Automatic ControlLaplace Transformation Dr. Aly Mousaad Aly Department of Mechanical Engineering Faculty of Engineering, Alexandria University
Laplace Transformation The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), defined by: L F(s)
Why s-domain? • We can transform an ordinary differential equation into an algebraic equation which is easy to solve. • It is also easy to analyze and design interconnected (series, feedback etc.) systems.
Sine and cosine functions Exercise: show that and
Unit ramp (integration by parts) similarly
Integrated function 2nd shifting theorem
Solution of differential equations using Laplace Transformation Transformed Equation Differential Equation Laplace Transformation Algebraic Manipulation Particular Integral Complementary function Transformed Solution Solution Inverse Laplace
Example Transformed Equation Transformed Solution