CSS260S
Control Systems
Chapter 2
Signals and Control Systems
By
M.Esmail
Email:esmailmoh19@gmail.com
Office: 01.83 Mechatronic
Contents.
In this chapter, you will learn:
• Complex Plane Transformations
(Laplace Transforms)
• Common Input Signals
• Laplace Transform Properties
• Partial Fraction Decomposition
• Inverse Laplace Transforms
• Computing of System Output
response.
M.Esmaill
System analysis steps
Introduction
➢ Dynamic systems are defined by mathematical model
approximations i.e. Differential equations.
➢ The mathematical model captures the system’s inputs and
outputs behaviour.
➢ For the simulation purpose/ better understanding, the solutions
of system models act as transformations on their inputs to yield
outputs.
➢
➢ Differential equations (ODEs) solutions can be obtained by:
• Using direct mathematic solving
• Using analytical approaches (Phase Plane analysis)
• Using form of transformation from time to the complex domain (the Z's
and Laplace's Transforms).
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Revision of complex number and complex
functions
Complex number: has a real part and Imaginary part both of which are
constant. It considered to be a point in the complex plane, or a directed
line to the point
z = x + jy
= z (cos + j sin  )
= z 
= z e j
where : z = x 2 + y 2
 y
x
x = real component
y = imaginary
Complex
variable component
: z = x + jy
 = tan −1  
Complex − conjugate : z = x − jy
Complex algebra
➢Given two complex numbers:
z = x + jy = z 
w = u + jv = w 
➢Sum/difference is given by:
➢Product is given by:
z  w = (x + jy )  (u + jv )
= (x  u ) + j ( y  v )
z  w = (x + jy )  (u + jv )
= (xu − yv) + j (xv + yu )
or z  w = z w ( +  )
➢Quotient is given by:
z 
z
z
=
=
( −  )
w w 
w
or
z x + jy (x + jy )(u − jv ) (xu + yv) + j ( yu − xv)
=
=
=
w u + jv (u + jv )(u − jv )
u 2 + v2
Complex Function
The Laplace Transform
• Is a linear Transformation of
parameters from real time to
complex domain.
• Converts Differential
Equations to Algebraic ones.
• Used to solve systems of
Differential Equations
Definition:
Laplace Transforms of Derivatives &
Anti-derivatives
❖ Laplace Transforms of Derivatives:
• Occur in system models
• Initial conditions assumed –zero
• Differentiation Multiply by s
❖ Laplace Transform of Anti-derivatives(Integrals):
• Reverse differentiation
• Integration Division by s
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Common Signals
❖ Singularity Set
• Functions derived from each
other by differentiation/
integration.
Set 2:
• Exponentials
• sine
• cosine
Set 1:
• Impulse
• Step
• Ramp
• Parabola
These functions together with the Exponential function are of
particular interest in Control Systems, as they are used in the
analysis of systems responses.
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The Exponential
❖ Definition:
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…cont…
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The Unit Impulse
❖Also called the Dirac delta
❖Formulations of important
theories.
❖Difficult to realize in practice
❖Can be approximated by a pulse
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Unit Step
❖Constant amplitude
❖Called unit step if normalized to amplitude = 1.
❖Easy to generate and use.
❖POSITION function
❖Integral of unit impulse
❖Definition:
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From M.S. Tsoeu
Ramp
❖Integral of a Step function
❖Constant gradient
❖VELOCITY function
❖Integral of a ramp.
❖ACCELERATION function
❖Defined as:
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From M.S. Tsoeu
Parabola
Definition:
❖Integral of a ramp.
❖ACCELERATION function
Sinusoids
❖Periodic functions of time
❖Period, amplitude, phase
❖Definition:
❖DeMoivre’sTheorem:
Laplace Transform:
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…cont…
Hence,
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Properties of Laplace Transforms
❖Linearity
❖Multiplication & Convolution
❖Initial Value Theorem
❖Final Value Theorem
❖Time Delay
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Linearity
❖The function is linear if it obeys Superposition
•Scalar Multiplication
•Distribution property
❖ Foundation of Important Control Theories.
❖Definition:
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Convolution
Multiplication
❖Output = convolution of plant model and Input.
❖Challenging to evaluate in time domain
❖Becomes a product in Laplace domain.
Product in Time Domain becomes Complex
Convolution
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Initial & Final Value Theorems
❖Evaluating the initial value of a
system’s output.
❖Evaluating the output final value.
❖Definition:
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Time Delay (Dead Time)
❖Delay between changing
input and seeing corresponding
output change.
❖Conveyor belts.
❖Data transmission over
network.
❖Pipelines
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… cont…
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Inverse Laplace Transform
❖Linear transformation from complex to real
domain.
❖Definition:
❖Seldom used for engineers
❖Alternatives are:
•Direct substitution
•Partial Fractions
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…cont…
❖Direct substitution
•Laplace function contains additive
terms.
•Each term is substituted by its
known time function
•Example:
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Partial Fractions
•Laplace function is a complex fraction.
•Fraction decomposed into partial fractions (additive)
•Direct replacement follows (i.e. exponential function)
❖Process:
•
Factor the denominator and split the rational function into partial
fractions and equate it with the original function.
• Multiplying both sides by the denominator and simplifying to
obtain to obtain an equation form of the problem.
• Factor out similar powers of the fraction variable on both sides
• Compare coefficients of similar powers and solve simultaneously
for each
❖ Residuals
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Examples…
Example1:
Example 2: Compute the time response of the function f(s) below
Example 3:
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Common Laplace Transforms
•Provided in Textbooks, Notes and
Exams.
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Summary of Laplace Transform Properties
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Systems and Input /Output Characteristics
❖ Using Laplace Transforms and its inverse to
evaluate the response of a given system.
❖ In time domain the output y(t) of a system
denoted by the mathematical model g(t) is given
by the convolution of the system model and the
input:
y(t) = g(t)*u(t)
❖ Hence in Laplace domain it becomes multiplication
y(s) = g(s)×u(s)
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Example: Computing a System’s Output
(a)
(b)
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..cont…
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• Thank You!
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