National Instruments 370755B-01

Chapter 2 Additive Error Reduction

Thus, the penalty for not being allowed to include Gu in the approximation

is an increase in the error bound, by σni + 1 + ... + σns. A number of

theoretical developments hinge on bounding the Hankel singular values of

Gr(s) and Gu(–s) in terms of those of G(s). With Gr(s) of order ni–1

, there

holds:

The transfer function matrix Gu(s), being unstable, does not have Hankel

singular values; however, Gu(–s) (which is stable) does have Hankel

singular values. They satisfy:

In most cases, the Hankel singular values of G(s) are distinct. If,

accordingly,

then Gr has degree (i–1), Gu has degree ns–i and

(2-4)

Observe that the bound (Equation2-3 or Equation 2-4), which is not

necessarily obtained, is one half that applying for both balanced truncation

(as implemented by balmoore( ) or, effectively, by redschur( )); it

also is one half that obtained when applying mreduce to a balanced

realization. In general, the D matrices of G and Gr are different, that is,

G(∞) ≠ Gr(∞) (in contrast to balmoore( ) and redschur( )). Similarly,

G(0) ≠ Gr(0) in general (in contrast to the result when mreduce is applied

to a balanced realization). The price paid for obtaining a smaller error

bound overall through Hankel norm reduction is that one no longer

(normally) secures zero error at ω = ∞ or ω = 0.

Two special cases should be noted. If nsr = 0 then Gr(s) is a constant only.

This constant can be added onto Gu(s), so that G(s) is then being

approximated by a totally unstable transfer function matrix, with error σ1;

this type of approximation is known as Nehari approximation. The second

special case arises when nsr = nm–1

(or NS –1 if the smallest Hankel

singular value has multiplicity 1). In this case, Gu(s) becomes a constant,

which can then be lumped in with Gr(s), so that G(s), of degree NS, is then

σkGr)σ

kG()k≤( 12…ni1–

,, ,=

σkGus–()[]σ

nik+G()≤

GGr–Gu

–∞σi

–∞σiσi1+... σns

+++=