Laplace transform

Level AdvancedDifficulty ★★★★★Concept⌖ Open in the map

What is it?

$F(s) = \int_0^\infty f(t)\,e^{-st}\,\dd t$ : like Fourier but with complex $s$ , it handles growth, decay and initial conditions. Linear ODEs become algebraic equations in $s$ , and a system becomes its transfer function $G(s)$ .

Formulas

$\mathcal L\{f'\}(s) = s F(s) - f(0), \qquad \mathcal L\{e^{at}\} = \frac{1}{s - a}$; $G(s) = \frac{Y(s)}{U(s)}, \qquad C_{\text{PID}}(s) = K_p + \frac{K_i}{s} + K_d\,s$
transfer function; a PID controller in the $s$ -domain

Where it shows up in computing

Control theory★★★★★fundamentalRobotics and control
Classical control (Bode, Nyquist, root locus) works with transfer functions in the Laplace domain.
PID control★★★★★frequentRobotics and control
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★★★★★
PID control★★★★★

∿ Signals, media and vision

Z-transform→Digital filters★★★★★

What depends on it

Z-transform

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