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144 CHAPTER 8. H 2 MODEL-ORDER REDUCTION
of d(s) as follows:
⎛
⎜
⎝
ã(−δ 1 )
ã(−δ 2 )
.
ã(−δ N )
⎞
⎛
⎟
⎠ = V (−δ 1,...,−δ N )V (δ 1 ,...,δ N ) −1 ⎜
⎝
x 1
x 2
. .
x N
⎞
⎟
⎠ . (8.30)
or:
Hence, Equation (8.23) attains the following form:
⎛ ⎞
ρ(δ 1)
⎛
e(δ 1) x2 1
ρ(δ 2)
e(δ 2) x2 2
= V (−δ
⎜
1 ,...,−δ N ) V (δ 1 ,...,δ N ) −1 . ⎟
⎜
⎝ ⎠
⎝
ρ(δ N )
e(δ N ) x2 N
⎛
⎜
⎝
ρ(δ 1)
e(δ 1) x2 1
ρ(δ 2)
e(δ 2) x2 2
.
ρ(δ N )
e(δ N ) x2 N
⎞
⎛
= M(δ 1 ,...,δ N )
⎟
⎜
⎠
⎝
x 1
x 2
.
x N
⎞
⎟
⎠
x 1
x 2
.
x N
⎞
⎟
⎠
(8.31)
(8.32)
in which the matrix M is given by M(δ 1 ,..., δ N )=V (−δ 1 ,..., −δ N ) V (δ 1 ,...,
δ N ) −1 . Note that the quantities e(δ i ), for i =1,...,N, are all non-zero because e(s)
and d(s) are co-prime. The matrix M is entirely available as a (numerical) matrix
since the entries can easily be computed explicitly in terms of the poles δ 1 ,...,δ N of
H(s) only. The coefficients in the equations in system (8.32) can be complex. This
is the case when H(s) has a complex conjugate pair of poles.
Theorem 8.1. The entry m i,j in row i and column j of the matrix M(δ 1 ,..., δ N )
is given by the non-zero quantity:
where d(s) =(s − δ 1 )(s − δ 2 ) ···(s − δ N ).
d(−δ i )
m i,j = −
(δ i + δ j ) d ′ (δ j ) , (8.33)
Proof. The entries in row i of the matrix M(δ 1 ,...,δ N ) are obtained by solving the
linear system of equations:
( )
mi1 m i2 ... m iN V (δ1 ,...,δ N )= ( 1 −δ i ... (−δ i ) ) N−1 (8.34)
According to Cramer’s rule, the entry m i,j is given by:
m i,j = det V (δ 1,δ 2 ,...,δ j−1 , −δ i ,δ j+1 ,...,δ N )
. (8.35)
det V (δ 1 ,δ 2 ,...,δ N )