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

CHAPTER 6 GEOMETRY OF DESIGN SPECIFICATIONS

6.2.2

Quasiconvex Functionals

A functional on is quasiconvex if for each

, the func-

De

nition

6.5:

H

2

R

tional inequality specication

( )

is convex.

fH

j

H

g

An equivalent de nition of quasiconvexity, that has a form similar to the de -

nition of convexity, is: whenever , ~

and

0 1],

H

H

2

H

2

( + (1 ) ~) max ( ) ( ~)

H

;

H

f

H

H

g:

From de nition 6.4 we can see that every convex functional is quasiconvex.

The values of a quasiconvex functional along a line in is plotted in gure 6.4

H

note that this functional is not convex. A quasiconvex function of one variable is

called unimodal, since, roughly speaking, it cannot have two separate regions where

it is small.

~

(H

)

~ )

;

)H

;

;

(1;+

(H

)

(

;

H

;

=

0

=

1

@

;

@

R

;

A functional is quasiconvex if for every pair of transfer ma-

Figure

6.4

trices and ~ the graph of along the line

+(1 ) ~ lies on or below

H

H

H

;

H

the larger of ( ) and ( ~), i.e., in the shaded region.

H

H

A positive weighted maximum of quasiconvex functionals is quasiconvex, but a

positive-weighted sum of quasiconvex functionals need not be quasiconvex.

There is a natural correspondence between quasiconvex functionals and nested

families of convex speci cations, i.e., linearly ordered parametrized sets of speci ca-

tions. Given a quasiconvex functional , we have the family of functional inequality

speci cations given by

=

( )

. This family is linearly ordered:

H

fH

j

H

g

H

is stronger than

if

.

H

6.2 AFFINE AND CONVEX SETS AND FUNCTIONALS

133

Conversely, suppose we are given a family of convex speci cations , indexed

H

by a parameter

, such that

is stronger than

if

(so the family of

2

R

H

H

speci cations is linearly ordered). The functional

family( ) = inf

(6.1)

H

f

j

H

2

H

g

( family( ) = if

for all ) simply assigns to a transfer matrix the index

H

1

H

62

H

H

corresponding to the tightest speci cation that it satis es, as shown in gure 6.5.

This functional family is easily shown to be quasiconvex, and its sub-level sets are

essentially the original speci cations:

family( )

+

H

fH

j

H

g

H

for any positive .

(5)

(3)

(4) H

H

H

(2)

H

(1)

H

q

@

I

@

H

Five members of a nested family of convex sets are shown.

Figure

6.5

Such nested families dene a quasiconvex function by (6.1) for the given

transfer matrix , we have family( ) = 2.

H

H

6.2.3

Linear Transformations

Convex subsets of and convex functionals on are often de ned via linear trans-

H

H

formations. Suppose that is a vector space and :

is a linear function.

V

L

H

!

V

If is a convex (or a ne) subset of , then the subset of de ned by

V

V

H

1 =

( )

H

f

H

j

L

H

2

V

g

is convex (or a ne). Similarly if is a convex (or quasiconvex or a ne) functional

on , then the functional on de ned by

V

H

( ) = ( ( ))

H

L

H

is convex (or quasiconvex or a ne, respectively).

These facts are easily established from the de nitions above we mention a few

important examples.

134