Control Engineering

Note: I do not claim any originality in these lectures. The contents of this presentation are mostly
taken from the book of Ogatta, Norman Nise, Bishop and B C. Kuo various other internet sources.
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 Syllabus
1. Control Systems: Open and closed loop control systems; Feedback and
feedforward control architectures, their basics and performance evaluation,
limitations, robustness and stability; Fundamentals of modeling dynamic
systems using the laws of physics and differential equations, linear
approximation using Taylor series.
2. Block Diagrams: Fundamentals of block diagram representations of control
systems, their simplifications and applications in designing control system
architecture; Signal Flow graph models; Simulation of control systems using
MATLAB.
3. Mass-Spring-Damper Systems: Analogies of single and multi-body systems,
natural and forced responses, damping ratios, resonant peaks and band
widths; Applications in real world including active vehicle suspension system
control with demonstration, and simulation via MATLAB.
4. RLC Circuit based Control: Concept, mathematical models and control
applications of RLC circuits including Operational Amplifiers, Demonstration,
MATLAB simulation.

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 Syllabus (Continued..)

5. State Variable Approach: State variables of a dynamic system, state

differential equation, system response using state transition matrix, simulation
of state variable models of control systems using MATLAB.
6. Inputs and Responses of Control Systems: Standard inputs (unit impulse,
rectangular, step, ramp, parabolic etc.); Responses of dynamic systems
(natural, forced, transient, steady-state etc.); Percentage overshoot, Lead-Lag.
7. Stability Analysis: Basic concept for linear systems using the Routh array
test, marginal stability, control design constraints, applications in feedback
systems.
8. Evans Root Locus techniques: Mathematical basis and application in control
design for real world systems.
9. Gain and Phase margins: Basic concept, polar plots, computation from Bode
diagrams and Nyquist plots, implications in terms of robust stability of control
systems.

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 Syllabus (Continued..)
10. Actuator Control: Pneumatic, hydro-pneumatic, electro-hydro-pneumatic
actutators, study of pneumatic circuits with physical demonstration, electro-
hydro-pneumatic control system demonstration and mathematical modeling for
4 post car lift, simulation using MATLAB; D.C. and servo motors control methods
and mathematical models, their analysis using block diagrams and transfer
functions.
11. Design of Feedback Control Systems: Phase Lead and Lag-Design using
Bode diagrams and root locus; Lead-Lag compensators based on frequency data
for open-loop linear systems; PLC based control fundamentals, physical
demonstration using trainer and MATLAB simulation; PID controller basics,
algorithms for control including ladder diagrams, designing PID controllers
based on empirical tuning rules, physical demonstration and modeling of water
level control in water reservoir and temperature control in heating set-ups.

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 Syllabus (Continued..)
12. Automotive control systems: Integration of engine management and
transmission control systems, power train control systems, automatic clutch
and throttle system, chassis control systems: Antilock Braking systems (ABS),
electronic damping control systems, power assisted steering systems, traction
systems. Cruise control.
13. Electromechanical system: mathematical modelling and designing of
electromechanical systems. Air bag and seat pre-tensioner systems. Servo-
mechanism.
14. Thermostatic control systems, electromechanical, hydraulic and pneumatic
positioner systems.

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 Outline
• Course Content
• Recommended Books
• Prerequisites of the subject
• Basic Definitions
• Types of Control Systems
• Transfer function
• Stability of Control System

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Classical Control Modern Control
• System Modelling • State Space Modelling
•Transfer Function • Eigenvalue Analysis
•Block Diagrams • Observability and Controllability
•Signal Flow Graphs • Solution of State Equations (state Transition Matrix)
• System Analysis • State Space to Transfer Function
•Time Domain Analysis • Transfer Function to State Space
•Frequency Domain Analysis •Direct Decomposition of Transfer Function
• Bode Plots, Nyquist Plots, Nichol’s Chart •Cascade Decomposition of Transfer Function
• Root Locus •Parallel Decomposition of Transfer Function
• System Design • State Space Design Techniques
•Compensation Techniques
•PID Control

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 Text Books
1. Modern Control Engineering, (5th Edition)
By: Katsuhiko Ogata.
(Prof Emeritus)
Mechanical Engineering
University of Minnesota

2. Control Systems Engineering, (6th Edition)

By: Norman S. Nise. (Professor Emeritus)
Electrical and Computer
Engineering Department
at California State Polytechnic University

8
 Reference Books

1. Modern Control Systems, (12th Edition)

By: Richard C. Dorf and Robert H. Bishop.

2. Automatic Control Systems, (9th Edition)

By: Golnaraghi and B. C. Kuo.

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 Prerequisites
• For Classical Control Theory
– Differential Equations
– Laplace Transform
– Basic Physics
– Ordinary and Semi-logarithimic graph papers
• For Modern Control theory above &
– Linear Algebra
– Matrices

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 What is Control System?
• A system Controlling the operation of another
system.
• A system that can regulate itself and another
system.
• A control System is a device, or set of devices to
manage, command, direct or regulate the
behaviour of other device(s) or system(s).

