A common exercise for students learning the **function of PID controllers** is to practice graphing a controller’s output with given input (PV and SP) conditions, either qualitatively or quantitatively.

This can be a frustrating experience for some students, as they struggle to accurately combine the effects of P, I, and/or D responses into a single output trend. Here, I will present a way to ease the pain.

## PID Controllers

Suppose for example you were tasked with graphing the response of a PD (proportional + derivative) controller to the following PV and SP inputs over time.

You are told the controller has a gain of 1, a derivative time constant of 0.3 minutes, and is reverse-acting:

My first recommendation is to qualitatively sketch the individual **P and D responses**. Simply draw two different trends, each one right above or below the given PV/SP trends, showing the shapes of each response over time.

You might even find it easier to do if you re-draw the original PV and SP trends on a piece of non-graph paper with the qualitative P and D trends also sketched on the same piece of non-graph paper.

The purpose of the **qualitative sketches** is to separate the task of determining shapes from the task of determining numerical values, in order to simplify the process.

After sketching the separate P and D trends, label each one of the “features” (changes either up or down) in these qualitative trends. This will allow you to more easily combine the effects into one output trend later:

Now, you may qualitatively sketch an output trend combining each of these “features” into one graph.

Be sure to label each ramp or step originating with the separate **P or D trends**, so you know where each “feature” of the combined output graph originates from:

Once the general shape of the output has been qualitatively determined, you may go back to the separate P and D trends to calculate numerical values for each of the labeled “features.”

Note that each of the PV ramps is 15% in height, over a time of 15 seconds (one-quarter of a minute). With a controller gain of 1, the proportional response to each of these ramps will also be a ramp that is 15% in height.

Taking our given derivative time constant of 0.3 minutes and multiplying that by the PV’s rate-of-change ( dPV/dt ) during each of its ramping periods (15% per one-quarter minute, or 60% per minute) yields a derivative response of 18% during each of the ramping periods. Thus, each derivative response “step” will be 18% in height.

Going back to the qualitative **sketches of P and D actions**, and to the combined (qualitative) output sketch, we may apply the calculated values of 15% for each proportional ramp and 18% for each derivative step to the labeled “features.”

We may also label the starting value of the output trend as given in the original problem (35%), to calculate actual output values at different points in time.

Calculating output values at specific points in the graph becomes as easy as cumulatively adding and subtracting the P and D “feature” values to the starting output value:

Now that we know the output values at all the critical points, we may quantitatively **sketch the output trend** on the original graph:

Credits : Tony R. Kuphaldt – Creative Commons Attribution 4.0 License