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216 CHAPTER 11. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-3
properties. First, the transformation matrix W in this approach contains only one
variable ρ 2 , whereas it contained both the variables ρ 1 and ρ 2 in the previous subsection.
Because the matrix W only contains one variable, it is (i) less challenging in
a computational point of view to compute the inverse of W and (ii) much easier to
compute the values which make the matrix W singular.
Consider the transpose of the matrix A(ρ 1 ,ρ 2 ) in Equation (11.39) and let us
denote its dimensions by m × n. The variable Ñ is the total degree of ρ 1 and ρ 2 in
the polynomial matrix A(ρ 1 ,ρ 2 ) T . In the first step, Ñ is therefore equal to N − 1
where N is the order of the original system H(s). Let us start by expanding the
polynomial matrix as follows:
A(ρ 1 ,ρ 2 ) T =
(
)
B(ρ 2 )+ρ 1 C(ρ 1 ,ρ 2 )
(11.59)
where the matrix B(ρ 2 ) is:
B(ρ 2 )=B 0 + ρ 2 B 1 + ρ 2 2 B 2 + ...+ ρÑ2 BÑ (11.60)
and where C(ρ 1 ,ρ 2 ) is:
C(ρ 1,ρ 2)=
(
)
1 C 0,0 + ρ 2C 0,1 + ... + ... + ρÑ−1 2 C 0, Ñ−1 +
)
ρ 1
(C 1,0 + ρ 2C 1,1 + ... + ρÑ−2 2 C 1, Ñ−2
+
.
. . .. . ) .
(CÑ−2,0 + ρ 2C N−3,1 +
ρÑ−2 1
ρÑ−1 1
(CÑ−1,0
)
(11.61)
The rank r of the matrix A(ρ 1 ,ρ 2 ) T will be r