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44 Introduction to Control Theory

larger than or equal to the maximum value of f(t). L 1 denotes the set of signals
with finite absolute area, while L 2 denotes the set of signals with finite total
energy.
The following definition of the norms of vector functions is not unique. A
discussion of these norms is found in [Boyd and Barratt].

DEFINITION 2.5–6 Let denote the set of n×1 vectors of functions f i ,
each of which belonging to L p. The norm of is

for p [1, ∞) and

Some common norms of scalar signals u(t) that are persistent (i.e. lim t→ ∞
u(t)≠0) are the following:

1. which is valid for signals with
finite steady-state power.
2. ||u||∞=sup t≥0 |u(t) which is valid for bounded signals but is dependent on
outliers.
3. which measures the steady-state average
resource consumption.

For vector signals, we obtain:

1.

2.

3.

Note that

On the other hand, if signals do not persist, we may find their L 2 or L 1
norms as follows:

1. which measures the total resource consumption

Copyright © 2004 by Marcel Dekker, Inc.