AER 372

1) Frequency
Response
2) Bode Plot
3) System Stability
4) Closed-loop
Control Systems
Frequency
Response
5) Design for
Dynamic Chapter 6
Compensation
6) Case Study Frequency Response Design Method

Prof. M.R. Emami

Winter 2025

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 M.R. Emami, 2025
 Outline
1) Frequency
Response
2) Bode Plot
 Frequency Response
3) System Stability
4) Closed-loop  Bode Plot
Frequency
Response
5) Design for  System Stability
Dynamic
Compensation
6) Case Study  Closed-loop Frequency Response
 Design for Dynamic Compensation
 Case Study

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 Frequency Response
 The steady-state response of a (linear,
time-invariant) system to sinusoidal
1) Frequency
inputs.
Response Steady State
2) Bode Plot 𝑢 𝑡 = 𝐴𝑠𝑖𝑛 𝜔 𝑡 1 𝑡 𝑦 𝑡 = 𝐴 𝐺 𝑗𝜔 sin 𝜔 𝑡 + ∠𝐺 𝑗𝜔
3) System Stability
PROOF:
4) Closed-loop
Frequency  Frequency response of a system can be obtained from the location of its
Response poles and zeros.
5) Design for
Dynamic 𝑌 𝑠
Compensation 𝐺 𝑠 = 𝐼. 𝐶. = 0 𝐴𝜔
𝑈 𝑠 𝑌 𝑠 =𝐺 𝑠
6) Case Study 𝐴𝜔 𝑠 +𝜔
𝑢 𝑡 = 𝐴𝑠𝑖𝑛 𝜔 𝑡 1 𝑡 𝑈 𝑠 =
𝑠 +𝜔
 Partial Fraction Expansion: (All 𝐺 𝑠 poles 𝑝 , 𝑝 , ⋯ , 𝑝 are distinct.)
𝛼 𝛼 𝛼 𝛼 𝛼∗
𝑌 𝑠 = + +⋯+ + +
𝑠−𝑝 𝑠−𝑝 𝑠−𝑝 𝑠 + 𝑗𝜔 𝑠 − 𝑗𝜔
𝑦 𝑡 =𝛼 𝑒 +𝛼 𝑒 +⋯+𝛼 𝑒 +𝛼 𝑒 + 𝛼∗𝑒 𝑡≥0 𝐼𝑚 𝛼
𝜓 = 𝑡𝑎𝑛
𝑅𝑒 𝛼
𝑦 𝑡 =𝛼 𝑒 +𝛼 𝑒 + ⋯+ 𝛼 𝑒 + 𝛼 𝑒 𝑒 + 𝛼 𝑒 𝑒
𝑦 𝑡 =𝛼 𝑒 +𝛼 𝑒 +⋯+𝛼 𝑒 + 𝛼 𝑒 +𝑒

𝑦 𝑡 =𝛼 𝑒 +𝛼 𝑒 +⋯+𝛼 𝑒 + 2 𝛼 cos −𝜔 𝑡 + 𝜓
stable LTI system 3
 M.R. Emami, 2025 𝑅𝑒 𝑝 , ⋯ , 𝑅𝑒 𝑝 <0 𝑦 𝑡 = 2 𝛼 cos 𝜔 𝑡 − 𝜓
 Frequency Response
PROOF (cont.):
𝐴𝜔
𝑌 𝑠 =𝐺 𝑠
𝑠 +𝜔
1) Frequency
Response 𝛼 𝛼∗
𝑡→∞ 𝑌 𝑠 = +
2) Bode Plot 𝑠 + 𝑗𝜔 𝑠 − 𝑗𝜔
3) System Stability (Amplitude Ratio)
𝐴𝜔 𝛼 𝛼∗
4) Closed-loop 𝐺 𝑠 = + 𝑀 = 𝐺 𝑗𝜔 Gain
Frequency 𝑠 + 𝑗𝜔 𝑠 − 𝑗𝜔 𝑠 + 𝑗𝜔 𝑠 − 𝑗𝜔
Response
Phase
𝐴𝜔 𝛼 𝑠 − 𝑗𝜔 𝐼𝑚 𝐺 𝑗𝜔
5) Design for 𝐺 𝑠 = + 𝛼∗ 𝜙 = 𝑡𝑎𝑛 = ∠𝐺 𝑗𝜔
Dynamic 𝑠 + 𝑗𝜔 𝑠 + 𝑗𝜔 𝑅𝑒 𝐺 𝑗𝜔
Compensation
𝐴𝜔 𝛼 𝑗𝜔 − 𝑗𝜔
6) Case Study Let 𝑠 = 𝑗𝜔 𝐺 𝑗𝜔 = + 𝛼∗ = 𝛼∗
2𝑗𝜔 𝑗𝜔 + 𝑗𝜔
𝐴
𝐺 𝑗𝜔 = 𝛼∗ = 𝛼 𝑒
2𝑗
𝐼𝑚 𝐺 𝑗𝜔
𝐺 𝑗𝜔 = 𝑀𝑒 ; 𝑀 = 𝐺 𝑗𝜔 and 𝜙 = ∠𝐺 𝑗𝜔 = 𝑡𝑎𝑛
𝑅𝑒 𝐺 𝑗𝜔
𝐴
𝐴 𝐴 𝛼 = 𝑀
− 𝑗𝑀𝑒 = 𝛼 𝑒 𝑀𝑒 = 𝛼 𝑒 2
2 2 𝜋
𝜓= −𝜙
2
 For a stable LTI system: 𝑡≥0
𝜋
𝑢 𝑡 = 𝐴𝑠𝑖𝑛 𝜔 𝑡 𝑦 𝑡 = 2 𝛼 cos 𝜔 𝑡 − 𝜓 = 𝐴𝑀 cos − 𝜔 𝑡+𝜙
2
𝑦 𝑡 = 𝐴 𝐺 𝑗𝜔 sin 𝜔 𝑡 + ∠𝐺 𝑗𝜔 4
 M.R. Emami, 2025 𝑦 𝑡 = 𝐴𝑀 sin 𝜔 𝑡 + 𝜙
 Frequency Response
Example: RC Circuit

1) Frequency
𝑑𝑦 𝑑𝑦 1 𝒦
𝑅𝐶 + 𝑦 𝑡 = 𝒦𝑢 𝑡 + 𝑦 𝑡 = 𝑢 𝑡 𝒦
Response 𝑑𝑡 𝑑𝑡 𝑅𝐶 𝑅𝐶
2) Bode Plot 𝑑𝑦
3) System Stability
𝒦 = 𝑅𝐶 + 𝑘𝑦 𝑡 = 𝑢 𝑡 ; 𝑘 = 1 𝑅𝐶
𝑑𝑡
4) Closed-loop 𝑌 𝑠 1
Frequency =𝐺 𝑠 = ; 𝑘=1
Response 𝑈 𝑠 𝑠+𝑘 1 10
𝑌 𝑠 =𝐺 𝑠 𝑈 𝑠 =
5) Design for 10 𝑠 + 1 𝑠 + 100
Dynamic 𝑢 𝑡 = 𝑠𝑖𝑛 10𝑡 1 𝑡 𝑈 𝑠 =
Compensation 𝑠 + 100
6) Case Study 10 𝑗 −𝑗
∗
𝛼 𝛼 𝛼 2 1 − 𝑗10 2 1 + 𝑗10
𝑌 𝑠 = + + = 101 + +
𝑠 + 1 𝑠 + 𝑗10 𝑠 − 𝑗10 𝑠 + 1 𝑠 + 𝑗10 𝑠 − 𝑗10
10 1
𝑦 𝑡 = 𝑒 + sin 10𝑡 + 𝜙 = 𝑦 𝑡 + 𝑦 𝑡 ; 𝜙 = −𝑡𝑎𝑛 10 = −84.2°
101 101

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 Frequency Response
𝑇𝑠 + 1
Example: Lead Network 𝐷 𝑠 =𝐾 ; 𝛼<1
𝛼𝑇𝑠 + 1
1) Frequency 𝑇𝑗𝜔 + 1
Response Frequency Response: 𝐷 𝑗𝜔 = 𝐾
𝛼𝑇𝑗𝜔 + 1
2) Bode Plot
3) System Stability
0
1 + 𝜔𝑇
4) Closed-loop 𝑀 = 𝐷 𝑗𝜔 = 𝐾 and 𝜙 = ∠𝐾 + ∠ 1 + 𝑗𝜔𝑇 − ∠ 1 + 𝑗𝛼𝜔𝑇
Frequency 1 + 𝛼𝜔𝑇
Response = 𝑡𝑎𝑛 𝜔𝑇 − 𝑡𝑎𝑛 𝛼𝜔𝑇
5) Design for
𝑀→ 𝐾
𝜔→0
Dynamic 𝜙→0
Compensation 𝐾 = 1; 𝛼 = 0.1; 𝑇 = 1
6) Case Study
𝑀→ 𝐾 𝛼

