frequency domain modelling
TRANSCRIPT
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AMME 3500/9501 :
System Dynamics and Control
Frequency Domain Modelling
Dr. Ian R. Manchester
A/Prof Ian R. Manchester Amme 3500 : Introduction
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Slide 3
Overview
• Linear Time-Invariant (LTI) Models
– Linearisation
– The Principle of Superposition
• Laplace transform
• Transfer Functions
A/Prof Ian R. Manchester Amme 3500 : Introduction
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Slide 4Dr. Ian R. Manchester Amme 3500 : Introduction
Mathematical Modelling
• A mathematical model is one or more equations
that describe the relationship between the system
variables
• These models are usually derived from basic
physical principles, often with some parameters
that need to be determined experimentally
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Slide 5
Linearity
• The net response to a sum of inputs is
the sum of the output responses of each
input considered in isolation
• Mathematically: = ()
• Linear: 1 + 2 = 1) + (2
A/Prof Ian R. Manchester Amme 3500 : Introduction
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Slide 6Dr. Ian R. Manchester Amme 3500 : Introduction
Dynamic Models
• A linear differential equation for a single-input, single-
output system has the form
Plant
u(t) y(t)
• (an-1, …, a0, bm, … b0) are the system’s parameters
• An LTI system has parameters that are time-invariant
• n is the order of the system
1
1 0 01
( ) ( ) ( )( ) ( )
n n m
n mn n m
d y t d y t d u t a a y t b b u t
dt dt dt
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Slide 7Dr. Ian R. Manchester Amme 3500 : Introduction
A Familiar Mechanical Example
• Spring damper system
• Mass M is suspended by
a spring and damper
• Force f is applied
• How do we predict the
motion of this system?M
f(t)
y(t)
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Slide 8Dr. Ian R. Manchester Amme 3500 : Introduction
Translation Mechanical Elements
• Force-velocity,
force-
displacement, and
impedancetranslational
relationships
for springs,
viscous dampers,and mass
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Slide 9Dr. Ian R. Manchester Amme 3500 : Introduction
Rotational Mechanical Elements
• Torque-angular
velocity, torque-
angular
displacement,and impedance
rotational
relationships for
springs, viscousdampers, and
inertia
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Slide 10Dr. Ian R. Manchester Amme 3500 : Introduction
Free Body Diagram
• Draw an FBD
• Insert forces acting on
body
• Sum forces
d K yM
f(t)
y(t)
Ky(t ) K d y(t )
F =myå
my = f (t )- Ky(t )- K d y(t )my+ K d y(t )+ Ky(t ) = f (t )
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Slide 11
Nonlinear Systems
• Most real systems are nonlinear:
– A spring has only so much travel available
– Wind resistance is a quadratic function of
velocity
– Trigonometric terms show up for rotational
joints
A/Prof Ian R. Manchester Amme 3500 : Introduction
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Slide 12
Linearisation (Taylor Series)
• A local approximation to a nonlinear
function can be found by linearisation:
≈ 0 +
− 0 + ℎ. . .
A/Prof Ian R. Manchester Amme 3500 : Introduction
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Slide 13
Pendulum Example
• + sin =
A/Prof Ian R. Manchester Amme 3500 : Introduction
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Slide 14
Hydraulic Pump Example
A/Prof Ian R. Manchester Amme 3500 : Introduction
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Slide 15Dr. Ian R. Manchester Amme 3500 : Introduction
Impulse Response
• vid
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Slide 16
Impulse Response
Dr. Ian R. Manchester Amme 3500 : Introduction
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Slide 17
Examples
• Response of a car suspension system to a pothole
• Flexing “modes” of an aircraft wing
• Or a high-precision industrial robot arm
• The response of blood glucose concentration, insulin
production, etc after eating a meal
Dr. Ian R. Manchester Amme 3500 : Introduction
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Slide 18Dr. Ian R. Manchester Amme 3500 : Introduction
The Unit Impulse
• An impulse is an infinitely short pulse at t =0
• Any signal can be thought of as
the summation (integral) of
many impulses at different points
in time
• By the principal of
superposition, if we can find the
response of the system to one
impulse, we will be able to find
the response to an arbitrary
input
( ) ( ) ( ) f t d f t
f(t)
t
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Slide 19
Impulse Response
• Suppose the input consisted of just three
impulses at times t = 0, 1, 2 of size 7, 8, 9.
• What is the value of y(t ) at time t = 5?
