FUMcpanel - Multivariable Control...
Transcript of FUMcpanel - Multivariable Control...
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Multivariable Control Multivariable Control SystemsSystemsSystemsSystems
Ali KarimpourpAssistant Professor
Ferdowsi University of MashhadFerdowsi University of Mashhad
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Chapter 5Chapter 5Controllability, Observability and Realization
Topics to be covered include:
• Controllability and Observability of Linear Dynamical Equations
• Output Controllability and Functional Controllability• Output Controllability and Functional Controllability
• Canonical Decomposition of Dynamical Equation
• Realization of Proper Rational Transfer Function Matrices
• Irreducible RealizationsIrreducible realization of proper rational transfer functions
Irreducible Realization of Proper Rational Transfer Function Vectors
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2Irreducible Realization of Proper Rational Matrices
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Chapter 5
Controllability, Observability and Realization
• Controllability and Observability of Linear Dynamical EquationsControllability and Observability of Linear Dynamical Equations
• Output Controllability and Functional Controllability
• Canonical Decomposition of Dynamical Equation
• Realization of Proper Rational Transfer Function Matrices
• Irreducible RealizationsIrreducible realization of proper rational transfer functions
Irreducible Realization of Proper Rational Transfer Function Vectors
Irreducible Realization of Proper Rational Matrices
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Chapter 5
Controllability and Observability of Linear Dynamical Equations
Definition 5-1
D fi iti 5 2Definition 5-2
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Chapter 5
Controllability and Observability of Linear Dynamical Equations
Theorem 5-1
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Chapter 5
Controllability and Observability of Linear Dynamical Equations
[ ]nn bAbAAbAbbbS 11 −−= [ ]mmm bAbAAbAbbbS 111 ............=
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Chapter 5
Controllability and Observability of Linear Dynamical Equations
Theorem 5-2
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Chapter 5
Controllability, Observability and Realization
• Controllability and Observability of Linear Dynamical EquationsControllability and Observability of Linear Dynamical Equations
• Output Controllability and Functional Controllability
• Canonical Decomposition of Dynamical Equation
• Realization of Proper Rational Transfer Function Matrices
• Irreducible RealizationsIrreducible realization of proper rational transfer functions
Irreducible Realization of Proper Rational Transfer Function Vectors
Irreducible Realization of Proper Rational Matrices
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Chapter 5
Output Controllability
Definition 5-3 Output Controllability
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Chapter 5
Functional Controllability
Definition 5-4 Functional controllability.
An m-input l-output system G(s) is functionally controllable if thenormal rank of G(s), denoted r, is equal to the number of outputs; that is, if G(s) has full row rank. A plant is functionally uncontrollable if r < l.if G(s) has full row rank. A plant is functionally uncontrollable if r l.
Remark 1: The minimal requirement for functional controllability is
that we have at least many inputs as outputs i e m ≥ lthat we have at least many inputs as outputs, i.e. m ≥ l
Remark 2: A plant is functionally uncontrollable if and only if
ωωσ ∀= 0))(( jG ωωσ ∀= ,0))(( jGlRemark 3: For SISO plants just G(s)=0 is functionally uncontrollable.
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Remark 4: A MIMO plant is functionally uncontrollable if the gain is identically zero in some output direction at all frequencies.
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Chapter 5
Functional Controllability
⎤⎡ 21
Consider following transfer function:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡++=
11
11
32
11
)( sssG It is Functionally controllable
⎦⎣ ++ 11 ss
Its state space representation is:
)(2123
)(00410030
)( tutxtx⎥⎥⎥⎤
⎢⎢⎢⎡
+⎥⎥⎥⎤
⎢⎢⎢⎡
−−
=&
0010
)(
1111
)(
21001000
)( tutxtx
⎤⎡
⎥⎥
⎦⎢⎢
⎣
+
⎥⎥
⎦⎢⎢
⎣ −−
It is not state controllable
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11)(
10000010
)( txty ⎥⎦
⎤⎢⎣
⎡=
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Chapter 5
Functional Controllability
10000 ⎤⎡⎤⎡
Consider following state space form:
It is state controllable and output controllable
)(000110
)(010001000
)( tutxtx⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=&
p
)(100010
)(
0000
txty ⎥⎦
⎤⎢⎣
⎡=
⎥⎦⎢⎣⎥⎦⎢⎣
Its transfer function is:
⎤⎡ 11It is not functionally controllable.
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=
32
2
11
11
)( sssG
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⎦⎣ 32 ss
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Chapter 5
Functional Controllability
� An Example: dc-dc boost converter
6
2 51
2498 3.049 10ˆ 609 3 207 10 ˆ
sx s s
⎡ ⎤⎢ ⎥⎡ ⎤ ⎢ ⎥
+ ×+ + × 1x1
15 82
2 5
609 3.207 10 ˆ ,ˆ 2.5 10 1.217 10
609 3.207 10
s sx s
s s
ω⎡ ⎤ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎣ ⎦
⎢ ⎥⎣ ⎦
+ + ×− × + ×
+ + ×
12x
1ω
functionally uncontrollable
6
:
2498 3 049 10 12 5 s + 7 44 0
or new system design
s + ×⎡ ⎤
functionally uncontrollable
2 51
2
2498 3.049 10ˆ 609 3.207 1
12.5 s + 7
ˆ
40
4 0sx s s sx⎡ ⎤
=
+ ×+ + ×
⎢ ⎥⎣ ⎦
2 51
5 8 5
2 5 2 5
ˆ609 3.207 10ˆ2.5 10 1.217 10 10
609 3 207 10 66.2
09 3 207 15
0in
svs
s s s s
ω⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥
+ + ×− × + × ×
+ + × + + ×⎣ ⎦
⎢ ⎥⎣ ⎦
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1313
609 3.207 10 609 3.207 10s s s s+ + × + + ×⎣ ⎦
functionally controllable
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Chapter 5
Functional Controllability
BAsICsG 1)()( −−= is functionally uncontrollable ifAn m-input l-output system
lBrank
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Chapter 5
Functional Controllability
Example 5-1
⎥⎥⎥⎤
⎢⎢⎢⎡
++= 421
21
1
)( sssG⎥⎦
⎢⎣ ++ 2
42
2ss
Thi i il i l 2 f G( ) i i l 1This is easily seen since column 2 of G(s) is two times column 1.
The uncontrollable output directions at low and high frequencies are, i lrespectively,
⎥⎤
⎢⎡
∞⎥⎤
⎢⎡ 21)(
11)0( yy
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⎥⎦
⎢⎣−
=∞⎥⎦
⎢⎣−
=15
)(,12
)0( 00 yy
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Chapter 5
Controllability, Observability and Realization
• Controllability and Observability of Linear Dynamical EquationsControllability and Observability of Linear Dynamical Equations
• Output Controllability and Functional Controllability
• Canonical Decomposition of Dynamical Equation
• Realization of Proper Rational Transfer Function Matrices
• Irreducible RealizationsIrreducible realization of proper rational transfer functions
Irreducible Realization of Proper Rational Transfer Function Vectors
Irreducible Realization of Proper Rational Matrices
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Chapter 5
Canonical Decomposition of a Linear Time-invariant Dynamical Equation
ECBuAxx
++=& uBxAPBuxPAPx +=+= −1&Pxx =EuCxy += uExCEuxCPy +=+= −1
Theorem 5-3 The controllability and observability of a linear time-invariantd i l i i i d i l f idynamical equation are invariant under any equivalence transformation.
