Laplace Transform Definition

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

2. Input signals

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

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

12. Zero initial values Proof:

13. Integration theorem

14. Translation theorem Proof:

17. Final value theorem • Initial value theorem

18. Complex translation theorem • Complex differentiation theorem

19. Example 4.1 Solution:

21. Example 4.2 (S1)

22. (S2)

23. ＊ Laplace transform procedure for differential equations Steps:

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

25. Algebraic rearrangement Zero initials (2) Transfer function

26. (3) Laplace Inversion Where

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

28. (ii) Partial fractions where

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

30. where Inversion

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

32. and

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

37. The step response: Example 4.3

38. (S1)

39. (S3) Find coefficients s=0 Inversion

40. Example 4.4 (S1) Laplace transformation

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

42. (S3) Inversion and using the identity

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

44. Example:

45. Input function f(t)

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

50. S1. Energy balance