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136 CHAPTER 8. H 2 MODEL-ORDER REDUCTION
In this thesis the approach of [50] is generalized and extended and, as a result, the
H 2 model reduction problem from order N to N − 1, N − 2, and N − 3 is solved. For
computing the real approximations of orders N − 1, N − 2, and N − 3, the algebraic
Stetter-Möller matrix method for solving systems of polynomial equations of Section
3.3 is used. This approach guarantees that the globally best approximation to the
given system is found (while typically many local optima exist).
For compact notation we introduce the concept of a co-order: the case where the
order of the given system of order N is reduced to order N − k is called the co-order
k case.
Remark 8.1. In the algebraic approach to global H 2 -approximation of [50], it is convenient
to impose the technical condition that all the N zeros of d(s), the poles of
H(s), are distinct for the co-order k = 1 case. Extensions may be developed in the
future to handle the case with poles of multiplicities larger than 1 too, but this will
complicate notation and the numerical procedures. In the setup presented in this and
in the next chapters we shall impose a similar constraint for ease of exposition for all
co-orders k.
This chapter is structured as follows: in Section 8.2 the H 2 model-order reduction
problem is introduced and formulated. In Section 8.3 a system of quadratic equations
is constructed for the co-order k case, by reparameterizing the H 2 model-order
reduction problem introduced in Section 8.2. This is a generalization of the algebraic
approach taken in [50]. The system of equations here contains the variables
x 1 ,...,x N , and the additional real-valued parameters ρ 1 ,ρ 2 ,...,ρ k−1 . It is in a particular
Gröbner basis form from which it is immediately clear that it admits a finite
number of solutions. In Section 8.4 a homogeneous polynomial of degree three is given
for the co-order k case which coincides with the H 2 model-order reduction criterion
for the difference between H(s) and G(s) at the stationary points. This is again a
generalization of the third order polynomial for the co-order k = 1 case presented in
[50]. Section 8.5 shows how the Stetter-Möller matrix method of Section 3.3 is used
to solve the system of quadratic equations in the co-order k case.
Applying the Stetter-Möller matrix method transforms the problem into an associated
eigenvalue problem. From the solutions of this eigenvalue problem, the
corresponding feasible solutions for the real approximation G(s) of order N − k can
be selected. The H 2 -criterion is used to select the globally best approximation: the
smallest real criterion value yields the globally optimal approximation.
Following this approach for the co-order k = 1 case, leads to a large conventional
eigenvalue problem. The transformation into an eigenvalue problem and computing
its solutions is the subject of Chapter 9. This largely builds on [50] but in Chapter 9
this technique is extended to a matrix-free version by using an nD-system approach
as presented in Chapter 5. In this way we are able to compute the globally optimal
approximation G(s) of order N − 1 of a given system H(s) of order N.
In the Chapters 10 and 11 the more difficult problems of computing a globally
optimal real H 2 -approximation G(s) for the co-order k = 2 and k = 3 case are