Bili Near Transformation
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Transcript of Bili Near Transformation
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Page 1
Bilinear Transformation
Control Engineering
by Dr. L. K. Wong
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Page 2
A Control System
Most plants are continuous-time systems
Power supply, power amplifier, motor Digital controllers are in discrete-time
Implemented by micro-controller
Controller PlantReference Output
+
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Page 3
Continuous-time Signals
f(t)
t
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Page 4
Discrete-time Signals
f*(t)
tT2T
3T4T
=
= Otherwise0),(
)(*nTttf
tf
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Page 5
Transformation
Convert a continuous-time transfer functionto a discrete-time transfer function
H(s) H(z)
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Page 6
Methods of Transformation
Backward difference Forward difference
Bilinear transformation z-transform
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Page 7
Theoretical Background
as
b
sRsYsH
+
=
=)()()(
)()()( tbrtayty +=!
)()()(1 tbrtayty +=
dttytyty
t
t+=2
1 )()()( 112
LetTkt )1(1 =
kTt =2
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Page 8
Theoretical Background
[ ])1()(2
)1()( 11 ++= kykyTkyky
y1(t)
tt1=(k1)T t2=kT
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Page 9
Theoretical Background
[ ])1()(2
)1()2
1()()2
1( +=+ krkrbTkyaTkyaT
[ ]11 )()(2
)()2
1()()2
1( +=+ zzRzRbTzzYaTzYaT
[ ])1()(2
)1()( 11 ++= kykyT
kyky
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Page 10
Theoretical Background
azz
T
b
aT
z
z
bT
aT
z
zz
bT
z
zaT
bT
zzaTaT
zbT
zaTaT
zbT
zRzY
++
=
++
=
+++
=
++
=
+++
+=
++=
)1(12
2)1(
12
2)1(
212
)1(
2)
21(
2
2)2
1()2
1(
)1(2
)2
1()2
1(
)1(2
)()(
1
1
1
1
1
11
1
1
11
1
1
1
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Page 11
Theoretical Background
az
z
T
bzRzYzH
++
==
)1(
12)()()(
1
1
as
b
sR
sYsH
+==
)(
)()(
1
12
+
=z
z
Ts
Compare
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Page 12
Example 1
Find a digital replacement of the followingcontinuous-time plant by bilineartransformation with sampling period of
T = 0.1s.
10010
1002.0
)( 2
2
1 ++++
= ssss
sH
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Page 13
Answer
1
120
112
+
=
+=
z
z
zz
Ts
4286.08571.0
7086.08571.07200.0300600700
496600504)(
2
2
2
2
1
++
=+
+=
zz
zzzz
zzzH
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Page 14
Frequency (rad/sec)
Phase(deg);Magn
itude(dB)
Bode Diagrams
-50
-40
-30
-20
-10
0
From: U(1)
10-1 100 101 102-100
-80
-60
-40
-20
0
To
:Y(1)
Frequency Warping10
5)(2 +=
ssH
9048.0
)1(0238.0)(2
+=
z
zzH
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Page 15
Frequency Warping
Approximation has been taken place use a trapezoidal to approximate the area under
a curve
Frequency response ofH(s) deviates fromthat ofH(z)
Significant if it lies in critical frequency e.g. 3dB cut-off frequency
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Page 16
Analytical Derivation
1
12
+
=
z
z
TsTj
A
Dezjs == andSubstitue
T
Tj
T
ee
ee
T
e
e
Tj
D
D
TjTj
TjTj
Tj
Tj
A
DD
DD
D
D
2
1cos
21sin
2
2
1
12
2
1
2
1
21
21
=
+
=
+
=
into
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Page 17
Analytical Derivation
TT
DA
2
1tan
2=
DA
DD
D
TT
2
1
2
1tan
small,isIf
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Page 18
Frequency Pre-warping
ModifyH(s) before applying transform Cancel out the warping effect exactly at a
frequency
Same frequency response ofH(s) andH(z)
at
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Page 19
Step 1
Calculate the pre-warped frequency
TTP 2
1
tan
2
=
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Page 20
Step 2
ReplacebyP
p
p
p
p
ss
ss
=
=
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Page 21
Step 3
Applying bilinear transformation
112
+=
zz
TsP
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Page 22
Example 2
Apply bilinear transformation withfrequency pre-warping at= 10 rad s1 tothe following continuous-time plant with
sampling period ofT = 0.1s. Calculate themagnitude and phase angle at for bothcontinuous-time and discrete-time plant.
10010
1002.0)(
2
2
3 ++++
=ss
sssH
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Page 23
Answer
Step 1
Step 2
926.10
1.01021tan
1.02
=
=P
P
p ss
s 915.0926.10
10==
10015.984.0
100183.084.0)(
2
2
3 ++++
=PP
PPP
ss
sssH
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Page 24
Answer
Step 3
Substitutes=j toH3(s) andz= ejT toH3(z),
Magnitude = 0.02
Phase = 0 rad
4077.07606.0
6979.07606.07098.0)(
2
2
3 ++
=zz
zzzH
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Page 25
Question
Can we select two frequencies to pre-warp?
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Page 26
w-transform
Transform az-plane transfer function into aso-calledw-plane transfer function
Inverse process of bilinear transformation
wwz += 11
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Page 27
Design on thew-plane
Employs-plane design velocity error constant
gain and phase margin
No need to tackle the irrational functionz= ejT
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Page 28
Further Modification
Large frequency distortion inw-transform Modify thew-transform as follows
2
1
21
wT
wT
z
+=
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Page 29
Example 3
TransformH4(z) into a transfer functionH4(w) inw-plane by the given bilineartransformation. Sketch the bode plot of
H4(w).
)8187.0)(1(
9356.003746.0)(4
+=
zz
zzH
w
wz
1.01
1.01
+
=
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Page 30
Answer
+
+
=
997.01
3001
1012
)(4ww
ww
wH
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Page 31
Conclusion
Bilinear transformation Frequency pre-warping
w-transform
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Page 32
Reference
M. Gopal, Digital Control Engineering.John Wiley & Sons.
I.J. Nagrath and M. Gopal, Control Systems
Engineering. 2nd edition. John Wiley &Sons.