Proof: Let we first consider controllability
[ ] [ ][ ] PSBABAABBP
PBPPAPBPPAPBPAPPBBABABABSn
nn
==
==−
−−−−−
12
1112112 ..........[ ] PSBABAABBP == .....
Similarly we can consider observability
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Chapter 5
Canonical Decomposition of a Linear Time-invariant Dynamical Equation
ECBuAxx +=&Theorem 5-4
Consider the n-dimensional linear time –invariant dynamical equation EuCxy +=
y q
If the controllability matrix of the dynamical equation has rank n1 (where n1
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Chapter 5
Canonical Decomposition of a Linear Time-invariant Dynamical Equation
Theorem 5-4 (Continue)
1Furthermore P=[q1 q2 … qn1 … qn]-1 where q1, q2, …, qn1 be any n1 linearly
independent column of S (controllability matrix) and the last n-n1 column of P
are entirely arbitrary so long as the matrix [q1 q2 … qn1 … qn] is nonsingular.
Proof: See “Linear system theory and design” Chi-Tsong Chen
ldimensiona+=+=
nEuCxyBuAxx&
ldimensiona≤+=
+=
EuxCyuBxAx
cc
cccc&Pxx =
ldimensiona−n ldimensiona1 −≤ nn
G(s)
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19Hence, we derive the reduced order controllable equation.
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Chapter 5
Canonical Decomposition of a Linear Time-invariant Dynamical Equation
ECBuAxx +=&Theorem 5-5
Consider the n-dimensional linear time –invariant dynamical equation EuCxy +=
y q
If the observability matrix of the dynamical equation has rank n2 (where n2
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Chapter 5
Canonical Decomposition of a Linear Time-invariant Dynamical Equation
Theorem 5-5 (Continue)
Furthermore the first n row of P are any n linearly independent rows of VFurthermore the first n2 row of P are any n2 linearly independent rows of V
(observability matrix) and the last and the last n-n2 row of P is entirely arbitrary
so long as the matrix P is nonsingularso long as the matrix P is nonsingular.
Proof: See “Linear system theory and design” Chi-Tsong Chen
ldimensiona−+=+=
nEuCxyBuAxx& Pxx =
+=
+=
EuxCyuBxAx
oo
oooo&
ldimensiona−n
G(s)
ldimensiona2 −≤ nn
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21Hence, we derive the reduced order observable equation.
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Chapter 5
Canonical Decomposition of a Linear Time-invariant Dynamical Equation
ECBuAxx
++=&Theorem 5-6 (Canonical decomposition theorem)
Consider the n-dimensional linear time –invariant dynamical equation EuCxy +=Consider the n-dimensional linear time invariant dynamical equation There exists an equivalence transformation
Pxx = Pxxwhich transform the dynamical equation to
[ ] Ex
CCBBx
AAAAAx ocococococ
⎥⎤
⎢⎡
⎥⎤
⎢⎡
⎥⎤
⎢⎡⎥⎤
⎢⎡
⎥⎤
⎢⎡
001312
&
&
d th d d di i l b ti
[ ] EuxxCCyuB
xx
AAA
xx
c
coccoco
c
co
c
co
c
co +⎥⎥⎥
⎦⎢⎢⎢
⎣
=⎥⎥⎥
⎦⎢⎢⎢
⎣
+⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣
=⎥⎥⎥
⎦⎢⎢⎢
⎣
0000
0 23&
and the reduced dimensional sub-equation
EuxCyuBxAx cocococo
+=
+=&
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22is observable and controllable and has the same transfer function matrix
as the first system.
EuxCy coco +
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Chapter 5
Canonical Decomposition of a Linear Time-invariant Dynamical Equation
Definition 5-5
A linear time-invariant dynamical equation is said to be reducible if and only if thereA linear time invariant dynamical equation is said to be reducible if and only if there
exist a linear time-invariant dynamical equation of lesser dimension that has the same
transfer function matrix. Otherwise, the equation is irreducible.q
Theorem 5-7
A linear time invariant dynamical equation is irreducible if and only if it is controllable
and observable.
Theorem 5-8
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Chapter 5
Controllability, Observability and Realization
• Controllability and Observability of Linear Dynamical EquationsControllability and Observability of Linear Dynamical Equations
• Output Controllability and Functional Controllability
• Canonical Decomposition of Dynamical Equation
• Realization of Proper Rational Transfer Function Matrices
• Irreducible RealizationsIrreducible realization of proper rational transfer functions
Irreducible Realization of Proper Rational Transfer Function Vectors
Irreducible Realization of Proper Rational Matrices
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Chapter 5
Realization of Proper Rational Transfer Function Matrices
Dynamical equation (state-space) description This transformation
The input-output description (transfer function matrix)
EuCxyBuAxx
+=+=& is unique
EBAsICsG +−= −1)()(
( )
The input-output description (transfer function matrix) Realization
BA&
Dynamical equation (state-space) description
This transformationEBAsICsG +−= −1)()(
EuCxyBuAxx
+=+=This transformation
is not unique
1 Is it possible at all to obtain the state-space description from the transfer function1. Is it possible at all to obtain the state-space description from the transfer function matrix of a system?
2. If yes, how do we obtain the state space description from the transfer function matrix?
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y , p p
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Chapter 5
Realization of Proper Rational Transfer Function Matrices
Theorem 5-9
A transfer function matrix G(s) is realizable by a finite dimensional
linear time invariant dynamical equation if and only if G(s) is a propery q y ( ) p p
rational matrix.
Proof: See “Linear system theory and design” Chi-Tsong Chen
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Chapter 5
Irreducible realizationsDefinition 5-6
Theorem 5-10
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Chapter 5
Irreducible realizations
Before considering the general case
(irreducible realization of proper rational matrices)
we start the following parts:
1. Irreducible realization of Proper Rational Transfer Functions
2 I d ibl R li ti f P R ti l T f F ti V t2. Irreducible Realization of Proper Rational Transfer Function Vectors
3. Irreducible Realization of Proper Rational Matrices
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Chapter 5
Irreducible realization of proper rational transfer functions
0,ˆ......ˆˆ
ˆ......ˆˆ)( 01
10
110 ≠
++++++
= −−
ααααβββ
nnn
nn
ssss
sg0
01
1
22
11
ˆ
ˆ
............
)(αβ
ααβββ
+++++++
= −−−
nnn
nn
ssss
sg
)()()(ˆ)(ˆˆ
)(ˆ)(ˆˆ
)(......
