trk(0) = 0, we can ignore the o set in the command signal. Since
c 250
H
kw
;
k
1
150 C and _c
300 C/Hr, we can specify
kw
k
1
pk trk slew =
trk wc 30 C
(8.13)
H
fH
j
kH
k
g
where we take ampl = 150 C and slew = 300 C/Hr in the de nition of the
M
M
norm
wc (see section 5.2.4). The speci cation (8.13) is tighter than the envelope
k
k
speci cation in gure 8.9: the speci cation (8.13) requires a peak tracking error of
no more than 30 C for any command input that is between 100 C and 400 C, and
slew limited by 300 C/Hr, while the speci cation in gure 8.9 requires the same
peak tracking error for a particular input that is between 100 C and 400 C, and
slew limited by 300 C/Hr.
8.1.3
Model Reference Formulation
An extension of the tracking error formulation consists of specifying a desired closed-
loop I/O transfer matrix ref des, called the reference or model transfer matrix,
H
and the goal is to ensure that cc
ref des. Instead of forming the tracking error
H
H
as trk = c
c, we form the model reference error mre = c
ref des c: the
e
z
;
w
e
z
;
H
w
di erence between the actual response c and the desired response, ref des c. This
z
H
w
is shown in gure 8.12. Note that the tracking error is just the model reference error
when the model transfer matrix is the identity.
c
9
;
q
r
w
ref des
mre
e
H
>
=
q
w
d
+
w
c
z
etc
w
a
zo
z
>
zetc
z
~P
q
y
u
P
K
An architecture for expressing I/O specications in terms of
Figure
8.12
the error from a desired transfer matrix ref des.
H
We will assume that the model reference error mre is contained in . Let mre
e
z
H
denote the submatrix of that is the closed-loop transfer matrix from the command
H
8.2 REGULATION SPECIFICATIONS
187
signal c to the model reference error mre. In the model reference formulation of
w
e
I/O speci cations, we constrain mre to be small in some appropriate sense:
H
mre =
mre mre
(8.14)
H
fH
j
kH
k
g
:
The general model reference error speci cation (8.14) can take a wide variety of
forms, depending on the norm used we refer the reader to section 8.1.2 for a partial
list, and chapter 5 for a general discussion.
8.2
Regulation Specifications
In this section we consider the e ect on c of d only, just as in the previous
z
w
sections we considered the e ect on c of the command inputs only. The response
z
of commanded variables to disturbances is determined by the closed-loop submatrix
cd regulation speci cations require that cd be \small". It is not surprising, then,
H
H
that regulation speci cations can usually be expressed in the form of norm-bound
inequalities, i.e.,
cd reg
(8.15)
kH
k
where
reg is some appropriate norm that depends, for example, on the model
k
k
of the disturbances, how we measure the size of the undesired deviation of the
commanded variables, and whether we limit the average or worst case deviation.
In the following sections we describe a few speci c forms the general regulation
speci cation (8.15) can take. Because these speci cations have a form similar to
I/O speci cations such as limits on tracking error or model reference error, we will
give a briefer description. For convenience we shall assume that d is a scalar
w
disturbance and c is a single commanded variable, since the extension to vector
w
disturbance signals and regulated variables is straightforward.
8.2.1
Rejection of Specific Disturbances
The simplest model for a disturbance is that it is constant, with some unknown
value. The speci cation that this constant disturbance be asymptotically rejected
at c is simply
z
asympt rej =
cd(0) = 0
H
fH
j
H
g
:
This speci cation has the same form as the asymptotic decoupling speci cation (8.5)
(and is therefore closed-loop a ne), but it has a very di erent meaning. The speci-
cation asympt rej can be tightened by limiting the step response of cd to lie in a
H
H
given envelope, as in the command response speci cations discussed in section 8.1.
For example, we may require that the e ect of a unit step input at d on c should
w
z
188