6020 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD
10-1 100101102
ω (rad/sec)
Bode plot for Prob. 6.5 (c)
10-1 100101102
-150
ω (rad/sec)
10-1 100101102
ω (rad/sec)
Bode plot for Prob. 6.5 (c)
10-1 100101102
-150
ω (rad/sec)
6021
10-1 100101
Frequency (rad/sec)
10-1 100101
-150
150
Frequency (rad/sec)
6022 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD
10-1 100101
Frequency (rad/sec)
Magnitude
Bode Plot for Prob 6.5(e)
10-1 100101
-300
0
Frequency (rad/sec)
6. Multiple poles at the origin Sketch the asymptotes of the Bode plot mag-
nitude and phase for each of the following open-loop transfer functions.
After completing the hand sketches verify your result using MATLAB.
Turn in your hand sketches and the MATLAB results on the same scales.
(a) L(s) = 1
s2(s+ 10)
(b) L(s) = 1
s3(s+ 8)
(c) L(s) = 1
s4(s+ 10)
(d) L(s) = (s+ 3)
s2(s+ 10)
(e) L(s) = (s+ 3)
s3(s+ 5)
(f) L(s) = (s+ 1)2
s3(s+ 10)
6023
10-1 100101102
10-5
100
ω (rad/sec)
Magnitude
Bode Plot for 6.6(a)
10-1 100101102
-300
-250
-200
-150
ω (rad/sec)
Phase (deg)
(g) L(s) = (s+ 1)2
s3(s+ 10)2
Solution:
1
10
6024 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD
10-1 100101102
ω (rad/sec)
10-1 100101102
-400
Frequency (rad/sec)
6025
10-1 100101102
ω (rad/sec)
Bode Plot for 6.6(c)
10-1 100101102
-100
ω (rad/sec)
Phase + 360 (deg)
6026 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD
10-1 100101102
10-4
ω (rad/sec)
Bode Plot for 6.6(d)
10-1 100101102
-200
-50
ω(rad/sec)
6027
10-1 100101102
ω (rad/sec)
Bode Plot for 6.6(e)
10-1 100101102
-300
-150
ω (rad/sec)
6028 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD
105
Bode Plot for 6.6(f)
10-2 10-1 100101102
-300
-50
Frequency (rad/sec)
6029
10-2 10-1 100101102
10-5
ω(rad/sec)
Bode plot for 6.6(g)
10-2 10-1 100101102
-300
-50
ω (rad/sec)
7. Mixed real and complex poles Sketch the asymptotes of the Bode plot mag-
nitude and phase for each of the following open-loop transfer functions. Af-
ter completing the hand sketches verify your result using Matlab. Turn
in your hand sketches and the Matlab results on the same scales.
(a) L(s) = (s+ 2)
s(s+ 10)(s2+ 2s+ 2)
(b) L(s) = (s+ 2)
s2(s+ 10)(s2+ 6s+ 25)
(c) L(s) = (s+ 2)2
s2(s+ 10)(s2+ 6s+ 25)
(d) L(s) = (s+ 2)(s2+ 4s+ 68)
s2(s+ 10)(s2+ 4s+ 85)
6030 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD
(e) L(s) = [(s+ 1)2+ 1]
s2(s+ 2)(s+ 3)
Solution:
10-5
Bode plot for Prob. 6.7 (a)
10-5
Bode plot for Prob. 6.7 (a)
10-2 10-1 100101102
-300
ω (rad/sec)
10-5
Bode plot for Prob. 6.7 (a)
10-5
Bode plot for Prob. 6.7 (a)
10-2 10-1 100101102
-300
ω (rad/sec)
10-2 10-1 100101102
10-5
ω (rad/sec)
Magnitude
10-2 10-1 100101102
10-5
ω (rad/sec)
Magnitude
10-2 10-1 100101102
-400
-200
ω (rad/sec)
10-2 10-1 100101102
10-5
ω (rad/sec)
