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46 Introduction to Control Theory

where T<∞. We also define the norm on L p e as

Similar definitions are available for and the interested reader is referred to
[Boyd and Barratt], [Desoer and Vidyasagar 1975].

EXAMPLE 2.5–7: Extended L p Spaces

The function f(t)=t belongs to L pe for any p∈[1, ∞] but not to L p .

System Norms
We would like next to study the effect of a multi-input-multi-output (MIMO)
system on a multidimensional signal. In other words, what happens to a
time-varying vector u(t) as it passes through a MIMO system H? Let H be a
system with m inputs and l outputs, so that its output to the input u(t) is
given by

y(t)=(Hu)(t)

We say that H is L p stable if Hu belongs to whenever u belongs to and
there exists finite constants >0 and b such that

If p=∞, the system is said to be bounded-input-bounded-output (BIBO) stable.

DEFINITION 2.5–7 The L p gain of the system H is denoted by p (H) and is
the smallest such that a finite b exists to verify the equation.

Therefore, the gain p characterizes the amplification of the input signal as
it passes through the system. The following lemma characterizes the gains of
linear systems and may be found in [Boyd and Barratt].

LEMMA 2.5–2: Given the linear system H such that an input u(t) results in
an output and suppose H is BIBO
stable, then

Copyright © 2004 by Marcel Dekker, Inc.