......)(0
0
0
01
1
22
11 seususgsusysusu
sssssy
nnn
nnn
+=+=+++++++
= −−−
αβ
αβ
ααβββ
......10 +++ ααα nss 01 ...... ααα +++ nss
)()(ˆ)(ˆ
susysg =
00n
)(ˆ sg
)(su
)()(ˆ)(ˆ
susysg =
ucxybuAxx0ˆ +=+=&
)()( sysg =buAxx +=
β̂
&
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)()(
susg =
eucxuyy +=+=0
0
ˆˆ
αβ
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Chapter 5
Irreducible realization of proper rational transfer functions
nnn
nnn
ssss
susysg
ααβββ
++++++
== −−−
............
)()(ˆ)(ˆ 1
1
22
11
0
01
1
22
11
ˆ
ˆ
............
)(αβ
ααβββ
+++++++
= −−−
nnn
nn
ssss
sgn)( 1
There are different forms of realization
01 ...... ααα +++ nss
Ob bl i l f li ti
There are different forms of realization
• Observable canonical form realization
• Controllable canonical form realization• Controllable canonical form realization
• Realization from the Hankel matrix
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• Realization from the Hankel matrix
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Chapter 5
Observable canonical form realization of proper rational transfer functions
nnn
nn sssg βββ +++= −−− ......)(ˆ 1
22
110
1
22
11
ˆ
ˆ......)(
ββββ+
++++++
= −−−
nnn
nn sssg
nnn ss αα +++ ......11
uuuyyy nnn
nnn βββαα +++=+++ −−− ............ )2(2
)1(1
)1(1
)( )))
)()( )
01
1 ...... ααα +++ nnn ss
)()( tytxn)=
)()()()()()()( 11)1(
11)1(
1 tutxtxtutytytx nnn βαβα −+=−+=−))
)()()()()()(.................................
)2()2()1( tttttt nnn −−− +++ ββ )))
)()()()()()()()()( 22)1(
122)1(
1)1(
1)2(
2 tutxtxtutytutytytx nnn βαβαβα −+=−+−+= −−)))
)()()()()()( )1()1()1()1()()1( nnn ββ )))
)()()(
)()(...)()()()(
11)1(
2
11)2(
1)2(
1)1(
1
tutxtx
tutytutytytx
nnn
nnnnn
−−
−−
−+=
−++−+=
βα
βαβα
Ali Karimpour June 2012
31)()()()()()(...)()()()( )1(1
)1(1
)1(1
)1(1
)()1(1
tutxtutytutytutytytx
nnnnn
nnnnn
βαβαβαβα
+−=+−=−++−+= −−
−−
)
)))
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Chapter 5
)()( tytx = )Observable canonical form realization of proper rational transfer functions
)()()()()()()()()(
)()()()()()()(
)()(
22)1(
122)1(
1)1(
1)2(
2
11)1(
11)1(
1
tutxtxtutytutytytx
tutxtxtutytytx
tytx
nnn
nnn
n
−−
−
−+=−+−+=
−+=−+=
βαβαβα
βαβα)))
))
)()()(
)()(...)()()()(.................................
)1(11
)2(1
)2(1
)1(1
tutxtx
tutytutytytx nnnnn
−−−−−
+=
−++−+=
βα
βαβα )))
)()()( 112 tutxtx nnn −− −+= βα
)()()()()()(...)()()()( )1(1
)1(1
)1(1
)1(1
)()1(1
tutxtutytutytutytytx
nnnnn
nnnnn
βαβαβαβα
+−=+−=−++−+= −−
−−
)
)))
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
−−
⎥⎥⎤
⎢⎢⎡
−− n
n
n
n
xx
xx
xx
0...010...00
2
1
12
1
12
1
ββ
αα
&
&
)()()()(y nnnnn ββ
[ ]
⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣
=
⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣
+
⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣
−=
⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣
−− n
n
n
n
xyuxx.
10...00ˆ..
100.......
0...10.
3
2
2
1
3
2
2
1
3
2
β
βα
&
&
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⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣ −⎥⎦⎢⎣ nnn xxx 1...00 11 βα
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Chapter 5
Observable canonical form realization of proper rational transfer functions
nnn
nnn
sssssg
ααβββ
++++++
= −−−
............)(ˆ 1
1
22
11
⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡
[ ]⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
=⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
+⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
−−−
=⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
−
−
−
−
n
n
n
n
n
n
xxx
yuxxx
xxx
10...00ˆ0...100...010...00
3
2
1
2
1
3
2
1
2
1
3
2
1
βββ
ααα
&
&
&
21 β̂βββ −− nn
[ ]
⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣ −⎥⎥⎥
⎦⎢⎢⎢
⎣ n
n
n
n
n x
y
xx...
1...00........
3
1
23
1
23
β
β
α&
0
01
1
22
11
ˆ............)(
αβ
ααβββ
+++++++
= −−−
nnn
nnn
sssssg
xxx 000 βα ⎤⎡⎤⎡⎤⎡⎤⎡ −⎤⎡ &
[ ] uxxx
yuxxx
xxx
n
n
n
n
n
n
0
03
2
1
2
1
3
2
1
2
1
3
2
1
ˆˆ
10...00ˆ0...100...010...00
αββ
ββ
ααα
+⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
=⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
+⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
−−−
=⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
−
−
−
−
&
&
Ali Karimpour June 2012
33xxx nnn
0
11
...1...00
........α
βα ⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣ −⎥⎥⎥
⎦⎢⎢⎢
⎣ &
-
Chapter 5
Observable canonical form realization of proper rational transfer functions
02
21
1ˆ......)( ββββ ++++=
−−n
nn sssg nnn sssg βββ +++=−− ......)(ˆ
22
11
01
1 ˆ......)(
ααα+
+++= −
nnn ss
sgn
nn sssg
αα +++= − ......
)( 11
xxx nn 111 0...00 βα⎥⎤
⎢⎡
⎥⎤
⎢⎡
⎥⎤
⎢⎡⎥⎤
⎢⎡ −
⎥⎤
⎢⎡ &
[ ] uxx
yuxx
xx
n
n
n
n
0
03
2
2
1
3
2
2
1
3
2
ˆˆ
.10...00ˆ
.........0...100...01
.αββ
βαα
+
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
=
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
+
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
−−
=
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
−
−
−
−
&
&
xxx nnn 11
...1...00
........βα ⎥
⎥⎦⎢
⎢⎣⎥
⎥⎦⎢
⎢⎣⎥
⎥⎦⎢
⎢⎣⎥⎥⎦⎢
⎢⎣ −⎥
⎥⎦⎢
⎢⎣ &
Th d i d d i l ti i b bl E i 1 Wh ?
The derived dynamical equation controllable as well if numerator and
The derived dynamical equation is observable. Exersise 1: Why?
Ali Karimpour June 2012
34
denominator of g(s) are coprime. Exersise 2: Why?
