President UniversityErwin SitompulSMI 1/1 Lecture 1 System Modeling and Identification Dr.-Ing. Erwin Sitompul President University

President University Erwin Sitompul SMI 1/1

Lecture 1System Modeling and Identification

Dr.-Ing. Erwin SitompulPresident University

http://zitompul.wordpress.com2 0 1 5


Textbook:“Process Modelling, Identification, and Control”, Jan Mikles, Miroslav Fikar, Springer, 2007.Syllabus: Chapter 1: IntroductionChapter 2: Mathematical Modeling

of ProcessesChapter 3: Analysis of Process ModelsChapter 4: Dynamical Behavior

of Processes Chapter 5: Discrete-Time Process ModelsChapter 6: Process Identification

Textbook and SyllabusSystem Modeling and Identification


Grade PolicySystem Modeling and Identification

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.


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.

●Heading of Written Homework Papers (Required)

System Modeling and IdentificationHomework 2

Rudi Bravo0029201800058

21 March 2021

No.1. Answer: . . . . . . . .

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.



Chapter 1Introduction



ControlChapter 1 Introduction

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


ControlChapter 1 Introduction

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.


Chapter 1 Introduction

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.

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.

Process, System, Model



Process, System, Model Process

System

Flow of matter

Flow of information

Flow of energy

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.


ProcessChapter 1 An Example of Process Control

A simple heat exchanger

Ti, qi

w To, qo

V T

Inlet

Outlet

V : volume of liquid in heat exchanger [m3]

q : Volume flow rate [m3/s]T : Temperature [K]w : Heat input [W]

Assumptions: • Ideal mixing• No heat loss• Constant heating rate• Exchanger has no heat capacity• T = To


Steady-StateChapter 1 An Example of Process Control


Ti, qi

w To, qo

V T

Inlet

Outlet

Input variables: • Inlet temperature Ti• Heat input wOutput variables: • Outlet temperature T

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:

o i( )pq c T T w q : Volume flow rate [m3/s]ρ : Liquid specific density [kg/m3]cp : Liquid specific heat capacity

[J/(kgK)]


Process ControlChapter 1 An Example of Process Control


Ti, qi

w To, qo

V T

Inlet

Outlet

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.

Input variables: • Inlet temperature Ti• Heat input wOutput variables: • Outlet temperature T


Dynamical Properties of the ProcessChapter 1 An Example of Process Control

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 coming (heat going (heat accumulation) from inlet and from outlet)heating element)

i i o o( ) ( ) ( )p p pd mc T q c T w q c Tdt


Assuming qi = qo and T = To,Dynamical Properties of the Process

Chapter 1 An Example of Process Control

ip p pdTmc q c T q c T wdt

ip p pdTV c q c T q c T wdt

The heat balance in the steady-state may be derived from the last equation, in the case that dT/dt = 0.

o i( )pq c T T w


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:

Feedback Process ControlChapter 1 An Example of Process Control

We speak about proportional control, and P is called the proportional gain.

w i w( ) ( ) ( ( ))pw t q c T T P T T t


Chapter 1 An Example of Process Control

Feedback Control of the Heat Exchanger

The scheme

The block diagram


Chapter 2Mathematical Modeling of

Processes



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


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.


General Principles of Modeling



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.


Balance equation in integral form:



0

0 in out( ) ( ) ( ) ( )t

t

m t m t m m d

Balance equation can be written in differential form:

in out( ) ( ) ( )dm t m t m tdt

The variable m in the equations above can be mass, energy, entropy, ..., impulse.


Mass balance in an unsteady-state is given by the law of mass conservation:



1 1

( ) m n

i i ji j

d V q qdt

ρ, ρi : Specific densities [kg/m3] V : Volume [m3]q, qi : Volume flow rates [m3/s]m : Number of inletsn : Number of outlets

In

Out

m V


Energy balance follows the general law of energy conservation:



, j1 1 1

( ) m n sp

i i p i i p li j l

d Vc Tq c T q c T Q

dt

ρ, ρi : Specific densities [kg/m3] V : Volume [m3]q, qi : Volume flow rates [m3/s]cp, cp,i : Specific heat capacities [J/(kgK)T, Ti : Temperatures [K]Q : Heat per unit time [W]m : Number of inletsn : Number of outletss : Number of heat sources and consumptions


Let us examine a liquid storage system shown below:

Chapter 2 Examples of Dynamic Mathematical Models

Single-Tank System

The mass balance for this process yields:

qi

qo

V h

ρ : Specific densities [kg/m3] V : Volume [m3]qi, qo : Volume flow rates [m3/s]A : Cross-sectional area of the

tank [m2]h : Height of liquid in the tank

[m]

i o( )d Ah q qdt

With A and ρ assumed to be constant,

i odhA q qdt


v1

Applying the law of mechanical energy conservation to the liquid near the outlet:


