Second-Order Systems
Unit 3: Time Response,
Part 2: Second-Order Responses
Engineering 5821:
Control Systems I
Faculty of Engineering & Applied Science
Memorial University of Newfoundland
January 28, 2010
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Second-Order Systems
Second-order systems (systems described by second-order DE’s)
have transfer functions of the following form:
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Second-Order Systems
Second-order systems (systems described by second-order DE’s)
have transfer functions of the following form:
G (s) =
ENGI 5821
b
s 2 + as + b
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Second-Order Systems
Second-order systems (systems described by second-order DE’s)
have transfer functions of the following form:
G (s) =
b
s 2 + as + b
(This TF may also be multiplied by a constant K , which affects
the exact constants of the time-domain signal, but not its form).
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Second-Order Systems
Second-order systems (systems described by second-order DE’s)
have transfer functions of the following form:
G (s) =
b
s 2 + as + b
(This TF may also be multiplied by a constant K , which affects
the exact constants of the time-domain signal, but not its form).
Depending upon the factors of the denominator we get four
categories of responses.
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Second-Order Systems
Second-order systems (systems described by second-order DE’s)
have transfer functions of the following form:
G (s) =
b
s 2 + as + b
(This TF may also be multiplied by a constant K , which affects
the exact constants of the time-domain signal, but not its form).
Depending upon the factors of the denominator we get four
categories of responses. If the input is the unit step, a pole at the
origin will be added which yields a constant term in the
time-domain.
ENGI 5821
Unit 3: Time Response
Category
Overdamped
Poles
Two real: −σ1 , −σ2
c(t)
K1 e −σ1 t + K2 e −σ2 t
Category
Overdamped
Underdamped
Poles
Two real: −σ1 , −σ2
Two complex: −σd ± jωd
c(t)
K1 e −σ1 t + K2 e −σ2 t
Ae −σd t cos(ωd t − φ)
Category
Overdamped
Underdamped
Undamped
Poles
Two real: −σ1 , −σ2
Two complex: −σd ± jωd
Two imaginary: ±jωn
c(t)
K1 e −σ1 t + K2 e −σ2 t
Ae −σd t cos(ωd t − φ)
A cos(ωn t − φ)
Category
Overdamped
Underdamped
Undamped
Critically damped
Poles
Two real: −σ1 , −σ2
Two complex: −σd ± jωd
Two imaginary: ±jωn
Repeated real: −σd
c(t)
K1 e −σ1 t + K2 e −σ2 t
Ae −σd t cos(ωd t − φ)
A cos(ωn t − φ)
K1 e −σd t + K2 te −σd t
We can characterize the response of second-order systems using
two parameters: ωn and ζ
We can characterize the response of second-order systems using
two parameters: ωn and ζ
Natural Frequency, ωn : This is the frequency of oscillation
without damping.
We can characterize the response of second-order systems using
two parameters: ωn and ζ
Natural Frequency, ωn : This is the frequency of oscillation
without damping. For example, the natural frequency of an RLC
circuit with the resistor shorted, or of a mechanical system without
dampers.
We can characterize the response of second-order systems using
two parameters: ωn and ζ
Natural Frequency, ωn : This is the frequency of oscillation
without damping. For example, the natural frequency of an RLC
circuit with the resistor shorted, or of a mechanical system without
dampers. An undamped system is described by its natural
frequency.
We can characterize the response of second-order systems using
two parameters: ωn and ζ
Natural Frequency, ωn : This is the frequency of oscillation
without damping. For example, the natural frequency of an RLC
circuit with the resistor shorted, or of a mechanical system without
dampers. An undamped system is described by its natural
frequency.
Damping Ratio, ζ: This measures the amount of damping.
We can characterize the response of second-order systems using
two parameters: ωn and ζ
Natural Frequency, ωn : This is the frequency of oscillation
without damping. For example, the natural frequency of an RLC
circuit with the resistor shorted, or of a mechanical system without
dampers. An undamped system is described by its natural
frequency.
Damping Ratio, ζ: This measures the amount of damping. For
underdamped systems ζ lies in the range [0, 1]:
We can characterize the response of second-order systems using
two parameters: ωn and ζ
Natural Frequency, ωn : This is the frequency of oscillation
without damping. For example, the natural frequency of an RLC
circuit with the resistor shorted, or of a mechanical system without
dampers. An undamped system is described by its natural
frequency.
