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ME451: Control SystemsME451: Control Systems
Prof. Clark Radcliffe and Prof.Prof. Clark Radcliffe and Prof. Jongeun ChoiJongeun Choi
Department of Mechanical EngineeringDepartment of Mechanical Engineering
Michigan State University Michigan State University
Lecture
Lecture
Time response of 2nd-order systems
Time response of 2nd-order systems
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Course roadmap
Course roadmap
Laplace transformLaplace transform
Transfer functionTransfer function
Models for systemsModels for systems
- • electricalelectrical
- • mechanicalmechanical
- • electromechanicalelectromechanical
Block diagramsBlock diagrams
LinearizationLinearization
ModelingModeling AnalysisAnalysis DesignDesign
Time responseTime response
- • TransientTransient
- • Steady stateSteady state
Frequency responseFrequency response
StabilityStability
- • Routh-HurwitzRouth-Hurwitz
- • NyquistNyquist
Design specsDesign specs
Root locusRoot locus
Frequency domainFrequency domain
PID & Lead-lagPID & Lead-lag
Design examplesDesign examples
((MatlabMatlab simulations &) laboratoriessimulations &) laboratories
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nd
nd
Order Systems
Order Systems
thth
Order Systems: 1 parameterOrder Systems: 1 parameter
““Strength of responseStrength of response”” calledcalled ““GainGain””
stst
Order Systems: 2 parametersOrder Systems: 2 parameters
““Strength of responseStrength of response”” calledcalled ““GainGain””
““Time to respondTime to respond”” calledcalled ““Time ConstantTime Constant””
ndnd
Order Systems: 3 parametersOrder Systems: 3 parameters
““Strength of responseStrength of response”” calledcalled ““GainGain””
““Time to respondTime to respond”” calledcalled ““Time ConstantTime Constant””
““Oscillation rateOscillation rate”” calledcalled ““FrequencyFrequency””
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Second-order systemsSecond-order systems
AA standard formstandard form of the second-order systemof the second-order system
DC motor position control exampleDC motor position control example
Closed-loop TFClosed-loop TF
AmplifierAmplifier MotorMotor
G ( s ) =
k!
n
2
s
2
n
( )
s +!
n
2
" : damping ratio
n
: undamped natural frequency
Note: the book does not include gain “ k ” in its standard form
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Step response for 2nd-order systemStep response for 2nd-order system
Input aInput a unit step functionunit step function to a 2nd-order system.to a 2nd-order system.
What is the output?
What is the output?
00
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u(t)u(t) y(t)y(t)
00
DC gain for a unit step (ifDC gain for a unit step (if G(s)G(s) is stable)is stable)
lim
t !"
( y ( t )) = lim
s! 0
( s G ( s ) U ( s )) = lim
s! 0
sG ( s )
s
= lim
s! 0
( G ( s )) = G ( 0 ) = k
k! n
2
s
2
n
( ) s +! n
2
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Step response for 2nd-order systemStep response for 2nd-order system
for various damping ratiofor various damping ratio
UndampedUndamped
UnderdampedUnderdamped
Critically dampedCritically damped
OverdampedOverdamped
0 5 10 15
0
0. 5
1
1. 5
2
y ( t )
k
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Step response for 2nd-order systemStep response for 2nd-order system
Underdamped Underdamped casecase
Math expression of y(t) for
Math expression of y(t) for
underdamped
underdamped
case
case
Damped natural frequencyDamped natural frequency
Y ( s ) = G ( s ) U ( s ) =
k!
n
2
s
2
n
s +!
n
2
s
y ( t ) = k 1!
e
! "# n
t
2
sin
d
( t + $)
! = tan
" 1
2
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Step response for 2nd-order systemStep response for 2nd-order system
UnderdampedUnderdamped casecase
Math expression of y(t) for
Math expression of y(t) for
underdamped
underdamped
case
case
Damped response frequencyDamped response frequency
Response time constant
Response time constant
y ( t ) = k 1!
e
! "# n
t
2
sin
d
( t + $)
! = tan
" 1
2
n
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Transfer Function Poles
Transfer Function Poles
For the 2nd order transfer functionFor the 2nd order transfer function
The denominator ==> Characteristic PolynomialThe denominator ==> Characteristic Polynomial
The Characteristic Polynomial roots areThe Characteristic Polynomial roots are thethe ““polespoles””
2nd order ==> 2 roots ==> two poles2nd order ==> 2 roots ==> two poles
G ( s ) =
k!
