output
w
k
k
kz
k
H
The size of a transfer matrix H can be measured by applying
Figure
5.2
a particular signal wpart, and measuring the size of the output with some
suitable signal norm
output.
k
k
5.1.2
Average Response Norm
A general method for measuring the size of a system, that directly takes into account
the response of the system to many input signals (and not just one particular input
signal), is to measure the average size of the response of to a speci c probability
H
distribution of input signals. If
output measures the size of the response, we
k
k
de ne
avg =
output
kH
k
E
kH
w
k
w
where
denotes expectation with respect to the distribution of input signals.
E
w
5.1.3
Worst Case Response Norm
Another general method for measuring the size of a system, that takes into account
the response of the system to many input signals, is to measure the worst case or
largest norm of the response of to a speci c collection of input signals. If output
H
k
k
measures the size of the response, we de ne
wc = sup
output
kH
k
kH
w
k
w
2W
where denotes the collection of input signals.
W
5.1.4
Gain of a System
An important special case of a worst case norm is a gain, de ned as the largest ratio
of the norm of the output to the norm of the input. If
is used to measure the
k
k
size of both the input and output signals, we de ne
gn = sup kHwk
(5.1)
kH
k
=0
:
kw
k
kw
k6
5.2 NORMS OF SISO LTI SYSTEMS
95
The gain
gn is therefore the maximum factor by which the system can scale the
kH
k
size (measured by the norm
) of a signal owing through it. The gain can also
k
k
be expressed as a worst case response norm:
gn = sup
kH
k
1 kHwk:
kw
k
If the transfer matrix is not square, we cannot really use the same norm to
H
measure the input and output signals, since they have di erent numbers of com-
ponents. In such cases we rely on our naming conventions to identify the \same"
norm to be used for the input and the output. For example, the RMS gain of a
2 3 transfer matrix is de ned by (5.1), where the norm in the numerator is the
RMS norm of a vector signal with 2 components, and the norm in the denominator
is the RMS norm of a vector signal with 3 components. It is also possible to de ne
a more general gain with di erent types of norms on the input and output, but we
will not use this generalization.
5.2
Norms of SISO LTI Systems
In this section we describe various norms for single-input, single-output (SISO)
systems.
5.2.1
Peak-Step
Our rst example of a norm of a system is from the rst paradigm: the size of
its response to a particular input. The particular input signal is a unit step we
measure the size of the response by its peak norm. We de ne:
pk step =
kH
k
ksk
1
where ( ) denotes the step response of we will refer to
pk step as the peak-step
s
t
H
kH
k
norm of . This would be an appropriate measure if, say, represents a set-point
H
w
command in a control system (a signal that might be expected to change values
only occasionally), and represents some actuator signal, say, a motor voltage. In
z
this case,
pk step (multiplied by the maximum possible changes in the set-point
kH
k
command) would represent a good approximation of the maximum motor voltage
(due to set-point changes) that might be encountered in the operation of the system.
5.2.2
RMS Response to a Particular Noise Input
A common measure of the size of a transfer function is the RMS value of its output
when its input is some particular stationary stochastic process. Suppose that the
particular input has power spectral density ( ), and is stable, meaning that
w
S
!
H
w
96