SEE 2113

CONTROL: MODELLING AND SIMULATION

LECTURER: DR. SHAHDAN SUDIN

P08-328
07-5535328
shahdan@fke.utm.my,
shahdan@utm.my

OBJECTIVE: To help the students understand the

basic concept of control system

WEBSITE: http://elearning.utm.my

SYLLIBUS: 1 Introduction to control systems

2 Mathematical modeling
3 System representation
4 Time domain response analysis
5 Introduction to computer simulation

ASSESSMENT: Test 1 : 15%

Test 2 : 15%
Assignment + Tut : 20%
Final Exam : 50%
REFERENCES: - Notes given in class or provided in website
Brief notes on SEE2113, Shaharum Sulaiman
Control Systems Engineering, 2008. N.S. Nise
Modern Control Engineering, 1997. K.Ogata
Modern Control Systems. R. C. Dorf & R. H. Bishop

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CHAPTER 1: INTRODUCTION

A) Introduction

- Control systems are important and are present almost everywhere

in our daily lives
- Examples of man-made control systems: CD player, radio
antenna, rockets/missiles, robots, oven, room air condition
- Examples of God-created control systems: level of adrenalin in
the human body, entry of light through the human eye, holding
and carrying things using hands, human riding a bicycle
- Malfunction of control systems: accidents (e.g: alcohol),
difficulties (e.g: water toilet pumps)
- Control engineers: are normally involved at the highest level in a
particular project, determining the suitable system performance,
functions of subsystems and their interconnections (interfacing
needs, hardware and software design, testing and simulation)
- Control engineers: are involved in several fields of science and
technology, and are normally required to communicate with
experts in other fields

B) Benefit or use of control systems

- Power amplifications: With control systems, heavy things can be

moved precisely (e.g: using robots, etc)
- Convenience: Operators only need to push buttons to move things
- Remote control: Robots are always used for dangerous operations
such as moving radioactive objects or bombs
- Noise filtering: Control systems are always faced with unwanted
signals/noise. For example, an aeroplane in a level flight is faced
with air turbulence. The aeroplane control systems can keep the
aeroplane on its course by filtering the ‘noise’ caused by the air
turbulence.

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- Repetition: Control systems can be used continuously for a long
period of time.

C) History of control systems

- One of the earliest control systems known is the water clock

invented by Ktesibios in 300 BC which is shown below

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- In 1681, Denis Papin introduced the steam pressure control
systems, where he invented the safety valve (very similar to the
present pressure cooker)

- During the same century, Cornelis Drebbel of Holland invented a

control system to regulate temperature for hatching chicken eggs
- On the following century, Edmund Lee was able to control the
speed of a windmill while James Watt invented the fly-ball speed
governor to control the speed of a steam engine
- The field of classical control systems known today began in the
later half of the mid 19th century when J.C. Maxwell wrote about
the stability of 3rd order systems based on the coefficients of the
differential equations
- In 1874, Edward John Routh, using the suggestion from William
Kingdon Clifford was able to extend the stability criterion to the
fifth order systems. In 1877, Routh submitted a paper entitled “A
treatise on the stability of a given state of motion”, which contains
what is known as the Routh-Hurwitz criterion
- During the second half of the 1800s, the development of control
systems focused on the steering and stabilizing of ships. In 1922,

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 the Sperry Gyroscope Company installed an automatic steering
system that used the elements of compensation and adaptive
control to improve the performance
- The general control theory was much influenced by N. Minorsky
who introduced the PID controller in 1885
- In 1948, W.R. Evans who worked in the aircraft industry,
developed a graphical technique for designing controllers, which
is called the Root Locus technique
- Today, control systems find widespread applications, from the
simplest household tools, to the most sophisticated military
equipment such as fighter planes and orbital satellites
- Theories in the field on control systems are continuously being
developed nowadays with the aid of advanced computer
technology

D) Control system definition

- A control system provides an output or response for a given input

or stimulus
- A controlled variable normally determines the input and output
of a control system

Input Control Output

Desired System Actual response
response

- For example: Elevator button and the desired level (input), actual
level of elevator (output)

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- Input is the desired level
- Output is the actual elevator level (control variable)

