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1:2
1
0:8
()ts
0:6
0:4
0:2
rise
settle
;
;
;
;
0
0
1
2
3
4
5
6
7
8
9
10
t
The value of the rise-time functional, rise, is the earliest time
Figure
8.3
after which the step response always exceeds 0 8. The value of the settling-
:
time functional, settle, is the earliest time after which the step response is
always within 5% of 1 0.
:
settle =
settle( cc)
max
H
fH
j
H
T
g
are convex: if two step responses each settle to with 5% within some time limit
max, then so does their average. Thus, the rise-time and settling-time functionals
T
rise and settle are quasiconvex, i.e.,
rise( cc + (1
) ~cc) max rise( cc) rise( ~cc)
H
;
H
f
H
H
g
for all 0
1, but we do not generally have
rise( cc + (1
) ~cc)
rise( cc) + (1
) rise( ~cc)
H
;
H
H
;
H
so they are not convex. Figure 8.4 demonstrates two step responses for which this
inequality, with = 0 5, is violated.
:
We mention that there are other de nitions of rise-time functionals that are not
quasiconvex. One example of such a de nition is the time for the step response to
rise from the signal level 0.1 (10%) to 0.9 (90%):
10 90( cc) = inf
( ) 0 9 inf
( ) 0 1
H
ft
j
s
t
:
g
;
ft
j
s
t
:
g:
;
While this may be a useful functional of cc in some contexts, we doubt the utility
H
of the (nonconvex) speci cation 10 90( cc)
max, which can be satis ed by
H
T
;
a step response with a long initial delay or a step response with very large high
frequency oscillations.
8.1 INPUT/OUTPUT SPECIFICATIONS
177
1:2
1
@
I
@
0:8
~s
0:6
@
I
@
0:4
@
( + ~) 2
s
s
=
0:2
@
I
@
s
0
0
1
2
3
4
5
6
7
8
9
10
t
The rise times of the step responses and ~ are 5.0 and 0.8
Figure
8.4
s
s
seconds respectively. Their average, ( +~) 2, has a rise time of 3 38 5 8 2
s
s
=
:
>
:
=
seconds. This example shows that rise time is not a convex functional of ,
H
although it is quasiconvex.
General Step Response Envelope Specifications
Many of the step response speci cations considered so far are special cases of general
envelope constraints on the step response:
env =
min( )
( ) max( ) for all
0
(8.3)
H
f
H
j
s
t
s
t
s
t
t
g
where min( ) max( ) for all
0. An example of an envelope constraint is shown
s
t
s
t
t
in gure 8.5. The envelope constraint env is convex, since if two step responses lie
H
in the allowable region, then so does their average.
We mention one useful way that a general envelope constraint env can be
H
expressed as a functional inequality speci cation with a convex functional. We
de ne the maximum envelope violation,
max env viol( cc) = sup max ( )
max( ) min( )
( ) 0
H
t 0
fs
t
;
s
t
s
t
;
s
t
g
:
The envelope speci cation env can be expressed as
H
env =
max env viol( cc) 0
H
f
H
j
H
g
:
General Response-Time Functional
The quasiconvex functionals rise and settle are special cases of a simple, general
paradigm for measuring the response time of a unit step response. Suppose that we
178