Nonlinear Control Sy.. - Free

1.4. FIRST-ORDER AUTONOMOUS NONLINEAR SYSTEMS
Example 1.4 Consider the system
Figure 1.2: The system i = cos x.
th = cosx
2 / \ 5ir/2 x
To analyze the trajectories of this system, we plot i versus x as shown in Figure 1.2. From
the figure we notice the following:
The points where ± = 0, that is where cos x intersects the real axis, are the equilibrium
points o f (1.11). Thus, all points o f the form x = ( 1 + 2k)7r/2, k = 0, +1, ±2, are
equilibrium points of (1.11).
Whenever ± > 0, the trajectories move to the right hand side, and vice versa. The
arrows on the horizontal axis indicate the direction of the motion.
From this analysis we conclude the following:
1. The system (1.11) has an infinite number of equilibrium points.
2. Exactly half of these equilibrium points are attractive or stable, and the other half are
unstable, or repellers.
The behavior described in example 1.4 is typical of first-order autonomous nonlinear systems.
Indeed, a bit of thinking will reveal that the dynamics of these systems is dominated
by the equilibrium points, in the sense that the only events that can occur to the trajectories
is that either (1) they approach an equilibrium point or (2) they diverge to infinity.
In all cases, trajectories are forced to either converge or diverge from an equilibrium point
monotonically. To see this, notice that ± can be either positive or negative, but it cannot
change sign without passing through an equilibrium point. Thus, oscillations around an
equilibrium point can never exist in first order systems. Recall from (1.10) that a similar
behavior was found in the case of linear first-order systems.
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