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9.3. EXAMPLE 155
displayed in numerical format:
H(s) = 8.00769 × 10−6 +1.71974 s + 260.671 s 2 + 1254.63 s 3 + 899.401 s 4 + ...
1.60154 × 10 −7 +0.0541886 s +17.9691 s 2 + 147.766 s 3 + 138.541 s 4 + ...
... + 1246.85 s 5 + 181.506 s 6 + 276.659 s 7 − 1.22269 s 8 +15.77 s 9
... + 252.726 s 5 +99.6262 s 6 +71.4313 s 7 +19.6362 s 8 +5.15548 s 9 + s 10 .
(9.4)
The poles of this transfer function are located at:
−4.0242 −0.1807 + 0.8543i
−0.2962 + 3.0209i −0.1807 − 0.8543i
−0.2962 − 3.0209i −0.1313
(9.5)
−0.0215 − 2.1720i −2.960 × 10 −6
−0.0215 + 2.1720i −3.091 × 10 −3
and the zeros at:
0.5866 + 3.5000i −0.3625 − 1.0208i
0.5866 − 3.5000i −0.2274
−0.0683 + 2.1804i −6.815 × 10 −3
−0.0683 − 2.1804i −4.660 × 10 −6
−0.3625 + 1.0208i
(9.6)
To compute an approximation G(s) of order 9, a system of 10 quadratic equations
in x 1 ,...,x 10 needs to be solved. This system has a finite number of complex solutions
(x 1 ,...,x 10 ) and the associated complex homogeneous polynomial V H of degree 3,
which coincides with the H 2 -criterion function at these solutions, is computed as:
V H = 3.83344 × 10 −20 x 3 1 +(1.36467 × 10 −14 − 1.00376 × 10 −15 i) x 3 2+
(1.36467 × 10 −14 +1.00376 × 10 −15 i) x 3 3 +(3.58175 × 10 −7 +
2.48189 × 10 −7 i) x 3 4 +(3.58175 × 10 −7 − 2.48189 × 10 −7 i) x 3 5+
0.415865 x 3 6 − (9.03372 × 10 −10 − 1.13998 × 10 −9 i) x 3 7−
(9.7)
(9.03372 × 10 −10 +1.13998 × 10 −9 i) x 3 8 +1.93482 × 10 7 x 3 9+
1.97345 × 10 13 x 3 10.
Note that the polynomial V H is highly ill-conditioned since the values of the coefficients
range between 10 −20 and 10 13 , whereas an imaginary part of 10 −7 seems to be
significant.
To compare the results of the present approach with the approach of [50], we first
construct likewise the 2 10 × 2 10 = 1024 × 1024 matrices A T x i
, i =1,...,10, explicitly
with Mathematica. Then the built-in function Eigensystem is used to calculate all the
eigenvalues and all the eigenvectors of these matrices. For each matrix A T x i
this takes
between 440 and 540 seconds together with the eigensystem computation (which takes
between 70 and 100 seconds). To compute and store all these matrices an amount of
925 MB of internal memory is required.