Nonlinear Control Sy.. - Free

1.6. PHASE-PLANE ANALYSIS OF LINEAR TIME-INVARIANT SYSTEMS 15
Figure 1.8: System trajectories for the system of Example 1.10.
CASE 3: Systems with Complex Conjugate Eigenvalues
The most interesting case occurs when the eigenvalues of the matrix A are complex conjugate,
.\1,2 = a ±13. It can be shown that in this case a similarity transformation M can be
found that renders the following similar matrix:
M-1AM=Q=
Thus the transform system has the form
Q
/3 a
y1 = aY1 -)3Y2 (1.20)
y2 = i3y1 + aye
(1.21)
The solution of this system of differential equations can be greatly simplified by introducing
polar coordinates:
P - yi + y2
0 =
Converting (1.20) and (1.21) to polar coordinates, we obtain
which has the following solution:
yl
tan-1(1).
p = Poe"
(1.22)
0 = 0o + Qt. (1.23)