Lecture 20

MECE 3350U
Control Systems
Lecture 20
Stability Margins
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MECE 3350 - C. Rossa 1 / 37 Lecture 20
Videos in this lecture
Lecture: https://youtu.be/IXSLZ-B0Zn0
Exercise 117: https://youtu.be/HUtiPTA8Mq0
Exercise 118: https://youtu.be/cpr9s0k1aMQ
Exercise 119: https://youtu.be/96mQDIujFEU
Exercise 120: https://youtu.be/PDJ-VSWXfng
Exercise 121: https://youtu.be/_j88ycAA0Fs
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Outline of Lecture 20
By the end of today’s lecture you should be able to
• Calculate the gain and phase margin of a system
• Obtain the gain and phase margin from a Bode plot
• Quantify the stability of an open-loop transfer function
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Applications
We wish to develop a closed-loop controller for a system whose dynamics is

unknown. The frequency response of the open-loop system has been obtained
experimentally using an oscilloscope.
What does it tell us about its closed-loop stability?
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Applications
The controller gain k has been specified to the process shown.
If b changes during operation, how can we ensure that the system remains
stable?
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Bode vs Nyquist plots
The closed loop system
kG(s)
T (s) =
1 + kG(s)
might be stable for only a range of values of k.

The proximity of the L(jω) locus to −1 + j0 is a measure of the relative
stability of the system.
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Bode vs Nyquist plot
Consider the open-loop transfer function
k
L(jω) =
jω(jωτ1 + 1)(jωτ2 + 1)
0.5
Imaginary Axis
-0.5
-1 Real Axis 0
As k is increased, the Nyquist plot approaches −1 + 0j and eventually encircles

the 1 point.
The point −1 + 0j can also be expressed in polar form as 1∠ − 180◦
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Bode vs Nyquist plot
k
L(jω) =
jω(jωτ1 + 1)(jωτ2 + 1)
50
Magnitude (dB)
0
-50
-100
-150
-90
-135
Phase (deg)
-180
-225
-270
10-1 100 101 102 103
Frequency (rad/s)
Gain margin: The increase in the loop gain when φ = −180◦ that results in
|L(jω)| = 1 or 0 dB.
Phase margin: The amount of phase shift at the crossover frequency that
results in ∠L(jω) = −180◦ .
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Gain and phase margins
Magnitude (dB)
0
Phase (deg)
-180
10-1 Frequency (rad/s) 103
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Gain and phase margins
0 0
Magnitude (dB)
-180 -180
Phase (deg)
Frequency (rad/s) Frequency (rad/s)
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True or false?
The following open-loop transfer function is closed-loop stable for any k > 0.
s 2 + 0.1s + 0.5
L(s) = k
s(s + 1)(s 2 + 0.05 + 0.5)
1
Imaginary Axis
-1
-1 Real Axis 0
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True or false?
The following open-loop transfer function is closed-loop stable for any k > 0.
s 2 + 0.1s + 0.5
L(s) = k
s(s + 1)(s 2 + 0.05s + 0.5)
50
Magnitude (dB)
-100
-90
-135
Phase (deg)
-180
10-2 10-1 100 101 102
Frequency (rad/s)
12/37
Phase margin
As an example, consider the open-loop second-order system
ωn2 ωn2
G(s) = → (1)
s(s + 2ζωn ) jω(jω + 2ζωn )
Step 1 - Find the crossover frequency (0 dB)
At the crossover frequency ω = ωc , the magnitude is 1. Find ωc that gives
ωn2
p = 1.
ωc ωc2 + 4ζ 2 ωn2
Step 2 - Find the phase of G(jω) at ωc for ωc found in Step 1, i.e. ∠G(jωc )

◦ ωc
φ = −90 − tan
2ζωn
Step 3 - The margin phase is PM = 180 − |φ|
If PM < 0, the system is closed-loop unstable.
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Gain margin
Consider the same open-loop second-order system
ωn2 ωn2
G(s) = →
s(s + 2ζωn ) jω(jω + 2ζωn )
Step 1 - Find the frequency ωf where ∠|G(jω)| = −180◦

