Enough of the technical stuff -- let's look at PID control from a common sense / practical approach. On this page we will build a PID controller using Microsoft Excel so that you can see, and experiment with, the results that we are talking about.
We focus on the difference (error) between the process variable (PV) and the setpoint (SP). There are three ways we can view the error.
One of the first ideas people usually have about designing an automatic process controller is what we call "proportional". Meaning, if the difference between the PV and SP is small -- then let's make a small correction to the output. If the difference between the PV and SP is large -- then let's make a larger correction to the output. This idea certainly makes sense.
We simulated a proportional only controller in Microsoft Excel (if you click on this link then you will download the Excel spreadsheet). Here is the chart showing the results of the first simulation:
In this first example, we assumed that there was no dead time, meaning, that if we made a change in the output of the controller, the input immediately changed. For example, zero dead time on our oven means that if we changed the temperature setpoint on the oven, then the temperature inside the oven instantly changed to the new setpoint (the oven did not require time to heat up or cool down).
The blue line represents a proportional constant of .1, the magenta lines represents a proportional constant of .2, the yellow line represents a proportional constant of .4, and the white line represents the setpoint (SP). From this graph, hopefully two things jump out at you. First, once the output settles out, the output (blue, magenta, and yellow lines) are no where near the setpoint (SP) (the white line). Therefore, some offset has to be added to the output to make the PV reach the SP. Second, the greater the proportional constant, the less the offset needs to be. For example the yellow line, with a proportional constant = .4 is closer to the white line than the blue line with a proportional constant of .1.
If you download the Excel spreadsheet of the PID controller simulator and look at the effects of increasing dead time you will notice that the outputs settle at the same output level -- it simply takes longer for the output to reach its final level.
In summary, automatic proportional (only) controllers are not very good because there is an offset that has to be continually adjusted.
The integral portion of the PID controller accounts for the offset problem in a proportional only controller. We have another Excel spreadsheet that simulates a PID controller with proportional and integral control. Here is a chart of the first simulation with proportional and integral:
As you can tell, the PI controller is much better than just the P controller. However, dead time of zero (as shown in the above graph) is not common. So let's take a look when the dead time equals two.
Now this graph is starting to look more typical of a PID controller. Notice how the dark blue line quickly goes up to the SP (50) and cycles around 50 a little but quickly settles down. In contrast, the dark purple line way overshoots the SP of 50, going above 80, back down to 30, then over 50, and back and forth until it eventually settles down.
If you download the Excel spreadsheet and look through the different scenarios you will notice that the P & I parameters that look good for one dead time do not look optimal for another dead time. In other words, for each process element (valve, motor, pump, heater, chiller, etc) you are trying to control -- you will have different process characteristics and will have to determine the optimal P, I, and possibly D constants. Determining what these constants should be is called "tuning". Theoretically, you want to minimize the sum of absolute errors, as given in the spreadsheets.
Let's show one other graph to warn you about a very dangerous condition:
We wanted to show this graph to illustrate what can happen if you choose the wrong parameters. The green line illustrates an unstable or "out-of-control" controller. Notice how it continues to get worse and worse. This is not good. This is why you want to start with very small P, I, and D constants and increase them to improve performance. If you start with large constants, bad things can happen.
Derivative control takes into consideration that if you change the output, then it takes time for that change to be reflected in the input (PV). For example, let's take heating of the oven. If we start turning up the gas flow, it will take time for the heat to be produced, the heat to flow around the oven, and for the temperature sensor to detect the increased heat. Derivative control sort of "holds back" the PID controller because some increase in temperature will occur without needing to increase the output further. Setting the derivative constant correctly, allows you to become more aggressive with the P & I constants.
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