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Ch. 13Kalman Filters

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IntroductionIn 1960, Rudolf Kalman developed a way to solve some of the

practical difficulties that arise when trying to apply Weiner filters.

There are D-T and C-T versions of the Kalman Filter we will

only consider the D-T version.

The Kalman filter is widely used in:

Control Systems

Navigation Systems Tracking Systems

It is less widely used in signal processing applications

KF initially arose in thefield of control systems

in order to make a

system do what you

want, you must know

what it is doing now

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The Three Keys to Leading to the Kalman Filter

Wiener Filter: LMMSE of a Signal (i.e., a Varying Parameter)

Sequential LMMSE: Sequentially Estimate a FixedParameter

State-Space Models: Dynamical Models for Varying Parameters

Kalman Filt er : Sequent ial LMMSE Est imat ion f or a t ime-var ying par amet er vect or but t he t ime var iat ion is

const r ained t o f ollow a st at e- space dynamical model.

Aside: There are many ways to mathematically model dynamical systems

Differential/Difference Equations

Convolution Integral/Summation

Transfer Function via Laplace/Z transforms

State-Space Model

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13.3 State-Variable Dynamical ModelsSystem State: the collection of variables needed to know how to determine how

the system will exist at some future time (in the absence of an input)

For an RLC circuit you need to know all of its current capacitor voltages and all

of its current inductor currents

Motivational Example: ConstantVelocity Aircraft in 2-D

=

)(

)(

)(

)(

)(

tv

tv

tr

tr

t

y

x

y

x

s

A/C positions (m)For t he const ant velocit y model we

would const r ain vx(t) & vy(t) t o beconst ant s Vx & Vy.A/C velocities (m/s)

If we know s(to) and there is no input we know how the A/Cbehaves for all future times: rx(to + ) = Vx + rx(to)

rx(to + ) = rx(to) + Vxry(to + ) = ry(to) + Vy

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D-T State Model for Constant Velocity A/CBecause measurements are often taken at discrete times we often

need D-T models for what are otherwise C-T systems(This is the same as using a difference equation to approximate a differential equation)

If every increment ofn corresponds to a duration of sec and

there is no driving force then we can write a D-T State Model as:

=

1000

0100

010

001

A

]1[][ = nn Ass

State Transition Matrix

rx[n] = rx[n-1] + vx[n-1]

ry [n] = ry[n-1] + vy[n-1]

vx[n] = vx[n-1]

vy[n] = vy[n-1]

We can include the effect of a vector input:

][]1[][ nnn BuAss +=

Input could be deterministic and/or random.

Matrix B combines inputs & distributes them to states.

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Thm 13.1 Vector Gauss-Markov Model Dont confusewith the G-M

Thm. of Ch. 6This theorem characterizes the probability model for a

specific state-space model with Gaussian Inputs

][]1[][ nnn BuAss +=Linear St at e Model: n 0

p1 ppknown

prknown

r1

s[n]: state vector is a vector Gauss-Markov processA: state transition matrix; assumed |i| < 1 for stability

B: input matrix

u[n]: driving noise is vector WGN w/ zero means[-1]: initial state ~N( s,Cs) and independent ofu[n]

eigenvalues

u[n] ~N(0,Q)E{u[n] uT[m]} = 0, n m

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Theorem:

s[n] for n 0 is Gaussian with the following characteristics

Mean of state vector is { } sAs 1][ += nnE diverges if e-valueshave |i| 1

Covariance between state vectors at m and n is

{ }( )

=

+++ +=

=m

nmk

Tkn-mTkTnm

TnEnmEmEnmnm

)(

]}][{][]}][[{][[],[:for

11 ABQBAACA

ssssC

s

s

],[],[:for mnnmnm Tss CC =< State Process is Not WSS!

Covariance Matrix: C[n] = Cs[n,n] (this is just notation)

Propagation of Mean & Covariance:

{ } { }TT

nn

nEnE

BQBAACC

sAs

+=

=

]1[][

]1[][

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Pr oof : (only for the scalar case:p = 1)

For the scalar case the model is: s[n] = a s[n-1] + b u[n] n 0

differs a bit from (13.1) etc.

