National Instruments NI MATRIXx Xmath - Manual (Page 33)

Chapter 2 Additive Error Reduction
Xmath Model Reduction Module 2-10 ni.com
The actual approximation error for discrete systems also depends on
frequency, and can be large at
ω = 0. The error bound is almost never tight,
that is, the actual error magnitude as a function of
ω almost never attains
the error bound, so that the bound can only be a guide to the selection of the
reduced system dimension.
In principle, the error bound formula for both continuous and discrete
systems can be improved (that is, made tighter or less likely to overestimate
the actual maximum error magnitude) when singular values occur with
multiplicity greater than one. However, because of errors arising in
calculation, it is safer to proceed conservatively (that is, work with the error
bound above) when using the error bound to select
nsr , and examine the
actual error achieved. If this is smaller than required, a smaller dimension
for the reduced order system can be selected.
mreduce( ) provides an alternative reduction procedure for a balanced
realization which achieves the same error bound, but which has zero error
at
ω = 0. For continuous systems there is generally some error at ω = ∞,
because the D matrix is normally changed. (This means that normally the
approximation of a strictly proper system through
mreduce( ) will not be
strictly proper, in contrast to the situation with balmoore( ) .) For discrete
systems the D matrix is also normally changed so that, for example, a
system which was strictly causal, or guaranteed to contain a delay (that is,
D = 0), will be approximated by a system
SysR without this property.
The presentation of the Hankel singular values may suggest a logical
dimension for the reduced order system; thus if
, it may be
sensible to choose nsr = k.
With mreduce( ) and a continuous system, the reduced order system
SysR is internally balanced, with the grammian , so
that its Hankel Singular Values are a subset of those of the original system
Sys . Provided , SysR also is controllable, observable, and
stable. This is not guaranteed if , so it is highly advisable to
avoid this situation. Refer to the balmoore( ) section for more on the
balmoore( ) algorithm.
With mreduce( ) and discrete systems, the reduced order system SysR is
not in general balanced (in contrast to balmoore( ) ), and its Hankel
singular values are not in general a subset of those of Sys . Provided
, the reduced order system SysR also is controllable,
observable and stable. This is not guaranteed if , so it is
highly advisable to avoid this situation. For additional information about
the
balmoore( ) function, refer to the Xmath Help.
σ
k
σ
k 1 +
»
diag σ
1
σ
2
..., σ
nsr
, , [ ]
σ
nsr
σ
nsr 1 +
>
σ
nsr
σ
nsr 1 +
=
σ
nsr
σ
srn 1 +
>
σ
nsr
σ
nsr 1 +
=