Chapter 2 Additive Error Reduction
Xmath Model Reduction Module 2-16 ni.com
By abuse of notation, when we say that G is reduced to a certain order, this
corresponds to the order of Gr(s) alone; the unstable part of Gu(s) of the
approximation is most frequently thrown away. The number of eliminated
states (retaining Gu) refers to:
(# of states in G) – (# of states in Gr) – (# of states in Gu)
This number is always the multiplicity of a Hankel singular value. Thus,
when the order of Gr is ni–1
the number of eliminated states is ni–ni–1
or
the multiplicity of σni – 1 + 1 = σni.
For each order ni–1
of Gr(s), it is possible to find Gr and Gu so that:
(Choosing i = 1 causes Gr to be of order zero; identify n0 = 0.) Actually,
among all “approximations” of G(s) with stable part restricted to having
degree ni–1
and with no restriction on the degree of the unstable part, one
can never obtain a lower bound on the approximation error than σni; in the
scalar or SISO G(s) case, the Gr(s) which achieves the previous bound is
unique, while in the matrix or MIMO G(s) case, the Gr(s) which achieves
the previous bound may not be unique [Glo84]. The algorithm we use to
find Gr(s) and Gu(s) however allows no user choice, and delivers a single
pair of transfer function matrices.
The transfer function matrix Gr(jω) alone can be regarded as a stable
approximation of G(jω). If the D matrix in Gr(jω) is approximately
chosen, (and the algorithm ensures that it is), then:
(2-3)
Table 2-1. Orders of G
Order of
Gr nsr
Order of
Gu nsu
Number of
Eliminated States
(Retaining Gu)
Number of
Eliminated States
(Discarding Gu)
0ns–n1n1ns
n1ns–n2n2–n1ns –n1
n2ns–n3n3–n2ns –n2
⇓ ⇓ ⇓ ⇓
nm–1 0ns–nm–1 ns –nm–1
Gjω()Grjω()–Gujω()–∞σni
≤
Gjω()Grjω()–∞σniσni1+... σns
+++≤