You tell your car how fast you want to go by pushing a button, but how does it know when to speed up? Cruise control may seem like magic, unless you know the physics behind it…
Cruise control systems use proportional integral derivative control (PID) to determine adjustments in speed that need to be made. By calculating the integral of speed (distance) and the derivative of speed (acceleration). Remember these from the velocity versus time graphs we looked at, where the distance is the area under the graph and the acceleration at any one time is the slope of the graph at that point.
The car calculated the difference between the distance it travels and the distance it would have traveled at the velocity you want in a given period of time. Because of the changes in the slope of the road (forces exerted on car due to gravity) and air resistance, the car’s speed can slow down. The cruise control system takes this actual speed into account and notes the difference between that and your desired speed. It then makes the proper adjustments to the car’s velocity by influencing the throttle (force exerted by the engine).
From: Control System Design by Karl Johan Astrom, 2002
**Another way the car could make these calculations of distance would be to use the angular distance via the car wheels. Calculating the number of rotations of the wheel and multiplying by the radius of the wheel would give the distance traveled.**
(2πr*ϴ = angular distance)
We have a number of equations that will allow us to calculate the speed of the car, and according to the rules of rolling without slipping, our translational speed has the same magnitude as our velocity (rw=v). From here it becomes easier to understand how the car knows the difference between the speed you are going and the speed you want to be going.
Say you start heading up a hill, the car can calculate the rate of acceleration (would be negative in this case), and make adjustments to the amount of gas being released to maintain the speed that you want.