The Laplace transform f(s) contains no information about the behavior of f(t) for t | 14 THE LAPLACE TRANSFORM Example . Find the Laplace transform of the function i 1 According to Eq. 2. 1 00 t 00 f S f e-stdt - e-S Jo S t 0 1 s Thus 1 1 5 There are several facts worth noting at this point 1. The Laplace transform f s contains no information about the behavior of f t for t 0. This is not a limitation for control system study because t will represent the time variable and we shall be interested in the behavior of systems only for positive time. In fact the variables and systems are usually defined so that f t 0 for t 0. This will become clearer as we study specific examples. 2. Since the Laplace transform is defined in Eq. by an improper integral it will not exist for every function f t . A rigorous definition of the class of functions possessing Laplace transforms is beyond the scope of this book but readers will note that every function of interest to us does satisfy the requirements for possession of a transform. 3. The Laplace transform is linear. In mathematical notation this means aL h f bLfaif where a and b are constants and f i and 2 are two functions of t. Proof. Using the definition L afi t bf2 t i a i i bf2 t e sldt Jo a fi j e stdt b f2 f e stdl Jo Jo bL f2 t 4. The Laplace transform operator transforms a function of the variable t to a function of the variable s. The variable is eliminated by integration. Transforms of Simple Functions We now proceed to derive the transforms of some simple and useful functions. For details on this and related mathematical topics see Churchill 1972 . THE LAPLACE TRANSFORM 15 1. The step function f f i 0 1 t 0 This important function is known as the unit-step function and will henceforth be denoted by u t . From Example it is clear that - As expected the behavior of the function for t 0 has no effect on its Laplace transform. Note that as a consequence of linearity the transform of any constant A that is f j Au t isjust f s A s. 2. The exponential function x 0 t 01 r -at -at U g I e at t 0J .
