Ask Dr Bob: What is PI Control?
Hi, I’m Bob Rice. And today we’re going to talk about PI control with a focus on the I part, the integral part of PI control. So let’s start with the equation. Right. So with a PI controller, we have a controller output, it’s equal to the bias value, plus the proportional part, the P only part of it, which is going to be your controller gain times your error. Now we introduced the integral portion, right? This is going to be KC over reset time times the integral times E of t d t. All right, this is a PI equation. Now, this is also what’s known as a dependent form of the PI equation, this is going to be the form of the equation that I’m going to use most frequently in this video series. What makes it dependent is that the controller gain is actually distributed across both the proportional and the integral portions, right, this is a dependent form, the controller gain is distributed.
In this particular example, the dependent form uses also what’s called a reset time, which is the value located here in the denominator. Okay, if I make this reset time smaller, it’ll effectively make this weighting factor larger and give the controller more integral action. So in this form, the dependent form of the equation, the reset time is in the denominator, a smaller value gives it a more integral action. So let’s take a look at what the integral action does. So again, we’re going to start with a very simple example, we’re going to make a set point change on a process. And we’re actually going to remove the proportional term, right, we’re going to get rid of this. And we’re going to look at just what the integral is going to do to this right. So this is an I-only controller, just so we can see the influence of what the integral term does. Right. So again, we have some sort of process variable, and we have a controller output, that starts at some value. Okay. And then right, when we make that setpoint change, right, we introduce an error into the system.
Now, whereas a P-only controller would see a big step in error, because it’s looking at whatever the error is, at this moment, the integral is an accumulation of air. And so it’s actually summing up the area in between the process variable in the setpoint. So it actually starts very small. And then as the process variable starts to collect that air, the integral will accumulate that, right, and it’s gonna accumulate, and it’ll slowly push the process variable up because we’re accumulating. Now, notice I’m getting smaller and smaller amounts of air. And so the amount accumulating, become smaller and smaller, right? And eventually, I’m going to actually, probably crossover, right, because this integral is accumulating and going and going and going, it’ll eventually slow down. But if I actually accumulate too much, it’s going to overshoot, and it’ll kind of ramp back down again. And it’ll come back down and probably do something, maybe along the lines of that, right, just as an example, right?
Obviously, the value of the reset time will change how much it ramps, if I give my controller a more integral action, which is a smaller reset time, you’ll find that you’ll accumulate your ramp that error a lot quicker, right? If you ramped this error a lot quicker, you’re gonna find your process variable is going to move quicker, but odds are, you’re going to overshoot more, right? And so you may end up with something that looks like this. If you don’t integrate enough, you’re gonna find that the integral will accumulate very, very slowly. And the process variable may not do much at all. Okay, so the integral action is an accumulation of error it builds based off of how much error you’ve accumulated over time. Now, how do you add this to the proportional term? All right. So if you remember from the last web series, we had a set point change. And we had a process variable that kind of responded like this, right? For a flow loop. And the controller output contribution of the P-only part was something like it started here, and then it got large, and then it kind of dropped back down again. Right, so this is your proportional contribution in here. Right? The integral is going to accumulate on to this, right? And so if this is the proportional contribution, the integral contribution is going to be something like this. It’s going to accumulate, and then it’s going to kind of stop accumulating and what you put those two terms together, and what you are allowing it to do is have that controller output go up
and then steadily add a new value.
Instead of your process variable having an offset because this error exists, that error is actually driving your integral to keep moving. And it’s going to move that process variable to get up to the setpoint. So the integral removes the offset that was caused by a P-only controller. All right, the integral is accumulating, it’s building up its momentum, it’s going to continue to accumulate, as long as there’s an error. If there’s an error, that means your process variables, are not at your setpoint. And you’re going to get a little bit of contribution to your integral to keep accumulating to it. Now, if you have too much integral, you tend to accumulate too quickly. And you start to introduce instability and overshoot in your processes, right? If you have too much integral action, right, you’re going to accumulate too much. And you’re going to cause overshoot, and oscillations, too little integral action, and you’re going to end up with a process that dies before it gets to the setpoint.
And it takes forever, for it to get that last little bit because it’s not accumulating that error. It’s not shifting that controller output enough, right. So if you see a controller that changes quickly, and then dies before it gets there, not enough integral, if you see a controller that just carries on and has lots of oscillations and waves, could be too much integral could be too much proportional, that can also cause overshoot, and oscillations, right. So there’s a little bit of a given a take with a PID controller, yes, you get rid of the overshoot, but the integral action does increase the likelihood that your process will start to oscillate. Alright, so proportional and integral, you add those together, you get the PI controller, which is the most common form of the PID equation. Okay, proportional and integral. And this web series where we reviewed a little bit of the impact of the integral tuning parameter on the shape of the controller output, and what the integral action does to the PI controller to be able to get rid of the offset.
If you have a particular topic or an idea that you would like us to cover, please email us at firstname.lastname@example.org. Thank you, and I hope you enjoy this video series.