Download - Sistem Kontrol I Kuliah II : Transformasi Laplace Imron Rosyadi, ST Email: [email protected] 1
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From Lecture #1
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Control System Design Process
• Diagram on the next page gives a flowchart of the control system design process
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4Original System
- Plant - Sensors
- Actuators
New System
Math Model of Controller
Key Activities of the“MAD” Control Engineer:
- Modeling - Analysis - Design
- ImplementationMath Model
of Plant
MeasurementModeling
Implementation - Physical controller - Coupling controller
with plant
Desired Performance
DevelopPerformance Specifications
Analysis
Design
Simulation
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Control System Design Process
• Hidden in this chart are three important elements :
1. Modeling the system (using mathematics)2. Analysis techniques for describing and
understanding the system’s behavior3. Design techniques for developing control
algorithms to modify the system’s behavior
• Modeling, analysis, and design = the MAD control theorist
• A fourth key element is Implementation
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Modeling is the key!
• The single most important element in a control system design and development process is the formulation of a model of the system.
• A framework for describing a system in a precise way makes it possible to develop rigorous techniques for analyzing the system and designing controllers for the system
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Modeling
• Key Point: most systems of interesting in engineering can be described (approximately) by▫Linear▫Ordinary▫Constant-coefficient▫Differential equations
• Call these LODEs• Where we are going looks like this:
PhysicalReality
LODELaplace
Transform
Requirescalculus to solve
Requiresalgebra to solve
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Review of Complex Number
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Complex Numbers: Notation and Properties (1)• A complex number:
• The complex plane
Rectangular (Cartesian) coordinates
Polar coordinates
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• Transformation between coordinates
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Complex Numbers: Notation and Properties (2)
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Complex Numbers: Notation and Properties (3)• Euler’s Formula:
• Note differentiation property
)sin()cos( xjxe jx
jx
jx
je
xjxj
xjx
xdx
djx
dx
de
dx
d
)sin()cos(
)cos()sin(
)sin()cos(
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Exercise
• Show how Euler’s Formula is a parameterization of the unit circle
sincos je j
)tan(tantan
1sincos1
cossin1
22r
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Complex Numbers: Notation and Properties (4)
• Alternate notation for polar coordinates using Euler’s Formula
Compare to
Note: keep track of degrees and radians!
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Complex Math – Review
• Complex multiplication and division: the hard way
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Complex Math – Review• Complex multiplication and division: the easy
way Given:
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Exercise
7.1099.0
0.767.331.4
6.30.761.4
7.336.3
41
23
j
js
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Complex Math – Review• Complex conjugate:
• Some key results:
Given:
Define complex conjugate as
jbas
jbas
22 bass
22
222
)(
)(2
))(())((
bax
baaxx
jbaxjbaxsxsx
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Complex Math – Review• A function of a complex number is also a complex
number
• Example
Given: )}(Im{)}(Re{)( sGjsGsG
10
10)(
s
sG
45 js
415
10
1045
10)45(
jjjG
2606.06442.0
1660.06224.0 j
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Complex Math – Review• Derivatives of a function of complex numbers, G(s),
can be computed in the usual way
• Poles/Zeros
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Complex Math – Review• Poles/Zeros at infinity
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Laplace Transform
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Laplace Transform Motivation
• Differential equations model dynamic systems
• Control system design requires simple methods for solving these equations!
• Laplace Transforms allow us to▫systematically solve linear time invariant (LTI)
differential equations for arbitrary inputs.▫easily combine coupled differential equations
into one equation.▫use with block diagrams to find
representations for systems that are made up of smaller subsystems.
uxbxm
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The Laplace Transform Definition
• Laplace Transform exists if integral converges for any value of s▫Region of convergence is not as important
for inverting “one-sided” transforms
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Laplace Transform Example (1)
• Example:
• Show that
Notation for “unit step”
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Laplace Transform Example (2)
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Laplace Transform of a Unit Step
• Find the Laplace Transform for the following function
otherwise0
01)(
ttus
s
s
es
dtesF stst
1
101
11)(
00
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Exercise
• Find the Laplace Transform for the following function
otherwise0
103)(
ttf
s
s
stst
es
es
es
dtesF
13
13
33)(
1
0
1
0
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The Laplace Transform Definition (Review)• Recall:
• The easiest way to use the Laplace Transform is by creating a table of Laplace Transform pairs. We can use several Laplace Transform properties to build the table.
