The c components of c are the regulated variables that the commands c are
n
z
w
intended to control or regulate|the \output" in classical control terminology. The
a components of a are the actuator signals, which we remind the reader must be
n
z
included in in any sensible formulation of the controller design problem. The o
z
n
components of o are other critical signals such as sensor signals or state variables.
z
The vector signal etc contains all remaining regulated variables.
z
We conformally partition the closed-loop transfer matrix :
H
2
c 3 2 cc
cd
3
2
3
z
H
H
?
c
w
6
a 7 6 ac
ad
7
z
= H
H
?
6
7
6
7
4
d 5
w
:
4
o 5 4 oc
od
5
z
H
H
?
etc
etc
w
z
?
?
?
The symbol is used to denote a submatrix of that is not used to formulate
?
H
performance speci cations (some of these submatrices will be used in chapter 10).
In the next few sections we consider speci cations on each of the other submatrices
of . We remind the reader that a convex or a ne speci cation or functional on a
H
submatrix of corresponds to a convex or a ne speci cation or functional on the
H
entire matrix (see section 6.2.3).
H
8.1
Input/Output Specifications
In this section we consider speci cations on cc, the closed-loop transfer matrix
H
from the command or set-point inputs to the commanded variables, i.e. the variables
the commands are intended to control. In classical control terminology, cc consists
H
of the closed-loop I/O transfer functions. Of course, it is possible for a control
system to not have any command inputs or commanded variables ( c = 0), e.g. the
n
classical regulator.
The submatrix cc determines the response of the commanded variables c to
H
z
the command inputs c only c will in general also be a ected by the disturbances
w
z
( d) and the other exogenous input signals ( etc) (these e ects are considered in the
w
w
next section). Thus, the signal cc c is the noise-free response of the commanded
H
w
variables. In this section, we will assume for convenience that d = 0, etc = 0,
w
w
so that c = cc c. In other words, throughout this section c will denote the
z
H
w
z
noise-free response of the commanded variables.
8.1.1
Step Response Specifications
Speci cations are sometimes expressed in terms of the step response of cc, espe-
H
cially when there is only one command signal and one commanded variable ( c = 1).
n
The step response gives a good indication of the response of the controlled variable
to command inputs that are constant for long periods of time and occasionally
change quickly to a new value (sometimes called the set-point). We rst consider
the case c = 1. Let ( ) denote the step response of the transfer function cc.
n
s
t
H
8.1 INPUT/OUTPUT SPECIFICATIONS
173
Asymptotic Tracking
A common speci cation on cc is
H
lim
s
( ) = cc(0) = 1
s
t
H
!1
which means that for c constant (and as mentioned above, d = 0, etc = 0),
w
w
w
c( ) converges to c as
, or equivalently, the closed-loop transfer function
z
t
w
t
!
1
from the command to the commanded variable is one at = 0. We showed in
s
section 6.4.1 that the speci cation
asympt trk =
cc(0) = 1
H
fH
j
H
g
is ane, since the functional
( ) = cc(0)
H
H
is a ne.
A strengthened version of asymptotic tracking is asymptotic tracking of order
: cc(0) = 1, (j)
cc (0) = 0, 1
. This speci cation is commonly encountered
k
H
H
j
k
for = 1 and = 2, and referred to as \asymptotic tracking of ramps" or \zero
k
k
steady-state velocity error" (for = 1) and \zero steady-state acceleration error"
k
(for = 2). These higher order asymptotic tracking speci cations are also a ne.
k
Overshoot and Undershoot
We de ne two functionals of cc: the overshoot,
H
os( cc) = sup ( ) 1
H
t 0 s t ;
and the undershoot,
us( cc) = sup ( )
H
t 0 ;s t :
Figure 8.1 shows a typical step response and the values of these functionals.
These functionals are convex, so the speci cations
os =
os( cc)
H
fH
j
H
g
us =
us( cc)
H
fH
j
H
g
are convex: for example, if each of two step responses does not exceed 10% overshoot
( = 0 1) then neither does their average (see section 6.4.2 and gure 6.6).
:
174