n
2
s
2
n
s +!
n
2
s
1 , 2
n
± j
n
2
s
2
n
s + "
n
2
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Transfer Function Poles Transfer Function Poles
Poles for the 2nd order transfer function
Poles for the 2nd order transfer function
Step Response
Step Response
The system poles set the response characteristicsThe system poles set the response characteristics
Re(Re( ss
1,21,
) =1/) =1/ττ andand ImIm(( ss
1,21,
dd
G ( s ) =
k!
n
2
s
2
n
( ) s +!
n
2
s
1 , 2
n
± j
n
2
y ( t ) = k 1!
e
! "# n
t
2
sin
d
( t + $)
n
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Step response for 2nd-order systemStep response for 2nd-order system
for various damping ratiosfor various damping ratios
UndampedUndamped
UnderdampedUnderdamped
Critically dampedCritically damped
OverdampedOverdamped
0 5 10 15
0
0. 5
1
1. 5
2
y ( t )
k
s
1 , 2
n
± j
n
2
= 0 ± j
n
2
= ± j
n
Complex conjugate imaginary poles
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Step response for 2nd-order systemStep response for 2nd-order system
for various damping ratiosfor various damping ratios
UndampedUndamped
UnderdampedUnderdamped
Critically dampedCritically damped
OverdampedOverdamped
0 5 10 15
0
0. 5
1
1. 5
2
y ( t )
k
s
1 , 2
n
± j
n
2
Complex conjugate complex poles
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Settling Time
Settling Time
Measures how long it takes to stay within a set
Measures how long it takes to stay within a set
percentage of steady statepercentage of steady state
Example 2% = 0.
Example 2% = 0.
So a system settlesSo a system settles
within 2% of final valuewithin 2% of final value
in about tin about t ≅≅ 44 ττ
2% settling time =
2% settling time = 4
τ
τ
y ( t ) = k 1!
e
! "# n
t
2
sin #
d
t + $
0.02 = e
! "# n
t
= e
! t $
t $ =! ln 0.
( )
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Properties of 2nd-order system
Properties of 2nd-order system
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Influence of real part of poles
Influence of real part of poles
Settling time
Settling time
ts
ts
decreases.
decreases.
tsts
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Influence ofInfluence of imagimag. part of poles. part of poles
Oscillation frequencyOscillation frequency ωω
dd
increases.increases.
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Influence of angle of polesInfluence of angle of poles
Over/under-shoot decreases.Over/under-shoot decreases.
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An exampleAn example
Require 5% settling timeRequire 5% settling time tsts << tsmtsm (given):(given):
ReRe
ImIm
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An example (cont
An example (cont
d)
d)
Require PO <
Require PO <
POm
POm
(given):
(given):
ReRe
ImIm
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An example (cont
An example (cont
d)
d)
Combination of two requirements
Combination of two requirements
ReRe
ImIm
&&
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Some remarks
Some remarks
Percent overshoot depends on
Percent overshoot depends on
, but NOT
, but NOT
ωnn
From 2nd-order transfer function, analytic
From 2nd-order transfer function, analytic
expressions of delay & rise time are hard to expressions of delay & rise time are hard to
obtain. obtain.
Time constant is 1/(Time constant is 1/(ζωζωnn), indicating), indicating
convergence speed. convergence speed.
ForFor ζζ>1, we cannot define peak time, peak>1, we cannot define peak time, peak
value, percent overshoot. value, percent overshoot.
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Performance measures (review)Performance measures (review)
Transient responseTransient response
Peak valuePeak value
Peak timePeak time
Percent overshoot
Percent overshoot
Delay timeDelay time
Rise time
Rise time
Settling timeSettling time
Steady state responseSteady state response
Steady state gainSteady state gain
Next, we will connectNext, we will connect
these measuresthese measures
with s-domain.
with s-domain.
(Today(Today’’s lecture)s lecture)
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Summary Summary
Transient response of 2nd-order system isTransient response of 2nd-order system is
characterized by
characterized by
Damping ratioDamping ratio ζζ && undampedundamped natural frequencynatural frequency ωωnn
Pole locationsPole locations
Delay time and rise time are not so easy toDelay time and rise time are not so easy to
characterize, and thus not covered in this course.characterize, and thus not covered in this course.
For transient responses of high order systems,For transient responses of high order systems,
we need computer simulations.
we need computer simulations.
