---
author: Akbar Rahman
date: \today
title: MMME2046 // Control
tags: [ mmme2046, uon, uni, control ]
uuid: 
---

# System and Block Diagrams

# Laplace Transform

$$F(s) = \mathscr L {F(t)} = \int^\infty_0 f(t)e^{-st} \mathrm{d}t$$

where $s = \alpha + j\omega$

The function $F(s)$ is often much easier to manipulate than periodic function $f(t)$.

## Final Value Theorem

As $f(t)$ tends to infinity, $sF(s)$ tends to 0.

## Example

$$\dot x_o = ax_o = ax_i$$

where $x_o$ is the output and $x_i$ is the input

Take the Laplace transform:

$$sX_o(s) + aX_o(s) = aX_i(s)$$

Rearrange to get equation for the transfer function:

$$G(s) = \frac{X_o}{X_i} = \frac{a}{s+a}$$

$$ X_o = GX_i $$

If $X_i$ is a unit step, then:

$$X_i = \frac1s$$

and

$$X_o = \frac{a}{s(s+a)}$$

Taking the inverse gives:

$$X_0 = 1 - e^{-at}$$