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Automatic Control Laplace Transformation

Automatic Control Laplace Transformation

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Automatic Control Laplace Transformation

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  1. Automatic ControlLaplace 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)

  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.

  4. Unit step function

  5. Exponential function

  6. Frequency shift

  7. Sine and cosine functions Let

  8. Sine and cosine functions Exercise: show that and

  9. Impulse function

  10. Unit ramp (integration by parts) similarly

  11. Differentiated function

  12. Integrated function 2nd shifting theorem

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

  14. Example Transformed Equation Transformed Solution

  15. Convolution integral Ex: