Transfer Function Models of Dynamical Processes

Process Dynamics and Control

1 LinearLinear SISOSISO ControlControl SystemsSystems

 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 TransferTransfer FunctionFunction

 Heated stirred-tank model (constant flow, )

 Taking the Laplace transform yields:

or letting

Transfer functions

3 TransferTransfer FunctionFunction

 Heated stirred tank example

+ +

e.g. The block is called the transfer function relating Q(s) to T(s)

4 ProcessProcess ControlControl

Time Domain Laplace Domain

Process Modeling, Transfer function Experimentation and Modeling, Controller Implementation Design and Analysis

Ability to understand dynamics in Laplace and time domains is extremely important in the study of process control

5 TransferTransfer functionsfunctions

 Transfer functions are generally expressed as a ratio of polynomials

Where

 The polynomial is called the characteristic polynomial of

 Roots of are the zeroes of  Roots of are the poles of

6 TransferTransfer functionfunction

 Order of underlying ODE is given by degree of characteristic polynomial e.g. First order processes

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 TransferTransfer FunctionFunction

 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 TransferTransfer functionfunction

 Transfer function is the unit impulse response

e.g. First order process,

Unit impulse response is given by

In the time domain,

9 TransferTransfer FunctionFunction

 Unit impulse response of a 1st order process

10 DeviationDeviation VariablesVariables

 To remove dependence on initial condition e.g.

Compute equilibrium condition for a given and

Define deviation variables

Rewrite linear ODE 0 or

11 DeviationDeviation VariablesVariables

Assume that we start at equilibrium

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 ProcessProcess ModelingModeling

 Gravity tank

h F

Objectives: height of liquid in tank L 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

13 TransferTransfer FunctionsFunctions

From mass balance and Newton’s law,

A system of simultaneous ordinary differential equations results

Linear or nonlinear?

14 NonlinearNonlinear ODEsODEs

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(x0) !f (x0) !x

x0 x 15 NonlinearNonlinear systemssystems

 First order Taylor series expansion

1. Function of one variable

2. Function of two variables

3. ODEs

16 TransferTransfer FFunctionunction

 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 BlockBlock DiagramsDiagrams

 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 BlockBlock DiagramsDiagrams

 Transfer functions in series

 Overall operation is the multiplication of transfer functions

 Resulting overall transfer function

19 BlockBlock DiagramsDiagrams

 Transfer functions in series (two first order systems)

 Overall operation is the multiplication of transfer functions

 Resulting overall transfer function

20 TransferTransfer FunctionsFunctions

 DC Motor example:  In terms of angular velocity

 In terms of the angle

21 TransferTransfer FunctionsFunctions

 Transfer function in parallel

 Overall transfer function is the addition of TFs in parallel

22 TransferTransfer FunctionsFunctions

 Transfer function in parallel

 Overall transfer function is the addition of TFs in parallel

23 TransferTransfer FunctionsFunctions

 Transfer functions in (negative) feedback form

 Overall transfer function

24 TransferTransfer FunctionsFunctions

 Transfer functions in (positive) feedback form

 Overall transfer function

25 TransferTransfer FunctionFunction

 Example 3.20

26 TransferTransfer FunctionFunction

 Example

 A positive feedback loop

27 TransferTransfer FunctionFunction

 Example 3.20

 Two systems in parallel  Replace by

28 TransferTransfer FunctionFunction

 Example 3.20

 Two systems in parallel

29 TransferTransfer FunctionFunction

 Example 3.20

 A negative feedback loop

30 TransferTransfer FunctionFunction

 Example 3.20

 Two process in series

31 FirstFirst OrderOrder SystemsSystems

 First order systems are systems whose dynamics are described by the transfer function

where

 is the system’s (steady-state) gain

 is the time constant

 First order systems are the most common behaviour encountered in practice

32 FirstFirst OrderOrder SystemsSystems

Examples, Liquid storage

 Assume:  Incompressible flow  Outlet flow due to gravity  Balance equation:  Total  Flow In  Flow Out

33 FirstFirst OrderOrder SystemsSystems

 Balance equation:

 Deviation variables about the equilibrium

 Laplace transform

 First order system with

34 FirstFirst OrderOrder SystemsSystems

Examples: Cruise control

DC Motor

35 FirstFirst OrderOrder SystemsSystems

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 FirstFirst OrderOrder SystemsSystems

 Step response of first order process

Step input signal of magnitude M

 The ultimate change in is given by

37 FirstFirst OrderOrder SystemsSystems

 Step response

38 FirstFirst OrderOrder SystemsSystems

 What do we look for?  System’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 FirstFirst OrderOrder SystemsSystems

 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 FirstFirst OrderOrder SystemsSystems

 Settling time

41 FirstFirst OrderOrder SystemsSystems

 Process initially at equilibrium subject to a step of magnitude 1

42 FirstFirst orderorder processprocess

Ramp response:

Ramp input of slope a

5 4.5

4 3.5

2.5

2 1.5 1

0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

43 FirstFirst OrderOrder SystemsSystems

Sinusoidal response

Sinusoidal2 input Asin(ωt)

1.5

0.5 AR

-0.5

-1 φ

-1.5 0 2 4 6 8 10 12 14 16 18 20

44 FirstFirst OrderOrder SystemsSystems

0 Bode Plots 10 High Frequency Asymptote -1 Corner Frequency 10

10 -2 -2 -1 0 1 2 10 10 10 10 10

0 -20 -40 -60 -80 -100 10 -2 10 -1 10 0 10 1 10 2

Amplitude Ratio Phase Shift

45 IntegratingIntegrating SystemsSystems

Example: Liquid storage tank

F Laplace domain dynamics

If there is no outlet flow,

46 IntegratingIntegrating SystemsSystems

 Example  Capacitor

 Dynamics of both systems is equivalent

47 IntegratingIntegrating SystemsSystems

 Step input of magnitude M

Slope =

Input

Output

Time Time

48 IntegratingIntegrating SystemsSystems

 Unit impulse response

Input

Output

Time Time

49 IntegratingIntegrating SystemsSystems

 Rectangular pulse response

Input

Output

Time Time

50 SecondSecond orderorder SystemsSystems

 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 SecondSecond OrderOrder SystemsSystems

Three types of second order process:

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 SecondSecond orderorder SystemsSystems

 Multicapacity Second Order Processes  Naturally arise from two first order processes in series

 By multiplicative property of transfer functions

53 TransferTransfer FunctionsFunctions

 First order systems in parallel

 Overall transfer function a second order process (with one zero)

54 SecondSecond OrderOrder SystemsSystems

 Inherently second order process: e.g. Pneumatic Valve

By Newton’s law p

55 SecondSecond OrderOrder SystemsSystems

 Feedback Control Systems

56 SecondSecond orderorder SystemsSystems

 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 SecondSecond OrderOrder SystemsSystems

 Roots of the characteristic polynomial

Case 1) Two distinct real roots System has an exponential behavior

Case 2) One multiple real root Exponential behavior

Case 3) Two complex roots System has an oscillatory behavior

58 SecondSecond OrderOrder SystemsSystems

 Step response of magnitude M

2 ξ=0 1.8 1.6 ξ=0.2 1.4 1.2 1 0.8 0.6 ξ=2 0.4 0.2 0 0 1 2 3 4 5 6 7 8 9 59 10 SecondSecond OrderOrder SystemsSystems

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 SecondSecond OrderOrder SystemsSystems

Characteristics of underdamped second order process

1. Rise time, 2. Time to first peak, 3. Settling time, 4. Overshoot:

5. Decay ratio:

61 SecondSecond OrderOrder SystemsSystems

 Step response

62 SecondSecond OrderOrder SystemsSystems

 Sinusoidal Response

where

63 SecondSecond OrderOrder SystemsSystems

64 MoreMore ComplicatedComplicated SystemsSystems

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 PolesPoles andand zeroeszeroes

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 PolesPoles 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 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: 1 ------4 s^4 + 6 s^3 + 3 s^2 + s » pole(sys) ans = 0 -1.0000 -0.2500 + 0.4330i MATLAB -0.2500 - 0.4330i

68 PolesPoles

 Function PZMAP

» pzmap(sys) MATLAB

69 PolesPoles

 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 PolesPoles

 Step Response

 Integrating factor dominates the dynamic behaviour

71 PolesPoles

 Example

 Poles  -1.0000 -0.0000 + 1.0000i -0.0000 - 1.0000i

 One negative real pole  Two purely complex poles

What is the dominant feature?