Damping Ratio, ζ: This measures the amount of damping. For
underdamped systems ζ lies in the range [0, 1]:
Second-Order Systems
Characteristics of Underdamped Systems
Damping ratio ζ is defined as follows:
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Damping ratio ζ is defined as follows:
ζ =
Exponential decay frequency
Natural frequency
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Damping ratio ζ is defined as follows:
ζ =
=
Exponential decay frequency
Natural frequency
|σd |
ωn
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Damping ratio ζ is defined as follows:
ζ =
=
Exponential decay frequency
Natural frequency
|σd |
ωn
The exponential decay frequency σd is the real-axis component of
the poles of a critically damped or underdamped system.
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Damping ratio ζ is defined as follows:
ζ =
=
Exponential decay frequency
Natural frequency
|σd |
ωn
The exponential decay frequency σd is the real-axis component of
the poles of a critically damped or underdamped system.
We now describe the general second-order system in terms of ωn
and ζ.
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Damping ratio ζ is defined as follows:
ζ =
=
Exponential decay frequency
Natural frequency
|σd |
ωn
The exponential decay frequency σd is the real-axis component of
the poles of a critically damped or underdamped system.
We now describe the general second-order system in terms of ωn
and ζ.
b
G (s) = 2
s + as + b
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Damping ratio ζ is defined as follows:
ζ =
=
Exponential decay frequency
Natural frequency
|σd |
ωn
The exponential decay frequency σd is the real-axis component of
the poles of a critically damped or underdamped system.
We now describe the general second-order system in terms of ωn
and ζ.
b
G (s) = 2
s + as + b
In other words we want to get the relationships from ωn and ζ to a
and b.
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Damping ratio ζ is defined as follows:
ζ =
=
Exponential decay frequency
Natural frequency
|σd |
ωn
The exponential decay frequency σd is the real-axis component of
the poles of a critically damped or underdamped system.
We now describe the general second-order system in terms of ωn
and ζ.
b
G (s) = 2
s + as + b
In other words we want to get the relationships from ωn and ζ to a
and b. Why? Because ωn and ζ are more meaningful and useful
for design.
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
If there were no damping, we would have a pure sinusoidal
response.
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
If there were no damping, we would have a pure sinusoidal
response. Thus, the poles would be on the imaginary axis and the
TF would have the form,
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
If there were no damping, we would have a pure sinusoidal
response. Thus, the poles would be on the imaginary axis and the
TF would have the form,
G (s) =
ENGI 5821
s2
b
+b
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
If there were no damping, we would have a pure sinusoidal
response. Thus, the poles would be on the imaginary axis and the
TF would have the form,
G (s) =
s2
b
+b
√
The poles are at s = ±j b.
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
If there were no damping, we would have a pure sinusoidal
response. Thus, the poles would be on the imaginary axis and the
TF would have the form,
G (s) =
s2
b
+b
√
The poles are at s = ±j b. The natural frequency is governed by
the position of the poles on the imaginary axis.
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
If there were no damping, we would have a pure sinusoidal
response. Thus, the poles would be on the imaginary axis and the
TF would have the form,
G (s) =
s2
b
+b
√
The poles are at s = ±j b. The natural frequency is governed by
the position
of the poles on the imaginary axis. Therefore,
√
ωn = b.
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
If there were no damping, we would have a pure sinusoidal
response. Thus, the poles would be on the imaginary axis and the
TF would have the form,
G (s) =
s2
b
+b
√
The poles are at s = ±j b. The natural frequency is governed by
the position
of the poles on the imaginary axis. Therefore,
√
ωn = b.
b = ωn2
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider an underdamped system with poles −σd ± jωd .
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider an underdamped system with poles −σd ± jωd . The
exponential decay frequency is σd .
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider an underdamped system with poles −σd ± jωd . The
exponential decay frequency is σd . For a general second-order
system the denominator is s 2 + as + b and the roots have real part
σd = −a/2.
