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Laplace Transform Definition

Chapter 4 Modelling and Analysis for Process Control. Laplace Transform Definition. Input signals. (c) A unit impulse function (Dirac delta function). * Properties of the Laplace transform. Linearity Differentiation theorem. Zero initial values. Proof:. Integration theorem.

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Laplace Transform Definition

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  1. Chapter 4Modelling and Analysis for Process Control • Laplace Transform • Definition

  2. Input signals

  3. (c) A unit impulse function (Dirac delta function)

  4. * Properties of the Laplace transform • Linearity • Differentiation theorem

  5. Zero initial values Proof:

  6. Integration theorem

  7. Translation theorem Proof:

  8. Final value theorem • Initial value theorem

  9. Complex translation theorem • Complex differentiation theorem

  10. Example 4.1 Solution:

  11. Example 4.2 (S1)

  12. (S2)

  13. * Laplace transform procedure for differential equations Steps:

  14. Exercises: a second-order differential equation (1) Laplace transform

  15. Algebraic rearrangement Zero initials (2) Transfer function

  16. (3) Laplace Inversion Where

  17. Inversion method: Partial fractions expansion (pp.931) (i) Fraction of denominator and

  18. (ii) Partial fractions where

  19. (iii) Inversion *Repeated roots If r1=r2, the expansion is carried out as

  20. where Inversion

  21. *Repeated roots for m times If the expansion is carried out as

  22. and

  23. and A3=2 as (a) case.

  24. The step response: Example 4.3

  25. (S1)

  26. (S3) Find coefficients s=0 Inversion

  27. Example 4.4 (S1) Laplace transformation

  28. (S2) Find coefficients s=0 s=1-j s=-1+j

  29. (S3) Inversion and using the identity

  30. Time delays: Consider Y(s)=Y1(s)e-st0 and

  31. Example:

  32. Input function f(t)

  33. *Input-Output model and Transfer Function Ex.4.5 Adiabatic thermal process example

  34. S1. Energy balance

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