1:2
1
os
0:8
() 0:6
t
s
0:4
0:2
us
0
;0:2
;0:4
0
1
2
3
4
5
6
7
8
9
10
t
A typical step response and its overshoot ( os) and under-
Figure
8.1
s
shoot ( us). The asymptotic tracking specication asympt trk requires that
H
the step response converge to one as
.
t
!
1
The functionals os and us are usually used together with the asymptotic track-
ing constraint asympt trk otherwise we might use the relative overshoot and relative
H
undershoot de ned by
8
sup ( )
1
if
0
<
cc(0)
cc(0)
ros( cc) =
t 0 s t =H
;
H
>
H
:
+
if cc(0) 0
1
H
8
sup ( )
0
<
cc(0)
if cc(0)
rus( cc) =
t 0 ;s t =H
H
>
H
:
+
if cc(0) 0
1
H
:
It is less obvious that the speci cations
ros =
ros( cc)
(8.1)
H
fH
j
H
g
rus =
rus( cc)
(8.2)
H
fH
j
H
g
are convex. To see that the relative overshoot constraint ros is convex, we rewrite
H
it as
ros =
cc(0) 0 ( ) (1 + ) cc(0) 0 for all
0
H
fH
j
H
>
s
t
;
H
t
g
:
If
~
ros and 0
1, then
=
+ (1
) ~ satis es cc(0) 0,
H
H
2
H
H
H
;
H
H
>
and for each
0 we have ( ) (1 + ) cc(0) 0. Hence,
ros.
t
s
t
;
H
H
2
H
8.1 INPUT/OUTPUT SPECIFICATIONS
175
Since the functional inequality speci cations (8.1{8.2) are convex for each ,
the relative overshoot and relative undershoot functionals are quasiconvex they are
not, however, convex. If one step response, ( ), has a relative overshoot of 30%,
s
t
and another step response ~( ) has a relative overshoot of 10%, then their average
s
t
has a relative overshoot not exceeding 30% but it may exceed 20%, the average
of the two relative overshoots. An example of two such step responses is shown in
gure 8.2.
3
s
;
;
2:5
( + ~) 2
s
s
=
;
2
;
;
1:5
~s
;
;
1
0:5
0
0
1
2
3
4
5
6
7
8
9
10
t
The relative overshoot of the step responses and ~ are 30%
Figure
8.2
s
s
and 10% respectively. Their average, ( + ~) 2, has a relative overshoot of
s
s
=
23%. This example shows that relative overshoot is not a convex functional
of . It is, however, quasiconvex.
H
Rise Time and Settling Time
There are many de nitions of rise time and settling time in use we shall use
rise( cc) = inf
( ) 0 8 for
H
fT
j
s
t
>
:
t
T
g
settle( cc) = inf
( ) 1 0 05 for
H
fT
j
js
t
;
j
<
:
t
T
g
as illustrated in gure 8.3. The functional rise is usually used together with the
asymptotic tracking speci cation asympt trk we can also de ne relative or normal-
H
ized rise time.
The functional inequality speci cations
rise =
rise( cc)
max
H
fH
j
H
T
g
176