decay to no more than 0 05 within some given time rej. Such a speci cation en-
:
T
sures that the closed-loop system will counteract the e ects of a rapidly applied (or
changed) constant disturbance on the commanded variable.
In most cases, however, disturbances cannot be so easily described. In the next
few sections we discuss speci cations that limit the e ect of disturbances about
which less is known.
8.2.2
RMS Regulation
A common model for a disturbance is a stochastic process with a known power
spectral density dist. The speci cation
S
rms reg =
cd
2
(8.16)
H
fH
j
kH
W
k
g
where is a spectral factor of dist, limits the RMS deviation of the commanded
W
S
variable (due to the disturbance) to be less then . This speci cation has exactly
the same form as the RMS mistracking limit (8.8): a weighted 2 norm-bound.
H
The power spectral density of the disturbance is rarely known precisely dist
S
is usually meant to capture only a few key features of the disturbance, perhaps its
RMS value and bandwidth. The power spectral density
2
dist( ) = 2
bw
a
!
S
!
2 + 2bw
!
!
for example, might be used to model a disturbance with an RMS value and a
a
bandwidth bw.
!
8.2.3
Classical Frequency Domain Regulation
We may not be willing to model the disturbance with a speci c power spectral
density. Instead, we may model d as having an unknown power spectral density,
w
but some given maximum RMS value. A limit on the worst case RMS response of
c can be expressed as the
norm-bound
z
H
1
hinf reg =
cd
H
fH
j
kH
k
g
1
which limits the RMS gain of the closed-loop transfer function cd. Often, this
H
speci cation is modi ed by frequency domain weights, re ecting the fact that either
a maximum possible weighted-RMS value for the disturbance is assumed, or a limit
on some weighted-RMS value of the commanded variable must be maintained. Such
a frequency-weighted
norm-bound can be cast in the more classical form:
H
1
hinf reg =
cd( )
reg( )
cd is stable
(8.17)
H
fH
j
jH
j
!
j
l
!
H
g
:
The classical interpretation is that reg( ) is a frequency-dependent limit on the
l
!
disturbance to commanded variable transfer function, and the speci cation (8.17)
ensures that the \disturbance to commanded variable transfer function is small at
those frequencies where the disturbance has signi cant energy".
8.2 REGULATION SPECIFICATIONS
189
8.2.4
Regulation Bandwidth
The classical frequency domain regulation speci cation (8.17) is often expressed as
minimum regulation bandwidth for the closed-loop system. One typical de nition
of the regulation bandwidth of the closed-loop system is
bw( cd) = sup
cd( )
0 1 for all
H
f
j
jH
j
!
j
:
!
g
which is the largest frequency below which we can guarantee that the disturbance
to commanded variable transfer function is no more than 20dB, as shown in
;
gure 8.13.
20
10
0
(dB))j( ;10
j!
cd
;20
H
j
;30
bw( cd)
H
;
;
;40
0:1
1
10
100
1000
!
The value of the regulation bandwidth functional, bw, is
Figure
8.13
the largest frequency below which the disturbance to commanded variable
transfer function, cd, is no more than 20dB.
H
;
The minimum bandwidth speci cation
min bw =
bw( cd)
min
H
f
H
j
H
g
is convex, since it is a frequency-dependent bound on the magnitude of , so
H
the bandwidth functional bw is quasiconcave, meaning that bw is quasiconvex.
;
Alternatively, we note that the inverse of the regulation bandwidth, i.e., 1 bw,
=
is quasiconvex. The inverse bandwidth 1 bw can be interpreted as a regulation
=
response time.
A generalized de nition of bandwidth, analogous to the generalized response
time, is given by
gbw( cd) = sup
cd( )
(
) for all
H
f
j
jH
j
!
j
M
!
=
!
g
190