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider an underdamped system with poles −σd ± jωd . The
exponential decay frequency is σd . For a general second-order
system the denominator is s 2 + as + b and the roots have real part
σd = −a/2.
We apply the definition for ζ:
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider an underdamped system with poles −σd ± jωd . The
exponential decay frequency is σd . For a general second-order
system the denominator is s 2 + as + b and the roots have real part
σd = −a/2.
We apply the definition for ζ:
ζ=
Exponential decay frequency
Natural frequency
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider an underdamped system with poles −σd ± jωd . The
exponential decay frequency is σd . For a general second-order
system the denominator is s 2 + as + b and the roots have real part
σd = −a/2.
We apply the definition for ζ:
ζ=
|σd |
Exponential decay frequency
=
ωn
Natural frequency
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider an underdamped system with poles −σd ± jωd . The
exponential decay frequency is σd . For a general second-order
system the denominator is s 2 + as + b and the roots have real part
σd = −a/2.
We apply the definition for ζ:
ζ=
|σd |
a/2
Exponential decay frequency
=
=
ωn
ωn
Natural frequency
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider an underdamped system with poles −σd ± jωd . The
exponential decay frequency is σd . For a general second-order
system the denominator is s 2 + as + b and the roots have real part
σd = −a/2.
We apply the definition for ζ:
ζ=
|σd |
a/2
Exponential decay frequency
=
=
ωn
ωn
Natural frequency
Thus, a = 2ζωn .
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider an underdamped system with poles −σd ± jωd . The
exponential decay frequency is σd . For a general second-order
system the denominator is s 2 + as + b and the roots have real part
σd = −a/2.
We apply the definition for ζ:
ζ=
|σd |
a/2
Exponential decay frequency
=
=
ωn
ωn
Natural frequency
Thus, a = 2ζωn . We can now describe the second-order system as
follows:
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider an underdamped system with poles −σd ± jωd . The
exponential decay frequency is σd . For a general second-order
system the denominator is s 2 + as + b and the roots have real part
σd = −a/2.
We apply the definition for ζ:
ζ=
|σd |
a/2
Exponential decay frequency
=
=
ωn
ωn
Natural frequency
Thus, a = 2ζωn . We can now describe the second-order system as
follows:
ωn2
G (s) = 2
s + 2ζωn s + ωn2
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider an underdamped system with poles −σd ± jωd . The
exponential decay frequency is σd . For a general second-order
system the denominator is s 2 + as + b and the roots have real part
σd = −a/2.
We apply the definition for ζ:
ζ=
|σd |
a/2
Exponential decay frequency
=
=
ωn
ωn
Natural frequency
Thus, a = 2ζωn . We can now describe the second-order system as
follows:
ωn2
G (s) = 2
s + 2ζωn s + ωn2
p
Poles: s1,2 = −ζωn ± ωn ζ 2 − 1
ENGI 5821
Unit 3: Time Response
Second-Order Systems
ENGI 5821
Characteristics of Underdamped Systems
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
e.g. Describe the category of the following systems:
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
e.g. Describe the category of the following systems:
ωn =
√
b,
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
e.g. Describe the category of the following systems:
ωn =
√
b, ζ =
a/2
ωn
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
e.g. Describe the category of the following systems:
ωn =
√
b, ζ =
a/2
ωn
=
a
√
2 b
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
e.g. Describe the category of the following systems:
ωn =
√
b, ζ =
(a) ζ = 1.155
a/2
ωn
=
a
√
2 b
=⇒ Overdamped
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
e.g. Describe the category of the following systems:
ωn =
√
b, ζ =
(a) ζ = 1.155
(b) ζ = 1
a/2
ωn
=
a
√
2 b
=⇒ Overdamped
=⇒ Critically damped
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
e.g. Describe the category of the following systems:
ωn =
√
b, ζ =
(a) ζ = 1.155
(b) ζ = 1
a/2
ωn
=
a
√
2 b
=⇒ Overdamped
=⇒ Critically damped
(c) ζ = 0.894
=⇒ Underdamped
ENGI 5821
Unit 3: Time Response
Characteristics of Underdamped Systems
Underdamped systems are very common and we will focus in
particular on designing compensators for underdamped systems
later in the course.
Characteristics of Underdamped Systems
Underdamped systems are very common and we will focus in
particular on designing compensators for underdamped systems
later in the course.
Characteristics of Underdamped Systems
Underdamped systems are very common and we will focus in
particular on designing compensators for underdamped systems
later in the course. Consider the step response for a general
second-order system:
Characteristics of Underdamped Systems
Underdamped systems are very common and we will focus in
particular on designing compensators for underdamped systems
later in the course. Consider the step response for a general
second-order system:
C (s) =
ωn2
s(s 2 + 2ζωn s + ωn2 )
Characteristics of Underdamped Systems
Underdamped systems are very common and we will focus in
particular on designing compensators for underdamped systems
later in the course. Consider the step response for a general
second-order system:
C (s) =
=
ωn2
s(s 2 + 2ζωn s + ωn2 )
K1
K2 s + K3 s
+
s
s(s 2 + 2ζωn s + ωn2 )
Characteristics of Underdamped Systems
Underdamped systems are very common and we will focus in
particular on designing compensators for underdamped systems
later in the course. Consider the step response for a general
second-order system:
C (s) =
=
ωn2
s(s 2 + 2ζωn s + ωn2 )
K1
K2 s + K3 s
+
s
s(s 2 + 2ζωn s + ωn2 )
We solve for K1 , K2 , K3 then take the ILT:
Characteristics of Underdamped Systems
Underdamped systems are very common and we will focus in
particular on designing compensators for underdamped systems
later in the course. Consider the step response for a general
second-order system:
C (s) =
=
ωn2
s(s 2 + 2ζωn s + ωn2 )
K1
K2 s + K3 s
+
s
s(s 2 + 2ζωn s + ωn2 )
We solve for K1 , K2 , K3 then take the ILT:
"
#
p
p
ζ
sin(ωn 1 − ζ 2 t)
c(t) = 1 − e −ζωn t cos(ωn 1 − ζ 2 t) + p
1 − ζ2
Characteristics of Underdamped Systems
Underdamped systems are very common and we will focus in
particular on designing compensators for underdamped systems
later in the course. Consider the step response for a general
second-order system:
C (s) =
=
ωn2
s(s 2 + 2ζωn s + ωn2 )
K1
K2 s + K3 s
+
s
s(s 2 + 2ζωn s + ωn2 )
We solve for K1 , K2 , K3 then take the ILT:
"
#
p
p
ζ
sin(ωn 1 − ζ 2 t)
c(t) = 1 − e −ζωn t cos(ωn 1 − ζ 2 t) + p
1 − ζ2
p
1
= 1− p
e −ζωn t cos(ωn 1 − ζ 2 t − φ)
1 − ζ2
Characteristics of Underdamped Systems
Underdamped systems are very common and we will focus in
particular on designing compensators for underdamped systems
later in the course. Consider the step response for a general
second-order system:
C (s) =
=
ωn2
s(s 2 + 2ζωn s + ωn2 )
K1
K2 s + K3 s
+
s
s(s 2 + 2ζωn s + ωn2 )
We solve for K1 , K2 , K3 then take the ILT:
"
#
p
p
ζ
sin(ωn 1 − ζ 2 t)
c(t) = 1 − e −ζωn t cos(ωn 1 − ζ 2 t) + p
1 − ζ2
p
1
= 1− p
e −ζωn t cos(ωn 1 − ζ 2 t − φ)
1 − ζ2
p
where φ = tan−1 ζ/ 1 − ζ 2 .
Although the two parameters ωn and ζ completely characterize the
form of the underdamped response, we usually specify the response
with the following derived parameters:
Although the two parameters ωn and ζ completely characterize the
form of the underdamped response, we usually specify the response
with the following derived parameters:
Peak time, Tp : The time required to reach the first
(maximum) peak.
Although the two parameters ωn and ζ completely characterize the
form of the underdamped response, we usually specify the response
with the following derived parameters:
Peak time, Tp : The time required to reach the first
(maximum) peak.
Percent overshoot, %OS: The amount that the response
exceeds the final value at Tp .
Although the two parameters ωn and ζ completely characterize the
form of the underdamped response, we usually specify the response
with the following derived parameters:
Peak time, Tp : The time required to reach the first
(maximum) peak.
