National Instruments 370755B-01

Chapter 3 Multiplicative Error Reduction

The error will be overbounded by the error

, and Gr will contain the same zeros in Re[s]≥0 as G.

If there is no zero (or rank reduction) of G(s) at the origin, one can take

a=0 and b–1= bandwidth over which a good approximation of G(s) is

needed, and at the very least b–1 sufficiently large that the poles of G(s)

lie in the circle of diameter [–b–1+j0, –a+j0]. If there is a zero or rank

reduction at the origin, one can replace a=0 by a=b. It is possible to take

b too small, or, if there is a zero at the origin, to take a too small. In these

cases an error message results, saying that there is a jω-axis zero and/or that

the Riccati equation solution may be in error. The basic explanation is that

as b → 0, and thus a → 0, the zeros of approach those of G(s). Thus,

for sufficiently small b, one or more zeros of may be identified as

lying on the imaginary axis. The remedy is to increase a and/or b above the

desirable values.

The previous procedure for handling jω-axis zeros or zeros at infinity will

be deficient if the number of such zeros is the same as the order of G(s); for

example, if G(s) = 1/d(s), for some stable d(s). In this case, it is possible

again with a bilinear transformation to secure multiplicative

approximations over a limited frequency band. Suppose that

Create a system that corresponds to with:

gtildesys=subs(gsys,(makep([-eps,1])/makep([1,-]))

bilinsys=makep([eps,1])/makep([1,0])

sys=subsys(sys,bilinsys)

Under this transformation:

•Values of G(s) along the jω-axis correspond to values of around

a circle in the left half plane on diameter (–ε–1 + j0, 0).

• Values of along the jω-axis correspond to values of G(s) around

a circle in the right half plane on diameter (0, ε–1 + j0).

G1–GG

–()

∞

˜1–G

˜G

˜r

–()

∞

˜s()

˜s() Gs

εs1+

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



˜s()