1
@
I
@
0:8
@
~13
s
@
I
@
@
0:6
()
( 13 + ~13) 2
s
s
=
t
@
I
@
s
@
0:4
13
s
0:2
0
;0:2
0
0:1
0:2
0:3
0:4
0:5
0:6
0:7
0:8
0:9
1
t
The set in (6.6) is convex, since if the step responses 13 and
Figure
6.6
V
s
~13 do not exceed 1.1, then their average, ( 13 +~13) 2, also does not exceed
s
s
s
=
1.1. By the argument in section 6.2.3, the specication os is convex.
H
6.4.3
A Quasiconvex Functional
We will see in chapter 8 that several important functionals are quasiconvex, for
example, those relating to settling time and bandwidth of a system. We describe
one such example here.
Consider the stability degree of a transfer matrix, de ned by
stab deg( ) = max
is a pole of
H
f<p
j
p
H
g
:
The functional stab deg is quasiconvex since, for each ,
stab deg( )
=
for
fH
j
H
g
fH
j
kH
k
<
1
<
;
g
1
(recall that
is the -shifted
norm described in section 5.2.7), and we
k
k
H
1
1
saw above that the latter speci cation is a ne. stab deg is not convex, however:
for most values of , we have
stab deg(
+ (1 ) ~) = maxn stab deg( ) stab deg( ~)o
H
;
H
H
H
:
6.5
Implications for Tradeoffs and Optimization
When the design speci cations are convex, and more generally, when a family of
design speci cations is given by convex functional inequalities, we can say much
more about the concepts introduced in chapter 3.
6.6 CONVEXITY AND DUALITY
139
6.5.1
Performance Space Geometry
Suppose that in the multicriterion optimization problem described in section 3.5,
the hard constraint hard and the objective functionals 1 ...
are convex. Then
D
L
the region of achievable speci cations in performance space,
= n
a
a
L
hard
...
is achievable o
(6.7)
1
L
A
a
2
R
D
^
D
^
^
D
1
L
is also convex.
To see this, suppose that and ~ each correspond to achievable speci cations.
a
a
Then there are transfer matrices and ~ that satisfy hard and also, for each ,
H
H
D
k
1
,
k
L
( )
( ~) ~
H
a
H
a
:
k
k
k
k
Now suppose that 0
1. The transfer matrix
+(1 ) ~ satis es the hard
H
;
H
constraint hard, since hard is convex, and also, for each , 1
,
D
D
k
k
L
( + (1 ) ~)
+ (1 )~
H
;
H
a
;
a
k
k
k
since the functional is convex. But this means that the speci cation +(1 )~
a
;
a
k
corresponds to an achievable speci cation, which veri es that is convex.
A
An important consequence of the convexity of the achievable region in perfor-
mance space is that every Pareto optimal speci cation is optimal for the classical
optimization problem with weighted-sum objective, for some nonnegative weights
(c.f. gure 3.8). Thus the speci cations considered in multicriterion optimization or
classical optimization with the weighted-sum or weighted-max objectives are exactly
the same as the Pareto optimal speci cations.
In the next section we discuss another important consequence of the convexity
of (which in fact is equivalent to the observation above).
A
6.6
Convexity and Duality
The dual function de ned in section 3.6.2 is always concave, meaning that
is
;
convex, even if the objective functionals and the hard constraint are not convex. To
see this, we note that for each transfer matrix that satis es hard, the function
H
D
of de ned by
H
( ) = 1 1( )
( )
;
H
;
;
H
H
L
L
is convex (indeed, linear). Since
can be expressed as the maximum of this family
;
of convex functions, i.e.,
( ) = max
( )
satis es hard
(6.8)
;
f
j
H
D
g
H
it is also a convex function of (see section 6.2.1).
140