National Instruments NI MATRIXx Xmath - Manual (Page 45)

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 H
r
(s) and unstable H
u
(s) such that:
Here, the s
ni
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 G
r
and G
u
are the same as those of H
r
and H
u
,
respectively. The error formulas are as follows:
Impulse Response Error
If G
r
is determined by the first (single-pass) algorithm, the impulse
response error (for t > 0) between the impulse responses of G and G
r
can
be bounded. As shown in Corollary 9.9 of [Glo84], if G
r
is of degree i – 1
and the multiplicity of the ith larger singular value
σ
i
of G is r, then:
H H
r
H
u
– –
σ
i
=
H H
r
–
σ
i
σ
n
i
1 +
... σ
ns
+ + + =
G
r
z ( ) H
r
α
z 1 –
z 1
+
-----------
⎝ ⎠
⎛ ⎞
=
G
u
z ( ) H
u
α
z 1 –
z 1
+
-----------
⎝ ⎠
⎛ ⎞
=
G e
j ω
( ) G
r
e
j ω
( )
– G
u
e
j ω
( )
–
∞
σ
n
i
=
G e
j ω
( ) G
r
e
j ω
( )
–
∞
σ
n
i
σ
n
i
1 +
... σ
ns
+ +
≤
σ
j
G G
r
–
[ ] σ
i
G for j ≤ 1 2 ... 2i 2 r + –
, , ,
=
σ
j i – 1 +
G ( ) for j ≤ 2i 1 – r ...,ns i 1 – +
,
+ =