Mar
26

# Linear in a Nonlinear World – Part II

In my December 18 blog “Linear in a Nonlinear World” we discussed the use of signal characterization to compensate for the installed characteristic of the control valve where the valve gain depends upon the operating point on the control valve characteristic. In part II we are looking at the use of signal characterization to compensate for a nonlinear process gain by translation of the original nonlinear process variable to a new linear one to enable adaptive controllers to better focus on other nonlinearities such as feed flow. Here in part II the process gain depends upon the operating point of the process variable. Examples of this translation to a new controlled variable are:

(1) Conductivity to % acid, base, or salt concentration

(2) Column top temperature to % reflux demand

(3) pH to % reagent demand

For conductivity, there is a peak in a plot of conductivity versus the acid, base, or salt concentration. The new process variable scale must be on one side or other of the peak. There is uncertainty in the exact location of the peak. If the operating point were to cross the peak, the process gain would go to zero and then change sign, which is disastrous to a control loop. The operating point must steer well clear of the peak.

For all of these examples, concentrations of other components in the feed can shift or change the shape of the curve but often the translation is better than no compensation at all for the process nonlinearity. For conductivity and pH, the effect of process temperature based on lab samples should be part of the calculation. For temperature, the effect of column pressure should be included (e.g. shift in boiling point with pressure).

The implementation involves first plotting the original versus the new process variable. For the examples noted this would be conductivity versus ion concentration for various temperatures, column temperature versus % reflux to feed ratio for various pressures, and pH versus % reagent to feed ratio for various temperatures. Since you are getting the X axis from the Y axis (the opposite of what is being done by the process), the data points for signal characterization are entered as Y,X pairs with a nonlinear bias to Y from a fit to the shift in the family of curves. Since the Fieldbus signal characterizer allows variable space of data points, closer points are used in the area of greatest curvature near the set point. This translation must be done for both the set point and the process variable. The original and new set points and process variables must be displayed and historized.

The benefits are most noticeable in pH loops because of their extreme sensitivity nonlinearity, and rangeability where changes in process gain of 100:1 and of reagent demand of 1000:1 are routine. Signal characterization has been shown to make dramatic reductions in startup time by the loop’s recognition that the acid or base reagent flow is really decades away from set point. It also prevents pH from zipping right through the neutral point (e.g. 7 pH) and banging between the flat portions of the titration curve, offering a settling time where there was none. The characterization restores the process time constant by slowing down the excursion rate and helps a continuous pH loop look more self-regulating by removing the acceleration from movement to steeper slopes on the titration curve. Thus, you see and realize the benefit from an investment in a well mixed vessel where the residence time is a process time constant that slows down concentration disturbances as discussed in blogs from the past few weeks.

There are some issues besides inaccurate curves and confusion in the operator interface. If your set point is always on a flat portion of the curve and the control system can keep the operating point close to the set point for the largest disturbance, the benefit from linearization is minimal. Additionally, if an excursion to the steep portion of the curve represents an extremely undesirable situation for equipment or environmental protection, then the elimination of the overreaction of the loop by removal of the acceleration through linearization may be the wrong thing to do even though it reduces overshoot and wasted reagent when returning the pH to its set point.

While you increase the dead time from valve dead band and resolution limits when the set point is on the steep part of the curve because you are slowing down the rate of change of the process variable and the overreaction of the controller output, this normally is much less important then the suppression of oscillations. The increase in dead band for operating points on the steep portions of the valve characteristic can be a concern for control valves because there is usually no stability issue from the much less severe nonlinearity of a valve.

These and other considerations and an application for pH control are shown in the attached file on “Linear Reagent Demand Control” which is an excerpt from my ISA pH Web Seminar at 2:00 EDT on May 16.

Linear Reagent Demand Control

I conclude with a top ten list that will appear in a future “Control Talk” column.

Top Ten Reasons Not to Use Linear Reagent Demand Control

(10) How do you know it is a pH loop if it is not oscillating

(9) You can better see if the pH sensor is still alive

(8) You can better tell if the operator is still alive

(7) You like bang-bang control

(6) Gives you chance to try out the manual mode

(5) The titration curve from the lab shows a straight line through the set point

(4) You like seeing the full effect of valve stick-slip

(3) Retuning loops is job security

(2) You can eat more doughnuts while waiting for a loop to startup and settle out

(1) Linear loops are for wimps