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

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Presentation on theme: " Laplace Transform 1.Definition Chapter 4 Modelling and Analysis for Process Control."— Presentation transcript:

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

2 2.Input signals

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

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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:

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17 5.Final value theorem 6.Initial value theorem

18 7.Complex translation theorem 8.Complex differentiation theorem

19 Example 4.1 Solution:

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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

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

28 (ii) Partial fractions where

29 (iii) Inversion * Repeated roots If r 1 =r 2, the expansion is carried out as

30 where Inversion

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

32 and

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36 and A 3 =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)=Y 1 (s)e -st 0 and

44 Example:

45 Input function f(t)

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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 Step change ( )

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

56 where

57 S3. Laplace 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

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

64 Ans.: (1)Steady-state gain: (2) Final value of the reactant concentration in the second reactor:

65 ※ Block diagr ams

66 @ Block diagram for

67 Example 4.7 Block diagram for

68 * Rules for block diagram

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71 Example 4.8 Determine the transfer functions

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73 Solution: ◎◎◎◎

74 Example Determine the transfer functions =?=?

75 @ Reduced block

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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=Y 0 /X 0 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 -t 0 s

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

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

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104 * Process control for a chemical reactor

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


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