Transfer Function Models of Dynamical Processes. Process Dynamics and Control

Transcription

1 Transfer Function Models of Dynamical Processes Process Dynamics and Control 1

2 Linear SISO Control Systems General form of a linear SISO control system: this is a underdetermined higher order differential equation the function must be specified for this ODE to admit a well defined solution 2

3 Heated stirred-tank model (constant flow, ) Transfer Function Taking the Laplace transform yields: or letting Transfer functions 3

4 Transfer Function Heated stirred tank example e.g. The block is called the transfer function relating Q(s) to T(s) 4

5 Process Control Time Domain Laplace Domain Process Modeling, Experimentation and Implementation Transfer function Modeling, Controller Design and Analysis Ability to understand dynamics in Laplace and time domains is extremely important in the study of process control 5

6 Transfer functions Transfer functions are generally expressed as a ratio of polynomials Where The polynomial is called the characteristic polynomial of Roots of Roots of are the zeroes of are the poles of 6

7 Order of underlying ODE is given by degree of characteristic polynomial e.g. First order processes Transfer function Second order processes Order of the process is the degree of the characteristic (denominator) polynomial The relative order is the difference between the degree of the denominator polynomial and the degree of the numerator polynomial 7

8 Transfer Function Steady state behavior of the process obtained form the final value theorem e.g. First order process For a unit-step input, From the final value theorem, the ultimate value of is This implies that the limit exists, i.e. that the system is stable. 8

9 Transfer function Transfer function is the unit impulse response e.g. First order process, Unit impulse response is given by In the time domain, 9

10 Transfer Function Unit impulse response of a 1st order process 10

11 To remove dependence on initial condition e.g. Deviation Variables Compute equilibrium condition for a given and Define deviation variables Rewrite linear ODE 0 or 11

12 Assume that we start at equilibrium Deviation Variables Transfer functions express extent of deviation from a given steady-state Procedure Find steady-state Write steady-state equation Subtract from linear ODE Define deviation variables and their derivatives if required Substitute to re-express ODE in terms of deviation variables 12

13 Process Modeling Gravity tank F o h F Objectives: height of liquid in tank Fundamental quantity: Mass, momentum Assumptions: Outlet flow is driven by head of liquid in the tank Incompressible flow Plug flow in outlet pipe Turbulent flow L 13

14 Transfer Functions From mass balance and Newton s law, A system of simultaneous ordinary differential equations results Linear or nonlinear? 14

15 Nonlinear ODEs Q: If the model of the process is nonlinear, how do we express it in terms of a transfer function? A: We have to approximate it by a linear one (i.e.linearize) in order to take the Laplace. f(x) f(x 0 )! f! x ( x 0 ) x 0 x 15

16 Nonlinear systems First order Taylor series expansion 1. Function of one variable 2. Function of two variables 3. ODEs 16

17 Transfer unction Procedure to obtain transfer function from nonlinear process models Find an equilibrium point of the system Linearize about the steady-state Express in terms of deviations variables about the steady-state Take Laplace transform Isolate outputs in Laplace domain Express effect of inputs in terms of transfer functions 17

18 Block Diagrams Transfer functions of complex systems can be represented in block diagram form. 3 basic arrangements of transfer functions: 1. Transfer functions in series 2. Transfer functions in parallel 3. Transfer functions in feedback form 18

19 Block Diagrams Transfer functions in series Overall operation is the multiplication of transfer functions Resulting overall transfer function 19

20 Block Diagrams Transfer functions in series (two first order systems) Overall operation is the multiplication of transfer functions Resulting overall transfer function 20

21 Transfer Functions DC Motor example: In terms of angular velocity In terms of the angle 21

22 Transfer Functions Transfer function in parallel + + Overall transfer function is the addition of TFs in parallel 22

23 Transfer Functions Transfer function in parallel + + Overall transfer function is the addition of TFs in parallel 23

24 Transfer Functions Transfer functions in (negative) feedback form Overall transfer function 24

25 Transfer Functions Transfer functions in (positive) feedback form Overall transfer function 25

26 Transfer Function Example

27 Transfer Function Example A positive feedback loop 27

28 Transfer Function Example 3.20 Two systems in parallel Replace by 28

29 Transfer Function Example 3.20 Two systems in parallel 29

30 Transfer Function Example 3.20 A negative feedback loop 30

31 Transfer Function Example 3.20 Two process in series 31

32 First Order Systems First order systems are systems whose dynamics are described by the transfer function where is the time constant is the system s (steady-state) gain First order systems are the most common behaviour encountered in practice 32

33 First Order Systems Examples, Liquid storage Assume: Incompressible flow Outlet flow due to gravity Balance equation: Total Flow In Flow Out 33

34 First Order Systems Balance equation: Deviation variables about the equilibrium Laplace transform First order system with 34

