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
( ) =