E) Block diagrams

- A Control System consists of subsystems and processes (or plants)

that are interconnected to control the system output
- For a system having one or more components, it is easier to
represent the components or the subsystems using block
diagrams, where the signal transfer function can be visualized
clearly
- 3 main characteristics of Block Diagrams:
a. Fundamental blocks
b. Components/subsystems
c. Signals

- Fundamental blocks

Input Output
Components /
Subsystems
Desired Actual
response response

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 - Represents components or subsystems such as controller,
amplifier, etc
- Each block may have one or more outputs
- The input and output signals may have the same form or they
may be changed into a different form depending on the
function of the component or subsystem
- Example:

Voltage Displacement
Valve

Voltage Speed
Motor

- Components/subsystems

- Summing junction

allows 2 or more signals to be added/subtracted. The + and –

signs indicate whether the signals are added or subtracted.

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 - Take-off point

allows the signals to be taken from any component’s output.

Assuming that it does not ‘load’ any components output (the
signals are not changed)
- Controllers, plants
- Actuators
- Sensors
- Amplifiers

- Signals in Control Systems

- Input / reference [r(t), R(s)]
- Output [c(t), C(s)]
- Error [e(t)=r(t)-c(t)]
- Feedback, disturbances / noise

- Example:

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F) Open loop control systems (OLCS)

- The output signal of an OLCS is not fed back to influence the

control action

- The control action of an OLCS depends only on the input signal

- OLCS are not capable of filtering disturbances or noise

- Examples: toaster, washing machine (the washing process),

electric fan, rice cooker, photocopy machine, etc

- OLCS are suitable when input signal for satisfactory system

performance can be estimated/approximated and does not change

- Advantages of OLCS are that its structure is simple compared to

a closed loop control system and is cheaper to build.

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G) Closed Loop Control Systems (CLCS)

- The output signal of a CLCS is fed back to influence the control

action and improve the overall system performance

The advantages of CLCS include:

- Ability to compensate for / filter disturbances and/or noise

- Not sensitive to noise and changes to system parameters or

environment

- Can be used repeatedly for a long period of time

- Easier to design a controller for a CLCS to achieve the desired

transient and steady state response

- Examples: position control systems (robot arms), velocity control

systems (vehicle cruise control), temperature control systems (air-
conditioner)

- However, in practical control systems, a combination of both

OLCS and CLCS is normally used.

- A simple example is the washing machine: the process of filling up

the tank with water is a CLCS operation, while the process of
washing and rinsing is an OLCS operation.

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 H) Transfer Function

- A control system with input r(t) and output c(t) can be

represented using a differential equation:
d n c (t ) d n1c(t ) d m r (t ) d m1r (t )
an  an 1  ...  a0 c(t )  bm  bm 1  ...  b0 r (t )
dt n dt n1 dt m dt m1

- This time domain equation can be transformed into an equation

in the Laplace domain using Laplace Transforms:

L[ f (t )]   e st f (t )dt  F ( s)
0

- The function F(s) is a function in the Laplace domain where s is a

complex number :
s    j

- The differential equation that has been mapped to the Laplace

domain can be written as:
a n s n C (s )  a n1 s n 1C ( s)  ...  a0 C ( s )  bm s m R ( s)  bm 1 s m 1 R( s)  ...  b0 R( s )

- The same equation can be re-written as:

C ( s) bm s m  bm1 s m1  ...  b0
  G ( s)
R( s ) a n s n  a n1s n1  ...  a0

- The input and the output has been separated although still related
using an equation

- This equation, G(s), is called the transfer function

- The transfer function can also be represented using a block

diagram:

R(s) C(s)
G(s)

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 C ( s)
- C ( s )  G ( s) R( s ) or G ( s ) 
R( s )

- G(s) is also the open loop transfer function in this case

- Important terms/definitions:
s 2  4s  3 Zeros (-1, -3)
G(s)  5
s  8s 4  20 s 3  16 s 2
order
( s  1)(s  3)
G( s) 
s ( s  2) 2 ( s  4)
2

type

Poles (0, 0, -2, -2, -4)

- For a 2nd order system:

n 2
G( s)  2
s  2 n s   n 2
where  n = natural frequency
 = damping ratio

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- The closed loop transfer function:

