Transfer Function Models
of Dynamical Processes
Process Dynamics and Control
1
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
Transfer Function

Heated stirred-tank model (constant flow,
)
 Taking the Laplace transform yields:
or letting
Transfer functions
3
Transfer Function

Heated stirred tank example
+
+
+
e.g. The block
is called the transfer function relating Q(s) to T(s)
4
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
Transfer functions

Transfer functions are generally expressed as a ratio of
polynomials
Where
 The polynomial
 Roots of
 Roots of
is called the characteristic polynomial of
are the zeroes of
are the poles of
6
Transfer function
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
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
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
Transfer Function

Unit impulse response of a 1st order process
10
Deviation Variables
To remove dependence on initial condition
e.g.

Compute equilibrium condition for a given
and
Define deviation variables
Rewrite linear ODE
0
or
11
Deviation Variables
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
Process Modeling

Gravity tank
Fo
h
Objectives: height of liquid in tank
Fundamental quantity: Mass, momentum
Assumptions:




F
L
Outlet flow is driven by head of liquid in the tank
Incompressible flow
Plug flow in outlet pipe
Turbulent flow
13
Transfer Functions
From mass balance and Newton’s law,
A system of simultaneous ordinary differential equations results
Linear or nonlinear?
14
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(x0)
!f
( x0 )
!x
x0
x
15
Nonlinear systems

First order Taylor series expansion
1. Function of one variable
2. Function of two variables
3. ODEs
16
Transfer Function

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
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
Block Diagrams

Transfer functions in series
 Overall operation is the multiplication of transfer functions
 Resulting overall transfer function
19
Block Diagrams

Transfer functions in series (two first order systems)
 Overall operation is the multiplication of transfer functions
 Resulting overall transfer function
20
Transfer Functions

DC Motor example:
 In terms of angular velocity
 In terms of the angle
21
Transfer Functions

Transfer function in parallel
+
+
 Overall transfer function is the addition of TFs in parallel
22
Transfer Functions

Transfer function in parallel
+
+
 Overall transfer function is the addition of TFs in parallel
23
Transfer Functions

Transfer functions in (negative) feedback form
 Overall transfer function
24
Transfer Functions

Transfer functions in (positive) feedback form
 Overall transfer function
25
Transfer Function

Example 3.20
26
Transfer Function

Example
 A positive feedback loop
27
Transfer Function

Example 3.20
 Two systems in parallel
 Replace
by
28
Transfer Function

Example 3.20
 Two systems in parallel
29
Transfer Function

Example 3.20
 A negative feedback loop
30
Transfer Function

Example 3.20
 Two process in series
31
First Order Systems

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
First Order Systems
Examples,
Examples Liquid storage
 Assume:
 Incompressible flow
 Outlet flow due to gravity
 Balance equation:
 Total
 Flow In
 Flow Out
33
First Order Systems

Balance equation:
 Deviation variables about the equilibrium
 Laplace transform

First order system with
34
First Order Systems
Examples:
Examples Cruise control
DC Motor
35
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
First Order Systems

Step response of first order process
Step input signal of magnitude M
 The ultimate change in
is given by
37
First Order Systems

Step response
38
First Order Systems

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
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
First Order Systems

Settling time
41
First Order Systems

Process initially at equilibrium subject to a step of
magnitude 1
42
First order process
Ramp response:
Ramp input of slope a
5
4.5
4
3.5
3
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
First Order Systems
Sinusoidal response
Sinusoidal
input Asin(ωt)
2
1.5
1
AR
0.5
0
-0.5
φ
-1
-1.5
0
2
4
6
8
10
12
14
16
18
20
44
First Order Systems
10
10
10
Bode Plots
0
High Frequency
Asymptote
Corner Frequency
-1
-2
10
-2
10
-1
10
0
10
1
10
2
0
-20
-40
-60
-80
-100
10
-2
10
-1
Amplitude Ratio
10
0
10
1
10
2
Phase Shift
45
Integrating Systems
Example: Liquid storage tank
Fi
h
F
Laplace domain dynamics
If there is no outlet flow,
46
Integrating Systems

