1 2
=
rms u( ) =
21 proc 2
22 sensor 22
(11.10)
H
kH
W
k2 + kH
W
k
:
Recall from section 8.2.2 that the RMS response to independent stochastic inputs
with known power spectral densities is a convex functional of
therefore rms u
H
is a convex function of , and (11.9) is a convex function of and .
H
2
1:75
1:5
1:5
1:25
1
1:0
0:75
0:5
0:5
0:25
0
0:25
;0:5
0:5
0:75
;1
;1
;0:5
0
0:5
1
1:5
2
Level curves of the RMS value of the actuator signal , with
Figure
11.11
u
sensor and actuator noises, given by (11.9).
11.3.3
RMS Gain Limit
Figure 11.12 shows the level curves of the worst case RMS actuator signal for any
u
reference input with RMS value bounded by 1, ,
r
i.e.
(a)
(b)
(c)
23 +
23 + (1
) 23
(11.11)
H
H
;
;
H
:
1
11.3 ACTUATOR EFFORT
259
1:5
275
225
1
175
125
0:5
0
25
75
;0:5
125
175
;1
225
;1:5
275
325
;2
;20
;15
;10
;5
0
5
10
15
20
Level curves of the worst case RMS actuator signal for
Figure
11.12
u
any reference input with RMS value bounded by 1, given by (11.11).
r
260