11.1.2
Some Worst Case Tracking Error Specifications
We now consider the tracking error. For our example, the tracking error transfer
function is given by
13 1
H
;
(we have not set up our example with the tracking error explicitly included in
the regulated variables). Figure 11.4 shows the level curves of the weighted peak
tracking error, ,
i.e.
pk trk(
) =
(a)
(b)
(c)
13 +
13 + (1
) 13 1
(11.2)
'
W
H
H
;
;
H
;
pk gn
where the weight is
( ) = 1
W
s
2 + 1:
s
(We use the symbol to denote the restriction of a functional to a nite-dimensional
'
domain.) Recall from section 8.1.2 that the weighted peak tracking error is a convex
functional of , and therefore pk trk is a convex function of and . From
H
'
gure 11.4 it can be seen that the level curves bound convex subsets.
Figure 11.5 shows the level curves of the peak tracking error, for reference inputs
bounded and slew-rate limited by 1, ,
i.e.
(a)
(b)
(c)
13 +
13 + (1
) 13 1
(11.3)
H
H
;
;
H
;
wc :
In section 6.3.2 we showed that a function of the form (11.3) is convex as expected,
the level curves in gure 11.5 bound convex subsets of slice.
H
In section 5.5.2 we showed that, for any transfer function ,
H
pk gn
wc 3
pk gn
kH
W
k
kH
k
kH
W
k
:
The reader should compare the level curves in gures 11.4 and 11.5 with this relation
in mind.
11.1 I/O SPECIFICATIONS
253
2
1:5
1
0:5
0:6
0:7
0:8
0
0:9
1:0
;0:5
1:1
1:2
1:3
;1
;1
;0:5
0
0:5
1
1:5
2
Level curves of the weighted peak tracking error, given
Figure
11.4
by (11.2).
2
1:5
1
0:5
1:0
0
1:2
1:4
1:6
1:8
;0:5
2:0
2:2
;1
;1
;0:5
0
0:5
1
1:5
2
Level curves of the peak tracking error, for reference inputs
Figure
11.5
bounded and slew-rate limited by 1, given by (11.3).
254