National Instruments Xmath Behaviors

Chapter 2 Additive Error Reduction

Algorithm

The algorithm does the following. The system Sys and the reduced order

system SysR are stable; the system SysU has all its poles in Re[s] > 0. If

the transfer function matrices are G(s), Gr(s) and Gu(s) then:

•Gr(s) is a stable approximation of G(s).

•Gr(s)+Gu(s) is a more accurate, but not stable, approximation of G(s),

and optimal in a certain sense.

Of course, the algorithm works with state-space descriptions; that of G(s)

can be minimal, while that of Gr(s) cannot be.

These statements are explained in the Behaviors section. If onepass is

specified, reduction is calculated in one pass. If onepass is not called or is

set to 0 {onepass=0}, reduction is calculated in (number of states of

Sys - nsr) passes. There seems to be no general rule to suggest which

setting produces the more accurate approximation Gr. Therefore, if

accuracy of approximation for a given order is critical, both should be tried.

As noted previously, if an approximation involving an unstable system is

desired, the default {onepass=1} is specified.

Behaviors

The following explanation deals first with the keyword {onepass}.

Suppose that σ1, σ2,..., σns are the Hankel Singular values of S, which has

transfer function matrix G(s). Suppose that the singular values are ordered

so that:

Thus, there are n1 equal values, followed by n2–n1 equal values, followed

by n3–n2 equal values, and so forth.

The order nsr of Gr(s) cannot be arbitrary when there are equal Hankel

singular values. In fact, the orders shown in Table2-1 for the strictly stable

Gr (all poles in Re[s]<0) and strictly unstable Gu (all poles Re[s]>0) are

possible (and there are no other possibilities).

σ1σ2... σn1

=== σn11+...>σ

n11+... σn2σn21+...>==

σnm1–1+σnm1–2+σnm=σns

()0≥==>