Linear Controller Design: Limits of Performance by Stephen Boyd and Craig Barratt - HTML preview

Chapter 6

Geometry of Design

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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