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138 CHAPTER 8. H 2 MODEL-ORDER REDUCTION
In the H 2 -approximation problem in continuous time, this enveloping space is
taken to be the Hardy space H 2 (Π + ) of complex functions that are square integrable
on the boundary iR of Π + = {s ∈ C | Re(s) > 0} and analytic on the closure of
the right-half plane Π + . Such functions are in one-to-one relationship with square
integrable functions on the imaginary axis ∂Π + = iR, which occur as their limits for
s → iω with ω ∈ R. The associated H 2 -inner product of two functions G(s) and H(s)
is given by:
〈G, H〉 = 1 ∫
G(s)H(−s ∗ ) ∗ ds = 1 ∫ ∞
G(iω)H(iω) ∗ dω, (8.1)
2πi i R
2π −∞
where the asterisk denotes complex conjugation. An important subset of H 2 (Π + )is
formed by S, the class of strictly proper rational functions which have all their poles
in the open left-half plane Π − = {s ∈ C | Re(s) < 0}. Such functions correspond to
transfer functions of continuous time stable and strictly causal LTI systems of finite
order.
When a state-space realization (A, B, C) is employed (thus D = 0) for H(s) (i.e.,
it holds that H(s) =C(sI − A) −1 B), then it is well-known [102] that its squared
H 2 -norm can be computed as:
‖H‖ 2 H 2
= CPC ∗ , (8.2)
in which P satisfies the continuous time Lyapunov equation:
AP + PA ∗ = −BB ∗ . (8.3)
In discrete time, an analogous framework is constituted by the enveloping Hardy
space H 2 (E) of complex functions which are square integrable on the boundary T =
{z ∈ C |z| =1} of E = {z ∈ C |z| > 1} and analytic on the closure of E. These
functions are in one-to-one correspondence with square integrable functions on the
unit circle T = {z ∈ C ||z| =1}, which occur as the radial limits of these functions in
H 2 (E). The associated H 2 -inner product is now given by:
∮
1
〈Ĝ, Ĥ〉 =
2πi T
Ĝ(z)Ĥ( 1 dz
)∗
z∗ z = 1
2π
∫ 2π
0
Ĝ(e iΩ )Ĥ(eiΩ ) ∗ dΩ. (8.4)
An important subset of H 2 (E) is the class of strictly proper rational functions which
have all their poles inside the unit disk D = {z ∈ C ||z| < 1}. When represented as
a Laurent series about infinity, a function Ĥ(z) in this class attains the form:
of which the corresponding H 2 -norm satisfies:
Ĥ(z) =Ĥ0 + Ĥ1z −1 + Ĥ2z −2 + ... (8.5)
‖Ĥ‖2 H 2
=
∞∑
|Ĥk| 2 < ∞. (8.6)
k=0