Many of the goals of controller design can be expressed in terms of the size of
various signals, e.g., tracking error signals should be made \small", while the
actuator signals should not be \too large". In this chapter we explore some of the
ways this notion of the size of a signal can be made precise, using norms, which
generalize the notion of Euclidean length.
4.1
Definition
There are many ways to describe the size of a signal or to express the idea that a
signal is small or large. For example, the fraction of time that the magnitude of a
signal exceeds some given threshold can serve as a measure of the size of the signal
we could de ne \small" to mean that the threshold is exceeded less than 1% of the
time. Among the many methods to measure the size of a signal, those that satisfy
certain geometric properties have proven especially useful. These measures of size
are called norms.
The geometric properties that norms satisfy are expressed in the framework of
a vector space roughly speaking, we have a notion of how to add two signals, and
how to multiply or scale a signal by a scalar (see the Notes and References).
Suppose is a vector space and :
+
. is a norm
De
nition
4.1:
V
V
!
R
f1g
on if it satises
V
Nonnegativity: ( ) 0,
v
Homogeneity: for ( )
, ( ) =
( ),
v
<
1
v
j
j
v
Triangle inequality: ( + )
( ) + ( ),
v
w
v
w
for all
and
.
2
R
v
w
2
V
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