Chapter 3 Multiplicative Error Reduction
© National Instruments Corporation 3-15 Xmath Model Reduction Module
The conceptual basis of the algorithm can best be grasped by considering
the case of scalar G(s) of degree n. Then one can form a minimum phase,
stable W(s) with |W(jω)|2 = |G(jω)|2 and then an all-pass function (the phase
function) W–1(–s) G(s). This all-pass function has a mixture of stable and
unstable poles, and it encodes the phase of G(jω). Its stable part has
nHankel singular values σi with σi ≤ 1, and the number of σi equal to 1
isthe same as the number of zeros of G(s) in Re[s]>0. State-variable
realizations of W,G and the stable part of W–1(–s)G(s) can be connected in
a nice way. The algorithm computes an additive Hankel norm reduction of
the stable part of W–1(–s)G(s) to cause a degree reduction equal to the
multiplicity of the smallest σi. The matrices defining the reduced order
object are then combined in a new way to define a multiplicative
approximation to G(s); as it turns out, there is a close connection between
additive reduction of the stable part of W–1(–s)G(s) and multiplicative
reduction of G(s). The reduction procedure then can be repeated on the new
phase function of the just found approximation to obtain a further reduction
again in G(s).
right and left
A description of the algorithm for the keyword right follows. It is based
on ideas of [Glo86] in part developed in [GrA86] and further developed
in[SaC88]. The procedure is almost the same when {left} is specified,
except the transpose of G(s) is used; the following algorithm finds an
approximation, then transposes it to yield the desired Gr(s).
1. The algorithm checks that G(s) is square, stable, and that the transfer
function is nonsingular at infinity.
2. With G(s) = D+C(sI–A)–1B square and stable, with D nonsingular
[rank(d) must equal number of rows in d] and G(jω) nonsingular for
all finite ω, this step determines a state variable realization of a
minimum phase stable W(s) such that,
W´(–s)W(s)= G(s)G´(–s)
with:
W(s) = Dw+C
w(sI–Aw)–1Bw
The various state variable matrices in W(s) are obtained as follows. The
controllability grammian P associated with G(s) is first found from
AP+ PA´ + BB´ = 0, then:
Aw=AB
w=PC´+BD´D
w=D´
The algorithm checks to see if there is a zero or singularity of G(s)
close to the jω-axis. The zeros are determined by calculating the