Single-Tank System

v1 : Outlet flow velocity [m/s] a1 : Cross-sectional area of the

outlet pipe [m2]

=(potential energy) (kinetic energy)1 2

12mgh mv

1 2v gh

o 1 1q v a

qi

qo

V h


Inserting q0 = v1a1 into the mass balance equation of the system:


Single-Tank System

i 1 1dhA q v adt

i 11

q adh vdt A A

or

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: i 1 1q a v

i 1 2q a gh2

is

1

12

qh

ag



Simulation of Single-Tank System The dynamic mathematical model of the single-tank system will

now be simulated using Matlab Simulink.a1 = 20 cm2

= 2010–4 m2 = 210–3 m2

A = 2500 cm2

= 250010–4 m2 = 0.25 m2

2i

s1

12

qh

ag

23

3

1 5 102 9.8 2 10

0.319 m

Manual calculation of steady-state liquid height yields:

g = 9.8 m/s2

qi = 5 liters/s = 510–3 m3/s

tsim = 200 s


Simulation with Matlab-SimulinkChapter 2 Examples of Dynamic Mathematical Models

Matlab-Simulink provides the best simulation environment for control engineers.


After construction, the Matlab Simulink block diagram is given as:


Simulation of Single-Tank System

i 11

q adh vdt A A


The simulation result, from transient until steady-state, can be observed by clicking the Scope.


Simulation of Single-Tank System

s 0.3185 mh


Impulse and Charge Balance Equations

1

( )

n

ii

d mv dvm ma Fdt dt


The impulse balance equation can be related to Newton’s law by:

1

n

jj

dq Idt

The charge balance equation can be related to Kirchhoff's law by:


A Two-Mass System: Suspension Model

m1 : mass of the wheelm2 : mass of the carx,y : displacements from equilibriumr : distance to road surface

s w 1( ) ( ) ( )k x y b x y k x r m x Equation for m1:

Equation for m2:s 2( ) ( )k y x b y x m y

Rearranging:s w w

1 1 1 1

( ) ( )k k kbx x y x y x rm m m m

s

2 2

( ) ( ) 0kby y x y xm m



A Two-Mass System: Suspension ModelUsing the Laplace transform:

( ) ( )

( ) ( )

x t X sd x t sX sdt

L

L

2

1s w w

1 1 1

( ) ( ) ( )

( ) ( ) ( ) ( )

bs X s s X s Y sm

k k kX s Y s X s R sm m m

2 s

2 2

( ) ( ) ( ) ( ) ( ) 0kbs Y s s Y s X s Y s X sm m

to transfer from time domain to frequency domain yields:



A Two-Mass System: Suspension ModelEliminating X(s) yields a transfer function:

( ) ( )( )

Y s F sR s

output transfer functioninput


w s

1 2

4 3 2s w w w s

1 2 1 2 1 1 2 1 2

( )( )

k b ksm m bY s

R s k k k b k kb b ks s s sm m m m m mm mm


Bridged Tee Circuit

1 o1 i1 1

1 2

0V VV V sCV

R R

v1

Resistor Inductor Capacitor

dvi C dtv Ri div L dt

( ) ( )V s R I s ( ) ( )V s sL I s ( ) ( )I s sC V s


o 12 o i

2

( ) 0V V sC V VR


RL Circuit1 o1 i 1 0

1 1V VV V Vs

1 o o1 o

0 ( 1)1

V V VV V s

s

o1 i

12 VV Vs s

oo i

11 2 VV s Vs s

o i2 3V s V

v1

Further calculation and eliminating V1,


o

i

12 3

VV s


Homework 1

v1

qi

qo

h1


Derive a dynamic mathematical model for the interacting tank-in-series system as shown below.

h2

v2 q1

a1 a2


Homework 1AChapter 2 Examples of Dynamic Mathematical Models

Derive a dynamic mathematical model for the tank with the form of a triangular prism as shown below.

v

qi1

qo

a

qi2

hmaxh ρ : Specific densities [kg/m3]

V : Volume of liquid in the tank [m3]h : Height of liquid in the tank [m]hmax : Height of the tank [m] A : Cross-sectional area of the liquid

surface [m2] Amax : Cross-sectional area of the tank

at the top [m2]qi1, qi2 : Volume flow rates of inlets [m3/s]qo : Volume flow rates of outlet[m3/s]v : Outlet flow velocity [m/s]a1 : Cross-sectional area of the outlet

pipe [m2]

Documents

President UniversityErwin SitompulSMI 1/1 Lecture 1 System Modeling and Identification Dr.-Ing. Erwin Sitompul President University