Example
 Capacitor
 Dynamics of both systems is equivalent
47
Integrating Systems
Output
Step input of magnitude M
Input

Time
Slope =
Time
48
Integrating Systems
Output
Unit impulse response
Input

Time
Time
49
Integrating Systems
Output
Rectangular pulse response
Input

Time
Time
50
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
Second Order Systems
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
Second order Systems

Multicapacity Second Order Processes
 Naturally arise from two first order processes in series
 By multiplicative property of transfer functions
53
Transfer Functions

First order systems in parallel
+
+
 Overall transfer function a second order process (with one zero)
54
Second Order Systems
Inherently second order process:
e.g. Pneumatic Valve

By Newton’s law
p
x
55
Second Order Systems

Feedback Control Systems
56
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
Second Order Systems

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
Second Order Systems

Step response of magnitude M
2
ξ=0
1.8
ξ=0.2
1.6
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
Second Order Systems
Observations
 Responses exhibit overshoot
 Large
when
yield a slow sluggish response
 Systems with
 As
(with
oscillatory
yield the fastest response without overshoot
) becomes smaller, system becomes more
60
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
Second Order Systems

Step response
62
Second Order Systems

Sinusoidal Response
where
63
Second Order Systems
64
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
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

If any

in
vanishes to a unique point
then there is at least one term of the form
does not vanish
66
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
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
-0.2500 - 0.4330i
MATLAB
68
Poles
Function PZMAP
» pzmap(sys)

MATLAB
69
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
Poles

Step Response
 Integrating factor dominates the dynamic behaviour
71
Poles

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
Poles

Step Response
 Purely complex poles dominate the dynamics
73
Poles

Example
 Poles

-8.1569
-0.6564
-0.1868
 Three negative real poles
What is the dominant dynamic feature?
74
Poles

Step Response
 Slowest exponential (
) dominates the dynamic
feature (yields a time constant of 1/0.1868= 5.35 s
75
Poles

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
Poles

Step Response
77
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
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
Poles
Oscillatory
Behaviour
Integrating
Behaviour
Pure Exponential
Stable
Unstable
Marginally
Stable
80
Poles

Example
Stable Poles
Unstable Poles
Fast Poles
Slow Poles
81
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
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
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
Zeros

Example
 Poles

 Zeros

-8.1569
-0.6564
-0.1868
-10.000
What is the effect of the zero on the dynamic behaviour?
85
Zeros

Poles and zeros
Fast, Stable
Zero
Dominant
Pole
86
Zeros

Unit step response:
 Effect of zero is negligible
87
Zeros

Example
 Poles

 Zeros

-8.1569
-0.6564
-0.1868
-0.1000
What is the effect of the zero on the dynamic behaviour?
88
Zeros

Poles and zeros:
Slow (dominant)
Stable zero
89
Zeros

Unit step response:
 Slow dominant zero yields an overshoot.
90
Zeros

Example
 Poles

 Zeros

-8.1569
-0.6564
-0.1868
10.000
What is the effect of the zero on the dynamic behaviour?
91
Zeros

Poles and Zeros
Dominant
Pole
Fast Unstable
Zero
92
Zeros

Unit Step Response:
 Effect of unstable zero is negligible
93
Zeros

Example
 Poles

 Zeros

-8.1569
-0.6564
-0.1868
0.1000
What is the effect of the zero on the dynamic behaviour?
94
Zeros

Poles and zeros:
Slow
Unstable Zero
95
Zeros

Unit step response:
 Slow unstable zero causes undershoot (inverse reponse)
96
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
Zeros

Example: System with complex zeros
Dominant
Pole
Slow
Stable Zeros
98
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
Zeros

Unit step response:
 Effect of slow zero is significant and yields oscillatory behaviour
100
Zeros

Example: System with zero at the origin
Zero at
Origin
101
Zeros

Unit step response
 Zero at the origin: eliminates the effect of the step
102
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
Delay
Fi
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
Delay
Delayed transfer functions
e.g. First order plus dead-time
Second order plus dead-time
105
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
Delay

Delayed step response:
107
Delay

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
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
Pade Approximations

Step response of Pade Approximation
 Pade approximation introduces inverse response
110