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11.5. EXAMPLE 227
of solutions (ρ 1 ,ρ 2 ,x 1 ,x 2 ,x 3 ,x 4 ), the polynomials a(s) and b(s) can be determined.
This yields five corresponding transfer functions of order one:
3.37127
G(s) =
−6.48665 + s ,
3.87128
G(s) =
−0.129578 + s ,
396.044
G(s) =
−0.409222 + s , (11.85)
G(s) = 2.97122
4.43913 + s ,
G(s) = −3.1894
0.174884 + s ,
with corresponding H 2 -criterion values 8.24643, 11.1784, 437.85, 8.13223, and 6.16803,
respectively.
The H 2 -criterion values are computed by substituting the values of ρ 1 , ρ 2 , and
x 1 ,...,x 4 into the third order polynomial:
V H = (−0.121789 + 0.00691205i)(1 − (0.666667 + 0.5i)ρ 1 +
(0.194444 + 0.666667i)ρ 2 ) 2 (1+(0.666667 + 0.5i)ρ 1 +
(0.194444 + 0.666667i)ρ 2 ) x 3 1 − (0.121789 + 0.00691205i)
(1 − (0.666667 − 0.5i)ρ 1 +(0.194444 − 0.666667i)ρ 2 ) 2
(1+(0.666667 − 0.5i)ρ 1 +(0.194444 − 0.666667i)ρ 2 ) x 3 2−
1.60977(1 − 0.6ρ 1 +0.36ρ 2 ) 2 (1+0.6ρ 1 +0.36ρ 2 ) x 3 3+
9.74309(1 − 0.333333ρ 1 +0.111111ρ 2 ) 2
(1+0.333333ρ 1 +0.111111ρ 2 ) x 3 4.
(11.86)
It is easy to see that only the last two approximations G(s), are feasible because
only those two have a pole in the open left half plane Π − . Since the last approximation
has the smallest real criterion value, this approximation G(s) is the globally optimal
H 2 -approximation of order one. This is the same transfer function as computed before
in Equation (11.77):
G(s) = −3.1894
0.174884 + s . (11.87)
The corresponding values of the solution of the system of equations (11.78) and
the two additional constraints in (11.79) are:
ρ 1 =4.39521
ρ 2 =1.70287
x 1 =2.56458 + 1.52372i
x 2 =2.56458 − 1.52372i
x 3 =2.36141
x 4 =1.54876.
(11.88)