A feedback adventure in twelve notebooks

Who Keeps the Drone Up?

How does a machine keep itself where it's supposed to be?

This started in a shower. One of us turned the knob, got scalded, turned it back, froze, and turned it again. Then we wondered: a little drone can hover perfectly still in the wind with no brain at all. How?

These are our notebooks from working it out, including every wrong turn. We start from zero: no calculus needed. By the end you'll know why the Laplace transform exists, what poles really are, and you'll tune a PID controller that beats your own hands.

Start with the shower →

The notebooks

1. The Shower ProblemWhy can't I just get the shower to 38 °C?
2. Open Loop vs. Closed LoopCan we just tell the drone what thrust to use?
3. Proportional ControlHow hard should the drone push when it's too low?
4. The Language of ChangeHow do we describe how fast something changes?
5. Why Exponentials Are MagicIs there a curve whose slope is a copy of itself?
6. Spinning NumbersWhat number squares to −1, and why would it wiggle?
7. The Map of sHow does a spinning number make a real wiggle, and what does every s look like?
8. The Drone's Real PersonalityWhy does the drone bounce like a spring?
9. The Laplace ProbeCan one measured area tell us where a signal lives?
10. Calculus into AlgebraCan we turn calculus into algebra?
11. Transfer Functions, Poles and ZerosCan we read a system's whole personality off one fraction?
12. PID: Fixing EverythingHow do we kill the droop and stop the wiggle?
13. Back to the ShowerWhy did we oscillate, and how close to the edge are we?
14. The Drone ChallengeCan you keep the drone at 2 m through everything?