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8.5. STETTER-MÖLLER FOR H 2 MODEL-ORDER REDUCTION 147
which can also be denoted by V H (x 1 ,...,x N ,ρ 1 ,...,ρ k−1 ) as:
This proves the theorem.
V H (x 1 ,...,x N ,ρ 1 ,...,ρ k−1 )=
N∑
i=1
ρ(δ i ) 2 ρ(−δ i )
e(δ i )d ′ (δ i )d(−δ i ) x3 i . (8.43)
For k = 1 and ρ(s) ≡ 1, Equation (8.43) specializes to the known results of
Theorem 5.2 in [50].
Note that for fixed values of ρ 1 ,ρ 2 ,...,ρ k−1 , the coefficients of this homogeneous
polynomial in x 1 ,x 2 ,...,x N of total degree 3 are all computable from the given
transfer function H(s) of order N.
8.5 Stetter-Möller matrix method for H 2 model-order reduction
In this section an approach is given to solve the system of equations (8.32) for
the quantities x 1 ,x 2 ,...,x N and ρ 1 ,...,ρ k−1 by applying the Stetter-Möller matrix
method as described in Section 3.3. This method gives rise to an eigenvalue
problem in terms of the k − 1 coefficients of ρ(s). For k>1, there will be k − 1
additional constraints because the solutions for x 1 ,x 2 ,...,x N are only feasible when
the coefficients ã N−1 ,...,ã N−k+1 of the polynomial ã(s) are all zero. In other words,
we intend to find values for ρ 1 ,...,ρ k−1 for which the system of equations admits
non-zero solutions for x 1 ,x 2 ,...,x N which make all the coefficients ã N−1 ,...,ã N−k+1
equal to zero. As said before, the coefficients ã N−1 ,...,ã N−k+1 , can be expressed
linearly in terms of the quantities x 1 ,x 2 ,...,x N , with coefficients that constitute the
last k − 1 rows of the matrix V (δ 1 ,δ 2 ,...,δ N ) −1 .
Thus, by applying the Stetter-Möller matrix method as introduced in Chapter 3,
we intend to transform the system of N equations:
⎛ ρ(δ 1) ⎞
⎛ ⎞
e(δ 1) x2 1
x 1
ρ(δ 2)
e(δ 2) x2 2
x 2
= M(δ
⎜
⎝ .
⎟ 1 ,...,δ N )
⎜
, (8.44)
⎠
⎝ .
⎟
⎠
ρ(δ N )
e(δ N ) x2 N
subject to the k − 1 additional constraints:
⎛ ⎞
⎛
ã N−1
ã N−2
⎜ . ⎟
⎝ . ⎠ = Γ(δ 1,...,δ N ) ⎜
⎝
ã N−k+1
x 1
x 2
.
.
x N
x N
⎞ ⎛
⎟
⎠ = ⎜
⎝
0
0
.
.
0
□
⎞
⎟
⎠ , (8.45)
into a structured polynomial eigenvalue problem of size 2 N × 2 N involving the k − 1
parameters in the polynomial ρ(s). Here the matrix Γ(δ 1 ,...,δ N ) is defined to consist
of the last k − 1 rows of the matrix V (δ 1 ,...,δ N ) −1 in reversed order: