Open Loop vs. Closed Loop

Meet the drone

June: I borrowed a drone! Its job: carry small packages and hover at exactly 2 metres so someone can grab them.

Mika: It's like the shower, right? One number we want (2 m), one thing we can change (how hard the propellers push).

Here are the facts. The drone weighs 0.5 kg, so gravity pulls it down with $0.5 \times 9.81 \approx 4.9$ newtons. The propellers push up with a thrust $\eff{T}$ that we choose. When the drone moves, air pushes back a little: about 1 newton for every metre per second of speed.

Don’t worry about the dots over $\out{h}$ yet. They describe speed and acceleration; we will build both in Chapter 3. When the drone is stationary, 4.9 N balances gravity: more thrust starts it upward, less lets it fall. When it is moving, air drag also pushes against its motion, so 4.9 N slows it toward rest.

Just tell it what to do

Theo: We know the physics exactly. So we don't need to look at anything. We can calculate a plan: push hard for a bit to climb, then push exactly 4.9 N to hover.

That kind of plan is called open loop: the plan decides the thrust, and nobody checks what actually happens. Try it. Choose a climb thrust and how long to climb, then launch.

Chapter 1: Open Loop vs. Closed Loop ↗

Theo: I worked it out. While climbing, the drone soon settles into a steady speed of (extra thrust ÷ drag). The height you end up at is just (extra thrust ÷ drag) × climb time. For 6 N that's $(6 - 4.905)/1 = 1.095$ m/s, so climb for $2 \div 1.095 \approx 1.83$ seconds.

There is a subtlety in Theo’s calculation: when he switches to 4.9 N, the drone is still climbing. Drag gradually stops it. His final-height formula includes that extra distance after the switch.

Press Theo's plan. It lands on 2.00 m. Beautiful.

Mika: Theo, that's amazing! We're done! Chapter over!

June: Let's just check what happens outside… there's a bit of a breeze today.

Run Theo's plan again, and while it's hovering press Gust (a push downward of 1 N for one second).

Theo: It dropped about a metre… and it's staying down there. My numbers were right!

June: Your numbers were perfect. The wind just wasn't in your plan. And the drone has no idea it's in the wrong place.

That was our mistake: we believed "if you get the numbers right, you don't need feedback." The calm-air success was real, and the plan was still useless the moment the world did something it didn't expect. After the gust, the thrust is exactly 4.9 N again, so the drone just stays wherever the gust left it.

The package test

Now the real job. Below, the drone is hovering at 2 m with a fixed 4.9 N of thrust, exactly balancing its weight. Someone clips on a 0.2 kg package.

Predict first! The package makes the drone heavier, but the thrust stays at 4.9 N. What happens?

Chapter 1: Open Loop vs. Closed Loop ↗

Crash. Nothing in an open-loop plan can notice that the drone is falling. Now switch the mode to Feedback: look at the height, reset, and try the package and the gust again.

In feedback mode the drone uses a simple rule: push with the hover thrust, plus an extra 8 N for every metre it is too low.

Mika: Whoa. The gust knocks it down less than 20 cm, and it climbs right back. The package makes it dip, but it doesn't crash!

Theo: It doesn't need to know what happened. It just sees it's too low and pushes harder. That's the trick.

June: But look: with the package it settles at about 1.75 m, not 2. Why doesn't it go all the way back?

Drawing it

Engineers draw these systems as boxes and arrows called block diagrams. Each box does something; each arrow carries a signal. We drew ours on the whiteboard one piece at a time.

Chapter 1: Open Loop vs. Closed Loop ↗

The key difference is that one arrow, looping back from the output to the start. That's why feedback control is called closed loop: information goes all the way round. Scroll back up to the package test: the diagram under the drone lights up whichever part is busy right now.

Same picture, different machine

Here's the same diagram for a car's cruise control. The setpoint is a speed, the plant is the car, and the disturbance is a hill. The controller presses the accelerator harder when the car is slower than it should be.

Chapter 1: Open Loop vs. Closed Loop ↗

June: Thermostats, cruise control, the shower, our drone… it's the same loop every time, with different labels!

What we know so far

Our loop has a mystery number in it: 8 newtons per metre. Why 8? Why not 2, or 50? And why does the drone settle too low?