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2.7 Lyapunov Stability Theorems 67

and verify that z e =0 is the desired equilibrium point of the modified system if
x e =x d is the desired equilibrium trajectory of the robot.

2.7 Lyapunov Stability Theorems

Lyapunov stability theory deals with the behavior of unforced nonlinear
systems described by the differential equations

(2.7.1)

where without loss of generality, the origin is an equilibrium point of (2.7.1).
It may seem to the reader that such a theory is not needed since all we had
to do in the examples of the previous section is to solve the differential
equations, and study the time evolution of a norm of the state vector. There
are at least two reasons why Lyapunov theory is needed. The first is that
Lyapunov theory will allow us to determine the stability of a particular
equilibrium point without actually solving the differential equations. This,
as is well known to any student of nonlinear differential equations, is a
large saving. The second and related reason for using Lyapunov theory is
that it provides us with qualitative results to the stability questions, which
may be used in designing stabilizing controllers of nonlinear dynamical
systems.
We shall first assume that any necessary conditions for (2.7.1) to have a
unique solution are satisfied [Khalil 2001], [Vidyasagar 1992]. The unique
solution corresponding to x(t 0)=x 0 is x(t, t 0, x 0) and will be denoted simply as
x(t). Before we actually introduce Lyapunov’s theorems, we review certain
classes of functions which will simplify the statement of Lyapunov theorems.

Functions of Class K
Consider a continuous function α:ℜ→ℜ

DEFINITION 2.7–1 We say that a belongs to class K, if

1. α(0)=0
2. α(x)>0, for all x>0
3. α is nondecreasing, i.e. α(x 1) α(x 2) for all x 1>x 2.

Copyright © 2004 by Marcel Dekker, Inc.