link to my thesis

218 CHAPTER 11. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-3
terms in ρ 1 and ρ 2 are worked out again, the following system of equations emerges:
⎛ ⎞
z 0,0
.
z 0,η2
M η1,η 2 .
= 0 (11.67)
z η1,0
⎜
⎝
⎟
. ⎠
z η1,η 2
where the matrix M η1,η 2
is a block matrix containing the coefficient matrices B 0 ,...,
BÑ, and C 0,0 ,...,CÑ−1,0 from the Equations (11.60) and (11.61). The dimension of
this matrix is 1 2 (η 1 + 2)(η 2 + Ñ + 1)(η 2 + Ñ +2)m × (η 1 + 1)(η 2 +1)n. Note that
this matrix is independent of ρ 1 and ρ 2 . Every non-zero vector in the kernel of the
matrix M η1,η 2
provides a vector z(ρ 1 ,ρ 2 ) which is a solution of (11.59).
Remark 11.6. The system of equations (11.67) in general is too restrictive in comparison
with the approach of the previous section. This is due to the fact that the
powers of ρ 1 and ρ 2 play a simultaneous role in the equations in (11.67). This is the
reason why this approach does not work in general. However, this approach is useful
because when the system of equations (11.65) admits a solution, we have a solution
for the system of equations (11.44) of the previous subsection too. In this way we can
still split off a singular block of the matrix pencil. Note that the system of equations
(11.44) may have solutions which are not of this more ‘restrictive’ form.
Example 11.1 (continued). Using this approach, A(ρ 1 ,ρ 2 ) T z(ρ 1 ,ρ 2 ) = 0 is written
as: (
)
B(ρ 2 )+ρ 1 C(ρ 1 ,ρ 2 ) z(ρ 1 ,ρ 2 ) = 0 (11.68)
where
and
B(ρ 2 )=B 0 + ρ 2 B 1 + ρ 2 2 B 2 + ρ 3 2 B 3 (11.69)
C(ρ 1 ,ρ 2 )= ( C 0,0 + ρ 2 C 0,1 + ρ 2 2 C 0,2
)
+ ρ1 (C 1,0 + ρ 2 C 1,1 )+ρ 2 1 (C 2,0 ) . (11.70)
We know that every solution z(ρ 1 ,ρ 2 ) should take the form z 0 (ρ 2 ) − ρ 1 z 1 (ρ 2 )
because η 1 is chosen as 1. If such a solution is substituted into (11.68), and the
relationships of the coefficients of the powers of ρ 1 are equated to zero, it can be
written as:
⎛
⎞
B(ρ 2 ) 0 ( )
( )
⎝ C(ρ 1 ,ρ 2 ) B(ρ 2 ) ⎠ z0 (ρ 2 )
z0 (ρ
= M 1,2 (ρ 1 ,ρ 2 )
2 )
= 0 (11.71)
−z 1 (ρ 2 )
−z 1 (ρ 2 )
0 C(ρ 1 ,ρ 2 )
where the dimension of the block matrix M 1,2 (ρ 1 ,ρ 2 )is(η 1 +2) m × (η 1 +1) n =
3 m × 2 n.