1:2
0:8
0:8
() 0:4
() 0:4
t
t
11
12
s
s
0
0
0:4
0:4
;
;
0
0:2
0:4
0:6
0:8
1
0
0:2
0:4
0:6
0:8
1
t
t
(a)
(b)
1:2
1:2
0:8
0:8
() 0:4
() 0:4
t
t
21
22
s
s
0
0
0:4
0:4
;
;
0
0:2
0:4
0:6
0:8
1
0
0:2
0:4
0:6
0:8
1
t
t
(c)
(d)
Design specications requiring the decoupling of responses to
Figure
8.7
step commands.
the inputs one at a time. Figure 8.8 shows the response of the commanded variables
to a particular command signal that has a set-point change in 1 at = 0 5 and
w
t
:
then a set-point change in 2 at = 2 0. The perturbation in 2 right after = 0 5
w
t
:
z
t
:
and the perturbation in 1 right after = 2 0 are due to command interaction.
z
t
:
The speci cation mimo env limits this perturbation, and guarantees that after the
H
set-point change for 2, for example, the e ect on 1 will fade away.
z
z
An extreme form of mimo env is to require that the o -diagonal step responses
H
be zero, or equivalently, that cc be diagonal. This is called exact or complete
H
decoupling:
dcpl =
cc is diagonal
H
fH
j
H
g
:
This speci cation forbids any command interaction at all, regardless of the com-
mand signals. dcpl is a ne.
H
8.1 INPUT/OUTPUT SPECIFICATIONS
181
2
1
w
@
@
R
1:5
@
I
@
1
1
z
0:5
0
2
z
;
;
;0:5
;
;
2
w
;1
0
0:5
1
1:5
2
2:5
3
3:5
4
t
An example of command interaction in a two-input, two-output
Figure
8.8
system. The individual step responses of the system are shown in gure 8.7.
Output 1 tracks a step change in the command signal 1. However, 1
z
w
z
is perturbed by a step change in the command signal 2. Similarly, 2 is
w
z
perturbed by a step change in 1.
w
Miscellaneous Step Response Specifications
Other speci cations often expressed in terms of the step response of cc include
H
monotonic step response and I/O slew-rate limits. The monotonic step response
constraint is
mono sr =
( ) is nondecreasing for all
0
H
f
H
j
s
t
t
g
=
( ) 0 for all
0
f
H
j
h
t
t
g
where ( ) is the impulse response of cc. Thus, mono sr requires that the com-
h
t
H
H
manded variable move only in the direction of the new set-point command in re-
sponse to a single abrupt change in set-point. mono sr is a stricter speci cation
H
than requiring that both the undershoot and overshoot be zero. mono sr is convex.
H
A related speci cation is a slew-rate limit on the step response:
slew sr =
d
( ) = ( )
slew for all
0
H
H
s
t
jh
t
j
M
t
:
dt
Response to Other Inputs
We close this section by noting that the response of c to any particular command
z
signal, and not just a unit step, can be substituted for the step response in all of
182