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196 CHAPTER 11. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-3
The system of equations (11.1) and the two additional constraints in (11.2) represent
N + 2 equations in N + 2 unknowns. The N + 2 unknowns show up as the N
quantities x 1 ,...,x N together with the unknown parameters ρ 1 and ρ 2 in (11.1).
When a solution (x 1 ,...,x N ,ρ 1 ,ρ 2 ) of the system of equations (11.1) is known,
and this solution also satisfies the additional constraints ã N−1 = 0 and ã N−2 =0
in (11.2), then an approximation G(s) of order N − 3 can be computed. To obtain
the globally optimal approximation G(s), all the solutions (x 1 ,...,x N ,ρ 1 ,ρ 2 ) are
required to select the best one. Section 11.1 describes how solutions of (11.1), which
satisfy both the additional constraints (11.2), can be obtained by using the Stetter-
Möller matrix method. This approach leads to a two-parameter eigenvalue problem
involving two matrices and one common eigenvector. Linearizing and solving such a
two parameter polynomial eigenvalue problem are the subjects of the Sections 11.2
and 11.3. Section 11.4 shows how to arrive at an approximation G(s) once these
solutions of the system of equations (11.1) and the constraints (11.2) are known. As
a proof of principle, an example is worked out in Section 11.5.
11.1 Solving the system of quadratic equations
Suppose a system of the form (11.1) together with the constraints (11.2) is given.
The idea now is to use the Stetter-Möller method in the same way as in the previous
chapters to find solutions x 1 ,...,x N , and ρ 1 and ρ 2 which lead to approximations
G(s) of order N − 3.
Using the polynomials in (11.2) we apply the Stetter-Möller matrix method and
consider the linear multiplication operators AãN−1 (ρ 1 ,ρ 2 ) and AãN−2 (ρ 1 ,ρ 2 ) within
the quotient space C(ρ 1 ,ρ 2 )[x 1 ,...,x N ]/I(ρ 1 ,ρ 2 ). Here I(ρ 1 ,ρ 2 ) denotes the ideal
generated by the parameterized set of polynomials of the quadratic system of equations
(11.1). With respect to an identical monomial basis B as in (8.47), these linear
operators are represented by the matrices AãN−1 (ρ 1 ,ρ 2 ) T and AãN−2 (ρ 1 ,ρ 2 ) T , which
denote multiplication by ã N−1 and ã N−2 .
We know from Section 3.3 that the eigenvalues λ in the eigenvalue problem AãN−i
(ρ 1 ,ρ 2 ) T v = λv, coincide with the values of the polynomial ã N−i at the solutions
of the system of equations (11.1), for i = 1, 2. Because the values of ã N−1 and
ã N−2 are required to be zero, the resulting problem is: AãN−1 (ρ 1 ,ρ 2 ) T v = 0 and
AãN−2 (ρ 1 ,ρ 2 ) T v = 0. Thus, actually we are searching here for values of ρ 1 and ρ 2
which make the matrices AãN−1 (ρ 1 ,ρ 2 ) T and AãN−2 (ρ 1 ,ρ 2 ) T simultaneously singular.
Both the matrices AãN−1 (ρ 1 ,ρ 2 ) T and AãN−2 (ρ 1 ,ρ 2 ) T are rational in ρ 1 and
ρ 2 and can be made polynomial in ρ 1 and ρ 2 . Using Theorem 10.1 the matrices
AãN−1 (ρ 1 ,ρ 2 ) T and AãN−2 (ρ 1 ,ρ 2 ) T are made polynomial in ρ 1 and ρ 2 by multiplying
elementwise the rows of the matrices with well chosen quantities. The polynomial matrices
in ρ 1 and ρ 2 are denoted by the matrices Ãã N−1
(ρ 1 ,ρ 2 ) T and Ãã N−2
(ρ 1 ,ρ 2 ) T .
These matrices have dimensions 2 N × 2 N and the total degree of ρ 1 and ρ 2 occurring
in the polynomial matrices is N − 1 (analogous to the result in Corollary 10.2).