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

CHAPTER 6 GEOMETRY OF DESIGN SPECIFICATIONS

Selecting a Submatrix or Entry of H

A simple but important example is the linear transformation that \selects" a sub-

matrix or entry from . More precisely, is the vector space of

transfer

H

V

p

q

matrices and is given by

L

( ) = T

L

H

E

H

E

w

z

where

n

p

and

n

q

the columns of

and

are unit vectors.

z

w

E

2

R

E

2

R

E

E

z

w

z

w

Thus ( ) is a submatrix of (or an entry of if = = 1) the unit vectors in

L

H

H

H

p

q

E

z

and

select the subsets of regulated variables and exogenous inputs, respectively.

E

w

If is a functional on

transfer matrices, a functional on is given by

p

q

H

( ) = ( ( )) = (

)

(6.2)

T

H

L

H

E

H

E

:

w

z

Informally, results from applying to a certain submatrix of

a convex (or

H

quasiconvex or a ne) functional of an entry or submatrix of yields a convex (or

H

quasiconvex or a ne) functional of .

H

To avoid cumbersome notation, we will often describe functionals or speci ca-

tions that take as argument only an entry or submatrix of , relying on the reader

H

to extend the functional to via (6.2).

H

Time Domain Responses

Let consist of scalar signals on +, and let be the transformation that maps

V

R

L

a transfer matrix into the unit step response of its

entry:

i

k

( ) =

L

H

s

where, for

0,

t

Z

( ) = 1 1

( )

H

j

!

ik

j

!

t

s

t

2

e

d!

:

j

!

;1

Since is linear, we see that a convex constraint on the step response of the

L

i

k

entry of a transfer matrix is a convex speci cation on .

H

A similar situation occurs when maps into its response to any particular

L

H

input signal part:

w

( ) =