-
Chapter 5
Controllable canonical form realization of proper rational transfer functions
nnn sssNsg
βββ +++==
−− ......)()( 12
21
1)01
22
11
ˆ
ˆ......)(
ββββ+
+++= −
−−
nnn
nn sssg
Let us introduce a new variable)()()()()()(
svsNsysusvsD
==
)
nnn sssD
gαα +++ − ......)(
)( 110
11 ...... ααα +++ n
nn ss
)()()( svsNsy =
We may define the state variable as: ⎥⎥⎥⎤
⎢⎢⎢⎡
=⎥⎥⎥⎤
⎢⎢⎢⎡
=)(
)()()(
)(
)1(2
1
tvtv
txtx
txWe may define the state variable as:
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
=
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
=
− )(....
)(....)(
)1( tvtx
tx
nn
nn xxxxxx === −13221 ,.....,, &&&Clearly
)()()()()( )1()1()( tvtvtvtutvx nn ααα −−−−== −&
Ali Karimpour June 2012
35)(...)()()(
)(...)()()()(
1211
11
txtxtxtutvtvtvtutvx
nnn
nnn
αααααα
−−−−===
−
−
-
Chapter 5
Controllable canonical form realization of proper rational transfer functions
02
21
1 ......)(ββββ)
+++ −− nnn ss n
nn sssN βββ +++ −− ......)()(2
21
1)
0
01
1
21
............
)(αβ
ααβββ
)++++
= −n
nnn
sssg
nnn
n
sssDsNsg
ααβββ
+++== − ......)(
)()( 11
21)
nn xxxxxx === −13221 ,.....,, &&&
)(...)()()()(...)()()()(
1211
)1(1
)1(1
)(
txtxtxtutvtvtvtutvx
nnn
nnn
nn
αααααα
−−−−=−−−−==
−
−−&
[ ] ⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
xx
xx
xx
00
0...1000...010
2
1
2
1
2
1
ββββ&
&
[ ]
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
=
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
+
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ −−−−
=
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
−−
−− n
nnn
nnnnn x
xyu
x
x
x
x.
...
1.0
....
1...000.......
.31213
121
3 ββββ
αααα&
&
Ali Karimpour June 2012
36
-
Chapter 5
Controllable canonical form realization of proper rational transfer functions
nnn
nnn
ssss
sDsNsg
ααβββ
++++++
== −−−
............
)()()( 1
1
22
11)
⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡
[ ]⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
=⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
+⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
=⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
−− nnn xxx
yuxxx
xxx
...000
.......0...1000...010
3
2
1
1213
2
1
3
2
1
ββββ&&
&
⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣ −−−−⎥⎥⎥
⎦⎢⎢⎢
⎣ −− nnnnnn xxx.
1..
...1...000.
121 αααα&
0
01
1
22
11
ˆˆ
............)(
αβ
ααβββ
+++++++
= −−−
nnn
nnn
sssssg
xxx 00010 ⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡ &
[ ] uxxx
yuxxx
xxx
nnn0
03
2
1
1213
2
1
3
2
1
ˆˆ
...000
1000.......0...1000...010
αβββββ +
⎥⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎢⎡
=
⎥⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎢⎡
+
⎥⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎢⎡
⎥⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎢⎡
=
⎥⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎢⎡
−−&
&
Ali Karimpour June 2012
37xxx nnnnnn
0
121
.1..
...1...000.αααα ⎥
⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣ −−−−⎥⎥⎥
⎦⎢⎢⎢
⎣ −−&
-
Chapter 5
Controllable canonical form realization of proper rational transfer functions
0
01
1
22
11
ˆˆ
............)(
αβ
ααβββ
+++++++
= −−−
nnn
nnn
sssssg nn
nnn
ssss
sDsNsg
ααβββ
++++++
== −−−
............
)()()( 1
1
22
11)
01 ...... ααα +++ nss nsss αα +++ ......)( 1
xx
xx
xx 111
00
01000...010
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡&
&
[ ] uxx
yuxx
xx
nnn0
03
2
1213
2
3
2
ˆˆ
....
1.00
.1...000.......0...100
.αβββββ +
⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣
=
⎥⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢⎢
⎣
+
⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣
=
⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣
−−
&
&
Th d i d d i l ti i t ll bl E i 3 Wh ?
xxx nnnnnn 121 1... αααα ⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣ −−−−⎥⎦⎢⎣ −−
The derived dynamical equation observable as well if numerator and
The derived dynamical equation is controllable . Exersise 3: Why?
Ali Karimpour June 2012
38
denominator of g(s) are coprime. Exersise 4: Why?
-
Chapter 5
Controllable and observable canonical form realization of proper rational transfer functions
Example 5-2 Derive controllable and observable canonical realization for following system.
3248182 23 +++ sss
2202663248182)(223
++++++ ssssssg
61163248182)( 23 +++
+++=
sssssssg
261166116
)( 2323 ++++=
+++=
sssssssg
Observable canonical form realization is:
20600 ⎤⎡⎤⎡⎤⎡⎤⎡ &Controllable canonical form realization is:
0010 ⎤⎡⎤⎡⎤⎡⎤⎡ &u
xxx
xxx
62620
6101101600
3
2
1
3
2
1
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
&
&
&
uxxx
xxx
100
6116100010
3
2
1
3
2
1
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
&
&
[ ] uxxx
y 2100 21
+⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡= [ ] u
xxx
y 262620 21
+⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
Ali Karimpour June 2012
39
x3 ⎥⎦⎢⎣ x3 ⎥⎦⎢⎣It is not controllable. It is not observable.Why? Why?
-
Chapter 5
Irreducible realization of proper rational transfer functions
Example 5-3 Derive irreducible realization for following transfer function.
3248182 23 +++ sss
2202663248182)(223
++++++ ssssssg
61163248182)( 23 +++
+++=
sssssssg
2206 ++= s261166116
)( 2323 ++++=
+++=
sssssssg
Observable canonical form realization is: Controllable canonical form realization is:
2652+
++=
ss
uxx
xx
620
5160
2
1
2
1
⎤⎡
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡−−
=⎥⎦
⎤⎢⎣
⎡&
&u
xx
xx
10
5610
2
1
2
1
⎤⎡
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡−−
=⎥⎦
⎤⎢⎣
⎡&
&
[ ] uxx
y 2102
1 +⎥⎦
⎤⎢⎣
⎡= [ ] u
xx
y 26202
1 +⎥⎦
⎤⎢⎣
⎡=
Ali Karimpour June 2012
40
It is controllable too. It is observable too.Why? Why?
-
Chapter 5
Irreducible realization of proper rational transfer functions
Realization from the Hankel matrix
nn βββ −1
nnn
nnn
sssssg
ααβββ
++++++
= − ............)( 1
1
110 ......)3()2()1()0()( 321 ++++= −−− shshshhsg
The coefficients h(i) will be called Markov parameters.
⎥⎥⎤
⎢⎢⎡
+ )1(...)4()3()2()(...)3()2()1(
ββ
hhhhhhhh
⎥⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢⎢
+=
.......
.......)2(...)5()4()3(
),(β
βαhhhh
H
⎥⎥⎦⎢
⎢⎣ −+++ )1(...)2()1()(
.......βαααα hhhh
Ali Karimpour June 2012
41
-
Chapter 5
Irreducible realization of proper rational transfer functions
Realization from the Hankel matrix
Theorem 5-11 Consider the proper transfer function g(s) as
nnn
nnnn
sssss
sgαα
ββββ+++
++++= −
−−
............