Percent overshoot, %OS: The amount that the response
exceeds the final value at Tp .
Settling time, Ts : The time required for the oscillations to die
down and stay within 2% of the final value.
Although the two parameters ωn and ζ completely characterize the
form of the underdamped response, we usually specify the response
with the following derived parameters:
Peak time, Tp : The time required to reach the first
(maximum) peak.
Percent overshoot, %OS: The amount that the response
exceeds the final value at Tp .
Settling time, Ts : The time required for the oscillations to die
down and stay within 2% of the final value.
Rise time, Tr : The time to go from 10% to 90% of the final
value.
Although the two parameters ωn and ζ completely characterize the
form of the underdamped response, we usually specify the response
with the following derived parameters:
Peak time, Tp : The time required to reach the first
(maximum) peak.
Percent overshoot, %OS: The amount that the response
exceeds the final value at Tp .
Settling time, Ts : The time required for the oscillations to die
down and stay within 2% of the final value.
Rise time, Tr : The time to go from 10% to 90% of the final
value.
Although the two parameters ωn and ζ completely characterize the
form of the underdamped response, we usually specify the response
with the following derived parameters:
Peak time, Tp : The time required to reach the first
(maximum) peak.
Percent overshoot, %OS: The amount that the response
exceeds the final value at Tp .
Settling time, Ts : The time required for the oscillations to die
down and stay within 2% of the final value.
Rise time, Tr : The time to go from 10% to 90% of the final
value.
Second-Order Systems
Characteristics of Underdamped Systems
Consider determining Tp , the time required to reach the first peak.
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider determining Tp , the time required to reach the first peak.
At the peak, the derivative is zero.
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider determining Tp , the time required to reach the first peak.
At the peak, the derivative is zero. Thus, we can solve for the
value of t for which ċ(t) = 0.
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider determining Tp , the time required to reach the first peak.
At the peak, the derivative is zero. Thus, we can solve for the
value of t for which ċ(t) = 0. We do this differentiation in the FD:
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider determining Tp , the time required to reach the first peak.
At the peak, the derivative is zero. Thus, we can solve for the
value of t for which ċ(t) = 0. We do this differentiation in the FD:
C (s)
=
ωn2
s(s 2 + 2ζωn s + ωn2 )
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider determining Tp , the time required to reach the first peak.
At the peak, the derivative is zero. Thus, we can solve for the
value of t for which ċ(t) = 0. We do this differentiation in the FD:
ωn2
s(s 2 + 2ζωn s + ωn2 )
d
ωn2
c(t) → sC (s) = 2
dt
s + 2ζωn s + ωn2
C (s)
=
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider determining Tp , the time required to reach the first peak.
At the peak, the derivative is zero. Thus, we can solve for the
value of t for which ċ(t) = 0. We do this differentiation in the FD:
ωn2
s(s 2 + 2ζωn s + ωn2 )
d
ωn2
c(t) → sC (s) = 2
dt
s + 2ζωn s + ωn2
C (s)
=
We now find the ILT to obtain ċ(t) and proceed to find the times
at which ċ(t) = 0.
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider determining Tp , the time required to reach the first peak.
At the peak, the derivative is zero. Thus, we can solve for the
value of t for which ċ(t) = 0. We do this differentiation in the FD:
ωn2
s(s 2 + 2ζωn s + ωn2 )
d
ωn2
c(t) → sC (s) = 2
dt
s + 2ζωn s + ωn2
C (s)
=
We now find the ILT to obtain ċ(t) and proceed to find the times
at which ċ(t) = 0.
COVERED ON BOARD
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Consider determining Tp , the time required to reach the first peak.
At the peak, the derivative is zero. Thus, we can solve for the
value of t for which ċ(t) = 0. We do this differentiation in the FD:
ωn2
s(s 2 + 2ζωn s + ωn2 )
d
ωn2
c(t) → sC (s) = 2
dt
s + 2ζωn s + ωn2
C (s)
=
We now find the ILT to obtain ċ(t) and proceed to find the times
at which ċ(t) = 0.
