System Modeling and Identification
Lecture 1
Dr.-Ing. Erwin Sitompul
President University
http://zitompul.wordpress.com
2 0 1 5
President University
Erwin Sitompul
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System Modeling and Identification
Textbook and Syllabus
Textbook:
“Process Modelling, Identification, and Control”,
Jan Mikles, Miroslav Fikar, Springer, 2007.
Syllabus:
Chapter 1: Introduction
Chapter 2: Mathematical Modeling
of Processes
Chapter 3: Analysis of Process Models
Chapter 4: Dynamical Behavior
of Processes
Chapter 5: Discrete-Time Process Models
Chapter 6: Process Identification
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System Modeling and Identification
Grade Policy
Final Grade = 10% Homework + 20% Quizzes +
30% Midterm Exam + 40% Final Exam +
Extra Points
 Homeworks will be given in fairly regular basis. The average of
homework grades contributes 10% of final grade.
 Written homeworks are to be submitted on A4 papers, otherwise
they will not be graded.
 Homeworks must be submitted on time, on the day of the next
lecture, 10 minutes after the class starts. Late submission will be
penalized by point deduction of –10·n, where n is the total
number of lateness made.
 There will be 3 quizzes. Only the best 2 will be counted. The
average of quiz grades contributes 20% of final grade.
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System Modeling and Identification
Grade Policy
 Midterm and final exams follow the schedule released by AAB
(Academic Administration Bureau).
 Make up of quizzes must be held within one week after the
schedule of the respective quiz.
 Make up for mid exam and final exam must be requested directly
to AAB.
System Modeling and Identification
Homework 2
Rudi Bravo
0029201800058
21 March 2021
No.1. Answer: . . . . . . . .
● Heading of Written Homework Papers (Required)
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Erwin Sitompul
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System Modeling and Identification
Grade Policy
 Extra points will be given if you solve a problem in front of the
class. You will earn 1 or 2.
 Lecture slides can be copied during class session. It is also
available on internet. Please check the course homepage regularly.
http://zitompul.wordpress.com
 The use of internet for any purpose during class sessions is
strictly forbidden.
 You are expected to write a note along the lectures to record your
own conclusions or materials which are not covered by the lecture
slides.
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System Modeling and Identification
Chapter 1
Introduction
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Chapter 1
Introduction
Control
 Control is the purposeful influence on an object (process) to
ensure the fulfillment of a required objectives.
 The objectives can be to satisfy the safety and optimal operation
of the technology, the product specifications under constraints of
disturbance, process stability, and other technical related matters.
 Control systems in the whole consist of technical devices and
human factor. Control systems must satisfy:
 Disturbance attenuation
 Stability guarantee
 Optimal process operation
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Chapter 1
Introduction
Control
 There are two main methods of control:
 Feedback control, where the information about process
output is used to calculate the control (manipulated) signal 
process output is fed back to process input
 Feedforward control, where the effect of control is not
compared with the desired result
 Practical control experience confirms the importance of
assumptions about dynamical behavior of processes.
 This behavior is described using mathematical models of
processes, which can be constructed from a physical or chemical
nature of processes or can be abstract.
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Chapter 1
Introduction
Process, System, Model
 The purpose of this course is to learn how to model a process,
