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156 CHAPTER 9. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-1
Following the approach of [50], all the 1024 tuples x 1 ,...,x 10 are determined from
the mutual eigenvectors of the matrices A T x 1
,...,A T x 10
. Each such tuple yields an
approximation of order N − 1 which is either feasible or infeasible. By substituting
these tuples into the polynomial V H , we compute the H 2 -criterion value for each
feasible solution. The solution with the smallest real H 2 -criterion value is used to
compute the polynomials b(s) and a(s) which yield the feasible globally optimal stable
approximation G(s) = b(s)
a(s) of order 9 with an H 2-criterion value of 0.000489593:
G(s) = 1.69597 + 260.566 s + 1254.59 s2 + 899.268 s 3 + 1246.84 s 4 + ...
0.0538714 + 17.9585 s + 147.756 s 2 + 138.523 s 3 + 252.714 s 4 + ...
... +181.474 s 5 + 276.659 s 6 − 1.22433 s 7 +15.7699 s 8
... +99.6209 s 5 +71.4287 s 6 +19.6358 s 7 +5.15532 s 8 + s 9 .
(9.8)
The solution used to compute this approximation is:
x 1 = 78.6020 x 6 = 0.000289407
x 2 = −0.180875 − 0.343180i x 7 = −0.00193041 + 0.00371561i
x 3 = −0.180875 + 0.343180i x 8 = −0.00193041 − 0.00371561i
x 4 = −0.00155184 − 0.00377204i x 9 = 5.99264 × 10 −6
x 5 = −0.00155184 + 0.00377204i x 10 = 2.91654 × 10 −6 (9.9)
The Hankel singular values of the approximation G(s) are now changed to the
values: 9.68, 8, 7,...,1. Note that the coefficients of the approximation G(s) are very
similar to the coefficients of the original problem H(s). Effectively, one pole/zero pair
of the original transfer function at s = 0 is removed by the model-order reduction
algorithm and has left the other poles and zeros nearly unchanged.
Another computational approach is by working with the matrix A T V H
. First, we
explicitly construct this matrix and compute all its eigenvalues. Second, we do not
use the explicit matrix A T V H
but we use the matrix-free approach for the associated
operator of the matrix A T V H
together with an iterative eigenvalue solver and an nDsystem,
to compute only the smallest real eigenvalue of A T V H
which immediately will
lead to a feasible globally optimal approximation of order 9.
To construct the matrix A T V H
explicitly and to compute all its eigenvalues and
eigenvectors is computationally a hard job: it takes, respectively, 27164 and 82 seconds.
Furthermore, it takes 275 MB of internal memory to store all the data. The
matrix A T V H
has 45 real eigenvalues, including the zero eigenvalue. The smallest positive
real eigenvalue λ min is computed as 0.000489593. This smallest real eigenvalue
yields the solution x 1 ,...,x 10 since it is read off from the corresponding eigenvector.
The solution x 1 ,...,x 10 computed here turns out to be the same solution as found
before in (9.9). This leads to the same globally optimal approximation G(s) of order
9 as in (9.8). The H 2 -criterion is again 0.000489593, which exactly equals the
eigenvalue λ min =0.000489593.
In Figure 9.1 all the eigenvalues of the matrix A T V H
, excluding the zero eigenvalue
(as it will not lead to a feasible approximation), are plotted in the complex plane.