35 First Order Systems Examples: Cruise control DC Motor 35

36 First Order Systems Liquid Storage Tank Speed of a car DC Motor First order processes are characterized by: 1. Their capacity to store material, momentum and energy 2. The resistance associated with the flow of mass, momentum or energy in reaching their capacity 36

37 Step response of first order process First Order Systems Step input signal of magnitude M The ultimate change in is given by 37

38 First Order Systems Step response 38

39 First Order Systems What do we look for? System s s Gain: Steady-State Response Process Time Constant: Time Required to Reach 63.2% of final value What do we need? System initially at equilibrium Step input of magnitude M Measure process gain from new steady-state Measure time constant 39

40 First Order Systems First order systems are also called systems with finite settling time The settling time is the time required for the system comes within 5% of the total change and stays 5% for all times Consider the step response The overall change is 40

41 First Order Systems Settling time 41

42 First Order Systems Process initially at equilibrium subject to a step of magnitude 1 42

43 First order process Ramp response: Ramp input of slope a

44 First Order Systems Sinusoidal response Sinusoidal input Asin(ωt) AR φ

45 First Order Systems 10 0 Bode Plots 10-1 Corner Frequency High Frequency Asymptote Amplitude Ratio Phase Shift 45

46 Integrating Systems Example: Liquid storage tank F i h Laplace domain dynamics F If there is no outlet flow, 46

47 Integrating Systems Example Capacitor Dynamics of both systems is equivalent 47

48 Integrating Systems Step input of magnitude M Input Output Slope = Time Time 48

49 Integrating Systems Unit impulse response Input Output Time Time 49

50 Integrating Systems Rectangular pulse response Input Output Time Time 50

51 Second order Systems Second order process: Assume the general form where = Process steady-state gain = Process time constant = Damping Coefficient Three families of processes Underdamped Critically Damped Overdamped 51

52 Three types of second order process: Second Order Systems 1. Two First Order Systems in series or in parallel e.g. Two holding tanks in series 2. Inherently second order processes: Mechanical systems possessing inertia and subjected to some external force e.g. A pneumatic valve 3. Processing system with a controller: Presence of a controller induces oscillatory behavior e.g. Feedback control system 52

53 Second order Systems Multicapacity Second Order Processes Naturally arise from two first order processes in series By multiplicative property of transfer functions 53

54 Transfer Functions First order systems in parallel + + Overall transfer function a second order process (with one zero) 54

55 Inherently second order process: e.g. Pneumatic Valve Second Order Systems p By Newton s law x 55

56 Second Order Systems Feedback Control Systems 56

57 Second order Systems Second order process: Assume the general form where = Process steady-state gain = Process time constant = Damping Coefficient Three families of processes Underdamped Critically Damped Overdamped 57

58 Second Order Systems Roots of the characteristic polynomial Case 1) Case 2) Case 3) Two distinct real roots System has an exponential behavior One multiple real root Exponential behavior Two complex roots System has an oscillatory behavior 58

59 Second Order Systems Step response of magnitude M ξ=0 ξ= ξ=

60 Second Order Systems Observations Responses exhibit overshoot when Large yield a slow sluggish response Systems with yield the fastest response without overshoot As (with ) becomes smaller, system becomes more oscillatory 60

61 Second Order Systems Characteristics of underdamped second order process 1. Rise time, 2. Time to first peak, 3. Settling time, 4. Overshoot: 5. Decay ratio: 61

62 Second Order Systems Step response 62

63 Second Order Systems Sinusoidal Response where 63

64 Second Order Systems 64

65 More Complicated Systems Transfer function typically written as rational function of polynomials where and can be factored following the discussion on partial fraction expansion s.t. where and appear as real numbers or complex conjuguate pairs 65

66 Poles and zeroes Definitions: the roots of are called the zeros of G(s) the roots of are called the poles of G(s) Poles: Directly related to the underlying differential equation If, then there are terms of the form in vanishes to a unique point If any then there is at least one term of the form does not vanish 66

67 Poles e.g. A transfer function of the form with can factored to a sum of A constant term from A from the term A function that includes terms of the form Poles can help us to describe the qualitative behavior of a complex system (degree>2) The sign of the poles gives an idea of the stability of the system 67

68 Poles Calculation performed easily in MATLAB Function POLE, PZMAP e.g.» s=tf( s );» sys=1/s/(s+1)/(4*s^2+2*s+1); Transfer function: s^4 + 6 s^3 + 3 s^2 + s» pole(sys) ans = i i MATLAB 68

69 Poles Function PZMAP» pzmap(sys) MATLAB 69

70 Poles One constant pole integrating feature One real negative pole decaying exponential A pair of complex roots with negative real part decaying sinusoidal What is the dominant feature? 70

