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180 CHAPTER 10. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-2
Remark 10.3. A disadvantage of this approach, from the perspective of computational
efficiency, is that in this way one can only decide about global optimality of a solution
when all the solutions of the polynomial eigenvalue problem have been computed and
further analyzed.
In the co-order one case, the third order polynomial V H (x 1 ,x 2 ,...,x N ) coincides
at the solutions (x 1 ,...,x N ) of the system of equations with the H 2 -criterion. In that
case, the third order polynomial offers an additional computational advantage, as seen
before in Chapter 9: if V H (x 1 ,x 2 ,...,x N ) denotes the third order polynomial, then
the globally optimal approximation of order N − 1 can be computed by computing
the smallest real eigenvalue of the matrix A T V H
which avoids the computation of all
the eigenvalues of the matrix A T V H
. This is achieved by using iterative eigenproblem
solvers. In the co-order k = 2 case the situation is more difficult: the eigenvalues of
the matrix Ãã N−1
(ρ 1 ) T do not correspond to H 2 -criterion values directly.
Currently it is investigated whether the third order polynomial specified in Equation
(10.42) can still be employed by iterative polynomial eigenproblem solvers to limit
the number of eigenvalues and solutions that require computation. Possibly, one could
modify the available Jacobi–Davidson methods for a polynomial eigenvalue problem,
see [61], [62], in such a way that it iterates with the matrix Ãã N−1
(ρ 1 ) T but targets on
small positive real values of the polynomial criterion function V H (x 1 ,x 2 ,...,x N ,ρ 1 )
first (in the same spirit as the JDCOMM method developed in Section 6.5 for conventional
eigenvalue problems with commuting matrices). This approach is currently
under investigation.
10.5 Algorithm for H 2 model-order reduction for the co-order k =2
case
For implementation purposes, the techniques discussed in the previous sections are
collected into a single algorithm to perform globally optimal H 2 model-order reduction
for the co-order k = 2 case.
1. For the given transfer function H(s) of order N to be reduced, construct the
system of quadratic equations (10.2).
2. Construct the matrix AãN−1 (ρ 1 ) T by applying the Stetter-Möller matrix method
using the ideal I(ρ 1 ) generated by the polynomials in (10.2). This matrix is
rational in ρ 1 . Construct the polynomial matrix Ãã N−1
(ρ 1 ) T by multiplication
of the rows with suitable scalar expressions as given by Theorem 10.1.
3. Use a linearization technique which transforms the polynomial eigenvalue problem
Ãã N−1
(ρ 1 ) T v = 0 into the generalized eigenvalue problem (B + ρ 1 C)z =0.
4. Partly compute the Kronecker canonical form of the matrix pencil (B+ρ 1 C) and
deflate it to the regular pencil (F +ρ 1 G) by splitting off the blocks corresponding
to the indeterminate, infinite and zero eigenvalues.