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

CHAPTER 6 GEOMETRY OF DESIGN SPECIFICATIONS

~ )

( )

)

H

H

;

;

(1;+

( ~)

H

;

(

;

H

= 0

= 1

;

;

;

;

A functional is ane if for every pair of transfer matrices

Figure

6.2

H

and ~ the graph of ( + (1

) ~) versus is a straight line passing

H

H

;

H

through the points (0 ( ~)), (1 ( )).

H

H

~ ))

( )

H

H

;

;

(1;+

( ~)

H

;

(

;

H

= 0

= 1

@

;

@

R

;

A functional is convex if for every pair of transfer matrices

Figure

6.3

and ~ the graph of along the line

+ (1

) ~ lies on or below a

H

H

H

;

H

straight line through the points (0 ( ~)), (1 ( )), i.e., in the shaded

H

H

region.

6.2 AFFINE AND CONVEX SETS AND FUNCTIONALS

131

Ane implies convex. If a set or functional is a ne, then it is convex: being

a ne is a stronger condition than convex. If the functionals and

are

;

both convex, then is a ne.

Intersections. Intersections of a ne or convex sets are a ne or convex, re-

spectively.

Weighted-sum Functional. If the functionals 1 ...

are convex, and 1

L

0, ...,

0, then the weighted-sum functional

L

wt sum( ) = 1 1( ) +

+

( )

H

H

H

L

L

is convex (see section 3.6.1).

Weighted-max Functional. If the functionals 1 ...

are convex, and 1

L

0, ...,

0, then the weighted-max functional

L

wt max( ) = max 1 1( ) ...

( )

H

f

H

H

g

L

L

is convex (see section 3.6.3).

The last two properties can be generalized to the integral of a family of convex

functionals and the maximum of an in nite family of convex functionals. Suppose

that for each

( is an arbitrary index set), the functional is convex. Then

2

I

I

the functional

( ) = sup

( )

H

f

H

j

2

I

g

is convex.

We now describe some of the relations between sets and functionals that are

convex or a ne. A functional equality speci cation formed from an a ne functional

de nes an a ne set: if is a ne and

, then

2

R

( ) =

fH

j

H

g

is a ne. Similarly, if is convex and

, then the functional inequality speci -

2

R

cation

( )

fH

j

H

g

called a sub-level set of , is convex.

The converse is not true, however: there are functionals that are not convex,

but every sub-level set is convex. Such functionals are our next topic.

132