Nonlinear Control Sy.. - Free

94 CHAPTER 3. LYAPUNOV STABILITY I. AUTONOMOUS SYSTEMS
which can be satisfied choosing a = 12, b = 1, c = 6, and d = 6. Using these values, we
obtain
V (x) = 12x2 - xi + 6x1x2 + 6x2
= 3(x1 + 2x2)2 + 9x2 + 3x2 - x1 (3.15)
V(x) = -6x2 - 30x2 + 6xi. (3.16)
So far, so good. We now apply Theorem 3.2. According to this theorem, if V(x) > 0 and
V < 0 in D - {0}, then the equilibrium point at the origin is "locally" asymptotically stable.
The question here is: What is the meaning of the word "local"? To investigate this issue,
we again check (3.15) and (3.16) and conclude that defining D by
D = {xEIIF2:-1.6 C2,
thus suggesting that any trajectory initiating within D will converge to the origin.
To check these conclusions, we plot the trajectories of the system as shown in Figure
3.6. A quick inspection of this figure shows, however, that our conclusions are incorrect.
For example, the trajectory initiating at the point x1 = 0, x2 = 4 is quickly divergent from
the origin even though the point (0, 4) E D. The problem is that in our example we tried
to infer too much from Theorem 3.2. Strictly speaking, this theorem says that the origin is
locally asymptotically stable, but the region of the plane for which trajectories converge to
the origin cannot be determined from this theorem alone. In general, this region can be a
very small neighborhood of the equilibrium point. The point neglected in our analysis is as
follows: Even though trajectories starting in D satisfy the conditions V(x) > 0 and V < 0,
thus moving to Lyapunov surfaces of lesser values, D is not an invariant set and there are
no guarantees that these trajectories will stay within D. Thus, once a trajectory crosses the
border xiI = f there are no guarantees that V(x) will be negative.
In summary: Estimating the so-called "region of attraction" of an asymptotically stable
equilibrium point is a difficult problem. Theorem 3.2 simply guarantees existence of a
possibly small neighborhood of the equilibrium point where such an attraction takes place.
We now study how to estimate this region. We begin with the following definition.
Definition 3.13 Let 1li(x, t) be the trajectories of the systems (3.1) with initial condition x
at t = 0. The region of attraction to the equilibrium point x, denoted RA, is defined by
RA={xED:1/i(x,t)-->xei as t --goo}.