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190 CHAPTER 10. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-2
Of the 321 eigenvalues ρ 1 , only 26 are real-valued. The values for x 1 ,...,x 7
are read off from the corresponding eigenvectors. The 26 tuples (x 1 ,...,x 7 ,ρ 1 ) are
substituted into the available third order polynomial V H (x 1 ,...,x 7 ,ρ 1 ), which coincides
with the H 2 -criterion at the solutions of the system of equations. It turns
out that there is only one solution which yields a real and feasible approximation
G(s) of order five: ρ 1 =2.03804, x 1 = −0.0583908, x 2 =0.0960792 − 0.0601039i,
x 3 = 0.0960792 + 0.0601039i, x 4 = −0.0207708, x 5 = 0.0179519 − 0.0274311i,
x 6 = 0.0179519 + 0.0274311i, x 7 = −0.00383800 with an H 2 -criterion value of
0.228778. This system G(s) is the H 2 globally optimal approximation of order 5
of the original system H(s) of order 7 and is computed as:
6.109 − 12.02s +14.93s 2 − 14.41s 3 +1.257s 4
G(s) =
0.007068 + 0.10286s +0.4956s 2 +1.210s 3 +1.573s 4 + s 5 . (10.54)
The system G(s) has its poles at −0.366841± 0.192807i, −0.358033± 0.454934i,
and −0.122789 and its zeros at 0.161729± 0.781139i, 0.734038, and 10.4062.
Computing the globally optimal approximation of order 5 of the original system
H(s) of order 7 by applying the co-order one technique twice, results in virtually the
same (numerical) approximation G(s).
When comparing the poles of the system H(s) and G(s) one finds that effectively
two real poles and two real zeros are removed and the remaining poles and zeros are
only adapted slightly. Figure 10.8 shows the poles of the transfer function H(s) and
its approximation G(s) obtained by the co-order k = 2 technique. Figure 10.9 shows
the poles of the transfer function H(s) and the poles of its approximations G 1 (s) of
order 6 and G 2 (s) of order 5 obtained by co-order k = 1 techniques.
For completeness the impulse responses and Bode diagrams of the systems H(s)
(blue) and G(s) (green) are given in the Figures 10.10 and 10.11, which show highquality
approximations.
10.6.3 Example 3
In this example a system H(s) of order 4 is given which has four complex poles. The
globally optimal approximations of order 2 are computed by applying the co-order
one technique twice, which results in the systems G 1 (s), and G 2 (s), and the co-order
two technique once, which results in the system G(s). The results are given in Table
10.1 which presents the locations of the poles and zeros, and the H 2 -criterion values
(V H = ||H(s) − G(s)|| 2 H 2
) of the systems H(s), G 1 (s), G 2 (s) and G(s).
Applying the co-order one technique twice, yields a significantly worse performance
than applying the co-order two technique once: the H 2 -error of G(s) is9.612,
whereas the H 2 -error of G 2 (s) is10.92. Moreover, the system G 2 (s) has a completely
different behavior from the system G(s), as shown in Figures 10.13 and 10.14, which
show the impulse responses and the Bode diagrams of the systems H(s) (blue), G 2 (s)
(green), and G(s) (red).