Optimal Estimation of Dynamic Systems, 2nd Edition

56 Optimal Estimation of Dynamic Systems

A

−1

=



B

−1

11

−A

−1

11

A

12

B

−1

22

−A

−1

22

A

21

B

−1

11

B

−1

22



where B

ii

is the Schur complement of A

ii

,givenby

B

11

= A

11

−A

12

A

−1

22

A

21

B

22

= A

22

−A

21

A

−1

11

A

12

Also, prove the matrix inversion lemma from these matrix inverses.

1.10 Create 101 synthetic measurements ˜y at 0.1 second intervals of the following:

˜y

j

= asint

j

−bcost

j

+ v

j

where a = b = 1,andv is a zero-mean Gaussian noise process with standard

deviation given by 0.01. Determine the unweighted least squares estimates

for a and b. Using the same measurements, find a value of

˜

y that is near

zero (near time

π



4), and set that “measurement” value to 1. Compute the

unweighted least squares solution, and compare it to the original solution.

Then, use weighted least squares to “deweight” ...