Dr. Ian R. Manchester Amme 3500 : Introduction
y = f (u1 + u2 + u3)
= f (u1)+ f (u2)+ f (u3)
= 7h(5)+ 8h(4)+ 9h(3)
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Slide 20Dr. Ian R. Manchester Amme 3500 : Introduction
Convolution Integral
• The output is the sum (integral) of each impulse
response of the system to each individual impulse of
the input , delayed by the appropriate time
• We call this convolution, and it is written as
y(t ) = h(t )u(t -t )d t 0
¥
ò
y(t ) = h(t )*u(t )
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Slide 21Dr. Ian R. Manchester Amme 3500 : Introduction
Example : Solving in time
• Assume we have a system described by the
following differential equation
• The impulse response for this system is
( ) kt h t e
˙ y(t )+ ky(t ) = u(t )
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Slide 22Dr. Ian R. Manchester Amme 3500 : Introduction
• If we want to find the response of the system
to a sinusoidal input, sin(wt )
Example : Solving in time
( ) ( ) ( )
sin( ( ))k
y t h u t d
e t d
w
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Slide 23Dr. Ian R. Manchester Amme 3500 : Introduction
Cascaded systems
• Now what happens if
we have multiple
components in thesystem?
h (t)u(t) y1(t)
h1(t)y(t)
1
1 1
1
( ) ( ) ( )
( ) ( ) ( )
( ) ( ( ) * ( )) * ( )
y t u t h d
y t y t h d
y t u t h t h t
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Slide 24Dr. Ian R. Manchester Amme 3500 : Introduction
Step Response
• Often we are interested inthe response of a system to astep change, e.g. the change
of reference signal
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Slide 25
Paradox?
• How can we have a step response for a
system defined by derivatives of the input?
Dr. Ian R. Manchester Amme 3500 : Introduction
1
1 0 01
( ) ( ) ( )( ) ( )
n n m
n mn n m
d y t d y t d u t a a y t b b u t
dt dt dt
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Slide 26Dr. Ian R. Manchester Amme 3500 : Introduction
The input signal as an Exponential
• This provides us with a rich basis function for
describing functions
• Through Euler ’s formula, we find
• Fourier analysis tells us that this is sufficient for
representing any signal
( )
(cos sin )
st j t
t j t
t
e e
e e
e t j t
w
w
w w
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Slide 27
The “Frequency Domain”
• We can also consider the response of a system
to sinusoidal inputs
• Fourier theory tells us that all signals can be
decomposed into a sums of sinusoids
• Our “basis signals” are complex exponentials:
Dr. Ian R. Manchester Amme 3500 : Introduction
u(t ) = e st = e(s + jw )t
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Slide 28Dr. Ian R. Manchester Amme 3500 : Introduction
Laplace Transform
• The Laplace transform is defined as:
• The inverse transform is:
F ( s ) = f (t )e - st dt 0
¥
ò
1( ) ( )
2
j st
j f t F s e ds
j
Where
s =s + jw
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Slide 29
Laplace Transform of a Step
Dr. Ian R. Manchester Amme 3500 : Introduction
F ( s ) = f (t )e - st dt 0
¥
ò
f (t )=1
Step input
Laplace Trans.
If s > 0
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Slide 30Dr. Ian R. Manchester Amme 3500 : Introduction
Table of Laplace Transforms
• What about that nastyintegral in the Laplaceoperation?
• We normally use tablesof Laplace transformsrather than solving the
preceding equationsdirectly
• This greatly simplifiesthe transformation
process
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Slide 31Dr. Ian R. Manchester Amme 3500 : Introduction
Laplace Transform Theorems
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Slide 32Dr. Ian R. Manchester Amme 3500 : Introduction
How does this help us?
• Convolution in time is
equivalent to
multiplication in theLaplace domain
( ) ( ) * ( )
( ) ( ) ( )
y t u t h t
Y s U s H s
H(s)U(s) Y1(s)
H1(s)Y(s)
1( ) ( ) ( ) ( )Y s U s H s H s
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Slide 33Dr. Ian R. Manchester Amme 3500 : Introduction
Comparing Solution Methods
• Starting with an impulse response, h(t), and an
input, u(t), find y(t)
u(t), h(t)
U(s), H(s)
convolution
y(t)
L L -1
Multiplication,
algebraic manipulation
Y(s)
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Slide 34Dr. Ian R. Manchester Amme 3500 : Introduction
Transfer Function
• The transfer function H(s) of a system isdefined as the ratio of the Laplace transforms
output and input with zero initial conditions
• It is also Laplace transform of the impulse
response ( ) ( ) * ( )
( ) ( ) ( )
y t u t h t
Y s U s H s
H ( s) = Y ( s)U ( s)
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Slide 35Dr. Ian R. Manchester Amme 3500 : Introduction
Example : Transfer Function
• Recall the system we examined earlier
• To find the transfer function for this system,we perform the following steps
sY ( s) - y(0)+ kY ( s) =U ( s)
Y ( s)( s+k )=U ( s)
H ( s) =Y ( s)
U ( s)=
1
s+ k
( ) kt h t eor
˙ y(t )+ ky(t ) = u(t )
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Slide 36Dr Ian R Manchester Amme 3500 : Introduction
Conclusions
The “Linear Time Invariant” abstraction
allows us to completely understand system
response by looking at certain basic responses
(impulse, step, frequency)
The Laplace transform (of signals) and transfer
functions (of systems) are a very convenient
representations for analysis