db
𝜔→∞
𝜙→0
s = tf (‘s’);
K = 1;
T = 1;
alpha = 0.1;
sysD = K*((T*s+1)/(alpha*T*s+1));
w = logspace(-1 , 2);
[mag , phase] = bode (sysD , w)
loglog (w , squeeze(mag)); grid;
axis ([0.1 100 1 10]);
semilogx (w , squeeze(phase)); grid;
axis ([0.1 100 0 60]);
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 Frequency Response
 The frequency response, i.e., 𝑀 𝜔 and 𝜙 𝜔 , of a (linear, time-invariant) system
can convey information about the system dynamics and its response to any input.
1) Frequency
Response 𝑌 𝑠 𝐾𝐺 𝑠 𝑅 𝑠 + 𝑈 𝑠 𝑌 𝑠
=𝒯 𝑠 = 𝐾 𝐺 𝑠
2) Bode Plot 𝑅 𝑠 1 + 𝐾𝐺 𝑠 _
3) System Stability Bandwidth (𝜔 ):
4) Closed-loop
Frequency  The maximum frequency at which the
Response output of the system will track a
5) Design for sinusoid input in a satisfactory manner.

db
Dynamic
Compensation  In most control systems, the
6) Case Study satisfactory manner is before when the
output is attenuated to a factor of 0.707
times the input (half-power point).
 A higher bandwidth corresponds to a faster
response (shorter rise and peak times).
Resonant Peak (𝑀 ):
 The maximum value of the frequency-response magnitude ratio.
 The overshot of system’s response can be estimated from resonant peak.
Typical Behaviour of Closed-loop Systems (Low-pass Filter Behaviour):
 The output follows the input ( 𝒯 𝑠 ≅ 1) at the lower excitation frequencies.
 The output ceases to follow the input ( 𝒯 𝑠 < 1) at the higher excitation
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 M.R. Emami, 2025 frequencies.
 Frequency Response
Example: Second–order System
1
𝒯 𝑠 =
𝑠 ⁄𝜔 + 2𝜁 𝑠⁄𝜔 + 1
1) Frequency  The damping of the system can be determined Step Response
Response
from peak overshoot in time domain or from
2) Bode Plot
resonant peak in frequency domain. So, the
3) System Stability
peak overshoot in time domain can be
4) Closed-loop
Frequency estimated from resonant peak in frequency
Response domain (which is about 1⁄ 2𝜁 for 𝜁 < 0.5).
5) Design for
Dynamic  Natural frequency 𝜔 in time domain is typically
Compensation near the bandwidth 𝜔 (within a factor of 2) in
6) Case Study frequency domain (they are equal for 𝜁 = 0.707).
So, the rise time can be estimated from the
bandwidth.
Frequency Response

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 Bode Plot
 A pair of graphs representing the frequency response of a system 𝐺 𝑠 ,
i.e., gain 𝐺 𝑗𝜔 and phase ∠𝐺 𝑗𝜔 , as a function of frequency 𝜔.
1) Frequency
Response  The phase plot is a semi-log graph showing 𝜙 (in degrees)
2) Bode Plot
versus 𝜔 (in 𝑟𝑎𝑑 ⁄𝑠).
3) System Stability  The gain plot is usually a log-log graph, since it turns it more convenient
Hendrick Wade Bode
4) Closed-loop to “sum up” different parts of a composite transfer function, e.g.,
Frequency
(1905-1982)
Response 𝐺 𝑠 𝐺 𝑠 𝐺 𝑗𝜔 𝐺 𝑗𝜔
5) Design for
𝐺 𝑠 = 𝐺 𝑗𝜔 =
𝐺 𝑠 𝐺 𝑠 𝐺 𝑗𝜔 𝐺 𝑗𝜔
Dynamic
Compensation
6) Case Study
log 𝐺 𝑗𝜔 = log 𝐺 𝑗𝜔 + log 𝐺 𝑗𝜔 − log 𝐺 𝑗𝜔 − log 𝐺 𝑗𝜔

Also, 𝜙 and the log scale of 𝐺 𝑗𝜔 together show both real and imaginary parts of the
transfer function:
log 𝐺 𝑗𝜔 = log 𝐺 𝑗𝜔 𝑒 = log 𝐺 𝑗𝜔 + 𝑗𝜙 log 𝑒

 Traditionally, the system gain is sometimes expressed in “decibel” units:

𝐺 𝑗𝜔 = 20 log 𝐺 𝑗𝜔
If the gain is presented in db (db graph), then the gain axis will be linear (not logarithmic).