Now we can just iterate this model and surmise its general form:

]0[]1[]0[ buass +=

]1[]0[]1[

]1[]0[]1[

2buabusa

buass

++=

+=

]2[]1[]0[]1[

]2[]1[]2[

23 buabubuasa

buass

+++=

+=

=

+ +=n

k

kn knbuasans

0

1 ][]1[][

!Now easy to find the mean:

{ } { } { }

sn

n

k

kn

a

knuEbasEansE

s

1

0 0

1][]1[][

+

= ==

+

=

+= "#"$%"#"$%

as claimed!

z.i. response

exponential

z.s. response

convolution

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Covariance between s[m] and s[n] is:

{ }

= = +=

++

=

+

=

+

++

+=

+

+=

=

m

k

m

l

l

kmnl

ks

nm

n

l

ls

n

m

k

ks

m

Ts

ns

ms

balnukmuEbaaa

lnbuansa

kmbuamsaE

ansamsEnmC

u0 0

)]([

211

0

1

0

1

11

2

]}[][{

][]][[

][]][[

]][][][[],[

}

{

)(

)(

""" #""" $%

Must use different

dummy variables!!

Cross-terms will

be zero why?

=

+++ +=m

nmk

kmnu

ks

nms babaaanmC

2211],[ For m n:

For m < n: ],[],[ mnCnmC ss =

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For mean & cov. propagation: from s[n] = a s[n 1] + b u[n]

"#"$%"""" #"""" $% 0theoreminaspropagates

]}[{]}1[{]}[{

=

+= nuEbnsaEnsE

{ }

{ }

{ } { }bnuEbansEnsEa

nsaEnbunasE

nsEnsEns

uns

"#"$%""""" #""""" $%2

][]})1[{]1[(

]})1[{][]1[(

]})[{][(]}[var{

2

]}1[var{

2

2

2

==+=

+=

=

which propagates as in theorem< End of Proof >

So we now have:

Random Dynamical Model (A St at e Model) St at ist ical Char act er izat ion of it

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Random Model for Constant Velocity A/C

+

=

=

][

][

0

0

]1[

]1[

]1[

]1[

1000

0100

010

001

][

][

][

][

][

nu

nu

nv

nv

nr

nr

nv

nv

nr

nr

n

y

x

y

x

y

x

y

x

y

x

s

Deterministic Propagationof Constant-Velocity

Random Perturbation

of Constant Velocities

=

2

2

000

000

0000

0000

]}[cov{

u

u

n

u

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0 2000 4000 6000 8000 10000 12000 14000 16000 180000

2000

4000

6000

8000

10000

12000

14000

X position (m)

Ypos

ition(m)

m/s100]1[]1[

m0]1[]1[

m/s5

sec1

==

==

=

=

yx

yx

u

vv

rr

Ex. Set of Constant-Velocity A/C TrajectoriesRed Line is Non-Random

Constant Velocity Trajectory

Acceleration of

(5 m/s)/1s = 5m/s2

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Observation ModelSo we have a random state-variable model for the dynamics of

the signal ( the signal is often some true A/C trajectory)

We need to have some observations (i.e., measurements) of the

signal

In Navigation Systems inertial sensors make noisy measurements

at intervals of time

In Tracking Systems sensing systems make noisy measurements(e.g., range and angles) at intervals of time

][][][][ nnnn wsHx +=Linear Observat ion Model:

Measured observation

vector at each time

allows multiple

measurements at each time

Observation Matrix

can change w/ time

State Vector Process

being observed

Vector Noise

Process

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The Estimation ProblemObserve a Sequence of Observation Vectors {x[0], x[1], x[n]}

Compute an Estimate of the State Vector s[n]

]|[ nns

estimate state at nusing observation up to n

Notation: ]|[ mns = Estimate ofs[n] using {x[0], x[1], x[m]}

Want Recur sive Solut ion:

Given: ]|[ nns and a new observation vector x[n + 1]

Find: ]1|1[ ++ nns

Thr ee Cases of I nt er est : Scalar St at e Scalar Obser vat ion Vect or St at e Scalar Obser vat ion Vect or St at e Vect or Obser vat ion