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2.004 - Spring 2018
2.004Dynamics & Control II
Kamal Youcef-ToumiProfessor
Department of Mechanical Engineering Massachusetts Institute of Technology
[email protected] 9th, 2018
Review: Laplace & Fourier Transforms – History, Concepts,
2.004 - Laplace Transforms -spring 2018
2.004 - Spring 2018
Laplace Transforms:History
2.004 - Spring 2018
Pierre-Simon LaplaceDeveloped mathematics in astronomy, physics, and statistics
Began work in calculus which led to the Laplace Transform
Focused later on celestial mechanics
One of the first scientists to suggest the existence of black holes 1749 – 1827
2.004 - Spring 2018
A historical note
Euler considered integrals as solutions to differential equations in the mid 1700’s:
Lagrange extended this while working on probability density functions and considered the following forms:
In 1785, Laplace used a transformation to solve finitedifference equations which led to the current transform
2.004 - Spring 2018
Laplace Transforms:Concepts
2.004 - Spring 2018
Laplace Transform – Operational method used to
Convert
Algebraic functions of s
Complex DomainComplex variable : s
Time DomainReal variable : t
Functions of time t
Differentiation and integration
OperationsAlgebraicoperations
Linear differentialEquations Algebraic equations
2.004 - Spring 2018
Laplace Transform – Operational method used to
§ Analyze general signals§ Provide a spectral representation for signals for which a
Fourier Transform does not exist§ Solve differential equations using inverse Laplace
transform or tables.§ Allow the use of graphical methods to predict system
performance without solving the differential equations of the system. These include response, steady state behavior, transient behavior.
§ …
2.004 - Spring 2018
§ f(t) a function of time t such that f(t)=0 for t<0. Thus our interest is in signals defined for t ≥ 0.
§ t is a time variable with sec as units§ s is a complex variable, complex frequency variable with
units § L is Laplace Transform operator§ F(s), F in capital letter, is the Laplace transform of f(t), f
in lower case.§ Then
Laplace Transforms
L [f(t)] =
1sec
s =σ + jω
2.004 - Spring 2018
Laplace Transforms
§ This is the one-sided or unilateral Laplace transform of f(t)
§ The two sided or bilateral Laplace transform is obtained by setting the lower limit of the integral to -∞.
§ In engineering applications, we are concerned with causal systems and thus in general we use the one sided form.
L [f(t)] = F(s) = f (t)0
∞
∫ e−stdts =σ + jω
2.004 - Spring 2018
Laplace & Fourier Transforms
2.004 - Spring 2018
Laplace Transform & Fourier Transform
F(jω) = f (t)0
∞
∫ e− jωtdt
Joseph Fourier 1768 - 1830
2.004 - Spring 2018
Laplace Transform & Fourier Transform
§ When s is replaced by jω, the Laplace transform becomes the one-sided or unilateral Fourier transform of f(t).
§ The two sided or bilateral Fourier transform is obtained by setting the lower limit of the integral to -∞.
§ There is no difference between the two for causal signals.§ The Fourier transform is then a special case of the Laplace
transform.§ Also, the Fourier transform is obtained by evaluating the
Laplace transform along the jω axis of the s-plane.
F(jω) = f (t)0
∞
∫ e− jωtdt
2.004 - Spring 2018
Laplace Transform & Fourier Transform
§ The Laplace transform can also be viewed as the Fourier transform of an exponentially windowed input signal.
§ …§ The Fourier Transform can be used to study the steady-state
response of a system
F(s) = f (t)0
∞
∫ e−(σ+ jω )tdt = [ f (t)e−σ t0
∞
∫ ]e− jωtdt
2.004 - Spring 2018
See handwritten notes. On Fourier series and Fourier transform.
Joseph Fourier introduced the series for thepurpose of solving the heat equation in a metalplate, publishing his initial results in his 1807.
Laplace Transform & Fourier Transform
2.004 - Spring 2018
End.
