Specifications
In this chapter we explore some geometric properties that design specications
may have, and dene the important notion of a closed-loop convex design spec-
ication. We will see in the sequel that simple and eective methods can be
used to solve controller design problems that are formulated entirely in terms of
closed-loop convex design specications.
6.1
Design Specifications as Sets
will denote the set of all
closed-loop transfer matrices we may think
H
n
n
z
w
of as the set of all conceivable candidate transfer matrices for the given plant.
H
Recall from chapter 3 that design speci cations are boolean functions or predicates
on . With each design speci cation we will associate the set
of all transfer
H
D
H
i
i
matrices that satisfy it:
=
satis es
H
fH
2
H
j
H
D
g
:
i
i
Of course, there is a one-to-one correspondence between subsets of (i.e., sets
H
of transfer matrices) and design speci cations. For this reason we will also refer to
subsets of as design speci cations. Whether the predicate (e.g., os( ) 6%)
H
H
or subset (
os( )
6% ) is meant should be clear from the context, if it
fH
j
H
g
matters at all.
The boolean algebra of design speci cations mentioned in chapter 3 corresponds
exactly to the boolean algebra of subsets, with some of the correspondences listed
in table 6.1.
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