 In this course, the gain plot is often a log-log graph showing 𝐺 versus 𝜔, but the decibel
scale is also shown on the right-hand-side of the graph. (The base 10 is omitted for
convenience.)
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 Bode Plot
 To draw the Bode plot, first turn the transfer function from the regular form:
𝑠+𝑧 𝑠 + 𝑧 ⋯ 𝑠 + 2𝜁 𝜔 𝑠 + 𝜔 ⋯
𝐺 𝑠 =𝐾
1) Frequency 𝑠+𝑝 𝑠 + 𝑝 ⋯ 𝑠 + 2𝜁 𝜔 𝑠 + 𝜔 ⋯
Response
2) Bode Plot into the Bode form:
3) System Stability 𝜏 𝑠 + 1 (𝜏 𝑠+1) ⋯ 𝑠 𝜔 + 2𝜁 𝑠 𝜔 +1 ⋯
4) Closed-loop 𝐺 𝑠 =𝐾 𝑠
Frequency 𝜏 𝑠+1 𝜏 𝑠+1 ⋯ 𝑠 𝜔 + 2𝜁 𝑠 𝜔 +1 ⋯
Response
5) Design for
Dynamic 𝑗𝜔 𝑗𝜔
𝑗𝜔𝜏 + 1 𝑗𝜔𝜏 + 1 ⋯ 𝜔 + 2𝜁 𝜔 +1 ⋯
Compensation
𝐺 𝑗𝜔 = 𝐾 𝑗𝜔
6) Case Study 𝑗𝜔 𝑗𝜔
𝑗𝜔𝜏 + 1 𝑗𝜔𝜏 + 1 ⋯ 𝜔 + 2𝜁 𝜔 +1 ⋯

 The Bode form can be a composite of any of the 3 classes:

Class 1: 𝐾 𝑗𝜔 , 𝑛 = ⋯ , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ⋯

Class 2: 𝑗𝜔𝜏 + 1 or 𝑗𝜔𝜏 + 1

Class 3: + 2𝜁 +1 or + 2𝜁 +1

 The Bode plot for each class can be considered separately, and then “added
up” to form the composite Bode plot of the transfer function 𝐺 𝑗𝜔 .
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 Bode Plot
Class 1: 𝐾 𝑗𝜔 , 𝑛 = ⋯ , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ⋯
log 𝐾 𝑗𝜔 = log 𝐾 + 𝑛 log 𝑗𝜔 = log 𝐾 + 𝑛 log 𝜔
1) Frequency
Response  The gain (magnitude) plot is a straight line with a slope of 𝑛 in the log-log
2) Bode Plot graph (or 𝑛 × 20 𝑑𝑏⁄𝑑𝑒𝑐𝑎𝑑𝑒 in db graph).
3) System Stability 𝐾 =1
 In the gain plot, locate 𝐾 at the
4) Closed-loop
Frequency frequency 𝜔 = 1, and then draw a
Response line with a slope 𝑛
5) Design for (or 𝑛 × 20 𝑑𝑏⁄𝑑𝑒𝑐𝑎𝑑𝑒 passing
Dynamic
Compensation 𝐾 at 𝜔 = 1.
6) Case Study
 This is the only class affecting the

𝐾
Bode Plot’s slope at lowest
frequencies, since the other two
classes are nearly constant at very
low frequencies.
𝑛 = +1
90
 The phase of 𝑗𝜔 is 𝜙 = 𝑛 × 90°
and independent from 𝜔, thus a 0
𝑛=0
horizontal line. So, in the phase
𝑛 = −1
plot for 𝑛 = −1, −90° ; 𝑛 = −2, -90

−180° ; 𝑛 = +1, 90° ; 𝑛 = +2, 180° ; 𝑛 = −2

etc. -180

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 M.R. Emami, 2025
 Bode Plot
Class 2: 𝑗𝜔𝜏 + 1 or 𝑗𝜔𝜏 + 1
 The magnitude approaches one asymptote at very low frequencies and another
asymptote at very high frequencies:
1) Frequency
Response
o For 𝜔𝜏 ≪ 1, 𝑗𝜔𝜏 + 1 ± ≅ 1. o For 𝜔𝜏 ≫ 1, 𝑗𝜔𝜏 + 1 ± ≅ 𝑗𝜔𝜏 ± .
2) Bode Plot  Calling 𝜔 = 1⁄𝜏 the break point, the gain plot is approximately constant (=1)
3) System Stability below the break point; and above the break point it behaves approximately like
4) Closed-loop Class 1 term 𝐾 𝑗𝜔 ± .
Frequency
Response
 Actual magnitude curve is a factor of 1.4
+1 power
(+3 db) at (above) break point for
5) Design for 𝜏 = 10
Dynamic 𝑗𝜔𝜏 + 1 and a factor of 0.707 (-3 db)
Compensation