which may have one of these objectives:
 Synthesis: Modeling as the fundamentals to influence a
process through a controller
 Analysis: Modeling as the fundamentals to a deep
understanding of the process and further to optimization of
the process
 Simulation: Modeling as the fundamentals to emulative
calculation under a given boundary condition
 Process is the entire activities where matter and/or energy are
stored, transported, and converted; whereas information is
stored, transported, converted, created, or destroyed.
 System is a part of a process, which is defined by the user, and
has an interconnection with the environment regarding the flow of
matter, energy, and information.
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Erwin Sitompul
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Chapter 1
Introduction
Process, System, Model
 An automobile may represent a process, consisting:
 Engine and driveline system
 Suspension system
 Braking System
 Climate control system
 The variables of interest depend on the user, i.e., engine
technician requires the relation between transmission and speed,
while aircon technician wants to know the relation between speed
and cooling performance.
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Chapter 1
Introduction
Process, System, Model
Flow of
matter
Flow of
energy
Process
System
Flow of
information
Environment
 Model is an appropriate description of the flows of a system to its
environment, using physical, chemical nature of the system, or
using abstract mathematical equations.
 A mathematical model is a description of a system using
mathematical concepts and language. A model may help to
explain a system and to study the effects of different components,
and to make predictions about behavior.
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Chapter 1
An Example of Process Control
Process
 A simple heat exchanger
Ti, qi
Inlet
V
T
V : volume of liquid in heat
exchanger [m3]
q : Volume flow rate [m3/s]
T : Temperature [K]
w : Heat input [W]
Outlet
w
To, qo
Assumptions:
• Ideal mixing
• No heat loss
• Constant heating rate
• Exchanger has no heat capacity
• T = To
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Chapter 1
An Example of Process Control
Steady-State
 A simple heat exchanger
Ti, qi
Inlet
V
Input variables:
• Inlet temperature Ti
• Heat input w
Output variables:
• Outlet temperature T
T
Outlet
w
To, qo
 The process is said to be in steady-state if the input and output
variables remain constant in time.
 The heat balance in the steady-state is of the form:
q  c p (To  Ti )  w
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q : Volume flow rate [m3/s]
ρ : Liquid specific density [kg/m3]
cp : Liquid specific heat capacity
[J/(kgK)]
Erwin Sitompul
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Chapter 1
An Example of Process Control
Process Control
 A simple heat exchanger
Ti, qi
Inlet
V
Input variables:
• Inlet temperature Ti
• Heat input w
Output variables:
• Outlet temperature T
T
Outlet
w
To, qo
 Control of the heat exchanger in this case means to influence the
process so that T will be kept close to Tw.
 This influence is realized with changes in w, which is called
manipulated variable.
 A thermometer must be placed on the outlet of the exchanger and
we may choose between manual control or automatic control.
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Chapter 1
An Example of Process Control
Dynamical Properties of the Process
 In the case that the control is realized automatically, the
knowledge about process response to changes of input variables is
required.
 This is the knowledge about dynamical properties of the
process, which is the description of the process in unsteady-state.
 The heat balance for the heat transfer process in a very short time
interval Δt converging to zero is given by:
(heat accumulation)
d
(mc pT )
dt
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=
(heat coming
from inlet and
heating element)