E(s) = R(s) - H(s)C(s) and C(s) = G(s)E(s) giving

C(s) = G(s)[R(s) - H(s)C(s)] and by re-arranging gives
C (s) G(s)

R(s) 1  G (s) H (s )
which is the closed loop transfer function

I) The effects of feedback on open loop systems

- Feedback is used to reduce the error between the input and

output of the system

- It effects the system performance characteristics such as stability,

overall system gain, sensitivity and bandwidth

- Effect on the system sensitivity

- Feedback can make the system’s response less sensitive to

external disturbances, parameter changes and noise

- Effect on the system stability

- An unstable system can be stabilized using feedback

- Stability refers to the ability of a system to follow its input

signal

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 - A system that cannot control its output, or its output increases
infinitely is an unstable system

- However, by adding feedback may also cause instability to an

already stable system

- Effect on the system gain

- A feedback influences system’s transfer function and

consequently, the overall system gain

- The letter ‘s’ in the Laplace domain represents a complex

number

- Therefore, the system’s gain can be obtained by finding the

magnitude of the system’s transfer function
C (s)
- In other words, the system gain =
R( s )
C (s)
- For the OLCS, its system gain =  G ( s)
R( s )
C (s) G (s )
- For the CLCS, its system gain = 
R( s ) 1  G ( s) H ( s )

- The system gain can be increased or decreased when adding

the feedback depending on whether the feedback is + or -

J) Analysis and design objectives of control systems

- Control systems are dynamic. They respond to the input by going

through a transient phase before settling to the steady state phase.
Normally, we would like the steady state output signal to be the
same as the input signal

Initial value  transient response  steady state value

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- Three main objectives in designing a control system

- Transient response: to make sure that the transient response of

a control system is within the specification needed. For
example, in an elevator operation, a very slow movement of the
elevator could be a boring long travel for the people in a hurry,
while a very fast movement could be uncomfortable for them.

- Stability: to make sure that the system can produce consistent

or steady output. An unstable system is harmful to the plant
and may cause serious accidents

- Steady state response: steady state response only exists for

stable systems. An important characteristic for design is the
steady state error. Hence, the steady state error has to be as
minimum as possible. Example, an elevator that does not stop
at the same level at the floor may cause serious accidents to its
passengers

- Other design objectives include the sensitivity to parameter

changes and cost.

- The design process of a control system: A systematic design process

has to go through a sequence of steps to achieve the objectives or
specifications

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- Step 1: Transforming the requirements into physical system
and design specifications. For example, in the antenna azimuth
position control system, the requirement would state the desire
to position the antenna from a remote location and describe
such features as weight and physical dimensions. Using the
requirements, the design specification such as the desired
transient response and steady-state accuracy are determined.

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- Step 2: At this step, the designer will interconnect the relation
between system components, such as electrical components and
mechanical components

- Step 3: Schematic diagram displays further information on the

components used.

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- Step 4: The development of a mathematical modeling. The
schematic diagram helps the development of mathematical
model for the system. Physical laws such as Kirchhoff’s laws
for electrical networks, Newton’s laws for mechanical systems,
along with simplifying assumptions, are used to model the
system mathematically. The mathematical model can be of
several forms such as differential equations, transfer functions
and state-space equations.

- Step 5: Block diagram reduction. Normally, a practical system

has complex structure. Hence, the block diagram algebra and
the signal flow graph methods (among others) are used to
simplify the block diagram structure of the system.

- Step 6: Analysis and design. At this stage, the designer has to

analyze and evaluate the system specification. In some cases, a
simple gain adjustment would solve the problem of getting the
required performance. On the other hand, in most cases,
controllers and compensator in many forms need to be
designed.

- Step 7: The development of system’s prototyped. For a real

system, which needs might be expensive to build, a small
system’s prototyped could be build to analyze on its practical
aspects. The real system can be build once any problems
encountered are satisfactorily solved.

