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5.5.1
Some General Inequalities
For convenience we will consider norms of SISO systems, i.e., norms of transfer
functions.
Since
pk step and
are each worst case peak norms over input signal
k
k
k
k
1
sets that have peaks no greater than one (a unit step in the rst case, and unit
amplitude sinusoids in the second), it follows that these norms will be no larger
than the peak gain of the system,
pk gn
kH
k
kH
k
1
pk gn
pk step
kH
k
kH
k
:
From the de nition of the Hankel norm, we can see that it cannot exceed the 2
L
gain, which we saw is the
norm, so we have
H
1
hankel
kH
k
kH
k
:
1
It is possible for a system to have a small RMS gain, but a large peak gain.
However, if has poles then the peak gain of can be bounded in terms of the
H
n
H
Hankel norm, and therefore, the RMS gain:
pk gn (2 + 1)
hankel (2 + 1)
(5.33)
k H
k
kH
k
n
kH
k
n
kH
k
:
1
1
This means that for low order systems, at least, the peak gain, RMS gain, and
Hankel norm cannot di er too much.
5.5.2
Approximating Norms: an Example
Consider the worst case norm described in section 5.1.3, with the amplitude bound
and slew-rate limit each equal to one:
wc = sup
1 _
1
kH
k
fkH
w
k
j
kw
k
kw
k
g
:
1
1
1
Roughly speaking, the bound and slew-rate limit establish a bandwidth limit of
about one for the input signal . We might therefore suspect that we can approxi-
w
mate
wc by a weighted peak gain, where the weight is some appropriate lowpass
kH
k
lter with a bandwidth near one:
wc
pk gn
kH
k
kH
W
k
:
We will show that this intuition is correct: for ( ) = 1 (2 + 1), we have
W
s
=
s
pk gn
wc 3
pk gn
(5.34)
kH
W
k
kH
k
kH
W
k
for all transfer functions . Thus, p3
pk gn approximates
wc to within
H
kH
W
k
kH
k
4 8dB.
:
To establish (5.34), suppose that 0 is a signal with
w
0
= 1
0
=
pk gn
kw
k
kH
W
w
k
kH
W
k
1
1
5.6 STATE-SPACE METHODS FOR COMPUTING NORMS
119
(such a 0 can be shown to exist). Let 1 =
0. Then we have
w
w
W
w
1
pk gn = 1
kw
k
kW
k
1
and
_1 =
s
0
s
= 1
kw
k
:
1
2 + 1w
2 + 1 pk gn
s
s
1
Therefore 1 satis es the amplitude limit
1 and slew-rate limit _1
1,
w
kw
k
kw
k
1
1
so we must have
1
wc. Since
1
=
pk gn, this means that
kH
w
k
kH
k
kH
w
k
kH
W
k
1
1
pk gn
wc
kH
W
k
kH
k
:
We now establish the right-hand inequality in (5.34). Consider 2 such that
w
2
1
_2
1
2
=
wc
kw
k
kw
k
kH
w
k
kH
k
1
1
1
(such a 2 can be shown to exist). De ne 3 by
w
w
3 = 2 _2 + 2 =
1
;
2
w
w
w
W
w
:
Then 3
2 _2 + 2
= 3, and
3
=
2
=
wc.
kw
k
kw
k
kw
k
kH
W
w
k
kH
w
k
kH
k
1
1
1
1
1
Hence
pk gn
3
wc
kH
W
w
k
1
= kHk
kH
W
k
3
3
kw
k
1
which establishes the right-hand inequality in (5.34).
This example illustrates an interesting tradeo in the selection of a norm. While
wc may better characterize the \size" of
in a given situation, the approxima-
kH
k
H
tion
pk gn (or even
see the previous section) may be easier to work
kH
W
k
kH
W
k
1
with, e.g., compute. If
1 and _
1 is only an approximate model of
kw
k
kw
k
1
1
possible 's, and the speci cation
need only hold within a factor of two
w
kz
k
1
or so, then
pk gn p3
kH
W
k
would be an appropriate speci cation.
5.6
State-Space Methods for Computing Norms
The 2, Hankel, and
norms, and the -entropy of a transfer matrix are readily
H
H
1
computed from a state-space realization methods for computing some of the other
norms we have seen are described in the Notes and References. In this section we
assume that
_ =
+
=
x
Ax
B
w
z
C
x
120