(qi  c pTi  w)
Erwin Sitompul

(heat going
from outlet)
 (qo  c pTo )
SMI 1/15
Chapter 1
An Example of Process Control
Dynamical Properties of the Process
 Assuming qi = qo and T = To,
mc p
dT
 q  c pTi  q  c pT  w
dt
V cp
dT
 q  c pT  q  c pTi  w
dt
 The heat balance in the steady-state may be derived from the last
equation, in the case that dT/dt = 0.
q  c p (To  Ti )  w
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Chapter 1
An Example of Process Control
Feedback Process Control
 In case of choosing an automatic control, the control device
performs the control actions which is described in a control law.
 The task of a control device is to minimize the difference between
Tw and T, which is defined as control error. (Tw is the set point)
 Suppose that we choose a controller that will change the heat
input proportionally to the control error, the control law can be
given as:
w(t )  q  c p (Tw  Ti )  P(Tw  T (t ))
 We speak about proportional control, and P is called the
proportional gain.
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Chapter 1
An Example of Process Control
Feedback Control of the Heat Exchanger
The scheme
The block diagram
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System Modeling and Identification
Chapter 2
Mathematical Modeling of
Processes
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Chapter 2
Mathematical Modeling of Processes
General Principles of Modeling
 A system is expressed through mathematical descriptions.
 These descriptions are called “mathematical models.”
 The behavior of the system with regard to certain inputs can be
characterized by using the mathematical model.
 Mathematical models can be divided into three groups, depending
on how they are obtained:
 Theoretical model, developed using physical, chemical
principles/laws
 Empirical model, obtained from mathematical analysis of
measurement data of the process/ system or through
experience
 Empirical-theoretical model, obtained from a combination
of theoretical and empirical modeling approach
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Chapter 2
Mathematical Modeling of Processes
General Principles of Modeling
 Theoretical models are derived from the so called “balance
equation of conserved quantity” that may include:
 Mass balance equation
 Energy balance equation
 Entropy balance equation
 Enthalpy balance equation
 Charge balance equation
 Heat balance equation
 Impulse balance equation
 A conserved quantity is a quantity whose total amount is
maintained constant and is understood to obey the principle of
conservation, which states that such a quantity can be neither
created nor destroyed.
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Chapter 2
Mathematical Modeling of Processes
General Principles of Modeling
 Alternatively, a conserved quantity is one whose total amount
remains constant in an isolated system, regardless of what
changes occur inside the system.
 An isolated system is a hypothetical system that has zero
interaction with its surroundings, i.e., zero transfer of material,
heat, work, radiation, etc. across the boundary.
 The balance equations in an unsteady-state are used to obtain the
dynamical model, which is expressed using differential equations.
 In most cases, ordinary differential equations are chosen to keep
the mathematical model simple.
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Chapter 2
Mathematical Modeling of Processes
General Principles of Modeling
 Balance equation in integral form:
t
m(t )  m(t0 )    min ( )  mout ( )d
t0
 Balance equation can be written in differential form:
dm(t )
 min (t )  mout (t )
dt
 The variable m in the equations above can be mass, energy,
entropy, ..., impulse.
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Chapter 2
Mathematical Modeling of Processes
General Principles of Modeling
 Mass balance in an unsteady-state is given by the law of mass
conservation:
n
d ( V ) m
  i qi    q j
dt
i 1
j 1
ρ, ρi
V
q, qi
m
n
:
:
:
:
:
Specific densities [kg/m3]
Volume [m3]
Volume flow rates [m3/s]
Number of inlets
Number of outlets
In
m  V
Out
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Chapter 2
Mathematical Modeling of Processes
General Principles of Modeling
 Energy balance follows the general law of energy conservation:
d ( Vc pT )
dt
m
n
s
i 1
j 1
l 1
  i qi c p ,iTi    q jc pT   Ql
ρ, ρi
V
q, qi
cp, cp,i
T, Ti
Q
m
n
s
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:
:
:
:
:
:
:
:
:
Specific densities [kg/m3]
Volume [m3]
Volume flow rates [m3/s]
Specific heat capacities [J/(kgK)
Temperatures [K]
Heat per unit time [W]
Number of inlets
Number of outlets
Number of heat sources and consumptions
Erwin Sitompul
SMI 1/25
Chapter 2
Examples of Dynamic Mathematical Models
Single-Tank System
 Let us examine a liquid storage system shown below:
qi
ρ
V
qi, qo
A
V
h
h
:
:
:
:
Specific densities [kg/m3]
Volume [m3]
Volume flow rates [m3/s]
Cross-sectional area of the
tank [m2]
: Height of liquid in the tank
[m]
qo
 The mass balance for this process yields:
d ( Ah  )
 qi   qo 
dt
 With A and ρ assumed to be constant,
A
dh
 qi  qo
dt
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Chapter 2
Examples of Dynamic Mathematical Models
Single-Tank System
 Applying the law of mechanical energy conservation to the liquid
near the outlet:
qi
V
v1
a1
h
v1
(potential energy)
: Outlet flow velocity [m/s]
: Cross-sectional area of the
outlet pipe [m2]
qo
= (kinetic energy)
mgh  12 mv12
v1  2 gh
qo  v1a1
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Chapter 2
Examples of Dynamic Mathematical Models
Single-Tank System
 Inserting q0 = v1a1 into the mass balance equation of the system:
A
dh
 qi  v1a1
dt
or
dh qi a1
  v1
dt A A
 The initial condition (i.e., the initial height of the liquid) can be
arbitrary, h(0) = h0.
 The tank will be in steady-state if dh/dt = 0.
 For a constant inlet flow rate qi, the steady-state liquid height hs is
given by:
qi  a1v1
qi  a1 2 gh
1  qi 
hs 
2 g  a1 
2
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Chapter 2
Examples of Dynamic Mathematical Models
Simulation of Single-Tank System
 The dynamic mathematical model of the single-tank system will
now be simulated using Matlab Simulink.
a1
A
=
=
=
=
=
=
20 cm2
2010–4 m2
210–3 m2
2500 cm2
250010–4 m2
0.25 m2
g
qi
tsim
=
=
=
=
9.8 m/s2
5 liters/s
510–3 m3/s
200 s
 Manual calculation of steady-state liquid height yields:
2
1  qi 
1  5  103 
 0.319 m
hs 






