Non Linear Saturation, 2nd and 3rd order systems

Cuthbert Nyack
If a controller uses high gain or has a large error as input, then the output can saturate i.e. reach a fixed value rather than being proportional to the input.
The applets below illustrate the effect of saturation on second and third order systems. In both cases the output saturates at an upper level Ul and a lower level Ll.
The effect of saturation is to modify the shape of the response and the effective gain but does not add any phase delay to the system. A second order system cannot therefore be made unstable by saturation effects.
The effect of saturation on a second order system is illustrated in the applet below.
eg parameters (5.0, 0.2, 5.0, -5.0, 1.0, 1.0) show that the saturation levels are high enough so as not to affect the response. With parameters (5.0, 0.2, 2.0, -2.0, 1.0, 1.0) the effect of saturation can be seen as a reduction in the gain, a lowering of the transient frequency and a "flattening" of the transient oscillations.
eg parameters (5.0, 0.2, 3.0, -1.0, 1.0, 1.0) and (5.0, 0.2, 1.0, -3.0, 1.0, 1.0) show that radical changes can be produced in the shape of the transient oscillations if the saturation levels are asymmetric.

The applet below show the effect of saturation on a third order system. Since this system can be unstable than saturation effects will affect the stability behaviour of the system.
The eg parameters (2.0, 1.0, 2.0, 1.0, -2.0, 2.0, 1.0) show that the saturation levels are high enough so as not to affect the response. With these parameters, the complex conjugate poles of the closed loop lie on the imaginary axis and the system is oscillatory.
The eg parameters (2.0, 1.0, 1.0, 1.0, -1.0, 2.0, 1.0) show saturation effects reducing the gain, lowering the transient oscillation frequency and "flattening" the shape of the waveform.
The eg parameters (K, 1.0, 3.0, 1.0, -3.0, 6.0, 0.45) show that for K up 2.0, the linear and saturated response are identical. For K greater than 2, the responses increasingly diverge, with the saturation one remaining almost constant at around 1.6pp.

Gif image below shows how applet should appear.

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COPYRIGHT 2006 Cuthbert Nyack.