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82 Introduction to Control Theory

EXAMPLE 2.8–2: BIBO Stability

Consider the system

y(t)=u (t)
2
It is BIBO stable since for any input u(t) such that |u(t)|<M<∞, the output is
2
bounded by M .

2.9 Advanced Stability Results

In this section we review some advanced stability concepts. These results
will be used in showing the closed-loop stability of systems when robust or
adaptive controllers are used. If the reader is only interested in implementing
these controllers, this section may be skipped. On the other hand, anyone
interested in designing new controllers should be aware of the results presented
here.

Passive Systems

Given a nonlinear system shown in Figure 2.9.1, we are interested in studying
the stability of such a system based on input-output measurements only.
Motivated by energy concepts in network theory, such as

[Stored Energy]=[External Power Input]+[Internal power Generation]

One can study the internal stability of all kinds of systems. In general,
T
the external power input is the scalar product y u of an input effort or
flow u and an output flow or effort y. The last equation then takes on the
form

(2.9.1)

In many cases (e.g. isolated system) g(t)=0 and one can use the stored
Energy

(2.9.2)

Copyright © 2004 by Marcel Dekker, Inc.