2.004 Dynamics & Control IImeche2004.mit.edu/sites/default/files/documents/2.004_Laplace... ·...

2.004 - Spring 2018

2.004Dynamics & Control II

Kamal Youcef-ToumiProfessor

Department of Mechanical Engineering Massachusetts Institute of Technology

Youcef@mit.eduFebruary 9th, 2018

Review: Laplace & Fourier Transforms – History, Concepts,

2.004 - Laplace Transforms -spring 2018

2.004 - Spring 2018

Laplace Transforms:History

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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

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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

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Laplace Transforms:Concepts

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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

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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.

§ …

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§ 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

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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

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Laplace & Fourier Transforms

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Laplace Transform & Fourier Transform

F(jω) = f (t)0

∞

∫ e− jωtdt

Joseph Fourier 1768 - 1830

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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

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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

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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

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End.

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2.004Dynamics & Control II

Kamal Youcef-ToumiProfessor

Department of Mechanical Engineering Massachusetts Institute of Technology

Youcef@mit.eduFebruary 9th, 2018

Review: Laplace Transforms - Properties

2.004 - Laplace Transforms -spring 2018

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Laplace Transforms:Properties

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§ 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

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• 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)

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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

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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.

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0)0()(lim)(lim

®¥®==

tsftfssF

Allows finding the initial condition withoutcomputing the inverses Laplace ofF(s).

Laplace Transform: Initial value theorem 1-2

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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

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Laplace Transform: Differentiation 1-6

.

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)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

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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

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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.)

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Laplace Transform: Differentiation 5-6 - Example

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Laplace Transform: Differentiation 6-6 - Example

-

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Laplace Transform: Integration 1-5

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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

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After substitutions:

)(1

)(1)(00

sFs

dtetfs

dttfL st

=

=úû

ùêë

éòò¥

-¥

Laplace Transform: Integration 3-5 – proof (Cont.)

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Laplace Transform: Integration – 4-5 – Example 1

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Laplace Transform: Integration 5-5 – Example 2

-

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Laplace Transform: Convolution Integral – 1-5

The star “*” is used to represent the convolution operation

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Laplace Transform: Convolution – 2-5

-

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Laplace Transform: Convolution – 3-5 - proof

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Laplace Transform: Convolution – 4-5 – Example 1

-

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Laplace Transform: Convolution – 5-5 - Example 2

-

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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

-

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Laplace Transform: Time Shift 1-3

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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

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Laplace Transform: Time Shift 3-3 - Example

-

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Summary of Laplace Transforms Properties

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End.

2.004 - Spring 2018

2.004Dynamics & Control II

Kamal Youcef-ToumiProfessor

Department of Mechanical Engineering Massachusetts Institute of Technology

Youcef@mit.eduFebruary 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

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§ 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

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End.

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2.004Dynamics & Control II

Kamal Youcef-ToumiProfessor

Department of Mechanical Engineering Massachusetts Institute of Technology

Youcef@mit.eduFebruary 9th, 2018

Review: Laplace Transforms – Common Signals

2.004 - Laplace Transforms -spring 2018

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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:

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An exponential function

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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

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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

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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) .

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Inverse Laplace

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Inverse Laplace

This formula is rarely used!

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End.