Note that for the matrix inverses given in (3-11) to exist, conditions
(1), (2), and (3) are needed. In addition, if HEXI - 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
N
N { X (-l)j-'II.l-]j.
(s-X) j=1 (S N+-j
+ { [sI-F]'- [F-XI]-N (3-13)
where it is assumed that x is not an eigenvalue of F. Using (3-13) and (3-10) it is then possible to write (3-5) as
R*(s) = H[sI-F]1 { x* + [F-Xll]'IGw + [T-XlI]-2Gv + [F-X21] -1 Gr
+ HEX IF]-IGw 1 + HEX I-I s1 HEX I-F]-2Gv 1
1 ~ (S-X) +Hx1-]G 1 (s-Xx12)
HEX I-F]- 1Gr 1 (3-14)
L2IF (s-X 2)
When (3-11) and (3-12) are substituted into (3-14) we obtain the desired
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) = [XlI-F-1Gws + [xII-1-IGv 1
(S-Xl)2'
1 +(3-15)
- [ExI-F]2Gv ( + X21I-F]'IGr1