Michael Brown Control Engineering CC 

Practical Process Control Training & Loop Optimisation




The D Term



I would like to mention at the outset, that the next three articles on the D term in the digital controllers were previously published some years ago in South African Instrumentation and Control magazine.  However I feel strongly that this section on the digital controllers in the Loop Signature series would hardly be complete if these were left out, so I must apologise in advance if you have previously read them.

Derivative is hardly ever used in feedback control loops, and most people have very little practical understanding of the subject.  Opinions on its use vary from one extreme to the other.  A Professor of control at a leading South African teaching establishment once told me that he had proved conclusively that derivative not only was of no help, but that it actually slowed down control response.  At the other end of the spectrum, a grizzled senior control technician in a large paper mill with many years of experience, stated that he insisted that his people tuned derivative into virtually every loop in the plant, as he was convinced that it not only speeded up response, but it also was a major contributor to loop stability.  Both of theses views could be said to be partially correct in some aspects, but do not reflect the whole truth.


In actual practice, derivative is generally found employed in less than one in several hundreds of control loops.  Reasons for this will be given in this, and in the next two articles.
The objective of the derivative term is to speed up the control response in very slow processes as often encountered in some temperature controls.  A good example of where derivative could be used was found at one of the plants in Secunda.   A practical control course was being held. The class had just finished tuning a very slow temperature loop, and were trying the calculated settings using only P and I control.  On making a 10% setpoint change, the valve opened up to about 30% under the proportional action, and the temperature started rising very slowly.  The integral action then also started ramping the output up and the temperature eventually got to setpoint about 45 minutes later.

The Operator watching this laughed, and mentioned that he could do the same thing in manual, but much faster.  To demonstrate this, and after the temperature had been reset to its original value, he placed the loop in automatic, and immediately opened the valve fully to 100%.  The temperature then started rising much more quickly than it had done the previous time.  When it had risen a certain amount, and based on his experience and judgement, but still some way from setpoint, he started closing down the valve to prevent the temperature from overshooting.  Eventually he got the temperature to the new setpoint about 40% faster than it had in the previous automatic PI control.  The same result would have occurred in automatic if derivative had been used correctly.



It should be noted that the D term is ineffective on self-regulating processes with only a single lag, irrespective of the size of the lag.  The I term cancels the poles effectively on its own, and D will actually not help the response at all.
The second type of process where the D term is particularly effective, is on an integrating process with a large lag.  These process types are typified in batch reactor temperature control.  A schematic of a simple batch reactor control scheme is shown in Figure 2.

The response of such a process to a manual step change in controller output is shown in Figure 3.  Note how the process variable curves slowly into the ramp.  In automatic, the D term will cancel out the lag, and the process will respond to changes very much faster.  It really works well in these applications.  In a pharmaceutical factory in the UK one plant consisted of only batch reactors.  Previously all the temperature control had been performed by using P only.  By adding the D term the reaction times were so much faster that production through the plant was eventually increased by a staggering 17%!


The reason that I action is not used in this particular type of process, is because a setpoint change on an integrating process that employs the I term in the controller will always result in an overshoot.  If there is no cooling on a batch reactor, then overshoot is not acceptable as there is no way to reduce the temperature back to setpoint in a reasonable time after an overshoot.

How is the D term applied in a controller?  Figure 4 shows the principle.  Derivative is used in Calculus to measure the slope of a line.  On a continuous system as seen in control applications, the error signal (or sometimes the process variable (PV) signal, as will be discussed in the next article in this series) is fed into a derivative calculation block.  The output represents the rate of change of the input signal. The derivative of a constant input signal is zero.  The derivative of a ramp input is a step.  The steeper the ramp, the bigger the output step.  The derivative of a vertical change in input, however big or small, gives a theoretical output of infinity.  This one of the main reasons why D is so seldom used.  Most process variable signals are noisy.  The small fast changing variations of PV cause the output of the controller to jump about too much.



Figure 5 is a schematic of a simple controller configured with a "parallel" PI algorithm.  For simplicity no I action is included.  Therefore a manual bias is provided to allow "manual reset" which allows one to manually eliminate offset between PV and setpoint.  A ramp change of setpoint is shown.  The resultant error signal, and the proportional action also ramp.  The output of the D unit is a step.  The final output of the controller is then a step followed by a ramp.


Figure 6 is a schematic of a simple controller configured with a "parallel" PI algorithm.  For simplicity no I action is
This response is enlarged in Figure 6, and is compared also with a P only response. It can be clearly seen that the controller output reacts to a step change in error T seconds faster with P+D control, as opposed to P only control.
In the next article, the application and use of derivative in modern digital controllers will be discussed.