Final_1.pdf

Multiplying the above by and evaluating the expected value, we have

Multiplying by and evaluating the expected value, we have

Kalman Filter Design: Lag d ( d >= 3)

To enhance readability, we use and interchangeably to denote the
posteriori state estimate. The slope is estimated as the mean of the last d
slope samples. This is given as

The state equation is therefore

The state variance is given by

1

11

2

+ 1 2 1











 ()−−−+ ()

d

X

d

var ˆttdvarXˆ

ˆˆ

ˆˆ

ˆˆˆ

|

|

XX

XX

d

X

d

X

d

X

tt t

ttd

tt t t d

−−

−−−

−−−−

=+

()−

=+











 −

11

11

1111

11

XXˆˆ

d

ttd−−− 11 −

Xˆ

t

ˆ

Xt|t

−−() 1 rXXttt−−− 134 cov()ˆ ,ˆ ]

cov()XXˆtt−− − − 13 ,ˆ =+gt 1 [() 112 rt 1 cov()XXˆt t−− 23 ,ˆ +−()rt− 1 var()Xˆt− 3 −

Xˆ

t− 3

−−() 1 rXXttt−−− 124 cov()ˆ ,ˆ ]

cov()XXˆtt−− − − 12 ,ˆ =+gt 1 [() 112 rt 1 var()Xˆt− 2 +−()rt− 1 cov()XXˆt t−− 23 ,ˆ −

Xˆ

t− 2

rXˆ gY

−−() (^11) tt−− 14 ]+−()tt−− 11

ˆˆˆ

Xtt−− 11 | =+gt− 1 [() 112 rXt− − 1 t 2 +−()rXt− − 1 t 3 −

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