Nonlinear Control Sy.. - Free

220 CHAPTER 8. PASSIVITY
Example 8.4 Consider the linear time-invariant system H(s) = (s + c)/[(s + b)(s + b)],
and let H'(s) = H(s)/[1 - eH(s)]. We first investigate the SPR condition on the system
H(s). We have
=
t
H( 3w )
II(
e[ 7w)] =
lim w2 te[H(,w)] =
W-ioo
from here we conclude that
(i) H(s) is SPR if and only if a + b > c.
(ii) H(s) is weak SPR if a + b = c.
(iii) H(s) is not SPR if a + b < c.
c+,yw
(ab-w2)+,yw(a+b)
c(ab - w2) + w2(a + b)
(ab-w2)2+w2(a+b)2
a + b - c
We now consider the system H'(s), after the loop transformation. We need to see whether
H'(s) is passive (i.e., Re[H'(3w)] > 0). To analyze this condition, we proceed as follows
if and only if
1
[H (7w) + H(-7w)] (abc - ec2) + w2(a + b - c - e) > 0
= Z (ab-eC-w2)2+(a+b-e)2
abc - ec2 > 0 (8.31)
a+b-c-e > 0. (8.32)
If a + b > c, we can always find an e > 0 that satisfies (8.31)-(8.32). However, if fl(s) is
weak SPR, then a + b = c, and no such e > 0 exists.
8.7 Exercises
(8.1) Prove Theorem 8.4 in the more general case when Q # 0 in Definitions 8.2 and 8.3.
(8.2) Prove Theorem 8.5 in the more general case when 0 $ 0 in Definitions 8.2 and 8.3.