National Instruments Xmath

Chapter 3 Multiplicative Error Reduction

Xmath Model Reduction Module 3-24 ni.com

Multiplicative approximation of (along the jω-axis) corresponds to

multiplicative approximation of G(s) around a circle in the right half plane,

touching the jω-axis at the origin. For those points on the jω-axis near the

circle, there will be good multiplicative approximation of G(jω). If a good

approximation of G(s) over an interval [–jΩ, jΩ] it is desired, then

ε–1 = 5Ω or 10 Ω are good choices. Reductio n then proceeds as follows:

1. Form .

2. Reduce through

bst( ).

3. Form with:

gsys=subsys(gtildesys(gtildesys,

makep([-eps,-1])/makep[-1,-0]))

Notice that the number of zeros of G(s) in the circle of diameter (0, ε–1 + j0)

sets a lower bound on the degree of Gr(s)—for such zeros become right half

plane zeros of , and must be preserved by bst( ). Zeros at s=∞ are

never in this circle, so a procedure for reducing G(s)=1/d(s) is available.

There is one potential source of failure of the algorithm. Because G(s) is

stable, certainly will be, as its poles will be in the left half plane circle

on diameter . If acquires a pole outside this circle

(butstill in the left ha lf pl ane of course)— and this appears possible in

principle—Gr(s) will then acquire a pole in Re [s] >0. Should this difficulty

be encountered, a smaller value of ε should be used.

Related Functions

singriccati(), ophank(), bst(), hankelsv()

˜s()

Grs() G

rs1εs–()⁄()–=

˜s()

ε–1–j00,=()

˜rs()