SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
4-14
Second-Order Systems
• More parameters that describe the response.
• Varying ‘a’ for first order systems simply changes
the speed of the response.
• Varying the parameters of a second order response
changes the speed and the form of the response.
Overdamped Response
• Consider the step response for the following
system:
9
G(s) = 2
s + 9s + 9
9
⇒ C step ( s ) =
s ( s 2 + 9 s + 9)
9
=
s ( s + 7.854)( s + 1.146)
• The output has a pole at p1=0 resulting from u(t)
and p2=-7.854 and p3=-1.146 coming from the
system itself.
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
4-15
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
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• The input pole generates the constant forced
response. The two system poles on the real axis
generates the exponential response:
K1e −7.854t , K 2 e −1.146t
• Known as overdamped response.
Underdamped Response
• Given
C ( s) =
9
s ( s 2 + 2 s + 9)
c(t ) = 1 − e − t (cos 8t +
8
sin 8t )
8
• Poles at:
s = 0 (due to input step)
s = −1 ± j 8 (natural response)
o Real part of pole matches the exponential
decay frequency of the sinusoid’s amplitude.
o Imaginary part matches frequency of the
sinusoidal oscillation. Define this frequency as
ωd , i.e. damped frequency of oscillation,
equals 8 in this case.
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
4-17
Undamped Response
• Given
C ( s) =
9
s ( s 2 + 9)
c(t ) = 1 − cos 3t
• Two imaginary poles at ± j 3 generate a sinusoidal
natural response with frequency ω = 3 rad/s .
• No real part implies an exponential that does not
decay.
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
4-18
Critically Damped Response
• Given
C ( s) =
9
s ( s 2 + 6 s + 9)
=
9
s ( s + 3) 2
c(t ) = 1 − 3te − 3t − e − 3t
• Pole at origin due to the step input and two real
poles at p1, p2=-3 from the system itself.
• Response consists of exponential and exponential
times t. Exponential parameter equals the location
of the real poles.
• Responses
overshoot.
are
fastest
possible
without
the
Unstable
• Poles are located on the right half plane.
• Results in positive exponentials. Response goes to
infinity as time increases.
• Example:
C ( s) =
9
3
−3
+
=
s ( s − 3)
s s−3
c(t ) = −3 + e 3t
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
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Summary of Natural Responses
1. Overdamped responses:
o Poles: two real at -σ1, -σ2.
c(t ) = K1e −σ 1t + K 2 e −σ 2 t
2. Underdamped responses:
o Poles: two complex at −σ d± jωd
c(t ) = K1e −σ d t ( K 2 cos ωd t + K 3 sin ωd t )
3. Undamped responses:
o Poles: 2 imaginary at ± jω1
c(t ) = A cos ω1t
4. Critically damped responses:
o Poles: Two real at − σ 1
c(t ) = K1e −σ 1t + K 2te −σ 2 t
5. Unstable:
o Poles: Any right half plane poles, e.g. at
σ d ± jω d .
c(t ) = K1eσ d t ( K 2 cos ωd t + K 3 sin ωd t )
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SEE 2113 KAWALAN: PEMODELAN DAN SIMULASI
4-20
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