Page 278 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

OBSERVABILITY, CONTROLLABILITY AND STABILITY 267

An approach to find out whether the system is observable is to con-
struct the observability Gramian (Bar-Shalom and Li, 1993). From (8.14):

2 3 2 3 2 3
zðiÞ HðiÞxðiÞ HðiÞ
zði þ 1Þ Hði þ 1Þxði þ 1Þ Hði þ 1ÞFðiÞ
6 7 6 7 6 7
6 7 6 7 6 7
¼ ¼ xðiÞð8:15Þ
6 7 6 7 6 7
6 zði þ 2Þ7 6 Hði þ 2Þxði þ 2Þ7 6 Hði þ 2ÞFðiÞFði þ 1Þ7
. . .
4 5 4 5 4 5
. . .
. . .
Equation (8.15) is of the type z ¼ Hx. The least squares estimate is
T
T
1
^ x x ¼ (H H) H z. See Section 3.3.1. The solution exists if and only if
T
T
the inverse of H H exists, or in other words, if the rank of H H is equal
to the dimension M of the state vector. Equivalent conditions are that
T
T
T
H H is positive definite (i.e. y H Hy > 0 for every y 6¼ 0), or that the
T
eigenvalues of H H are all positive. See Appendix B.5.
Translated to the present case, the requirement is that for at least one
n 0 the observability Gramian G, defined by:
! T !
n j 1 j 1
T X Y Y
G ¼ H ðiÞHðiÞþ Hði þ jÞ Fði þ kÞ Hði þ jÞ Fði þ kÞ
j¼1 k¼0 k¼0
ð8:16Þ
has rank equal to M. Equivalently we check whether the Gramian is
positive definite. For time-invariant systems, F and H do not depend on
time, and the Gramian simplifies to:
n
G ¼ X HF j T HF j ð8:17Þ
j¼0

If the system F is stable (the magnitude of all eigenvalues of F are less
than one), we can set n !1 to check whether the system is observable.
A second approach to determine the observability of a time-invariant,
deterministic system is to construct the observability matrix:

H
2 3
HF
6 7
6 7
HF
6 2 7 ð8:18Þ
. 7
M ¼ 6
.
6 7
4 . 5
HF M 1