The following content has been provided by the University of Erlangen-Nürnberg.

Let me give you a brief introduction of what control theory is.

So if I take from Wikipedia, control theory is a branch of mathematics

and the idea is that it deals with dynamical systems and uncertainty.

And the main point of control theory is that it takes a physical model of something

and it makes sure that that something behaves in the way we expect it to behave.

So the main idea here is that you get a plan which is whatever you're trying to control.

So I will take one example from the control domain which is cruise control.

It's very easy to explain things using cruise control.

So you have your car and you have the throttle and the amount of power you're putting into your engine.

And you want your car to maintain a specific speed running on a highway.

Now let's assume the easiest condition, the highway is completely free

and you have no troubles as no other cars, no problems.

It is very easy to compute a control signal for that.

So it is very easy, you power up your car and then at some point you click a button

and the car knows how much throttle is putting in the engine.

So in principle you could just say why do you need control for that?

You have already told me the value that I should inject into the motor.

And that is correct.

So in general you have actuators and measurements.

And in this case you measure the current speed and you actuate on the throttle for the engine.

We usually take the plant and we take a linear model of it.

And the idea is that the world is non-linear and most of the things that you deal with are non-linear.

But linear models are easy approximation to describe what is happening now.

And I will go back to this later.

But now let me go back to why do you need the control there.

Now my road is going uphill and of course I have no measure of that.

But I can see that my speed is suddenly decreasing.

I am putting the same throttle in the engine and my speed is suddenly decreasing.

So what my controller would do is that it knows that my speed there, my objective is not reached.

I get an error and then the controller will say,

okay I will increase the throttle that I put, I will speed up the car.

That is a very easy example.

Now let me add something more.

What if there is ice on the road?

Then it becomes much more complicated to do all this because there is something else which is unpredictable that is acting on your system.

So usually what you need to do for these things is that your model is not linear

because now your car is not any more behaving in the same way as before.

But you can get data from what is happening right now and adjust your model so that it reflects better what is happening.

And for this you do usually linearization.

So you take the model, you create a new linear model that describes the current situations

and then you are able to change your controller to fit this new requirements that you have.

So now how do you build this model online?

Let's say you have an initial guess.

If I increase the power in my engine the speed is going to increase.

And if I decrease it the speed is going to decrease.

That is a good initial guess.

But now let's say that my engine is broken and at some point even if I increase the speed,

even if I increase the throttle it is not going to increase the speed.

That is something that could happen.