the speci cations above. This specic input tracking requirement is convex.
For example, we may require that in response to the particular command signal
c = pgm, shown in gure 8.9, the commanded variable c lies in the envelope
w
w
z
shown. pgm might represent an often repeated temperature cycle in an industrial
w
oven (the mnemonic abbreviates \program"). The speci cation
pgm trk =
cc pgm
pgm
30
H
f
H
j
kH
w
;
w
k
g
1
shown in gure 8.9, requires that the actual temperature, c, always be within 30 C
z
of the commanded temperature, c.
w
450
400
350
300
(C) 250
T
200
150
100
50
0
1
2
3
4
5
6
7
8
9
10
(hours)
t
An example of a temperature command signal that might be
Figure
8.9
used in a plastics process. Powder is slowly melted, and then sintered for
3 hours. It is then rapidly cooled through the melt point. The envelope
constraints on the actual temperature require the temperature error to be
less than 30 C.
8.1.2
Tracking Error Formulation
Step response speci cations constrain the response of the system to speci c com-
mands: a step input at each command. By linearity and time-invariance, this
constrains the response to commands that are constant for long periods and change
abruptly to new values, which is sometimes a suitable model for the commands that
will be encountered in practice. In many cases, however, the typical command sig-
nals are more diverse|they may change frequently in a way that is not completely
predictable. This is often the case in a command-following system, where the goal
8.1 INPUT/OUTPUT SPECIFICATIONS
183
is to have some system variables follow or track a continuously changing command
signal.
In the tracking error formulation of I/O speci cations, we de ne the tracking
error as trk = c c: the di erence between the actual response of the commanded
e
z
;
w
variables and the commands, as shown in gure 8.10.
;
r
+
trk 9
e
c
>
=
q
q
w
c
z
w
d
w
a
z
etc
z
>
w
~
o
zetc
z
P
q
y
u
P
K
An architecture for expressing I/O specications in terms of
Figure
8.10
the tracking error trk = c
c.
e
z
;
w
We will assume that trk is available as a part of , and trk will denote the
e
z
H
submatrix of that is the closed-loop transfer matrix from the commands c to
H
w
the tracking errors trk. A general tracking error speci cation has the form
e
trk =
trk trk err
(8.7)
H
fH
j
kH
k
g
i.e., the closed-loop transfer matrix from commands to tracking error should be
small, as measured with the norm trk err. Using any of the norms from chapter 5
k
k
allows a wide variety of I/O speci cations to be formed from the general tracking
error speci cation trk, all of which are convex. We will brie y list some of these
H
speci cations and their interpretations.
RMS Mistracking Limit
One simpli ed model of the command signal is that it is a stochastic process with a
known power spectrum cmd. Of course this model is quite crude, and only intended
S
to capture a few key features of the command signal, such as size and bandwidth:
the command signal may in fact be generated by a human operator. If we accept
this model, and take the RMS value as our measure of the size of the tracking
184