Optimal Estimation of Dynamic Systems, 2nd Edition

Least Squares Approximation 11

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

x

1

x

2

(a) Observable System

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

x

1

x

2

(b) Unobservable System

Figure 1.6: Contour Plots for an Observable and Unobservable System

If the rank of H is n (i.e., there are at least n independent observation equations), then

H

T

H is strictly positive definite and can be inverted to obtain the explicit solution

for the optimal estimate:

ˆ

x =(H

T

H)

−1

H

T

˜

y

(1.26)

Equation (1.17) is the matrix equivalent of Gauss’ original “equations of condition”

which he wrote in index/summation notation.

2

Equation (1.26) serves as the most

common basis for algorithms ...