dr. / mohamed ahmed ebrahim mohamed automatic control by dr. / mohamed ahmed ebrahim mohamed e-mail:...
TRANSCRIPT
Dr. / Mohamed Ahmed Ebrahim Mohamed
Automatic Control
By
Dr. / Mohamed Ahmed Ebrahim
MohamedE-mail: [email protected]
Web site: http://bu.edu.eg/staff/mohamedmohamed033
BODE PLOT
1• Introduction.
2• Frequency Response Definition.
3• Bode Plot Definition.
4• Frequency Response Plot.
5• Viewpoints of analyzing control system behavior.
6• Logarithmic coordinate.
7• Bode Plot Construction.
Frequency Response Definition
What is frequency response of a system?
The frequency response of a system is defined as the steady-state response of the system to a sinusoidal input signal.
The sinusoid is a unique input signal, and the resulting output signal for a linear system, as well as signals throughout the system, is sinusoidal in the steady-state.
Bode Plot Definition
What is Bode Plot?
Bode Plot is a (semi log) plot of the transfer function magnitude and phase angle as a function of frequency.
Polar Plots
Frequency Response Plots
• The transfer function can be separated into magnitude and phase angle information
H(j) = |H(j)| Φ(j)
e.g., H(j)=Z(jw)
Frequency Response
• Routh-Hurwitz • Root locus• Bode diagram (plots)• Nyquist plots• Nicols plots• Time domain
)( js
)( js
)( js
)( js
)( js
Viewpoints of analyzing control system behavior
L.T.I systemtAtr sin)( )sin()( tBty
Magnitude: Phase:
A
B
G(s)
H(s)
＋－
)(ty)(tr
)()(1
)(
)(
)(
sHsG
sG
sR
sY
jsjs
Magnitude: Phase:
)()(1
)(
jHjG
jG
)]()(1[
)(
jHjG
jG
Steady state response
1
210log
decDecade :
1
22log
octOctave :
1 10 1002 3 4 20
dB
Logarithmic coordinate
• 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
))()((
))((
)(
)(2
21
21
basspsps
zszsk
sR
sY
Case I : k
Magnitude:
Phase:
)(log20 dBkkdB
0,180
0,0
k
kk
o
o
)(dBGH
GH
1.0 1 10
090
0180
Bode Plot Construction
Case II :
Magnitude:
Phase:
)(log20)(
1dBp
jdB
p
pj
op
)90()(
1
)(dBGH
GH
1.0 1 10
0900180
ps
1
090
1p
1p
2p
2p
Case III :
Magnitude:
Phase:
)(log20)( dBpjdB
p
pj op )90()(
)(dBGH
GH
1.0 1 10
0900180
ps
0901p
1p
2p
2p0180
Case IV :1)1
1(
)(
sa
oras
a
Magnitude:
Phase:
])(1log[10
)(1log20)1(
2
21
a
aaj
dB
aaj
10 tan0)1(
)(dBGH
GH
1.0 1 10
0900180
090
0180
01log100 dBa
a
]log20log20[
log201
adBa
dBaa
ja
oGHa
a 00tan0 1
oGHa
a 90tan 1
01.32log1011 dBja
045a
1a
Case V :
Magnitude:
Phase:
])(1log[10
)(1log20)1(
2
2
a
aaj
dB
aaj
1tan)1(
)(dBGH
GH
1.0 1 10
0900180
090
0180
)11
()(
sa
ora
as
01log100 dBa
a
adBa
dBaa
ja
log20log20
log201
oGHa
a 00tan0 1
oGHa
a 90tan 1
01.32log1011 dBja
1a
045a
Case VI :22
2
2)(
nn
n
sssT
2
1
2
221
22
2
)(1
2
tan)(2))(1(
1)(
)(
2tan)(
2)()(
n
n
nn
n
n
nn
n
jTj
jT
jTj
jT
1,)log(40
1,)2log(20
1,0
)(
nn
n
n
jT
1,
1,
1,
180
90
0
)( 0
0
n
n
n
o
jT
n
Example : 50( 2)( )
( 10)
sT s
s s
)10
10)(
2
2)(
1(10)(
s
s
ssT
0,0,)(
)()(
1
1
iinpz
pss
zsksT
Minimum phase system
Type 0 : (i.e. n=0)
)()(
1
1
ps
pksT p
)(dBGH
11.0 p 1p 110p
A
AK p log20
0dB/dec
Type I : (i.e. n=1)
)()(
1
1
pss
pksT v
)(dBGH
11.0 p 1p
110p
A
-20dB/dec
-40dB/dec
1AKv log20 0
dBj
Kv 0log200
vk0
Type 2 : (i.e. n=2)
)()(
12
1
pss
pksT a
)(dBGH
11.0 p 1p 110p
A
-40dB/dec
-60dB/dec
1AKa log200
dBj
Ka 0)(
log202
0
ak20
A transfer function is called minimum phase when all the poles and zeroes are LHP and non-minimum-phase when there are RHP poles or zeroes.