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 K) Examples of control systems

- Prob 1.2

Desired diff. Voltage diff. Fuel flow

Desired temp.
Amplifiers Actual temp.
+ Thermostat Heater
and valves
-

- Prob 1.5
Desired
radiation
Transducer

Input voltage Rod position

Actual
+ Motor & radiation
Amplifier Reactor
drive sys.
-
Radiation
error voltage

Sensor
Voltage prop. to
actual radiation

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 Graduating
& drop-out
- Prob 1.6 rate

Actual
Desired Student Net rate
student rate of influx Actual stud.
Desired stud. Population rate popln.
popln.+ error -
Admin. Admissions + Integrate
-

- Prob 1.7
Voltage
prop. Voltage
to representing
Desired desired actual
volume volume Volume volume Actual
+ error volume
Transducer Volume Radio
control cct.
-

Effective
volume +
error
Transducer
-
Voltage prop.
to speed

Transducer

Speed

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L) Classification of control systems

- A control system can be classified according to several criteria

depending on the purpose of the system and the relevant classes it
belongs to
1. Classification based on type of signals used
- Continuous control systems: signals that are used in subsystems
and components in the control system is in time domain, t.

- Discrete type control systems: signals that are used in

subsystems and components in the control system is in discrete
or digital form.

2. Classification based on the mathematical model of the system

- Linear control systems

- An ideal system
- Is always used during the analysis and design phase
- Several tool/techniques can be used to solve the control
problems
- Example: dc motor

- Non-linear control systems

- Systems that normally exist in the real world

- Difficult to analyze
- No general method to find the control solution
- Approximations are normally made so that the system
behaves like a linear system to allow analysis and design
- Example: ac motor, inverted pendulum

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- Control systems can also be grouped into two groups

1. Kinetic (tracking) control systems

- Control variables: displacement/position, speed,
acceleration
- Fast system response with small delay time
- Input may be fixed or changing
- Normally involves electrical or hydraulic manipulators

2. Process (regulating) control systems

- Control variables: temperature, flow, level, pressure

- Slow system response with large delay time
- System responds to a fixed set point
- Examples of manipulators include pneumatic manipulators
or heating elements

M) Common terminology in control systems (English  Malay)

- Automatic: Kawalan yang menggunakan peralatan yang

beroperasi secara sendiri berdasarkan kaedah tertentu untuk
menghasilkan sambutan yang dikehendaki.

- Disturbance: Isyarat yang tidak diingini yang berpunca dari

dalaman atau luaran sistem. Ia mengganggu sistem dan
memberikan kesan buruk terhadap isyarat keluaran sistem.

- Plant: Peralatan seperti gabungan komponen-komponen mesin

yang digunakan untuk melaksanakan sesuatu kerja yang hendak
dikawal.

- Process: Operasi yang berturutan dan berterusan yang

menghasilkan perubahan antara satu operasi dengan operasi
seterusnya. Kadangkala, istilah ini digunakan bagi loji.

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- Design: Langkah-langkah mencipta komponen-komponen sistem.
Setiap komponen tersebut mempunyai peranan tertentu dalam
sistem keseluruhan.

- Synthesis: Langkah-langkah menggabungkan komponen-

komponen sistem bagi menghasilkan sebuah sistem yang padu.

- System: Gabungan komponen-komponen yang beraksi bagi

mencapai sesuatu tujuan. Ia terdiri daripada sistem fizikal
(motor, ketuhar, dsb.) dan bukan fizikal (ekonomi, kemasukan
pelajar ke universiti, dsb.)

- Control system: Gabungan komponen-komponen bagi membentuk

sebuah sistem yang menghasilkan keluaran yang dikehendaki.

- Open-loop control sistem: Sistem yang menggunakan alat untuk

mengawak proses tanpa menggunakan suapbalik. Oleh itu,
isyarat keluaran tidak mempengaruhi tindakan kawalan sistem

- Closed-loop control system: Sistem yang mengukur isyarat

keluaran sebenar dan membandingkannya dengan isyarat yang
dikehendaki.

- Regulating control system: Sistem kawalan suapbalik yang isyarat

masukannya tetap atau berubah perlahan.

- Tracking control system: Sistem kawalan yang isyarat

masukannya berubah secara pantas.

- Servomechanism: Sistem kawalan suapbalik yang keluarannya

berbentuk mekanikal seperti kedudukan, kelajuan dan pecutan.

- Negative feedback: Isyarat keluaran yang disuapbalik supaya ia

dibandingkan dengan isyarat masukan

- Positive feedback: Isyarat keluaran yang disuapbalik dan

dicampurkan dengan isyarat masukan

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