01 - Laplace Transform for Process Control

The Laplace transform of a function, \(f(t)\), is defined as \[F(s) = \mathcal{L}[f(t)] = \int_0^\infty f(t) e^{-st} dt\] where \(F(s)\) is the symbol for the Laplace transform, \(\mathcal{L}\) is the Laplace transform operator, and \(f(t)\) is some function of time, \(t\).

The \(\mathcal{L}\) operator transforms a time domain function \(f(t)\) into an ‘\(s\)’ domain function, \(F(s)\).

\(s\) is a complex variable: \(s = x + iy\).