National Instruments Xmath right and left

Chapter 3 Multiplicative Error Reduction

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

is the 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