Nonlinear Control Sy.. - Free

1.6. PHASE-PLANE ANALYSIS OF LINEAR TIME-INVARIANT SYSTEMS 17
Figure 1.10: Trajectories for the system of Example 1.12
±2l_r0.1
A1,2 0.5j -1 X2 X,
The eigenvalues of the A matrix are = 0.5 ± j; thus the origin is an unstable focus.
Figure 1.10 shows the spiral behavior of the trajectories. The system in this case is also
oscillatory, but the amplitude of the oscillations grow exponentially with time, because of
the presence of the nonzero o: term.
The following table summarizes the different cases:
Eigenvalues Equilibrium point
A1, A2 real and negative stable node
A1, A2 real and positive unstable node
A1, A2 real, opposite signs saddle
A1, A2 complex with negative real part stable focus
A1, A2 complex with positive real part unstable focus
A1, A2 imaginary
center
As a final remark, we notice that the study of the trajectories of linear systems about
the origin is important because, as we will see, in a neighborhood of an equilibrium point
the behavior of a nonlinear system can often be determined by linearizing the nonlinear
equations and studying the trajectories of the resulting linear system.