ii. ObserverForm Realization (continue)
(1) Rowreduced NonSingular Matrix of Polynomials
─
given a coprime LMFD,
G
(
s
)=
Q
L

1
P
L
.
─
Q
L
can be made rowreduced through
elementary row operations
.
Let
l
i
be the degree of the
i
th
row of
P
L
and
p
i
be the degree of the
i
th
row of
Q
L
,
i
= 1, 2, ...,
p
.
G
(
s
) is proper if and only
if
.
,
,
2
,
1
,
p
i
p
l
i
i
L
=
≤
G
(
s
) is strictly proper if and
only if
.
,
,
2
,
1
,
p
i
p
l
i
i
L
=
<
(2) Properness of Matrices of Rational Functions
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(3) Coefficient Matrices of a LMFD
be strictly proper, coprime, and with
Q
L
being
rowreduced.
Let
Q
L
be represented by its rows as shown:
L
L
P
Q
s
G
1
)
(
−
=
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
Lp
L
L
L
Q
Q
Q
Q
r
M
r
r
2
1
Assume that
{
}
.
,
,
2
,
1
,
deg
p
i
Q
p
Let
Li
i
L
r
=
=
We have
{
}
{
}
p
p
p
p
p
p
p
p
hr
Lhr
hr
Lhr
p
p
p
p
rpp
p
rp
p
rp
p
p
r
p
r
p
r
p
p
r
p
r
p
r
r
r
r
L
s
s
s
diag
s
D
where
Q
s
D
Q
s
s
s
diag
s
h
s
h
s
h
s
h
s
h
s
h
s
h
s
h
s
h
s
H
where
s
L
s
H
s
Q
L
L
L
M
O
M
M
L
L
2
1
2
1
2
2
2
1
1
1
)
(
)
(
)
(
),
(
)
(
)
(
2
1
2
22
21
1
12
11
=
⋅
=
⋅
=
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
+
=
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