db
at (below) break point for 𝑗𝜔𝜏 + 1 .
6) Case Study
 For the phase graph:
o For 𝜔𝜏 ≪ 1, ∠1 = 0° .
o For 𝜔𝜏 ≫ 1, ∠ 𝑗𝜔𝜏 ± = ±90° .
o For 𝜔𝜏 ≅ 1, ∠ 𝑗𝜔𝜏 + 1 ± ≅ ±45° .
 Middle asymptote connects 5 times below
the break point to 5 times above. +1 power

 Actual phase curve deviates from 𝜏 = 10

asymptotes by ~11° at their intersections.
 Both composite magnitude and phase
curves are unaffected by this class at low
𝜔 below the break point (by more than a
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 M.R. Emami, 2025 factor of 10), since its magnitude is 1 and its phase is < 5° .
 Bode Plot
Class 3: + 2𝜁 +1 or + 2𝜁 +1

 The behaviour is similar to Class 2. But, the break point for this Class is 𝜔 = 𝜔 ,
1) Frequency
Response and the slope of the asymptote for high frequencies is +2 (+40 𝑑𝑏⁄𝑑𝑒𝑐𝑎𝑑𝑒 ) for
2) Bode Plot +1 power and −2 (−40 𝑑𝑏⁄𝑑𝑒𝑐𝑎𝑑𝑒) for -1 power.
3) System Stability  The transition of the magnitude curve through
4) Closed-loop the break point region mainly depends on the
Frequency
Response damping ratio 𝜁. The magnitude curve is a
5) Design for factor of 2𝜁 times (for power +1) or

db
Dynamic 2𝜁 times (for power -1) at the break point.
Compensation
-1 power
6) Case Study  The peak of the magnitude curve is:
1
𝐺 𝑗𝜔 = at 𝜔 = 𝜔 1 − 2𝜁
2𝜁 1 − 𝜁
 For the phase graph, similar to Class 2:
o For 𝜔 ≪ 𝜔 , ∠1 = 0° .
o For 𝜔 ≫ 𝜔 , ∠ 𝑗𝜔 ± = ±180° .
o For 𝜔 ≅ 𝜔 , ∠ ± 𝑗2𝜁 ≅ ±90° .
 The transition of the phase curve through the
break point is between the two extremes for -1 power
𝜁 = 0, where the phase curve is a step function
jumping from 0 to ±180° at 𝜔 = 𝜔 , and 𝜁 = 1,
where 𝐺 𝑗𝜔 is a composite of two Class 2
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 M.R. Emami, 2025 terms with the same break point frequency.
 Bode Plot
Composite Graph
 Manipulate the transfer function into the Bode form, and identify all the break point frequencies.

1) Frequency  Determine the value of 𝑛 for Class-1 terms. Plot the low-frequency magnitude asymptote
Response through the point 𝐾 at 𝜔 = 1 with a slope of 𝑛 (𝑛 × 20 𝑑𝑏⁄𝑑𝑒𝑐𝑎𝑑𝑒) .
2) Bode Plot
3) System Stability  Complete the composite magnitude asymptotes: Extend the low-frequency asymptote until the
first break point frequency. Then, step the slope by ±1 or ±2, depending on whether the break
4) Closed-loop
Frequency
point is from a Class-2 or Class-3 term in the numerator or denominator. Continue through all
Response break points in ascending order.
5) Design for
Dynamic  The approximate magnitude curve is increased from the asymptote value by a factor of
Compensation 1.4 (+3 𝑑𝑏) at Class-2 numerator break points, and decreased by a factor of 0.707 (−3 𝑑𝑏) at
6) Case Study Class-2 denominator break points. At Class-3 break points, the resonant peak (or valley) occurs,
a magnitude ratio of 𝐺 𝑗𝜔 = 2𝜁 below the numerator break point and 𝐺 𝑗𝜔 = 2𝜁
above the denominator break point. (These values may change if the difference between the
adjacent break points is less than one decade.)

