SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
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The General Second-Order System
• Two important quantities that describes the
response of second order systems:
o Natural frequency, ωn : The frequency of
oscillation of the system without damping.
o Damping ratio, ζ : Parameter that describes
the damped oscillations of the 2nd order
response.
Bigger, means more ‘damped’
response, i.e. less oscillations.
ζ =
=
Exponential decay frequency
Natural frequency (rad/second )
1 Natural period (seconds)
2π Exponential time sonstant
• Define general 2nd order response in terms of ωn
and ζ as:
• Hence the pole is given as:
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
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• Four time responses based on ζ :
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
Example:
Given
the
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transfer
function
ωn 2
G(s) = 2
, find ζ and ωn .
2
s + 2ζω n s + ωn
Example: Find the value of ζ , and sketch the
kind of response expected.
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
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Underdamped Second-Order Systems
• A common model for physical problems.
• A detailed description of the underdamped
response is necessary for both analysis and design.
ωn 2
C ( s) =
s ( s 2 + 2ζω n s + ωn 2 )
=
K 2 s + K3
K1
+ 2
, ζ <1
2
s s + 2ζω n s + ωn
1
= −
s
( s + ζω n ) +
ζ
1−ζ
2
ω
1
−
ζ
n
2
( s + ζω n ) 2 + ωn 2 1 − ζ 2
• Taking the inverse Laplace transform,
c(t ) = 1 − e
⎛
−ζω n t ⎜
⎜
⎝
2
cos ωn 1 − ζ t +
ζ
1−ζ
2
sin ωn
⎞
1−ζ t ⎟
⎟
⎠
2
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
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• plot of c(t)
• Less ζ implies more oscillation.
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
Second
order
specifications
underdamped
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response
• Peak time, Tp: The time required to reach the first,
or maximum, peak.
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
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• Percent overshoot, %OS: The amount that the
response overshoots the final value at the peak
time, expressed as a percentage of the steady-state
value.
• We can also find the inverse of the equation
allowing us to find ζ given %OS.
• Relationship between ζ and %OS can be used:
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
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• Rise time, Tr: The time required for the response
to go from 0.1 to 0.9 of the final value.
• Relationship between Trωn and ζ can be used.
• Settling time, TS: The time required for the
response to reach and stays within ± 2% of the
steady-state value/final value.
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
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Example:
Given a transfer function, G ( s ) =
Tp, %OS, TS, and Tr.
100
2
s + 15s + 100
, find
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
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Relation between Tp, %OS, TS, and Tr to
the system poles.
• Consider the pole plot for an underdamped secondorder system:
• Notice that the distance from the origin to the pole
equals ωn.
• cos θ =
− ζω n
ωn
=ζ .
• Before we already defined:
Tp =
π
ωn 1 − ζ 2
,
TS =
4
ζω n
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
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• ωd = ωn 1 − ζ 2 is the imaginary part of the pole,
called the damped frequency of oscillation.
• TP is inversely proportional to the imaginary part of
the pole. ⇒ Horizontal lines are lines of constant
peak time.
• TS is inversely proportional to the magnitude of the
real part of the pole. ⇒ Vertical lines are lines of
constant settling time.
• Radial lines are lines of constant ζ . ⇒ Radial lines
are lines of constant %OS.
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
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• Step responses of second order underdamped
systems as pole moves: (a) with constant real part,
(b) with constant imaginary part, (c) with constant
damping ratio.
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
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Example:
Given the pole plot shown, find ζ , ω n , Tp, %OS, and
TS.
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
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Example:
For the system shown, find J and D to yield 20%
overshoot and a settling time of 2 seconds for a step
input of torque T(t).
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
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Example:
For the unit step response shown, find the transfer
function of the system.
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