11.2.3
RMS Regulation
Suppose that proc and sensor are independent, zero-mean stochastic processes with
n
n
power spectral densities
proc( ) = 2proc
S
!
W
sensor( ) = 2sensor
S
!
W
where
proc = 0 04
W
:
sensor = 0 01
W
:
( , scaled white noises). Figure 11.9 shows the level curves of the RMS value of
i.e.
p due to these noises,
, the level curves of the function
y
i.e.
rms yp(
) = rms yp
(a) + (b) + (1
) (c)
(11.6)
'
H
H
;
;
H
where
1 2
=
rms yp( ) =
11 proc 2
12 sensor 22
(11.7)
H
k
H
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 yp
H
is a convex function of , and rms yp is a convex function of and .
H
'
11.3
Actuator Effort
11.3.1
A Particular Disturbance
We consider again the particular actuator-referred disturbance part( ) shown in
d
t
gure 11.7. Figure 11.10 shows the peak actuator signal due to the actuator-
u
referred disturbance part, ,
d
i.e.
(a)
(b)
(c)
21 +
21 + (1
) 21 part
(11.8)
H
H
;
;
H
d
1
which is a convex function on 2.
R
11.3.2
RMS Limit
Figure 11.11 shows the level curves of the RMS value of due to the noises described
u
in section 11.2.3, , the level curves of the function
i.e.
rms u
(a) + (b) + (1
) (c)
(11.9)
H
H
;
;
H
11.3 ACTUATOR EFFORT
257
2
0:07
0:06
1:5
0:05
0:04
1
0:5
0:04
0:05
0
0:06
0:07
;0:5
0:08
0:09
;1
;1
;0:5
0
0:5
1
1:5
2
Level curves of the RMS value of , with sensor and actuator
Figure
11.9
y
p
noises, given by (11.6).
2
0:6
0:55
0:5
1:5
0:45
0:4
0:35
1
0:3
0:25
0:5
0:2
0:2
0
0:25
0:3
;0:5
0:35
0:4
;1
;1
;0:5
0
0:5
1
1:5
2
Level curves of the peak actuator signal , due to the par-
Figure
11.10
u
ticular actuator-referred disturbance
( ) shown in gure 11.7, given
d
t