Sampled data design by log gain diagrams

The bilinear transformation
z = (1+w)/(1-w)
converts a
z
-transform function
G(z)
of a sampled-data system into a new function
G(w)
, called the
w
-transform function, which is a rational function in variable
w
. This bilinear transformation maps the unit circle on the
z
- plane onto the imaginary axis of the
w
-plane. Consequently, it is now possible to readily draw log magnitude and phase diagrams against a frequency scale of the open-loop
w
-transform function of a sampled-data system by use of asymptotic techniques. Then, by use of a Nichols chart and correlation information available from continuous systems, it is possible to predict the approximate time domain performance. Design by modification of the open-loop transfer function can be made on the diagram in the same manner as employed for continuous systems on the Bode diagram. The resulting
w
-transform can be converted to its equivalent Laplace transform. The ratio of this transform function and the original Laplace transform function of the system's equipment gives the required compensator. Remote s-plane poles may have to be added to have the compensator physically realizable. Restricting the modifying
w
-plane poles to lie between (0) and (-1) permits the compensator to be realizable as an RC network.