# Deltas Rule

## Deltas Rule in the Equation for the Digital Implementation of the PID Algorithm

09/15/2008

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Greg McMillan and Stan Weiner bring their wits and more than 66 years of process control experience to bear on your questions, comments, and problems. Write to them at controltalk@putman.net.

Stan: After six years of writing this column, we are not running out of ideas; we are just running out of time. Being retired has taken its toll. There is not enough time for doing everything, which includes saving the Everglades.

Greg: I am just trying to save my yard from a Texas-size drought and an onslaught of armadillos. I am told to just shoot these armored invaders, but I am the only Texan without a gun, and I am not much of a hacker with my rattlesnake hoe, although I enthusiastically chopped up the snake that nearly killed my dog.

Stan: Maybe we need to take a lesson here from process control and concentrate not so much on our current states, such as Texas and Florida, and more on deltas, such as how much value we get out of each day.

Greg: I guess students are suppose to realize naturally that all the variables shown in process control texts were really deltas or deviations, whether you are talking about transforms or differential equations. You always need to add the current value (initial condition) to the result. If I ever teach another class on process control, it will be clear that “deltas rule.”

Stan: It is obvious deltas rule in the equation for the digital implementation of the PID algorithm. The delta in controller output from the proportional mode is the delta in the error or in the process variable multiplied by the controller gain, for proportional action on error and proportional action on PV (PID structure), respectively. The delta in output from the other PID modes also depends on these same deltas.

The matrices for model-predictive control are based on deltas of process inputs and outputs. The process gain is the delta in the process output divided by the delta in the process input. For example, if you plot composition, pH or temperature versus the ratio of manipulated flow to feed flow, the process gain is the slope of the curve divided by feed flow.  For these loops, you can also say “ratios rule,” but that is another story.

Stan: The valve gain is the delta in the flow divided by the delta in the valve signal (slope of the valve’s installed characteristic). The measurement gain is the delta in the signal as a percent of scale for the delta in the process variable. The open-loop gain is the product of the valve gain, process gain and measurement gain.

Greg: The whole game in process control is dealing with deltas. If there were no deltas, you wouldn’t need a controller. The process would be drawing a straight line. There are always deltas, whether it is set-point changes from start-ups or shutdowns, grade transitions or batch sequences or disturbances in flows, compositions and temperatures. The job of a controller is to transfer deltas in the controlled variable to the manipulated variables. Slow tuning transfers much less variability than fast tuning. When deltas in the manipulated flows upset other loops, then slow tuning is preferable. If the deltas in the controlled variables are costly or dangerous, then fast, but still stable tuning is used. For runaway reactors, an increase in the divergence rate can become a point of no return.

Stan: Most of the really interesting processes, such as column or reactor temperature and composition, have such a slow response that they appear to be always ramping in the control region. Before they settle out, they are hit by another disturbance. They are called “near integrators” or “pseudo integrators” because their lined-out value (steady-state value) is beyond the practical time frame or control range.

Greg: The short-cut tuning method does not need the process to be lined-out, since it looks at the delta of the ramp rate of the process output for a delta in process input. If you want to know more about this method for integrators and loops that look like integrators, consider helping pay for my next evening out by buying Good Tuning: A Pocket Guide 2005, 2nd ed., published by ISA. If ISA sells five copies, I can go to WhataBurger. Royalties on technical books don’t amount to a hill of beans. I am sitting on a hill here in Texas, but I don’t see no beans.

Stan: For the identification of model parameters, you need to use deltas. You need to match up the deltas in the process outputs between the plant and model for deltas in the process inputs. Also, the size or richness of the deltas determine the choice of variables. Just dumping historical data of controlled variables of continuous processes tightly regulated at fixed set points will lead to poor models. A design of experiments where set points are changed generally is needed.

Greg: If you want to control a batch profile, you need to use the rate of change of the temperature or concentration of interest. In other words, you need to control the slope of the batch profile (delta in process output divided by delta in batch time) as discussed last month in “Unlocking the Secret Profiles of Batch Reactors.”. An integrating response can then be modeled as the self-regulating responses of the slope. This transformation of the controlled variables to slopes increases the robustness and understandability besides eliminating the one direction response of a batch profile.

One of my better successes with increasing reaction rates was to use the derivative of the temperature as the controlled variable so that the PID controller was controlling the increase in the reaction rate. I have seen process engineers forget this in the identification of reaction rates, the ultimate delta. For example, research by a graduate student to identify the kinetic parameters of a bioreactor model showed an impressive match between the model’s profile and the plant’s profile. Unfortunately, several parameters were way off. Wrong values of parameters were compensating for wrong value of other parameters. This was scary because you wouldn’t generally know the parameters were wrong for an actual plant. When the identification was switched to localized changes in the slope of the batch profiles (deltas), the identification program found the right values.

Stan: If we historized the deltas of key process variables over appropriate time intervals for start-ups, shutdowns, disturbances, transitions and batches, maybe we would start to recognize the importance of dynamics and the paths of disturbances. The time intervals need to be large enough to show a change in the process variable beyond the noise and the resolution limit of  the measurement. Often you can improve control even after tuning the loop by simply slowing down an upset, feed change, or set point change. Speed kills. Many disturbances can be tracked down to an overzealous level controller and many processes can be calmed by slowing down the response of a culprit loop.

Greg: Process engineers are taught to think steady-state and compute static values. Most people don’t realize we operate in a delta world. I have had chemical engineering students ask why we need controllers. They want to know why you can’t just set the flow to the value on the process flow diagram. Control engineers have an edge because they recognize and deal with dynamics, a skill that can serve them well in other arenas, which leads me to the Control Talk Election Special “Top Ten List.”

#### “Top Ten Reasons a Control Engineer Should Be President”

10. A computer in every pocket
9. A nuclear reactor in every home
8. Completely automated and remotely controlled combat
7. Dynamic programs
6. Models of the economy
5. No overshoot of the budget
4. Regulatory control of financial markets
3. Feed-forward control of Congress
2. Model-predictive control of energy
1. Real-time optimization of manufacturing operations.

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