National Instruments 370755B-01

Chapter 3 Multiplicative Error Reduction

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).

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 it is

desired to have a good approximation of G(s) over an interval [–jΩ, jΩ],

then a choice such as ε–1 = 5 Ω or 10 Ω needs to be made. Reduction 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

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( ). Obviously, zeros at

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

available.

˜s() Gs

εs1+

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



˜s()

Grs() G

˜rs1εs–()⁄()–=

0ε1–

,j0+()s

˜s()