72 PolesPoles

 Step Response

 Purely complex poles dominate the dynamics

73 PolesPoles

 Example

 Poles  -8.1569 -0.6564 -0.1868

 Three negative real poles

What is the dominant dynamic feature?

74 PolesPoles

 Step Response

 Slowest exponential ( ) dominates the dynamic feature (yields a time constant of 1/0.1868= 5.35 s

75 PolesPoles

 Example

 Poles -4.6858 0.3429 + 0.3096i 0.3429 - 0.3096i

 One negative real root  Complex conjuguate roots with positive real part

 The system is unstable (grows without bound)

76 PolesPoles

 Step Response

77 PolesPoles

 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 PolesPoles

 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 PolesPoles

Oscillatory Integrating Behaviour Behaviour

Pure Exponential

Stable Unstable

Marginally Stable 80 PolesPoles

 Example

Stable Poles

Unstable Poles

Fast Poles Slow Poles

81 PolesPoles andand zeroeszeroes

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 ZerosZeros

 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, , are stable zeros  Slow stable zeros lead to overshoot

 Zeros with positive real parts, , are unstable zeros  Slow unstable zeros lead to undershoot (or inverse response)

83 ZerosZeros

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 ZerosZeros

 Example

 Poles  -8.1569 -0.6564 -0.1868

 Zeros  -10.000

What is the effect of the zero on the dynamic behaviour?

85 ZerosZeros

 Poles and zeros

Dominant Fast, Stable Pole Zero

86 ZerosZeros

 Unit step response:

 Effect of zero is negligible 87 ZerosZeros

 Example

 Poles  -8.1569 -0.6564 -0.1868

 Zeros  -0.1000

What is the effect of the zero on the dynamic behaviour?

88 ZerosZeros

 Poles and zeros:

Slow (dominant) Stable zero

89 ZerosZeros

 Unit step response:

 Slow dominant zero yields an overshoot. 90 ZerosZeros

 Example

 Poles  -8.1569 -0.6564 -0.1868

 Zeros  10.000

What is the effect of the zero on the dynamic behaviour?

91 ZerosZeros

 Poles and Zeros

Dominant Fast Unstable Pole Zero

92 ZerosZeros

 Unit Step Response:

 Effect of unstable zero is negligible

93 ZerosZeros

 Example

 Poles  -8.1569 -0.6564 -0.1868

 Zeros  0.1000

What is the effect of the zero on the dynamic behaviour?

94 ZerosZeros

 Poles and zeros:

Slow Unstable Zero

95 ZerosZeros

 Unit step response:

 Slow unstable zero causes undershoot (inverse reponse) 96 ZerosZeros

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 ZerosZeros

 Example: System with complex zeros

Dominant Pole

Slow Stable Zeros

98 ZerosZeros

 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 ZerosZeros

 Unit step response:

 Effect of slow zero is significant and yields oscillatory behaviour 100 ZerosZeros

 Example: System with zero at the origin

Zero at Origin

101 ZerosZeros

 Unit step response

 Zero at the origin: eliminates the effect of the step 102 ZerosZeros

 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 DelayDelay

Control loop

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 DelayDelay

Delayed transfer functions

e.g. First order plus dead-time

Second order plus dead-time

105 DelayDelay

 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 DelayDelay

 Delayed step response:

107 DelayDelay

 Problem  use of the dead time approximation makes analysis (poles and zeros) more difficult

 Approximate dead-time by a rational (polynomial) function  Most common is Pade approximation

108 PadePade ApproximationsApproximations

 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 PadePade ApproximationsApproximations

 Step response of Pade Approximation

 Pade approximation introduces inverse response 110