link to my thesis

236 CHAPTER 12. CONCLUSIONS & DIRECTIONS FOR FURTHER RESEARCH
k = 1 case, it is convenient to impose the technical condition that all the poles of
the original given system H(s) are distinct. We have imposed a similar constraint for
ease of exposition for all co-orders k in this thesis. Extensions may be developed in
the future to handle the case with poles of multiplicities larger than 1 too.
Moreover, the algebraic techniques presented in this part of the thesis may be
combined with numerical local search techniques to arrive at practical algorithms,
for instance to save on hardware requirements, computation time, or to deal with
numerical accuracy issues. To deal with numerical conditioning issues is an important
research topic, to be able to deal with problems that are of a somewhat larger size.
A disadvantage of the co-order k = 2 and k = 3 case, from the perspective of
computational efficiency, is that one can only decide about global optimality of a
solution G(s) when all the solutions of the polynomial eigenvalue problem have been
computed and further analyzed. Currently it is investigated for the co-order k =2
case whether the third order polynomial V H can be employed by iterative polynomial
eigenproblem solvers to limit the number of eigenvalues that require computation.
Possibly, one could modify the available Jacobi–Davidson methods for a polynomial
eigenvalue problem in such a way that it iterates with the involved matrix but targets
on small positive real values of the criterion function V H first (in the same spirit as
the JDCOMM method).
When we compute the Kronecker canonical forms of the singular matrix pencils
in the co-order k = 2 and k = 3 case, we use exact arithmetic to split off the singular
blocks of these pencils. This is only possible when the given transfer function H(s) is
given in exact format. Exact computations are computationally hard and, moreover,
they become useless when the transfer function H(s) is given in numerical format.
To overcome this computational limitation it is advisable to use numerical algorithms
to compute the Kronecker canonical forms of the singular matrix pencils as described
for example in [34].
Finally, H 2 model-order reduction of the multi-input, multi-output case along
similar lines as the approach followed here will lead to further generalizations too.
Furthermore, the discrete time case can be handled too when using the isometric
isomorphism mentioned in Remark 8.2 which transforms the continuous time case
into the discrete time case.