Since you asked about PV, from a "more than you wanted to know" perspective:
A PID loop basically has three control variables: the process variable (PV) which is the measured variable (such as temperature), the set point (SP) which is where you want the measured variable to be, and the output, which is the calculated result used to control something (such as a heating element) designed to move the PV towards the SP.
The calculated value is generally obtained by an algorithm that includes a proportional component, an integral component, and a derivative component (hence: PID). Each of the components have a gain factor that you can increase or decrease to increase or decrease the weight of that component in the calculation for the output value.
The proportional component is the error (difference between SP and PV) multiplied by a settable “gain” factor. The farther out the error value the larger the proportional value. The integral value adds time to the calculation, which causes it to grow larger based on the duration of the error. The derivative component is calculated using the slope of the error over time; i.e., it notices how fast the error is changing. Both the integral and derivative values are also multiplied by a gain factor that you set.
In very general terms (aimed at a brew pot for this example), the proportional component corrects *now*; my temperature is low, so I need to add heat; the farther out, the greater the heat. Once PV=SP, the proportional term will be zero and yield, by itself, a zero output. We would then begin to lose heat until the PV was below the SP, at which time we would add heat; we would never maintain the PV at SP for very long. The integral component helps correct for this as it examines the necessary correction over time and creates an output component that will keep some heat on even if at temperature. The derivative term realizes that, without looking at the speed at which the error is changing (the slope), we can easily overshoot our SP; it attempts to avoid by throttling back if we approach the SP rapidly.
Note that the output is a variable value (usually expressed as 0% to 100%) but can control either a variable device or an on/off device. For example, in my work I’m generally dealing with PIDs controlling valves which can be opened to varying positions; I can have it open 25%, 50%, etc., so the output directly relates as I can adjust from 0% (closed) to 100% (fully open) to reach a flow rate that achieves my desired SP. With the temperature loops I work with, it’s normally increasing or decreasing gas flow. However, I’ve also worked with PIDs where the output is controlling a fully on or fully off device, not a variable device; this is probably more the norm for home brewing. For those you vary the length of time the output is on during a predetermined period. I can set the period for, say, ten seconds and vary the amount of time the output is on during each of the ten-second periods. If the temperature is way too low, it’s on the full ten seconds. At temperature, it may only be on for half a second out of each ten-second period. The output of 0% to 100% in this example equates to being on from zero seconds out of ten to being on for ten seconds out of ten. (Also, as an aside, the control can actually consist of two different devices, for example something that adds heat or something that adds coolant.)
There are other variables that can come in play with most PIDs; you can set a dead band, which can be thought of as a “close enough” region; if I set a dead band of plus/minus two degrees with a set point of 165, then from 163 to 167 I won’t worry about changing the output. There is also a bias setting that I have seen used (and used) in various ways; basically, it’s a direct add-in to the final calculated value that biases the calculated value (and can be either positive or negative, so can bias in either direction). For example, you could have a bias value based on the current outside temperature.
There are, I’m sure, better explanations out there but the above is an off-the-cuff, in a nutshell, down-and-dirty, choose-another-euphemism description.
Rick
