Nonlinear Control Sy.. - Free

22 CHAPTER 1. INTRODUCTION
Figure 1.14: (a) Three-dimensional view of the trajectories of Lorenz' chaotic system; (b)
two-dimensional projection of the trajectory of Lorenz' system.
Theorem 1.1 [76] Consider the two dimensional system
x=f(x)
where f : IR2 -> j 2 is continuously differentiable in D C 1R2, and assume that
(1) R C D is a closed and bounded set which contains no equilibrium points of x = f (x).
(2) There exists a trajectory x(t) that is confined to R, that is, one that starts in R and
remains in R for all future time.
Then either R is a closed orbit, or converges toward a closed orbit as t - oo.
According to this theorem, the Poincare-Bendixson theorem predicts that, in two dimensions,
a trajectory that is enclosed by a closed bounded region that contains no equilibrium
points must eventually approach a limit cycle. In higher-order systems the new dimension
adds an extra degree of freedom that allows trajectories to never settle down to an
equilibrium point or closed orbit, as seen in the Lorenz system.
1.9 Examples of Nonlinear Systems
We conclude this chapter with a few examples of "real" dynamical systems and their
nonlinear models. Our intention at this point is to simply show that nonlinear equation
arise frequently in dynamical systems commonly encountered in real life. The examples in
this section are in fact popular laboratory experiments used in many universities around
the world.