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The function with the simplest Laplace Transform (1)
• A special input (class) has a very simple Laplace Transform
• The impulse function:▫Has unit “energy”
▫Is zero except at t=0
Think of pulse in the limit
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The function with the simplest Laplace Transform (2)
1t
sFtf
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LT Properties: Scaling and Linearity
• Proof: Both properties inherited from linearity of integration and the Laplace Transform definition
sFsFtftf
saFtaf
sFtf
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Example 1• Find the following Laplace Transforms
▫Hint: Use Euler’s Formula
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Example 1 (2) tjtj eej
t 2
1sin
22
11
2
1sin
s
jsjsjtL
tjtj eet 2
1cos
22
11
2
1cos
s
s
jsjstL
22
22
cos
sin
s
st
st
sFtf
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LT Properties: Time and Frequency Shift
• Proof of frequency shift: Combine exponentials
sFtfe
sFetutf
sFtf
t
s
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Example 2
• Find the following Laplace Transforms
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Example 2 (2)
22
22
)(
cos
as
as
s
ste
ass
atL
22
22
)(
sin
as
ste
ass
atL
22
22
cos
sin
as
aste
aste
sFtf
at
at
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LT Properties: Integration & Differentiation
• Proof of Differentiation Theorem: Integration by parts
vduuvudv
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LT Properties: Integration & Differentiation (2)
01
0
dfss
sFdf
fssFtfdt
dsFtf
t
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Example 3
• Find Laplace Transform for
• What is the Laplace Transform of▫Derivative of a step?▫Derivative of sine?
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Example 3 (2)
2
111)(
ssstdtut LL
1)0(1)(
u
ss
dt
tduL Impulse!
10
1
1)sin(22
s
s
ss
dt
tdL Cosine!
22
11
asste
ass
at
L
1
sin
1
1
1
2
2
2
s
st
dt
d
tudt
das
te
st
sFtf
at
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Exercise• What is the Laplace Transform of
tdt
dcos
1
1
1
1
11
1
)cos(22
2
2
2
2
ss
s
s
s
s
ss
dt
tdL -Sine!
1
1cos
2 s
tdt
dsFtf
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Initial Value Theorem
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Final Value Theorem
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Inverse Laplace and LODE solution
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Inverse Laplace Transform
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Partial Fraction Idea -1
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Partial Fraction Idea -2
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Partial Fraction Idea -3
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Recall: Laplace differentiation theorem (1)
• The differentiation theorem
• Higher order derivatives
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Differentiation Theorem (revisited)
• Differentiation Theorem when initial conditions are zero
000
1
121 f
dt
df
dt
dsfssFstf
dt
dn
nnnn
n
n
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Solving differential equations: a simple example (1)• Consider
0,1 tdt
dx
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Solving differential equations: a simple example (2)
• Solution Summary▫Use differentiation theorem to take Laplace
Transform of differential equation▫Solve for the unknown Laplace Transform
Function▫Find the inverse Laplace Transform
0,0 txttx
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Example 1• Find the Laplace Transform for the solution to
120300
00...0
2
121
sXxssXxsxsXs
fsffssFstfdt
dL nnnn
n
n
Notation:
30,100,123 xxtxxx
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- Partial Fraction Expansions• In general, LODEs can be transformed into a function
that is expressed as a ratio of polynomials• In a partial fraction expansion we try to break it into its
parts, so we can use a table to go back to the time domain:
• Three ways of finding coefficients▫ Put partial fraction expansion over common
denominator and equate coefficients of s (Example 1)▫ Residue formula ▫ Equate both sides for several values of s (not
covered)
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- Partial Fraction Expansions• Have to consider that in general we can
encounter:
▫Real, distinct roots▫Real repeated roots▫Complex conjugate pair roots (2nd order
terms)▫Repeated complex conjugate roots
222
2
22211 ))(()()()(
)()(
bas
GFsEs
bas
DCs
ps
B
ps
AK
sD
sNsX
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Example 1, Part 2• Given X(s), find x(t).
