National Instruments Xmath Discrete-Time Systems

Chapter 2 Additive Error Reduction

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+=

∑

≤

Gjω()Gnsr jω()–Gujω()–∞σnsr 1+

Hs() Gαs+

αs–

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