Nonlinear Two-mode Controllers

It has been demonstrated that a loop whose gain varies inversely with amplitude is prone to limit-cycle. Any controller with similar characteristics can promote limit cycling in an otherwise linear loop. On-off controllers are in this categoiy. So any nonlinear device that is purpose inserted into a loop for the sake of engendering stability must have the opposite characteristic gain increasing with amplitude. The only stabilizing nonlinear devices discussed up to this point have this property -it was manifested as a dead zone in the three-state controller and as the linear mode in the dual mode system. 

It is possible to create a controller with a continuous nonlinear function whose gain increases with amplitude. In contrast to the three-state controller, its gain in the region of zero error would be greater than zero, with integrating action to avoid offset. But its change in gain with amplitude should be less severe than that of a dual-mode system. Thus it would be more tolerant of inaccuracy in the control parameters. 

The continuous nonlinear controller could be mathematically described by the expression 

III this way its gain varies with the absolute magnitude of the error. A suitable linear function can be used. 

FIG 5 19. The proportional characteristic of a continuous nonlinear controller displays variable damping. 

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A nonlinear two-mode controller seems generally to outperform a linear two-mode controller. The nonlinear function provides an extra margin of stability similar to what can be attained with derivative. In cases where so much noise is superimposed on the measurement that derivative cannot be used, a nonlinear function can be quite valuable. 

FIG 5.22. The nonlinear two-mode controller is superior in all respects on a noisy flow loop. 

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