link to my thesis

8.2. THE H 2 MODEL-ORDER REDUCTION PROBLEM 137
addressed. The practical relevance of the co-order two and three problem derives
from the fact that the poles of a real transfer function are either real or they show up
as complex conjugate pairs. These poles determine the characteristic modes of the
system. Model-order reduction techniques often tend to remove unimportant modes,
but in the co-order one case only one mode gets discarded. To remove a complex
conjugate pair of poles requires a new technique other than provided by the co-order
one case. Indeed, repeated application of the co-order one technique to achieve a
larger reduction of the model order, can easily be shown to be non-optimal.
The co-order two and three problems are approached by generalizing and extending
the ideas for the co-order k = 1 case. Using again the reparameterization
technique from Section 8.3 a quadratic system of equations is set up. In the co-order
k ≥ 2 case, the coefficients in the complex system of equations involve k − 1 additional
real variables. In addition, any feasible complex solution of this system must
satisfy k − 1 linear constraints. When taking the additional linear condition(s) in
the co-order 2 and co-order 3 case into account, the Stetter-Möller matrix method
approach yields eigenvalue problems other than the conventional eigenvalue problem
in the co-order one case:
• Chapter 10 describes how the Stetter-Möller matrix method in the co-order
k = 2 case yields a polynomial eigenvalue problem. Such a polynomial eigenvalue
problem can be rewritten as a generalized eigenvalue problem by using
linearization techniques. This generalized eigenvalue problem is finally solved
accurately by Kronecker canonical form techniques.
• In Chapter 11 the co-order k = 3 case is studied in which application of the
Stetter-Möller matrix method leads to a two-variable polynomial eigenvalue
problem involving rectangular matrices. This problem is reducible to a polynomial
eigenvalue problem in one variable involving a single square matrix. To
perform this reduction a generalization of the Kronecker Canonical form technique
for a one-variable polynomial matrix is developed and used.
From the eigenvalues and corresponding eigenvectors of these eigenvalue problems,
the feasible solutions for the approximation problem can be selected and constructed.
To perform the selection of the globally optimal solution G(s), the H 2 model-order
reduction criterion mentioned in Theorem 8.2 in Section 8.4 is evaluated for k =2
and k =3.
Each of the Chapters 9, 10, and 11 contain several worked examples.
8.2 The H 2 model-order reduction problem
When studying model-order reduction and system approximation from a geometric
point of view, it is convenient to start from an enveloping inner product space which
contains both the system to be approximated and the subset of systems in which an
approximation is to be found.