History

The origin of Laplace transforms dates back to the era of a British civil engineer named Oliver Heaviside, who lived from 1850 to 1925. An accomplished mathematician, Heaviside was experimenting with what came to be called mathematical operators. In operational calculus, for example, the letter p might replace the operation of taking a derivative, so that if x = f(t), then px was equivalent to

Oliver Heaviside's contemporaries ridiculed his work, not so much because of the operators themselves, but because he actually moved operators around in algebraic expressions as if they were ordinary terms. This did not deter him, however, because as far as he was concerned, the method worked.

Eventually, results prevailed, and from Oliver Heaviside's beginnings, mathematicians developed a set of operational transforms that came to be known as the Laplace transforms. Just as the use of logarithms can reduce multiplication and division to addition and subtraction, Laplace transforms, where they can be applied, can reduce the problem of solving a differential equation to one of solving an algebraic equation.

In the analysis of control systems, process variables vary with time. The observed behavior may be described by a differential equation, which has time as the independent variable. In such cases, what is required is an expression x = f(t), which describes the behavior of the dependent variable on a time basis, and which is clear of any derivatives. Laplace transforms is a mathematical technique through which this may be achieved.

In Laplace transforms, time...