• This Laplace Transform function is not immediately familiar, but it is made up of parts that are.
• Factor denominator, then use partial fraction expansion:
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Finding A, B, and C
• To solve, re-combine RHS and equate numerator coefficients (“Equate coefficients” method)
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Final Step• Example 1 completed:
• Since
• By inspection,
s
KtKe t 0,
0,2
52
2
1 2 teetx tt
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Residue Formula (1)• The residue formula allows us to find one
coefficient at a time by multiplying both sides of the equation by the appropriate factor.
• Returning to Example 1:
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Residue Formula (2)• For Laplace Transform with non-repeating roots,
• The general residue formula is:
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Example 2• Find the solution to the following differential
equation:
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Example 2 (2)
)2(
1
)1(
2)(
ss
sX
0,2)( 2 teetx tt
0)(21)(3)(
0)(20)(300)(2
2
sXssXssXs
sXxssXxsxsXs
3)()23( 2 ssXss
)2()1()2)(1(
3)(
s
B
s
A
ss
ssX
221
311
1
s
sXsA 112
322
2
s
sXsB
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Inverse Laplace Transform with Repeated Roots• We have discussed taking the inverse Laplace
transform of functions with non-repeated, real roots using partial fraction expansion.
• Now we will consider partial fraction expansion rules for functions with repeated (real) roots:▫# of constants = order of repeated roots
• Example:
23223
4
)3()3(3)3(
)1(
s
E
s
D
s
C
s
B
s
A
ss
s
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Repeated real roots in Laplace transform table• The easiest way to take an inverse Laplace
transform is to use a table of Laplace transform pairs.
222
222
1
2
)(
)(2)sin(
2)sin(
)(
1
!
)(
1)()(
as
astte
s
stt
asn
etas
te
sFtf
at
n
atn
at
Repeated Real Roots
Repeated Imaginary Roots(also use cosine term)
Repeated Complex Roots(also use cosine term)
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Example with repeated roots• Example: find x(t)
• Take Laplace Transform of both sides:
tef 2
21
1x
10,00
2 2
xx
exxx t
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Example with repeated roots (2)
•Terms with repeated roots:
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Example with repeated roots (3) C = 1B = 2
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Exercise 1• Find the solution to the following differential
equation 0)0(1)0(044 xxxxx
tt
ss
teetx
Ass
sA
s
s
ssXsB
s
B
s
A
s
ssX
ssssX
sXssXssXs
sXxssXxsxsXs
22
22
22
2
22
2
2
2
2
1:2
22
2
4
242
222
4
444
04440
040400
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Above …
Inverse Laplace and LODE solutions
- Partial fraction expansions - LODE solution examples
* Real roots* Real, repeated roots
Next:* Complex roots
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NOTE:
A complex conjugate pair is actually two distinct, simple first order poles, so can find residues and combine in the usual way:
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Inverse Laplace Transform with Complex Roots
• To simplify your algebra, don’t use first-order denominators such as
• Instead, rename variables
• So that
21 KKB 21 11 KjKjC
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Laplace Transform Pairs for Complex Roots
• More Laplace transform pairs (complex roots):
• Also, see the table in your textbook and most other control systems textbooks.
22
22
)()sin(
)()cos(
)()(
ste
s
ste
sFtf
t
t
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Return to example from above:74
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Example with complex roots• Example: find x(t)
• Laplace Transform
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Example with complex roots (2)
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Example with complex roots (3)
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Example with complex roots (5)
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Exercise 2• Find solution to the following differential
equation
0)0(1)0(084 xxxxx
tetetx
ss
s
s
sss
ssX
sXssXssXs
sXxssXxsxsXs
tt 2sin2cos
22
2
22
2
42
484
4
0844
080400
22
2222
2
2
2
2
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Resume Lecture #2
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•Review of Complex Number•Laplace Transform•Inverse Laplace Transform•Solving LODE
82