 Plot the low-frequency asymptote of the phase curve, 𝜙 = 𝑛 × 90° .

 As a guide, the approximate phase curve changes by ±90° or ± 180° at each break point in
ascending order. For Class-2 terms in the numerator, the change of phase is +90°; for those in
the denominator the change is -90°. For Class-3 terms the change is ±180°.

 Locate the asymptotes for each individual phase curve so their phase change corresponds to the
steps in the phase toward or away from the approximate curve indicated by Step 6.

 Graphically, add each phase curve. Keep in mind that the curve will start at the lowest-frequency
asymptote, and end on the highest frequency asymptote, and will approach the intermediate
 M.R. Emami, 2025 asymptotes to an extent that is determined by how close the break points are to each other. 14
 Bode Plot
2000 𝑠 + 0.5
Example: 𝐺 𝑠 =
𝑠 𝑠 + 10 𝑠 + 50 0.5 𝑟𝑎𝑑⁄𝑠
2 𝑗𝜔⁄0.5 + 1
1) Frequency
 Bode form: 𝐺 𝑗𝜔 =
𝑗𝜔 𝑗𝜔⁄10 + 1 𝑗𝜔⁄50 + 1
; Break Points: 10 𝑟𝑎𝑑 ⁄𝑠
Response 50 𝑟𝑎𝑑⁄𝑠
First
2) Bode Plot  Class-1 Term: 2 𝑗𝜔
Asymptote
3) System Stability
4) Closed-loop
Frequency  3 Class-2 Terms 3 Break
Points
Response
5) Design for Step the asymptote slope at 𝜔 = 0.5 by +1,
Dynamic then at 𝜔 = 10 by −1, then at 𝜔 = 50 by −1.
Compensation
6) Case Study  Draw the magnitude curve with a factor 1.4
above the first break point, and 0.707 below
the second and third break points.

 Low-frequency asymptote of phase curve 𝜙 = −90° .

𝑠⁄0.5 + 1

 Step the phase curve asymptote by 90°

at 𝜔 = 0.5, and −90° at 𝜔 = 10 and 50.

 Draw the phase curve for individual

terms.

 Graphically add the individual

phase curves at each value of 𝜔 to
obtain the composite phase graph.
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 Bode Plot
10
Example: 𝐺 𝑠 =
𝑠 𝑠 + 0.4𝑠 + 4
2.5
1) Frequency
Response
 Bode form: 𝐺 𝑗𝜔 =
𝑗𝜔 𝑗𝜔⁄2 + 2 0.1 𝑗𝜔⁄2 + 1
; Break Point: 2 𝑟𝑎𝑑⁄𝑠

2) Bode Plot  Class-1 Term: 2.5 𝑗𝜔

3) System Stability
4) Closed-loop
Frequency  Class-3 Term: denominator, 𝜔 = 2 , 𝜁 = 0.1
Response
First asymptote with −1 slope intersecting
5) Design for
Dynamic
𝐺 = 2.5 , 𝜔 = 1. Step the asymptote slope
Compensation at 𝜔 = 2 by −2.
6) Case Study
 Draw the magnitude curve with a factor of
1⁄ 2 × 0.1 = 5 above the break point.

 Low-frequency asymptote of phase curve

𝜙 = −90° .

 Step the phase curve asymptote by −180°

at 𝜔 = 2.

 Draw the phase curve for individual

terms.

