PID control

Level UniversityDifficulty ★★★★★Application⌖ Open in the map

What is it?

The most used controller in the world: correct proportionally to the error (P), to its accumulated integral (I, removes steady offset) and to its derivative (D, anticipates and damps). Calculus in one line, running in thermostats, drones, cruise control and hard-disk heads.

Formulas

$u(t) = K_p\,e(t) + K_i\int_0^t e(s)\,\dd s + K_d\,\frac{\dd e}{\dd t}$; $u_k = K_p e_k + K_i h\sum_{j\le k} e_j + K_d\,\frac{e_k - e_{k-1}}{h}$
discrete version on a microcontroller (Riemann sum + finite difference)

The mathematics behind it

Derivative★★★★★frequent
The D term reacts to the derivative of the error, anticipating where the system is going.
Second-order linear equations (oscillations)★★★★★frequent
Tuning a controller is choosing the damping ratio $\zeta$ of the closed loop: fast without overshoot.
Laplace transform★★★★★frequent
Derivative and integral actions become $s$ and $1/s$ : PID design is pole placement in $s$ .

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⚙ Robotics and control

Control theory★★★★★
Control theory→Trajectory optimization and MPC★★★★★

What depends on it

Control theory

Exercises

1Applied

A drone hovers 2 m below its target height. Explain what each of P, I and D contributes, and what goes wrong with only P.

Solution

P pushes up in proportion to the 2 m error; D brakes as the error shrinks quickly, preventing overshoot; I accumulates any persistent error. With only P, gravity needs a non-zero thrust, which requires a non-zero error: the drone settles below the target (steady-state error) and may oscillate if $K_p$ is large.

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.