)( 11
22
110
( ) ( ) ....,3,2,1,everyfor),(),( =++= lklmkmHmmH ρρ
then g(s) has degree m if and only if
Ali Karimpour June 2012
42
-
Chapter 5
Irreducible realization of proper rational transfer functions Realization from the Hankel matrixRealization from the Hankel matrix
buAxx +=&Now consider the dynamical equation
ebAsIcsebAsIcsg +−=+−= −−−− 1111 )()()(eucxy +=
g )()()(
.....3221 ++++= −−− bscAcAbscbse......,3,2,1)( 1 == − ibcAih i)(
⎥⎥⎥⎤
⎢⎢⎢⎡
+ )1(.....)2()(.....)2()1(
nhhnhhh
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
−+
=+
)12(.....)1()(................
),1(
nhnhnh
nnH
⎥⎥⎦⎢
⎢⎣ ++ )2(.....)2()1( nhnhnh
Let the first σ rows be linearly independent and the (σ+1) th row of H(n+1,n) be linearly dependent on its previous rows. So
Ali Karimpour June 2012
43
y p p
0),1(]0.....01.....[ 21 =+ nnHaaa σ
-
Chapter 5
Irreducible realization of proper rational transfer functions Realization from the Hankel matrixRealization from the Hankel matrix
0),1(]0.....01.....[ 21 =+ nnHaaa σWe claim that the σ-dimensional dynamical equationWe claim that the σ dimensional dynamical equation
hhh
)3()2()1(
0000000.....10000.....010
⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
uh
xx..
)3(
..........
..........00.....000
.
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
+
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
=& (I)
[ ] uhxyh
haaaaa
)0(00.....001)(
)1(.....
10.....000
1321
+=
⎥⎥⎥
⎦⎢⎢⎢
⎣
−⎥⎥⎥
⎦⎢⎢⎢
⎣ −−−−− − σσ
σσ
is a controllable and observable (irreducible realization).
[ ]
Exercise 5: Show that (I) is a controllable and observable (irreducible realization) of
Ali Karimpour June 2012
44
Exercise 5: Show that (I) is a controllable and observable (irreducible realization) of
eucxybuAxx
+=+=&
-
Chapter 5
Irreducible realization of proper rational transfer functions
Example 5-4 Derive irreducible realization for following transfer function.3248182)(
23 +++ ssssg6116
)( 23 +++=
ssssg
.....2303410141062)( 654321 ++−−+−+= −−−−−− sssssssg
⎤⎡
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
−−−−
−
=341014101410
14106
)3,4(H
⎥⎦
⎢⎣ −− 2303410
We can show that the rank of H(4,3) is 2. So [ ] 0)3,4(0156 =H
Hence an irreducible realization of g(s) is610⎥⎤
⎢⎡
+⎥⎤
⎢⎡
&
Ali Karimpour June 2012
45[ ] uxy
uxx
2011056
+=
⎥⎦
⎢⎣−
+⎥⎦
⎢⎣ −−
=
-
Chapter 5
Realization of Proper Rational Transfer Function Vectors
⎤⎡ )(
Consider the rational function vector
⎤⎡⎤⎡ )()
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
= .)()(
)(2
1
sgsg
sG⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
+⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
= .)()(
.)(2
1
2
1
sgsg
ee
sG
)
)
⎥⎥⎥
⎦⎢⎢⎢
⎣ )(.sgq ⎥
⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣ )(..sge qq
)
⎥⎥⎤
⎢⎢⎡
++++++
⎥⎥⎤
⎢⎢⎡
−−
−−
nnn
nnn
ssss
ee
ββββββ
....
....
1 22
221
21
12
121
11
2
1
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
++++
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
= −
n
nnn ss
sGαα
.
......
1
.
.)(
21
11
Ali Karimpour June 2012
46
⎥⎦
⎢⎣ +++
⎥⎦⎢⎣−−
qnn
qn
qq sse βββ ....2
21
1
-
Chapter 5
Realization of Proper Rational Transfer Function Vectors
⎥⎥⎥⎤
⎢⎢⎢⎡
++++++
⎥⎥⎥⎤
⎢⎢⎢⎡
−−
−−
nnn
nnn
ssss
ee
ββββββ
....
....
1 22
221
21
12
121
11
2
1
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ +++
++++
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
=
−−
−
qnn
qn
q
nnn
q ss
ss
e
sG
βββ
αα
......
.....1
.
.)(
22
11
11
ee
yy
nnn
nnn
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
−−
−−
...
...00
0...1000...010
2
1
21)2(2)1(22
11)2(1)1(11
2
1
ββββββββ
u
e
x
y
yuxx
nnn
⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣
+
⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣
=
⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
+
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
=..
.................
.
.
0..
1...000.............. 2
1)2()1(
21)2(2)1(222
ββββ
ββββ&
ey qqnqnqqnqnnn
⎦⎣⎦⎣⎦⎣⎥⎥⎦⎢
⎢⎣⎥
⎥⎦⎢
⎢⎣ −−−−
−−−−
...1... 1)2()1(121
ββββαααα
This is a controllable form realization of G(s).
Ali Karimpour June 2012
47We see that the transfer function from
u to yi is equal to nnn
inn
in
ii ss
sse
ααβββ
++++++
+ −−−
.........
11
22
11
-
Chapter 5
Realization of Proper Rational Transfer Function Vectors
Example 5-5 Derive a realization for following transfer function vector.
⎤⎡ + 3s
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
++
++=
34
)2)(1(3
)(
ss
sss
sG
⎥⎦
⎤⎢⎣
⎡
+++
++++⎥
⎦
⎤⎢⎣
⎡=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
++
+++
=)2)(1(
)3()3)(2)(1(
110
34
)2)(1(3
)(2
sss
ssssss
s
sG
⎥⎦
⎤⎢⎣
⎡
++++
++++⎥
⎦
⎤⎢⎣
⎡=
⎥⎦⎢⎣ +
2396
61161
10
3
2
2
23 ssss
sss
s
H i i l di i l li i f G( ) i i bHence a minimal dimensional realization of G(s) is given by
⎥⎤
⎢⎡
+⎥⎤
⎢⎡⎥
⎤⎢⎡
+⎥⎤
⎢⎡
016900
100010
&
Ali Karimpour June 2012
48
uxyuxx ⎥⎦
⎢⎣
+⎥⎦
⎢⎣
=⎥⎥⎥
⎦⎢⎢⎢
⎣
+⎥⎥⎥
⎦⎢⎢⎢
⎣ −−−=
113210
6116100
-
Chapter 5
Realization of Proper Rational Matrices
There are many approaches to find irreducible realizations for proper i l irational matrices.
1. One approach is to first find a reducible realization and then applypp pp ythe reduction procedure to reduce it to an irreducible one.
