1 Spring 2010 1 ME451: Control Systems ME451: Control Systems Prof. Clark Radcliffe, Prof. Clark Radcliffe, Prof. Prof. Jongeun Choi Jongeun Choi Department of Mechanical Engineering Department of Mechanical Engineering Michigan State University Michigan State University Lecture 2 Lecture 2 Laplace transform Laplace transform Spring 2010 2 Course roadmap Course roadmap Laplace transform Laplace transform Transfer function Transfer function Models for systems Models for systems • electrical electrical • mechanical mechanical • electromechanical electromechanical Block diagrams Block diagrams Linearization Linearization Modeling Modeling Analysis Analysis Design Design Time response Time response • Transient Transient • Steady state Steady state Frequency response Frequency response • Bode plot Bode plot Stability Stability • Routh Routh-Hurwitz -Hurwitz • ( (Nyquist Nyquist) Design specs Design specs Root locus Root locus Frequency domain Frequency domain PID & Lead-lag PID & Lead-lag Design examples Design examples (Matlab Matlab simulations &) laboratories simulations &) laboratories Spring 2010 3 Laplace transform Laplace transform One of most important math tools in the course! One of most important math tools in the course! Definition: For a function f(t) (f(t)=0 for t<0), Definition: For a function f(t) (f(t)=0 for t<0), We denote Laplace transform of f(t) by F(s). We denote Laplace transform of f(t) by F(s). f(t f(t) t 0 F(s F(s) (s: complex variable) Spring 2010 4 Example of Laplace transform Example of Laplace transform Step function Step function 0 f(t) f(t) t 5 Remember L(u(t)) = 1/s f (t ) = 5u(t ) = 5 t ! 0 0 t < 0 " # $ % $ F(s) = f (t )e ! st dt 0 " # = 5e ! st dt 0 " # = 5 e ! st dt 0 " # = 5 ! 1 s e ! st $ % & ' 0 " $ % ( & ' ) = 5 1 s $ % ( & ' ) = 5 s Spring 2010 5 Integration is Hard Integration is Hard Tables are Easier Tables are Easier Spring 2010 6 Laplace Laplace transform table transform table (Table B.1 in Appendix B of the textbook) (Table B.1 in Appendix B of the textbook) Inverse Laplace Transform Inverse Laplace Transform
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Spring 2010 1
ME451: Control SystemsME451: Control Systems
Prof. Clark Radcliffe,Prof. Clark Radcliffe, Prof. Prof. Jongeun ChoiJongeun ChoiDepartment of Mechanical EngineeringDepartment of Mechanical Engineering
Michigan State UniversityMichigan State University
Laplace transformLaplace transform One of most important math tools in the course!One of most important math tools in the course! Definition: For a function f(t) (f(t)=0 for t<0),Definition: For a function f(t) (f(t)=0 for t<0),
We denote Laplace transform of f(t) by F(s).We denote Laplace transform of f(t) by F(s).
f(tf(t))
tt00F(sF(s))
(s: complex variable)
Spring 2010 4
Example of Laplace transformExample of Laplace transform Step functionStep function
00
f(t)f(t)
tt55
Remember L(u(t)) = 1/s
f (t) = 5u(t) =5 t ! 00 t < 0
"#$
%$
F(s) = f (t)e! st dt0
"
# = 5e! st dt0
"
# = 5 e! st dt0
"
#
= 5 !1se! st$% &'0
"$%(
&')= 5 1
s$%(
&')=5s
Spring 2010 5
Integration is HardIntegration is Hard
Tables are EasierTables are Easier
Spring 2010 6
Laplace Laplace transform tabletransform table(Table B.1 in Appendix B of the textbook)(Table B.1 in Appendix B of the textbook)
Properties of Laplace transformProperties of Laplace transform
LinearityLinearity
Ex.Ex.
Proof.Proof.
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Properties of Laplace transformProperties of Laplace transform
DifferentiationDifferentiation
Ex.Ex.
Proof.Proof.
t-domaint-domain s-domains-domain
Spring 2010 9
Properties of Laplace transformProperties of Laplace transform
IntegrationIntegration
Proof.Proof.
t-domaint-domain s-domains-domain
Spring 2010 10
Properties of Laplace transformProperties of Laplace transform
Final value theoremFinal value theorem
Ex.Ex.
if all the poles of if all the poles of sFsF(s) are in(s) are inthe left half planethe left half plane (LHP) (LHP)
Poles of Poles of sF(ssF(s) are in LHP) are in LHP, so final value , so final value thmthm applies. applies.
Ex.Ex.
Some poles of Some poles of sF(ssF(s) are not in LHP) are not in LHP, so final value, so final valuethmthm does does NOTNOT apply. apply.
Spring 2010 11
Properties of Laplace transformProperties of Laplace transform
Initial value theoremInitial value theorem
Ex.Ex.
Remark: In this theorem, it does not matter ifRemark: In this theorem, it does not matter ifpole location is in LHS or not.pole location is in LHS or not.
if the limits exist.if the limits exist.
Ex.Ex.
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Properties of Laplace transformProperties of Laplace transform
ConvolutionConvolution
IMPORTANT REMARKIMPORTANT REMARK
ConvolutionConvolution
L!1 F1(s)F2 (s)( ) " f1(t) f2 (t)
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An advantage of An advantage of Laplace Laplace transformtransform We can transform an ordinary differentialWe can transform an ordinary differential
equation (ODE) into an algebraic equation (AE).equation (ODE) into an algebraic equation (AE).
ODEODE AEAE
Partial fraction Partial fraction expansionexpansionSolution to ODESolution to ODE
t-domaint-domain s-domains-domain
1122
33
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Example 1Example 1 1st Order ODE with input and Initial Condition1st Order ODE with input and Initial Condition
Take Take Laplace Laplace TransformTransform
Solve for Solve for Y(s)Y(s)
5 !y(t) +10y(t) = 3u(t) y(0) = 1
5 sY (s) ! y(0)[ ] +10 Y (s)[ ] = 3 1s
"#$
%&'
5s +10( )Y (s) = 5y(0) + 3 1s
!"#
$%&
Y (s) = 55s +10( ) +
3s 5s +10( ) =
1s + 2( ) +
0.6s s + 2( )
(Initial Condition) + (Input)
Spring 2010 15
Example 1 (cont)Example 1 (cont) Use table to Invert Use table to Invert Y(s)Y(s) term by term to find term by term to find y(t)y(t)
If we are interested in only the final value of y(t), applyIf we are interested in only the final value of y(t), applyFinal Value Theorem:Final Value Theorem:
Example 3 (contExample 3 (cont’’d)d)
Spring 2010 21
Example: NewtonExample: Newton’’s laws law
We want to know the trajectory of x(t). By We want to know the trajectory of x(t). By Laplace Laplace transform,transform,
EX. Air bag and accelerometerEX. Air bag and accelerometer Tiny MEMS accelerometerTiny MEMS accelerometer
Microelectromechanical Microelectromechanical systems (MEMS)systems (MEMS)
(Pictures from various websites)
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In this way, we can find a rathercomplicated solution to ODEs easily by
using Laplace transform table!
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Summary & ExercisesSummary & Exercises Laplace transform (Important math tool!)Laplace transform (Important math tool!)
DefinitionDefinition Laplace transform tableLaplace transform table Properties of Laplace transformProperties of Laplace transform Solution to Solution to ODEs ODEs via Laplace transformvia Laplace transform
ExercisesExercises Read Appendix A, B.Read Appendix A, B. Solve Quiz problemsSolve Quiz problems……