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
- how the conversion is done is simply through the rules of the Transfer Function
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 .
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 .