the kalman filterpeople.stat.sfu.ca/~dac5/courses/stats_for_ode_models...kalman filter an efﬁcient...

The Kalman Filteran introduction

© Dave Campbell 2009

Friday, June 12, 2009

Kalman Filter

an efficient recursive filter that estimates the state of a dynamic system from noisy data.

also called linear quadratic estimation


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We wish to estimate the state

xt is the actual value taken by the system which is governed by a linear stochastic difference equation with control inputs

The stochastic term (process noise) is

We don’t actually observe xt instead, for some matrix H and measurement noise vt we observe zt=Hxt+vt

Discrete Kalman Filterx ∈ℜn

xt = αxt−1 + βut−1 + wt−1

u ∈ℜ

wt


I observe (with a lot of noise) Z=D2X+err

I need the H matrix: zt=Hxt+vt

Then I will need to describe the update:α



Model:

Where measurement noise

And process noise

Parameters (all assumed constant for now)

is an n x n matrix and ß is n x l

H is n x n and may be sparse and/ or rank deficient

wt ~ N(0,Ω)

vt ~ N(0,Σ)


zt = Hxt + vt

α


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The filtering idea:

1. Take a guess about the next state step ahead xt+1 (Prediction)

2. Make an observation zt+1

3. Update the estimate for xt+1 (Correction)

4. return to step 1

This makes it suited to online state estimation

Model: xt = αxt−1 + βut−1 + wt−1

zt = Hxt + vt



zt = Hxt + vt

We have estimates of xt before ( ) and after ( ) observing zt accounting for the past.

We estimate using and zt:

best new = best old + weight(new obs - old guess at obs)

residual or innovation: (new obs - new obs guess before observed)

x̂t−

x̂t

x̂t−x̂t

x̂t = x̂t− + K(zt − Hx̂t

− )



zt = Hxt + vtModel:

The a priori state estimate is a prediction based on out most recent observation and state estimate

We estimate using and ut-1:

We just use our best current information and update by assuming no noise

x̂t− x̂t−1x̂t− = α x̂t−1 + But−1

x̂t−1


K is the gain or blending factor that minimizes posteriori covariance.

a posteriori estimate error

Find K by substituting

into

Differentiate and set = 0, then solve for K

xt = αxt−1 + βut−1 + wt−1 zt = Hxt + vt

et = xt − x̂t

Δ t = E(etetT )


− ) Δ t = E(etetT )

x̂t− = α x̂t−1 + But−1 x̂t = x̂t

− + K(zt − Hx̂t− )


a priori estimate error

a priori estimate error covariance

Then

et− = xt − x̂t

−

Δ−t = E(et

−[et− ]T)

Kt = Δ t−HT(HΔ t

−HT + Σ)−1

xt = αxt−1 + βut−1 + wt−1 zt = Hxt + vt

x̂t− = α x̂t−1 + But−1 x̂t = x̂t

− + K(zt − Hx̂t− )



zt = Hxt + vt

We have estimates of xt before ( ) and after ( ) observing zt accounting for the past.

We estimate using and zt:

best new = best old + weight(new obs - old guess at obs)

residual or innovation: (new obs - new obs guess before observed)

x̂t−

x̂t

x̂t−x̂t


− )


a priori estimate error

a priori estimate error covariance

Then

as -> 0 then Kt->H-1 weighs the resids more heavily (we trust the observation more)

as ->0 Kt->0 and the prediction is considered very good (ignore the zt)

et− = xt − x̂t

−

Kt = Δ t−HT(HΔ t

−HT + Σ)−1Δ−t = E(et

−[et− ]T)

Σ


− )

Δ t−


Assume that K and are time varying and unknown

We need to estimate K and project (and update) forward

This just comes from var(xt-)

Δ

ΔΔ t

− = αΔ t−1αT +Ω

x̂−t = E(αxt−1 + βut−1 + wt−1)

Δ−t = E xt − x̂

−t⎡⎣ ⎤⎦ xt − x̂

−t⎡⎣ ⎤⎦T( )


After projecting forward we update it when more recent information becomes available:

This is just the variance of the updated estimate (note that zt is an observed value not a rv)

Δ t

Δk = (I − KtH)Δ t−



− )


The Kalman Filter maintains the 1st and 2nd moments of the state distribution

If process noise and observation noise are normal then the filter is actually Bayesian:

E(xt ) = x̂tE (xt − x̂t )(xt − x̂t )[ ] = Δ t

P(xt | zt ) ~ N(x̂t ,Δ t )


The estimate is recursive and only uses the most recent information and the future prediction.

It doesn’t deal with all of the data directly.

But in this basic Kalman filter:

we need estimates of and in advance. Also they should be constant

Σ Ω


Kalman Filter

This is the ‘linear regression’ of the measurement error and process noise state space model.


Back to Matlab to see how it turns out


Extended Kalman FilterFor nonlinear systems

The basic idea:

linearize the system using a Taylor expansion around the current estimate

basics: http://www.cs.unc.edu/~welch/kalman/kalmanIntro.html


http://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf




the kalman filterpeople.stat.sfu.ca/~dac5/courses/stats_for_ode_models...kalman filter an efﬁcient...

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