8
7
1:5
6
5
1
4
3
0:5
0
1:6
1:8
;0:5
2
2:2
2:4
2:6
;1
;1
;0:5
0
0:5
1
1:5
2
The level curves of the sensitivity transfer function magni-
Figure
11.17
tude, given by (11.18).
Figure 11.18 shows the subset of slice that meets the generalized gain margin
H
speci cation, together with the inner approximation (11.19), i.e.,
1
H(a)
13 + H(b)
13 + (1
)H(c)
13
2:02:
(11.20)
;
;
;
0:2
1
The generalized gain margin speci cation is not in general convex, even though in
this case the subset of slice that satis es the generalized gain margin speci cation
H
is convex. The inner approximation (11.20) is a norm-bound on H, and is therefore
convex (see section 6.3.2).
11.5.3
Robust Stability with Relative Plant Uncertainty
Figure 11.19 shows the subset of slice that meets the speci cation
H
rob stab( )
(11.21)
D
P
where is the plant perturbation set (10.10), i.e., robust stability with the relative
P
plant uncertainty Wrel err described in section 10.2.3. The speci cation (11.21) is
equivalent to the convex inner approximation
Wrel err H(a)
13 + H(b)
13 + (1
)H(c)
13
< 1
(11.22)
;
;
1
derived from the small gain theorem in section 10.4.4, i.e., in this case the small
gain theorem is not conservative (see the Notes and References in chapter 10).
11.5 ROBUSTNESS SPECIFICATIONS
265
1:5
1
1
13
0 2 2 02
k
;
H
k
:
1
:
0:5
?
0
;0:5
@
I
@
exact
;1
;1:5
;2
;10
;8
;6
;4
;2
0
2
4
6
8
10
The boundary of the exact region where the generalized gain
Figure
11.18
margin specication is met is shown, together with the boundary of the inner
approximation (11.20). This specication is tighter than the specication
+4 3 5db gm shown in gure 11.16.
D
;
:
1:5
1
0:5
0
;0:5
;1
;1:5
;2
;20
;15
;10
;5
0
5
10
15
20
The region where the robust stability specication (11.21) is
Figure
11.19
met is shaded. This region is the same as the inner approximation (11.22).
266