Chapter 3 Multiplicative Error Reduction
Xmath Model Reduction Module 3-8 ni.com
state-variable representation of G. In this case, the user is effectively asking
for Gr = G. When the phase matrix has repeated Hankel singular values,
they must all be included or all excluded from the model, that is,
νnsr =νnsr+ 1 is not permitted; the algorithm checks for this.
The number of νi equal to 1 is the number of zeros in Re[s]>0 of G(s), and
as mentioned already, these zeros remain as zeros of Gr(s).
If error is specified, then the error bound formula (Equation3-2) in
conjunction with the νi values from step 3 is used to define nsr for step 4.
For nonsquare G with more columns than rows, the error formula is:
If the user is presented with the νi, the error formula provides a basis for
intelligently choosing nsr. However, the error bound is not guaranteed to
be tight, except when nsr = ns–1.
Securing Zero Error at DCThe error G–1(G–Gr) as a function of frequency is always zero at ω = ∞.
When the algorithm is being used to approximate a high order plant by a
low order plant, it may be preferable to secure zero error at ω = 0. A method
for doing this is discussed in [GrA90]; for our purposes:
1. We need a bilinear transformation of sys= 1/z. Given G(s) we generate
H(s) through:
bilinsys=makepoly([b3,b4]/makepoly([b1,b2])
sys=subsys(sys,bilinsys)
2. Reduce with the previous algorithm:
[sr,nsr,hsv] = bst(sys)
3. Use the bilinear transformation s= 1/z again:
[sr1,nsr1] = bilinear(sr,nsr,[0,1,1,0])
The νi are the same for G(s) and H(s)=G(s–1). The error bound formula is
the same; H is stable and H(jω)H'(–jω) of full rank for all ω including
ω=∞ if and only if G has the same property; right half plane zeros of G are
still preserved by the algorithm. The error G–1(G–Gr), though now zero at
ω=0 , is in general nonzero at ω=∞.
GG
r
–()
*G*G()
1–GG
r
–()
∞
12⁄2vi
1vi
–
-------------
insr1+=
ns
∑
≤