Final_1.pdf

APPENDIX

Kalman Filter Design: Lag 1

The state equation of the Kalman filter is given as

The observation equation is given as

Yt=Xt+ht

The variance of Ytis calculated as described in the discussion on the obser-
vation equation. We now define

wheregt= 1 – Kt,Ktis the Kalman gain as described in the standard predictor-
corrector framework. The posteriori estimate of the state, and its variance is
given as

We note that the a posterori estimate is actually a convex combination of the
a priori estimate and the observation. The value of gtas computed here en-
sures that the variance of the resulting combination is a minimum. We now
proceed to obtain a recursive relation for. The Kalman equa-
tion for the subscript t– 1 is

Substituting for , we have

ˆˆˆ

XgXXtt−− 11 |||=−t− 1 () (^21) t t−−2 2 tt−− 33 +−()gYt− − 1 t 1
Xˆtt−− 12 |

ˆˆ

XgXtt−− 11 ||=+−t tt− −−1 12() 1 gYt− − 1 t 1

cov(XXˆˆtt−− 12 )

ˆˆ

var ˆ

var ˆ var

var ˆ var

||

|

|

|

XgX gY

X

XY

XY

tt t tt t t

tt

tt t

tt t

=+−()

()=

()()

()+ ()

−

−

−

1

1

1

1

g

Y

YX

t

t

ttt

=

()

()+ ()−

var

var var ˆ|1

ˆˆˆ

var ˆ var ˆ var ˆ cov ˆ ,ˆ

|||

|||||

XXX

XXX XX

tt t t t t

tt tt t t tt t t

−−−−−

−−−−−−−−−

=−

()= ()+ ()− ()

11122

111221122

2

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