To show how the phase plane looks for linear and nonlinear systems, 2 applets are shown below, the first is for a linear 3rd order system and the second for a nonlinear 3rd order system with cubic nonlinearity.

for the linear system the eg parameters (K, 1.0, 1.0, 1.0, na, 3.0) show the effect of K on the phase plane. For K = 8 2 of the closed loop poles are on the imaginary axis, the system is oscillatory and the phase plane plot(green) appears as an ellipse.

For (7.2, 1.0, 1.0, 1.0, na, 8.0), the system is stable and the phase plane plot converges from the ellipse to the origin.

For (8.5, 1.0, 1.0, 1.0, na, 5.0), the system is unstable and the phase plane plot diverges from ellipse outwards.

The applet below shows the phase plane for a 3rd order system with a cubic nonlinearity.

The parameters (8.0, 1.0, 1.0, 1.0, 0.0, 3.0) show the oscillatory response of the linear system.

(8.0, 1.0, 1.0, 1.0, -0.5, 3.0) show the cubic nonlinearity has stabilised the response of the system.

(K = 10.4 to 29, 1.0, 1.0, 1.0, -0.5, 3.0) show the response of the system settling to a limit cycle because of the nonlinearity.

The amplitude of the cycle depends on z.

Gif image below shows how applet should appear.

COPYRIGHT © 2006 Cuthbert Nyack.