Classical and Modern Controls v2

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Graduate Research Assistant
Aeronautics & Astronautics Department
Institute of Space Technology
Islamabad
Feb 18, 2016

Outline
1 Introduction
2 Overview
3 Signal and Systems
4 Modeling
5 Frequency (continous)
6 Time (Continous)
7 Software
8 Optional
9 Labs
10 Quiz

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
kalman Filter
3/242
Introduction
Contact
• Ofﬁce phone: 051-907-5504 ¤
• E-mail: qejaz@cae.nust.edu.pk 1
• Ofﬁce hours: After 11:00 am
1
ejaz.rehman@ist.edu.pk

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
kalman Filter
4/242
Introduction
Text Book

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
kalman Filter
5/242
Introduction
Text Book

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
kalman Filter
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Introduction
Branches of Electrical Engineering 2
1 Signal Processing
2 Systems and Controls
3 Electronic Design
4 Microelectronics
5 VLSI
6 Electrical Energy
7 Electromagnetics
8 Optics and photonics
9 Telecommunications
10 Computer Systems and Software
11 Bioengineering
2 Link

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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Introduction
A Brief Introduction
The only mechanical device that existed for
numerical computation at the beginning of human
history was the abacus, invented in Sumeria circa
2500 BC
And is still widely used by merchants, traders and
clerks in Asia, Africa, and elsewhere

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
kalman Filter
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Introduction
Antikythera mechanism
The Antikythera mechanism, some time around 100 BC
in ancient Greece, is the ﬁrst known analog computer
(mechanical calculator)
Designed to predict astronomical positions and eclipses
for calendrical and astrological purposes as well as the
Olympiads, the cycles of the ancient Olympic Games

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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Introduction
Badi al − Zaman Ab ¯u al − Izz Ism¯a ¯il
ibnal − Raz ¯azal − Jazar¯i
The Kurdish medieval scientist Al-Jazari built
programmable automata3 in 1206 AD.
• Born: 1136 CE
• Era: Islamic GOlden Age
• Died: 1206 CE
3
Same Idea as in Movie Automata (2014)

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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Introduction
Johann Bernoulli 4
• 1667: Born in Switzerland, son of an apothecary (in
medical profession)
• 1738: His son, Daniel Bernoulli published
Bernoulli’s principle
• Students include his son Daniel, EULER, L’Hopital
• 1748: Death
4
http://en.wikipedia.org/wiki/JohannBernoulli

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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Introduction
Leonhard Euler 5
• 1707: Born in Switzerland, son of a pastor
• Among several other things, developed Euler’s identity,
ejω
= cos(ω) + jsin(ω)
• Also developed marvelous polyhedral fromula, nowadays
written as v − e + f = 2.
• Friend of his doctoral advisor’s son, Daniel Bernoulli,
who developed Bernoulli’s principle
• 1783: Death
5
http://en.wikipedia.org/wiki/LeonhardEuler

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
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Introduction
Pierre-Simon Laplace 6
• 1749: Born in France, son of a laborer
• 1770-death: Worked on probability, celestial mechanics,
heat theory
• 1785: Examiner, examined and passed Napoleon in
exam
• 1790: Paris Academy of Sciences, worked with
Lavoisier, Coulomb
• 1827: Died
6
http://en.wikipedia.org/wiki/Pierre-Simon Laplace

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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Introduction
Joseph Fourier 7
• 1768: Born in France, son of a tailor
• 1789-1799: Promoted the French Revolution
• 1798: Went with Napoleon to Egypt and made governor
of Lower Egypt
• 1822: Showed that representing a function by a
trigonometric series greatly simpliﬁes the study of heat
propagation
• 1830: Fell from stairs and died shortly afterward
7
http://en.wikipedia.org/wiki/JosephFourier

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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Introduction
Charles Babbage
Babbage is credited with inventing the ﬁrst mechanical
computer that eventually led to more complex designs.
• Born: 26 December 1791 London, England
• Considered by some to be a father of the computer
• Died: 18 October 1871 (aged 79) Marylebone, London,
England

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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Introduction
John Vincent Atanasoff (1903-1995)
Figure: Atanasoff, in the 1990s.
Built ﬁrst digital computer in the 1930s.

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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Introduction
Howard Hathaway Aiken (1901-1980)
• Built Mark I, during 1939-1944
• Presented to public in 1944
• Reaction was great
• Although Mark I meant a great deal for the
development in computer science, it’s not
recognised greatly today.
• The reason for this is the fact that Mark I (and also
Mark II) was not electronic - it was
electromagnetical

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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Introduction
J. Presper Eckert (1919-1995) and
Mauchly (1907-1980)
Built ENIAC (Electronic Numerical Integrator and
Computer), the ﬁrst electronic general-purpose
computer during 1943-1945 at a cost of $468,000.

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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Introduction
Alan Mathison Turing8
• Born: 23 June 1912
• Turing is widely considered to be the father of
theoretical computer science and artiﬁcial
intelligence
• Famous for Breaking Enigma Machine Code
• Died: 7 June 1954 (aged 41)
8
The Imitation Game: A 2014 Movie biographied on turing

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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Turing Machine
. . . b b a a a a . . . Input/Output Tape
q0q1
q2
q3 ...
qn
Finite Control
q1
Reading and Writing Head
(moves in both directions)

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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History
FORTRAN
• Inventor: John Backus
FORTRAN, derived from Formula Translating
System
• It is a general-purpose, imperative programming
language that is especially suited to numeric
computation and scientiﬁc computing. Originally
developed by IBM
• First Appeared: 1957; 59 years ago

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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History
C++
• Inventor: Bjarne Stroustrup (at Bell Labs)
• It is a general-purpose programming language. It has imperative,
object-oriented and generic programming features, while also providing
facilities for low-level memory manipulation
• C++ is standardized by the International Organization for
Standardization (ISO)
• First Appeared: 1983; 33 years ago

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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Excellence of Human
Equations: Changed The World
17 Equations That Changed The World
Pythagora.s Theorem a2
+ b2
= c2
Pythagoras,530 BC
Logarithms logxy = logx + logy John Napier, 1610
Calculus df
dt
= limh→0
f(t+h)−f(t)
h
Newton, 1668
Law of Gravity F = G m1m2
r2 Newton, 1687
Complex Identity i2
= −1 Euler, 1750
Polyhedra Formula V − E + F = 2 Euler, 1751
Normal Distribution φ(x) = 1√
2πρ
e
(x−µ)2
2ρ2
C.F. Gauss, 1810
Wave Equation ∂2
u
∂t2 = c2 ∂2
u
∂x2 J. d’Almbert,1746
Fourier Transform f(ω) =
∞
−∞
f(x)e−2πixω
dx J. Fourier, 1822
Navier-Stokes Equation ρ(∂v
∂t
+ v. v) = − p + .T + f C. Navier, G. Stokes,
1845
Maxwell’s Equations
E =
ρ
0
.H = 0
× E = − 1
c
∂H
∂t
× H = 1
c
∂E
∂t
J.C. Maxwell, 1865
Second Law of Ther-
mosynamics
dS ≥ 0 L. Boltzmann, 1874
Relativity E = mc2
Einstein, 1905
Schrodinger’s Equation i ∂
∂t
= H E. Schrodinger, 1927
Information Theory H = − p(x)logp(x) C. Shannon, 1949
Chaos Theory xt+1 = kxt (1 − xt ) Robert May,1975
Black-Scholes
Equation 1
2
σ2
S2 ∂2
V
∂S2 + rS ∂V
∂S
+ ∂V
∂t
− rV = 0 F. Black, M. Scholes,
1990

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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Introduction
Modern programming
Whatever the approach to development may be, the ﬁnal
program must satisfy some fundamental properties. The
following properties are among the most important
S Reliability
S Robustness
S Usability
S Portability
S Maintainability
S Efﬁciency/performance

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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History
4GL
Some fourth generation programming language
• Matlab/Simulink
• LabVIEW
• Python
• Wolfram
• Unix Shell

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
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Introduction
Matlab
• Initial Release: 1984; 32 years ago
• MATLAB is a multi-paradigm numerical computing
environment and fourth-generation programming
language
• Widely Used for Academic, Research Development
• Cross-Platform Software
• Latest Stable Release: Matlab R2015b

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
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Introduction
LabVIEW
• Initial Release: 1983; 33 years ago
• LabVIEW (short for Laboratory Virtual Instrument
Engineering Workbench) is a system-design
platform and development environment for a visual
programming language from National Instruments
• The graphical language is named G used by
LabVIEW
• Cross-Platform Software
• Latest Stable Release: 2015/ August 2015

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
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Calculus
Integration by parts
9
9 Link

Control Systems
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Linear Algebra
Partial fraction expansion
10
10 Link

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
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Matlab
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Linear Algebra
Determinants
Here’s an easy illustration that shows why the
determinant of a matrix with linear dependent rows is 0
M =
a b
2a 2b
⇒ |M| = a(2b) − 2b(a) = 0
Let’s look at a 3x3 example.

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
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Linear Algebra
Determinants
M =



a b c
2a 2b c
d e f



⇒ |M| = a(2bf − 2ce) − b(2af − 2cd) + c(2ae − 2bd) = 0
Let’s change the order of rows
M =



d e f
a b c
2a 2b c



⇒ |M| = d(2bc − 2bc) − e(2ac − 2ac) + f(2ab − 2ab) = 0
Let’s change the order of rows again
M =



d e f
2a 2b c
a b c



⇒ |M| = a(2ce − 2bf) − b(2dc − 2af) + c(2db − 2ae) = 0
In other words, if we have dependent rows, then the
determinant of the matrix is 0

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
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Linear Algebra
Adjoint of matrix
11
11 Link

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
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Linear Algebra
Inverse of matrix
A−1 = 1
|A| (Adjoint of A)

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
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Basic Math
Laplace
Overview
Signal and
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Laplace Transform
Tables

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
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Laplace Transform
Tables
12
12 Link

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
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Laplace Transform
Tables
13
13 Link

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Administration
Matlab
LabVIEW
Basic Math
Laplace
Overview
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Parabolic Graph
3/2
0
x2
dx
x
f(x)
1 11
2
2 3
1
2
21
4
3
x2

Control Systems
Qazi Ejaz Ur Rehman
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Fast Fourier Transform

Control Systems
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Flow Chart
Start
Input
Process 1
Decision 1
Process 2a
Process 2b
Output
Stop
yes
no

Control Systems
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Flow Chart
initialize
model
expert system
identify
candidate
models
evaluate
candidate
models
update
model
is best
candidate
better?
stop
yes
no

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
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(continous)
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Laplace Transform
Plot of simple ﬁrst order equation
Let H(s) = 1
s+10 ,We’ve plotted the magnitude of H(s)
below, i.e., |H(s)|. Other possible 3D plots are ∠ H(s),
Re(H(s)) and Im(H(s)) respectively. Notice that |H(s)|
goes to ∞ at the pole s = -10.

Control Systems
Qazi Ejaz Ur Rehman
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Laplace Transform
3-D code for Transfer functions
1 a=−40;c=40;b=1;
2 g=a : b : c ;
3 h=g ;
4 [ r , t ]= meshgrid (g , h ) ;
5 s=r +1 i ∗ t ;
6 Hs = 1 . / ( s+10) ; %t r a n s f e r function
7 mesh( r , t , abs (Hs) ) ;

Control Systems
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Laplace Transform
Plot of Second order equation
Let H(s) = s+5
(s+10)(s−5) . We’ve plotted the magnitude of
H(s) below, i.e., |H(s)|. Other possible 3D plots are
∠H(s), Re(H(s)) and Im(H(s)), respectively. Notice that
|H(s)| goes to ∞ at the pole s = -10 and 5 while it
converges to down at the zero s=-5.

Control Systems
Qazi Ejaz Ur Rehman
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Laplace Transform
Laplace Transform of integration and derivative
For more details on how the Laplace transform for
integration is 1/s and Laplace transform for derivative is
s then see
http://www2.kau.se/yourshes/AB28.pdf

Control Systems
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Control Systems
FREQUENCY (Continous) and Time

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Control Systems
Introduction
Among mechanistic systems, we are interested in linear
systems. Here are some examples:
1 Filters (analog and digital
2 Control sysytems
• A control system is an interconnection of
components forming a system conﬁguration that will
provide a desired system response
• An open loop control system utilizes an actuating
device to control the process directly without using
feedback uses a controller and an actuator to obtain
the desired response
• A closed loop control system uses a measurement
of the output and feedback of this signal to compare
it with the desired output (reference or command)

Control Systems
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Control Systems
Introduction
• To understand and control complex systems, one
must obtain quantitative mathematical models of
these systems
• It is therefore necessary to analyze the
relationships between the system variables and to
obtain a mathematical model
• Because the systems under consideration are
dynamic in nature, the descriptive equations are
usually differential equations
• Furthermore, if these equations are linear or can be
linearized, then the Laplace Transform can be used
to simplify the method of solution

Control Systems
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Control Systems
Introduction
Control system analysis and design focuses on three
things:
1 transient response
2 stability
3 steady state errors
For this, the equation (model), impulse response and
step response are studied. Other important parameters
are sensitivity/robustness and optimality. Control system
design entails tradeoffs between desired transient
response, steady-state error and the requirement that
the system be stable.

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Control Systems
Analysis
The analysis of control systems can be done in the
following nine ways:
1 equation
2 poles/zeros/controllability/observability
3 stability
4 impulse response
5 step response
6 steady-state response
7 transient response
8 sensitivity
9 optimality
The design of control systems can be done in the
following ways:
1 Pole placement (PID in frequency and time, state
feedback in time)
Back to immediate slide

Control Systems
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Matrix reconstruction
Image reconstruction
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Linear algebra
Eigen-decomposition
A: NxN square matrix with
N linearly independent eigenvectors
A = S ∧ S−1 S: eigen vectors in the columns
if A is symmetric ∧: Diagonal eigen value matrix
the eigenvectors Q: orthonormal
are orthonormal eigenvectors in the columns
A = Q ∧ Q−1 = Q ∧ QT
the eigenvalues are real
all matrices are NxN

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SVD

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Linear algebra
SVD (example)

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SVD: successive matrix reconstruction
• Z = sin(xy )
• Original matrix size: 63×63
x and y axes, 0 : 0.1 : 2π
• Max N: 63
6
5
4
3
2
1
N = 63
0
6
5
4
3
2
1
2
0
-2
0

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Linear Algebra
SVD: successive matrix reconstruction cont.
Z = sin(xy)
= λk uk vT
k

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Z = sin(xy)
= λk uk vT
k

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Linear Algebra
Z = sin(xy)
= λk uk vT
k

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Z = sin(xy)
= λk uk vT
k

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Linear Algebra
Z = sin(xy)
= λk uk vT
k

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Linear Algebra
Z = sin(xy)
= λk uk vT
k

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Linear Algebra
Z = sin(xy)
= λk uk vT
k

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Linear Algebra
Z = sin(xy)
= λk uk vT
k

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Linear Algebra
Z = sin(xy)
= λk uk vT
k

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Linear Algebra
Z = sin(xy)
= λk uk vT
k

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Linear Algebra
Z = sin(xy)
= λk uk vT
k

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Linear Algebra
Z = sin(xy)
= λk uk vT
k

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Linear Algebra
Z = sin(xy)
= λk uk vT
k

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Linear Algebra
Z = sin(xy)
= λk uk vT
k

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Linear Algebra
SVD: successive image reconstruction
• Original image size: 339×262
• Max N: 339

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Image = λk uk vT
k

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Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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k

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Image = λk uk vT
k

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k

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k

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k

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Image = λk uk vT
k

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k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Image = λk uk vT
k

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Linear Algebra
Image = λk uk vT
k

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Modeling
RLC Circuit
L
di(t)
dt
+ Ri(t) +
1
C
q(t) = v(t) (1)
i(t) =
dq(t)
dt
(2)
⇒ L
d2q(t)
dt2
+ R
dq(t)
dt
+
1
C
q(t) = v(t)
⇒ L¨q(t) + R ˙q(t) +
1
C
q(t) = v(t)

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Modeling Series RLC Circuit
State Space Representation
Let,
x1 = q(t)
x2 = ˙x1 = ˙q(t)
˙x2 = ¨q(t)
Substituting,
L¨q(t) + R ˙q(t) +
1
C
q(t) = v(t)
L ˙x2 + Rx2 +
1
C
x1 = v(t)
Now Write,
˙x1 = x2
˙x2 = −
1
LC
x1 −
R
L
x2 +
1
L
v(t)
˙x1
˙x2
=
0 1
− 1
LC −R
L
x1
x2
+
0
1
L
v(t)

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Modeling
C parallel with RL circuit
iC = −iL + u(t)
⇒ C
dvC
dt
= −iL + u(t)
⇒
dvC
dt
= −
1
C
iL +
1
C
u(t)
VC = VL + iLR
= L
diL
dt
+ iLR
solve for diL
dt

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Modeling
C parallel with RL circuit
Starting off with differential equations, we go to state
space
dvC
dt
= − 1
C iL +
1
C
u(t)
diL
dt
= 1
L vC −
1
L
iLR
˙VC
˙iL
=
0 − 1
C
1
L −R
L
VC
iL
+
1
C
0
u(t)

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Modeling
Constant acceleration model
¨s(t) = a
t
t0
¨s(τ) dτ =
t
t0
a dτ
˙s(τ)|
t
t0
= a τ|
t
t0
˙s(t) − ˙s(t0) = at − at0
t
t0
˙s(τ)dτ −
t
t0
˙s(t0)dτ =
t
t0
aτdτ −
t
t0
at0dτ
s(τ)|
t
t0
− ˙s(t0)τ|
t
t0
=
1
2
a τ
2
|
t
t0
− at0τ|
t
t0
s(t) − s(t0) − ˙s(t0)t + ˙s(t0)t0 =
1
2
at
2
−
1
2
at0
2
− at0t + at0
2
let initial time t0 = 0, initial distance s(t0) = 0, and some initial velocity ˙s(t0) = vi , to get the familiar
equation,
s(t) = vi t +
1
2
at
2
If we take the derivative with respect to t, we get vf = vi + at

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Modeling
Constant acceleration model
• The equations s = vit + 1
2 at2 and vf = vi + at can be
written in state space as,
s
vf
=
0 t
0 1
si
vi
+
1
2 t2
t
f
m
and writing in terms of states x and input u, we get,
xt =
xt
˙xt
=
0 t
0 1
xt−1
˙xt−1
+
1
2
t2
m
t
m
u
• Note that we have used f = ma, and the input u is
the force f

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Modeling
DC Motor cont..
vb = Kbω
= Kb
˙θ
Differential equations
L
di
dt
+ Ri = v − vb 1
J ¨θ + b ˙θ = Km i 2
R: electrical resistance 1
ohm
L: electrical inductance 0.5H
J: moment of inertia 0.01 kg.m2
b: motor friction constant 0.1 N.m.s
Kb: emf constant 0.01 V/rad/sec
Km: torque constant 0.01 N.m/Amp
Lab 2
Laplace Domain
LsI(s) + RI(s) = V(s) − Vb(s) (3)
Js
2
θ + bsθ = Km I(s) (4)
where Vb(s) = Kbω(s) = Kb sθ
solving equation 3 and 4 simultaneously
angular distance (rad)
G1(s) =
θ(s)
V(s)
=
Km
[( Ls + R)( Js + b) + KbKm ]
1
s
angular rate (rad/sec)
Gp(s) =
ω(s)
V(s)
= sG1(s) =
= Kb
L Js2 + ( L b + R J)s + ( R b + KbKm )

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Modeling
DC Motor cont..
G1(s) =
θ(s)
V(s)
=
1
s
Km
[(Ls + R)(Js + b) + KbKm]
Gp(s) =
˙θ(s)
V(s)
=
Km
[(Ls + R)(Js + b) + KbKm]
Note that we have set Td , TL, TM = 0 for calculating G1(s) and Gp(s).

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Modeling
DC Motor cont..
A motor can be represented simply as an integrator. A
voltage applied to the motor will cause rotation. When
the applied voltage is removed, the motor will stop and
remain at its present output position. Since it does not
return to its initial position, we have an angular
displacement output without an input to the motor.

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Frequency (continous) : analysis
Introduction
• Known as classical control, most work is in Laplace
domain
• You can replace s in Laplace domain with jω to go
to frequency domain

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Frequency (continous): analysis
Test Waveform

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Systems: 1st order
dy
dt
+ a0y = b0r
sY(s) − y(¯0) + a0Y(s) = b0R(s)
sY(s) + a0Y(s) = b0R(s) − y(¯0)
Y(s) =
b0
s + a0
R(s) +
y(¯0)
s + a0
• It is considered stable if the natural response
decays to 0, i.e., the roots of the denominator must
lie in LHP, so a0 0
• The time constant τ of a stable ﬁrst order system is
1/a0
• In other words, the time constant is the negative of
the reciprocal of the pole

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Systems: 1st order

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Systems: 1st order

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Systems: 2nd order
Let G(s) =
ω2
n
s(s+2ζωn)
Y(s) = X(s) − Y(s) G(s)
Y(s) = E(s)G(s)
Y(s) + Y(s)G(s) = X(s)G(s)
⇒
Y(s)
X(s)
=
G(s)
1 + G(s)
=
ω2
n
s2 + 2ζω2
ns + ω2
n
=
b0
s2 + 2ζω2
ns + ω2
n
ζ is dimensionless damping ratio and ωn is the natural frequency or undamped frequency

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Systems: 2nd order
The poles can be found by ﬁnding the roots of the
denominator of Y(s)
X(s)
s1,2 =
−(2ζωn) ± (2ζωn)2 − 4ω2
n
2
=
−(2ζωn) ± (4ζ2ω2
n) − 4ω2
n
2
=
−(2ζωn) ± 2ωn ζ2 − 1
2
= −ζωn ± ωn ζ2 − 1
= −ζωn ± jωn 1 − ζ2

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Systems: 2nd order
Formulas:
%OS = e−ζπ/
√
1−ζ2
× 100
Notice that % OS only depends on the damping ratio ζ

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Systems: 2nd order: Damping

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Systems: 2nd order: Damping
Underdamped system
• Pole positions for an underdamped (ζ 1) second
order system s1, s2 = −ζωn ± jωn 1 − ζ2
when plotted on the s-plane

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Systems: 2nd order
1 Rise Time Tr : The time required for the waveform to go from 0.1 of the
final value to 0.9 of the final value
2 Peak Time Tp: The time required to reach the first, or maximum, peak
• % overshoot: The amount that the waveform
overshoots the steady-state or final, value at the
peak time, expressed as a percentage of the
steady-state value
3 The time required for the transient’s damped oscillations to reach and
stay within 2% of the steady-state value

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Systems: types
Relationships between input, system type, static error
constants and steady-state errors.

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The Characteristics of P, I, and D Controllers
• A proportional controller (Kp) will have the effect of
reducing the rise time and will reduce but never
eliminate the steady-state error.
• An integral control (Ki) will have the effect of
eliminating the steady-state error for a constant or
step input, but it may make the transient response
slower.
• A derivative control (Kd ) will have the effect of
increasing the stability of the system, reducing the
overshoot, and improving the transient response.
The effects of each of controller parameters, Kp, Kd , and
Ki on a closed-loop system are summarized in the table
below.

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The Characteristics of P, I, and D Controllers
CL RESPONSE RISE TIME OVERSHOOT SETTLING TIME S-S ERROR
Kp Decrease Increase Small Change Decrease
Ki Decrease Increase Increase Eliminate
Kd Small Change Decrease Decrease No Change
Note that these correlations may not be exactly accurate, because Kp, Ki , and Kd are dependent on
each other. In fact, changing one of these variables can change the effect of the other two. For this
reason, the table should only be used as a reference when you are determining the values for Ki , Kp
and Kd .
u(t) = Kpe(t) + Ki e(t)dt + Kp
de
dt

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Effect of poles and zeros

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• The zeros of a response affect the residue, or
amplitude, of a response component but do not
affect the nature of the response, exponential,
damped, sinusoid, and so on
Starting with a two-pole system with poles at -1 ±
j2.828, we consecutively add zeros at -3, -5 and -10.
The closer the zero is to the dominant poles, the greater
its effect on the transient response.

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T(s) =
(s + a)
(s + b)(s + c)
=
A
s + b
+
B
s + c
=
(−b + a)/(−b + c)
s + b
+
(−c + a)/(−c + b)
s + c
if zero is far from the poles, then a is large compared to
b and c, and
T(s) ≈ a
1/(−b + c)
s + b
+
1/(−c + b)
s + c
=
a
(s + b)(s + c)
If the zero is far from the poles, then it looks like a
simple gain factor and does not change the relative
amplitudes of the components of the response.

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Root locus
Representation of paths of closed loop poles as the gain
is varied.

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Root locus
• The root locus graphically displays both transient
response and stability information
• The root locus can be sketched quickly to get an
idea of the changes in transient response
generated by changes in gain
• The root locus typically allows us to choose the
proper loop gain to meet a transient response
speciﬁcation
• As the gain is varied, we move through different
regions of response
• Setting the gain at a particular value yields the
transient response dictated by the poles at that
point on the root locus
• Thus, we are limited to those responses that exist
along the root

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Nyquist
Determine closed loop system stability using a polar plot
of the open-loop frequency responseG(jω)H(jω) as ω
increases from -∞ to ∞

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Routh Hurwitz
Find out how many closed-loop system poles are in LHP
(left half-plane), in RHP (right half-plane) and on the jω
axis

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Performance Indeces cont.
• A performance index is a quantitative measure of
the performance of a system and is chosen so that
emphasis is given to the important system
speciﬁcations
• A system is considered an optimal control system
when the system parameters are adjusted so that
the index reaches an extremum, commonly a
minimum value

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ISE =
T
0
e2
(t)dt integral of square of error
ITSE =
T
0
te2
(t)dt integral of time multiplied by square of error
IAE =
T
0
|e(t)|dt absolute magniture of error
ITAE =
T
0
t|e(t)|dt integral of time multiplied by absolute of errorr
• The upper limit T is a ﬁnite time chosen somewhat
arbitrarily so that the integral approaches a
steady-state value
• It is usually convenient to choose T as the settling
time Ts

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Optimum coefﬁcients of T(s) based on the ITAE criterion
for a step input
s − ωn
s2
+ 1.4ωns + ω2
n
s3
+ 1.75ωns2
+ 2.15ω2
ns + ω3
n
s4
+ 2.1ωns3
+ 3.4ω2
ns2
+ 2.7ω3
ns + ω4
n
s5
+ 2.8ωns4
+ 5.0ω2
ns3
+ 5.5ω3
ns2
+ 3.4ω4
ns + ω5
n
s6
+3.25ωns5
+6.60ω2
ns4
+8.60ω3
ns3
+7.45ω4
ns2
+3.95ω5
ns+ω6
n

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Optimum coefﬁcients of T(s) based on the ITAE criterion
for a ramp input
s2
+ 3.2ωns + ω2
n
s3
+ 1.75ωns2
+ 3.25ω2
ns + ω3
n
s4
+ 2.41ωns3
+ 4.93ω2
ns2
+ 5.14ω3
ns + ω4
n
s5
+ 2.19ωns4
+ 6.50ω2
ns3
+ 6.30ω3
ns2
+ 5.24ω4
ns + ω5
n

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Block diagram
Open loop transfer function
T1(s) =
Y1(s)
R(s)
= Gc(s) Gp(s) H(s)
Closed loop transfer function
T1(s) =
Y(s)
R(s)
= Gc(s) Gp(s) H(s)

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Time (Continous)
Introduction
Write your models in the form below:
˙x(t) = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t)
Here,
A is called the system matrix
B is called the Input matrix
C is called the output matrix
D is is called the Disturbance matrix
A B are also called as Jacobin matrix

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Time (Continous)
Overview
In the next few slides, let’s look at some aspects of
analysis in TIME (continuous). During this analysis, the
relationship between classical control vs modern control
will also become clear:
w classical control vs modern control
v transfer function vs state space (matrix)
v poles vs eigen values
v asymptotic stability vs BIBO stability
w Other aspects, only possible in modern control
include:
v controllability
v observability
v senstivity

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Time (Continous)
Overview
1. transfer function vs state space
˙x(t) = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t)
sX = AX + BU TAKE LAPLACE TRANSFORM
sX − AX = BU
(sI − A)X = BU
⇒ X = (sI − A)−1
BU
⇒ Y = C(sI − A)−1
BU + DU
G(s) =
Y
U
= C(sI − A)−1
BU + D
= C
adjoint(sI − A)
det(sI − A)
B + D

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Time (Continous)
Overview
2. poles vs eigen values
Normally, D = 0, and therefore,
G(s) = C
adjoint(sI − A)
det(sI − A)
B
• The poles of G(s) come from setting its
denominator, equal to 0, i.e., let det(sI-A) = 0 and
solve for roots
• But this is also the method for ﬁnding the
eigenvalues of A!
• Therefore, (in the absence of pole-zero
cancellations), transfer function poles are identical
to the system eigenvalues

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Time (Continous)
Overview
3. asymptotic stability vs BIBO stability
• In classical control, we say that a system is stable if
all poles are in LHP (left-half plane of Laplace
domain)
• This is called Asymptotic stability
• In modern control, a system is stable if the system
output y(t) is bounded for all bounded inputs u(t)
• This is called BIBO stability
• Considering the relationship between poles and
eigenvalues, then eigenvalues of A must be
negative

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Time (Continous)
Overview
4. controllability
The property of a system when it is possible to take the state
from any initial state x(t0) to any ﬁnal state x(tf ) in a ﬁnite
time, tf − t0 by means of the input vector u(t), t0 ≤ t ≤ tf
A system is completely controllable if the system state x(tf ) at
time tf can be forced to take on any desired value by applying
a control input u(t) over a period of time from t0 to tf

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Time (Continous)
Overview
4. controllabilitycont..

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Time (Continous)
Overview
4. controllability cont..

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Time (Continous)
Overview
• The Solution to u(t), ˙u(t), ..., un−2(t), un−1(t) can
only be found if Pc is invertible
• Another way to say this is that Pc is full rank
• x(n)(t) is the state that results from n transitions of
the state with input present
• Anx(t) is the state that results from n transitions of
the state with no input present
• PC is therefore called the controllability matrix

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Time (Continous)
Overview
Simple example with 2 states, i.e., n = 2,
A =
−2 1
−1 −3
, B =
1
0
PC = [B AB] =
1 −2
0 −3
|PC| = −1 = 0 ⇒ controllable
In Matlab,
1 A=input ( ’A= ’ ) ;
2 B=input ( ’B= ’ ) ;
3 P= ctrb (A,B) ; %rank (P)
4 unco=length (A)−rank (P) ;
5 i f unco == 0
6 disp ( ’ System i s c o n t r o l l a b l e ’ )
7 else
8 disp ( ’ System i s uncontrollable ’ )
9 end

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Intro. to Matlab
1 The name MATLAB stands for MATrix LABoratory.
2 MATLAB was written originally to provide easy
access to matrix software developed by the
LINPACK (linear system package) and EISPACK
(Eigen system package) projects.
3 MATLAB has a number of competitors. Commercial
competitors include Mathematica, TK Solver,
Maple, and IDL.
4 There are also free open source alternatives to
MATLAB, in particular GNU Octave, Scilab,
FreeMat, Julia, and Sage which are intended to be
mostly compatible with the MATLAB language.
5 MATLAB was ﬁrst adopted by researchers and
practitioners in control engineering.

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Intro. to Simulink
An essential part of Matlab
1 The name Simulink stands for Simulations and links
2 Old name was Simulab
3 Simulink is widely used in automatic control and
digital signal processing for multidomain simulation
and Model-Based Design.

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Intro. to Matlab
Toolboxes to be used in this course are
1 Simulink
2 Mupad (Symbolic math toolbox)
3 Control System toolbox
• Sisotool / rltool
• PID tunner
• LtiView
4 System Identiﬁcation toolbox
5 Aerospace toolbox
6 Simulink Control Design
7 Simulink Design Optimization
8 Simulink 3D animation
9 GUI development
10 Report Generation

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Plotting step response manually
Instead of using the step command to plot step
response, the following manual method can be used for
better understanding:
1 %time
2 t s t a r t = 0 ;
3 tstep = 0.01 ;
4 tstop = 3 ;
5 t = t s t a r t : tstep : tstop ;
6 %step response
7 CF.SR_b = CF. TF_b ;
8 CF.SR_a = [CF. TF_a 0 ] ; %notice that we’ ve added a zero as the l a s t
term to cater f o r m u l t i p l i c a t i o n with 1/ s
9 CF. SR_eqn = t f (CF.SR_b,CF.SR_a) ;
10 [CF. SR_r , . . .
11 CF.SR_p, . . .
12 CF. SR_k ] = residue (CF.SR_b,CF.SR_a) ;
13 %amplitude of step response
14 CF. SR_y =CF. SR_r (1)∗exp (CF.SR_p(1)∗t )+ . . .
15 CF. SR_r (2)∗exp (CF.SR_p(2)∗t ) + . . .
16 CF. SR_r (3)∗exp (CF.SR_p(3)∗t ) ;
17
18 p l o t ( t ,CF. SR_y) ;
19 grid on
20 xlabel ( ’ Time ( sec ) ’ )
21 ylabel ( ’ Wheel angularvelocity rad / sec ) ’ ) ;
22 t i t l e ( ’DC motor step response ’ ) ;
Step Response
♠
Back to the slide 58

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FIR filter design
1 In the Matlab command window, type fdatool
2 Set parameters
3 Export filter to Matlab workspace
4 Set variable name to b for FIR filter
5 Check your design, plot frequency response

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Frequency Response
freqz(b,a,f,fs) % this one line does it all

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FIR filter design
1 In the Matlab command window, type fdatool
2 Set parameters
• Convert to single section
3 Export filter to Matlab workspace
4 Set variable name to b for FIR filter
5 Check your design, plot frequency response

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Frequency Response
freqz(b,a,f,fs) % this one line does it all

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DTFS/DFT/FFT
1 function [ f ,X] = c r e a t e f f t ( x , fs ,NFFT)
2 N=length ( x ) ;
3 X_temp= f f t ( x ,NFFT) /N;
4 f =fs /2∗ linspace (0 ,1 ,NFFT/2+1) ;
5 X=2∗abs ( X_temp ( 1 :NFFT/2+1) ) ;
1 function y=myDFT( x ,N) %DFT, same as f f t
2 n=0:N−1;
3 k=0:N−1;
4 W=exp(− j ∗2∗pi /N) ;
5 Wnk=W. ^ ( n’∗ k ) ; %DFS matrix
6 y=Wnk∗x ;
1 function y=myIDFT( x ,N) %DFT, same as f f t
2 n=0:N−1;
3 k=0:N−1;
4 W=exp(− j ∗2∗pi /N) ;
5 Wnk=W.^(−n’∗ k ) ; %DFS matrix
6 y=Wnk∗x /N;

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DTFS/DFT/FFT
1 x =[0 ,1 ,2 ,3]; %input
2 N=4; %time period
3 n=0:N−1; %t i m e i n d e x
4 k=0:N−1; %f r e q u e n c y i n d e x
5 W=exp(− j ∗2∗ pi /N) ;
6 nk=n ’ ∗ k ;
7 Wnk=W. ^ nk ;
8 f =0.1;
9 X=x∗Wnk

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Noise generation
1 fs =8000;
2 Ts=1/ fs ;
3 t =0:Ts : 4 ;
4 f =1000; %f s =2 f
5 x=sin (2∗ pi ∗ f ∗ t ) ;
6 sound ( x , fs )
7 [ f ,X]= c r e a t e f f t ( x , fs ,2^ nextpow2 ( length ( x )
) ) ;
8 subplot (2 ,1 ,1) ;
9 p l o t ( t (1:150) , x (1:150) ) ;
10 xlabel ( ’ rightarrow sec ’ ) ;
11 subplot (2 ,1 ,2) ;
12 p l o t ( f ,X) ; xlabel ( ’Hz ’ ) ;
13 audiowrite ( ’ na . wav ’ ,x ,44100) ;

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Applications of signal processing
N=10000;
NFFT =2^nextpow2(N);
fs=1000;
Ts=1/fs;
t=(0:N-1)*Ts;
xclean=5*sin(2*pi*220*t) + 2.2*cos(2*pi*120*t);
x=xclean + 20*randn(size(t));
[f,Xclean]=createfft(xclean, fs, NFFT);
[f,X]=createfft(x,fs,NFFT);
subplot(2,2,1); plot(1000*t(1:350),xclean(1:350))
title(’Clean signal’)
xlabel(’time (milliseconds)’)
subplot(2,2,2); plot(1000*t(1:350),x(1:350))
title(’Signal Corrupted with Zero-Mean Random Noise’)
xlabel(’time (milliseconds)’)
% plot fft
subplot(2,2,3); plot(f,Xclean);%x2 because single sided
title(’Single-Sided Amplitude Spectrum of x_clean (t)’)
xlabel(’Frequency(Hz)’)
ylabel(’|Xclean(f)|’)
subplot(2,2,4); plot(f,X) ;%2 is multiplied because single sided
title(’Single-Sided Amplitude Spectrum of x(t)’)
xlabel(’Frequency (Hz)’)
ylabel(’|X(f)|’)
Signal Noise

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Signal Noise

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Remove noise from speech
1 % read wave f i l e
2 x_orig=wavread ( t e s t i n g . wav)
3
4 %create tone
5 fs =44100;
6 Ts=1/ fs ;
7 t =:Ts : ( length ( x_orig ) −1) / fs ;
8 f =8000;
9 tone=sin (2∗ pi ∗ f ∗ t ) ;
10
11 %make noisy signal
12 x=x_orig+tone ;
13 %create low pass f i l t e r ( fpass =4500, fstop =7000, fs
=441 ,00)
14 b
15 a = 1

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Remove noise from speech cont...
1 x _ f i l t = f i l t e r (b , a , x ) ;
2 [ f , X_orig ]= c r e a t e f f t ( x_orig ,44100 ,2^ nextpow2 ( length ( x_orig ) ) ) ;
3 [ f ,X]= c r e a t e f f t ( x ,44100 ,2^ nextpow2 ( length ( x ) ) ) ;
4 [ f , X _ f i l t ]= c r e a t e f f t ( x _ f i l t ,44100 ,2^ nextpow2 ( length ( x _ f i l t ) ) ) ;
5
6 subplot 321; p l o t ( x_orig ) ; t i t l e ( ’ o r i g i n a l voice , x ’ ) ; xlabel ( ’Hz ’ ) ;
7 subplot 322; p l o t ( f , X_orig ) ; axis ( [ 0 20000 0 0.004]) ;
8 xlabel ( ’Hz ’ ) ; t i t l e ( ’ o r i g i n a l signal , x , f f t ’ )
9
10 subplot 323; p l o t ( x ) ; t i t l e ( ’ o r i g i n a l voice + noise ’ )
11 subplot 325; p l o t ( x _ f i l t ) ; t i t l e ( ’ f i l t e r e d x ’ )
12 subplot 324; p l o t ( f ,X) ; axis ( [ 0 20000 0 0.004]) ;
13 xlabel ( ’Hz ’ ) ; t i t l e ( ’ noisy signal , x , f f t ’ )
14 subplot 326; p l o t ( f , X _ f i l t ) ; axis ( [ 0 20000 0 0.004]) ;
15 xlabel ( ’Hz ’ ) ; t i t l e ( ’ f i l t e r e d , x , f f t ’ )
16
17 gk=audioplayer ( x_orig ,44100) ;
18 play ( gk ) ;
19
20 xsc=x /(4∗(max( x )−min ( x ) ) ) ;
21 gtk=audioplayer ( x ,44100) ;
22 play ( gtk ) ;
23 audiowrite ( ’ noisy . wav ’ , xsc ,44100) ;
24
25
26 xscn= x _ f i l t /(4∗(max( x _ f i l t )−min ( x _ f i l t ) ) ) ;
27 gtk=audioplayer ( xscn ,44100) ;
28 play ( gtk ) ;
29 audiowrite ( ’ f i l t e r e d . wav ’ , xscn ,44100) ;

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Filter Response

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Computational Complexity Classes
LOG
Time
LOG
Space
PTIME
N
PTIM
E
NPC
co-N
PTIM
E
PSPACE
EXPTIME
EXPSPACE
.
.
.
ELEMENTARY
.
.
.
2EXPTIME
MATLAB

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Lab1
Learn to Record and Share Your
Results Electronically
• Learn how to make a website and put your results
on it
• Website ﬁles must not be path dependent, i.e, if I
copy them to any location such as a USB, or
different directory, the website must still work
• The main ﬁle of the website must be index.html
• Many tools are available, but a good cross-platform
open source software is kompozer available from
http://www.kompozer.net/

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Lab1
Learn to Extend Existing Work in a
Controls Topic
Make groups, pick a research topic, create a website
with following headings:
1 Introduction
2 Technical Background
3 Expected Experiments
4 Expected Results
5 Expected Conclusions
Present your website. Every group member will be
quizzed randomly. Your ﬁnal work will count towards
your lab exam.

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
kalman Filter
213/242
Generic Block for Control Systems
Controller System
Disturbances
u
Measurements
r e y
−
ym

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
kalman Filter
214/242
Generic Block
Diagram Analogy to Simulink BLocks
1
s
v0
1
s
d0
a v d
i1 f1
i2 f2
i3 f3
i4 f4
i5 f5
f6

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
kalman Filter
215/242
Simulink
Diagram Analogy to Simulink BLocks
1
s
v0
1
s
d0
a v d
i1 f1
i2 f2
i3 f3
i4 f4
i5 f5
f6

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
kalman Filter
216/242
Generic Block
Example of Control System
Navigation
equations
Gyros
Accelero-
meters
ωb
ib
fb
IMU
INS
Velocity
vl
Attitude
Rb
l
Horizontal
position
Rl
e
Depth
z

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
kalman Filter
217/242
Lab1
Gain Effect on Systems

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
kalman Filter
218/242
Lab1
Second order Systems Vs Third order systems

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
kalman Filter
219/242
Lab2
Mathematical Modeling of Motor
• The model we will use will be for a DC motor as
given in this slides on Back to Motor Modelling Slide
• Use the following default values for the 6 constants
needed to model the DC motor:
Km 0.01 Nm/Amp
Kb 0.01 V/rad/s
L 0.5 H
R 1 ω
J 0.01 kg m2
b 0.1 N m s
• Enter in Matlab

Control Systems
Qazi Ejaz Ur Rehman
Avionics Engineer
Introduction
Overview
Signal and
Systems
Modeling
Frequency
(continous)
Time
(Continous)
Software
Optional
Labs
Quiz
kalman Filter
220/242
Lab2
Mathematical Modeling of Motor
1 %( a ) t r a n s f e r function
2 CF. TF_b = Km; %numerator
3 CF. TF_a = [ L∗J L∗b+R∗J R∗b+Kb∗Km] ; %
denominator
4 CF. TF_eqn = t f (CF. TF_b ,CF. TF_a) ; %
equation
5 %( b ) f i n d impulse response
6 impulse (CF. TF_eqn ) ;
7 %( c ) f i n d step response
8 step (CF. TF_eqn ) ;
Optionally, see this slide on plotting step response manually

Classical and Modern Controls v2

Classical and Modern Controls v2

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