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186 CHAPTER 10. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-2
The H 2 model-order reduction error between the original system H(s) and the
approximation G 2 (s) of order 4 is 3.742, whereas the model-order reduction error of
the approximation G(s) computed by the co-order k = 2 technique was 3.739.
Figure 10.4: The poles of H(s) (red circles), G 1 (s) (blue diamonds), and G 2 (s) (green
squares)
The Figures 10.5 and 10.6 show the impulse responses and the Bode diagrams
of the systems H(s) (blue), G(s) (green), and G 2 (s) (red). The plots of G(s) and
G 2 (s) virtually overlap in both figures (only G 2 (s) is visible as it is plotted last) and
differ only slightly from H(s). This shows the approximations of order 4 to be of
high-quality.
10.6.2 Example 2
In this example the system to be reduced is chosen to be a system of order N =7,
given by the following transfer function:
H(s) =
2+3s − 3s 2 + s 3 − 2s 4 − 12s 5 + s 6
(s + 1 2 )(s + 1 3 )(s + 1 8 )(s + 1 2 + i 2 )(s + 1 2 − i 2 )(s + 1 3 + i 3 )(s + 1 3 − i (10.50)
3
).
We now want to compute its globally optimal stable real approximation of order
5. Note that the poles of H(s) are δ 1 = − 1 2 , δ 2 = − 1 2 − i 2 , δ 3 = − 1 2 + i 2 , δ 4 = − 1 3 ,
δ 5 = − 1 3 − i 3 , δ 6 = − 1 3 + i 3 , and δ 7 = − 1 8 . Its zeros are −0.745632, −0.472807,
0.162586 − 0.780510i, 0.162586 + 0.780510i, 0.734023, and 12.1592.