11.5.4
Robust Performance
Figure 11.20 shows the subset of slice that meets the speci cation
H
rob(
H23
75)
(11.23)
D
P
k
k
1
where is the plant perturbation set (10.10), i.e., the plant perturbations described
P
in section 10.2.3 never cause the RMS gain from the reference input to the actuator
signal to exceed 75. In section 10.5.2 we showed that an inner approximation of the
speci cation (11.23) is
H(a)
13 (j!) + H(b)
13 (j!) + (1
)H(c)
13 (j!) < l(!) for !
(11.24)
;
;
2
R
(note that H23 = H13=Pstd
0 ), where
l(!) =
1
q
2; Wrel err(j!) 2 + 1=(75Pstd
0 (j!)) 2
j
j
j
j
which is also shown in gure 11.20. The exact region is not in general convex,
although in this case it happens to be convex (see the Notes and References in
chapter 10).
1:5
1
exact
;
;
0:5
0
@
I
@
;0:5
inner
;1
;1:5
;2
;20
;15
;10
;5
0
5
10
15
20
The boundary of the exact region where the robust perfor-
Figure
11.20
mance specication (11.23) is met is shown, together with the inner approx-
imation (11.24).
Figure 11.21 shows the exact speci cation (11.23), together with the convex inner
approximation (11.24) and a convex outer approximation (i.e., a speci cation that
11.5 ROBUSTNESS SPECIFICATIONS
267
is weaker than (11.23)). The outer approximation is the simultaneous satisfaction
of the speci cations
rob stab( ) and
H23
75
(11.25)
D
P
k
k
1
described in sections 11.5.3 and 11.3.3 respectively. The speci cations (11.25) re-
quire robust stability, and that the nominal system has an RMS gain from the
reference to actuator signal not exceeding 75. The robust performance speci ca-
tion (11.23) therefore implies (11.25), so (11.25) is an outer approximation of the
robust performance speci cation (11.23). The outer approximation in gure 11.21
is the set of , for which
Wrel err H(a)
13 + H(b)
13 + (1
)H(c)
13
< 1
;
;
1
(11.26)
H(a)
23 + H(b)
23 + (1
)H(c)
23
75
;
;
1
(see gure 10.21).
1:5
1
exact
;
;
0:5
0
@
I
@
;0:5
inner
@
I
@
;1
outer
;1:5
;2
;20
;15
;10
;5
0
5
10
15
20
The boundary of the exact region where the robust perfor-
Figure
11.21
mance specication (11.23) is met is shown, together with the inner approx-
imation (11.24) and the outer approximation (11.26). The outer approxi-
mation is the intersection of the nominal performance specication and the
robust stability specication (see gures 11.12 and 11.19).
268