Nonlinear Control Sy.. - Free

16 CHAPTER 1. INTRODUCTION
From here we conclude that
Figure 1.9: Trajectories for the system of Example 1.11
In the polar coordinate system p either increases exponentially, decreases exponentially,
or stays constant depending whether the real part a of the eigenvalues A1,2 is positive,
negative, or zero.
The phase angle increases linearly with a "velocity" that depends on the imaginary
part /3 of the eigenvalues \1,2-
In the yl - y2 coordinate system (1.22)-(1.23) represent an exponential spiral. If
a > 0, the trajectories diverge from the origin as t increases. If a < 0 on the other
hand, the trajectories converge toward the origin. The equilibrium [0, 0] in this case
is said to be a stable focus (if a < 0) or unstable focus (if a > 0).
If a = 0 the trajectories are closed ellipses. In this case the equilibrium [0, 0] is said
to be a center.
Example 1.11 Consider the following system:
0 -0 1
1 [ X2
(1.24)
The eigenvalues of the A matrix are A1,2 = 0 f 3f . Figure 1.9 shows that the trajectories
in this case are closed ellipses. This means that the dynamical system (1.24) is oscillatory.
The amplitude of the oscillations is determined by the initial conditions.
Example 1.12 Consider the following system: