Control Systems

EEE/ECE/INSTR F242
Tutorial 1
Dr. Alivelu M Parimi
Laplace transforms
Information
• Course handout will be discussed in the first class.
• Notices in CMS will be put in the common section
“Course name ECE/EEE/INSTR F242 CONTROL
SYSTEMS FIRST SEMESTER 2020-21 L”

• Today’s tutorial:

• Tutorial 1
• Introduction of Laplace transform
• Application of Laplace transform
25-Jan-22 2
 Control System: Analysis

A combination of various elements connected as a unit to direct or regulate itself

or any other system in order to provide a specific output is known as a Control
system.

• The input and output of a control system must have appropriate

mathematical relationship between them.
• When there exists linear proportionality between input and
output of the system then it is known as a linear control system,
otherwise a non-linear system.
25-Jan-22 3
Control System: Analysis
• Input: For every system to provide a specific result,
some excitation signal must be provided. This signal
is usually given through an external source. So, the
externally provided signal for the desired operation
is known as input.
• Output: The overall response of the system
achieved after application of the input is known as
output.

25-Jan-22 4
Control System: modeling
• The first step in the control design process is to develop
appropriate mathematical models of the system to be
controlled.
• These models may be derived either from physical laws
or experimental data.
• Using the Laplace transform, it is possible to convert a
system's time-domain representation into a frequency-
domain input/output representation, known as
the transfer function.
• In so doing, it also transforms the governing differential
equation into an algebraic equation which is often
easier to analyze.
25-Jan-22 5
Laplace transform: Introduction
• The Laplace transform is a particularly elegant way to solve
linear differential equations with constant coefficients.
• The Laplace transform describes signals and systems not as
functions of time, but as functions of a complex variable s.
• When transformed into the Laplace domain, differential
equations become polynomials of ’s’.
• Solving a differential equation in the time domain becomes
a simple polynomial multiplication and division in the
Laplace domain.
• However, the input and output signals are also in the
Laplace domain, and any system response must undergo
an inverse Laplace transform to become a meaningful time-
dependent signal.

25-Jan-22 6
Why use Laplace Transforms?
• Find solution to differential equation using algebra
• Relationship to Fourier Transform allows easy way
to characterize systems
• No need for convolution of input and differential
equation solution
• Useful with multiple processes in system

25-Jan-22 7
How to use Laplace
• Find differential equations that describe system
• Obtain Laplace transform
• Perform algebra to solve for output or variable of
interest
• Apply inverse transform to find solution

25-Jan-22 8
 What are Laplace transforms?

F(s)  L{f ( t )}   f ( t )e st dt
0
  j
1

1
f ( t )  L {F(s)}  F (s ) e st
ds
2j  j

• t is real, s is complex!
• Inverse requires complex analysis to solve
• Note “transform”: f(t)  F(s), where t is integrated and s is
variable
• Conversely F(s)  f(t), t is variable and s is integrated
• Assumes f(t) = 0 for all t < 0
25-Jan-22 9
 Evaluating F(s) = L{f(t)}

F ( s )  L{ f (t )}   f (t )e  st dt
• Hard Way – do the t)  1
f (integral 0

f (t )  1 
let f (t ) = 1 1
F ( s )   e dt   (0  1) 
 st 1

01
s1 s
F ( s )   e dt  at (0  1) 
 st

0 - atf (t )  e
let s s
f (t ) = eat  
f (t )  e F ( s )   e e dt   e
 at  st ( s  a ) t
dt 
1
 0  0
sa
1
F ( s )   e e dt   e
 at  st ( s  a ) t
dt 
0 0
sa
25-Jan-22 10
Table of selected Laplace
Transforms
1
f ( t )  u ( t )  F(s) 
s
 at 1
f ( t )  e u ( t )  F(s) 
sa
s
f ( t )  cos( t )u ( t )  F(s)  2
s 1
1
f ( t )  sin( t )u ( t )  F(s)  2
s 1

25-Jan-22 11
More transforms
n!
f ( t )  t u ( t )  F(s)  n 1
n

s
0! 1
n  0, f ( t )  u ( t )  F(s)  1 
s s
1!
n  1, f ( t )  tu ( t )  F(s)  2
s N=2?
5! 120
n  5, f ( t )  t 5 u ( t )  F(s)  6  6
s s

f ( t )  ( t )  F(s)  1

25-Jan-22 12
Note on step functions in Laplace
• Unit step function definition:
u ( t )  1, t  0
u ( t )  0, t  0
• Used in conjunction with f(t)  f(t)u(t) because of
Laplace integral limits:


L{f ( t )}   f ( t )e dt st

25-Jan-22 13
Laplace Transforms
•Some Laplace Transforms
•Wide variety of function can be transformed

•Inverse Transform

•Often requires partial

fractions or other
manipulation to find a form
that is easy to apply the
inverse

25-Jan-22 14
Laplace Transforms

25-Jan-22 15
Laplace Transform for ODEs
•Equation with initial conditions

•Laplace transform is linear

•Apply derivative formula

•Rearrange

•Take the inverse

25-Jan-22 16