Laplace Transform Definition

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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

6 See Also

If everything is included in 1 article then it gets very long. With all the TeX equations being converted to pngs the article gets too big for some browsers to edit. So the original article has been broken up into multiple articles.