Magnitude
10-2 10-1 100101102
10-5
ω (rad/sec)
Magnitude
10-2 10-1 100101102
-400
-200
ω (rad/sec)
10-2 10-1 100101102
10-2 10-1 100101102
10-2 10-1 100101102
-300
-100
ω (rad/sec)
10-2 10-1 100101102
10-2 10-1 100101102
10-2 10-1 100101102
-300
-100
ω (rad/sec)
6033
10-2 10-1 100101102
Magnitude
Bode plot for Prob. 6.7 (d)
10-2 10-1 100101102
Magnitude
Bode plot for Prob. 6.7 (d)
10-2 10-1 100101102
-100
ω (rad/sec)
Phase (deg)
10-2 10-1 100101102
Magnitude
Bode plot for Prob. 6.7 (d)
10-2 10-1 100101102
Magnitude
Bode plot for Prob. 6.7 (d)
10-2 10-1 100101102
-100
ω (rad/sec)
Phase (deg)
6034 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD
10-2 10-1 100101102
10-5
ω (rad/sec)
Magnitude
10-2 10-1 100101102
10-5
ω (rad/sec)
Magnitude
-200
10-2 10-1 100101102
10-5
ω (rad/sec)
Magnitude
10-2 10-1 100101102
10-5
ω (rad/sec)
Magnitude
-200
8. Right half plane poles and zeros Sketch the asymptotes of the Bode plot
magnitude and phase for each of the following open-loop transfer functions.
Make sure the phase asymptotes properly take the RHP singularity into
account by sketching the complex plane to see how the \L(s)changes as
sgoes from 0 to +j1:After completing the hand sketches verify your
(c) L(s) = s1
s2
(d) L(s) = s2+ 2s+ 1
s(s+ 20)2(s22s+ 2)
6035
(e) L(s) = (s+ 2)
s(s1)(s+ 6)2
(f) L(s) = 1
(s1)[(s+ 2)2+ 3]
Solution:
10-5
Bode plot for Prob. 6.8 (a)
10-5
Bode plot for Prob. 6.8 (a)
10-1 100101102
-200
-100
ω (rad/sec)
10-5
Bode plot for Prob. 6.8 (a)
10-5
Bode plot for Prob. 6.8 (a)
10-1 100101102
-200
-100
ω (rad/sec)
6036 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD
10-2 10-1 100101102
ω (rad/sec)
Bode plot for Prob. 6.8 (b)
10-2 10-1 100101102
-300
-150
ω (rad/sec)
Phase (deg)
10-2 10-1 100101102
ω (rad/sec)
Bode plot for Prob. 6.8 (b)
10-2 10-1 100101102
-300
-150
ω (rad/sec)
Phase (deg)
6037
10-2 10-1 100101
105
Bode plot for Prob. 6.8 (c)
10-2 10-1 100101
105
Bode plot for Prob. 6.8 (c)
10-2 10-1 100101
-100
ω (rad/sec)
10-2 10-1 100101
105
Bode plot for Prob. 6.8 (c)
10-2 10-1 100101
105
Bode plot for Prob. 6.8 (c)
10-2 10-1 100101
-100
ω (rad/sec)
6038 CHAPTER 6. THE FREQUENCY-RESPONSE DESIGN METHOD
10-2 10-1 100101102
10-5
100
ω (rad/sec)
Bode plot for Prob. 6.8 (d)
10-2 10-1 100101102
ω (rad/sec)
10-2 10-1 100101102
10-5
100
ω (rad/sec)
Bode plot for Prob. 6.8 (d)
10-2 10-1 100101102
ω (rad/sec)
6039
10-2 10-1 100101102
10-5
ω (rad/sec)
Magnitude
Bode plot for Prob. 6.8 (e)
10-2 10-1 100101102
10-5
ω (rad/sec)
Magnitude
Bode plot for Prob. 6.8 (e)
10-2 10-1 100101102
-300
ω (rad/sec)
10-2 10-1 100101102
10-5
ω (rad/sec)
Magnitude
Bode plot for Prob. 6.8 (e)
10-2 10-1 100101102
10-5
ω (rad/sec)
Magnitude
Bode plot for Prob. 6.8 (e)
10-2 10-1 100101102
-300
ω (rad/sec)