As it will be introduced later in this report, when dealing with
digital or hybrid control systems, one has to deal with a system which
has some of its components working in the discrete-time domain, some
others in the analog one. One therefore has to introduce a means in
order to get from one domain to the other.
At this point of the presentation it is of some interest to notice
that the designing of a digital control system can be viewed as the
design of an Infinite Impulse Response filter (which thus has some
poles).
So we are dealing with the design of a filter in which we will have to
deal with some information in the discrete-time domain and some other
in the analog domain: the choice here is to use the Bilinear transform
in order to meet our requirements.
Many analysis and design techniques for continuous-time linear time-invariant systems
are based on the property that in the s-plane the stability boundary
is the imaginary axis. Thus these techniques cannot be applied to
linear time-invariant discrete-time systems in the z-plane, since the
stability boundary is the unit circle. However, through the use of the
transformation
or, solving for s
the unit circle of the z-plane transforms into the imaginary axis of the s-plane.
