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8.2. THE H 2 MODEL-ORDER REDUCTION PROBLEM 139
When a state-space realization (Â, ˆB,Ĉ, ˆD) is employed for Ĥ(z) (i.e., it holds that
Ĥ(z) = ˆD + Ĉ(zI − Â)−1 ˆB), then it is well-known [102] that its squared H2 -norm
can be computed as:
‖Ĥ‖2 H 2
= Ĉ ˆP Ĉ∗ + ˆD ˆD ∗ , (8.7)
in which ˆP satisfies the discrete time Lyapunov-Stein equation:
ˆP − Â ˆP Â∗ = ˆB ˆB ∗ . (8.8)
Remark 8.2. Note that the presence of a constant term Ĥ(∞) = ˆD causes Ĥ(z) to
be proper rather than strictly proper, which constitutes a slight difference between
the discrete-time case and the continuous-time case (where D = 0). However, in H 2
model-order reduction the role played by ˆD is simple and easily understood – it can
be assigned an optimal value irrespective of Â, ˆB, and Ĉ and it does not affect the
order. Therefore it is convenient to restrict again to strictly proper rational transfer
functions also in the discrete-time case.
There exists an isometric isomorphism between the subspace of strictly proper
rational transfer functions of H 2 (E) in the discrete time case and the corresponding
subspace S of H 2 (Π + ) in the continuous time case which leaves the McMillan degree
unchanged. Such an isometry is for example given by:
√ ( )
2 1+s
Ĥ(z) ↦→ H(s) =
(8.9)
1 − sĤ 1 − s
of which the inverse mapping is given by:
√ ( )
2 z − 1
H(s) ↦→ Ĥ(z) = z +1 H . (8.10)
z +1
This mapping has the additional property that it transforms the associated discrete
time Lyapunov-Stein equation (8.8) into the corresponding continuous time Lyapunov
equation (8.3), leaving its solution ˆP = P unchanged, when taking a state-space
realization (Â, ˆB,Ĉ) ofĤ(z) into a corresponding realization (A, B, C) =((Â − I)
(Â + I)−1 , √ 2(Â + I)−1 ˆB , Ĉ)ofH(s).
In discrete time, the sequence of coefficients {Ĥk} constitutes the impulse response
of the system with transfer function Ĥ(z). Conversely, the transfer function Ĥ(z)
is the z-transform of the impulse response. The different expressions for computing
the H 2 -norm either in the frequency domain or in the time domain reflect Parseval’s
identity. Similarly, in continuous time it holds that the transfer function H(s) is the
Laplace transform of the impulse response Ce At B, of which the square of its H 2 -norm
is equivalently given by: ∫ ∞
|Ce At B| 2 dt. (8.11)
0
These properties make that the H 2 -norm has a highly relevant system theoretic interpretation.