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150 CHAPTER 8. H 2 MODEL-ORDER REDUCTION
• Feasibility requires furthermore q 0 to be non-zero, in which case a(−s) is determined
as ã(s)/q 0 . The polynomial a(s) of degree N − k should be Hurwitz.
Note that only solutions x 1 ,...,x N which yield a real-valued polynomial a(s)
are of importance, which requires the parameters x i to be either real or to occur
as complex conjugate pairs. Note also that feasibility requires ρ 1 ,...,ρ k−1 to be
real-valued.
• Finally, the polynomial b(s) is obtained from Equation (8.18) as follows:
b(s) = e(s)a(s) − q 0a(−s) 2 ρ(s))
, (8.49)
d(s)
where the polynomial ρ(s) is of degree ≤ k − 1 as in (8.20).
These steps yield an approximation G(s) = b(s)
a(s)
of order N−k. The approximation
which yields the smallest real value for the H 2 -criterion V H (x 1 ,...,x N ,ρ 1 ,...,ρ k−1 ),
as described in (8.43), is the globally optimal approximation of order N − k to the
given system H(s).
Proof. The proof is straightforward and therefore omitted.
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