1. Automatic Control Laplace Transformation Dr. Aly Mousaad Aly Department of Mechanical Engineering Faculty of Engineering, Alexandria University

2. 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) Slides 22

3. 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. Slides 23

4. Unit step function Slides 24

5. Exponential function Slides 25

6. Frequency shift Slides 26

7. Sine and cosine functions Slides 27 Let

8. Sine and cosine functions Slides 28 Exercise: show that and

9. Impulse function Slides 29

10. Unit ramp Slides 210 (integration by parts) similarly

11. Differentiated function Slides 211

12. Integrated function Slides 212 2 nd shifting theorem

13. Solution of differential equations using Laplace Transformation Slides 213 Differential Equation Transformed Equation Transformed Solution Laplace Transformation Algebraic Manipulation Inverse Laplace Particular Integral Complementary function

14. Example Slides 214 Transformed Solution Transformed Equation

15. Convolution integral Slides 215 Ex: