Nonlinear Control Sy.. - Free

1.3. EQUILIBRIUM POINTS 5
1.3 Equilibrium Points
An important concept when dealing with the state equation is that of equilibrium point.
Definition 1.1 A point x = xe in the state space is said to be an equilibrium point of the
autonomous system
x = f(x)
if it has the property that whenever the state of the system starts at xe1 it remains at xe for
all future time.
According to this definition, the equilibrium points of (1.6) are the real roots of the equation
f (xe) = 0. This is clear from equation (1.6). Indeed, if
dx
x _
= f(xe) = 0
dt
it follows that xe is constant and, by definition, it is an equilibrium point. Equilibrium
point for unforced nonautonomous systems can be defined similarly, although the time
dependence brings some subtleties into this concept. See Chapter 4 for further details.
Example 1.3 Consider the following first-order system
± = r + x 2
where r is a parameter. To find the equilibrium points of this system, we solve the equation
r + x2 = 0 and immediately obtain that
(i) If r < 0, the system has two equilibrium points, namely x = ff.
(ii) If r = 0, both of the equilibrium points in (i) collapse into one and the same, and the
unique equilibrium point is x = 0.
(iii) Finally, if r > 0, then the system has no equilibrium points.
1.4 First-Order Autonomous Nonlinear Systems
It is often important and illustrative to compare linear and nonlinear systems. It will
become apparent that the differences between linear and nonlinear behavior accentuate as
the order of the state space realization increases. In this section we consider the simplest