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is convex. Let and ~ be any two transfer matrices, and let 0
1. Now,
H
H
using the triangle inequality and homogeneity property for norms, together with
the fact that and 1
are nonnegative, we see that
;
( + (1 ) ~) =
xx + (1
) ~xx
des
xx
H
;
H
k
H
;
H
;
H
k
= ( xx
des
xx ) + (1
)( ~xx
des
xx )
k
H
;
H
;
H
;
H
k
( xx
des
xx ) + (1
)( ~xx
des
xx )
k
H
;
H
k
k
;
H
;
H
k
=
xx
des
xx + (1
) ~xx
des
xx
kH
;
H
k
;
kH
;
H
k
= ( ) + (1 ) ( ~)
H
;
H
so the functional is convex. Since is convex, the norm-bound speci cation (6.3)
is convex. In chapters 8{10 we will see that many speci cations can be expressed
in the form (6.3), and are therefore closed-loop convex. (We note that the entropy
functional de ned in section 5.3.5 is also convex, although it is not a norm.)
An important variation on the norm-bound speci cation (6.3) is the speci cation
xx
(6.4)
kH
k
<
1
which requires that an entry or submatrix of have a nite norm, as measured by
H
. This speci cation is a ne, since if xx and ~xx each satisfy (6.4) and
,
k
k
H
H
2
R
then we have
xx + (1
) ~xx
xx + 1
~xx
k
H
;
H
k
j
jkH
k
j
;
jkH
k
<
1
so that xx + (1 ) ~xx also satis es (6.4).
H
;
H
If we use the
norm for
, (6.4) is the speci cation that xx be stable,
H
k
k
H
1
i.e., the poles of xx have negative real parts. Similarly, if we use the -shifted
H
a
H
1
norm as
, (6.4) is the speci cation that the poles of xx have real parts less
k
k
H
than . These speci cations are therefore a ne.
;a
6.4
Some Examples
In chapters 7{10 we will encounter many speci cations and functionals that are
a ne or convex in most cases we will simply state that they are a ne or convex
without a detailed justi cation. In this section we consider a few typical speci ca-
tions and functionals for our standard example plant (see section 2.4), and carefully
establish that they are a ne or convex.
6.4.1
An Affine Specification
Consider the speci cation that the closed-loop transfer function from the reference
input to p have unity gain at = 0, so that a constant reference input results in
r
y
!
6.4 SOME EXAMPLES
137
p = in steady-state. The set of 2 3 transfer matrices that corresponds to this
y
r
speci cation is
dc =
13(0) = 1
(6.5)
H
f
H
j
H
g
:
We will show that this set is a ne. Suppose that
~
dc, so that 13(0) =
H
H
2
H
H
~13(0) = 1, and
. Then the transfer matrix
=
+ (1 ) ~ satis es
H
2
R
H
H
;
H
13(0) =
13(0) + (1
) ~13(0) = + (1 ) = 1
H
H
;
H
;
and hence
dc.
H
2
H
Alternatively, we note that the speci cation dc can be written as the functional
H
equality constraint
dc =
dc( ) = 1
H
f
H
j
H
g
where dc( ) = 13(0) is an a ne functional.
H
H
6.4.2
Convex Specifications
Consider the speci cation act e introduced in section 3.1: \the RMS deviation of
D
due to the sensor and process noises is less than 0.1". The corresponding set of
u
transfer matrices is
act e =
act e ( ) 0 1
H
fH
j
H
:
g
where we de ned the functional
1 2
Z
=
1
act e ( ) =
1
;
21( ) 2 proc( ) + 22( ) 2 sensor( )
H
2
jH
j
!
j
S
!
jH
j
!
j
S
!
d!
;1
and proc and sens are the power spectral densities of the noises proc and sensor,
S
S
n
n
respectively.
To show that act e is a convex functional we rst express it as a norm of the
submatrix 21 22] of :
H
H
H
2
1 0 3
T
act e ( ) =
0
0 1
proc
0
W
4
5
H
1
H
0 0
0
sensor
W
2
where proc( ) 2 = proc( ) and sensor( ) 2 = sensor( ). From the results
jW
j
!
j
S
!
jW
j
!
j
S
!
of sections 6.2.3 and 6.3.2 we conclude that act e is a convex functional, and
therefore act e is a convex speci cation.
H
As another example, consider the speci cation os introduced in section 3.1:
D
\the step response overshoot from the command to p is less than 10%". We
y
can see that the corresponding set of transfer matrices, os, is convex, using the
H
argument in section 6.2.3 with = 1, = 3 and the convex subset of scalar signals
i
k
= : +
( ) 1 1 for
0
(6.6)
V
f
s
R
!
R
j
s
t
:
t
g
:
This is illustrated in gure 6.6.
138