospace industry but found fewapplications in general industries until more recently.

! Notable successors for industrial applications of Kalman filters include shippositioning systems, fault monitoring and steel mill applications.


3/32

Expected ValueExpected Value! The expected value is a weighted average, weighted by the

probability of each sample.

! Assume functionf(x) is continuous,

! Iff(x) is discrete,

! IfP(x) is constant for allx, E{f(x)} is the average off(x).

( ){ } ( ) ( ) dxxPxfxfE =

( ){ } ( ) ( )=

xxPxfxfE


4/32

Gaussian probability densityGaussian probability densityThe Gaussian probability density function of the random vector of

dimension n is given by the expression:

( ) ( )11 1

( ) exp2(2 ) det( )

T

nx x P x x

P

=

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

For scalar system:

( )2 11 1( ) exp

2(2 )x x

=

where is the variance


5/32

MotivationMotivation


6/32

PreliminariesPreliminaries -- State EquationsState Equations

! Consider a system with TF:

! This has one possible state vector representation

)2)(1(

)3(2)(

)(

)(

++

+==

ss

ssG

su

sy

[ ]

=

+

=

2

1

2

1

2

1

24

1

1

20

01

x

xy

ux

x

x

x

&

&


7/32

Advantages of State EquationsAdvantages of State Equations

! Can be used for time-varying systems

! Can be used for nonlinear systems

! Can be used for discrete systems

)(C(t))(

)(B(t))(A(t))(

txty

tutxtx

=

+=&

( )

))(()(

),(),()(

txgty

ttutxftx

=

=&

1 A B

C

k k k k

kk

x x u G

y x

+

= + +

=


8/32

State EquationsState Equations

! Can be written in vector form,

==

+=

2

1whereC

BA

x

xxxy

uxx&

AA

CC+

+

u(t)y(t)

x(t)

BB

)(tx&


9/32

Need for a Kalman FilterNeed for a Kalman Filter

in Controlin Control

! Conventional design uses plant output for control purposes.

+

+

u(s) y(s)x(s)

(s)

+

+

+

-Measured

output

plant

BB

AA

CC

GG

ControllerController


10/32

State feedback controlState feedback control

! Has many advantages including, excellent robustness,optimal solutions or easy pole placement.

! Problem:- Cannot measure all the states.

BB

AA

CC++

y(t)x(t)

GG(t)

++

+-

KK


11/32

Need Black BoxNeed Black Box

State EstimatorState Estimator


12/32

!Kalman filter is the best linear estimator.

!Good results in practise due to optimalityand structure

!Convenient form foronline real-timeprocessing

!Easy to formulate and implement given abasic understanding

!Recursive form

Why is Kalman filteringWhy is Kalman filtering

so popular?so popular?


13/32

What is a Kalman filterWhat is a Kalman filter

and what can it Do?and what can it Do?

! Optimal estimator

"provides an estimate of some desired quantity such that

a specified cost metric is minimised, e.g.,

!Recursive" on-line data processing,

" low computational burden.

imisedminisxx.t.sxxopt =


14/32

Preliminaries:Preliminaries:

Typical Stochastic SystemTypical Stochastic System

! Plant Equations

( ) A ( ) B ( ) G ( )

( ) C ( ) ( )

x t x t u t t

y t x t t

= + +

= +

&

Measurement

noise Knowncontrol

Disturbance

x(t)

(t)

+

+u(t)

+

+

+y(t)BB

AA

CC

GG

)t(x&

V(t)


15/32

Typical stochastic systemTypical stochastic system

ResponsesResponses

tt

O/P without noise or

disturbance

t

O/P without noise

but including

disturbance

O/P with noise & disturbance

x(t)

(t)

+

+u(t)

+

+

+y(t)BB

AA

CC

GG

)t(x&

( )v t


16/32

Kalman FilterKalman Filter (First step)(First step)

BB

A

A

CC y(t)x(t)

0

+

+

u(t)+ +

+

0

t

t

AA

CC

+

+

)( tx)( tyu(t)

BB


17/32

Model Estimator StructureModel Estimator Structure

disturbance presentdisturbance present

BB

AA

CC

x(t)+

+u(t)

GG(t)

+

AA

CC

+

+

)( tx )( ty

BB

y(t)

t

t


18/32

Kalman Filter StructureKalman Filter Structure

( )xy

tytytutxtx

C

)()(K)(B)(A)(

=

++=&

BB

AA

CCx

GG

u +

AA

CC

)( tx )( ty

BB

y

KK

+

How do we calculate the gain K?

