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72 Introduction to Control Theory

4. Let

and pick a Lyapunov function candidate

so that

so that the origin is SL.

Lyapunov Theorems may be used to design controllers that will stabilize a
nonlinear system such as a robot. In fact, if one chooses a Lyapunov function
candidate V(t, x), then finding its total derivative V(t, x) will exhibit an
explicit dependence on the control signal. By choosing the control signal to
make V(t, x) negative definite, stability of the closed-loop system is
guaranteed. Unfortunately, it is not always easy to guarantee the global
asymptotic stability of an equilibrium point using Lyapunov Theorem. This
is due to the fact, that V(t, x) may be shown to be negative but not necessarily
negative-definite. If the open-loop system were autonomous, Lyapunov theory
is greatly simplified as shown in the next section.

The Autonomous Case

Suppose the open-loop system is not autonomous, i.e. is not explicitly
dependent on t, then a time-independent Lyapunov function candidate V(x)
may be obtained and the positive definite conditions are greatly simplified as
described next.

LEMMA 2.7–1: A time invariant continuous function V(x) is positive definite
if V(0)=0 and V(x)>0 for x≠0. It is locally positive definite if the above holds
in a neighborhood of the origin

Note that the condition that V(0)=0 is not necessary and that as long as
V(0) is bounded above the Lyapunov results hold without modification.

Copyright © 2004 by Marcel Dekker, Inc.