ωn2 ωn2 −ωf2 − j2ζωn ωf
G(jω) = = 2
×
jω(jω + 2ζωn ) −ωf + j2ζωn ωf −ωf2 − j2ζωn ωf
ωn2 ωf2 2ζωn3 ωf
G(jω) = − −j 4
ωf4 2 2 2
+ 4ζ ωn ωf ωf + 4ζ 2 ωn2 ωf2
At ωf , =[G(jωf )] = 0 (imaginary part is zero)
2ζωn3 ωf
− =0
ωf4 + 4ζ 2 ωn2 ωf2
ωf = 0 Not a valid frequency
ωf = ∞ What does it mean?
ωf = constant. Proceed to Step 2
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Gain margin
Step 2 - Find the gain of G(jω) at ω = ωc , i.e., |G(jωf )| = kMG

ωn2
kMG = p
ωf ωf2 + 4ζ 2 ωn2
Then gain margin in Decibels is
MG = −20 log(kMG )
→ MG > 0: Stable. The gain can be multiplied by kMG dB before the system
becomes marginally stable (or MG dB can be added before instability);
→ MG = 0 The system is marginally stable.
→ MG < 0: Unstable. The gain can be divided by kMG dB before the system
becomes marginally stable (or MG dB must be subtracted to achieve stability).
15/37
Exercise 117
A unit feedback control system has a loop transfer function
k
L(s) =
s(s + 2)(s + 10)
For k = 50, determine the cross over frequency, the gain margin, and the phase
margin.
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Exercise 117 - continued
k
L(s) =
s(s + 2)(s + 10)
17/37
k
L(s) =
s(s + 2)(s + 10)
18/37
Exercise 118
A unit feedback control system has a loop transfer function
k
L(s) =
(s + 1)2
Determine the gain k so that the phase margin is 60◦
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k
L(s) =
(s + 1)2
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k
L(s) =
(s + 1)2
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Bode plot for k = 4.
20
Magnitude (dB)
0
-20
-40
-60
0
-45
Phase (deg)
-90
-135
-180
10-2 10-1 100 101 102
Frequency (rad/s)
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Exercise 119
A system has a loop transfer function
1 + s/5
T (s) = 10.5
s(1 + s/2)(1 + s/10)
Show that the crossover frequency is 5 rad/s and that the phase margin is 40◦
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1 + s/5
T (s) = 10.5
s(1 + s/2)(1 + s/10)
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1 + s/5
T (s) = 10.5
s(1 + s/2)(1 + s/10)
25/37
Consider a unit feedback system with the loop transfer function
k
L(s) =
s(s + 1)(s + 4)
(a) For k = 5, show that the gain margin is 12 dB

(b) If we wish to achieve a gain margin of 20 dB, determine the value of k
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k
L(s) =
s(s + 1)(s + 4)
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k
L(s) =
s(s + 1)(s + 4)
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Bode diagram for k = 2.

Magnitude (dB) 50
-50
-100
-150
-90
-135
Phase (deg)
-180
-225
-270
10-2 10-1 100 101 102
Frequency (rad/s)
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Exercise 121 - Using Matlab
Consider a unit feedback system with a proportional controller such that the
loop transfer function is
s 2 + 0.1s + 0.5
L(s) = k
s(s + 1)(s + 2)(s 2 + 0.05s + 0.5)
with
Using Matlab, plot the phase and gain margins for 0 ≤ k ≤ 10. Specify the
maximum value of k that results in a stable closed loop system.
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Matlab code for Exercise 121
clear all; close all;

s = tf([1 0],[1]);
i = 1;
for k = 1:0.05:10;
H = k ∗ (s 2 + 0.1 ∗ s + 0.5)/(s ∗ (s + 1) ∗ (s 2 + 0.05 ∗ s + 0.5) ∗ (s + 2));
[Gm,Pm,Wcg,Wcp] = margin(H);
PhaseM(i) = Pm;
GainM(i) = Gm;
K(i) = k;
i = i+1;
end
yyaxis left
plot(K, PhaseM);
yyaxis right
plot(K, GainM);
yyaxis left
title(’Phase and gain margins’)
xlabel(’k’)
ylabel(’phase margin [deg]’)
yyaxis right
ylabel(’Gain Margin [dB]’)
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Skills check 50 - From 2018 final examination
The Bode and Nyquist plots of a given transfer function G(s) are shown below.
Identify the gain and phase margins of G(s) on each diagram (3 marks):
Bode Diagram Nyquist Diagram