Method I, Method II, Method III, Method IV and Method V
2. In the second approach irreducible realization will yield directly.pp y y
Ali Karimpour June 2012
49
-
Chapter 5
Realization of Proper Rational Matrices
Method I: Given a proper rational matrix G(s), if we first find
an irreducible realization for every element gij(s) of G(s) as jijjiijij
jijijijij
uexcy
ubxAx
+=
+=&
an irreducible realization for every element gij(s) of G(s) as jijjiijijy
⎥⎤
⎢⎡⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
112
11
12
11
12
11
12
11
00
000000
ubb
xx
AA
xx&
&
⎤⎡
⎥⎦
⎤⎢⎣
⎡
⎥⎥⎥
⎦⎢⎢⎢
⎣
+
⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣
=
⎥⎥⎥
⎦⎢⎢⎢
⎣2
1
22
21
12
22
21
12
22
21
12
22
21
12
00
000000 u
bb
xx
AA
xx&
&
yyy +
12111 yyy +=
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡+
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
1
2221
1211
21
12
11
2221
1211
2
1
0000
uu
eeee
xxx
cccc
yy
22222 yyy +=
⎥⎦
⎢⎣ 22x
Clearly this equation is generally not controllable and not observable.
Ali Karimpour June 2012
50
To reduce this realization to irreducible one requires the application of the reductionprocedure twice (theorems 5-4 and 5-5).
-
Chapter 5
Realization of Proper Rational Matrices
⎥⎤
⎢⎡⎥⎥⎤
⎢⎢⎡
+⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
=⎥⎥⎥⎤
⎢⎢⎢⎡
112
11
12
11
12
11
12
11
00
000000
ubb
xx
AA
xx&
&
⎤⎡
⎥⎦
⎢⎣⎥⎥⎥
⎦⎢⎢⎢
⎣
+
⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣
=
⎥⎥⎥
⎦⎢⎢⎢
⎣
11
2
22
21
22
21
22
21
22
21
00
000000
x
ub
bxx
AA
xx&
&
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
1
2221
1211
21
12
11
2221
1211
2
1
0000
uu
eeee
xxx
cccc
yy
⎥⎦
⎢⎣ 22x
00000)(
0011
1
111
bbAsI
⎤⎡⎥⎤
⎢⎡⎥⎤
⎢⎡ −⎤⎡
−
Proof:
00
0
)(0000)(0000)(0
0000
2221
1211
22
21
12
122
121
112
2221
1211
eeee
bb
b
AsIAsI
AsIcc
cc⎥⎦
⎤⎢⎣
⎡+
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣ −−
−⎥⎦
⎤⎢⎣
⎡
−
−
−
Ali Karimpour June 2012
51)(
)()()()(
)()()()(
2221
1211
22221
222221211
2121
12121
121211111
1111 sGsgsgsgsg
ebAsIcebAsIcebAsIcebAsIc
=⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
+−+−+−+−
=−−
−−
-
Chapter 5
Realization of Proper Rational Matrices
Method II: Given a proper rational matrix G(s), if we find the controllable canonical-form realization for the ith column, Gi(s), of G(s) say,
iiiii ubxAx +=&
iiiii
iiiii
uexCy +=
uu
bb
xx
AA
xx
0...00...0
0....00....0
2
1
2
1
2
1
2
1
2
1
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡&
&
This realization is always
[ ] [ ]ueeexCCCyubxAx ppppp
....00
............00
...........22222
+=⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
+
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
=
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ &
This realization is alwayscontrollable. It is however generally not observable.
[ ] [ ]ueeexCCCy pp ........ 2121 +=Proof:
[ ] [ ]....0...00...0
0...)(00...0)(
.... 212
11
2
11
21 eeeb
bAsI
AsI
CCC +⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
−−
−
−
[ ] [ ]
)()()()(...)()(
....
...00......
)(...00......
....
11211
21
1
21
sgsgsg
eee
bAsI
CCC
p
p
pp
p
⎥⎤
⎢⎡
+
⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣ −−
Ali Karimpour June 2012
52
[ ] )()(...)()(
......)(...)()(
)(.....)(
21
22221111
111 sG
sgsgsg
sgsgsgebAsICebAsIC
qpqq
ppppp =
⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣
=+−+−= −−
-
Chapter 5
Realization of Proper Rational Matrices
Method III: Let a proper rational matrix G(s), where consider )()()( ∞+= GsGsG)
mmm)( 21the monic least common denominator of G(s) as mmmm ssss αααψ ++++= −− ...)( 2211
Then we can write G(s) as [ ] )(...)(
1)( 221
1 ∞++++=−− GRsRsR
ssG m
mm
ψ )(sψ
Then the following dynamic equation is a realization of G(s).
I 0000 ⎤⎡⎤⎡
uxI
I
xp
p
pppp
pppp
..00
.......0...000...00
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
+⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
=&
[ ] uGxRRRRyIIIII
I
p
p
ppmpmpm
pppp
)(
0......000
121
∞+=
⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣ −−−− −− αααα
Ali Karimpour June 2012
53
[ ] uGxRRRRy mmm )(... 121 ∞+= −−Exercise 6: Show that the above dynamical equation is a controllable realization of G(s)
-
Chapter 5
Realization of Proper Rational Matrices
Method IV: Gilbert diagonal representation.
S di ti t l i lSuppose distinct real eigenvalues.
Reminders
qisBpisCBCG kkkkkkk ××= ρρ
{ }IIdiA λλ{ }r
IIdiagA r ρρ λλ ,...,11=
[ ] ⎥⎤
⎢⎡B1
Ali Karimpour June 2012
54
[ ]rCCC ...1=⎥⎥⎥
⎦⎢⎢⎢
⎣
=
rBB M
-
Chapter 5
Realization of Proper Rational Matrices
Gilbert diagonal representation.
⎥⎤
⎢⎡ 121
⎥⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢⎢
⎣ ++++
+
++
+++
=
0)3)(2)(1(
)3(21
14
12
10411
)(
sss
ssssG
[ ]02101
⎥⎥⎤
⎢⎢⎡
[ ]01010
⎥⎥⎤
⎢⎢⎡
[ ]32100⎥⎤
⎢⎡
[ ]10011
⎥⎥⎤
⎢⎢⎡
⎥⎦
⎢⎣ ++++ )3)(2)(1(1 ssss
⎥⎤
⎢⎡
⎥⎤
⎢⎡
⎥⎤
⎢⎡
⎥⎤
⎢⎡
100100
1000000
1010000
1000021
1)(G
[ ]02110⎥⎥⎦⎢
⎢⎣
[ ]0102
1⎥⎥⎦⎢
⎢⎣−
[ ]32100⎥⎥⎥
⎦⎢⎢⎢
⎣
[ ]10001⎥⎥⎦⎢
⎢⎣
⎥⎥⎥
⎦⎢⎢⎢
⎣+
+⎥⎥⎥
⎦⎢⎢⎢
⎣+
+⎥⎥⎥
⎦⎢⎢⎢
⎣ −+
+⎥⎥⎥
⎦⎢⎢⎢
⎣+
=000100
4000000
3020010
2021000
1)(
sssssG
⎥⎤
⎢⎡
⎥⎤
⎢⎡− 0210001 ⎤⎡⎤⎡− 021001
uxx
⎤⎡
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
+
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ −−
−=
100321010
400003000020
& uxx
⎥⎤
⎢⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−=
101
100010021
400020001
&
⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒nrealizatio minimal
Ali Karimpour June 2012
55xy⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
002110101001 xy
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
021110
-
Chapter 5
Realization of Proper Rational Matrices
Gilbert diagonal representation.
