The sets of transfer matrices that satisfy specications are generally innite-
dimensional. In this chapter we consider our standard example described in
section 2.4 with an additional two-dimensional ane specication. This allows us
to visualize a two-dimensional \slice" through the various specications we have
encountered. The reader can directly see, for this example, that specications we
have claimed are convex are indeed convex.
Recall from section 2.4 that (a), (b), and (c) are the closed-loop transfer ma-
H
H
H
trices resulting from the three controllers (a), (b), and (c) given there. The
K
K
K
closed-loop a ne speci cation
o
slice = n
=
(a) + (b) + (1
) (c) for some
H
H
H
H
H
;
;
H
2
R
requires to lie on the plane passing through these three transfer matrices. The
H
speci cation slice has no practical use, but we will use it throughout this chapter
H
to allow us to plot two-dimensional \slices" through other (useful) speci cations.
Figure 11.1 shows the subset 1
2, 1
2 of slice. Most plots that
;
;
H
we will see in this chapter use this range. Each point in gure 11.1 corresponds to
a closed-loop transfer matrix for example, (a) corresponds to the point = 1,
H
= 0, (b) corresponds to the point = 0, = 1, and (c) corresponds to the
H
H
point = 0, = 0. Also shown in gure 11.1 are the points
0 6 (a) + 0 3 (b) + 0 1 (c) and
0 2 (a) 0 6 (b) + 1 8 (c)
:
H
:
H
:
H
;
:
H
;
:
H
:
H
:
Each point in gure 11.1 also corresponds to a particular controller, although we
will not usually be concerned with the controller itself. The controller that realizes
the closed-loop transfer matrix
(a) + (b) + (1
) (c)
H
H
;
;
H
can be computed by two applications of equation (7.10) from chapter 7.
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