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8.3. A REPARAMETERIZATION OF THE MODEL REDUCTION PROBLEM 141
Remark 8.3. Note that in the MIMO case, the corresponding H 2 -inner product is
obtained by applying the trace operator to the expressions given above for the SISO
case. The state-space formulation needs to be adapted in a similar fashion, by including
the trace operator.
8.3 A reparameterization of the H 2 model-order reduction problem
From an algebraic point of view, note that the squared H 2 -norm of H(s) − G(s) in
(8.13) can be worked out to constitute a (rather complicated) rational expression in
terms of the 2n coefficients of a(s) and b(s), when
G(s) = b(s)
a(s) . (8.15)
In Section 8.4 of this chapter we show how this criterion can be evaluated at some
special points of interest by means of a third order homogeneous polynomial.
From a geometric point of view it is appropriate to regard the problem of minimizing
the criterion function in (8.13) as a projection problem within the Hardy space
H 2 (Π + ) of functions that are analytic on the open right-half plane Π + and which
satisfy the property that their H 2 -norm attains a well-defined finite value.
Proceeding as in [50], it is well known that the space S n constitutes a smooth
submanifold of H 2 (Π + ). Therefore, if b(s)/a(s) is an optimal approximation in S n to
e(s)/d(s) with respect to the H 2 -norm, then b(s)/a(s) is an orthogonal projection of
e(s)/d(s) ontoS n . The difference
e(s)
d(s) − b(s)
a(s)
(8.16)
is perpendicular to the tangent space to S n at the point G(s). The tangent space to S n
at G(s) is easily computed as the set {t(s)/a(s) 2 | t(s) polynomial of degree ≤ 2n−1}.
Now, from the theory of Hardy spaces it is known that the orthogonal complement of
this tangent space in H 2 (Π + ) is obtained as the set {a(−s) 2 R(s) | R(s) ∈H 2 (Π + )}.
Consequently, G(s) is a stationary point if and only if there exists a function R(s) ∈
H 2 (Π + ) for which:
e(s)
d(s) − b(s)
a(s) = a(−s)2 R(s) (8.17)
(see [8], [9], and [80]).
Clearly R(s) needs to be a real and rational function and its poles are in the open
left-half plane Π − . Since the factor a(−s) 2 has its zeros in the open right-half plane
Π + none of its factors cancel and it follows upon multiplication by a(s)d(s) that there
exists a non-vanishing real polynomial q(s) =a(s)d(s)R(s) such that:
e(s)a(s) − b(s)d(s) =a(−s) 2 q(s). (8.18)
Note that q(s) 0 since H(s) and G(s) have different degrees. Note also that
the degree of q(s) is≤ k − 1. The value k = N − n ≥ 1 is the co-order of the
approximation G(s).