COVERED ON BOARD
Tp =
π
p
ωn 1 − ζ 2
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Percent overshoot is defined as follows,
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Percent overshoot is defined as follows,
cmax − cfinal
%OS =
× 100
cfinal
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Percent overshoot is defined as follows,
cmax − cfinal
%OS =
× 100
cfinal
If the input is a unit step, cfinal = 1.
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Percent overshoot is defined as follows,
cmax − cfinal
%OS =
× 100
cfinal
If the input is a unit step, cfinal = 1.
"
#
p
p
ζ
c(t) = 1 − e −ζωn t cos(ωn 1 − ζ 2 t) + p
sin(ωn 1 − ζ 2 t)
1 − ζ2
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Percent overshoot is defined as follows,
cmax − cfinal
%OS =
× 100
cfinal
If the input is a unit step, cfinal = 1.
"
#
p
p
ζ
c(t) = 1 − e −ζωn t cos(ωn 1 − ζ 2 t) + p
sin(ωn 1 − ζ 2 t)
1 − ζ2
cmax
= c(Tp )
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Percent overshoot is defined as follows,
cmax − cfinal
%OS =
× 100
cfinal
If the input is a unit step, cfinal = 1.
"
#
p
p
ζ
c(t) = 1 − e −ζωn t cos(ωn 1 − ζ 2 t) + p
sin(ωn 1 − ζ 2 t)
1 − ζ2
√
2
cmax = c(Tp ) = 1 + e (−ζπ/ 1−ζ )
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Percent overshoot is defined as follows,
cmax − cfinal
%OS =
× 100
cfinal
If the input is a unit step, cfinal = 1.
"
#
p
p
ζ
c(t) = 1 − e −ζωn t cos(ωn 1 − ζ 2 t) + p
sin(ωn 1 − ζ 2 t)
1 − ζ2
√
2
cmax = c(Tp ) = 1 + e (−ζπ/ 1−ζ )
We obtain,
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Percent overshoot is defined as follows,
cmax − cfinal
%OS =
× 100
cfinal
If the input is a unit step, cfinal = 1.
"
#
p
p
ζ
c(t) = 1 − e −ζωn t cos(ωn 1 − ζ 2 t) + p
sin(ωn 1 − ζ 2 t)
1 − ζ2
√
2
cmax = c(Tp ) = 1 + e (−ζπ/ 1−ζ )
We obtain,
√
%OS = e (−ζπ/
ENGI 5821
1−ζ 2 )
× 100
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Percent overshoot is defined as follows,
cmax − cfinal
%OS =
× 100
cfinal
If the input is a unit step, cfinal = 1.
"
#
p
p
ζ
c(t) = 1 − e −ζωn t cos(ωn 1 − ζ 2 t) + p
sin(ωn 1 − ζ 2 t)
1 − ζ2
√
2
cmax = c(Tp ) = 1 + e (−ζπ/ 1−ζ )
We obtain,
√
%OS = e (−ζπ/
1−ζ 2 )
× 100
This relationship is invertible,
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
Percent overshoot is defined as follows,
cmax − cfinal
%OS =
× 100
cfinal
If the input is a unit step, cfinal = 1.
"
#
p
p
ζ
c(t) = 1 − e −ζωn t cos(ωn 1 − ζ 2 t) + p
sin(ωn 1 − ζ 2 t)
1 − ζ2
√
2
cmax = c(Tp ) = 1 + e (−ζπ/ 1−ζ )
We obtain,
√
%OS = e (−ζπ/
1−ζ 2 )
× 100
This relationship is invertible,
− ln(%OS/100)
ζ=q
π 2 + ln2 (%OS/100)
ENGI 5821
Unit 3: Time Response
Second-Order Systems
ENGI 5821
Characteristics of Underdamped Systems
Unit 3: Time Response
The settling time Ts is the time required for c(t) to reach and stay
within 2% of the final value.
