1:5
(b)
H
1
0:5
(a)
(b)
(c)
0:6H
+
0:3H
+
0:1H
(c)
(a)
H
H
0
(a)
(b)
(c)
;0:5
;0:2H
;
0:6H
+
1:8H
;1
;1
;0:5
0
0:5
1
1:5
2
Each point (
) corresponds to a closed-loop transfer matrix
Figure
11.1
that lies in the plane through
,
, and
.
(a)
(b)
(c)
H
H
H
H
Many of the gures in this chapter show level curves of functionals on slice,
H
,
i.e.
n
]
(a) + (b) + (1
) (c) = o
T
H
H
;
;
H
:
For quasiconvex functionals this set might not be a connected curve, so we de ne
the level curve to be the boundary of the convex set
n
]
(a)
(b)
(c)
o
T
+
+ (1
)
H
H
;
;
H
:
In most cases these two de nitions agree.
11.1
I/O Specifications
11.1.1
A Settling Time Limit
The step responses from the reference input to p for the three closed-loop systems
r
y
are shown in gure 11.2. Figure 11.3 shows the level curves of the reference to p
r
y
settling-time functional settle, ,
i.e.
settle
(a)
(b)
(c)
13 +
13 + (1
) 13
(11.1)
H
H
;
;
H
:
Recall from section 8.1.1 that settle, and therefore (11.1), are quasiconvex. From
gure 11.3 it can be seen that the level curves bound convex subsets.
11.1 I/O SPECIFICATIONS
251
1:6
(a)
s
13
1:4
;
;
(c)
s
13
1:2
;
;
1
@
I
@
(b)
0:8
s
13
0:6
0:4
0:2
0
;0:2
0
1
2
3
4
5
6
7
8
9
10
t
The step responses from the reference input, , to plant out-
Figure
11.2
r
put, , for the closed-loop transfer matrices
,
, and
.
(a)
(b)
(c)
y
H
H
H
p
2
1:5
9
1
8
0:5
7
0
5
6
7
8
9
;0:5
;1
;1
;0:5
0
0:5
1
1:5
2
Level curves of the step response settling time, from the
Figure
11.3
reference to , given by (11.1).
r
y
p
252