Presentation on theme: "CHAPTER III LAPLACE TRANSFORM"— Presentation transcript:
1 CHAPTER III LAPLACE TRANSFORM Process ControlCHAPTER IIILAPLACE TRANSFORM
2 The Laplace transform of a function f(t) is defined as; In the application of Laplace transform variable time is eliminated and a new domain is introduced.In the modeling of dynamic systems differential equations are solved by using Laplace transform.
3 Properties of Laplace Transform The Laplace transform contains no information about f(t) for t<0. (Since t represents time this is not a limitation)Laplace transform is defined with an improper integral. Therefore the required conditions arethe function f(t) should be piecewise continuousthe integral should have a finite value; i.e., the function f(t) does not increase with time faster than e -st decreases with time.
4 Laplace transform operator transforms a function of variable t to a function of variable s. e.g., T(t) becomes T(s)The Laplace transform is a linear operator.Tables for Laplace transforms are available. In those tables inverse transforms are also given.
5 Transforms of Some Functions 1. The constant function
12 Solution of Differential equations by Laplace Transform For the solution of linear, ordinary differential equations with constant coefficients Laplace Transforms are applied.The procedure involves:Take the Laplace Transform of both sides of the equation.Solve the resulting equation for the Laplace transform of the unknown function. i.e., evaluate x(s).Find the function x(t), which has the Laplace Transform obtained in step 2.