## Limitations Of The Nonlinear Controller

The dead-band or notch-gain controller attempts to compensate for the nonlinearity of the titration curve by matching a high controller gain with the leading and existing tails of the titration curve and a low controller gain with the steep center section of a titration curve. While this may appear to be a good fit when one looks at Figure 7.40.5, an expanded view of the region around the setpoint (neutrality) in Figure 7.40.6 reveals the inadequacy of the compensation. The nonlinear controllers also often assume the titration curve has a symmetrical S shape, which is not necessarily the case.

A more effective compensation technique uses line segments to approximate the curve, as shown in Figures 7.40.5 and 7.40.6. More line segments are used in the expanded view near the setpoint. Line segments are preferred to polynomials because smooth transitions from one polynomial to another are difficult to obtain and high order polynomials often have small humps and gain reversals which prevent the use of high controller gains or the use of rate action. The line segments are used to compute the abscissa (the ratio of reagent flow to influent flow) from the ordinate (pH). The abscissa is used as the controlled

variable. If the curve is constant and accurate, the process gain becomes 1. Combined with a linear final element and a constant feed, the result is a nearly linear loop. The curve can be adjusted for temperature effects. The pH setpoint also goes through the same line-segment calculation. The display for the operator can be kept in units of pH, by the use of an output tracking strategy which provides a dummy controller for the operator interface.

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