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 Terminologies

Manipulated Variable

Input
or Process Output
Set point Controller Or Or
or Plant Controlled Variable
reference

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 Types of Control System

• Natural Control System

– Universe
– Human Body

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 Types of Control System
• Manmade Control System
– Aeroplanes
– Chemical Process

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 Types of Control System
• Manual Control Systems
– Room Temperature regulation Via Electric Fan
– Water Level Control

• Automatic Control System

– Home Water Heating Systems (Geysers)
– Room Temperature regulation Via A.C
– Human Body Temperature Control
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 Types of Control System
Open-Loop Control Systems
Open-Loop Control Systems utilize a controller or control actuator to
obtain the desired response.
• Output has no effect on the control action.
• In other words output is neither measured nor fed back.

Input Output
Controller Process

Examples:- Washing Machine, Toaster, Electric Fan, microwave oven,

e.t.c 16
 Types of Control System
Open-Loop Control Systems

• Since in open loop control systems reference input is not

compared with measured output, for each reference input there
is fixed operating condition. Therefore, the accuracy of the
system depends on calibration.

• The performance of open loop system is severely affected by the

presence of disturbances, or variation in operating/
environmental conditions.

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 Types of Control System
Closed-Loop Control Systems

Closed-Loop Control Systems utilizes feedback to compare the actual

output to the desired output response.

Input Output
Comparator Controller Process

Measurement

Examples:- Refrigerator, Electric Iron, Air conditioner

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 Types of Control System
Multivariable Control System

Outputs
Temp
Humidity Comparator Controller Process
Pressure

Measurements

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 Types of Control System
Feedback Control System

• A system that maintains a prescribed relationship between the output

and some reference input by comparing them and using the difference
(i.e. error) as a means of control is called a feedback control system.

Input + error Output

Controller Process
-

Feedback

• Feedback can be positive or negative.

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 Types of Control System
Servo System

• A Servo System (or servomechanism) is a feedback control system in

which the output is some mechanical position, velocity or acceleration.

Antenna Positioning System Modular Servo System (MS150)

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 Transfer Function
• Transfer Function is the ratio of Laplace transform of the
output to the Laplace transform of the input.
Considering all initial conditions to zero.
u(t) y(t)
Plant

If u(t )  U ( S ) and
y ( t )  Y ( S )

• Where  is the Laplace operator.

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 Transfer Function
• Then the transfer function G(S) of the plant is given
as
Y (S )
G( S ) 
U (S )

U(S) G(S) Y(S)

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 Why Laplace Transform?
• By use of Laplace transform we can convert many
common functions into algebraic function of complex
variable s.
• For example

 sin t  2
s  2
Or
 at 1
e 
sa
• Where s is a complex variable (complex frequency) and
is given as
s    j 24
 Laplace Transform of Derivatives
• Not only common function can be converted into
simple algebraic expressions but calculus operations
can also be converted into algebraic expressions.

• For example
dx(t )
  sX ( S )  x( 0)
dt

d 2 x(t ) 2 dx( 0)
  s X ( S )  x( 0) 
dt 2 dt
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 Laplace Transform of Derivatives
• In general

d x(t )
n
n 1 n 1
 n
 s X (S )  s
n
x( 0)    x ( 0)
dt

• Where x(0) is the initial condition of the system.

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 Initial Condition Explained

• u is the input voltage applied at t=0

• y is the capacitor voltage

• If the capacitor is not already charged then

y(0)=0.

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 Laplace Transform of Integrals

1
  x(t )dt  X ( S )
s

• The time domain integral becomes division by

s in frequency domain.

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 Calculation of the Transfer Function
• Consider the following ODE where y(t) is input of the system and
x(t) is the output.
d 2 x(t ) dy(t ) dx(t )
A C B
• or dt 2 dt dt

Ax' ' (t )  Cy' (t )  Bx' (t )

• Taking the Laplace transform on either sides

A[ s 2 X ( s )  sx(0)  x' (0)]  C[ sY ( s )  y(0)]  B[ sX ( s )  x(0)]

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 Calculation of the Transfer Function
A[ s 2 X ( s )  sx(0)  x' (0)]  C[ sY ( s )  y(0)]  B[ sX ( s )  x(0)]

• Considering Initial conditions to zero in order to find the transfer

function of the system

As 2 X ( s )  CsY ( s )  BsX ( s )

• Rearranging the above equation

As 2 X ( s )  BsX ( s )  CsY ( s )
X ( s )[ As 2  Bs ]  CsY ( s )
X (s) Cs C
 
Y ( s ) As  Bs As  B
2
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 Examples
1. Find out the transfer function of the RC network shown in figure-1.
Assume that the capacitor is not initially charged.

Figure-1

2. u(t) and y(t) are the input and output respectively of a system defined by
following ODE. Determine the Transfer Function. Assume there is no any
energy stored in the system.

6u' ' (t )  3u(t )   y(t )dt  3 y' ' ' (t )  y(t )

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 Transfer Function
• In general

• Where x is the input of the system and y is the output of

the system.

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 Transfer Function

• When order of the denominator polynomial is greater

than the numerator polynomial the transfer function is
said to be ‘proper’.

• Otherwise ‘improper’

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 Transfer Function
• Transfer function helps us to check

– The stability of the system

– Time domain and frequency domain characteristics of the

system

– Response of the system for any given input

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 Stability of Control System
• There are several meanings of stability, in general
there are two kinds of stability definitions in control
system study.

– Absolute Stability

– Relative Stability

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 Stability of Control System

• Roots of denominator polynomial of a transfer

function are called ‘poles’.

• And the roots of numerator polynomials of a

transfer function are called ‘zeros’.

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 Stability of Control System

• Poles of the system are represented by ‘x’ and

zeros of the system are represented by ‘o’.
• System order is always equal to number of
poles of the transfer function.
• Following transfer function represents nth
order plant.

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 Stability of Control System
• Pole is also defined as “it is the frequency at which
system becomes infinite”. Hence the name pole
where field is infinite.

• And zero is the frequency at which system becomes

0.

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 Relation b/w poles and zeros and frequency
response of the system
• The relationship between poles and zeros and the frequency
response of a system comes alive with this 3D pole-zero plot.

Single pole system

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Relation b/w poles and zeros and frequency
response of the system
• 3D pole-zero plot
– System has 1 ‘zero’ and 2 ‘poles’.

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Relation b/w poles and zeros and frequency
response of the system

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 Example
• Consider the Transfer function calculated in previous
slides.
X (s) C
G( s )  
Y (s) As  B

the denominato r polynomial is As  B  0

• The only pole of the system is

B
s
A

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 Examples
• Consider the following transfer functions.
– Determine
• Whether the transfer function is proper or improper
• Poles of the system
• zeros of the system
• Order of the system

s3 G( s ) 
s
i) G( s )  ii)
s( s  2 ) ( s  1)( s  2)( s  3)

( s  3) 2 s 2 ( s  1)
iii) G( s )  iv) G( s ) 
s( s 2  10 ) s( s  10 )
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 Stability of Control Systems
• The poles and zeros of the system are plotted in s-plane
to check the stability of the system.
j

LHP RHP

Recall s    j


s-plane

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 Stability of Control Systems
• If all the poles of the system lie in left half plane the
system is said to be Stable.
• If any of the poles lie in right half plane the system is said
to be unstable.
• If pole(s) lie on imaginary axis the system is said to be
marginally stable. j

LHP RHP

s-plane
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 Stability of Control Systems
• For example
C
G( s )  , if A  1, B  3 and C  10
As  B
• Then the only pole of the system lie at

pole  3
j

LHP RHP

X 
-3

s-plane
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 Examples
• Consider the following transfer functions.
 Determine whether the transfer function is proper or improper
 Calculate the Poles and zeros of the system
 Determine the order of the system
 Draw the pole-zero map
 Determine the Stability of the system

s3 G( s ) 
s
i) G( s )  ii)
s( s  2 ) ( s  1)( s  2)( s  3)

( s  3) 2 s 2 ( s  1)
iii) G( s )  iv) G( s ) 
s( s 2  10 ) s( s  10 )
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 Another definition of Stability
• The system is said to be stable if for any bounded
input the output of the system is also bounded
(BIBO).
• Thus for any bounded input the output either
remain constant or decrease with time.
u(t) overshoot
y(t)
1
1
Plant
t
t
Unit Step Input
Output
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 Another definition of Stability
• If for any bounded input the output is not
bounded the system is said to be unstable.

u(t)
y(t)
1
e at
Plant
t
t
Unit Step Input
Output

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 BIBO vs Transfer Function
• For example
Y ( s) 1 Y (s) 1
G1 ( s)   G2 ( s)  
U ( s) s  3 U ( s) s  3
Pole-Zero Map Pole-Zero Map
4 4
unstable
3 stable 3

2 2

1 1
Imaginary Axis

Imaginary Axis
0 0

-1 -1

-2 -2

-3 -3

-4 -4
-4 -2 0 2 4 -4 -2 0 2 4
Real Axis Real Axis
 BIBO vs Transfer Function
• For example
Y ( s) 1 Y (s) 1
G1 ( s)   G2 ( s)  
U ( s) s  3 U ( s) s  3

1 Y ( s)
1 1 1 Y (s) 1
 G1 ( s)    1
 G2 ( s)   1
 1
U ( s) s3 U (s) s 3
 y (t )  e 3t u (t )  y (t )  e3t u (t )
 BIBO vs Transfer Function
• For example
Y (s) 1
G1 ( s) 
Y ( s)

1 G2 ( s)  
U ( s) s  3 U ( s) s  3
3t
y(t )  e u (t ) y (t )  e3t u (t )
12
exp(-3t)*u(t) x 10 exp(3t)*u(t)
1 12

10
0.8

8
0.6
6
0.4
4

0.2
2

0 0
0 1 2 3 4 0 2 4 6 8 10
 BIBO vs Transfer Function
• Whenever one or more than one poles are in
RHP the solution of dynamic equations
contains increasing exponential terms.
• Such as e3t .
• That makes the response of the system
unbounded and hence the overall response of
the system is unstable.