Final_1.pdf

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

Kalman Filter Design: Lag 2

In order to enhance readability, we use and interchangeably to de-
note the posteriori state estimate. The state equation of the Kalman filter is
obtained as a first order approximation of the Taylor expansion about the
current point. To do that, an estimate of the derivative is required. The pre-
vious design used the first difference of the previous step as an estimate of
the derivative. In this design, the derivative is estimated with two sample
points. The mean and variance of the first sample are as follows:

The mean and variance of the second sample are as follows:

The covariance between the two samples are as follows:

The minimum variance linear combination of the two samples is given by

wherertis given as

rt =

−

+−

var( ) cov( )

var( ) var( ) cov( )

sample 2 samples

sample 1 sample 2 2 samples

rXtt(ˆˆ−− 12 −+− −Xt)( )( 1 r Xt tˆˆ−− 23 Xt)

+cov()XXˆtt−− 23 ,ˆ

cov( ) cov ˆˆ, ˆˆ

cov( ) cov ˆ ,ˆ var ˆ cov ˆ ,ˆ

samples

samples

=−()()−



= ()− ()− ()+

−−−−

−− − −−

XXXX

XX X XX

ttt t

tt t tt

1223

12 2 13

EXX

XX XX

tt

tt tt

()ˆˆ

var( ) var ˆ var ˆ cov ˆ ,ˆ

sample 2

sample 2

=+

= ()+ ()− ()

−−

−− −−

23

232 23

EXX

XX XX

tt

tt tt

()ˆˆ

var( ) var ˆ var ˆ cov ˆ ,ˆ

sample1

sample1

=−

= ()+ ()− ()

−−

−− −−

12

122 12

ˆ

Xt|t

Xˆ

t

cov()XXˆtt−− −− 11 ||,ˆt t2 2=gt− (^1)  2 var()Xˆt t−−2 2|−cov()X Xˆt t−− −−2 2||,ˆtt 33

Xˆ

t− 2

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