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So recall, last time, we saw that. We designed a controller that was nice and

Â smooth. It didn't overreact to small errors. made a system stable. Yet didn't

Â achieve tracking. And this was the proportional regulator, or the p

Â regulator. and let's return to our performance objectives a little bit. We've

Â talked about them briefly. But a controller at the minimum should.

Â Stabilize the system. If it doesn't do that, we know nothing and I've written

Â this rather awkward looking acronym here, BIBO, which is something out of the Lord

Â of the Rings almost. What it stands for is, bounded in, bounded out which means

Â that if the control signal is bounded, the state of the system should also be

Â bounded. What this means is that, by doing. Reasonable things the system

Â doesn't blow up. And our system doesn't do that. Tracking means we should get to the

Â reference value we want. And robustness means we shouldn't have to know too much

Â about parameters that we really have no way of knowing. And preferably we should

Â be able to fight noise as. Well, so recall at this was the model and when I

Â introduced this wind resistant term here, we had a little bit of a problem.The

Â proportional regulator couldn't overcome it and lets have another controller done

Â one that explicitly cancels out the effect of the wind resistance. So here is my.

Â Attempt 3, I'm going to use this part, which is the proportional part that we

Â already talked about, and then I'm going to add this thing which is plus gamma

Â m/c*x. Well why did I do this? Well, I did this

Â for the following reason that if you reach steady state x is not equal to 0, then now

Â What you get is well this was the p part. This is the controller, the p controller.

Â And then the effect of this thing well you're going to multiply this by c/m. What

Â you're going get then is plus gamma x. And then you have wind resistance which is

Â negative gamma x. So the gamma x, the bad parts cancel out. And in fact all we're

Â left with then is that x. Has to be equal to r. So, voila, we've sol ved the

Â problem. We have perfect tracking. Or, have we?

Â dom, dom, dom. No, we have not. And, why is this? Well, we have stability and we

Â have tracking, but we don't have robustness. Here are three things that we

Â don't know. Gamma, m, and c. And our controller depends explicitly on, On these

Â coefficients. So all of a sudden we have to know all these physical parameters that

Â we don't know, so this is not a robust control design. So Attempt 3 is a failure.

Â Okay, let's go back to the P-Regulator and see what's going on there. What, what's

Â actually happening is that the proportional error is doing a fine job

Â pushing the system up to close to where it should be, but, then it kind of runs out

Â of steam, and it can't push hard enough to overcome The effect of the wind

Â resistance. So the proportional thing isn't hard enough, but take a look here.

Â This is the error, then the error starts accumulating over time, so if we somehow,

Â if we're able to collect All of these errors over time, even though they are

Â very small. Over time, that should be enough, so that we can use this now

Â accumulated error to push all the way up. So I wish there was some way of collecting

Â things over time in a plot like this. And, of course, there. There is, this is

Â something called an integral. So, if we take the integral over the error we're

Â collecting the error over time and over time as this errors going to accumulate

Â it's going to give us enough pushing power to actually overcome the wind resistance.

Â So attempt 4 is a pi. Regulator. So what I have here is the error at time

Â t. This is my kp, which is my proportional gain. So this is the p part that we

Â already saw. And now, I'm adding an integral that is integrating up the error

Â from. The beginning to the current time. And it's collecting this. And then we have

Â another term here, or another coefficient. The ki, where I stands for the integral

Â part. So this a pi regulator. And it is 2/3 of. The most common regulator found

Â anywhere in the world, and in fact it's almos t 2/3 of commercial grade cruise

Â controllers. So if I have a p and an i, what could possibly be missing to get to

Â all of them? 3/3 instead of just 2/3. Well, we take a derivative. Right, we have

Â proportion, we have integral, and we have a derivative. So, why not produce what's

Â called a PID-Regulator? So now we have a proportional term with a proportional

Â gain. We have an integral part with an integral gain. And then we have a

Â derivative part with a derivative gain, so this is. It's an extremely useful

Â controller that shows up a lot. And, in fact, I'm going to hand, have to hand out

Â a big sweetheart to the PID regulator. Because it's such an important type of

Â control structure that shows up all the time. And in fact we're going to get quite

Â good at designing the PID regulators. Now having said that, I can draw hearts all I

Â want, let's see it in action and see what it actually does. And if I use just the PI

Â regulator, not even a D component to the cruise controller, then all of a sudden I

Â get something that's getting up quickly, nice and slowly, I mean smoothly, to 70.

Â Miles per hour, which is my reference. So this solves the problem. I don't know

Â parameters, so it's robust. I'm achieving tracking, because I'm getting to 30 miles

Â per hour. And, I'm stable in the sense that I didn't crash. So, this seems like a

Â very useful design.

Â