Download presentation

Published byJaxson jay Fereday Modified over 4 years ago

1
**Chapter 4 Modelling 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
**＊ Repeated roots (iii) Inversion**

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 ( )

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)**

Physical systems, Transforms of the derivation of input and output variables Steady state responses

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

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

65
※ Block diagrams

66
@ Block diagram for

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**

P2. (p.69)

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

89
**S2. Amplitude ratio and phase angle**

Ex.4.9 Consider a second-order system

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

＃

91
**G(s)=K(1+τs) 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

96
**Ex. 5 Consider a first-order lag by Ex. 1**

To show Bode plot. S1. MR1 as ω 0 S2. As ω

97
＃ ＊ Types of Bode plots Gain element First-order lag Dead time Second-order lag First-order lead Integrator

104
**＊ Process control for a chemical reactor**

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

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google