20 2
Magnitude (dB)
0 1.5
-20 1
0.5
Imaginary Axis
-40 1
-60 0
0
-0.5
Phase (deg)
-90
-1
-180
-1.5
-270 -2
10-1 100 101 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Frequency (rad/s) Real Axis
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Consider the following transfer function:

200
G(s) =
(s + 1)(s 2 + 5s + 100)
(a) Calculate the phase and magnitude of G(s) at 1 rad/s.
(b) Calculate the phase margin.
(c) Draw the Bode plot of G(s). Based on the Bode plot, specify the
approximate gain at the cut-off frequencies and at 103 rad/s .
(d) Calculate the gain margin (4 marks).
Answers: (a) 3dB, −50◦ , (b) 144◦ , (d) 8.87 dB 33/37

The Bode and Nyquist plots of a given transfer function G(s) are shown below.
Identify the gain and phase margins of G(s) on each diagram (3 marks):
Bode Diagram Nyquist Diagram

20 2
Magnitude (dB)
0 1.5
-20 1
0.5
Imaginary Axis
-40 1
-60 0
0
-0.5
Phase (deg)
-90
-1
-180
-1.5
-270 -2
10-1 100 101 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Frequency (rad/s) Real Axis
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Skills check 53 - From 2018 deferred final examination
The Nyquist plot for a control system resembles the one shown below. What
are the gain and phase margins ? (3 marks).
Answers: GM = 1/α, and 1/β. P.M = φ 35/37

Student course feedback survey
https://www.cci-survey.ca/ontariotechu/ca/
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Next class...
• Space state models
37/37

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MECE 3350U

Exercise 117: https://youtu.be/HUtiPTA8Mq0

Exercise 118: https://youtu.be/cpr9s0k1aMQ

Exercise 119: https://youtu.be/96mQDIujFEU

Exercise 120: https://youtu.be/PDJ-VSWXfng

Exercise 121: https://youtu.be/_j88ycAA0Fs

By the end of today’s lecture you should be able to

• Calculate the gain and phase margin of a system

• Obtain the gain and phase margin from a Bode plot

• Quantify the stability of an open-loop transfer function

We wish to develop a closed-loop controller for a system whose dynamics is

What does it tell us about its closed-loop stability?

The controller gain k has been specified to the process shown.

might be stable for only a range of values of k.

Consider the open-loop transfer function

As k is increased, the Nyquist plot approaches −1 + 0j and eventually encircles

10-1 Frequency (rad/s) 103

Frequency (rad/s) Frequency (rad/s)

Step 1 - Find the crossover frequency (0 dB)

At the crossover frequency ω = ωc , the magnitude is 1. Find ωc that gives

Step 1 - Find the frequency ωf where ∠|G(jω)| = −180◦

Step 2 - Find the gain of G(jω) at ω = ωc , i.e., |G(jωf )| = kMG

Then gain margin in Decibels is

A unit feedback control system has a loop transfer function

A unit feedback control system has a loop transfer function

Determine the gain k so that the phase margin is 60◦

A system has a loop transfer function

Consider a unit feedback system with the loop transfer function

(a) For k = 5, show that the gain margin is 12 dB

Bode diagram for k = 2.

clear all; close all;

Bode Diagram Nyquist Diagram

Consider the following transfer function:

(a) Calculate the phase and magnitude of G(s) at 1 rad/s.

(b) Calculate the phase margin.

(d) Calculate the gain margin (4 marks).

Answers: (a) 3dB, −50◦ , (b) 144◦ , (d) 8.87 dB 33/37

Bode Diagram Nyquist Diagram

Answers: GM = 1/α, and 1/β. P.M = φ 35/37

• Space state models

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