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142 CHAPTER 8. H 2 MODEL-ORDER REDUCTION
One may now take the n coefficients of the monic polynomial a(s) together with
the n coefficients of b(s) and the N − n coefficients of q(s) as the N + n optimization
parameters in conjunction with the N + n equations associated with the terms in
(8.18). This leads to a polynomial system of equations, which is structured in the
sense that the coefficients of b(s) show up in a linear fashion, but which may still be
quite hard to solve algebraically because the system is not in Gröbner basis form. By
means of a suitable reparameterization this structure can be reworked.
When the model-order is reduced by k from order N to order n = N − k, then
the degree of q(s) is at most k − 1. We shall employ the notation:
q(s) =q 0 + q 1 s + ...+ q k−1 s k−1 0. (8.19)
A reparameterization is used which involves the N zeros δ 1 ,...,δ N of the given polynomial
d(s), which upon substitution into Equation (8.18) enables the elimination of
the polynomial b(s) from the problem. As stated before, we here require the technical
assumption that the N zeros δ 1 ,...,δ N of d(s) are all distinct.
We first focus on the generic situation where q 0 0. For i =0, 1,...,k − 1we
define ρ i = qi
q 0
and let the polynomial ρ(s) be:
ρ(s) = q(s) =1+ρ 1 s + ρ 2 s 2 + ...+ ρ k−1 s k−1 . (8.20)
q 0
Note that ρ(s) is of degree ≤ k − 1 and that ρ 0 = q0
q 0
=1.
Then Equation (8.18) can be manipulated as follows. Multiplying both sides by
q 0 and introducing:
ã(s) :=q 0 a(−s) (8.21)
yields:
e(s)ã(−s) − q 0 b(s)d(s) =ã(s) 2 ρ(s). (8.22)
Note that ã(s) is not monic, in contrast to a(s).
Evaluation of Equation (8.22) at the N distinct zeros δ 1 ,...,δ N of the polynomial
d(s) gives:
e(δ i )ã(−δ i )=ã(δ i ) 2 ρ(δ i ), (i =1,...,N). (8.23)
Note that in Equation (8.23) it holds that ã(δ i ) 0, and that the equations no longer
involve the polynomial b(s). This equation constitutes a system of N equations in N
unknowns.
When studying the H 2 model-order reduction problem for co-order k, the polynomial
ã(s) is, for future computational convenience, treated as a polynomial of degree
≤ N − 1, while its degree is actually N − k. Fork = 1 we write:
ã(s) =ã N−1 s N−1 +ã N−2 s N−2 + ...+ã 1 s +ã 0 . (8.24)
For k > 1 we use a similar notation, where we then have to add the additional
constraints:
ã N−1 = ···=ã N−k+1 =0. (8.25)