Page 70 -

2.6 Stability Theory 51

as a result of Rayleigh-Ritz theorem.

2.6 Stability Theory

The first stability concept we study, concerns the behavior of free systems, or
equivalently, that of forced systems with a given input. In other words, we
study the stability of an equilibrium point with respect to changes in the
initial conditions of the system. Before doing so however, we review some
basic definitions. These definitions will be stated in terms of continuous,
nonlinear systems with the understanding that discrete, nonlinear systems
admit similar results and linear systems are but a special case of nonlinear
systems.
Let x e be an equilibrium (or fixed) state of the free continuous-time, possibly
time-varying nonlinear system

(2.6.1)

i.e. f(x e , t)=0, where x, f are n×1 vectors.
We will first review the stability of an equilibrium point x e with the
understanding that the stability of the state x(t) can always be obtained with
a translation of variables as discussed later. The stability definitions we use
can be found in [Khalil 2001], [Vidyasagar 1992].

DEFINITION 2.6–1 In all parts of this definition x e is an equilibrium point
at time t 0 , and ||.|| denote any function norm previously defined.

1. Stability: x e is stable in the sense of Lyapunov (SL) at t 0 , if starting
close enough to x e at t 0 , the state will always stay close to x e at later
times. More precisely, x e is SL at t 0 , if for any given >0, there exists
a positive ( , t 0 ) such that if

then

Copyright © 2004 by Marcel Dekker, Inc.