3
2 g  a1  2  9.8  2  10 
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Chapter 2
Examples of Dynamic Mathematical Models
Simulation with Matlab-Simulink
 Matlab-Simulink provides the best simulation
environment for control engineers.
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Chapter 2
Examples of Dynamic Mathematical Models
Simulation of Single-Tank System
 After construction, the Matlab Simulink block diagram is given as:
dh qi a1
  v1
dt A A
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Chapter 2
Examples of Dynamic Mathematical Models
Simulation of Single-Tank System
 The simulation result, from transient until steady-state, can be
observed by clicking the Scope.
hs  0.3185 m
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Chapter 2
Examples of Dynamic Mathematical Models
Impulse and Charge Balance Equations
 The impulse balance equation can be related to Newton’s law by:
n
d (mv)
dv
 m  ma   Fi
dt
dt
i 1
 The charge balance equation can be related to Kirchhoff's law by:
dq n
 Ij
dt j 1
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Chapter 2
Examples of Dynamic Mathematical Models
A Two-Mass System: Suspension Model
m1
m2
x,y
r
:
:
:
:
mass of the wheel
mass of the car
displacements from equilibrium
distance to road surface
Equation for m1:
ks ( x  y)  b( x  y)  kw ( x  r )  m1 x
Equation for m2:
ks ( y  x)  b( y  x)  m2 y
Rearranging:
ks
kw
kw
b
x  ( x  y)  ( x  y) 
x
r
m1
m1
m1
m1
ks
b
y
( y  x) 
( y  x)  0
m2
m2
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Chapter 2
Examples of Dynamic Mathematical Models
A Two-Mass System: Suspension Model
Using the Laplace transform:
L
x(t ) 
 X (s)
d
L
x(t ) 
 sX ( s )
dt
to transfer from time domain to frequency domain yields:
b
s X ( s)  s  X (s)  Y ( s)  
m1
ks
kw
kw
 X ( s )  Y ( s )   X ( s )  R( s )
m1
m1
m1
2
ks
b
s Y ( s) 
s Y ( s )  X ( s )  
Y ( s )  X ( s )   0
m2
m2
2
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Chapter 2
Examples of Dynamic Mathematical Models
A Two-Mass System: Suspension Model
Eliminating X(s) yields a transfer function:
kw b 
ks 
s



m1m2 
b
Y (s)

R( s)
 b
k w ks
b  3  ks k k w  2  k w b 
4
s  
s



s

s






m
m
m
m
m
m
m
m1m2
 1
2 
 1
2
1 
 1 2
Y ( s)
output
 F ( s) 
 transfer function
R( s)
input
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Chapter 2
Examples of Dynamic Mathematical Models
Bridged Tee Circuit
v1
V1  Vi V1  Vo

 sC1V1  0
R1
R2
v  Ri
v  L di
dt
i  C dv
dt
V (s)  R  I (s) V (s)  sL  I (s) I (s)  sC V (s)
Resistor
Inductor
Capacitor
Vo  V1
 sC2 (Vo  Vi )  0
R2
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Chapter 2
Examples of Dynamic Mathematical Models
RL Circuit
V1  Vi V1 V1  Vo
 
0
1
1
s
V1  Vo Vo  0

 V1  Vo ( s  1)
s
1
v1
Further calculation and eliminating V1,
1  Vo

V1  2     Vi
s s

1  Vo

Vo  s  1  2     Vi
s s

Vo  2s  3  Vi
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Vo
1

Vi 2s  3
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Chapter 2
Examples of Dynamic Mathematical Models
Homework 1
Derive a dynamic mathematical model for the interacting tank-inseries system as shown below.
qi
h1
v1
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a1
q1
Erwin Sitompul
h2
a2
v2
SMI 1/39
qo
Chapter 2
Examples of Dynamic Mathematical Models
Homework 1A
Derive a dynamic mathematical model for the tank with the form of
a triangular prism as shown below.
qi2
qi1
h
a
qo v
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hmax
ρ
V
h
hmax
A
:
:
:
:
:
Amax
:
qi1, qi2
qo
v
a1
:
:
:
:
Specific densities [kg/m3]
Volume of liquid in the tank [m3]
Height of liquid in the tank [m]
Height of the tank [m]
Cross-sectional area of the liquid
surface [m2]
Cross-sectional area of the tank
at the top [m2]
Volume flow rates of inlets [m3/s]
Volume flow rates of outlet[m3/s]
Outlet flow velocity [m/s]
Cross-sectional area of the outlet
pipe [m2]
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