Linear Control Systems Design lecture number 9

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Lecture 9
Almughtaribeen University – [ Abusabah I. A. Ahmed]
ALMUGHTARIBEEN
UNIVERSITY
Almughtaribeen University
College of Engineering
Linear Control Systems Design
Department of Electrical Engineering
Bode Plot Design

❑ Introduction
❑ Frequency Domain Specifications
❑ Bode Plot
❑ Gain and Phase Margins
❑ Practice Problem
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Lecture Outline
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Introduction
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❑ The plot can be used to interpret how the input affects
the output in both magnitude and phase over
frequency.
❑ In Time-Domain analysis, Impulse, unit step, ramp,
etc. are used as input to the system.
❑ Frequency responses are generally derived by
using the standard Laplace transform of
sinusoidal forcing functions.
❑ We shall look at a convenient graphical technique for
obtaining frequency response of linear systems.

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Frequency Domain Specifications
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❑ Extraction of Transfer function from time domain is
difficult using differential equations.
❑ Using frequency response, Transfer Function can be
easily obtained from the experimental data.
❑ A system may be designed, so that effects of noise
are negligible.
❑ Analysis and Design are extended to certain non-
linear control systems.
❑ The design of controller can be easily done in the
frequency domain, as compared to time domain
design.

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1. Resonant Frequency (𝝎𝒓)
The frequency at which the system has
maximum magnitude is known as the resonant
frequency. At this, the slope of magnitude
curve is zero.
Note that resonant frequency is different than both
the undamped and damped natural frequencies.
2. Resonant Peak (𝑴𝒓)
This is the maximum value of the transfer function amplitude 𝑇(𝑗𝜔)
𝑴𝒓 depends on damping ratio 𝜉 only and indicates the relative stability of stable
closed loop system.
The value of 𝑴𝒓 should be from 1.1 to 1.5.

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3. Cut-off Frequency
The frequency at which the magnitude T(jω) is 0.707 times
less than its, maximum value is known as cut-off frequency.
4. Cut off Rate
Is the rate of change of slope of
magnitude at cut-off frequency.
Cut off rate indicates the ability of
a system to distinguish signal
from noise.

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5. Bandwidth
For feedback control systems, the range of frequencies over
which M is equal to or greater than 0.707𝑀𝑟 is defined as
bandwidth 𝜔𝑏, When 𝑀𝑟=1, Bandwidth = Cut-off frequency.
The bandwidth of a control system indicates the noise- filtering
characteristics of the system.

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6. Gain Margin
Amount of gain in decibels that can be added to the loop
before the closed loop system becomes unstable.
Gain Crossover: The Point at which the magnitude plot
crossover 0dB.
7. Phase Margin:
Amount of phase shift in degrees that can be added to the loop
before the closed loop system becomes unstable.
Phase Crossover: The Point at which the phase plot crosses is
180°

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Bode Plot
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❑ The Bode form is a method for the frequency domain analysis.
The Bode plot of the function G(jω) is composed Bode of two
plots:
✓One with the magnitude of G(jω) plotted in decibels (dB) versus
log10(jω).
✓The other with the phase of G(jω) plotted in degree versus
log10(ω).
❑ Feature of the Bode Plots:
✓ Since the magnitude of G(jω) in the Bode plot is expressed in dB,
products and division factors in G() become additions and subtraction,
respectively.
✓ The phase relations are also added and subtracted from each other
algebraically.
✓ The magnitude plot of Bode of G(jω) can be approximated by straight-
lines segments which allow the simple sketching of the bode plot
without detailed computation.

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Bode Plot
Basic Procedure
1. Replace s → 𝑗𝜔, to convert it into frequency domain.
2. Write the magnitude & Convert it into dB
Magnitude = 20log G(j ω).
3. Find the φ angle.
4. Vary ‘ω’ from min to max value & draw the
approximate magnitude and Phase plot.

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Bode Plot
L.T.I system
t
A
t
r 
sin
)
( = )
sin(
)
( 
 +
= t
B
t
y
Magnitude:
A
B Phase: 
G(s)
H(s)
＋
－
)
(t
y
)
(t
r
)
(
)
(
1
)
(
)
(
)
(
s
H
s
G
s
G
s
R
s
Y
+
= 

 j
s
j
s =

+
=
Magnitude:
)
(
)
(
1
)
(



j
H
j
G
j
G
+
Phase:
)]
(
)
(
1
[
)
(



j
H
j
G
j
G
+


Steady state response
Basic Procedure

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Bode Plot
❑ The gain magnitude is many times expressed in
terms of decibels (dB)
dB = 20 log10 A
where A is the amplitude or gain
a decade is defined as any 10-to-1 frequency range
an octave is any 2-to-1 frequency range
20 dB/decade = 6 dB/octave
Basic Procedure

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Bode Plot
Basic rules

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Bode Plot
Basic rules

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Bode Plot
Basic rules

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Bode Plot
Basic rules

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Bode Plot
Basic rules

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Bode Plot
Basic rules

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Bode Plot
Basic rules

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Bode Plot
Basic rules

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Gain and Phase Margins

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Gain margin: the difference
(in dB) between 0dB and
the system gain, computed
at the frequency where the
phase is 180°
Phase margin: the
difference (in °) between
the system phase and 180°,
computed at the frequency
where the gain is 1 (i.e.,
0dB)
A system is stable if the gain and phase margins are both
positive

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Example 9-1
Draw the Bode Plot of following Transfer function.
)
(
)
(
10
20
+
=
s
s
s
G
Solution
)
.
(
)
(
1
1
0
2
+
=
s
s
s
G
The transfer function contains
1. Gain Factor (K=2)
2. Derivative Factor (s)
3. 1st Order Factor in denominator (0.1s+1)-1

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)
.
(
)
(
1
1
0
2
+
=
s
s
s
G
1. Gain Factor (K=2)
dB
K dB
6
)
2
log(
20 =
=
2. Derivative Factor (s)
dB/decade
20
)
log(
20 =
= ω
dB
s
3. 1st Order Factor in denominator (0.1s+1)
0
)
1
log(
20
1
1
.
0
1
,
10
when =
=
+
<<
dB
jω
ω
dB/dec
20
)
1
.
0
log(
20
1
1
.
0
1
,
10
when =
=
+
>> ω
ω
ω
dB
j

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)
(
)
(
10
20
+
=
s
s
s
G

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)
(
)
(
10
20
+
=
s
s
s
G

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Relative Stability
A transfer function is called minimum phase when all the poles
and zeros are LHP and non-minimum-phase when there are
RHP poles or zeros.
Minimum phase system Stable
The gain margin (GM) is the distance on the bode magnitude plot
from the amplitude at the phase crossover frequency up to the 0 dB
point. GM= -(dB of GH measured at the phase crossover frequency)
The phase margin (PM) is the distance from -180 up to the phase
at the gain crossover frequency.
PM=180+phase of GH measured at the gain crossover frequency

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❑ Phase crossover frequency (ωp) is the frequency at which
the phase angle of the open-loop transfer function equals
–180°.
❑ The gain crossover frequency (ωg) is the frequency at
which the magnitude of the open loop transfer function, is
unity.
❑The gain margin (Kg) is the reciprocal of the magnitude of
G(jω) at the phase cross over frequency.
❑ The phase margin (γ) is that amount of additional phase
lag at the gain crossover frequency required to bring the
system to the verge of instability.

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Practice Problem
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Sketch the Bode plots for the given open loop transfer function of
a control system.

Thank You
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Linear Control Systems Design lecture number 9

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