2010 Spring ME854 - GGZ Page 1Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional Transformation Linear Fractional Transformation (LFT)(LFT)
.,, somefor
)1()(
zero,not is If (LFT).ation transformfractionallinear a called , and ,,,with
)(
form theof : mappingcomplex variableoneConsider
1
C
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Definition 9.1
exist. )( provided
)(),( with :),(
LFTupper an define ,exist )( provided
)(),( with :),(
LFTlower a Define matrices.complex other twobe and let and
as dpartitionematrix complex a be Let
1
11
12
1
112122
1
22
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1
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)()(
2221
1211
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−
−××
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−××
××
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∆−
∆−∆+=∆⋅
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∈∆∈∆
∈
=
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2010 Spring ME854 - GGZ Page 2Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional Transformation Linear Fractional Transformation (LFT)(LFT)
121
1
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122
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2010 Spring ME854 - GGZ Page 3Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional Transformation Linear Fractional Transformation (LFT)(LFT)
.invertible is )( if posed)-(or well defined- wellbe tosaid is , ),( LFT,An1
22
−∆−∆ MIMFl
Definition 9.2
Lemma 9.1
, ,
with
),()()(
),())((
Then .invertible is Suppose
11
11
11
11
1
1
−−=
−
−=
=++
=++
−−
−−
−−
−−
−
−
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111
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1
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))((
))((
)()(
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:Note
111
−
−−−
+
−−−−
++
−
−
++=
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2010 Spring ME854 - GGZ Page 4Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional Transformation Linear Fractional Transformation (LFT)(LFT)
Lemma 9.2
.matrix r nonsingulaany for
,00
0
,
0
00
0
isThat . , , ,with
))((),(
then,invertible is If )
.matrix r nonsingulaany for
,0
00
,
0
0
00
isThat . , , ,with
)()(),(
then,invertible is If )
. with LFTgiven a be ),(Let
1
2122
1112
1
21
22
1112
22
1
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1
2122
1
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−
−−
−−−−
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2010 Spring ME854 - GGZ Page 5Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional Transformation Linear Fractional Transformation (LFT)(LFT)
Note:
[ ]
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have we9.1, Lemma Using
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2010 Spring ME854 - GGZ Page 6Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional Transformation Linear Fractional Transformation (LFT)(LFT)
Note:
( )
( ) [ ] [ ]
11
)(
1
)(
1
1
1
)(0
)_(
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0
1
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Note
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0
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where
))(( ,
11
1
1
−−
+
−
+
−
−
−
+
−
+
−
−
++=+−−++=
+−+−+=
+
+−−=
−
−
−
−−
−
−+=∆
−
−=∆
−−=
++=∆
−−
−
−
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2010 Spring ME854 - GGZ Page 7Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional Transformation Linear Fractional Transformation (LFT)(LFT)
Lemma 9.3
.
by given is
where),,()],([ Then r.nonsingula is and Let
1
2221
1
22
1
221221
1
221211
1
22
2221
1211
−−=
∆=∆
=
−−
−−
−
MMM
MMMMMMN
N
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MMM uu
( )
( )
( )∆=
−∆−−∆+=
∆+∆−∆−=∆
−+=−
∆−∆+=∆
−−−−−
−−−−−−
−−−−−−−
−
−
,
)())((
)(],[
have we
))(())((
identity theUsing
)( , Note
12112122
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1
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1
21
1
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1
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1
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2010 Spring ME854 - GGZ Page 8Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional Transformation Linear Fractional Transformation (LFT)(LFT)
Simple Block Diagram and LFT
( ) [ ]
+−+−
+−+−=
−−+
+
=
−−−
=
=
=
−−
−−
−
FFPKIKWFPFPKIKW
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2010 Spring ME854 - GGZ Page 9Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional Transformation Linear Fractional Transformation (LFT)(LFT)
Parametric Uncertainty
[ ].1.01
where
),()1.01(1.01
)1.01(
1.01.01
m
1
. in LFTanby drepresenteeasily be can case, thisin that Note
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s variationfollowing thehave parameters that Suppose
.
by described be can motion system theof equation dynamic
The side.right theon systemdamper -spring-massa Consider
1.01
1
1
1
m1
ij
mm
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m
mm
m
kcm
mM
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−+=
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=++
− δδδδ
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δ
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ɺɺɺ
m
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2010 Spring ME854 - GGZ Page 10Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional Transformation Linear Fractional Transformation (LFT)(LFT)
Parametric Uncertainty (cont’d-1)
have we
equations, above into of elements Substitute
and
)(
2.0
3.0
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:equations dynamic following list the Now
1
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2
1
1212112
21
M
y
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=
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δ
δ
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ɺ
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m
F
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2010 Spring ME854 - GGZ Page 11Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional Transformation Linear Fractional Transformation (LFT)(LFT)
Parametric Uncertainty (cont’d-1)
−−−−−
=
∆=
−−−−−
=
−
−=
−−−−−
−−−−−
1.0111
00002.00
000003.0
000010
where,),(
Therefore
1.0111
00002.00
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havecan we,1.01
that Recall
1.0111
2
1
2
1
2
1
1.01112
1
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1
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2010 Spring ME854 - GGZ Page 12Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Basic PrincipleBasic Principle
matrix. transfer ingcorrespond the
gcalculatinby obtained becan matrix
theand s,' out the pulling :principles basic The
LFT plant andy uncertaintPlant b)
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2010 Spring ME854 - GGZ Page 13Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Basic PrincipleBasic Principle
.
w
u
u
u
u
M
w
u
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u
u
aadae
bcbdbe
de
z
y
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=
4
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4
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00
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1
2
1121112
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1
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212
2
121
2
212
yeydwywedy
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22
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where
δ
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2010 Spring ME854 - GGZ Page 14Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Basic PrincipleBasic Principle
−−−
−−
−−−−
=
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+
+
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State-Space
8
u2
7
u1
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=
2
1
2
1
2
1
2
1
2
1
2
1
2
1
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u
u
n
n
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2010 Spring ME854 - GGZ Page 15Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Redheffer Star ProductRedheffer Star Product
−
−=∗
−
=
=
−
−
),()(
)(),(
as defined is and ofproduct star theThen .invertible is
)(at further th assume square, and defined wellis product matrix that theSuch
,
matrces patitioned compatibly are and that Suppose
2221
1
112221
12
1
22111211
11221122
2221
1211
2221
1211
PKFPKPIK
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KP
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−+−
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+=
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+=
+=
−
−
−+=
−−
−
−+=
−
−
−
w
w
KPKPIKKPKPIK
KPKIPPKPKIPP
z
z
wKwPKPKIz
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ˆˆ
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Note
121
221122212222
211
112211121111
)(),(
1222
1
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1
112221
12
1
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)(),(
2111
1
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2010 Spring ME854 - GGZ Page 16Linear Fractional TransformationLinear Fractional Transformation
Linear Fractional TransformationLinear Fractional Transformation
Redheffer Star ProductRedheffer Star Product
where,tion representa a has ˆˆ
:matrix transfer Then the
,
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22212
12111
21
22212
12111
21
=∗
∗
=
=
DC
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z
z
w
wKP
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1
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1
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1
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121
122
1
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1
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1
12
1
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1
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1
111
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2
122
1
12
1
1
1
1
2211
1
2
~ , where
~
~
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−=−=
∗
=
+
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=
+
+=
∗
=
+
+=
∗
=
+
+=
−−
−−
−−
−−
−−
−−
−−
−−