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204 CHAPTER 11. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-3
with z 0 (ρ 2 ),z 1 (ρ 2 ),...,z η1 (ρ 2 ) all polynomial in ρ 2 with a maximal degree of η 2 .The
idea is to substitute this expression for z(ρ 1 ,ρ 2 ) into (11.22), which leads to:
(
) (z0
P (ρ 2 ) T + ρ 1 Q(ρ 2 ) T (ρ 2 ) − ρ 1 z 1 (ρ 2 )+ρ 2 1z 2 (ρ 2 )+...+(−1) η1 ρ η1
1 z η 1
(ρ 2 ) ) =
1
(
·
(
+P (ρ 2 ) T z 0 (ρ 2 )
)
)
+
ρ 1 ·
(
−P (ρ 2 ) T z 1 (ρ 2 ) +Q(ρ 2 ) T z 0 (ρ 2 )
)
+
ρ 2 1 · +P (ρ 2 ) T z 2 (ρ 2 ) −Q(ρ 2 ) T z 1 (ρ 2 ) +
.
.
.
(
)
.
ρ η1
1 ·
(
(−1) η1 P (ρ 2 ) T z η1 (ρ 2 ) (−1) η1−1 Q(ρ 2 ) T z η1−1(ρ 2 )
)
+
ρ η1+1
1 ·
(−1) η1 Q(ρ 2 ) T z η1 (ρ 2 ) =0
(11.24)
When the relationships for the coefficients are worked out, this generates a system
of equations which can be expressed in matrix-vector form as follows:
⎛
P (ρ 2 ) T 0 ... ... 0
⎞
. ⎛
Q(ρ 2 ) T P (ρ 2 ) T ..
0 Q(ρ 2 ) T . .. . ..
. 0 0 .. . .. . ..
⎜
⎝
⎜
⎝
.
.
.
.. P (ρ2 ) T ⎟
⎠
0 0 ... ... Q(ρ 2 ) T
⎛
M η1,η 2
(ρ 2 )
⎜
⎝
z 0 (ρ 2 )
−z 1 (ρ 2 )
z 2 (ρ 2 )
.
(−1) η1 z η1 (ρ 2 )
z 0 (ρ 2 )
−z 1 (ρ 2 )
z 2 (ρ 2 )
.
.
(−1) η1 z η1 (ρ 2 )
⎞
=
⎟
⎠
⎞
=0
⎟
⎠
(11.25)
where the block matrix M η1,η 2
(ρ 2 ) has dimensions (η 1 +2) m × (η 1 +1) n. The rank
of the matrix M η1,η 2
(ρ 2 ) is denoted by r η1,η 2
and it holds that r η1,η 2
< (η 1 +1)n.
Since the variable η 2 denotes the degree of ρ 2 in z(ρ 1 ,ρ 2 ), z i (ρ 2 ) can be written
as:
z i (ρ 2 )=z i,0 + ρ 2 z i,1 + ρ 2 2 z i,2 + ...+ ρ η2
2 z i,η 2
, (11.26)
for i =0,...,η 1 .
When (11.26) is substituted into the equations of (11.25) and the relationships
for the coefficients are worked out, a system of equations can be set up which only
involves constant coefficients. When P (ρ 2 ) T = B T + ρ 2 D T and Q(ρ 2 ) T = C T are
plugged into the expressions, the resulting system of equations can be written in