tive at is an element of ( ). (In fact, it is the only element of ( ).)
x
@
x
@
x
Scaling: If
0 and is convex, then a subgradient of
at is given by
w
w
x
, where is any subgradient of at .
w
g
g
x
Sum: If ( ) = ( ) + + m( ), where ... m are convex, then any
x
x
x
1
1
of the form = + + m is in ( ), where i
i( ).
g
g
g
g
@
x
g
2
@
x
1
Maximum: Suppose that
( ) = sup
( )
x
f
x
j
2
Ag
where each
is convex, and is any index set. Suppose that
is
A
2
A
ac
h
such that
( ) = ( ) (so that
( ) achieves the maximum). Then if
x
x
x
ac
h
ac
h
( ), we have
( ). Of course there may be several di erent
g
2
@
x
g
2
@
x
ac
h
indices that achieve the maximum we need only pick one.
A special case is when is the maximum of the functionals ... n, so that
1
= 1 ... . If ( ) = i( ), then any subgradient of i( ) is also a
A
f
ng
x
x
g
x
subgradient of ( ).
x
From these tools we can derive additional tools for determining a subgradient of
a weighted sum or weighted maximum of convex functionals. Their use will become
clear in the next section.
For quasiconvex functionals, we have the analogous tools:
Dierentiable functional: If is quasiconvex and di erentiable at , with
x
nonzero derivative, then its derivative at is a quasigradient of at .
x
x
Scaling: If
0 and is quasiconvex, then any quasigradient of at is
w
x
also a quasigradient of
at .
w
x
Maximum: Suppose that
( ) = sup
( )
x
f
x
j
2
Ag
where each is quasiconvex, and is any index set. Suppose that
A
2
A
ac
h
is such that
( ) = ( ). Then if is a quasigradient of
at , then
x
x
g
x
ac
h
ac
h
is a quasigradient of at .
g
x
Nested family: Suppose that is de ned in terms of a nested family of convex
sets, i.e., ( ) = inf
, where
whenever
(see
x
f
j
x
2
C
g
C
C
section 6.2.2). If T(
) = 0 de nes a supporting hyperplane to x at ,
(
)
g
z
;
x
C
x
then is a quasigradient of at .
g
x
(The sum tool is not applicable because the sum of quasiconvex functionals need
not be quasiconvex.)
13.4 COMPUTING SUBGRADIENTS
301
13.4
Computing Subgradients
In this section we show how to compute subgradients of several of the convex func-
tionals we have encountered in chapters 8{10. Since these are convex functionals
on , an in nite-dimensional space, the subgradients we derive will be linear func-
H
tionals on . In the next section we show how these can be used to calculate
H
subgradients in n when a nite-dimensional approximation used in chapter 15 is
R
made the algorithms of the next chapter can then be used.
In general, the convex functionals we consider will be functionals of some par-
ticular entry (or block of entries) of the closed-loop transfer matrix H. To simplify
notation, we will assume in each subsection that H consists of only the relevant
entry or entries.
13.4.1
An RMS Response
We consider the weighted 2 norm,
H
1=2
Z
(H) = 1 1
2
2
Sw(!) H(j!) d!
j
j
;1
with SISO H for simplicity (and of course, Sw(!) 0). We will determine a
subgradient of at the transfer function H0. If (H0) = 0, then the zero functional
is a subgradient, so we now assume that (H0) = 0. In this case is di erentiable
6
at H0, so our rst rule above tells us that our only choice for a subgradient is the
derivative of at H0, which is the linear functional sg given by
sg
Z
(H) =
1
1
2 (H
S
H
0)
w(!)
0(j!)H(j!) d!:
<
;1
(The reader can verify that for small H, (H0+H)
(H0)+ sg(H) the Cauchy-
Schwarz inequality can be used to directly verify that the subgradient inequal-
ity (13.3) holds.)
Using the subgradient for , we can nd a supporting hyperplane to the maxi-
mum RMS response speci cation (H)
.
There is an analogous expression for the case when H is a transfer matrix. For
(H) = H 2 and H0 = 0, a subgradient of at H0 is given by
k
k
6
sg
Z
(H) =
1
1
2 (H
(H
0)
0(j!) H(j!)) d!:
<
T
r
;1
13.4.2
Step Response Overshoot
We consider the overshoot functional,
(H) = sups(t) 1
t 0
;
302