So recall, last time, we saw that. We designed a controller that was nice and smooth. It didn't overreact to small errors. made a system stable. Yet didn't achieve tracking. And this was the proportional regulator, or the p regulator. and let's return to our performance objectives a little bit. We've talked about them briefly. But a controller at the minimum should. Stabilize the system. If it doesn't do that, we know nothing and I've written this rather awkward looking acronym here, BIBO, which is something out of the Lord of the Rings almost. What it stands for is, bounded in, bounded out which means that if the control signal is bounded, the state of the system should also be bounded. What this means is that, by doing. Reasonable things the system doesn't blow up. And our system doesn't do that. Tracking means we should get to the reference value we want. And robustness means we shouldn't have to know too much about parameters that we really have no way of knowing. And preferably we should be able to fight noise as. Well, so recall at this was the model and when I introduced this wind resistant term here, we had a little bit of a problem.The proportional regulator couldn't overcome it and lets have another controller done one that explicitly cancels out the effect of the wind resistance. So here is my. Attempt 3, I'm going to use this part, which is the proportional part that we already talked about, and then I'm going to add this thing which is plus gamma m/c*x. Well why did I do this? Well, I did this for the following reason that if you reach steady state x is not equal to 0, then now What you get is well this was the p part. This is the controller, the p controller. And then the effect of this thing well you're going to multiply this by c/m. What you're going get then is plus gamma x. And then you have wind resistance which is negative gamma x. So the gamma x, the bad parts cancel out. And in fact all we're left with then is that x. Has to be equal to r. So, voila, we've sol ved the problem. We have perfect tracking. Or, have we? dom, dom, dom. No, we have not. And, why is this? Well, we have stability and we have tracking, but we don't have robustness. Here are three things that we don't know. Gamma, m, and c. And our controller depends explicitly on, On these coefficients. So all of a sudden we have to know all these physical parameters that we don't know, so this is not a robust control design. So Attempt 3 is a failure. Okay, let's go back to the P-Regulator and see what's going on there. What, what's actually happening is that the proportional error is doing a fine job pushing the system up to close to where it should be, but, then it kind of runs out of steam, and it can't push hard enough to overcome The effect of the wind resistance. So the proportional thing isn't hard enough, but take a look here. This is the error, then the error starts accumulating over time, so if we somehow, if we're able to collect All of these errors over time, even though they are very small. Over time, that should be enough, so that we can use this now accumulated error to push all the way up. So I wish there was some way of collecting things over time in a plot like this. And, of course, there. There is, this is something called an integral. So, if we take the integral over the error we're collecting the error over time and over time as this errors going to accumulate it's going to give us enough pushing power to actually overcome the wind resistance. So attempt 4 is a pi. Regulator. So what I have here is the error at time t. This is my kp, which is my proportional gain. So this is the p part that we already saw. And now, I'm adding an integral that is integrating up the error from. The beginning to the current time. And it's collecting this. And then we have another term here, or another coefficient. The ki, where I stands for the integral part. So this a pi regulator. And it is 2/3 of. The most common regulator found anywhere in the world, and in fact it's almos t 2/3 of commercial grade cruise controllers. So if I have a p and an i, what could possibly be missing to get to all of them? 3/3 instead of just 2/3. Well, we take a derivative. Right, we have proportion, we have integral, and we have a derivative. So, why not produce what's called a PID-Regulator? So now we have a proportional term with a proportional gain. We have an integral part with an integral gain. And then we have a derivative part with a derivative gain, so this is. It's an extremely useful controller that shows up a lot. And, in fact, I'm going to hand, have to hand out a big sweetheart to the PID regulator. Because it's such an important type of control structure that shows up all the time. And in fact we're going to get quite good at designing the PID regulators. Now having said that, I can draw hearts all I want, let's see it in action and see what it actually does. And if I use just the PI regulator, not even a D component to the cruise controller, then all of a sudden I get something that's getting up quickly, nice and slowly, I mean smoothly, to 70. Miles per hour, which is my reference. So this solves the problem. I don't know parameters, so it's robust. I'm achieving tracking, because I'm getting to 30 miles per hour. And, I'm stable in the sense that I didn't crash. So, this seems like a very useful design.