HW3 – Nichols plots and frequency domain specifications FORMULAE FOR SECOND ORDER SPECIFICATIONS: If the higher order closed-loop system has two dominant poles (and no dominant zeros) it can be approximated by a second order system. Here are the formulae for transient specifications for second order systems or approximate second order systems: 1. Natural frequency: 𝜔𝑛 (distance of pole to origin) 2. Damping ratio: 𝜁 (related to angle of the pole) 3. The closed-loop dominant poles: 𝑝1, 𝑝2 = −𝜎 ± 𝑗𝜔𝑑 = −𝜔𝑛𝜁 ± 𝑗𝜔𝑛√1 − 𝜁 2 4. Angle of the poles with negative real axis: 𝛽 = arccos 𝜁 = cos−1 𝜁 5. The damped frequency of oscillations is the imaginary part of the roots: 𝜔𝑑 6. The exponential time constant is the reciprocal of the real part of the roots: 𝜏 = 1 𝜎 = 1 𝜔𝑛𝜁 7. Distance of the pole from origin is √𝜎2 + 𝜔𝑑 2 = 𝜔𝑛 = natural frequency. 8. Settling time: 𝑇𝑠 = 4 𝜎 = 4 𝜔𝑛𝜁 9. Peak time: 𝑇𝑝 = 𝜋 𝜔𝑑 10. Maximum overshoot: 𝑀𝑝 = 𝑒 − 𝜋𝜁 √1− 𝜁2 = 𝑒−𝜋 cot 𝛽 = 𝑒 −𝜋 𝜎 𝜔𝑑 11. To get percent overshoot multiply 𝑀𝑝 with 100. Similar characteristics in the frequency domain are: 1. Resonant frequency of the closed loop system: 𝜔𝑟 = 𝜔𝑛√1 − 2𝜁 2 2. Resonant peak amplitude the closed loop system: 𝑀𝑟 = 1 2𝜁√1−𝜁2 Note that there is no resonant frequency or peak when the damping ratio is greater than 1 √2 = 0.707 3. Cutoff frequency the closed loop system: 𝜔𝑐 = 𝜔𝑛√1 − 2𝜁 2 + √1 + (1 − 2𝜁2)2 Cutoff frequency tells about the bandwidth and is related to speed of the closed loop system. Higher cut-off frequency → higher bandwidth → lower settling time, rise-time, peak time. 4. Damping ratio of the closed loop system 𝜁 is related to the phase margin of the open loop system: 𝜙 = tan−1 2 √√4+ 1 𝜁4 −2 (𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠) ≈ 100𝜁 (𝑖𝑛 𝑑𝑒𝑔𝑟𝑒𝑒𝑠) The approximate formula for phase margin, 100𝜁, works only for phase margins up to about 60 degrees. Errors specifications: 1. For a typical system with a plant G and controller K. The error is 𝐸(𝑠) = 1 1+𝐺𝐾 𝑅 The steady state error is 𝑒(∞) = lim 𝑠→0 𝐸(𝑠)𝑠 = lim 𝑠→0 𝑠𝑅 1+𝐺𝐾 2. Steady state error for a step input R(s) = 1/s is: 𝑒(∞) = lim 𝑠→0 1 1 + 𝐺𝐾 = 1 1 + lim 𝑠→0 𝐺𝐾 = 1 1 + 𝑑𝑐 𝑔𝑎𝑖𝑛 𝑜𝑓 𝐺𝐾 = 1 1 + 𝐾𝑝 3. Steady state error for a ramp input R(s) = 1/s2 is: 𝑒(∞) = lim 𝑠→0 1 𝑠(1 + 𝐺𝐾) = 1 lim 𝑠→0 𝑠𝐺𝐾 = 1 𝐾𝑣 If GK has no integrator 𝐾𝑣 is 0 (ramp error is infinite). If GK has 1 integrator 𝐾𝑣 is finite, it is the x-intercept (frequency intercept) of the asymptote line from the left half of the bode(GK). If GK has more integrators 𝐾𝑣 is infinite. 4. To reduce the error due to reference command (and disturbance dy) of frequency 𝜔 by a factor of F: |𝐺(𝑗𝜔)| > 𝐹 + 1 . 5. To reduce the error due to a sinusoidal noise command of frequency 𝜔 by a factor of F: |𝐺(

1. HW3 – Nichols plots and frequency domain specifications
FORMULAE FOR SECOND ORDER SPECIFICATIONS:
If the higher order closed-loop system has two dominant poles
(and no dominant zeros) it can be approximated by a second
order
system. Here are the formulae for transient specifications for
second order systems or approximate second order systems:
1. Natural frequency: �� (distance of pole to origin)
2. Damping ratio: � (related to angle of the pole)
3. The closed-loop dominant poles: �1, �2 = −� ± ��� =
−��� ± ���√1 − �
2
4. Angle of the poles with negative real axis: � = arccos � =
cos−1 �
5. The damped frequency of oscillations is the imaginary part of
the roots: ��
6. The exponential time constant is the reciprocal of the real
part of the roots: � =
1
�
=
1

2. ���
7. Distance of the pole from origin is √�2 + ��
2 = �� = natural frequency.
8. Settling time: �� =
4
�
=
4
���
9. Peak time: �� =
�
��
10. Maximum overshoot: �� = �
−
��
√1− �2
= �−� cot � = �
−�
�

3. ��
11. To get percent overshoot multiply �� with 100.
Similar characteristics in the frequency domain are:
1. Resonant frequency of the closed loop system: �� = ��√1
− 2�
2
2. Resonant peak amplitude the closed loop system: �� =
1
2�√1−�2
Note that there is no resonant frequency or peak when the
damping ratio is greater than
1
√2
= 0.707
3. Cutoff frequency the closed loop system: �� = ��√1 − 2�
2 + √1 + (1 − 2�2)2
Cutoff frequency tells about the bandwidth and is related to
speed of the closed loop system.
Higher cut-off frequency → higher bandwidth → lower settling
time, rise-time, peak time.
4. Damping ratio of the closed loop system � is related to the
phase margin of the open loop system:
� = tan−1

4. 2
√√4+
1
�4
−2
(�� �������) ≈ 100� (�� �������)
The approximate formula for phase margin, 100�, works only
for phase margins up to about 60 degrees.
Errors specifications:
1. For a typical system with a plant G and controller K.
The error is �(�) =
1
1+��
�
The steady state error is �(∞) = lim
�→0
�(�)� = lim
�→0
��
1+��
2. Steady state error for a step input R(s) = 1/s is:

5. �(∞) = lim
�→0
1
1 + ��
=
1
1 + lim
�→0
��
=
1
1 + �� ���� �� ��
=
1
1 + ��
3. Steady state error for a ramp input R(s) = 1/s2 is:
�(∞) = lim
�→0
1
�(1 + ��)
=

6. 1
lim
�→0
���
=
1
��
If GK has no integrator �� is 0 (ramp error is infinite). If GK
has 1 integrator �� is finite, it is the x-intercept (frequency
intercept) of the asymptote line from the left half of the
bode(GK). If GK has more integrators �� is infinite.
4. To reduce the error due to reference command (and
disturbance dy) of frequency � by a factor of F: |�(��)| > � +
1 .
5. To reduce the error due to a sinusoidal noise command of
frequency � by a factor of F: |�(��)| <
1
1+�
.
1. Most of the specifications for the system of the unity
feedback architecture shown on the right are boiled down to

7. Bode/Nyquist/Nichols plots of the GK(s).
a. If there was a sensor block �(�) in the feedback
path, whose Bode/Nyquist/Nichols plots will you plot to check
stability of the closed loop system?
b. If there was a prefilter block �(�) after the reference
command, whose Bode/Nyquist/Nichols plots will you plot to
check stability of the closed loop system?
2. Here is a Bode plot of some open loop system GK, plot the
Nichols plot by hand. Show the gain and phase margins in there.
Is the closed loop system stable?

8. 3. Mark the following No-go regions in the Bode plot of GK in
Q2:
a. If you want error due to reference R in the frequency range 0
to 1 rad/s to be <1% of the R amplitude.
b. If you want error due to noise N in the frequency range 10 to
100 rad/s to be <1% of the N amplitude.
c. If you want error due to disturbance dY in the frequency
range 0 to 1 rad/s to be <1% of the dY amplitude.
d. If you want error due to disturbance dU in the frequency
range 0 to 1 rad/s to be <1% of the dU amplitude.
Here is the Bode of the plant G(s):
R(s) E(s)

9. -
+
�(�) Y(s
)
�
-100
-50
0
50
M
a
g
n
itu
d
e
(
d
B
)
10
-1
10

10. 0
10
1
10
2
-270
-180
-90
0
P
h
a
s
e
(
d
e
g
)
Bode Diagram
Gm = 2.92 dB (at 2.24 rad/sec) , Pm = 24.1 deg (at 1.63
rad/sec)
Frequency (rad/sec)

11. 4. A Controller � is used to control the angle �, of a motor by
providing the required voltage to the motor.
The bode plot of the openloop system �� is shown below:
a. What happens to the phase margin of the system as the time
delay increases? Explain why.
b. If there is time delay between the controller and the plant,
what is that maximum time delay this controller can
handle before it becomes unstable. (Do not do Pade
Approximations). Show all your calculations.
c. If there is time delay introduced by the sensor that measures
the angle, then what is the maximum time delay this
system can handle before it becomes unstable. (Do not do Pade
Approximations)
-100
-50
0
50

12. M
a
g
n
itu
d
e
(
d
B
)
10
-2
10
-1
10
0
10
1
10
2
-180
-135
-90

13. P
h
a
s
e
(
d
e
g
)
Bode Diagram
Gm = Inf dB (at Inf rad/sec) , Pm = 48.5 deg (at 0.882 rad/sec)
Frequency (rad/sec)
5. Practice: closed-loop specifications from open loop plots.
Figure on the side shows the Nichols chart of a transfer
function GK(s). Magnified view of a portion of the
graph is also shown below, with some points labelled
on it. Answer the questions below.

14. a. What is the gain margin as read from the Nichols chart?
b. What is the phase margin as read from the Nichols chart?
c. What is the resonant peak amplitude and the resonant
frequency of the closed loop system as read from the
Nichol’s chart?
d. What is �� , the cutoff frequency of the closed loop system
as read from the Nichol’s chart?
e. Calculate the damping ratio, � of the closed-loop poles from
the phase margin.
f. Using the � and �� calculate the % overshoot and settling
time of the step response of the closed loop system.
g. What is GK(s) in the limit as s goes to zero? Hence what is
the steady state error for a step input?
h. Extra credit: What is sGK in the limit as s goes to zero?
Hence what is the steady state error for a ramp input? You
are provided the info that at the top-right of the chart, the gain
is 43.2dB, the phase is -90 deg. and the frequency
is 0.01 rad/s.
R(s) E(s)
-
+
�(�) Y(s

15. )
