error, the general tracking speci cation (8.7) can be expressed as the weighted 2
H
norm-bound
rms trk =
trk
2
(8.8)
H
fH
j
kH
W
k
g
where is a spectral factor of cmd: cmd( ) = ( ) ( ).
W
S
S
!
W
j
!
W
j
!
Worst Case RMS Mistracking Limit
We may reduce our a priori assumptions about the command signal even further,
by assuming that we do not know the spectrum, but know only a maximum cmd-
W
weighted RMS value for the command signal c, where cmd is some appropriate
w
W
weight. If our measure of the size of the tracking error trk is the worst case trk-
e
W
weighted RMS value it might have, where trk is some appropriate weight, then the
W
appropriate norm in the general tracking error speci cation (8.7) is the weighted
norm:
H
1
hinf trk =
trk trk cmd
(8.9)
H
fH
j
kW
H
W
k
g
:
1
For c = 1 (meaning the weights are scalar, and the maximum singular value of the
n
transfer function is simply its magnitude), this speci cation can also be cast in the
more classical form:
hinf trk =
trk( )
trk( )
trk is stable
(8.10)
H
fH
j
jH
j
!
j
l
!
H
g
where
trk( ) =
l
!
cmd( ) trk( ) :
jW
j
!
W
j
!
j
The classical interpretation is that trk( ) is a frequency-dependent limit on the
l
!
tracking error transfer function, and the speci cation (8.10) ensures that the \com-
mand to tracking error transfer function is small at those frequencies where the
command has signi cant energy". An example is shown in gure 8.11.
Worst Case Peak Mistracking Limit
Another speci c form that the general tracking error speci cation (8.7) can take is
a worst case peak mistracking limit:
pk trk =
trk pk gn
(8.11)
H
fH
j
kH
k
g
:
This speci cation arises as follows. We use an unknown-but-bounded model of the
command signals: we assume only
c
(8.12)
kw
k
M
:
1
8.1 INPUT/OUTPUT SPECIFICATIONS
185
20
10
( )
l
!
;
;
0
;10
20dB
;
dB
;
;
;20
;30
@
I
@
~
;40
trk
jH
j
@
I
@
( trk + ~trk) 2
j
H
H
=
j
;50
@
I
@
trk
jH
j
;60
0:1
1
10
100
1000
(Hz)
f
Upper bounds on frequency response magnitudes are convex.
Figure
8.11
The bound ( ) on the tracking error transfer function ensures that the
l
!
tracking error transfer function is below 20dB at frequencies below 10Hz,
;
and rolls o below 1Hz. Two transfer functions trk and ~trk that satisfy
H
H
the specication (8.10) are shown, together with their average. Of course,
the magnitude of ( trk + ~trk) 2 is not the average of the magnitudes of
H
H
=
trk and ~trk, although it is no larger than the average.
H
H
Our measure of the tracking error is the worst case peak (over all command signals
consistent with (8.12)) of the tracking error trk:
e
trk
trk whenever c satis es (8.12)
ke
k
M
w
:
1
This constraint is precisely (8.11), with = trk .
M
=
M
Since most plants are strictly proper, trk pk gn will usually be at least one.
kH
k
This can be seen from the block diagram in gure 8.10: a step change in the
command input c will produce an immediate, equal sized change in the tracking
w
error. After some time, the closed-loop system will drive the tracking error to a
smaller value. For this reason, the speci cation (8.11) may not be useful.
A useful variation on this worst case peak tracking error limit is to assume more
about the command signals, for example, to assume a maximum slew rate as well
as a maximum peak for the command signal. In this case the appropriate norm in
the general tracking error limit (8.7) would be the worst case norm
wc, from
k
k
section 5.2.4.
For example, consider the temperature response envelope shown in gure 8.9.
Provided the system achieves asymptotic tracking of constant commands, so that
186