PID · controls

what's this demo?

The damped second-order plant G(s) = 1 / (s² + 0.4 s + 1) is the workhorse of every intro controls course. It lags. It rings. It's just hard enough to make a controller look smart and just simple enough to debug in your head.

Below is the same plant wearing two faces. On top: a needle gauge that sweeps from 0 toward the red target line at $y = 1$ in real time — the way an oven temperature dial or a motor-RPM tach would actually look on a real machine. Below it: the same response, but plotted vs. time so you can see rise, overshoot, and settle in retrospect. Hit ▶ play to watch a 20-second run.

PID step response

Plant G(s) = 1 / (s² + 0.4 s + 1). Controller u = K_p e + K_i ∫e dt + K_d ė. Setpoint steps from 0 → 1 at t = 0.

y 0.000 target 1.000 t 0.00 s u 0.00

target (setpoint r) 1.00 K_p 2.00 K_i 0.50 K_d 0.50

noise off

disturbance —

rise time —

overshoot —

settle (±2%) —

SSE (live) —

what to try

Hit ▶ play with defaults. The needle leaves 0, ramps up, overshoots past the target, then settles. The trace below records that whole story; the dot on it tracks the needle.
Pure proportional (K_i = K_d = 0): bump K_p. Faster needle, but it parks below the target — the steady-state offset never goes away.
Add integral: the needle eventually pulls up to the target, but overshoot grows and settling stretches out.
Add derivative: overshoot calms down — D acts on the rate of change, so it brakes before the needle hits the wall.
Now turn on noise and replay. The D-term explodes because it amplifies the derivative of the noisy measurement — watch the needle judder. This is why we'll need observers (Lesson 10).
Kick the plant. At t = 8 s the needle suddenly drops; only the I-term reliably brings it back to the target.
Ziegler–Nichols ballpark: K_p ≈ 3.0, K_i ≈ 1.5, K_d ≈ 0.7.

show the math

The plant is canonical damped second-order with natural frequency $o m e g a_{n} = 1$ and damping ratio $z e t a = 0.2$ :

dd o t y + 0.4, d o t y + y = u

The controller closes the loop on the error $e = r - y$ :

u (t) = K_{p}, e (t) + K_{i} in t_{0}^{t} e (a u), d a u + K_{d}, d o t e (t)

We integrate by forward-Euler at $d t = 0.01, e x t s$ for $T = 20, e x t s$ . Toggling noise adds Gaussian $s i g ma$ to the measurement before the error is computed — notice how the derivative term reacts to the noisy measurement, not the smooth true state.