National Instruments Xmath

Chapter 3 Multiplicative Error Reduction

Xmath Model Reduction Module 3-22 ni.com

The values of G(s) along the jω-axis are the same as the values of

around a circle with diameter defined by [a–j0, b–1+j0] on the positive

real axis (refer to Figure 3-2). Also, the values of along the jω-axis

are the same as the values of G(s) around a circle with diameter defined by

[–b–1+j0, –a+j0].

We can implement an arbitrary bilinear transform using the subsys( )

function, which substitutes a given transfer function for the s- or z-domain

operator, as previously shown.

To implement use:

gtildesys=subsys(gsys,makep([-b,1]/makep([1,-a])

To implement use:

gsys=subsys(gtildesys,makep([b,1]/makep([1,a])

Note The systems substituted in the previous calls to subsys invert the function

specification because these functions use backward polynomial rotation.

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, again 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.

˜s()

˜s() Gsa–

bs–1+

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

⎝

⎠

⎛

⎞

Gs() G

˜sa+

bs 1+

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

⎝

⎠

⎛

⎞

˜s()

˜s() Gsa–

bs–1+

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

⎝

⎠

⎛

⎞

˜s() G

˜rs() G

˜s()

Grs() G

˜r

sa+

bs 1+

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

⎝⎠

⎛⎞