Control Systems/Z Transform Mappings

§Z Transform Mappings

There are a number of different mappings that can be used to convert a system from the complex Laplace domain into the Z-Domain. None of these mappings are perfect, and every mapping requires a specific starting condition, and focuses on a specific aspect to reproduce faithfully. One such mapping that has already been discussed is the bilinear transform, which, along with prewarping, can faithfully map the various regions in the s-plane into the corresponding regions in the z-plane. We will discuss some other potential mappings in this chapter, and we will discuss the pros and cons of each.

§Bilinear Transform

The Bilinear transform converts from the Z-domain to the complex W domain. The W domain is not the same as the Laplace domain, although there are some similarities. Here are some of the similiarities between the Laplace domain and the W domain:

1. Stable poles are in the Left-Half Plane
2. Unstable poles are in the right-half plane
3. Marginally stable poles are on the vertical, imaginary axis

With that said, the bilinear transform can be defined as follows:

[Bilinear Transform]

$w = \frac{2}{T} \frac{z - 1}{z + 1}$

[Inverse Bilinear Transform]

$z = \frac{(T/2) + w}{(T/2) - w}$

Graphically, we can show that the bilinear transform operates as follows:

§Prewarping

The W domain is not the same as the Laplace domain, but if we employ the process of prewarping before we take the bilinear transform, we can make our results match more closely to the desired Laplace Domain representation.

Using prewarping, we can show the effect of the bilinear transform graphically:

The shape of the graph before and after prewarping is the same as it is without prewarping. However, the destination domain is the S-domain, not the W-domain.

§Matched Z-Transform

If we have a function in the laplace domain that has been decomposed using partial fraction expansion, we generally have an equation in the form:

$Y(s) = \frac{A}{s + \alpha_1} + \frac{B}{s + \alpha_2} + \frac{C}{s + \alpha_3} + ...$

And once we are in this form, we can make a direct conversion between the s and z planes using the following mapping:

[Matched Z Transform]

$s + \alpha = 1 - z^{-1}e^{-\alpha T}$
Pro
A good direct mapping in terms of s and a single coefficient
Con
requires the Laplace-domain function be decomposed using partial fraction expansion.

§Simpson's Rule

[Simpson's Rule]

$s = \frac{3}{T} \frac{z^2-1}{z^2+4z^1+1}$
CON
Essentially multiplies the order of the transfer function by a factor of 2. This makes things difficult when you are trying to physically implement the system. It has been shown that this transform produces unstable roots (outside of unit unit circle).

§(w, v) Transform

Given the following system:

$Y(s) = G(s, z, z^\alpha)X(s)$

Then:

$w = \frac{2}{T} \frac{z-1}{z+1}$
$v(\alpha) = 1 - \alpha(1 - z^{-1}) + \frac{\alpha(\alpha - 1)}{z}(1-z^{-1})^2$

And:

[(w, v) Transform]

$Y(z) = G(w, z, v(\alpha))\left[X(z) - \frac{x(0)}{1 + z^{-1}}\right]$
Pro
Directly maps a function in terms of z and s, into a function in terms of only z.
Con
Requires a function that is already in terms of s, z and α.