Cruise Control

Pretty interesting, the teacher in MATH213 gave a formal introduction to this.

We have a car to moves at a given velocity. We know Newton’s Laws of Motion , so . Taking into account drag, we’ll get this equation

  • is the mass
  • is a drag coefficient
  • is the velocity of the car
  • is the velocity forcing term

You should be very comfortable with the Transfer Function.

So if we want the car to go at a certain velocity, we can control .

Breaking down into 2 parts:

  • is the control input
  • is the disturbance input

The controller controls .

Now, how does this work exactly? Like if , and then , does the velocity instantly spike up? like the function has to be differentiable right. It’s a differential equation.

The teacher gets this problem to be solved in frequency space.

In time-domain In frequency-domain

We end up with our first Open Loop Controller.

It’s open-loop because there is no feedback into the system.

In standard form, the transfer function

Proportional Controller

Now, we design our first closed loop controller.

  • is the Laplace transform of desired velocity function

So instead of directly feeding in a control input blindly, we know first look at the error , and that is used to dynamically adjust the control input .

Now, we obviously don’t know , but we can define it . The control input is some proportion to the error term.

We can write

Therefore, We write it in Standard First Order System:

We know have our velocity in terms of some reference. But then how does it actually get working in real life??

  • I’m confused

Alright, so let’s go back into time domain by taking the .

Standard First Order System.

Integral Controller

For the integral controller, we get that where

  • is the car transfer function, the same as before
  • is the controller transfer function

Notice how this is where we compose a transfer function.

So we get

  • Actually, this is a more generic form. For proportional controller, the transfer function (because remember we defined , for integral controller its a .

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