controls
Control theory is how you make a real, sluggish, noisy thing track a set-point you actually want. Measure → compare → push → repeat. This page starts where most courses start — a damped second-order plant and a PID controller — but lets you turn the knobs first and read the math second.
PID step response
Plant G(s) = 1 / (s² + 0.4 s + 1). Controller u = Kp e + Ki ∫e dt + Kd ė. Setpoint steps from 0 → 1 at t = 0.
rise time —
overshoot —
settling (±2%) —
steady-state err —
what to try
- Pure proportional (Ki=Kd=0): bump Kp. Faster, but watch the steady-state offset linger.
- Add integral: offset disappears, but the overshoot grows and settling time blows up.
- Add derivative: overshoot calms down — D acts on the rate of change, so it brakes before you hit the wall.
- Ziegler-Nichols ballpark: Kp≈3.0, Ki≈1.5, Kd≈0.7.
where this is going
Ten short lessons, each with a sandbox like the one above. Built in order.
- 01Feedback. Why measure-and-correct beats open-loop, even when your model is wrong.
- 02PID. The three knobs that go further than they deserve. live above ↑
- 03Transfer functions. What a system is, written in s.
- 04Poles & zeros. Where stability lives. Move a pole, watch the response change.
- 05Bode plots. Gain and phase at every frequency, on log paper.
- 06Nyquist. The contour that tells you when the loop will sing.
- 07Root locus. How the poles slide as you turn up the gain.
- 08State-space. Vectors instead of polynomials. ẋ = Ax + Bu.
- 09Controllability & observability. Can you steer it? Can you see it?
- 10Observers. Estimating the state you can't measure — Luenberger, then Kalman.
Then three applied: motor speed, inverted pendulum, quadrotor altitude.