Week 3 - Laplace Transform

Context: when dealing with control systems, taking derivatives and integration is common, but is too error-prone, computation-intensive, and complicated. The Laplace transform of a function will allow us to deal with the system in frequency domain. As a result, derivation and integration turns into multiplication and division of \(s\), the complex frequency.

Definition

The definition of full Laplace transform is given as:

\[\mathcal L:\int_{-\infty}^\infty f(t)e^{-st}\mathrm dt\]

The function is integrated from time being \(-\infty\) to \(\infty\). However, we don’t care about what happens the time far far before. We care about the system after time 0. Thus we have the half Laplace transform:

\[\mathcal L:\int_{0^-}^\infty f(t)e^{-st}\mathrm dt\]

List of Laplace Transforms

Integration

\[\mathcal L\left\{\int x \mathrm dx\right\}=\frac 1 s x\]

Derivation

\[\mathcal L \left\{\frac {\mathrm d}{\mathrm dt} x\right\}=sx\]

Delta Function

\[\mathcal L \left\{\delta (t)\right\}=\mathcal L \left\{\frac{\mathrm d u(t)}{\mathrm dt}\right\}=s\cdot \frac 1 s=1\]

Where \(\delta(t)\) is a “infinite spike” at \(t=0\), and \(u(t)\) is the unit step function.