Optimal Estimation of Dynamic Systems, 2nd Edition

Probability Concepts in Least Squares 73

As in the case of estimation without aprioriestimates, the previous derivation can

also be shown in compact form. The following loss function to be minimized is

J =

1

2

Tr



E



(

ˆ

x −x)(

ˆ

x−x)

T



+ Tr[Λ(I −MH −N)] (2.80)

Substituting Equations (2.43) and (2.45) into Equation (2.54) leads to

ˆ

x = M

˜

y+ N

ˆ

x

a

=(MH + N)x + Mv+ Nw

(2.81)

Next, as before we assume that the true state x and error terms v and w are un-

correlated with each other. Using Equations (2.44) and (2.46) with the uncorrelated

assumption leads to

J =

1

2

Tr(MRM

T

+ NQN

T

)+Tr[Λ(I −MH −N)] (2.82)

Therefore, we have the following necessary conditions:

∇

M

J = MR−Λ

T

H