2.004 - Spring 2018 Add References
2.004Dynamics & Control II
Kamal Youcef-ToumiProfessor
Department of Mechanical Engineering Massachusetts Institute of Technology
[email protected] 9th, 2018
Review: Laplace Transforms - Properties
2.004 - Laplace Transforms -spring 2018
2.004 - Spring 2018 Add References
Laplace Transforms:Properties
2.004 - Spring 2018 Add References
§ f(t) a function of time t such that f(t)=0 for t<0. Thus our interest is in signals defined for t ≥ 0.
§ t is a time variable with sec as units§ s is a complex variable, complex frequency variable with
units § L is Laplace Transform operator§ F(s), F in capital letter, is the Laplace transform of f(t), f
in lower case.§ Then
Laplace Transforms
L [f(t)] =
1sec
2.004 - Spring 2018 Add References
• f(t) and F(s) are transform pairs• Each f(t) has a unique F(s) and each F(s) has a unique f(t).
Laplace Transforms: Uniqueness
f (t) ↔ F(s)
2.004 - Spring 2018 Add References
Laplace Transform: Linearity 1-2
the
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Laplace Transform: Linearity 2-2
.
.
.
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Laplace Transform: Final value theorem 1-4
)()(lim)(lim ¥== ftfssF0®s ¥®t
Allows finding f(∞) without computing the inverse of F(s).
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Laplace Transform: Final value theorem 2-4 - proof
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Example:
For
[ ] ttesFnotesssF t 3cos)(
3)2(3)2()( 2122
22-=
++-+
= -
Find )(¥f .
[ ] 03)2(3)2(lim)(lim)( 22
22=
++-+
==¥sssssFf
0®s0®s
Laplace Transform: Final value theorem 3-4
2.004 - Spring 2018 Add References
Laplace Transform: Final value theorem 4-4
mx..+ b x
.+ kx = F
1. Compute the steady state value via final value theorem
2. Compute the steady state value via setting all derivatives to zero.At steady state all derivatives are zero.
Example: for a constant F force input.
2.004 - Spring 2018 Add References
0)0()(lim)(lim
®¥®==
tsftfssF
Allows finding the initial condition withoutcomputing the inverses Laplace ofF(s).
Laplace Transform: Initial value theorem 1-2
2.004 - Spring 2018 Add References
forExample:
F(s)= (s+ 2)
(s+1)2+52
Find f(0)
1)26(2
2lim
25122lim
5)1()2(lim)(lim)0(
2222
222
2
2
22
=++
+=
úúû
ù
êêë
é
++++
=++
+==
sssssssss
ssss
sssssFf
¥®s¥®s ¥®s
¥®s
Laplace Transform: Initial value theorem 2-2
2.004 - Spring 2018 Add References
Laplace Transform: Differentiation 1-6
.
2.004 - Spring 2018 Add References
)0(...
)0(')0()()(
)0('')0(')0()()(
)0(')0()()(
)1(
21
233
3
22
2
-
--
--
--=úû
ùêë
é
---=úû
ùêë
é
--=úû
ùêë
é
n
nnnn
n
f
fsfssFsdttdfL
casegeneral
fsffssFsdttdfL
fsfsFsdttdfL
Laplace Transform: Differentiation – 2-6
In general
2.004 - Spring 2018 Add References
If L[f(t)] = F(s), then
)0()(])([ fssFdttdfL -=
Integrate by parts:
Laplace Transform: Differentiation – 3-6 - proof
u = e−st ↔ du=− se−stdt
dv= df (t)dt
dt=df (t)↔ v= f (t)
ò ò¥ ¥
¥-=
0 00| vduuvudv
2.004 - Spring 2018 Add References
After substitutions
[ ]
ò
ò¥
-
¥-¥-
+-=
--=úûù
êëé
0
00
)()0(0
)()( |
dtetfsf
dtsetfetfdtdfL
st
stst
thus
)0()()( fssFdttdfL -=úûù
êëé
Laplace Transform: Differentiation – 4-6 – proof (Cont.)
