feedback control systems ( fcs )
DESCRIPTION
Feedback Control Systems ( FCS ). Lecture-32-33 Closed Loop Frequency Response. Dr. Imtiaz Hussain email: [email protected] URL : http://imtiazhussainkalwar.weebly.com/. Introduction. - PowerPoint PPT PresentationTRANSCRIPT
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Feedback Control Systems (FCS)
Dr. Imtiaz Hussainemail: [email protected]
URL :http://imtiazhussainkalwar.weebly.com/
Lecture-32-33Closed Loop Frequency Response
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Introduction
• One of the important problems in analyzing a control system is to find all closed-loop poles or at least those closes to the jω axis (or the dominant pair of closed-loop poles).
• If the open-loop frequency-response characteristics of a system are known, it may be possible to estimate the closed-loop poles closest to the jω axis.
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Closed Loop Frequency Response• For a stable, unity-feedback closed-loop system, the closed-loop
frequency response can be obtained easily from that of the open loop frequency response.
• Consider the unity-feedback system shown in following figure. The closed-loop transfer function is
)()(
)()(
sG
sG
sR
sC
1
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Closed Loop Frequency Response• Following figure shows the polar plot of G(s).
• The vector OA represents G(jω1), where ω1 is the frequency at point A.
• The length of the vector OA is
• And the angle is
)( 1jG
)( 1jG
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Closed Loop Frequency Response
• The vector PA, the vector from -1+j0 point to Nyquist locus represents 1+G(jω1).
• Therefore, the ratio of OA, to PA represents the closed loop frequency response.
)()(
)()(
1
1
1
1
1
jR
jC
jG
jG
PA
OP
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Closed Loop Frequency Response• The magnitude of the closed loop
transfer function at ω=ω1 is the ratio of magnitudes of vector OA to vector PA.
• The phase of the closed loop transfer function at ω=ω1 is the angle formed by OA to PA (i.e Φ-θ).
• By measuring the magnitude and phase angle at different frequency points, the closed-loop frequency-response curve can be obtained.
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Closed Loop Frequency Response
• Let us define the magnitude of the closed-loop frequency response as M and the phase angle as α, or
jMejR
jC
)()(
ieZ
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Closed Loop Frequency Response
• Let us define the magnitude of the closed-loop frequency response as M and the phase angle as α, or
• From above equation we can find the constant-magnitude loci and constant-phase-angle loci.
• Such loci are convenient in determining the closed-loop frequency response from the polar plot or Nyquist plot.
jMejR
jC
)()(
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Constant Magnitude Loci (M circles)• To obtain the constant-magnitude loci, let us first note
that G(jω) is a complex quantity and can be written as follows:
• Then the closed loop magnitude M is given as
• And M2 is
jYXjG )(
jYX
jYXM
1 )(
)()()(
sG
sG
sR
sC
1
22
222
1 YX
YXM
)(
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Constant Magnitude Loci (M circles)
• Hence
• If M=1 then,
• This is the equation of straight line parallel to y-axis
and passing through (-0.5,0) point.
22
222
1 YX
YXM
)(
22222 21 YXYXXM
02 22222222 YXYMXMXMM
0121 222222 MYMXMXM )()(
012 X
2
1X
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Constant Magnitude Loci (M circles)
• If M≠1 then,
• Add to both sides
0121 222222 MYMXMXM )()(
011
22
22
2
22
M
MYX
M
MX
22
2
1M
M
22
2
22
2
2
22
2
22
1111
2
M
M
M
M
M
MYX
M
MX
22
22
22
4
2
22
111
2
M
MY
M
MX
M
MX
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Constant Magnitude Loci (M circles)
• This is the equation of a circle with
22
22
22
4
2
22
111
2
M
MY
M
MX
M
MX
22
22
2
2
22
11
M
MY
M
MX
1
01
2
2
2
M
Mradius
M
Mcentre ,
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Constant Magnitude Loci (M circles)
• The constant M loci on the G(s) plane are thus a family of circles.
• The centre and radius of the circle for a given value of M can be easily calculated.
• For example, for M=1.3, the centre is at (–2.45, 0) and the radius is 1.88.
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Constant Phase Loci (N circles)
• The phase angle of closed loop transfer function is
• The phase angle α is
jYX
jYX
jR
jC
1)(
)(
jYX
jYXe j
1
)(tan)(tanX
Y
X
Y
111
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Constant Phase Loci (N circles)
• If we define
• then
• We obtain
)(tan)(tanX
Y
X
Y
111
Ntan
)(tan)(tantan
X
Y
X
YN
111
X
Y
X
YX
Y
X
Y
N
11
1
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Constant Phase Loci (N circles)
X
Y
X
YX
Y
X
Y
N
11
1
22 YXX
YN
YYXXN )( 22
0122 YN
YXX
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Constant Phase Loci (N circles)
0122 YN
YXX
Adding to both sides 24
1
4
1
N
2222
4
1
4
1
4
11
4
1
NNY
NYXX
2
22
4
1
4
1
2
1
2
1
NNYX
This is an equation of circle with
24
1
4
1
2
1
2
1
Nradius
Ncentre
,
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Closed Loop Frequency Response
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Closed Loop Frequency Response
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END OF LECTURES-32-33
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