Optimal Estimation of Dynamic Systems, 2nd Edition

Batch State Estimation 335

where the gain matrix K

k

is defined as

K

k

≡ P

+

fk

Φ

T

k

(P

−

fk+1

)

−1

(5.52)

Note that Equation (5.51) is no longer a function of the backward covariance P

+

bk

or P

−

bk

. Therefore, the smoother covariance can be solved directly from knowledge

of the forward covariance alone, which provides a very computationally efficient

algorithm.

The RTS smoother state estimate equation is given by

ˆ

x

k

=

ˆ

x

+

fk

+ K

k

[

ˆ

x

k+1

−

ˆ

x

−

fk+1

] (5.53)

The proof of this form begins by comparing Equation (5.53) to Equation (5.21). From

this comparison we need to prove that the following relationship is true:

−K

k

ˆ

x

+

fk

+ P

k

ˆχ

−

bk

= K

k

[

ˆ

x

k+1

−

ˆ

x

−

fk+1

] (5.54)

Substituting Equations ...