Laplace Transform Definition
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## 1 Introduction

Laplace transforms is the tool controls engineers use to avoid dealing ith differential equations directly. Derivatives become a simple multiplication by s and integration is division by s. All of the messy tricks and manipulations required by differential equations is replaced with algebra.

Each linear time invariant component in a system can be turned into a transfer function in the Laplace domain. Once in the Laplace domain SISO system components can be multiplied, divided, summed, and subtracted using algebra. MIMO components can be manipulated using linear algebra.

Once a single transfer function for the system has been created the system response to an input can be easily determined. The Laplace domain system response can then be converted to time domain response by using the Inverse Laplace Transform and partial fractions.

## 2 Definition of the Laplace Transform

Definition[1]

 $LaTeX: \mathcal{L} \left[f\left( t \right)\right]=F\left(s\right)= \int_{0}^{\infty} f\left(t\right)e^{-st}dt$ Laplace 1

where

$LaTeX: \mathcal{L}$ denotes the Laplace transform,
$LaTeX: F\left(s\right)$ is the Laplace Transform,
$LaTeX: f\left(t\right)$ is the time function, and
$LaTeX: s=\sigma + j\omega$ is complex variable

The Laplace Transform is a linear operator which transforms $LaTeX: f \left( t \right)$ with a real argument t to a function $LaTeX: F \left( s \right)$ with a complex argument s.

## 3 Region of Convergence

The Laplace transform $LaTeX: F \left( s \right)$ typically exists for all complex numbers such that $LaTeX: Re\left\{s\right\} > a$, where $LaTeX: a$ is a real constant which depends on the growth behavior of $LaTeX: f \left( t \right)$. The subset of values of $LaTeX: s$ for which the Laplace transform exists is called the region of convergence (ROC) or the domain of convergence.

## 4 Final Value Theorem[2]

If all poles of $LaTeX: sF \left( s \right)$ lie in the left-half $LaTeX: s$ plane then,

 $LaTeX: f \left( \infty \right)= \lim_{t \to \infty}f \left( t \right)= \lim_{s \to 0}sF \left( s \right)$

Note if $LaTeX: sF\left(s\right)$ has poles on the imaginary axis or in the right-half s plane, $LaTeX: f\left(\infty\right)$ does not exist.

In other words, if $LaTeX: sF\left(s\right)$ is stable then the Final Value Theorem can be used to calculate the steady state error of the system $LaTeX: F\left(s\right)$.

## 5 References

1. ISBN 0 340 63183 X, Engineering Vibration Analysis with Application to Control Systems
2. ISBN 1-56347-261-9, Space Vehicle Dynamics and Control