The settling time Ts is the time required for c(t) to reach and stay
within 2% of the final value.
p
1
c(t) = 1 − p
e −ζωn t cos(ωn 1 − ζ 2 t − φ)
1 − ζ2
The settling time Ts is the time required for c(t) to reach and stay
within 2% of the final value.
p
1
c(t) = 1 − p
e −ζωn t cos(ωn 1 − ζ 2 t − φ)
1 − ζ2
Consider just the exponential envelope of c(t),
The settling time Ts is the time required for c(t) to reach and stay
within 2% of the final value.
p
1
c(t) = 1 − p
e −ζωn t cos(ωn 1 − ζ 2 t − φ)
1 − ζ2
Consider just the exponential envelope of c(t),
1
p
e −ζωn t
2
1−ζ
The settling time Ts is the time required for c(t) to reach and stay
within 2% of the final value.
p
1
c(t) = 1 − p
e −ζωn t cos(ωn 1 − ζ 2 t − φ)
1 − ζ2
Consider just the exponential envelope of c(t),
1
p
e −ζωn t
2
1−ζ
Solve for the time at which the envelope decays to 0.02
The settling time Ts is the time required for c(t) to reach and stay
within 2% of the final value.
p
1
c(t) = 1 − p
e −ζωn t cos(ωn 1 − ζ 2 t − φ)
1 − ζ2
Consider just the exponential envelope of c(t),
1
p
e −ζωn t
2
1−ζ
Solve for the time at which the envelope decays to 0.02
1
p
e −ζωn t = 0.02
2
1−ζ
The settling time Ts is the time required for c(t) to reach and stay
within 2% of the final value.
p
1
c(t) = 1 − p
e −ζωn t cos(ωn 1 − ζ 2 t − φ)
1 − ζ2
Consider just the exponential envelope of c(t),
1
p
e −ζωn t
2
1−ζ
Solve for the time at which the envelope decays to 0.02
1
p
e −ζωn t = 0.02
2
1−ζ
p
− ln(0.02 1 − ζ 2 )
Ts =
ζωn
The settling time Ts is the time required for c(t) to reach and stay
within 2% of the final value.
p
1
c(t) = 1 − p
e −ζωn t cos(ωn 1 − ζ 2 t − φ)
1 − ζ2
Consider just the exponential envelope of c(t),
1
p
e −ζωn t
2
1−ζ
Solve for the time at which the envelope decays to 0.02
1
p
e −ζωn t = 0.02
2
1−ζ
p
− ln(0.02 1 − ζ 2 )
4
Ts =
≈
ζωn
ζωn
The settling time Ts is the time required for c(t) to reach and stay
within 2% of the final value.
p
1
c(t) = 1 − p
e −ζωn t cos(ωn 1 − ζ 2 t − φ)
1 − ζ2
Consider just the exponential envelope of c(t),
1
p
e −ζωn t
2
1−ζ
Solve for the time at which the envelope decays to 0.02
1
p
e −ζωn t = 0.02
2
1−ζ
p
− ln(0.02 1 − ζ 2 )
4
Ts =
≈
ζωn
ζωn
Note that this is a conservative estimate since the sinusoid might
actually reach and stay within 2% earlier.
Second-Order Systems
Characteristics of Underdamped Systems
There is no analytical form for Tr (time to go from 10% to 90% of
final value).
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
There is no analytical form for Tr (time to go from 10% to 90% of
final value). This value can be calculated numerically and has been
formed into a table:
ENGI 5821
Unit 3: Time Response
Second-Order Systems
Characteristics of Underdamped Systems
There is no analytical form for Tr (time to go from 10% to 90% of
final value). This value can be calculated numerically and has been
formed into a table:
ENGI 5821
Unit 3: Time Response
Relationship to Pole Plot
The following is the pole plot for a general second-order system:
Relationship to Pole Plot
The following is the pole plot for a general second-order system:
Relationship to Pole Plot
The following is the pole plot for a general second-order system:
σd = ζωn is the real part of the pole and is called the exponential
decay frequency.
Relationship to Pole Plot
The following is the pole plot for a general second-order system:
σd = ζωn is the real part of the pole and is called the exponential
decay frequency.
p
ωd = ωn 1 − ζ 2 is the imaginary part and is called the damped
frequency of oscillation.
Relationship to Pole Plot
The following is the pole plot for a general second-order system:
σd = ζωn is the real part of the pole and is called the exponential
decay frequency.
p
ωd = ωn 1 − ζ 2 is the imaginary part and is called the damped
frequency of oscillation.