2.004 - Spring 2018 Add References
Laplace Transform: Differentiation 5-6 - Example
2.004 - Spring 2018 Add References
Laplace Transform: Differentiation 6-6 - Example
-
2.004 - Spring 2018 Add References
Laplace Transform: Integration 1-5
2.004 - Spring 2018 Add References
stst
t
stt
es
vdtedv
and
dttfdudxxfuLet
partsbyIntegrate
dtedxxfdttfL
--
-¥¥
-==
==
úû
ùêë
é=úû
ùêë
é
ò
ò òò
1,
)(,)(
:
)()(
0
0 00
Laplace Transform: Integration 2-5 - proof
2.004 - Spring 2018 Add References
After substitutions:
)(1
)(1)(00
sFs
dtetfs
dttfL st
=
=úû
ùêë
éòò¥
-¥
Laplace Transform: Integration 3-5 – proof (Cont.)
2.004 - Spring 2018 Add References
Laplace Transform: Integration – 4-5 – Example 1
2.004 - Spring 2018 Add References
Laplace Transform: Integration 5-5 – Example 2
-
2.004 - Spring 2018 Add References
Laplace Transform: Convolution Integral – 1-5
The star “*” is used to represent the convolution operation
2.004 - Spring 2018 Add References
Laplace Transform: Convolution – 2-5
-
2.004 - Spring 2018 Add References
Laplace Transform: Convolution – 3-5 - proof
2.004 - Spring 2018 Add References
Laplace Transform: Convolution – 4-5 – Example 1
-
2.004 - Spring 2018 Add References
Laplace Transform: Convolution – 5-5 - Example 2
-
2.004 - Spring 2018 Add References
Laplace Transform: Time scale
dτ=adt and dt= dτ/a
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Laplace Transform: Exponential scaling - 1-2
Replace s by s-a
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Laplace Transform: Exponential scaling 2-2
-
2.004 - Spring 2018 Add References
Laplace Transform: Time Shift 1-3
2.004 - Spring 2018 Add References
Laplace Transform: Time Shift 2-3 - proof
A time shift of f(t) by T in the time domain corresponds to multiplyF(s) by exp(-sT) in the s domain.
Use change of variables:τ=t-TSo t =τ + T And dτ= dt
2.004 - Spring 2018 Add References
Laplace Transform: Time Shift 3-3 - Example
-
2.004 - Spring 2018 Add References
Summary of Laplace Transforms Properties
2.004 - Spring 2018 Add References
End.
2.004 - Spring 2018
2.004Dynamics & Control II
Kamal Youcef-ToumiProfessor
Department of Mechanical Engineering Massachusetts Institute of Technology
[email protected] 9th, 2018
Review: Laplace Transforms – Region of Convergence
2.004 - Laplace Transforms -spring 2018
2.004 - Spring 2018
Laplace Transforms: Region of convergence
§ The Laplace transform F(s) of f(t) exists if the integral converges,
§ All complex values of s for which the integral converges form the region of convergence (ROC).
§ The ROC is a region in the s-plane§ A function f(t) will have a Laplace transform if it is of
exponential order, that is
For some real number . § ..
f (t)0
∞
∫ e−stdt s =σ + jωwhere
limt→∞| f (t)e−σ t |= 0
σ
2.004 - Spring 2018
§ In general functions that increase faster that the exponential function do not have a Laplace transform.
§ For example, the following functions do not have a Laplace transform
§ Note that can have a Laplace transform if it is for 0 ≤ t ≤ T < ∞ where T is a finite time, and that =0 for t < 0 and t > T.
f1(t) = et2
f2 (t) = tet2
for 0 ≤ t ≤ ∞
for 0 ≤ t ≤ ∞
f1(t)
f1(t)
Laplace Transforms: Region of convergence
2.004 - Spring 2018
End.
