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The manipulation of block diagrams of
sampled data systems are very similar to that for block diagrams of systems expressed in
the Laplace domain. This should not be surprising since the z-transform is a
special case of the Laplace transform.
However, because of the presence of
samplers, there are some extra rules to follow. Consider a continuous process under
digital control via the classical feedback framework. The detailed block diagram of this
system including the presence of the samplers is,
Always remember that the samplers determine the type of
signal propagating around the system. Hence the position of the samplers are important. In
particular, note that the transfer function between Y(z) and U(z) for the following two
systems (System A and System B) are different.
System A
Here, the relationship between Y(z) and U(z) is given by:
System B
With this system, the relationship between Y(z) and U(z)
is given by:
In general,  .
Summary
By learning this particular set of notes, you would have
the essential tools to enable you to design digital controllers and to analyse the
behaviour of discrete systems. Because of the programmable nature of the digital computer,
computer based control algorithms can range from the very simple to the very
sophisticated.
There is however, one set of controllers that will always
be in use, namely the three-term PID type controller. The next set of notes will examine
the ways this ubiquitous and versatile controller can be developed into various discrete PI/PID forms, and how modifications can be made to improve
its performance.
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