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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.