# Week 3 - Laplace Transform

Updated 2017-10-10

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.