Non Linear 3rd order system with Cubic.

Cuthbert Nyack
When the controller used in a control system has nonlinear characteristics, then that nonlinearity may be approximated as a cubic characteristic.
The applets below illustrate the changes in the dynamics of third order systems when the controller has cubic characteristics.
The extent of the nonlinearity is determined by the parameter b which can be +ve or -ve.

The first applet shows the step response and indicates that the effect of the cubic is to increase or decrease the gain and transient frequency depending on whether b is +ve or -ve. Since a third order system can become unstable then the cubic nonlinearity can increase or decrease the range of K for the system to be stable depending on whether b is -ve or +ve.
The response does show some modification of the waveform resulting from the introduction of the third harmonic in the response.
eg parameters (K, a=1.0, b, c=1.0, 2.0, 1.0)
for a = c = 1.0, the range of K for stability for the linear system is 0 to 2. With b = -0.5 the range increases to 0 to 2.74 for the system to reach constant amplitude oscillation while for b = +0.5, the range reduces to 0 to 1.52.
For -ve b the effect is more than just an increase in the range of K to reach constant amplitude oscillation. Beyond this value of K the response remains constant amplitude but increasing distortion for K ~ 6 for b = -0.5. For K = 3.27, the minimum of the oscillation is zero. This has the characterictic of a limit cycle. As K is increased, the output increases but the increasing output reduces K so a steady state is rapidly reached.

The applet below shows the response of a third order system with a cubic nonlinearity to a sine input.
eg. parameters (2.0, 1.0, 1.0 -0.1, 0.8, w, 6.0)
The applet shows that for w from 0.85 to 1.14 the frequency of the nonlinear response remains tied to the applied frequency while for the linear case the response is the sum of components at the natural frequency and applied frequency. This phenomenon is known as frequency entrainment.

Gif image below shows how applet should appear.

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COPYRIGHT 2006 Cuthbert Nyack.