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10.6. EXAMPLES 185
Figure 10.3: The poles of H(s) (red circles) and of G(s) (green squares)
(ii) The co-order k = 1 technique as described in Chapter 9 is now applied twice
to H(s) to arrive at an approximation of order 4 in another way and to compare
the results with the globally optimal approximation found by the co-order k =2
technique above. Using the co-order one technique, first an approximation G 1 (s) of
order 5 is computed:
G 1 (s) =
−10.2783 − 73.8312s + 107.911s 2 − 79.6939s 3 +29.515s 4
0.0328014 + 0.374281s +1.20205s 2 +2.1838s 3 +1.82557s 4 + s 5 . (10.48)
The poles of G 1 (s) are located at −0.508403− 0.877968i, −0.508403+ 0.877968i,
−0.338916− 0.358517i, −0.338916+ 0.358517i, and −0.130928, and its H 2 -criterion
value, ||H(s) − G(s)|| 2 H 2
,is2.488.
Applying the co-order one technique once again to the transfer function G 1 (s) to
find an approximation G 2 (s) of order 4 gives:
G 2 (s) =
−83.8768 + 112.212s − 79.2856s 2 +28.0997s 3
0.258479 + 0.903876s +1.94628s 2 +1.59856s 3 + s 4 . (10.49)
The poles of G 2 (s) have slightly different locations as the poles of G(s): they
are located at −0.496926− 0.930667i, −0.496926+ 0.930667i, −0.302353− 0.375237i,
and −0.302353+ 0.375237i. The zeros of G 2 (s) are located at 0.667769− 1.25010i,
0.667769+ 1.25010i, and 1.48604. In comparison to the poles of G 1 (s) only the slowest
mode at 0.130928 is discarded, whereas the other poles are slightly adapted. Figure
10.4 shows a plot of the poles of H(s), G 1 (s), and G 2 (s) in the complex plane.