The post What Are the Equivalent Methods to Eliminate Proportional Step and Derivative Kick? first appeared on the ISA Interchange blog site.
The following technical discussion is part of an occasional series showcasing the ISA Mentor Program, authored by Greg McMillan, industry consultant, author of numerous process control books, 2010 ISA Life Achievement Award recipient and retired Senior Fellow from Solutia Inc (now Eastman Chemicals). Greg will be posting questions and responses from the ISA Mentor Program, with contributions from program participants.
In the previous post on bumpless transfer, it was mentioned that there are ways to optimize the immediate change you get in the PID output for a setpoint change. For a structure of PID on error, there is a step from the proportional mode and a kick from the derivative mode in the PID output. This fast and immediate change in PID output makes the setpoint response faster but operators and other loops may get upset by the large abrupt change in PID output. A gradual and smooth response of the PID output for a setpoint change may increase the time to reach setpoint by a matter of seconds for pressure loops and minutes for temperature loops but this may be inconsequential.
Also, the prevention of setpoint overshoot is more easily achievable when there is no proportional step. For bioreactor temperature and pH control, overshoot must be totally eliminated and increases in setpoint response time are insignificant because the batch cycle times are days to weeks. Additionally, for many gas unit operations in hydrocarbon processes, abrupt changes and overshoot of final resting value by the PID output must be avoided at all costs due to severe interactions from heat integration and recycle streams.
Brian Hrankowsky, an ISA Mentor Program resource, has found several equivalent methods that give the user a lot of options to totally eliminate the proportional step and derivative kick for a setpoint change without affecting load response. His use of Laplace transforms not only provides conclusive proof but shows that this knowledge gained in university control theory courses has considerable practical value. Laplace transforms are effectively used by Brian here to provide a unified view and a valuable perspective showing that the user has a lot of ways of achieving this objective.
Brian Hrankowsky’s Derivation of Equivalent Methods
Vendors provide PID features in various combinations that often are intended to address the same control problem or objective. It is not always clear how to translate or compare these features from one vendor PID algorithm to another. The purpose of these derivations is to illustrate that several methods for eliminating or controlling the proportional step and derivative kick in a PID output resulting from a change of setpoint are equivalent by seeing they end up with the same Laplace transform equations.
All algorithms below are based on the ISA standard form. To identify control actions taken on process value vs. error, a hyphen is used to separate the two. Actions to the left of the hyphen are taken on error. Actions to the right are taken on PV. For example, a PID controller with a structure of proportional and integral on error and derivative on PV is shown as “PI-D” and a structure with integral on error with proportional and derivative on PV is shown as “I-PD”.
After setting beta and gamma to zero, the Laplace transform is the same as the one for “I-PD” structure previously shown so there is no need to repeat the remaining equations.
Some controllers use a setpoint lead and lag where the lag is automatically set to the RESET time and the lead is a factor of the reset time between zero and one inclusive. Below, I’ve used beta to represent the factor.
The resulting equation is the same as for the 2DOF with beta and gamma set equal to zero.
We see that there are four equivalent methods of eliminating the proportional step and derivative kick from a setpoint change. Since not all PID controllers have a built in option for adding a lead-lag or filter to the setpoint or a structure with Integral action on error and proportional and derivative action on the process variable or two degrees freedom, this recognition of equivalent functionality has significant practical value as it allows the control engineer to achieve the desired result with whichever of the options are available for the particular control system. The user can readily eliminate abrupt changes in the PID output and prevent overshoot of the setpoint by the PV and overshoot of the final resting value by the PID output when required.
Any change that is to be made to a control system must be thoroughly functionally tested by realistic simulations of the process’s dynamic response. The ability of the control system improvement to deal with abnormal besides normal operating conditions must be verified. The commissioning and performance of improvements should be closely monitored to ensure they meet plant requirements.