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8/7/2019 Chapter 11 Part I
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Chapter 11 Frequency-domain Analysis and
Design of Control System
r. . u
Department of Mechanical Engineering
Universit of Houston
Houston, TX 77204-4006
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Contents
Bode diagram
Nyquist stability criterion
-
control system
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11-2 Bode diagram representation of the frequencyresponse
A bode diagram consists of two graphs:Ma nitude in decibel dB versus fre uenc
Phase angle versus frequency.
.
The stand representation of magnitude of G(jW) is
)(log20 jG
where the base of logarithm is 10
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Advantage to use decibel
baba log20log20log20 +=
1
R(s)C(s)
1+s.
20
30
40
de(dB)
Bode Diagram
-20
-10
0
ude(dB)
Bode Diagram
-10
-5
0
tude(dB)
Bode Diagram
+ 010
Magnitu
45
90
(deg
)
-40
-30Magnit
-45
0
se(de
g)
-20
-15Magni
-30
0
e(deg)
=
10-1
100
101
102
103
0
Phas
Fre uenc rad/sec
10-2
10-1
100
101
102
-90
Pha
Fre uenc rad/sec10
-210
-110
010
110
210
3-60
Phas
Fre uenc rad/sec
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Bode Diagram
K=3
9.5
10
10.5
11
itude(dB)
8.5
9
.
M
ag
1
-0.5
0
.
Phase(deg)
100
101
-
Frequency (rad/sec)
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Bode diagram of integral
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Bode diagram of derivative
dBdB )20log20()10log20( +=
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(starting from 0 dB)
(Slope of -20 dB/dec)
Corner frequency : 1=
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-20 dB/dec
T
1
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-20dB/dec
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+
1
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0o
-45o
-90o
T
1
T
10
T10
1
Asymptote (phase frequency response)
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If > 1, G(j) can be expressed as a product of two first-order terms with real
poles.If 0
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For case that
For the asymptote, the curve at low frequency (
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For the case that
Note that
Slope = - 40 dB/ dec
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0dB
-40 dB/ dec
Asymptote (magnitude-frequency plot) for a second-order system (0
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=0, =0
=n, = -90o
o, - -
0o
-90o
-180o
n 10n10
n
As m tote hase-fre uenc lot for a second-order s stem 0<
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Resonant frequency
mp u e a max mum va ue a r
For case that
At r
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Transfer function having neither poles not zeros in the right-half s-plane are
called minimum-phase transfer functions.
Transfer function having poles and/or zeros in the right-half s-plane are called
nonminimum-phase transfer functions.
Minimum phase system Nonminimum phase system
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G1: minimum phase
(slow in response).
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- , -
characteristics are directly related.
If magnitude curve is specified, then the phase-angle curve is
un que y eterm ne an v ce versa.
This does not hold for a nonminimum-phase system.
Nonmimum-phase situations may arise (1) when a system
includes a nonmiimum-phase element or elements and (2) in
e case w ere a m nor oop s uns a e.
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System identification example
Experimental bode diagram of a second order system is
obtained. Identify numerical values of m, b, and k.
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Half power bandwidth method*
2
12 =
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Assume that the open-loop transfer function is given as
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Type 0 system
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For type 1 system
=1
-
20dB /dec (or extension) with the0-dB line has a frequency
.
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The frequency a at the intersection
of the initial -40 dB/dec segment (or
-
square root of ka numerically.
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Referring the figure below, the frequency b at which the
-
zero-frequency value is called cutoff frequency.
The fre uenc ran e in which the ma nitude of the0closed loop does not drop -3 dB is called bandwidth of the
system.
Bandwidth indicates frequency where gain starts to fall off from its
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n s ep response curves
Unit ramp response curves
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agn u e nee s o e conver e n o ec e .
Semilogx (w, magdB)
Or
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