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

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Presentation transcript:

Automatic Control Laplace 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) Slides 22

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

Unit step function Slides 24

Exponential function Slides 25

Frequency shift Slides 26

Sine and cosine functions Slides 27 Let

Sine and cosine functions Slides 28 Exercise: show that and

Impulse function Slides 29

Unit ramp Slides 210 (integration by parts) similarly

Differentiated function Slides 211

Integrated function Slides nd shifting theorem

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

Example Slides 214 Transformed Solution Transformed Equation

Convolution integral Slides 215 Ex:

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