71 Poles Step Response Integrating factor dominates the dynamic behaviour 71

72 Poles Example Poles i i One negative real pole Two purely complex poles What is the dominant feature? 72

73 Poles Step Response Purely complex poles dominate the dynamics 73

74 Poles Example Poles Three negative real poles What is the dominant dynamic feature? 74

75 Poles Step Response Slowest exponential ( ) dominates the dynamic feature (yields a time constant of 1/0.1868= 5.35 s 75

76 Poles Example Poles i i One negative real root Complex conjuguate roots with positive real part The system is unstable (grows without bound) 76

77 Poles Step Response 77

78 Poles Two types of poles Slow (or dominant) poles Poles that are closer to the imaginary axis Smaller real part means smaller exponential term and slower decay Fast poles Poles that are further from the imaginary axis Larger real part means larger exponential term and faster decay 78

79 Poles Poles dictate the stability of the process Stable poles Poles with negative real parts Decaying exponential Unstable poles Poles with positive real parts Increasing exponential Marginally stable poles Purely complex poles Pure sinusoidal 79

80 Poles Oscillatory Behaviour Integrating Behaviour Pure Exponential Stable Unstable Marginally Stable 80

81 Poles Example Stable Poles Unstable Poles Fast Poles Slow Poles 81

82 Poles and zeroes Definitions: the roots of are called the zeros of G(s) the roots of are called the poles of G(s) Zeros: Do not affect the stability of the system but can modify the dynamical behaviour 82

83 Zeros Types of zeros: Slow zeros are closer to the imaginary axis than the dominant poles of the transfer function Affect the behaviour Results in overshoot (or undershoot) Fast zeros are further away to the imaginary axis have a negligible impact on dynamics Zeros with negative real parts, Slow stable zeros lead to overshoot, are stable zeros Zeros with positive real parts,, are unstable zeros Slow unstable zeros lead to undershoot (or inverse response) 83

84 Zeros Can result from two processes in parallel + + If gains are of opposite signs and time constants are different then a right half plane zero occurs 84

85 Zeros Example Poles Zeros What is the effect of the zero on the dynamic behaviour? 85

86 Zeros Poles and zeros Fast, Stable Zero Dominant Pole 86

87 Zeros Unit step response: Effect of zero is negligible 87

88 Zeros Example Poles Zeros What is the effect of the zero on the dynamic behaviour? 88

89 Zeros Poles and zeros: Slow (dominant) Stable zero 89

90 Zeros Unit step response: Slow dominant zero yields an overshoot. 90

91 Zeros Example Poles Zeros What is the effect of the zero on the dynamic behaviour? 91

92 Zeros Poles and Zeros Dominant Pole Fast Unstable Zero 92

93 Zeros Unit Step Response: Effect of unstable zero is negligible 93

94 Zeros Example Poles Zeros What is the effect of the zero on the dynamic behaviour? 94

95 Zeros Poles and zeros: Slow Unstable Zero 95

96 Zeros Unit step response: Slow unstable zero causes undershoot (inverse reponse) 96

97 Zeros Observations: Adding a stable zero to an overdamped system yields overshoot Adding an unstable zero to an overdamped system yields undershoot (inverse response) Inverse response is observed when the zeros lie in right half complex plane, Overshoot or undershoot are observed when the zero is dominant (closer to the imaginary axis than dominant poles) 97

98 Zeros Example: System with complex zeros Dominant Pole Slow Stable Zeros 98

99 Zeros Poles and zeros Poles have negative real parts: The system is stable Dominant poles are real: yields an overdamped behaviour A pair of slow complex stable poles: yields overshoot 99

100 Zeros Unit step response: Effect of slow zero is significant and yields oscillatory behaviour 100

101 Zeros Example: System with zero at the origin Zero at Origin 101

102 Zeros Unit step response Zero at the origin: eliminates the effect of the step 102

103 Zeros Observations: Complex (stable/unstable) zero that is dominant yields an (overshoot/undershoot) Complex slow zero can introduce oscillatory behaviour Zero at the origin eliminates (or zeroes out) the system response 103

104 Delay F i Control loop h Time required for the fluid to reach the valve usually approximated as dead time Manipulation of valve does not lead to immediate change in level 104

105 Delay Delayed transfer functions e.g. First order plus dead-time Second order plus dead-time 105

106 Delay Dead time (delay) Most processes will display some type of lag time Dead time is the moment that lapses between input changes and process response Step response of a first order plus dead time process 106

107 Delay Delayed step response: 107

108 Problem use of the dead time approximation makes analysis (poles and zeros) more difficult Delay Approximate dead-time by a rational (polynomial) function Most common is Pade approximation 108

109 Pade Approximations In general Pade approximations do not approximate dead-time very well Pade approximations are better when one approximates a first order plus dead time process Pade approximations introduce inverse response (right half plane zeros) in the transfer function 109

110 Pade Approximations Step response of Pade Approximation Pade approximation introduces inverse response 110