Page 73 -

54 Introduction to Control Theory

GAS for any t 0 and the system is said to be GAS in this case, since it
can only have one equilibrium point x e . See Figure 2.6.4.

EXAMPLE 2.6–1: Stability of Various Systems
1. Consider the scalar time-varying system given by

the solution of this equation for all t t 0 is

The equilibrium point is located at x e=y e=0. Let us use the 1-norm
given by |y| and suppose that our aim is to keep |y(t)|< for all t t 0.
It can be seen that our objective is achieved if

The origin is therefore a stable equilibrium point of this system.
This is further illustrated in Figure 2.6.5

2. The damped pendulum system has many equilibrium points as
described in Example 2.3.1. It can be shown that the equilibrium
point located at the origin of the state-space is unstable. This is
illustrated in Figure 2.6.6 where it is seen that no matter how close
to the origin the initial state is, the norm of x(t) can not be pre-
specified. On the other hand, note that the two equilibrium points
at [ , 0] are stable.

Figure 2.6.4: Global asymptotic stability of x e at t 0

Copyright © 2004 by Marcel Dekker, Inc.