43
Note that for the matrix inverses given in (3-11) to exist, conditions
(1), (2), and (3) are needed. In addition, if H[xl F]~*G is not
square, any right inverse can be used.
We shall make use of the following formula which can be obtained
using a partial fraction expansion
Csl-F]"1
(s-X)'
N i
- { I (-1)J
j=l
L[AI-F]"j-
(S-A)
N+l-j
+ ( [sI-F]-1 [F-AirN }
(3-13)
where it is assumed that A is not an eigenvalue of F. Using (3-13) and
(3-10) it is then possible to write (3-5) as
R*(s) = HCSI-F]-1 { x* + [F-A^rtw + [F-A1I]"2Gv + [F-A^l'V }
t + H[X1I-F]_1Sv^-i- HCAjI-F^Gv^
T
(3-14)
When (3-11) and (3-12) are substituted into (3-14) we obtain the desired
k .
R (s).
It also follows from (3-4), (3-10), and (3-12) that the X*(s) which
occurs during tracking can be expressed as
X*(s) = [A1I-F]"1Gwtilrr + [A1I-F]1Gvy-^- ^
(s-Aj)
-,-2... 1
1
- [A.I-F]" Gv-^y-y + [A2I-F]-GrT^x-T
(3-15)