Laplace Transforms, Its History, Importance And Uses.

Introduction :

Laplace Transforms are very much essential in an automatic control. Based on an operational standpoint, the rest parts are about guidelines for a primer in automatic control. Beyond an undergraduate course, the two last parts, a little bit more technical, are devoted on the one hand to get the model of a computer controlled system and on the other hand to relate the operational standpoint to usual tables used in some cases.

The Laplace transform approach leads to dene the transfer function of a system. It is used to get the corresponding response signal of a system with respect to a given input signal. The Laplace transform is also important for the analysis and design of control systems.

This tool appears thus a necessary and unavoidable burden for students participating in automatic control courses.

The Laplace transform is one of the mathematical tools used for the solution of ordinary linear differential equations. The Laplace transform method has the following two attractive features:

1. The homogeneous equation and the particular integral are solved in one operation.
2. The Laplace transform converts the differential equation into an algebraic equation in seconds.

It is possible to manipulate the algebraic equation by simple algebraic rules to obtain the solution in the s-domain. The final solution is obtained by taking the inverse Laplace transform.

History of Laplace Transform :

Transformation in mathematics deals with the conversion of one function to another function that may not be in the same domain.The transform method finds its application in those problems which can’t be solved directly.

This transform is named after the mathematician and renowned astronomer Pierre Simon Laplace who lived in France. He used a similar transform on his additions to the probability theory. It became popular after World War Two.

This transform was made popular by Oliver Heaviside, an English Electrical Engineer. Other famous scientists such as Niels Abel, Mathias Lerch, and Thomas Bromwich used it in the 19th century. The complete history of the Laplace Transforms can be tracked a little more to the past, more specifically 1744.

This is when another great mathematician called Leonhard Euler was researching on other types of integrals. Euler however did not pursue it very far and left it. An admirer of Euler called Joseph Lagrange;made some modifications to Euler’s work and did further work.

LaGrange’s work got Laplace’s attention 38 years later, in 1782 where he continued topick up where Euler left off. But it was not 3 years later; in 1785 where Laplace had a stroke of genius and changed the way we solve differential equations forever. He continued to work on it and continued to unlock the true power of the Laplace transform until 1809, where he started to use infinity as an integral condition.

Importance of Laplace Transform :

The following is a summary of the areas where the Laplace transform is considered important by the teachers :

1.Importance in specific areas as
• Automatic Control
• Circuit theory
• Economics (from the statistical point of view)

2. Importance as a tool
• To solve differential equations• For static analysis • For continuous systems

3. To facilitate calculation working in the Laplace domain.
4. Is fundamental to understand the systems.
5. To solve problems eliminating noise, perturbations, etc.
6. As a way of describing development of processes.

Conclusion :

To teach a basic lecture in automatic control using the operational method offers some advantages. The integral or derivative operators allow to link every notion to its physical meaning. We should keep in mind that a transfer operator is always related to only a differential equation or a difference equation.

Moreover Laplace transform has also been applied to various problems: evaluation of payments, reliability and maintenance strategies, utility factions of analysis, choice of investments, assembly line and queuing system problems, theory of system and element behaviours, investigation of the dispatching aspect of job-shop scheduling, assessing econometric models and may others areas as well.