2.004 - Spring 2018 Add references
2.004Dynamics & Control II
Kamal Youcef-ToumiProfessor
Department of Mechanical Engineering Massachusetts Institute of Technology
[email protected] 9th, 2018
Review: Laplace Transforms – Common Signals
2.004 - Laplace Transforms -spring 2018
2.004 - Spring 2018 Add references
Laplace Transforms of some common signals
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A step function
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Example: A step functionf(t) = 0 for t < 0.f(t) = A for t 0Where A is a real constant.The Laplace transform of f(t) is found
The Laplace integral must converge for the transform to exist.
Note that the step function is undefined at t=0. This does not matter since
Laplace Transforms – Step function 1-2
L [f(t)] = As
A0−
0+
∫ e−stdt = 0
≥ Unit step
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The Laplace transform is valid for all s except at the pole s=0.
Laplace Transforms – Step function 2-2
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A ramp function
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Laplace Transforms – ramp function
u = At and dv = e-stdtdu=Adt and v=e-st/(-s)
Use integration by parts with
ò ò¥ ¥
¥-=
0 00| vduuvudv
t ↔1s2
L[At]= Ate−st dt0
∞
∫
And for the unit ramp f(t) = t:
2.004 - Spring 2018 Add references
An exponential function
2.004 - Spring 2018 Add references
Example: An exponential functionf(t) = 0 for t <0.f(t) = A e-αt for t ≥ 0Where A and α >0 are real constants.The Laplace transform of f(t) is found
The Laplace integral must converge for the transform to exist.
Laplace Transforms: Exponential function
L [f(t)] = Ae−αt0
∞
∫ e−stdt = As+α
2.004 - Spring 2018 Add references
Delta function or Dirac delta function
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Laplace Transforms: A pulse 1-4
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Laplace Transforms: Delta or Dirac delta function 2-4
0
d(t)
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0 t0
f(t)
d(t – t0)
d(t – t0) = 0 for t ¹ t0
01)(0
0
0 >=-ò+
-
ede
e
dtttt
t
Laplace Transforms: A unit impulse 3-4
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The sifting or sampling property of the unit impulse.
Laplace Transforms: A unit impulse 4-4
0 t0
f(t)
d(t – t0)
t1 t2
f (t)δ (t − t0 )dt=f (t0 ) t1 < t0 < t20 otherwise
⎧⎨⎪
⎩⎪t1
t2
∫
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Delta function:Sifting or sampling Property
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The sifting or sampling property of the unit impulse. Laplace Transforms: A unit impulse 1-2
Continuous f(t) T1 T2
• to process this functionin the computer it must be sampled and represented by a finite set of numbers• fixed sampling interval ΔT• Sampling by A/D• Set of delayed Dirac Delta functions
N: {fn} (n = 0 . . . N − 1), wherefn = f(T1 + nΔT).
s(t,ΔT ) = Σn=−∞
∞
δ(t − nΔT )
multiply
To get
2.004 - Spring 2018 Add references
s(t,ΔT ) = Σn=−∞
∞
δ(t − nΔT )
fn = f (nΔT ) = nΔT−
nΔT+∫ f *(t)dt
f *(t) = s(t,ΔT ) f (t) = Σn=−∞
∞
f (t)δ(t − nΔT )
The sifting or sampling property of the unit impulse. Laplace Transforms: A unit impulse 2-2
:set of delayed Dirac delta functions
discrete sample sequence byIntegration across each impulse
sampled waveform
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Sine & Cosine functions
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Laplace Transforms: Sine & Cosine functions 1-2
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Laplace Transforms: Sine & Cosine functions 2-2
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Summary:Transforms of some common functions
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Laplace Transforms
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Laplace Transforms
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Laplace Transforms: Exercises
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Match each time function to its corresponding Laplace transform:
Laplace Transforms: Exercise
f1(t) = t
f2 (t) = e3t
f3(t) = e−3t
f4 (t) = cost
f5(t) = te−2t
a) .
a) .
b) .
c) .
d) .
1s2
1s−3
1s+3
ss2 +1
1(s+ 2)2
1) .
2) .
3) .
4) .
5) .
2.004 - Spring 2018 Add references
Inverse Laplace
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Inverse Laplace
This formula is rarely used!
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End.