National Instruments 370755B-01

Chapter 3 Multiplicative Error Reduction

Xmath Model Reduction Module 3-12 ni.com

Any zero (or rank reduction) on the jω-axis of G(s) becomes a zero (or rank

reduction) in Re[s] > 0 of , and if G(s) has a zero (or rank reduction)

at infinity, this is shifted to a zero (or rank reduction) of at the point

b–1, (in Re[s] > 0). If all poles of G(s) are inside the circle of diameter

[–b–1+j0, a+j0], all poles of will be in Re[s] < 0, and if G(s) has no

zero (or rank reduction) on this circle, will have no zero (or rank

reduction) on the jω-axis, including ω=∞.

If G(s) is nonsingular for almost all values of s, it will be nonsingular or

have no zero or rank reduction on the circle of diameter [–b–1+j0, –a+j0]

for almost all choices of a,b. If a and b are chosen small enough, G(s) will

have all its poles inside this circle and no zero or rank reduction on it, while

then will have all poles in Re[s]< 0 and no zero or rank reduction on

the jω-axis, including s=∞. The steps of the algorithm, when G(s) has a

zero on the jω-axis or at s=∞, are as follows:

1. For small a,b with 0 < a < b–1, form as shown for

gtildesys.

2. Reduce to , this being possible because is stable and

has full rank on s=jω, including ω=∞.

3. Form as shown for gsys.

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. The user

will be presented with an error message 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 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

˜s()

˜s() Gsa–

bs–1+

-------------------









˜s() G

˜rs() G

˜s()

Grs() G

˜r

sa+

bs 1+

---------------





G1–GG

–()

∞

˜1–G

˜G

˜r

–()

∞

˜s()