( ) ⎤⎡ + 421 s( )( )( )
( )( ) ⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+++−
+++
+=
21
211
4142
11
)(
sss
sss
ssG
⎥⎤
⎢⎡⎥⎤
⎢⎡ 0121 [ ]110⎥
⎤⎢⎡ [ ]110⎥
⎤⎢⎡
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡=
001001211)(sG
⎥⎦
⎢⎣⎥⎦
⎢⎣− 1001
[ ]111⎥⎦⎢⎣
[ ]110⎥⎦⎢⎣
⎥⎦
⎢⎣+
⎥⎦
⎢⎣+
⎥⎦
⎢⎣−+ 004112011
)(sss
⎥⎤
⎢⎡
⎥⎤
⎢⎡− 010001
⎤⎡⎤⎡uxx
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
+
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ −−
−=
111110
400002000010
& uxx
⎤⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−=
021
111001
200010001
&
⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒⇒nrealizatio minimal
Ali Karimpour June 2012
56xy ⎥⎦
⎤⎢⎣
⎡−
=01010021 xy ⎥
⎦
⎤⎢⎣
⎡−
=101021
-
Chapter 5
Realization of Proper Rational Matrices
Gilbert diagonal representation.
⎥⎤
⎢⎡
⎥⎤
⎢⎡− 0000012
( ) ( ) ( ) ⎥⎤
⎢⎡ 1111
kd
uxx
⎤⎡
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
+
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ −−
−=
321000000
200012000120
& ( ) ( ) ( )
( ) ( )
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
⎥⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢⎢
+++
++++
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
1100
21
21
210
2222
)(32
432
sss
ssss
iflhebkgda
sG
xmifclhebkgda
y⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
( ) ⎥⎦⎢⎣
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ +
++⎥⎦⎢⎣ 321
21000
2200 2
s
ssmifc
⎤⎡⎤⎡⎤⎡⎤⎡ kgda
( )[ ]
( )[ ]
( )[ ] [ ]321
21321
21321
21321
21)( 234
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
++
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
++
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
++
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+=
mlk
sihg
sfed
scba
ssG
Ali Karimpour June 2012
57
-
Chapter 5
Realization of Proper Rational Matrices
Method IV: Gilbert diagonal representation.
R titi l i lRepetitive real eigenvalues.
r1 Jordan block of order 3r1 Jordan block of order 3
r2 - r1 Jordan block of order 2
r3 – r2 Jordan block of order 1
Ali Karimpour June 2012
58
-
Chapter 5
Realization of Proper Rational Matrices
Gilbert diagonal representation. Repetitive real eigenvalues.
2 Jordan block of order 2
0 J d bl k f d 10 Jordan block of order 1
Ali Karimpour June 2012
59
-
Chapter 5
Realization of Proper Rational Matrices
Gilbert diagonal representation. Repetitive real eigenvalues.
Ali Karimpour June 2012
60
-
Chapter 5
Realization of Proper Rational Matrices
Gilbert diagonal representation. Repetitive real eigenvalues.
:),4(B)1(:,C
:),4(B
:),4(B)2(:,C )5(:,C
Ali Karimpour June 2012
61
),(
:),7(B
-
Chapter 5
Realization of Proper Rational Matrices
Gilbert diagonal representation. Repetitive real eigenvalues.
⎤⎡⎤⎡ 211100:),4(B)3(:,C )5(:,C
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
101101211
351000100
3M :),7(B
:),9(B)8(:,C ),()8(:,C
:)4(B)4(:,C )7(:,C
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−=
101101211
010000100
4M
:),4(B)(
:),7(B
)(
)9(B)9(C
Ali Karimpour June 2012
62
:),9(B)9(:,C
-
Chapter 5
Realization of Proper Rational Matrices
Gilbert diagonal representation. Repetitive real eigenvalues.
Ali Karimpour June 2012
63
-
Chapter 5
Realization of Proper Rational Matrices
Method V: It is possible to obtain observable realization of a proper G(s). Let
)2()1()0()( 21 HHHG ...........)2()1()0()( 21 +++= −− sHsHHsG
Consider the monic least common denominator of G(s) as
mmmm ssss αααψ ++++= −− ...)( 22
11
Then after deriving H(i) one can simply show
Exercise 7: Proof equation (I)
)(1)(...)2()1()( 21 IiiHimHimHimH m ≥−−−+−−+−=+ ααα
)()( 32211 BCACABCBEBAICEG
Let {A, B, C and E} be a realization of G(s) then we have
Ali Karimpour June 2012
64
...........)()( 32211 ++++=−+= −−−− BsCACABsCBsEBAsICEsG
-
Chapter 5
Realization of Proper Rational Matrices
Then {A, B, C and D} be a realization of G(s) if and only if
210)1()0( iBCAiHdHE i ,....2,1,0)1()0( ==+= iBCAiHandHE i
Now we claim that the following dynamical equation is a realization of G(s).
HI )1(000 ⎤⎡⎤⎡
uHH
xI
I
xqqqq
qqqq
..)2()1(
.......0...000...00
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
+⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
=&
[ ] uHxIymH
mHIIII
I
q
qqmqmqm
qqqq
)0(0...00
)()1(
...
...000
121
+=
⎥⎥⎥
⎦⎢⎢⎢
⎣
−⎥⎥⎥
⎦⎢⎢⎢
⎣ −−−− −− αααα
[ ]y q )(We can readily verify that
⎥⎤
⎢⎡
++
⎥⎤
⎢⎡
⎥⎤
⎢⎡
)2()1(
)4()3(
)3()2(
iHiH
HH
HH
)1(iHBCAi
Ali Karimpour June 2012
65⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ +
+=
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ +
=
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ +
=
)(...
)2(,...,
)2(...
)4(,
)1(...
)3( 2
imH
iHBA
mH
HBA
mH
HAB i )1( += iHBCA
i
)0(HE =
-
Chapter 5
Realization of Proper Rational Matrices
Now we shall discuss in the following a method which will yield directly irreducible
realizations. This method is based on the Hankel matrices.
...........)2()1()0()( 21 +++= −− sHsHHsGLet G(s) be pq ×
Consider the monic least common denominator of G(s) as
mmmm ssss αααψ ++++= −− ...)( 22
11Define
⎥⎥⎥⎤
⎢⎢⎢⎡
qqqq
qqqq
II
0...000...00
⎥⎥⎥⎤
⎢⎢⎢⎡
−−
− pmppp
pmppp
IIII
I
1
000...00...00
αα
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ −−−−
=
−− qqmqmqm
qqqq
IIIII
M
121 ......000
.......
αααα⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ −
−= −
pppp
pmppp
II
IIN
1
2
...00.........
