11.6
Nonconvex Design Specifications
11.6.1
Controller Stability
Figure 11.22 shows the subset of slice that is achieved by an open-loop stable
H
controller, i.e.,
]T
H(a) + H(b) + (1
)H(c) is
;
;
achieved by a stable controller K
:
(11.27)
From gure 11.22, we see that (11.27) is a nonconvex subset of slice. We conclude
H
that a speci cation requiring open-loop controller stability,
P
1P
k stab = H
H = Pzw + PzuK(I yuK); yw
;
H
for some stable K that stabilizes P
is not in general convex.
3
2:5
2
1:5
1
0:5
0
;0:5
;1
;5
;4
;3
;2
;1
0
1
2
3
Region where the closed-loop transfer matrix is achieved
Figure
11.22
H
by a controller that is open-loop stable. It is not convex.
11.7
A Weighted-Max Functional
Consider the functional
'wt max( ) = max 'pk trk( ) 0:5'max sens( ) 15'rms yp(
)
f
g
11.7 A WEIGHTED-MAX FUNCTIONAL
269
where the functions 'pk trk, 'max sens, and 'rms yp are given by (11.2), (11.18),
and (11.6). The level curves of the function 'wt max(
) are shown in gure 11.23.
The function 'wt max will be used for several examples in chapter 14.
2
4:0
3:5
1:5
3:0
2:5
1
2:0
1:5
0:5
0
0:8
0:9
1:0
;0:5
1:1
1:2
1:3
1:4
;1
;1
;0:5
0
0:5
1
1:5
2
The level curves of wt max(
).
Figure
11.23
'
270