Chapter 2 Additive Error Reduction
Xmath Model Reduction Module 2-22 ni.com
We use sysZ to denote G(z) and define:
bilinsys=makepoly([-1,a]/makepoly([1,a])
as the mapping from the z-domain to the s-domain. The specification is
reversed because this function uses backward polynomial rotation. Hankel
norm reduction is then applied to H(s), to generate, a stable reduced order
approximation Hr(s) and unstable Hu(s) such that:
Here, the sni are the Hankel singular values of both G(z) and H(s); they are
the same:
We then implement the s-domain equivalent with:
sysS=subsys(sysZ,bilinsys)
There is no simple rule for choosing α; the choice α= 1 is probably as good
as any. The orders of Gr and Gu are the same as those of Hr and Hu,
respectively. The error formulas are as follows:
Impulse Response Error
If Gr is determined by the first (single-pass) algorithm, the impulse
response error (for t> 0) between the impulse responses of G and Gr can
be bounded. As shown in Corollary 9.9 of [Glo84], if Gr is of degree i–1
and the multiplicity of the ith larger singular value σi of G is r, then:
HH
rHu
–– σi
=
HH
r
–σiσni1+... σns
+++=
Grz() Hrαz1–
z1+
-----------
⎝⎠
⎛⎞
=
Guz() Huαz1–
z1+
-----------
⎝⎠
⎛⎞
=
Ge
jω
()Grejω
()–Guejω
()–∞σni
=
Ge
jω
()Grejω
()–∞σniσni1+...σns
++≤
σjGG
r
–[]σ
iGfor j≤1 2 ... 2i2r+–,, ,=
σji–1+G()for j≤2i1–r..., ns i 1–+,+=