@
I
@
max
s
1
@
I
@
2
s
0:8
0:6
()t
@
I
@
1
s
s
0:4
@
I
@
0:2
( 1 + 2) 2
s
s
=
min
s
;
;
0
;0:2
0
0:1
0:2
0:3
0:4
0:5
0:6
0:7
0:8
0:9
1
t
General envelope specication on a step response. The two step
Figure
8.5
responses 1 and 2 satisfy this constraint their average (shown in dashed
s
s
line) also satises the envelope constraint.
have upper and lower bounds for a general envelope constraint, max( ) and min( ).
s
t
s
t
Suppose that max( ) does not increase, and min( ) does not decrease, for increasing
s
t
s
t
. For each
0, we consider the time-scaled envelope speci cation
t
T
>
min(
) ( ) max( ) for all
0
(8.4)
s
t=T
s
t
s
t=T
t
:
(8.4) de nes a nested family of convex speci cations, parametrized by . For = 1
T
T
we have the original envelope speci cation with bounds min and max for
1
s
s
T
>
we have a weaker speci cation, and for
1 we have a stronger speci cation
T
<
roughly speaking, is the normalized response time. We de ne the generalized
T
response-time functional as
grt( cc) = inf
min(
) ( ) max( ) for all
0
H
fT
j
s
t=T
s
t
s
t=T
t
g
:
This construction is shown in gure 8.6. The comments at the end of section 6.2.2
show that grt is quasiconvex.
Step Response Interaction
We now consider the case where there are multiple commands (and multiple com-
manded variables) so that cc is an c
c transfer matrix, where c
1. Its
H
n
n
n
>
diagonal entries are the transfer functions from the command inputs to their associ-
ated commanded variables, which may be required to meet the various speci cations
8.1 INPUT/OUTPUT SPECIFICATIONS
179
1:6
max( 0 1)
s
t=
:
;
;
1:4
max( 0 2)
s
t=
:
;
max( 0 4)
s
t=
:
1:2
;
;
;
1
( )
;
s
t
@
@
R
() 0:8
@
I
t
@
I
@
@
min( 0 4)
s
t=
:
s
0:6
@
min( 0 2)
s
t=
:
@
I
0:4
@
@
0:2
@
min( 0 1)
s
t=
:
0
;0:2
0
0:1
0:2
0:3
0:4
0:5
0:6
0:7
0:8
0:9
1
t
A step response is shown together with the envelopes min( )
Figure
8.6
s
t=T
and max( ) for three values of , where min( ) = 1 1 2exp and
s
t=T
T
s
t
;
:
;t
max( ) = 1 + exp . For
= 0 2 the step response just lies inside the
s
t
;t
T
:
envelopes, so the value of grt is 0.2.
discussed above, e.g., limits on overshoot or rise time. The o-diagonal entries of
cc are the transfer functions from the commands to other commanded variables,
H
and are called the command interaction transfer functions. It is generally desir-
able that these transfer functions be small, so that each command input does not
excessively disturb the other commanded variables.
Let ( ) denote the step response matrix of cc. One mild constraint on com-
s
t
H
mand interaction is asymptotic decoupling:
asympt dcpl = n
cc(0) = lim ( ) is diagonal o
(8.5)
H
H
H
t
s
t
:
!1
This speci cation ensures that if the commands are constant, then the e ect on
each commanded variable due to the other commands converges to zero: there is
no steady-state interaction for constant commands.
A stronger speci cation that limits command interaction is an envelope con-
straint on each entry of ( ),
s
t
mimo env =
min( )
( ) max( ) for all
0
(8.6)
H
f
H
j
s
t
s
t
s
t
t
g
where min( ) and max( ) are matrices, and the inequalities in (8.6) are component
s
t
s
t
by component. The envelope speci cation mimo env is convex.
H
An example of mimo env is shown in gure 8.7, along with a step matrix ( )
H
s
t
that meets it. Of course, the responses shown in gure 8.7 are for steps applied to
180