Nonlinearities Recommendations Tips

We conclude our review of nonlinearities with an overview of applications and a detailed list of simple solutions to minimize the adverse effects of this everyday problem in nearly all control loops. It is impressive how setting lambda equal to 3x the largest deadtime value helps to solve most nonlinearity problems. Lambda is the closed loop time constant for self-regulating processes and the arrest time for integrating and runaway processes.


For composition and temperature control of continuous operations on well mixed liquid volumes, PID tuning does not change much with feed rate except for the effect of a nonlinear installed flow characteristic. The process dead time and hence the reset time and rate time is relatively constant. The effect on controller gain from an increase in process gain is cancelled out by the increase in process time constant as the feed rate decreases. The PID gain does not need to be decreased at low production rates if the PID manipulates a secondary flow controller, linear trim control valve with a high valve to system pressure drop ratio or a VSD with low static head. 

For pH control, the increase in reagent injection delay at low production rates results in a need to decrease the PID gain and increase the PID reset time at low production rates. The effect can be minimized by reducing the piping volume between the control valve and the injection point and eliminating dip tube volume by injecting the reagent into high flow feed and recirculation streams.

For composition, pH, and temperature control of continuous operations on plug flow volumes, the process gain and dead time both increase as the feed is decreased. The PID gain must be decreased and the reset time increased at low production rates. A nonlinear installed flow characteristic will enhance this effect of production rate. 

For batch operations, the integrating process gain is largest for composition control at the start of the batch when liquid level is the lowest. For temperature control, the integrating process gain is also proportional to the heat transfer coefficient area. Since the coil or jacket area covered by process fluid is proportional to level, the integrating process gain stays about the same when temperature is controlled by manipulating the coil or jacket temperature. If the jacket also covers the bottom of the vessel, for low liquid levels the integrating process gain changes with level. 

The fact that the process gain for operating point process nonlinearities, such as pH, varies with the slope of the line connecting thenew and old operating points makes gain scheduling problematic. Signal characterization where the process variable is translated from pH to the abscissa of the titration curve and scales 0-100% reagent demand inherently eliminates the nonlinearity. Even if characterization is not accurate, it frees up an adaptive controller to compensate for unknowns or for the effects of changes in production rate. 

For inverse response a smarter feedforward may be warranted that reduces the initial reaction of the process in the wrong direction. This has turned out to be particularly important on boiler drum level where the shrink or swell can cause a low or high drum level trip, respectively. The additional feedforward to decrease the inverse response decays out leaving the traditional feedforward based on material and energy balances.


  1. Use a flow loop and cascade control to isolate valve and VSD nonlinearities from a process loop such as composition and temperature.
  2. When a flow loop is not feasible possibly due to rangeability limits, schedule tuning settings preferably with an adaptive tuner as function of flow rate to account for the changes in the valve or VSD gain from a nonlinear installed flow characteristic.
  3. Schedule tuning settings preferably with an adaptive tuner as function of flow rate to account for the changes in transportation delays.
  4. Use signal characterization to reduce process operating point nonlinearities and to free up an adaptive tuner to handle unknowns and production rate nonlinearities.
  5. To reduce the valve and VSD 86% response time for small changes in PID output, decrease the deadband, resolution limit, and threshold sensitivity limit and speed up valve slewing rate and VSD setpoint rate limits.
  6. To decrease the dead time from the valve or VSD for setpoint changes, use a structure that has some proportional action on error or use a setpoint lead-lag.
  7. To reduce the effect of inverse response increase the lambda to at least 3 dead times. If tighter control is needed, add a smart feedforward that provides an initial correction in the opposite direction of the final correction that decays out before the traditional feedforward based on material and energy balances takes effect.
  8. For nonlinear time constants set the reset time equal to the smallest value of the primary time constant and use 3x the largest dead time value for lambda.