Chapter 2 Additive Error Reduction
© National Instruments Corporation 2-21 Xmath Model Reduction Module
2. Find a stable order ns– 2 approximation Gns –2
of Gns–1
(s), with
3. (Step ns–nr):
Find a stable order nsr approximation of Gnsr+1
,
with
Then, because for ,
for , ..., this being a property of the algorithm, there follows:
The only difference that arises when singular values have multiplicity in
excess of 1 is that the degree reduction at a given step is greater. Thus, if
σns(G) has multiplicity k, then G(s) is approximated by Gns–k(s) of degree
ns –k.
No separate optimization of the D matrix of Gnsr is required. The
approximation error bound is the same as for the first algorithm. The actual
approximation error may be different. Notice that this second algorithm
does not calculate an unstable Gu(s) such that
Discrete-Time Systems
No special algorithm is presented for discrete-time systems. Any stable
discrete-time transfer-function matrix G(z) can be used to define a stable
continuous-time transfer-function matrix by a bilinear transformation, thus
when α is a positive constant.
Gns 1–jω()Gns 2–jω()–∞σns 1–Gns 1–
()=
.
.
.
Gnsr 1+jω()Gnsr jω()–∞σnsr 1+Gnsr 1+
()=
σiGns 1–
()σ
iG()≤ins<
σiGns 2–
()σ
iGns 1–
()≤
isi–≤
Gjω()Gnsr jω()–σnsr 1+Gnsr 1+
()... σns
++ G()≤
σiG()
insr1+=
ns
∑
≤
Gjω()Gnsr jω()–Gujω()–∞σnsr 1+
=
Hs() Gαs+
αs–
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