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
Relative Stability
Open loop transfer function :
Closed-loop transfer function :
)()( sHsG
)()(1 sHsG
Open loop Stability poles of in LHP)()( sHsG
)0,0()0,1(
Re
Im
RHPClosed-loop Stability poles of in left side of (-1,0)
)()( sHsG
)(dBGH
GH
0900180
090
0180
0180
0)0,1(
dB
g
p
Gain crossover frequency: g
phase crossover frequency: p
P.M.>0
G.M.>0
Stable system
)(dBGH
GH
0900180
090
0180
g
pP.M.<0
G.M.<0
Stable system
Unstable system
Unstable system
• Straight-line approximations of the Bode plot may be drawn quickly from knowing the poles and zeroes– response approaches a minimum near the zeroes– response approaches a maximum near the poles
• The overall effect of constant, zero and pole terms
Term Magnitude Break
Asymptotic Magnitude Slope
Asymptotic Phase Shift
Constant (K) N/A 0 0
Zero upward +20 dB/decade + 90
Pole downward –20 dB/decade – 90
Bode Plot Summary
• Express the transfer function in standard form
• There are four different factors:– Constant gain term, K– Poles or zeroes at the origin, (j)±N
– Poles or zeroes of the form (1+ j)– Quadratic poles or zeroes of the form 1+2(j)+(j)2
2
22221
)()(21)1(
)()(21)1()(
bbba
N
jjj
jjjjKj
H
Bode Plot Summary
• We can combine the constant gain term (K) and the N pole(s) or zero(s) at the origin such that the magnitude crosses 0 dB at
• Define the break frequency to be at ω=1/ with magnitude at ±3 dB and phase at ±45°
NdB
N
NdBN
KjKZero
Kj
KPole
/10
/10
)/1()(:
)(:
Bode Plot Summary
Magnitude Behavior Phase Behavior
Factor Low Freq
Break Asymptotic Low Freq
Break Asymptotic
Constant 20 log10(K) for all frequencies 0 for all frequencies
Poles or zeros at origin
±20N dB/decade for all frequencies with a crossover of
0 dB at ω=1
±90 (N) for all frequencies
First order (simple) poles or zeros
0 dB ±3N dB at ω=1/
±20N dB/decade
0 ±45 (N) with slope ±45 (N) per decade
±90 (N)
Quadratic poles or zeros
0 dB see ζ at ω=1/
±40N dB/decade
0 ±90 (N) ±180 (N)
where N is the number of roots of value τ
Bode Plot Summary
Single Pole & Zero Bode Plots
ω
Pole at ωp=1/
Gain
Phase
ω
0°
–45°
–90°
One Decade
0 dB
–20 dB
ω
Zero at ωz=1/
Gain
Phase
ω
+90°
+45°
0°
One Decade
+20 dB
0 dB
ωp ωz
Assume K=120 log10(K) = 0 dB
• Further refinement of the magnitude characteristic for first order poles and zeros is possible sinceMagnitude at half break frequency: |H(½b)| = ±1 dB
Magnitude at break frequency: |H(b)| = ±3 dB
Magnitude at twice break frequency: |H(2b)| = ±7 dB
• Second order poles (and zeros) require that the damping ratio ( value) be taken into account; see Fig. 9-30 in textbook
Bode Plot Refinements
• We can also take the Bode plot and extract the transfer function from it (although in reality there will be error associated with our extracting information from the graph)
• First, determine the constant gain factor, K• Next, move from lowest to highest frequency noting the
appearance and order of the poles and zeros
Bode Plots to Transfer Function
Frequency Response Plots
Bode Plots – Real Poles (Graphical Construction)
Frequency Response Plots
Bode Plots – Real Poles
Frequency Response Plots
Bode Plots – Real Poles
Gain and Phase Margin Let's say that we have the following system: where K is a variable (constant) gain and G(s) is the plant under consideration.
The gain margin is defined as the change in open loop gain required to make the system unstable. Systems with greater gain margins can withstand greater changes in system parameters before becoming unstable in closed loop. Keep in mind that unity gain in magnitude is equal to a gain of zero in dB The phase margin is defined as the change in open loop phase shift required to make a closed loop system unstable. The phase margin is the difference in phase between the phase curve and -180 deg at the point corresponding to the frequency that gives us a gain of 0dB (the gain cross over frequency, Wgc). Likewise, the gain margin is the difference between the magnitude curve and 0dB at the point corresponding to the frequency that gives us a phase of -180 deg (the phase cross over frequency, Wpc).
Gain and Phase Margin
-180
Examples - Bode
Examples - Bode
Examples – Bode
Mohamed Ahmed Ebrahim