Notice the following:
Relationship to Pole Plot
The following is the pole plot for a general second-order system:
σd = ζωn is the real part of the pole and is called the exponential
decay frequency.
p
ωd = ωn 1 − ζ 2 is the imaginary part and is called the damped
frequency of oscillation.
Notice the following:
ωn is the distance to the origin
Relationship to Pole Plot
The following is the pole plot for a general second-order system:
σd = ζωn is the real part of the pole and is called the exponential
decay frequency.
p
ωd = ωn 1 − ζ 2 is the imaginary part and is called the damped
frequency of oscillation.
Notice the following:
ωn is the distance to the origin
cos θ = ζ
Relationship to Pole Plot
Relationship to Pole Plot
We can relate Tp , Ts , and %OS to the locations of the poles.
Relationship to Pole Plot
We can relate Tp , Ts , and %OS to the locations of the poles.
Tp =
π
p
ωn 1 − ζ 2
Relationship to Pole Plot
We can relate Tp , Ts , and %OS to the locations of the poles.
Tp =
π
π
p
=
ωd
ωn 1 − ζ 2
Relationship to Pole Plot
We can relate Tp , Ts , and %OS to the locations of the poles.
Tp =
π
π
p
=
ωd
ωn 1 − ζ 2
Ts =
4
ζωn
Relationship to Pole Plot
We can relate Tp , Ts , and %OS to the locations of the poles.
Tp =
π
π
p
=
ωd
ωn 1 − ζ 2
Ts =
4
4
=
ζωn
σd
Relationship to Pole Plot
We can relate Tp , Ts , and %OS to the locations of the poles.
Tp =
π
π
p
=
ωd
ωn 1 − ζ 2
Ts =
4
4
=
ζωn
σd
%OS = f (ζ)
Tp = π/ωd
Ts = 4/σd
Tp = π/ωd
Ts = 4/σd
Design Example
Given the system below, find J and D to yield 20% overshoot and
a settling time of 2 seconds for a step input torque T (t).
The transfer function must first be determined,
Design Example
Given the system below, find J and D to yield 20% overshoot and
a settling time of 2 seconds for a step input torque T (t).
The transfer function must first be determined,
G (s) =
s2
1/J
+ DJ s +
K
J
Relating to the standard form of a second-order systems we have,
Design Example
Given the system below, find J and D to yield 20% overshoot and
a settling time of 2 seconds for a step input torque T (t).
The transfer function must first be determined,
G (s) =
s2
1/J
+ DJ s +
K
J
Relating to the standard form of a second-order systems we have,
r
K
ωn =
J
Design Example
Given the system below, find J and D to yield 20% overshoot and
a settling time of 2 seconds for a step input torque T (t).
The transfer function must first be determined,
G (s) =
s2
1/J
+ DJ s +
K
J
Relating to the standard form of a second-order systems we have,
r
K
D
ωn =
2ζωn =
J
J
Design Example
Given the system below, find J and D to yield 20% overshoot and
a settling time of 2 seconds for a step input torque T (t).
The transfer function must first be determined,
G (s) =
s2
1/J
+ DJ s +
K
J
Relating to the standard form of a second-order systems we have,
r
K
D
ωn =
2ζωn =
J
J
The specification of 20% overshoot allows us to calculate
ζ = 0.456.
Design Example
Given the system below, find J and D to yield 20% overshoot and
a settling time of 2 seconds for a step input torque T (t).
The transfer function must first be determined,
G (s) =
s2
1/J
+ DJ s +
K
J
Relating to the standard form of a second-order systems we have,
r
K
D
ωn =
2ζωn =
J
J
The specification of 20% overshoot allows us to calculate
ζ = 0.456.
The specification of Ts = 2 allows us to calculate ζωn = 2.
Design Example
Given the system below, find J and D to yield 20% overshoot and
a settling time of 2 seconds for a step input torque T (t).
The transfer function must first be determined,
G (s) =
s2
1/J
+ DJ s +
K
J
Relating to the standard form of a second-order systems we have,
r
K
D
ωn =
2ζωn =
J
J
The specification of 20% overshoot allows us to calculate
ζ = 0.456.
The specification of Ts = 2 allows us to calculate ζωn = 2. From
these values we can easily calculate D = 1.04 and J = 0.26.