.0.....0
α
α
Ali Karimpour June 2012
66
-
Chapter 5
Realization of Proper Rational Matrices
mmmm ssss αααψ ++++= −− ...)( 22
11
⎥⎤
⎢⎡ qqqq I 0...00
⎥⎤
⎢⎡ − pmppp I0...00 α
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
=
qqqq
qqqq
I
IM
...000.......
0...00
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
−−
= −−
pmppp
pmppp
pppp
IIII
N 21
..........0.....0
0...0αα
⎥⎥⎦⎢
⎢⎣ −−−− −− qqmqmqm
qqqq
IIII 121 ... αααα⎥⎥⎦⎢
⎢⎣ − pppp II 1...00 α
We also define the two following Hankel matrices
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
+=
...)1()3()2(
)()2()1(mHHH
mHHH
T
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
++
=...
)2()4()3()1()3()2(
~ mHHHmHHH
T
⎥⎦
⎢⎣ −+ )12()1()( mHmHmH
⎥⎦
⎢⎣ ++ )2()2()1( mHmHmH
It can be readily verified that
Ali Karimpour June 2012
67TNMTT ==~ ,.....2,1,0== iTNTM ii
-
Chapter 5
Realization of Proper Rational Matrices
It can be readily verified that
TNMTT ==~ ,.....2,1,0== iTNTM iiTNMTT == ,.....2,1,0iTNTM
[ ]0II =lkI , )( kllk >×Let be as the form
[ ]0, klk II =Note that the left-upper-corner of M iT = TN i is H(i+1) so:
TT ....,2,1,0)1( ,,,, ===+ iITNITIMIiHT
pmpi
qmqT
pmpi
qmq
It can be readily verified that
Ali Karimpour June 2012
68But we want Irreducible Realization of Proper Rational Matrices
-
Chapter 5
Irreducible Realization of Proper Rational Matrices
Ali Karimpour June 2012
69
-
Chapter 5
Irreducible Realization of Proper Rational Matrices
Example 5-6 Derive an irreducible realization for the following proper rational function.
⎤⎡ −−− 1232 2 ss
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−−
+++=
153
154
1)1(
232
)(2
ss
ss
ssss
sG
⎦⎣ ++ 11 ss
2)1()( += sssψLeast common denominator of G(s), is
06050403021102 ⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎡.....
2106
2105
.2104
2103
2102
2111
3402
)( 654321 −−−−−− ⎥⎦
⎤⎢⎣
⎡−−
+⎥⎦
⎤⎢⎣
⎡−
+⎥⎦
⎤⎢⎣
⎡−−
+⎥⎦
⎤⎢⎣
⎡−
+⎥⎦
⎤⎢⎣
⎡−−
+⎥⎦
⎤⎢⎣
⎡−
+⎥⎦
⎤⎢⎣
⎡−
−= sssssssG
⎤⎡ 030211 Non-zero singular values of T
⎥⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎢⎡
−−−−−
−−−−
=⎥⎥⎥⎤
⎢⎢⎢⎡
=212121040302212121
030211
)4()3()2()3()2()1(
HHHHHH
T
gare 10.23, 5.79, 0.90 and 0.23.
So, r = 4.
Ali Karimpour June 2012
70⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ −−−−
⎥⎦⎢⎣
212121050403212121
)5()4()3( HHH,
-
Chapter 5
Irreducible Realization of Proper Rational Matrices
⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
0 10260 09780 10570 54960.5306-0.6022- 0.58720.10490.6574- 0.4905 0.0196-0.4003-
0 82640 10540 20780 51270.0071-0.0581-0.5238-0.2357-0.1621-0.8902-0.25450.3413-
⎥⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢⎢
⎣
−=
⎥⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢⎢
⎣ −−−
=
0.18880.38750.5432 0.13820.4522 0.2949-0.23110.6989- 0.1888-3875.00.5432-0.1382-
0.10260.0978-0.1057-0.5496
0071.00581.00.5238-0.2357-5392.04316.00.26270.6738-
0071.00581.0052380.23570.8264-0.1054-0.2078-0.5127
rr UY
⎥⎦⎢⎣⎥⎦⎢⎣
⎥⎥⎤
⎢⎢⎡
0.0034- 0.0551- 1.2598- 0.7539- 0.0770- 0.8443- 0.6121 1.0915-
⎥⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢⎢
==
0.2560- 0.4093 0.6317 2.1553- 0.0034 0.0551 1.2598 0.7539
0.3923- 0.1000- 0.4999- 1.6398 ˆ 2/1r SYY
⎥⎥⎥⎤
⎢⎢⎢⎡
==
⎥⎥⎦⎢
⎢⎣
1.3066 0.5557 1.3066- 0.2543- 1.4124 0.0471- 0.4421 2.2356- 0.4421- 1.7579 0.3355 1.2803-
ˆ
0.0034- 0.0551- 1.2598- 0.7539-
2/1 HUSU
Ali Karimpour June 2012
71⎥⎥
⎦⎢⎢
⎣ 0.0896 0.2147 0.0896- 0.0487 0.2519- 0.3121- 0.3675 0.2797- 0.3675- 0.0927- 0.5711- 0.4652 r
USU
-
Chapter 5
Irreducible Realization of Proper Rational Matrices
⎤⎡ 0 07370 21070 07370 16030 07370 1067
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
== −
0.0149- 1.1356- 0.0149 1.7406- 0.0149- 0.3415- 0.0613- 0.4551 0.0613 0.1112- 0.0613- 0.9386- 0.2178- 0.1092 0.2178 0.0864- 0.2178- 0.1058 0.0737- 0.2107- 0.0737 0.1603 0.0737- 0.1067-
ˆ 2/1† HrYSY
⎦⎣
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
== −0.2161 0.1031- 0.0440- 0.1718 1.1176- 0.6349- 0.2441 0.0328 1.3847- 0.5171 0.0081- 0.1251-
ˆ 2/1† SUU
⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣ 0.3976 0.4086 0.2259 0.0432 0.9525 0.3109- 0.0961 0.2185-
0.3976- 0.4086- 0.2259- 0.0432- SUU r
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
==0.8076 0.2888- 0.1800- 0.2227-
0.0772 0.1604- 1.0139- 0.1588 0.1904- 0.2155 0.0369 1.2497-
ˆ~ˆ †† UTYA
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
==0.5711- 0.4652 1.4124 0.0471- 0.3355 1.2803-
ˆ,
TpmpIUB
⎥⎦
⎢⎣ 0.4476- 0.1354 0.1181- 0.1246
⎥⎦
⎢⎣ 0.2519- 0.3121-
⎥⎤
⎢⎡
==0.0770- 0.8443- 0.6121 1.0915-
ŶIC ⎥⎤
⎢⎡−
==02
)0(HE
Ali Karimpour June 2012
72
⎥⎦
⎢⎣
==0.0034- 0.0551- 1.2598- 0.7539- ,
YIC qmq ⎥⎦
⎢⎣ −
==34
)0(HE
-
Chapter 5
Exercises
Ali Karimpour June 2012
73