Today’s goals
•
•
Monday
– Proof that the frequency response as function of frequency ω is simply the value
of the transfer function at s=jω
– Bode plots: amplitude and phase of the frequency response on a log-log plot
– Bode plots for elementary 1st order systems:
derivative; integrator; zero; pole
Today
– Frequency response and Bode plots of underdamped 2nd order systems
– Cascading sub-systems: rules for Bode plots of systems with
multiple poles and zeros
2.004 Fall ’07
Lecture 31 – Wednesday, Nov. 21
Elementary Bode plots: 1st order
Normalized and scaled
Bode plots for
a. G(s) = s;
b. G(s) = 1/s;
c. G(s) = (s + a);
d. G(s) = 1/(s + a)
Images removed due to copyright restrictons.
Please see: Fig. 10.9 in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, 2004.
2.004 Fall ’07
Lecture 31 – Wednesday, Nov. 21
Bode plot for underdamped 2nd order system
20
10
1
G(s) = 2
s + 2ζωn s + ωn2
0
20 log (Mω 2n)
1
G(jω) = 2
(ωn − ω2 ) + j2ζωn ω
Low-frequency
asymptote
ζ = 0.1
0.2
0.3
ζ = 0.5
1.0
0.7
-10
1.5
High-frequency
asymptote
-20
-30
Note: the Bode magnitude at ω = ωn is
-40
−20log2ζ.
-50
0.1
1
ω/ωn
This can be used as correction to the asymptotic plot.
10
Figure 10.16
0
-20
1.5
Phase (degrees)
-40
-60
ζ = 0.1
0.2
0.5 0.3
0.7
1.0
-80
-100
-120
1.0
0.7
0.5
0.3
ζ = 0.1
0.2
-140
-160
-180
0.1
Figure by MIT OpenCourseWare.
2.004 Fall ’07
Lecture 31 – Wednesday, Nov. 21
1
ω/ωn
Asymptote
1.5
10
Figure 10.17
Cascading 1st order subsystems
3
K(s + 3)
=
G(s) =
s(s + 1)(s + 2)
Magnitude plot
2
K
Ã
s(s + 1)
s
3
Ã
!
+1
s
+1
2
!
Image removed due to copyright restrictons.
Please see: Fig. 10.11 in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, 2004.
2.004 Fall ’07
Lecture 31 – Wednesday, Nov. 21
Cascading 1st order subsystems
3
K(s + 3)
=
G(s) =
s(s + 1)(s + 2)
Phase plot
K
2
Ã
s(s + 1)
s
3
Ã
!
+1
s
+1
2
!
Image removed due to copyright restrictons.
Please see: Fig. 10.12 in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, 2004.
2.004 Fall ’07
Lecture 31 – Wednesday, Nov. 21
Cascading 1st and 2nd order subsystems
!
Ã
Magnitude plot
3
K(s + 3)
Ã
=
G(s) =
(s + 2)(s2 + 2s + 25)
50 s
K
+1
2
s
+1
! Ã3
s2
25
+
2s
25
+1
!
Image removed due to copyright restrictons.
Please see: Fig. 10.18 in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, 2004.
2.004 Fall ’07
Lecture 31 – Wednesday, Nov. 21
Cascading 1st and 2nd order subsystems
!
Ã
Phase plot
3
K(s + 3)
Ã
=
G(s) =
(s + 2)(s2 + 2s + 25)
50 s
2
K
+1
s
+1
! Ã3
s2
25
+
2s
25
+1
!
Image removed due to copyright restrictons.
Please see: Fig. 10.19 in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, 2004.
2.004 Fall ’07
Lecture 31 – Wednesday, Nov. 21