 Graphically add the individual

phase curves at each value of 𝜔 to
obtain the composite phase graph.
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 Bode Plot
Example: Satellite Attitude Control (Collocated)
Θ 𝑠 0.01 𝑠 + 0.01𝑠 + 1
𝐺 𝑠 = =
1) Frequency 𝑇 𝑠 𝑠 𝑠 ⁄4 + 0.02 𝑠⁄2 + 1
Response
2) Bode Plot  One Class-1 term (second-order, denominator).
3) System Stability  One Class-3 term in numerator (𝜔 = 1 , 𝜁 = 0.005).
4) Closed-loop
Frequency  One Class-3 term in denominator
Response
(𝜔 = 2 , 𝜁 = 0.01).
5) Design for
Dynamic  Magnitude ratio for 𝜔 = 1 is
Compensation
2 × 0.005 = 0.01 below break point.
6) Case Study
 Magnitude ratio for 𝜔 = 2 is
1⁄ 2 × 0.01 = 50 above break point.
 The phase starts at −180° , then changes
for 180° to 0° at 𝜔 = 1, and changes for
−180° at 𝜔 = 2.
 Matlab Code:
s = tf (‘s’);
sysG = 0.01*(s^2+0.01*s+1) /
((s^2)*((s^2)/4+0.02*(s/2)+1));
w = logspace(0 , 1);
[mag , phase] = bode (sysG , w);
loglog (w , squeeze(mag)); grid;
semilogx (w , squeeze(phase)); grid;
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 Bode Plot
Nonminimum Phase System
Minimum 𝑠+1
𝐺 𝑠 = 10
Phase System 𝑠 + 10
1) Frequency
Response
Nonminimum 𝐺 𝑠 = 10 𝑠 − 1
2) Bode Plot
Phase System 𝑠 + 10
3) System Stability
4) Closed-loop  A zero in the RHP.
Frequency
Response  The above two systems have the
5) Design for same magnitude curve:
Dynamic
Compensation 𝐺 𝑗𝜔 = 𝐺 𝑗𝜔
6) Case Study  For a zero in RHP, the phase
decreases at its break point (instead
of the usual phase increase for a zero
in LHP).
 Minimum phase system (all zeros in
the LHP) has the smallest net change
in the associated phase compared to
its nonminimum phase counterpart.
 The more zeros in the RHP for the
nonminimum phase system, the greater
change in its phase curve compared to its
minimum phase counterpart.
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 System Stability 𝐺 𝑠
Neutral Stability
Open-loop TF: 𝐿 𝑠 = 𝐾𝐺 𝑠
1) Frequency
𝐾𝐺 𝑠 = 1
Response
Root-locus Conditions:
∠𝐺 𝑠 = 180° 𝑜𝑟 − 180°
2) Bode Plot
3) System Stability Onset of Stability: Root locus intersects the imaginary axis.
4) Closed-loop
Frequency 𝐾=2 𝑠 = 𝑗1.0 = 𝑗𝜔 Neutral Stability
Response
 The Bode plot of a system that is neutrally stable (i.e., 𝐾 has
5) Design for
Dynamic
a value so that closed-loop root falls on imaginary axis) will
Compensation satisfy:
6) Case Study 𝐾𝐺 𝑗𝜔 = 1 and ∠𝐺 𝑗𝜔 = −180°
 For most systems, decreasing 𝐾 from the value corresponding
to neutral stability will move the system toward further stability,
and increasing 𝐾 beyond the value will push the system
toward instability.
 In such cases, by looking at the Bode plot of the open-loop
𝐾𝐺 𝑠 frequency response, if the following condition holds,
the closed-loop system is stable:
𝐾𝐺 𝑗𝜔 < 1 at ∠𝐺 𝑗𝜔 = −180°
 The above stability condition may not hold for closed-loop
systems where in their open-loop Bode plot, 𝐾𝐺 𝑗𝜔 crosses
the unity more than once.

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 System Stability 𝐺 𝑠
Stability Margins
Closed-loop System Stability Condition (common case):
1) Frequency 𝐾𝐺 𝑗𝜔 < 1 at ∠𝐺 𝑗𝜔 = −180°
Response
2) Bode Plot  Two quantities measure the extent of system stability:
3) System Stability Gain Margin (GM) and Phase Margin (PM) .
4) Closed-loop
Frequency
Gain Margin (GM):
Response  The factor by which the gain can be increased before instability
5) Design for results. On Bode plot, the GM is the inverse of 𝐾𝐺 𝑗𝜔 value
Dynamic where ∠𝐺 𝑗𝜔 = −180° (or on a db scale, the vertical distance
Compensation
between 𝐾𝐺 𝑗𝜔 value where ∠𝐺 𝑗𝜔 = −180° and the
6) Case Study
magnitude = 0 line).
Example:
∠𝐺 𝑗𝜔 = −180°
𝐾 = 0.1 𝐾𝐺 𝑗𝜔 = 0.05 𝐺𝑀 = 20 26 𝑑𝑏 Stable
°
∠𝐺 𝑗𝜔 = −180 Neutrally
𝐾=2 𝐾𝐺 𝑗𝜔 =1 𝐺𝑀 = 1 0 𝑑𝑏
Stable
∠𝐺 𝑗𝜔 = −180°
𝐾 = 10 𝐾𝐺 𝑗𝜔 =5 𝐺𝑀 = 0.2 −14 𝑑𝑏 Unstable
 GM is the factor by which gain 𝐾 can be increased before instability
results. If 𝐺𝑀 < 1 0 𝑑𝑏 , system is unstable.
 From the root locus, GM is the ratio of 𝐾 values at where the
locus crosses 𝑗𝜔-axis and where the nominal closed-loop
poles are located on the locus.
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 System Stability 𝐺 𝑠
Stability Margins
Closed-loop System Stability Condition (common case):
1) Frequency 𝐾𝐺 𝑗𝜔 < 1 at ∠𝐺 𝑗𝜔 = −180°
Response
2) Bode Plot  Two quantities measure the extent of system stability:
3) System Stability
Gain Margin (GM) and Phase Margin (PM) .
4) Closed-loop Phase Margin (PM):
Frequency
Response  The amount by which the phase of 𝐺 𝑗𝜔 exceeds −180° when
5) Design for 𝐾𝐺 𝑗𝜔 = 1. This is the degree to which the above stability
Dynamic condition is met.
Compensation
6) Case Study
Example:
𝐾𝐺 𝑗𝜔 =1
𝐾 = 0.1 ∠𝐺 𝑗𝜔 = −100° 𝑃𝑀 = 80° Stable
𝐾𝐺 𝑗𝜔 =1 Neutrally
𝐾=2 ∠𝐺 𝑗𝜔 = −180° 𝑃𝑀 = 0°
Stable
𝐾𝐺 𝑗𝜔 =1
𝐾 = 10 ∠𝐺 𝑗𝜔 = −215° 𝑃𝑀 = −35° Unstable

