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208 CHAPTER 11. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-3
(11.31), the transformation matrices V (ρ 2 ) and W (ρ 2 ) are constructed as follows:
⎛
⎞
W (ρ 2 )=
P (ρ 2 ) T z 1 (ρ 2 ) w 2 ... w m ,
⎜
⎟
⎝
⎠
⎛
V (ρ 2 )=
⎜
⎝
z 0 (ρ 2 ) z 1 (ρ 2 ) v 3 ... v n ⎟
⎠
⎞
(11.33)
where P (ρ 2 ) T z 1 (ρ 2 ), z 0 (ρ 2 ) and z 1 (ρ 2 ) are the first columns of the matrices W (ρ 2 )
and V (ρ 2 ) and where w 2 ,...,w m and v 3 ,...,v n are basis column vectors to complete
these matrices up to square invertible matrices.
We know from the equations in (11.25) that the following holds: P (ρ 2 ) T z 0 (ρ 2 )
=0,(−1) η1 Q(ρ 2 ) T z η1 (ρ 2 ) = 0 and P (ρ 2 ) T z i (ρ 2 )=Q(ρ 2 ) T z i−1 (ρ 2 ) for i =1,...,η 1 .
In this example this means the following:
⎧
P (ρ 2 )
⎪⎨
T z 0 (ρ 2 )=0
Q(ρ 2 ) T z 1 (ρ 2 )=0
(11.34)
⎪⎩
P (ρ 2 ) T z 1 (ρ 2 )=Q(ρ 2 ) T z 0 (ρ 2 )
The required transformation here is:
W (ρ 2 ) −1( ) P (ρ 2 ) T + ρ 1 Q(ρ 2 ) T V (ρ 2 ). (11.35)
)
This yields for W (ρ 2 )
(P −1 (ρ 2 ) T V (ρ 2 )+ρ 1 Q(ρ 2 ) T V (ρ 2 ) the following:
⎛
⎜
⎝
P (ρ 2 ) T z 1 (ρ 2 ) w 2 ... w m ⎟
⎠
⎞
−1 ⎛
⎜
⎝
ρ 1 P (ρ 2 ) T z 1 (ρ 2 ) P (ρ 2 ) T z 1 (ρ 2 ) ...
⎞
⎟
⎠
(11.36)