What is the new controller?

)t(x&

Plant

Filter

+

+

++ -

++

+


19/32

ContinuousContinuous--Time Kalman FilterTime Kalman Filter

(Kalman(Kalman--Bucy Filter)Bucy Filter)

( ) ( ) ( ) ( ) ( )[ ]( ) xx

tytxtutxtx

=

++=

0

CK(t)BA&

K(t)K(t)

AA

u

B

B

yx

CC+ +

++-


20/32


21/32

Problem StatementProblem Statement

! Define the estimation error

!Define performance measure

! Problem- Determine the gainKto minimiseJ

)()()(~ txtxtx =

)(~)(~E))(~(JT

txtxtx =


22/32

Noise and DisturbanceNoise and Disturbance

CharacteristicsCharacteristics! Assume Gaussian White noise zero mean

"Uncorrelated instant to instant

"Constant spectral density

! Define Covariance of Noise

{ }

{ } )(.)(R)()(E

)(.)(Q)()(E

T

T

=

=

ttt

ttt

Expectation Average

Dirac delta is 1 iff t=and

0 otherwise


23/32

How to SolveHow to Solve

the Problemthe Problem

!ExpandJin terms of system equations

!Calculate gradient (Weiner-Hopf Equation)

!Set gradient = 0 at optimum

!CalculateKfrom resulting equations

J

K

Zero Gradient


24/32

Solution for the Gain MatrixSolution for the Gain Matrix

! The Kalman Gain matrix is given by,

! Where P(t) is the filtering error covariance given by thesolution of,

! Gain can be pre-computed unless system nonlinear or

system/noise varying in an unknown way use discreteequations and update knowledge of the system at eachinstant.

1T

R)CP()K(

=

tt

T T -1 TP( ) AP PA PC R CP GQGd tdt

= + +


25/32

Generic Nature of FilteringGeneric Nature of Filtering

ProblemsProblems

Steel

Ship


26/32

Separation Principle ofSeparation Principle ofStochastic ControlStochastic Control


27/32

KF in ControlKF in Control

Separation PrincipleSeparation Principle

!Plant

! Criterion

( ) A ( ) B ( ) G ( )( ) C ( ) ( )

x t x t u t t y t x t t

= + +

= +

&

}{J +=T

T

TTdtuRuxQxE


28/32

Separation PrincipleSeparation Principle

BB CC

AA

KfKf

AA

CC

BB

KcKc

u x y

Filter

x

Controller

xKu c=


29/32

Real ExamplesReal Examples

! Determination of planet orbit parameters from limitedearth observations

! Tracking targets eg aircraft, missiles using Radar & EOsystems

! Sensor data fusion

! The process of finding the best estimate from noisy data

amounts to filtering out the noise.! However a Kalman Filter doesnt just clean up the data

measurements, but also projects these measurements ontothe state estimate

Why use the word Filter?Why use the word Filter?


30/32

!If all noise is Gaussian, KF minimises the

mean square error of the estimatedparameters.

!Kalman filter is the best linear estimator.

Optimal in What Sense?Optimal in What Sense?

What if the noise is NOT Gaussian?What if the noise is NOT Gaussian?


31/32

ConclusionsConclusions

!Simple structure easy to motivate

!Kalman gives optimal solution

!For very wide range of systems and problems!Many applications-control only one.


32/32

Any Questions?Any Questions?

Documents

L13 KF Intuitive Intro