 A positive PM is required for stability.

(Gain) Crossover Frequency (𝜔 ):

 The frequency at which the (open-loop) magnitude value on
Bode plot is unity. The crossover frequency of an open-loop
frequency response is highly correlated with the closed-loop
system bandwidth, hence its response speed.
𝑃𝑀 = ∠𝐿 𝑗𝜔 − −180° (𝐿 𝑠 is open-loop TF.) 21
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 System Stability
Stability Margins 𝑅 𝑠 + 𝑈 𝑠 𝑌 𝑠
𝐾 𝐺 𝑠
_
 The PM for any value of gain 𝐾 can be obtained directly
1) Frequency from the Bode plot for 𝐺 𝑗𝜔 , i.e., 𝐾 = 1, without any need
Response to draw the plot for different gains.
2) Bode Plot
 For any given 𝐾, on the 𝐺 𝑗𝜔 plot identify where
3) System Stability 𝐺 𝑗𝜔 = 1⁄𝐾 . Then, find the corresponding phase on the
4) Closed-loop ∠𝐺 𝑗𝜔 plot. The difference between this phase and −180°
Frequency is the PM for 𝐾𝐺 𝑗𝜔 ,
Response
5) Design for Example:
Dynamic
Compensation 𝐾=5 𝐺 𝑗𝜔 = 0.2 ∠𝐺 𝑗𝜔 = −202° 𝑃𝑀 = −22°Unstable
6) Case Study 𝑃𝑀 = 45°
𝐾 = 0.5 𝐺 𝑗𝜔 =2 ∠𝐺 𝑗𝜔 = −135°
1
𝑃𝑀 = 70° 𝐺 𝑗𝜔 =5= 𝐾 = 0.2
0.2

 PM is more commonly used (than GM) for closed-loop system

frequency response analysis, because:
 For a typical 2nd-order system, 𝐺𝑀 = ∞, since the phase
curve reaches −180° at infinite frequency.

 PM is closely related to the damping ratio of the system.

 The value of 𝑃𝑀 = 30 is often regarded as the lowest

adequate value for a safe margin of system stability.
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 System Stability
Stability Margins
Example: Second-order System
𝜔 𝜔
1) Frequency Open-loop TF: 𝐺 𝑠 = Closed-loop TF: 𝒯 𝑠 =
Response 𝑠 𝑠 + 2𝜁𝜔 𝑠 + 2𝜁𝜔 𝑠 + 𝜔
2) Bode Plot  For such a system, the PM is a function of damping ratio:
3) System Stability 2𝜁 𝑃𝑀 < 65°

∠𝒯 𝑠
4) Closed-loop 𝑃𝑀° = tan 𝑃𝑀 °
Frequency 𝜁≅
Response 1 + 4𝜁 − 2𝜁 100
5) Design for  The approximate relation is often used as a rule of thumb for
Dynamic
Compensation other systems, as well.
6) Case Study  The relation between PM and resonant peak 𝑀 and step-response overshoot 𝑀 can be
determined from the graph, primarily for 2nd-order closed-loop systems, but also as a rough
estimate for other systems.

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