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

2. 2. Input signals

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

11. ＊ Properties of the Laplace transform
1. Linearity
2. Differentiation theorem

12. Zero initial values
Proof:

13. 3. Integration theorem

14. 4. Translation theorem
Proof:

17. 5. Final value theorem
6. Initial value theorem

18. 7. Complex translation theorem
8. 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
(2) Transfer function
Zero initials

26. (3) Laplace Inversion
Where
s
s
X
1
)
( 

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

51. S2. Under steady-state initial conditions
and define deviation variable

52. S3. Standard form
where

53. S4. Transfer function (Laplace form)
@ Step change ( )
s
M
s
i /
)
( 


55. ＊ Non-adiabatic thermal process example
S1. model
S2. Under deviation variables, the standard form

56. where

57. S3. Laplace form
@ Transfer functions

58. Ex. 4.6 Thermal process with transportation delay

59. @ Dead time

60. @ Transfer functions

61. ※ Transfer function (G(s))
Note: The transfer function defines the steady-state and
dynamic characteristic, or total response, of a system
described by a linear differential equation.

62. ＊Important properties of G(s)
1. Physical systems,
2. Transforms of the derivation of input and output
variables
3. Steady state responses
m
n 

63. ＊ Steady-state gain ( )
Ex. Consider two isothermal CSTRs in series
0
( ) s
G s 

64. Ans.:
(1) Steady-state gain:
(2) Final value of the reactant concentration in the
second reactor:
0
( ) p
s
G s K



65. ※ Block diagrams

66. @ Block diagram for
1
1
)
(
)
(




s
s
s
i 

67. Example 4.7 Block diagram for

68. ＊ Rules for block diagram

71. Example 4.8 Determine the transfer functions

73. Solution:
◎
◎

74. Example 3-4.3 Determine the transfer functions
=？

75. @ Reduced block

77. Example 4.9
=？

78. ◎ Answer

79. ◎ Design steps for transfer function

80. @ Review of complex number
c=a+ib

81. Polar notations

82. ※ Frequency response

83. ◎ Experimental determination of frequency response
S1. Process (valve, model, sensor/transmitter)

84. S2. Input signal
S3. Output response
where

85. P1. Amplitude of output signal
P2. Output signal ‘lags’ the input signal by θ.
P3. Amplitude ratio (AR): AR=Y0/X0
P4. Magnitude ratio (MR): MR=AR/K
P5. Phase angle (θ): if θ is negative, it is a lag angle.

86. Ex.4.7 A first-order transfer function G(s)=K/(τs+1)
＊ Consider a form of
If the input is set as
Then the output

87. ＊Through inverse Laplace transformation, the output
response is reduced as
P1.
P2. (p.69)

88. Ex.4.8 Consider a first-order system
S1.

89. S2. Amplitude ratio and phase angle
Ex.4.9 Consider a second-order system

90. S1. s=iω to decide amplitude ratio
＃

91. S2. Phase angle
＃
Ex.4.10 Consider a first-order lead transfer function
G(s)=K(1+τs)

92. Ex.4 Consider a pure dead time transfer function
G(s) =e-t0s

93. Ex.5 Consider an integrator
G(s)=1/s
G(i)=-(1/ )i

94. ＊ Expression of AR and θ for general OLTF

95. ※ Bode plot
•A common graphical representation of AR (MR) and θ functions.
•Bode plot consists: (1) log AR or (log MR) vs. log ω
(2) θ vs. log ω
* (3) 20 log AR (db) vs. log ω
2
k

96. Ex. 5 Consider a first-order lag by Ex. 1
To show Bode plot.
S1. MR1 as ω 0
S2. As ω 

97. ＃
＊ Types of Bode plots
(a) Gain element
(b) First-order lag
(c) Dead time
(d) Second-order lag
(e) First-order lead
(f) Integrator

104. ＊ Process control for a chemical reactor

106. Homework 2#
(1) Q4.6
(2) Q4.10
(3) Q4.16
(4) Q4.18 (※Difficulty)